![wonderful as they doubtless are, are indisputably eclipsed by the
structures formed by many insects; and the regular villages of the
beaver, by far the most sagacious architect amongst quadrupeds,
must yield the palm to a wasp's nest. You will think me here guilty of
exaggeration, and that, blinded by my attachment to a favourite
pursuit, I am elevating the little objects, which I wish to recommend
to your study, to a rank beyond their just claim. So far, however, am
I from being conscious of any such prejudice, that I do not hesitate
to go further, and assert that the pyramids of Egypt, as the work of
man, are not more wonderful for their size and solidity than are the
structures built by some insects.
To describe the most remarkable of these is my present object: and
that some method may be observed, I shall in this letter describe
the habitations of insects living in a state of solitude, and built each
by a single architect; and in a subsequent one, those of insects living
in societies, built by the united labours of many. The former class
may be conveniently subdivided into habitations built by the parent
insect, not for its own use, but for the convenience of its future
young; and those which are formed by the insect that inhabits them
for its own accommodation. To the first I shall now call your
attention.
The solitary insects which construct habitations for their future
young without any view to their own accommodation, chiefly belong
to the order Hymenoptera, and are principally different species of
wild bees. Of these the most simple are built by Colletes[758]
succincta, fodiens, c. The situation which the parent bee chooses,
is either the dry earth of a bank, or the vacuities of stone walls
cemented with earth instead of mortar. Having excavated a cylinder
about two inches in depth, running usually in a horizontal direction,
the bee occupies it with three or four cells about half an inch long,](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-56-320.jpg)
![and one-sixth broad, shaped like a thimble, the end of one fitting
into the mouth of another. The substance of which these cells are
formed is two or three layers of a silky membrane, composed of a
kind of glue secreted by the animal, resembling gold-beater's leaf,
but much finer, and so thin and transparent that the colour of an
included object may be seen through them. As soon as one cell is
completed, the bee deposits an egg within, and nearly fills it with a
paste composed of pollen and honey; which having done, she
proceeds to form another cell, storing it in like manner until the
whole is finished, when she carefully stops up the mouth of the
orifice with earth. Our countryman Grew seems to have found a
series of these nests in a singular situation—the middle of the pith of
an old elder-branch—in which they were placed lengthwise one after
another with a thin boundary between each[759].
Cells composed of a similar membranaceous substance, but placed
in a different situation, are constructed by Anthidium
manicatum[760]. This gay insect does not excavate holes for their
reception, but places them in the cavities of old trees, or of any
other object that suits its purpose. Sir Thomas Cullum discovered the
nest of one in the inside of the lock of a garden-gate, in which I
have also since twice found them. It should seem, however, that
such situations would be too cold for the grubs without a coating of
some non-conducting substance. The parent bee, therefore, after
having constructed the cells, laid an egg in each, and filled them
with a store of suitable food, plasters them with a covering of
vermiform masses, apparently composed of honey and pollen; and
having done this, aware, long before Count Rumford's experiments,
what materials conduct heat most slowly, she attacks the woolly
leaves of Stachys lanata, Agrostemma coronaria, and similar plants,
and with her mandibles industriously scrapes off the wool, which
with her fore legs she rolls into a little ball and carries to her nest.
This wool she sticks upon the plaster that covers her cells, and thus
closely envelops them with a warm coating of down impervious to
every change of temperature[761].](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-57-320.jpg)
![The bee last described may be said to exercise the trade of a
clothier. Another numerous family would be more properly compared
to carpenters, boring with incredible labour out of the solid wood
long cylindrical tubes, and dividing them into various cells. Amongst
these, one of the most remarkable is Xylocopa[762] violacea, a large
species, a native of Southern Europe, distinguished by beautiful
wings of a deep violet colour, and found commonly in gardens, in the
upright putrescent espaliers or vine-props of which, and occasionally
in the garden seats, doors and window-shutters, she makes her
nest. In the beginning of spring, after repeated and careful surveys,
she fixes upon a piece of wood suitable for her purpose, and with
her strong mandibles begins the process of boring. First proceeding
obliquely downwards, she soon points her course in a direction
parallel with the sides of the wood, and at length with unwearied
exertion forms a cylindrical hole or tunnel not less than twelve or
fifteen inches long and half an inch broad. Sometimes, where the
diameter will admit of it, three or four of these pipes, nearly parallel
with each other, are bored in the same piece. Herculean as this task,
which is the labour of several days, appears, it is but a small part of
what our industrious bee cheerfully undertakes. As yet she has
completed but the shell of the destined habitation of her offspring;
each of which, to the number of ten or twelve, will require a
separate and distinct apartment. How, you will ask, is she to form
these? With what materials can she construct the floors and ceilings?
Why truly God doth instruct her to discretion and doth teach her. In
excavating her tunnel she has detached a large quantity of fibres,
which lie on the ground like a heap of saw-dust. This material
supplies all her wants. Having deposited an egg at the bottom of the
cylinder along with the requisite store of pollen and honey, she next,
at the height of about three quarters of an inch, (which is the depth
of each cell,) constructs of particles of the saw-dust glued together,
and also to the sides of the tunnel, what may be called an annular
stage or scaffolding. When this is sufficiently hardened, its interior
edge affords support for a second ring of the same materials, and
thus the ceiling is gradually formed of these concentric circles, till](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-58-320.jpg)
![there remains only a small orifice in its centre, which is also closed
with a circular mass of agglutinated particles of saw-dust. When this
partition, which serves as the ceiling of the first cell and the flooring
of the second, is finished, it is about the thickness of a crown-piece,
and exhibits the appearance of as many concentric circles as the
animal has made pauses in her labour. One cell being finished, she
proceeds to another, which she furnishes and completes in the same
manner, and so on until she has divided her whole tunnel into ten or
twelve apartments.
Here, if you have followed me in this detail with the interest which I
wish it to inspire, a query will suggest itself. It will strike you that
such a laborious undertaking as the constructing and furnishing
these cells, cannot be the work of one or even of two days.
Considering that every cell requires a store of honey and pollen, not
to be collected but with long toil, and that a considerable interval
must be spent in agglutinating the floors of each, it will be very
obvious to you that the last egg in the last cell must be laid many
days after the first. We are certain, therefore, that the first egg will
become a grub, and consequently a perfect bee, many days before
the last. What then becomes of it? you will ask. It is impossible that
it should make its escape through eleven superincumbent cells
without destroying the immature tenants; and it seems equally
impossible that it should remain patiently in confinement below them
until they are all disclosed. This dilemma our heaven-taught architect
has provided against. With forethought never enough to be admired
she has not constructed her tunnel with one opening only, but at the
further end has pierced another orifice, a kind of back-door, through
which the insects produced by the first-laid eggs successively
emerge into day. In fact, all the young bees, even the uppermost, go
out by this road; for, by an exquisite instinct, each grub, when about
to become a pupa, places itself in its cell with its head downwards,
and thus is necessitated, when arrived at its last state, to pierce its
cell in this direction[763].](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-59-320.jpg)
![Ceratina albilabris of Spinola, who has given an interesting account
of its manners, forms its cell upon the general plan of the bee just
described, but, more economical of labour, chooses a branch of briar
or bramble, in the pith of which she excavates a canal about a foot
long and one line, or sometimes more, in diameter, with from eight
to twelve cells separated from each other by partitions of particles of
pith glued together[764].
Such are the curious habitations of the carpenter bees. Next I shall
introduce you to the not less interesting structures of another family
which carry on the trade of masons, (Megachile muraria,) building
their solid houses solely of artificial stone. The first step of the
mother bee is to fix upon a proper situation for the future mansion
of her offspring. For this she usually selects an angle, sheltered by
any projection, on the south side of a stone wall. Her next care is to
provide materials for the structure. The chief of these is sand, which
she carefully selects grain by grain from such as contains some
mixture of earth. These grains she glues together with her viscid
saliva into masses the size of small shot, and transports by means of
her jaws to the site of her castle[765]. With a number of these
masses, which are the artificial stone of which her building is to be
composed, united by a cement preferable to ours, she first forms the
basis or foundation of the whole. Next she raises the walls of a cell,
which is about an inch in length and half an inch broad, and before
its orifice is closed in form resembles a thimble. This, after
depositing an egg and a supply of honey and pollen, she covers in,
and then proceeds to the erection of a second, which she finishes in
the same manner, until the whole number, which varies from four to
eight, is completed. The vacuities between the cells, which are not
placed in any regular order, some being parallel to the wall, others
perpendicular to it, and others inclined to it at different angles, this
laborious architect fills up with the same material of which the cells
are composed, and then bestows upon the whole group a common
covering of coarser grains of sand. The form of the whole nest,
which when finished is a solid mass of stone so hard as not to be](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-60-320.jpg)
![easily penetrated with the blade of a knife, is an irregular oblong of
the same colour as the sand, and to a casual observer more
resembling a splash of mud than an artificial structure. These bees
sometimes are more economical of their labour, and repair old nests,
for the possession of which they have very desperate combats. One
would have supposed that the inhabitants of a castle so fortified
might defy the attacks of every insect marauder. Yet an Ichneumon
and a beetle (Clerus apiarius) both contrive to introduce their eggs
into the cells, and the larvæ proceeding from them devour their
inhabitants[766].
Other bees of the same family with that last described, use different
materials in the construction of their nests. Some employ fine earth
made into a kind of mortar with gluten. Another (Osmia[767]
cærulescens), as we learn from De Geer, forms its nest of
argillaceous earth mixed with chalk, upon stone walls, and
sometimes probably nidificates in chalk-pits. O. bicornis selects the
hollows of large stones for the site of its dwelling; while others
prefer the holes in wood.
The works thus far described require in general less genius than
labour and patience: but it is far otherwise with the nests of the last
tribe of artificers amongst wild bees, to which I shall advert—the
hangers of tapestry, or upholsterers—those which line the holes
excavated in the earth for the reception of their young, with an
elegant coating of flowers or of leaves. Amongst the most interesting
of these is Megachile[768] Papaveris, a species whose manners have
been admirably described by Reaumur. This little bee, as though
fascinated with the colour most attractive to our eyes, invariably
chooses for the hangings of her apartments the most brilliant
scarlet, selecting for its material the petals of the wild poppy, which
she dexterously cuts into the proper form. Her first process is to
excavate in some pathway a burrow, cylindrical at the entrance but
swelled out below, to the depth of about three inches. Having
polished the walls of this little apartment, she next flies to a
neighbouring field, cuts out oval portions of the flowers of poppies,](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-61-320.jpg)
![seizes them between her legs and returns with them to her cell; and
though separated from the wrinkled petal of a half-expanded flower,
she knows how to straighten their folds, and, if too large, to fit them
for her purpose by cutting off the superfluous parts. Beginning at
the bottom, she overlays the walls of her mansion with this brilliant
tapestry, extending it also on the surface of the ground round the
margin of the orifice. The bottom is rendered warm by three or four
coats, and the sides have never less than two. The little upholsterer,
having completed the hangings of her apartment, next fills it with
pollen and honey to the height of about half an inch; then, after
committing an egg to it, she wraps over the poppy lining so that
even the roof may be of this material; and lastly closes its mouth
with a small hillock of earth[769]. The great depth of the cell
compared with the space which the single egg and the
accompanying food deposited in it occupy, deserves particular
notice. This is not more than half an inch at the bottom, the
remaining two inches and a half being subsequently filled with earth.
—When you next favour me with a visit, I can show you the cells of
this interesting insect as yet unknown to British entomologists, for
which I am indebted to the kindness of M. Latreille, who first
scientifically described the species[770].
Megachile centuncularis, M. Willughbiella, and other species of the
same family, like the preceding, cover the walls of their cells with a
coating of leaves, but are content with a more sober colour,
generally selecting for their hangings the leaves of trees, especially
of the rose, whence they have been known by the name of the leaf-
cutter bees. They differ also from M. Papaveris in excavating longer
burrows, and filling them with several thimble-shaped cells
composed of portions of leaves so curiously convoluted, that, if we
were ignorant in what school they have been taught to construct
them, we should never credit their being the work of an insect. Their
entertaining history, so long ago as 1670, attracted the attention of
our countrymen Ray, Lister, Willughby, and Sir Edward King; but we](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-62-320.jpg)
![The process which one of these bees employs in cutting the pieces
of leaf that compose her nest is worthy of attention. Nothing can be
more expeditious: she is not longer about it than we should be with
a pair of scissors. After hovering for some moments over a rose-
bush, as if to reconnoitre the ground, the bee alights upon the leaf
which she has selected, usually taking her station upon its edge so
that the margin passes between her legs. With her strong mandibles
she cuts without intermission in a curve line so as to detach a
triangular portion. When this hangs by the last fibre, lest its weight
should carry her to the ground, she balances her little wings for
flight, and the very moment it parts from the leaf flies off with it in
triumph; the detached portion remaining bent between her legs in a
direction perpendicular to her body. Thus without rule or compasses
do these diminutive creatures mete out the materials of their work
into portions of an ellipse, into ovals or circles, accurately
accommodating the dimensions of the several pieces of each figure
to each other. What other architect could carry impressed upon the
tablet of his memory the entire idea of the edifice which he has to
erect, and, destitute of square or plumb-line, cut out his materials in
their exact dimensions without making a single mistake? Yet this is
what our little bee invariably does. So far are human art and reason
excelled by the teaching of the Almighty[771].
Other insects besides bees construct habitations of different kinds
for their young, as various species of burrowing wasps (Fossores),
Geotrupes, c., which deposit their eggs in cylindrical excavations
that become the abode of the future larvæ. In the procedures of
most of these, nothing worth particularizing occurs; but one species
called by Reaumur the mason-wasp, (Odynerus muraria,) referred to
in a former letter, works upon so singular a plan, that it would be
improper to pass it over in silence, especially as these nests may be
found in this country in most sandy banks exposed to the sun. This
insect bores a cylindrical cavity from two to three inches deep, in
hard sand which its mandibles alone would be scarcely capable of
penetrating, were it not provided with a slightly glutinous liquor
which it pours out of its mouth, that, like the vinegar with which](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-64-320.jpg)
![Hannibal softened the Alps, acts upon the cement of the sand, and
renders the separation of the grains easy to the double pickaxe with
which our little pioneer is furnished. But the most remarkable
circumstance is the mode in which it disposes of the excavated
materials. Instead of throwing them at random on a heap, it
carefully forms them into little oblong pellets, and arranges them
round the entrance of the hole so as to form a tunnel, which, when
the excavation is completed, is often not less than two or three
inches in length. For the greater part of its height this tunnel is
upright, but towards the top it bends into a curve, always however
retaining its cylindrical form. The little masses are so attached to
each other in this cylinder, as to leave numerous vacuities between
them, which give it the appearance of filagree-work. You will readily
divine that the excavated hole is intended for the reception of an
egg, but for what purpose the external tunnel is meant is not so
apparent. One use, and perhaps the most important, would seem to
be to prevent the incursions of the artful Ichneumons, Chrysidæ, c.
which are ever on the watch to insinuate their parasitic young into
the nests of other insects: it may render their access to the nest
more difficult; they may dread to enter into so long and dark a
defile. I have seen, however, more than once a Chrysis come out of
these tunnels. That its use is only temporary, is plain from the
circumstance that the insect employs the whole fabric, when its egg
is laid and store of food procured, in filling up the remaining vacuity
of the hole; taking down the pellets, which are very conveniently at
hand, and placing them in it until the entrance is filled[772].—Latreille
informs us, that a nearly similar tunnel, but composed of grains of
earth, is built at the entrance of its cell by a bee of his family of
pioneers[773].
Under this head, too, may be most conveniently arranged the very
singular habitations of the larvæ of the Linnæan genus Cynips, the
gall-fly, though they can with no propriety be said to be constructed
by the mother, who, provided with an instrument as potent as an
enchanter's wand, has but to pierce the site of the foundation, and](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-65-320.jpg)
![commodious apartments, as if by magic, spring up and surround the
germe of her future descendants. I allude to those vegetable
excrescencies termed galls, some of which resembling beautiful
berries and others apples, you must have frequently observed on the
leaves of the oak, and of which one species, the Aleppo gall, as I
have before noticed, is of such importance in the ingenious art de
peindre la parole et de parler aux yeux[774]. All these tumours owe
their origin to the deposition of an egg in the substance out of which
they grow. This egg, too small almost for perception, the parent
insect, a little four-winged fly, introduces into a puncture made by
her curious spiral sting, and in a few hours it becomes surrounded
with a fleshy chamber, which not only serves its young for shelter
and defence, but also for food; the future little hermit feeding upon
its interior and there undergoing its metamorphosis. Nothing can be
more varied than these habitations. Some are of a globular form, a
bright red colour, and smooth fleshy consistence, resembling
beautiful fruits, for which indeed, as you have before been told, they
are eaten in the Levant: others, beset with spines or clothed with
hair, are so much like seed-vessels, that an eminent modern chemist
has contended respecting the Aleppo gall, that it is actually a
capsule[775]. Some are exactly round; others like little mushrooms;
others resemble artichokes; while others again might be taken for
flowers: in short, they are of a hundred different forms, and of all
sizes from that of a pin's head to that of a walnut. Nor is their
situation on the plant less diversified. Some are found upon the leaf
itself; others upon the footstalks only; others upon the roots; and
others upon the buds[776]. Some of them cause the branches upon
which they grow to shoot out into such singular forms, that the
plants producing them were esteemed by the old botanists distinct
species. Of this kind is the Rose-willow, which old Gerard figures and
describes as not only making a gallant shew, but also yeelding a
most cooling aire in the heat of summer, being set up in houses for
the decking of the same. This willow is nothing more than one of
the common species, whose twigs, in consequence of the deposition
of the egg of a Cynips in their summits, there shoot out into](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-66-320.jpg)
![numerous leaves totally different in shape from the other leaves of
the tree, and arranged not much unlike those composing the flower
of a rose, adhering to the stem even after the others fall off. Sir
James Smith mentions a similar lusus on the Provence willows,
which at first he took for a tufted lichen[777]. From the same cause
the twigs of the common wild rose often shoot out into a beautiful
tuft of numerous reddish moss-like fibres wholly dissimilar from the
leaves of the plant, deemed by the old naturalists a very valuable
medical substance, to which they erroneously gave the name of
Bedeguar. None of these variations is accidental or common to
several of the tribe, but each peculiar to the galls formed by a single
and distinct species of Cynips.
How the mere insertion of an egg into the substance of a leaf or
twig, even if accompanied, as some imagine, by a peculiar fluid,
should cause the growth of such singular protuberances around it,
philosophers are as little able to explain, as why the insertion of a
particle of variolous matter into a child's arm should cover it with
pustules of small pox. In both cases the effects seem to proceed
from some action of the foreign substance upon the secreting
vessels of the animal or vegetable: but of the nature of this action
we know nothing. Thus much is ascertained by the observations of
Reaumur and Malpighi—that the production of the gall, which
however large attains its full size in a day or two[778], is caused by
the egg or some accompanying fluid: not by the larva, which does
not appear until the gall is fully formed[779]; that the galls which
spring from leaves almost constantly take their origin from
nerves[780]; and that the egg, at the same time that it causes the
growth of the gall, itself derives nourishment from the substance
that surrounds it, becoming considerably larger before it is hatched
than it was when first deposited[781].—When chemically analysed,
galls are found to contain only the same principles as the plant from
which they spring, but in a more concentrated state.](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-67-320.jpg)
![No productions of nature seem to have puzzled the ancient
philosophers more than galls. The commentator on Dioscorides,
Mathiolus, who agreeably to the doctrine of those days ascribed
their origin to spontaneous generation, gravely informs us that
weighty prognostications as to the events of the ensuing year may
be deduced from ascertaining whether they contain spiders, worms,
or flies. Other philosophers, who knew that except by rare accident
no other animals are to be found in galls, besides grubs of different
kinds which they rationally conceived to spring from eggs, were
chiefly at a loss to account for the conveyance of these eggs into the
middle of a substance in which they could find no external orifice.
They therefore inferred that they were the eggs of insects deposited
in the earth, which had been drawn up by the roots of trees along
with the sap, and after passing through different vessels had
stopped, some in the leaves, others in the twigs, and had there
hatched and produced galls! Redi's solution of the difficulty was even
more extraordinary. This philosopher, who had so triumphantly
combated the absurdities of spontaneous generation, fell himself
into greater. Not having been able to witness the deposition of eggs
by the parent flies in the plants that produce galls, he took it for
granted that the grubs which he found within them could not spring
from eggs: and he was equally unwilling to admit their origin from
spontaneous generation,—an admission which would have been fatal
to his own most brilliant discoveries. He therefore cut the knot, by
supposing that to the same vegetative soul by which fruits and
plants are produced, is committed the charge of creating the larvæ
found in galls[782]! An instance truly humiliating, how little we can
infer from a man's just ideas on one point, that he will not be guilty
of the most pitiable absurdity on another!
Though by far the greater part of the vegetable excrescencies
termed galls, are caused by insects of the genus Cynips, they do not
always originate from this tribe. Some are produced by weevils
belonging to Schüppel's genus Ceutorhynchus; as those on the roots
of kedlock (Sinapis arvensis), which I have ascertained to be
inhabited by the larvæ of Curculio contractus Marsh., Rhynchænus](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-68-320.jpg)
![assimilis, F. From the knob-like galls on turnips called in some places
the anbury, I have bred another of these weevils, (Curculio
pleurostigma, Marsh., Rhynchænus sulcicollis, Gyll.) and I have little
doubt that the same insects, or species allied to them, cause the
clubbing of the roots of cabbages. It seems to be a beetle of the
same family that is figured by Reaumur[783], as causing the galls on
the leaves of the lime-tree. Others owe their origin to moths, as
those resembling a nutmeg which Reaumur received from
Cyprus[784]; and others again to two-winged flies, as the woody
galls of the thistle caused by Trypeta Cardui[785], and the cottony
galls found on ground ivy, wild thyme, c. as well as a very singular
one on the juniper resembling a flower, described by De Geer[786],
all which are the work of minute gall-gnats (Cecidomyiæ, Latr.).
Some of these last convert even the flowers of plants into a kind of
galls, as T. Loti of De Geer[787], which inhabits the blossoms of Lotus
corniculatus; and one which I have myself observed to render the
flowers of Erysimum Barbarea like a hop blossom. A similar
monstrous appearance is communicated to the flowers of Teucrium
supinum by a little field-bug, Tingis Teucrii of Host[788], and to
another plant of the same genus by one of the same tribe described
by Reaumur[789]. In these two last instances, however, the
habitations do not seem strictly entitled to the appellation of galls,
as they originate not from the egg, but from the larva, which, in the
operation of extracting the sap, in some way imparts a morbid action
to the juices, causing the flower to expand unnaturally: and the
same remark is applicable to the gall-like swellings formed by many
Aphides, as A. Pistaciæ, which causes the leaves of different species
of Pistacia to expand into red finger-like cavities; A. Abietis, which
converts the buds or young shoots of the fir into a very beautiful
gall, somewhat resembling a fir-cone, or a pine-apple in miniature;
and A. Bursariæ, which with its brood inhabits angular utriculi on the
leafstalk of the black poplar, numbers of which I have observed on
those trees by the road-side from Hull to Cottingham.—The majority
of galls are what entomologists have denominated monothalamous,](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-69-320.jpg)
![or consisting of only one chamber or cell; but some are
polythalamous, or consisting of several.
Having thus described the most remarkable of the habitations
constructed by the parent insects for the accommodation of their
future young, I proceed to the second kind mentioned, namely,
those which are formed by the insect itself for its own use. These
may be again subdivided into such as are the work of the insects in
their larva state; and such as are formed by perfect insects.
Many larvæ of all orders need no other habitations than the holes
which they form in seeking for, or eating, the substances upon which
they feed. Of this description are the majority of subterranean larvæ,
and those which feed on wood, as the Bostrichi or labyrinth beetles;
the Anobia which excavate the little circular holes frequently met
with in ancient furniture and the wood work of old houses; and
many larvæ of other orders, particularly Lepidoptera. One of these
last, the larva of Cossus ligniperda differs from its congeners in
fabricating for its residence during winter a habitation of pieces of
wood lined with fine silk[790]. Under this division, too, come the
singular habitations of the subcutaneous larvæ, so called from the
circumstance of their feeding upon the parenchyma included
between the upper and under cuticles of the leaves of plants,
between which, though the whole leaf is often not thicker than a
sheet of writing-paper, they find at once food and lodging. You must
have been at some time struck by certain white zigzag or labyrinth-
like lines on the leaves of the dandelion, bramble, and numerous
other plants: the next-time you meet with one of them, if you hold it
up to the light you will perceive that the colour of these lines is
owing to the pulpy substance of the leaf having there been
removed; and at the further end you will probably remark a dark-
coloured speck, which, when carefully extricated from its covering,](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-70-320.jpg)
![you will find to be the little miner of the tortuous galleries which you
are admiring. Some of these minute larvæ, to which the parenchyma
of a leaf is a vast country, requiring several weeks to be traversed by
the slow process of mining which they adopt—that of eating the
excavated materials as they proceed—are transformed into beetles
(Cionus Thapsi, c.); others into flies; and a still greater number into
very minute moths, as Gracillaria? Wilkella, Clerkella, c. Many of
these last are little miracles of nature, which has lavished on them
the most splendid tints tastefully combined with gold, silver and
pearl: so that, were they but formed upon a larger scale, they would
far eclipse all other animals in richness of decoration.
Another tribe of larvæ, not very numerous, content themselves for
their habitations with simple holes, into which they retire
occasionally. Many of these are merely cylindrical burrows in the
ground, as those formed by the larvæ of field-crickets, Cicindelæ
and Ephemeræ. But the larvæ of the very remarkable lepidopterous
genus (Nycterobius of Mr. MacLeay) before alluded to[791], excavate
for themselves dwellings of a more artificial construction; forming
cylindrical holes in the trees of New Holland, particularly the
different species of Banksia, to which they are very destructive, and
defending the entrance against the attacks of the Mantes and other
carnivorous insects by a sort of trap-door composed of silk
interwoven with leaves and pieces of excrement, securely fastened
at the upper end, but left loose at the lower for the free passage of
the occupant. This abode they regularly quit at sun-set, for the
purpose of laying in a store of the leaves on which they feed. These
they drag by one at a time into their cell until the approach of light,
when they retreat precipitately into it, and there remain closely
secluded the whole day, enjoying the booty which their nocturnal
range has provided. One species lifts up the loose end of its door by
its tail, and enters backward, dragging after it a leaf of Banksia
serrata, which it holds by the footstalk[792].
A third description of larvæ, chiefly of the two lepidopterous tribes
Tortricidæ and Tineidæ, form into convenient habitations the leaves](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-71-320.jpg)
![of the plants on which they feed. Some of these merely connect
together with a few silken threads several leaves so as to form an
irregular packet, in the centre of which the little hermit lives. Others
confine themselves to a single leaf, of which they simply fold one
part over the other. A third description form and inhabit a sort of roll,
by some species made cylindrical, by others conical, resembling the
papers into which grocers put their sugar, and as accurately
constructed, only there is an opening left at the smaller extremity for
the egress of the insect in case of need. If you were to see one of
these rolls, you would immediately ask by what mechanism it could
possibly be made—how an insect without fingers could contrive to
bend a leaf into a roll, and to keep it in that form until fastened with
the silk which holds it together? The following is the operation. The
little caterpillar first fixes a series of silken cables from one side of
the leaf to the other. She next pulls at these cables with her feet;
and when she has forced the sides to approach, she fastens them
together with shorter threads of silk. If the insect finds that one of
the larger nerves of the leaf is so strong as to resist her efforts, she
weakens it by gnawing it here and there half through. What
engineer could act more sagaciously?—To form one of the conical or
horn-shaped rolls, which are not composed of a whole leaf, but of a
long triangular portion cut out of the edge, some other manœuvres
are requisite. Placing herself upon the leaf, the caterpillar cuts out
with her jaws the piece which is to compose her roll. She does not
however entirely detach it: it would then want a base. She detaches
that part only which is to form the contour of the horn. This portion
is a triangular strap, which she rolls as she cuts. When the body of
the horn is finished, as it is intended to be fixed upon the leaf in
nearly an upright position, it is necessary to elevate it. To effect this,
she proceeds as we should with an inclined obelisk. She attaches
threads or little cables towards the point of the pyramid, and raises it
by the weight of her body[793].
A still greater degree of dexterity is manifested in fabricating the
habitations of the larvæ of some other moths which feed on the
leaves of the rose-tree, apple, elm, and oak, on the under-side of](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-72-320.jpg)
![membranes on that side. Then putting out their heads they cut the
intermediate portions, carefully avoiding the larger nerves of the
leaf; afterwards they sew up the detached sides more closely, and
only intersect the nerves when their labour is completed[794].—The
habitation made by a moth, which lives upon a species of
Astragalus, is in like manner formed of the epidermis of the leaves,
but in this several corrugated pieces project over each other, so as to
resemble the furbelows once in fashion[795].
Other larvæ construct their habitations wholly of silk. Of this
description is that of a moth, whose abode, except as to the
materials which compose it, is formed on the same general plan as
that just described, and the larva in like manner feeds only on the
parenchyma of the leaf. In the beginning of spring, if you examine
the leaves of your pear-trees, you will scarcely fail to meet with
some beset on the under surface with several perpendicular downy
russet-coloured projections, about a quarter of an inch high, and not
much thicker than a pin, of a cylindrical shape, with a protuberance
at the base, and altogether resembling at first sight so many spines
growing out of the leaf. You would never suspect that these could be
the habitations of insects; yet that they are is certain. Detach one of
them, and give it a gentle squeeze, and you will see emerge from
the lower end a minute caterpillar with a yellowish body and black
head. Examine the place from which you have removed it, and you
will perceive a round excavation in the cuticle and parenchyma of
the leaf, the size of the end of the tube by which it was concealed.
This excavation is the work of the above-mentioned caterpillar, which
obtains its food by moving its little tent from one part of the leaf to
the other, and eating away the space immediately under it. It
touches no other part; and when these insects abound, as they
often do to the great injury of pear-trees[796], you will perceive
every leaf bristled with them, and covered with little withered
specks, the vestiges of their former meals. The case in which the
caterpillar resides, and which is quite essential to its existence, is
composed of silk spun from its mouth almost as soon as it is](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-74-320.jpg)
![excluded from the egg. As it increases in size, it enlarges its
habitation by slitting it in two, and introducing a strip of new
materials. But the most curious circumstance in the history of this
little Arab is the mode by which it retains its tent in a perpendicular
posture. This it effects partly by attaching silken threads from the
protuberance at the base to the surrounding surface of the leaf. But
being not merely a mechanician, but a profound natural philosopher
well acquainted with the properties of air, it has another resource
when any extraordinary violence threatens to overturn its slender
turret. It forms a vacuum in the protuberance at the base, and thus
as effectually fastens it to the leaf as if an air-pump had been
employed! This vacuum is caused by the insect's retreating on the
least alarm up its narrow case, which its body completely fills, and
thus leaving the space below free of air. In detaching one of these
cases you may easily convince yourself of the fact. If you seize it
suddenly while the insect is at the bottom, you will find that it is
readily pulled off, the silken cords giving way to a very slight force;
but if, proceeding gently, you give the insect time to retreat, the
case will be held so closely to the leaf as to require a much stronger
effort to loosen it. As if aware that, should the air get admission
from below, and thus render a vacuum impracticable, the strongest
bulwark of its fortress would be destroyed, our little philosopher
carefully avoids gnawing a hole in the leaf, contenting itself with the
pasturage afforded by the parenchyma above the lower epidermis;
and when the produce of this area is consumed, it gnaws asunder
the cords of its tent, and pitches it at a short distance as before.
Having attained its full growth, it assumes the pupa state, and after
a while issues out of its confinement a small brown moth, with long
hind legs, the Phalæna Tinea serratella of Linné[797].
Some larvæ, which form their covering of pure silk, are not content
with a single coating, but actually envelop themselves in another,
open on one side and very much resembling a cloak; whence
Reaumur called them Teignes à fourreau à manteau. What is very
striking in the construction of this cloak, is, that the silk, instead of
being woven into one uniform close texture, is formed into](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-75-320.jpg)
![numerous transparent scales over-wrapping each other, and
altogether very much resembling the scales of a fish[798]. These
mantle-covered cases, one of which I once had the pleasure of
discovering, are inhabited by the larva of a little moth apparently
first described by Dr. Zincken genannt Sommer, who calls it Tinea
palliatella[799].
Various substances besides silk are fabricated into habitations by
other larvæ, though usually joined together either with silk or an
analogous gummy material. Thus Diurnea? Lichenum forms of pieces
of lichen a dwelling resembling one of the turreted Helices, many of
which I observed in June 1812 on an oak in Barham. The larvæ of
another moth, which also feeds upon lichens, instead of employing
these vegetables in forming its habitation, composes it of grains of
stone eroded from the walls of buildings upon which its food is
found, and connected by a silken cement. These insects were the
subject of a paper in the Memoirs of the French Academy[800], by M.
de la Voye, who, from the circumstance of their being found in great
abundance on mouldering walls, attributed to them the power of
eating stone, and regarded them as the authors of injuries
proceeding solely from the hand of time: for the insects themselves
are so minute, and the coating of grains of stone composing their
cases is so trifling, that Reaumur observes they could scarcely make
any perceptible impression on a wall from which they had procured
materials for ages[801].—Another lepidopterous larva, but of a much
larger size and different genus, the case of which is preserved in the
cabinet of the President of the Linnean Society, who pointed it out to
me, employs the spines apparently of some species of Mimosa,
which are ranged side by side so as to form a very elegant fluted
cylinder. A similar arrangement of pieces of small twigs is observable
in the habitation of the females[802] of the larvæ of a moth referred
by Von Scheven to Bombyx vestita, F.; which Ochsenheimer regards
as synonymous with Psyche graminella, while P. Viciella of the
Wiener Verzeichniss covers itself with short portions of the stems of
grasses placed transversely, and united by means of silk into a five-](https://image.slidesharecdn.com/16380053-250518152536-e2544340/85/Quantum-Field-Theory-13-Basics-In-Mathematics-And-Physics-Quantum-Electrodynamics-Gauge-Theory-A-Bridge-Between-Mathematicians-And-Physicists-Eberhard-Zeidler-76-320.jpg)
Quantum Field Theory 13 Basics In Mathematics And Physics Quantum Electrodynamics Gauge Theory A Bridge Between Mathematicians And Physicists Eberhard Zeidler
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- 6. Quantum Field Theory I: Basics in Mathematics and Physics
- 7. Eberhard Zeidler Quantum Field Theory I: Basics in Mathematics and Physics A Bridge between Mathematicians and Physicists With 94 Figures and 19 Tables 123
- 8. Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Inselstrasse 22 04103 Leipzig Germany e-mail: ezeidler@mis.mpg.de Library of Congress Control Number: 2006929535 Mathematics Subject Classification (2000): 35Qxx, 58-xx, 81Txx, 82-xx, 83Cxx ISBN-10 3-540-34762-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34762-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer L ATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 46/3100/YL 5 4 3 2 1 0
- 9. TO THE MEMORY OF JÜRGEN MOSER (1928–1999)
- 10. Preface Daß ich erkenne, was die Welt im Innersten zusammenhält.1 Faust Concepts without intuition are empty, intuition without concepts is blind. Immanuel Kant (1724–1804) The greatest mathematicians like Archimedes, Newton, and Gauss have always been able to combine theory and applications into one. Felix Klein (1849–1925) The present comprehensive introduction to the mathematical and physical aspects of quantum field theory consists of the following six volumes: Volume I: Basics in Mathematics and Physics Volume II: Quantum Electrodynamics Volume III: Gauge Theory Volume IV: Quantum Mathematics Volume V: The Physics of the Standard Model Volume VI: Quantum Gravity and String Theory. Since ancient times, both physicists and mathematicians have tried to under- stand the forces acting in nature. Nowadays we know that there exist four fundamental forces in nature: • Newton’s gravitational force, • Maxwell’s electromagnetic force, • the strong force between elementary particles, and • the weak force between elementary particles (e.g., the force responsible for the radioactive decay of atoms). In the 20th century, physicists established two basic models, namely, • the Standard Model in cosmology based on Einstein’s theory of general relativity, and • the Standard Model in elementary particle physics based on gauge theory. 1 So that I may perceive whatever holds the world together in its inmost folds. The alchemist Georg Faust (1480–1540) is the protagonist of Goethe’s drama Faust written in 1808.
- 11. VIII Preface One of the greatest challenges of the human intellect is the discovery of a unified theory for the four fundamental forces in nature based on first principles in physics and rigorous mathematics. For many years, I have been fascinated by this challenge. When talking about this challenge to colleagues, I have noticed that many of my colleagues in mathematics complain about the fact that it is difficult to understand the thinking of physicists and to follow the pragmatic, but frequently non-rigorous arguments used by physicists. On the other hand, my colleagues in physics complain about the abstract level of the modern mathematical literature and the lack of explicitly formulated connections to physics. This has motivated me to write the present book and the volumes to follow. It is my intention to build a bridge between mathematicians and physicists. The main ideas of this treatise are described in the Prologue to this book. The six volumes address a broad audience of readers, including both under- graduate students and graduate students as well as experienced scientists who want to become familiar with the mathematical and physical aspects of the fascinating field of quantum field theory. In some sense, we will start from scratch: • For students of mathematics, I would like to show that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different math- ematical questions. • For students of physics, I would like to introduce fairly advanced mathe- matics which is beyond the usual curriculum in physics. For historical reasons, there exists a gap between the language of mathemati- cians and the language of physicists. I want to bridge this gap.2 I will try to minimize the preliminaries such that undergraduate students after two years of studies should be able to understand the main body of the text. In writing this monograph, it was my goal to follow the advise given by the poet Johann Wolfgang von Goethe (1749–1832): Textbooks should be attractive by showing the beauty of the subject. Ariadne’s thread. In the author’s opinion, the most important prelude to learning a new subject is strong motivation. Experience shows that highly motivated students are willing to take great effort to learn sophisticated sub- jects. I would like to put the beginning of Ariadne’s thread into the hands of the reader. 2 On November 7th 1940, there was a famous accident in the U.S.A. which was recorded on film. The Tacoma Narrows Bridge broke down because of unexpected nonlinear resonance effects. I hope that my bridge between mathematicians and physicists is not of Tacoma type.
- 12. Preface IX Remember the following myth. On the Greek island of Crete in ancient times, there lived the monster Minotaur, half human and half bull, in a labyrinth. Every nine years, seven virgins and seven young men had to be sacrificed to the Minotaur. Ariadne, the daughter of King Minos of Crete and Pasiphaë fell in love with one of the seven young men – the Athenian Prince Theseus. To save his life, Ariadne gave Theseus a thread of yarn, and he fixed the beginning of the thread at the entrance of the labyrinth. After a hard fight, Theseus killed the Minotaur, and he escaped from the labyrinth by the help of Ariadne’s thread.3 For hard scientific work, it is nice to have a kind of Ariadne’s thread at hand. The six volumes cover a fairly broad spectrum of mathematics and physics. In particular, in the present first volume the reader gets information about • the physics of the Standard Model of particle physics and • the magic formulas in quantum field theory, and we touch the following mathematical subjects: • finite-dimensional Hilbert spaces and a rigorous approach to the basic ideas of quantum field theory, • elements of functional differentiation and functional integration, • elements of probability theory, • calculus of variations and the principle of critical action, • harmonic analysis and the Fourier transform, the Laplace transform, and the Mellin transform, • Green’s functions, partial differential equations, and distributions (gener- alized functions), • Green’s functions, the Fourier method, and functional integrals (path in- tegrals), • the Lebesgue integral, general measure integrals, and Hilbert spaces, • elements of functional analysis and perturbation theory, • the Dirichlet principle as a paradigm for the modern Hilbert space approach to partial differential equations, • spectral theory and rigorous Dirac calculus, • analyticity, • calculus for Grassmann variables, • many-particle systems and number theory, • Lie groups and Lie algebras, • basic ideas of differential and algebraic topology (homology, cohomology, and homotopy; topological quantum numbers and quantum states). We want to show the reader that many mathematical methods used in quan- tum field theory can be traced back to classical mathematical problems. In 3 Unfortunately, Theseus was not grateful to Ariadne. He deserted her on the Is- land of Naxos, and she became the bride of Dionysus. Richard Strauss composed the opera Ariadne on Naxos in 1912.
- 13. X Preface particular, we will thoroughly study the relation of the procedure of renor- malization in physics to the following classical mathematical topics: • singular perturbations, resonances, and bifurcation in oscillating systems (renormalization in a nutshell on page 625), • the regularization of divergent infinite series, divergent infinite products, and divergent integrals, • divergent integrals and distributions (Hadamard’s finite part of divergent integrals), • the passage from a finite number of degrees of freedom to an infinite number of degrees of freedom and the method of counterterms in complex analysis (the Weierstrass theorem and the Mittag–Leffler theorem), • analytic continuation and the zeta function in number theory, • Poincaré’s asymptotic series and the Ritt theorem in complex analysis, • the renormalization group and Lie’s theory of dynamical systems (one- parameter Lie groups), • rigorous theory of finite-dimensional functional integrals (path integrals). The following volumes will provide the reader with important additional ma- terial. A summary can be found in the Prologue on pages 11 through 15. Additional material on the Internet. The interested reader may find additional material on my homepage: Internet: www.mis.mpg.de/ezeidler/ This concerns a carefully structured panorama of important literature in mathematics, physics, history of the sciences and philosophy, along with a comprehensive bibliography. One may also find a comprehensive list of math- ematicians, physicists, and philosophers (from ancient until present time) mentioned in the six volumes. My homepage also allows links to the lead- ing centers in elementary particle physics: CERN (Geneva, Switzerland), DESY (Hamburg, Germany), FERMILAB (Batavia, Illinois, U.S.A.), KEK (Tsukuba, Japan), and SLAC (Stanford University, California, U.S.A.). One may also find links to the following Max Planck Institutes in Germany: As- tronomy (Heidelberg), Astrophysics (Garching), Complex Systems in Physics (Dresden), Albert Einstein Institute for Gravitational Physics (Golm), Math- ematics (Bonn), Nuclear Physics (Heidelberg), Werner Heisenberg Institute for Physics (Munich), and Plasmaphysics (Garching). Apology. The author apologizes for his imperfect English style. In the preface to his monograph The Classical Groups, Princeton University Press, 1946, Hermann Weyl writes the following: The gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle. “Was das heissen will, weiss jeder, Der im Traum pferdlos geritten ist,”4 4 Everyone who has dreamt of riding free, without the need of a horse, will know what I mean.
- 14. Preface XI I am tempted to say with the Swiss poet Gottfried Keller (1819–1890). Nobody is more aware than myself of the attendant loss in vigor, ease and lucidity of expression. Acknowledgements. First of all I would like to thank the Max Planck So- ciety in Germany for founding the Max Planck Institute for Mathematics in the Sciences (MIS) in Leipzig in 1996 and for creating a superb scientific environment here. This treatise would have been impossible without the ex- tensive contacts of the institute to mathematicians and physicists all over the world and without the excellent library of the institute. My special thanks go to the intellectual fathers of the institute, Friedrich Hirzebruch (chairman of the Founder’s Committee) and Stefan Hildebrandt in Bonn, Karl-Heinz Hoff- mann and Julius Wess in Munich, and the late Jürgen Moser in Zurich who was an external scientific member of the institute. I would like to dedicate this volume to the memory of Jürgen Moser who was a great mathemati- cian and an amiable man. Moreover, I would like to thank Don Zagier (Max Planck Institute for Mathematics in Bonn and Collège de France in Paris), one of the greatest experts in number theory, for the kindness of writing a beautiful section on useful techniques of number theory in physics. I am very grateful to numerous colleagues in mathematics and physics from all over the world for illuminating discussions. It is not possible to men- tion the names of all of them, since the list is very long. In particular, I would like to thank the professors from the Institute of Theoretical Physics at Leipzig University, Bodo Geyer, Wolfhard Janke, Gerd Rudolph, Manfred Salmhofer, Klaus Sibold, Armin Uhlmann, and Rainer Verch for nice cooper- ation. For many stimulating discussions on a broad spectrum of mathematical problems, I would like to thank the co-directors of the MIS, Wolfgang Hack- busch, Jürgen Jost, and Stefan Müller. For getting information about new research topics, I am very grateful to my former and present collaborators: Günther Berger, Ludmilla Bordag, Friedemann Brandt, Friedemann Brock, Chand Devchand, Bertfried Fauser, Felix Finster, Christian Fleischhack, Jörg Frauendiener, Hans-Peter Gittel, Matthias Günther, Bruce Hunt, Konrad Kaltenbach, Satyanad Kichenas- samy, Klaus Kirsten, Christian Klein, Andreas Knauf, Alexander Lange, Roland Matthes, Johannes Maul†, Erich Miersemann, Mario Paschke, Hoang Xuan Phu, Karin Quasthoff, Olaf Richter†, Alexander Schmidt, Rainer Schu- mann, Friedemann Schuricht, Peter Senf†, Martin Speight, Jürgen Tolksdorf, Dimitri Vassilevich, Hartmut Wachter, and Raimar Wulkenhaar. For experienced assistance in preparing this book, I would like to thank Kerstin Fölting (graphics, tables, and a meticulous proof-reading of my entire latex-file together with Rainer Munck), Micaela Krieger–Hauwede (graphics, tables, and layout), and Jeffrey Ovall (checking and improving my English style). For supporting me kindly in various aspects and for helping me to save time, I am also very grateful to my secretary, Regine Lübke, and to the staff of the institute including the librarians directed by Ingo Brüggemann,
- 15. XII Preface the computer group directed by Rainer Kleinrensing, and the administration directed by Dietmar Rudzik. Finally, I would like to thank the Springer-Verlag for a harmonious collaboration. I hope that the reader of this book enjoys getting a feel for the unity of mathematics and physics by discovering interrelations between apparently completely different subjects. Leipzig, Fall 2005 Eberhard Zeidler
- 16. Contents Part I. Introduction Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1 The Revolution of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 Quantization in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.1 Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.2 The Fundamental Role of the Harmonic Oscillator in Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.2.3 Quantum Fields and Second Quantization . . . . . . . . . . . 52 1.2.4 The Importance of Functional Integrals . . . . . . . . . . . . . 57 1.3 The Role of Göttingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.4 The Göttingen Tragedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.5 Highlights in the Sciences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.5.1 The Nobel Prize. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.5.2 The Fields Medal in Mathematics . . . . . . . . . . . . . . . . . . 71 1.5.3 The Nevanlinna Prize in Computer Sciences . . . . . . . . . 72 1.5.4 The Wolf Prize in Physics . . . . . . . . . . . . . . . . . . . . . . . . . 73 1.5.5 The Wolf Prize in Mathematics . . . . . . . . . . . . . . . . . . . . 73 1.5.6 The Abel Prize in Mathematics . . . . . . . . . . . . . . . . . . . . 75 1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2. Phenomenology of the Standard Model for Elementary Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 The System of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2 Waves in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2.1 Harmonic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.2.3 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.2.4 Electromagnetic Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2.5 Superposition of Waves and the Fourier Transform . . . 86
- 17. XIV Contents 2.2.6 Damped Waves, the Laplace Transform, and Disper- sion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2.7 The Response Function, the Feynman Propagator, and Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Historical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.3.1 Planck’s Radiation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.3.2 The Boltzmann Statistics and Planck’s Quantum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.3.3 Einstein’s Theory of Special Relativity . . . . . . . . . . . . . . 109 2.3.4 Einstein’s Theory of General Relativity . . . . . . . . . . . . . 111 2.3.5 Einstein’s Light Particle Hypothesis . . . . . . . . . . . . . . . . 112 2.3.6 Rutherford’s Particle Scattering . . . . . . . . . . . . . . . . . . . . 113 2.3.7 The Cross Section for Compton Scattering . . . . . . . . . . . 115 2.3.8 Bohr’s Model of the Hydrogen Atom . . . . . . . . . . . . . . . . 120 2.3.9 Einstein’s Radiation Law and Laser Beams . . . . . . . . . . 124 2.3.10 Quantum Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2.4 The Standard Model in Particle Physics . . . . . . . . . . . . . . . . . . . 127 2.4.1 The Four Fundamental Forces in Nature . . . . . . . . . . . . 127 2.4.2 The Fundamental Particles in Nature . . . . . . . . . . . . . . . 130 2.5 Magic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 2.6 Quantum Numbers of Elementary Particles . . . . . . . . . . . . . . . . 143 2.6.1 The Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.6.2 Conservation of Quantum Numbers . . . . . . . . . . . . . . . . . 154 2.7 The Fundamental Role of Symmetry in Physics . . . . . . . . . . . . 162 2.7.1 Classical Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2.7.2 The CPT Symmetry Principle for Elementary Particles 170 2.7.3 Local Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 2.7.4 Permutations and Pauli’s Exclusion Principle . . . . . . . . 176 2.7.5 Crossing Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.7.6 Forbidden Spectral Lines in Molecules . . . . . . . . . . . . . . 177 2.8 Symmetry Breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 2.8.1 Parity Violation and CP Violation . . . . . . . . . . . . . . . . . . 178 2.8.2 Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.8.3 Splitting of Spectral Lines in Molecules . . . . . . . . . . . . . 179 2.8.4 Spontaneous Symmetry Breaking and Particles . . . . . . . 180 2.8.5 Bifurcation and Phase Transitions . . . . . . . . . . . . . . . . . . 182 2.9 The Structure of Interactions in Nature . . . . . . . . . . . . . . . . . . . 183 2.9.1 The Electromagnetic Field as Generalized Curvature . . 183 2.9.2 Physics and Modern Differential Geometry . . . . . . . . . . 184 3. The Challenge of Different Scales in Nature . . . . . . . . . . . . . . 187 3.1 The Trouble with Scale Changes . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.2 Wilson’s Renormalization Group Theory in Physics . . . . . . . . . 189 3.2.1 A New Paradigm in Physics . . . . . . . . . . . . . . . . . . . . . . . 191
- 18. Contents XV 3.2.2 Screening of the Coulomb Field and the Renormaliza- tion Group of Lie Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3.2.3 The Running Coupling Constant and the Asymptotic Freedom of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 3.2.4 The Quark Confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . 204 3.2.5 Proton Decay and Supersymmetric Grand Unification . 205 3.2.6 The Adler–Bell–Jackiw Anomaly . . . . . . . . . . . . . . . . . . . 205 3.3 Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 3.4 A Glance at Conformal Field Theories . . . . . . . . . . . . . . . . . . . . 207 Part II. Basic Techniques in Mathematics 4. Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1 Power Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 4.2 Deformation Invariance of Integrals . . . . . . . . . . . . . . . . . . . . . . . 212 4.3 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.4 Cauchy’s Residue Formula and Topological Charges. . . . . . . . . 213 4.5 The Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.6 Gauss’ Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . 215 4.7 Compactification of the Complex Plane . . . . . . . . . . . . . . . . . . . 217 4.8 Analytic Continuation and the Local-Global Principle . . . . . . . 218 4.9 Integrals and Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 219 4.10 Domains of Holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.11 A Glance at Analytic S-Matrix Theory . . . . . . . . . . . . . . . . . . . . 224 4.12 Important Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5. A Glance at Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.1 Local and Global Properties of the Universe . . . . . . . . . . . . . . . 227 5.2 Bolzano’s Existence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.3 Elementary Geometric Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.4 Manifolds and Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.5 Topological Spaces, Homeomorphisms, and Deformations . . . . 235 5.6 Topological Quantum Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.6.1 The Genus of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.6.2 The Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . 242 5.6.3 Platonic Solids and Fullerenes . . . . . . . . . . . . . . . . . . . . . . 244 5.6.4 The Poincaré–Hopf Theorem for Velocity Fields . . . . . . 245 5.6.5 The Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . 246 5.6.6 The Morse Theorem on Critical Points of Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.6.7 Magnetic Fields, the Gauss Integral, and the Linking Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 5.6.8 Electric Fields, the Kronecker Integral, and the Mapping Degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
- 19. XVI Contents 5.6.9 The Heat Kernel and the Atiyah–Singer Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.6.10 Knots and Topological Quantum Field Theory . . . . . . . 263 5.7 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 5.7.1 The Topological Character of the Electron Spin . . . . . . 265 5.7.2 The Hopf Fibration of the 3-Dimensional Sphere . . . . . 268 5.7.3 The Homotopy Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 5.7.4 Grassmann Manifolds and Projective Geometry . . . . . . 274 5.8 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 6. Many-Particle Systems in Mathematics and Physics . . . . . . 277 6.1 Partition Function in Statistical Physics . . . . . . . . . . . . . . . . . . . 279 6.2 Euler’s Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.3 Discrete Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.4 Integral Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.5 The Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.5.1 The Prime Number Theorem – a Pearl of Mathematics 291 6.5.2 The Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.5.3 Dirichlet’s L-Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier . . . . . . . . . . . . . . . . . . . . . . . 305 6.7.1 The Generalized Mellin Transformation . . . . . . . . . . . . . 305 6.7.2 Dirichlet Series and their Special Values . . . . . . . . . . . . . 309 6.7.3 Application: the Casimir Effect. . . . . . . . . . . . . . . . . . . . . 312 6.7.4 Asymptotics of Series of the Form f(nt) . . . . . . . . . . 317 7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 7.1 Geometrization of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 7.2 Ariadne’s Thread in Quantum Field Theory . . . . . . . . . . . . . . . 326 7.3 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 7.4 Finite-Dimensional Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 335 7.5 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 7.6 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 7.7 Lie’s Logarithmic Trick for Matrix Groups . . . . . . . . . . . . . . . . . 345 7.8 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.9 Basic Notions in Quantum Physics. . . . . . . . . . . . . . . . . . . . . . . . 349 7.9.1 States, Costates, and Observables . . . . . . . . . . . . . . . . . . 350 7.9.2 Observers and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 354 7.10 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces . . . . . . . . 359 7.12 The Trace of a Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 363 7.13 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
- 20. Contents XVII 7.14 Probability and Hilbert’s Spectral Family of an Observable . . 368 7.15 Transition Probabilities, S-Matrix, and Unitary Operators . . . 370 7.16 The Magic Formulas for the Green’s Operator . . . . . . . . . . . . . . 372 7.16.1 Non-Resonance and Resonance . . . . . . . . . . . . . . . . . . . . . 373 7.16.2 Causality and the Laplace Transform . . . . . . . . . . . . . . . 377 7.17 The Magic Dyson Formula for the Retarded Propagator . . . . . 381 7.17.1 Lagrange’s Variation of the Parameter . . . . . . . . . . . . . . 383 7.17.2 Duhamel’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 7.17.3 The Volterra Integral Equation. . . . . . . . . . . . . . . . . . . . . 386 7.17.4 The Dyson Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 7.18 The Magic Dyson Formula for the S-Matrix . . . . . . . . . . . . . . . 390 7.19 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 7.19.1 The Schrödinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 392 7.19.2 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 7.19.3 The Dirac Interaction Picture . . . . . . . . . . . . . . . . . . . . . . 394 7.20 Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 7.20.1 Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 7.20.2 Partial Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . 401 7.20.3 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . 409 7.20.4 Functional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 7.21 The Discrete Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . 416 7.21.1 The Magic Feynman Propagator Formula. . . . . . . . . . . . 417 7.21.2 The Magic Formula for Time-Ordered Products . . . . . . 422 7.21.3 The Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 7.22 Causal Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 7.22.1 The Wick Rotation Trick for Vacuum Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 7.22.2 The Magic Gell-Mann–Low Reduction Formula . . . . . . 427 7.23 The Magic Gaussian Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 7.23.1 The One-Dimensional Prototype . . . . . . . . . . . . . . . . . . . 428 7.23.2 The Determinant Trick. . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 7.23.3 The Zeta Function Trick . . . . . . . . . . . . . . . . . . . . . . . . . . 434 7.23.4 The Moment Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 7.23.5 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . 435 7.24 The Rigorous Response Approach to Finite Quantum Fields . 438 7.24.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 7.24.2 Discrete Space-Time Manifold . . . . . . . . . . . . . . . . . . . . . 441 7.24.3 The Principle of Critical Action . . . . . . . . . . . . . . . . . . . . 445 7.24.4 The Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 7.24.5 The Global Quantum Action Principle . . . . . . . . . . . . . . 447 7.24.6 The Magic Quantum Action Reduction Formula for Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 7.24.7 The Magic LSZ Reduction Formula for Scattering Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
- 21. XVIII Contents 7.24.8 The Local Quantum Action Principle . . . . . . . . . . . . . . . 452 7.24.9 Simplifying the Computation of Quantum Effects. . . . . 454 7.24.10 Reduced Correlation Functions . . . . . . . . . . . . . . . . . . . . 455 7.24.11 The Mean Field Approximation . . . . . . . . . . . . . . . . . . . 456 7.24.12 Vertex Functions and the Effective Action . . . . . . . . . . 457 7.25 The Discrete ϕ4 -Model and Feynman Diagrams . . . . . . . . . . . . 459 7.26 The Extended Response Approach . . . . . . . . . . . . . . . . . . . . . . . . 477 7.27 Complex-Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 7.28 The Method of Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . 487 7.29 The Formal Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 8. Rigorous Finite-Dimensional Perturbation Theory . . . . . . . . 497 8.1 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.1.1 Non-Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.1.2 Resonance, Regularizing Term, and Bifurcation . . . . . . 499 8.1.3 The Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . 502 8.1.4 The Main Bifurcation Theorem . . . . . . . . . . . . . . . . . . . . 503 8.2 The Rellich Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 8.3 The Trotter Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 8.4 The Magic Baker–Campbell–Hausdorff Formula . . . . . . . . . . . . 508 8.5 Regularizing Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 8.5.1 The Weierstrass Product Theorem . . . . . . . . . . . . . . . . . . 509 8.5.2 The Mittag–Leffler Theorem . . . . . . . . . . . . . . . . . . . . . . . 510 8.5.3 Regularization of Divergent Integrals. . . . . . . . . . . . . . . . 511 8.5.4 The Polchinski Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 513 9. Fermions and the Calculus for Grassmann Variables . . . . . . 515 9.1 The Grassmann Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 9.2 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 9.3 Calculus for One Grassmann Variable . . . . . . . . . . . . . . . . . . . . . 516 9.4 Calculus for Several Grassmann Variables . . . . . . . . . . . . . . . . . 517 9.5 The Determinant Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 9.6 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . 519 9.7 The Fermionic Response Model . . . . . . . . . . . . . . . . . . . . . . . . . . 519 10. Infinite-Dimensional Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . 521 10.1 The Importance of Infinite Dimensions in Quantum Physics. . 521 10.1.1 The Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . . . 521 10.1.2 The Trouble with the Continuous Spectrum . . . . . . . . . 524 10.2 The Hilbert Space L2(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.2.1 Measure and Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 10.2.2 Dirac Measure and Dirac Integral . . . . . . . . . . . . . . . . . . 529 10.2.3 Lebesgue Measure and Lebesgue Integral . . . . . . . . . . . . 530 10.2.4 The Fischer–Riesz Theorem . . . . . . . . . . . . . . . . . . . . . . . 531 10.3 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
- 22. Contents XIX 10.3.1 Gauss’ Method of Least Squares . . . . . . . . . . . . . . . . . . . . 532 10.3.2 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 533 10.3.3 Continuous Fourier Transform . . . . . . . . . . . . . . . . . . . . . 535 10.4 The Dirichlet Problem in Electrostatics as a Paradigm . . . . . . 540 10.4.1 The Variational Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 10.4.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 10.4.3 The Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 547 10.4.4 Weierstrass’ Counterexample. . . . . . . . . . . . . . . . . . . . . . . 549 10.4.5 Typical Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 10.4.6 The Functional Analytic Existence Theorem . . . . . . . . . 555 10.4.7 Regularity of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . 558 10.4.8 The Beauty of the Green’s Function . . . . . . . . . . . . . . . . 560 10.4.9 The Method of Orthogonal Projection . . . . . . . . . . . . . . 564 10.4.10 The Power of Ideas in Mathematics . . . . . . . . . . . . . . . . 567 10.4.11 The Ritz Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 10.4.12 The Main Existence Principle . . . . . . . . . . . . . . . . . . . . . 569 11. Distributions and Green’s Functions . . . . . . . . . . . . . . . . . . . . . . 575 11.1 Rigorous Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 11.1.1 The Discrete Dirac Delta Function . . . . . . . . . . . . . . . . . 580 11.1.2 Prototypes of Green’s Functions . . . . . . . . . . . . . . . . . . . . 581 11.1.3 The Heat Equation and the Heat Kernel . . . . . . . . . . . . 586 11.1.4 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 11.1.5 The Schrödinger Equation and the Euclidean Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 11.2 Dirac’s Formal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 11.2.1 Dirac’s Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 11.2.2 Density of a Mass Distribution . . . . . . . . . . . . . . . . . . . . . 591 11.2.3 Local Functional Derivative . . . . . . . . . . . . . . . . . . . . . . . . 591 11.2.4 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 11.2.5 Formal Dirac Calculus and the Fourier Transform . . . . 596 11.2.6 Formal Construction of the Heat Kernel . . . . . . . . . . . . . 606 11.3 Laurent Schwartz’s Rigorous Approach . . . . . . . . . . . . . . . . . . . . 607 11.3.1 Physical Measurements and the Idea of Averaging . . . . 607 11.3.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 11.3.3 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 11.3.4 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 11.4 Hadamard’s Regularization of Integrals . . . . . . . . . . . . . . . . . . . . 618 11.4.1 Regularization of Divergent Integrals. . . . . . . . . . . . . . . . 618 11.4.2 The Sokhotski Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 11.4.3 Steinmann’s Renormalization Theorem . . . . . . . . . . . . . . 620 11.4.4 Regularization Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 11.5 Renormalization of the Anharmonic Oscillator . . . . . . . . . . . . . 625 11.5.1 Renormalization in a Nutshell. . . . . . . . . . . . . . . . . . . . . . 625 11.5.2 The Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
- 23. XX Contents 11.5.3 The Nonlinear Problem and Non-Resonance . . . . . . . . . 629 11.5.4 The Nonlinear Problem, Resonance, and Bifurcation . . 630 11.5.5 The Importance of the Renormalized Green’s Function 632 11.5.6 The Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . 633 11.6 The Importance of Algebraic Feynman Integrals . . . . . . . . . . . . 634 11.6.1 Wick Rotation and Cut-Off . . . . . . . . . . . . . . . . . . . . . . . . 634 11.6.2 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . 636 11.6.3 Weinberg’s Power-Counting Theorem . . . . . . . . . . . . . . . 638 11.6.4 Integration Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 11.7 Fundamental Solutions of Differential Equations . . . . . . . . . . . . 644 11.7.1 The Newtonian Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 646 11.7.2 The Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 11.7.3 The Beauty of Hironaka’s Theorem . . . . . . . . . . . . . . . . . 647 11.8 Functional Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 11.8.1 The Feynman Path Integral for the Heat Equation. . . . 651 11.8.2 Diffusion, Brownian Motion, and the Wiener Integral . 654 11.8.3 The Method of Quantum Fluctuations . . . . . . . . . . . . . . 655 11.8.4 Infinite-Dimensional Gaussian Integrals and Zeta Function Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 11.8.5 The Euclidean Trick and the Feynman Path Integral for the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 658 11.9 A Glance at Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 660 11.9.1 The Fourier–Laplace Transform . . . . . . . . . . . . . . . . . . . . 660 11.9.2 The Riemann–Hilbert Problem . . . . . . . . . . . . . . . . . . . . . 662 11.9.3 The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 11.9.4 Symmetry and Special Functions . . . . . . . . . . . . . . . . . . . 664 11.9.5 Tempered Distributions as Boundary Values of Ana- lytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 11.10 The Trouble with the Euclidean Trick . . . . . . . . . . . . . . . . . . . . 666 12. Distributions and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 12.1 The Discrete Dirac Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 12.1.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 12.1.2 The Four-Dimensional Discrete Dirac Delta Function . 670 12.1.3 Rigorous Discrete Dirac Calculus . . . . . . . . . . . . . . . . . . . 673 12.1.4 The Formal Continuum Limit . . . . . . . . . . . . . . . . . . . . . . 673 12.2 Rigorous General Dirac Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 675 12.2.1 Eigendistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 12.2.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 12.2.3 The von Neumann Spectral Theorem . . . . . . . . . . . . . . . 678 12.2.4 The Gelfand–Kostyuchenko Spectral Theorem . . . . . . . 679 12.2.5 The Duality Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 12.2.6 Dirac’s Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 12.2.7 The Schwartz Kernel Theorem . . . . . . . . . . . . . . . . . . . . . 681 12.3 Fundamental Limits in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
- 24. Contents XXI 12.3.1 High-Energy Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 12.3.2 Thermodynamic Limit and Phase Transitions . . . . . . . . 682 12.3.3 Adiabatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 12.3.4 Singular Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 12.4 Duality in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 12.4.1 Particles and de Broglie’s Matter Waves . . . . . . . . . . . . . 690 12.4.2 Time and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 12.4.3 Time and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 12.4.4 Position and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 692 12.4.5 Causality and Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . 695 12.4.6 Strong and Weak Interaction . . . . . . . . . . . . . . . . . . . . . . 702 12.5 Microlocal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 12.5.1 Singular Support of a Distribution . . . . . . . . . . . . . . . . . . 704 12.5.2 Wave Front Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 12.5.3 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . 714 12.5.4 Short-Wave Asymptotics for Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 12.5.5 Diffraction of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 12.5.6 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . 728 12.5.7 Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . 728 12.6 Multiplication of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 12.6.1 Laurent Schwartz’s Counterexample . . . . . . . . . . . . . . . . 730 12.6.2 Hörmander’s Causal Product . . . . . . . . . . . . . . . . . . . . . . 732 Part III. Heuristic Magic Formulas of Quantum Field Theory 13. Basic Strategies in Quantum Field Theory . . . . . . . . . . . . . . . . 739 13.1 The Method of Moments and Correlation Functions. . . . . . . . . 742 13.2 The Power of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 13.3 The Relation Between the S-Matrix and the Correlation Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 13.4 Perturbation Theory and Feynman Diagrams . . . . . . . . . . . . . . 747 13.5 The Trouble with Interacting Quantum Fields. . . . . . . . . . . . . . 748 13.6 External Sources and the Generating Functional . . . . . . . . . . . . 749 13.7 The Beauty of Functional Integrals . . . . . . . . . . . . . . . . . . . . . . . 751 13.7.1 The Principle of Critical Action . . . . . . . . . . . . . . . . . . . . 752 13.7.2 The Magic Feynman Representation Formula . . . . . . . . 753 13.7.3 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 13.7.4 Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 13.7.5 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 13.7.6 The Magic Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . 756 13.8 Quantum Field Theory at Finite Temperature . . . . . . . . . . . . . 757 13.8.1 The Partition Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 13.8.2 The Classical Hamiltonian Approach . . . . . . . . . . . . . . . . 760
- 25. XXII Contents 13.8.3 The Magic Feynman Functional Integral for the Par- tition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 13.8.4 The Thermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . 763 14. The Response Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 14.1 The Fourier–Minkowski Transform . . . . . . . . . . . . . . . . . . . . . . . . 770 14.2 The ϕ4 -Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 14.2.1 The Classical Principle of Critical Action . . . . . . . . . . . . 774 14.2.2 The Response Function and the Feynman Propagator . 774 14.2.3 The Extended Quantum Action Functional . . . . . . . . . . 782 14.2.4 The Magic Quantum Action Reduction Formula for Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 14.2.5 The Magic LSZ Reduction Formula for the S-Matrix . . 785 14.2.6 The Local Quantum Action Principle . . . . . . . . . . . . . . . 787 14.2.7 The Mnemonic Functional Integral . . . . . . . . . . . . . . . . . 787 14.2.8 Bose–Einstein Condensation of Dilute Gases . . . . . . . . . 788 14.3 A Glance at Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 789 14.3.1 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 791 14.3.2 The Principle of Critical Action . . . . . . . . . . . . . . . . . . . . 792 14.3.3 The Gauge Field Approach . . . . . . . . . . . . . . . . . . . . . . . . 794 14.3.4 The Extended Action Functional with Source Term . . . 797 14.3.5 The Response Function for Photons . . . . . . . . . . . . . . . . 799 14.3.6 The Response Function for Electrons . . . . . . . . . . . . . . . 800 14.3.7 The Extended Quantum Action Functional . . . . . . . . . . 801 14.3.8 The Magic Quantum Action Reduction Formula . . . . . . 803 14.3.9 The Magic LSZ Reduction Formula . . . . . . . . . . . . . . . . . 803 14.3.10 The Mnemonic Functional Integral . . . . . . . . . . . . . . . . 804 15. The Operator Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 15.1 The ϕ4 -Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 15.1.1 The Lattice Approximation . . . . . . . . . . . . . . . . . . . . . . . . 815 15.1.2 Fourier Quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 15.1.3 The Free 2-Point Green’s Function . . . . . . . . . . . . . . . . . 820 15.1.4 The Magic Dyson Formula for the S-Matrix . . . . . . . . . 822 15.1.5 The Main Wick Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 824 15.1.6 Transition Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 15.1.7 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 15.1.8 Scattering Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . 839 15.1.9 General Feynman Rules for Particle Scattering . . . . . . . 843 15.1.10 The Magic Gell-Mann–Low Reduction Formula for Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845 15.2 A Glance at Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 846 15.3 The Role of Effective Quantities in Physics . . . . . . . . . . . . . . . . 847 15.4 A Glance at Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 15.4.1 The Trouble with the Continuum Limit . . . . . . . . . . . . . 850
- 26. Contents XXIII 15.4.2 Basic Ideas of Renormalization . . . . . . . . . . . . . . . . . . . . . 850 15.4.3 The BPHZ Renormalization . . . . . . . . . . . . . . . . . . . . . . . 853 15.4.4 The Epstein–Glaser Approach . . . . . . . . . . . . . . . . . . . . . 854 15.4.5 Algebraic Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . 858 15.4.6 The Importance of Hopf Algebras . . . . . . . . . . . . . . . . . . 859 15.5 The Convergence Problem in Quantum Field Theory . . . . . . . . 860 15.5.1 Dyson’s No-Go Argument . . . . . . . . . . . . . . . . . . . . . . . . . 860 15.5.2 The Power of the Classical Ritt Theorem in Quantum Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861 15.6 Rigorous Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 15.6.1 Axiomatic Quantum Field Theory . . . . . . . . . . . . . . . . . . 866 15.6.2 The Euclidean Strategy in Constructive Quantum Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 15.6.3 The Renormalization Group Method . . . . . . . . . . . . . . . . 872 16. Peculiarities of Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 16.1 Basic Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 16.2 The Principle of Critical Action . . . . . . . . . . . . . . . . . . . . . . . . . . 878 16.3 The Language of Physicists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 16.4 The Importance of the Higgs Particle . . . . . . . . . . . . . . . . . . . . . 886 16.5 Integration over Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 16.6 The Magic Faddeev–Popov Formula and Ghosts . . . . . . . . . . . . 888 16.7 The BRST Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 16.8 The Power of Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 16.8.1 Physical States, Unphysical States, and Cohomology . . 893 16.8.2 Forces and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 16.8.3 The Cohomology of Geometric Objects . . . . . . . . . . . . . . 896 16.8.4 The Spectra of Atoms and Cohomology . . . . . . . . . . . . . 899 16.8.5 BRST Symmetry and the Cohomology of Lie Groups . 900 16.9 The Batalin–Vilkovisky Formalism. . . . . . . . . . . . . . . . . . . . . . . . 903 16.10 A Glance at Quantum Symmetries . . . . . . . . . . . . . . . . . . . . . . 904 17. A Panorama of the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 17.1 Introduction to Quantum Field Theory . . . . . . . . . . . . . . . . . . . . 907 17.2 Standard Literature in Quantum Field Theory . . . . . . . . . . . . . 910 17.3 Rigorous Approaches to Quantum Field Theory . . . . . . . . . . . . 911 17.4 The Fascinating Interplay between Modern Physics and Math- ematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913 17.5 The Monster Group, Vertex Algebras, and Physics . . . . . . . . . . 919 17.6 Historical Development of Quantum Field Theory . . . . . . . . . . 924 17.7 General Literature in Mathematics and Physics . . . . . . . . . . . . 925 17.8 Encyclopedias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 17.9 Highlights of Physics in the 20th Century. . . . . . . . . . . . . . . . . . 926 17.10 Actual Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928
- 27. XXIV Contents Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 A.2 The International System of Units . . . . . . . . . . . . . . . . . . . . . . . . 934 A.3 The Planck System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936 A.4 The Energetic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942 A.5 The Beauty of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . 944 A.6 The Similarity Principle in Physics . . . . . . . . . . . . . . . . . . . . . . . 946 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995
- 28. Prologue We begin with some quotations which exemplify the philosophical underpin- nings of this work. Theoria cum praxi. Gottfried Wilhelm Leibniz (1646–1716) It is very difficult to write mathematics books today. If one does not take pains with the fine points of theorems, explanations, proofs and corollaries, then it won’t be a mathematics book; but if one does these things, then the reading of it will be extremely boring. Johannes Kepler (1571–1630) Astronomia Nova The interaction between physics and mathematics has always played an important role. The physicist who does not have the latest mathemati- cal knowledge available to him is at a distinct disadvantage. The mathe- matician who shies away from physical applications will most likely miss important insights and motivations. Marvin Schechter Operator Methods in Quantum Mechanics5 In 1967 Lenard and I found a proof of the stability of matter. Our proof was so complicated and so unilluminating that it stimulated Lieb and Thirring to find the first decent proof. Why was our proof so bad and why was theirs so good? The reason is simple. Lenard and I began with mathe- matical tricks and hacked our way through a forest of inequalities without any physical understanding. Lieb and Thirring began with physical under- standing and went on to find the appropriate mathematical language to make their understanding rigorous. Our proof was a dead end. Theirs was a gateway to the new world of ideas collected in this book. Freeman Dyson From the Preface to Elliott Lieb’s Selecta6 The state of the art in quantum field theory. One of the intellectual fathers of quantum electrodynamics is Freeman Dyson (born in 1923) who 5 North-Holland, Amsterdam, 1982. 6 Stability of Matter: From Atoms to Stars, Springer, New York, 2002.
- 29. 2 Prologue works at the Institute for Advanced Study in Princeton.7 He characterizes the state of the art in quantum field theory in the following way: All through its history, quantum field theory has had two faces, one looking outward, the other looking inward. The outward face looks at nature and gives us numbers that we can calculate and compare with experiments. The inward face looks at mathematical concepts and searches for a con- sistent foundation on which to build the theory. The outward face shows us brilliantly successful theory, bringing order to the chaos of particle in- teractions, predicting experimental results with astonishing precision. The inward face shows us a deep mystery. After seventy years of searching, we have found no consistent mathematical basis for the theory. When we try to impose the rigorous standards of pure mathematics, the theory becomes undefined or inconsistent. From the point of view of a pure mathematician, the theory does not exist. This is the great unsolved paradox of quantum field theory. To resolve the paradox, during the last twenty years, quantum field theo- rists have become string-theorists. String theory is a new version of quan- tum field theory, exploring the mathematical foundations more deeply and entering a new world of multidimensional geometry. String theory also brings gravitation into the picture, and thereby unifies quantum field the- ory with general relativity. String theory has already led to important advances in pure mathematics. It has not led to any physical predictions that can be tested by experiment. We do not know whether string theory is a true description of nature. All we know is that it is a rich treasure of new mathematics, with an enticing promise of new physics. During the coming century, string theory will be intensively developed, and, if we are lucky, tested by experiment.8 Five golden rules. When writing the latex file of this book on my com- puter, I had in mind the following five quotations. Let me start with the mathematician Hermann Weyl (1885–1930) who became a follower of Hilbert in Göttingen in 1930 and who left Germany in 1933 when the Nazi regime came to power. Together with Albert Einstein (1879–1955) and John von Neumann (1903–1957), Weyl became a member of the newly founded Insti- tute for Advanced Study in Princeton, New Jersey, U.S.A. in 1933. Hermann Weyl wrote in 1938:9 The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader an impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon. 7 F. Dyson, Selected Papers of Freeman Dyson with Commentaries, Amer. Math. Soc., Providence, Rhode Island, 1996. We recommend reading this fascinating volume. 8 In: Quantum Field Theory, A 20th Century Profile. Edited by A. Mitra, Indian National Science Academy and Hindustan Book Agency, 2000 (reprinted with permission). 9 H. Weyl, The Classical Groups, Princeton University Press, 1938 (reprinted with permission).
- 30. Prologue 3 For his fundamental contributions to electroweak interaction inside the Stan- dard Model in particle physics, the physicist Steven Weinberg (born 1933) was awarded the Nobel prize in physics in 1979 together with Sheldon Glashow (born 1932) and Abdus Salam (1926–1996). On the occasion of a conference on the interrelations between mathematics and physics in 1986, Weinberg pointed out the following: 10 I am not able to learn any mathematics unless I can see some problem I am going to solve with mathematics, and I don’t understand how anyone can teach mathematics without having a battery of problems that the student is going to be inspired to want to solve and then see that he or she can use the tools for solving them. For his theoretical investigations on parity violation under weak interaction, the physicist Cheng Ning Yang (born 1922) was awarded the Nobel prize in physics in 1957 together with Tsung Dao Lee (born 1926). In an interview, Yang remarked:11 In 1983 I gave a talk on physics in Seoul, South Korea. I joked “There exist only two kinds of modern mathematics books: one which you cannot read beyond the first page and one which you cannot read beyond the first sentence. The Mathematical Intelligencer later reprinted this joke of mine. But I suspect many mathematicians themselves agree with me. The interrelations between mathematics and modern physics have been pro- moted by Sir Michael Atiyah (born 1929) on a very deep level. In 1966, the young Atiyah was awarded the Fields medal. In an interview, Atiyah empha- sized the following: 12 The more I have learned about physics, the more convinced I am that physics provides, in a sense, the deepest applications of mathematics. The mathematical problems that have been solved, or techniques that have arisen out of physics in the past, have been the lifeblood of mathematics. . . The really deep questions are still in the physical sciences. For the health of mathematics at its research level, I think it is very important to maintain that link as much as possible. The development of modern quantum field theory has been strongly influ- enced by the pioneering ideas of the physicist Richard Feynman (1918–1988). In 1965, for his contributions to the foundation of quantum electrodynam- ics, Feynman was awarded the Nobel prize in physics together with Julian Schwinger (1918–1994) and Sin-Itiro Tomonaga (1906–1979). In the begin- ning of the 1960s, Feynman held his famous Feynman lectures at the Califor- nia Institute of Technology in Pasadena. In the preface to the printed version of the lectures, Feynman told his students the following: Finally, may I add that the main purpose of my teaching has not been to prepare you for some examination – it was not even to prepare you to 10 Notices Amer. Math. Soc. 33 (1986), 716–733 (reprinted with permission). 11 Mathematical Intelligencer 15 (1993), 13–21 (reprinted with permission). 12 Mathematical Intelligencer 6 (1984), 9–19 (reprinted with permission).
- 31. 4 Prologue serve industry or military. I wanted most to give you some appreciation of the wonderful world and the physicist’s way of looking at it, which, I believe, is a major part of the true culture of modern times.13 The fascination of quantum field theory. As a typical example, let us consider the anomalous magnetic moment of the electron. This is given by the following formula Me = − e 2me geS with the so-called gyromagnetic factor ge = 2(1 + a) of the electron. Here, me is the mass of the electron, −e is the negative electric charge of the electron. The spin vector S has the length /2, where h denotes Planck’s quantum of action, and := h/2π. High-precision experiments yield the value aexp = 0.001 159 652 188 4 ± 0.000 000 000 004 3. Quantum electrodynamics is able to predict this result with high accuracy. The theory yields the following value a = α 2π − 0.328 478 965 α π 2 +(1.175 62 ± 0.000 56) α π 3 −(1.472 ± 0.152) α π 4 (0.1) with the electromagnetic fine structure constant α = 1 137.035 989 500 ± 0.000 000 061 . Explicitly, a = 0.001 159 652 164 ± 0.000 000 000 108 . The error is due to the uncertainty of the electromagnetic fine structure constant α. Observe that 9 digits coincide between the experimental value aexp and the theoretical value a. The theoretical result (0.1) represents a highlight in modern theoretical physics. The single terms with respect to powers of the fine structure constant α have been obtained by using the method of perturbation theory. In order to represent graphically the single terms appearing in perturbation theory, Richard Feynman (1918–1988) invented the language of Feynman diagrams in about 1945.14 For example, Fig. 0.1 shows some simple Feynman diagrams 13 R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures in Physics, Addison-Wesley, Reading, Massachusetts, 1963. 14 For the history of this approach, see the quotation on page 27.
- 32. Prologue 5 (a) γ γ e− e− - - - (b) ? 6 e− e− - - - - - - - - γ e− e− Fig. 0.1. Feynman diagrams for the Compton scattering between electrons and photons. In higher order of perturbation theory, the Feynman diagrams become more and more complex. In particular, in order to get the α3 -term of (0.1), one has to use 72 Feynman diagrams. The computation of the α3 -term has taken 20 years. The α4 -term from (0.1) is based on 891 Feynman diagrams. The computation has been done mainly by numerical approximation methods. This needed years of su- percomputer time.15 The mathematical situation becomes horrible because of the following fact. Many of the Feynman diagrams correspond to divergent higher- dimensional integrals called algebraic Feynman integrals. Physicists invented the ingenious method of renormalization in order to give the apparently meaningless integrals a precise interpretation. Renormaliza- tion plays a fundamental role in quantum field theory. Physicists do not expect that the perturbation series (0.1) is part of a convergent power series expansion with respect to the variable α at the origin. Suppose that there would exist such a convergent power series expansion a = ∞ n=1 anαn , |α| ≤ α0 near the origin α = 0. This series would then converge for small negative values of α. However, such a negative coupling constant would correspond to a repelling force which destroys the system. This argument is due to Dyson.16 Therefore, we do not expect that the series (0.1) is convergent. In Sect. 15.5.2, we will show that each formal power series expansion is indeed the asymptotic expansion of some analytic function in an angular domain, by the famous 1916 Ritt theorem in mathematics. 15 See M. Veltman, Facts and Mysteries in Elementary Particle Physics, World Sci- entific, Singapore, 2003; this is a beautiful history of modern elementary particle physics. 16 F. Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85 (1952), 631–632.
- 33. 6 Prologue From the mathematical point of view, the best approach to renormaliza- tion was created by Epstein and Glaser in 1973. The Epstein–Glaser theory avoids the use of divergent integrals and their regularization, but relies on the power of the modern theory of distributions (generalized functions). Physicists have also computed the magnetic moment of the myon. As for the electron, the coincidence between theory and experiment is of fan- tastic accuracy. Here, the theory takes all of the contributions coming from electromagnetic, weak, strong, and gravitative interaction into account.17 It is a challenge for the mathematics of the future to completely un- derstand formula (0.1). Let us now briefly discuss the content of Volumes I through VI of this mono- graph. Volume I. The first volume entitled Basics in Mathematics and Physics is structured in the following way. Part I: Introduction • Chapter 1: Historical Introduction • Chapter 2: Phenomenology of the Standard Model in Particle Physics • Chapter 3: The Challenge of Different Scales in Nature. Part II: Basic Techniques in Mathematics • Chapter 4: Analyticity • Chapter 5: A Glance at Topology • Chapter 6: Many-Particle Systems • Chapter 7: Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory • Chapter 8: Rigorous Finite-Dimensional Perturbation Theory • Chapter 9: Calculus for Grassmann Variables • Chapter 10: Infinite-Dimensional Hilbert Spaces • Chapter 11: Distributions and Green’s Functions • Chapter 12: Distributions and Quantum Physics. Part III: Heuristic Magic Formulas of Quantum Field Theory • Chapter 13: Basic Strategies in Quantum Field Theory • Chapter 14: The Response Approach • Chapter 15: The Operator Approach • Chapter 16: Peculiarities of Gauge Theories • Chapter 17: A Panorama of the Literature. Describing the content of Volume I by a parable, we will first enter a mountain railway in order to reach easily and quickly the top of the desired mountain and to admire the beautiful mountain ranges. Later on we will try to climb to the top along the rocks. 17 See M. Böhm, A. Denner, and H. Joos, Gauge Theories of the Strong and Elec- troweak Interaction, Teubner, Stuttgart, 2001, p. 80.
- 34. Prologue 7 In particular, the heuristic magic formulas from Part III should help the reader to understand quickly the language of physicists in quantum field theory. These magic formulas are non-rigorous from the mathematical point of view, but they are extremely useful for computing physical effects. Modern elementary particle physics is based on the Standard Model in particle physics introduced in the late 1960s and the early 1970s. Before studying thoroughly the Standard Model in the next volumes, we will discuss the phenomenology of this model in the present volume. It is the goal of quantum field theory to compute • the cross sections of scattering processes in particle accelerators which char- acterize the behavior of the scattered particles, • the masses of stable elementary particles (e.g., the proton mass as a bound state of three quarks), and • the lifetime of unstable elementary particles in particle accelerators. To this end, physicists use the methods of perturbation theory. Fortunately enough, the computations can be based on only a few basic formulas which we call magic formulas. The magic formulas of quantum theory are extremely useful for describing the experimental data observed in particle accelerators, but they are only valid on a quite formal level. This difficulty is typical for present quantum field theory. To help the reader in understanding the formal approach used in physics, we consider the finite-dimensional situation in Chapter 6. In the finite-dimensional case, we will rigorously prove all of the magic formulas used by physicists in quantum field theory. Furthermore, we relate physics to the following fields of mathematics: • causality and the analyticity of complex-valued functions, • many-particle systems, the Casimir effect in quantum field theory, and number theory, • propagation of physical effects, distributions (generalized functions), and the Green’s function, • rigorous justification of the elegant Dirac calculus, • duality in physics (time and energy, time and frequency, position and mo- mentum) and harmonic analysis (Fourier series, Fourier transformation, Laplace transformation, Mellin transformation, von Neumann’s general op- erator calculus for self-adjoint operators, Gelfand triplets and generalized eigenfunctions), • the relation between renormalization, resonances, and bifurcation, • dynamical systems, Lie groups, and the renormalization group, • fundamental limits in physics, • topology in physics (Chern numbers and topological quantum numbers), • probability, Brownian motion, and the Wiener integral,
- 35. 8 Prologue • the Feynman path integral, • Hadamard’s integrals and algebraic Feynman integrals. In fact, this covers a broad range of physical and mathematical subjects. Volume II. The second volume entitled Quantum Electrodynamics con- sists of the following parts. Part I : Introduction • Chapter 1: Mathematical Principles of Natural Philosophy • Chapter 2: The Basic Strategy of Extracting Finite Information from Infinities • Chapter 3: A Glance at the Mathematical Structure Behind Renor- malization in Physics. Part II : Basic Ideas in Quantum Mechanics • Chapter 4: The Principle of Critical Action and the Harmonic Oscilla- tor as a Paradigm • Chapter 5: Quantization of the Harmonic Oscillator. Part III : Scattering Processes in Quantum Mechanics • Chapter 6: Quantum Particles on the Real Line – Ariadne’s Thread in Scattering Theory • Chapter 7: Three-Dimensional Motion of a Quantum Particle • Chapter 8: Observables and Operator Theory – the Trouble with Infi- nite Dimensions • Chapter 9: The Hydrogen Atom as a Paradigm in Functional Analysis • Chapter 10: Large Atoms and Molecules. Part IV : Relativistic Fields • Chapter 11: Einstein’s Theory of Special Relativity • Chapter 12: The Electromagnetic Field • Chapter 13: Dirac’s Relativistic Electron. Part V : Basic Ideas of Quantum Field Theory • Chapter 14: Chain of Quantized Harmonic Oscillators • Chapter 15: Quantum Electrodynamics. Part VI : Renormalization • Chapter 16: Radiative Corrections • Chapter 17: A Glance at the Bogoliubov–Parasiuk–Hepp–Zimmermann Renormalization • Chapter 18: The Beauty of the Epstein–Glaser Approach • Chapter 19: A Glance at Algebraic Renormalization • Chapter 20: The Renormalization Flow Method. The final goal of quantum field theory is the foundation of a rigorous math- ematical theory which contains the Standard Model as a special low-energy approximation. At present we are far away from reaching this final goal. From the physical point of view, the most successful quantum field theory is quan- tum electrodynamics. This will be studied in Volume II along with some
- 36. Prologue 9 applications to important physical processes like Compton scattering be- tween electrons and photons, the spontaneous emission of light by molecules, Cherenkov radiation of fast electrons, the Lamb shift in the hydrogen spec- trum, the anomalous magnetic moment of the electron, and the Hawking radiation of black holes. We also study the physics and mathematics behind the crucial phenomenon of renormalization and the change of scales in physics culminating in the modern theory of the renormalization group. Generally, we try to include both interesting mathematics and interesting physics. In partic- ular, we will discuss the relation of renormalization in physics to the following mathematical subjects: Euler’s gamma function, the Riemann–Liouville in- tegral, and dimensional regularization; Borel summation of divergent series; pseudo-convergence of iterative methods for ill-posed problems, Hopf alge- bras and Rota–Baxter algebras; theory of categories; wave front sets and the theory of distributions, Euler’s and Feynman’s mathemagics. Volume III. The fundamental forces in the universe are described by gauge field theories which generalize both Gauss’ surface theory and Maxwell’s theory of electromagnetism. The third volume entitled Gauge The- ories is divided into the following parts. Part I : The Euclidean Space as a Paradigm • Chapter 1: The Algebraic Structure of the Euclidean Space • Chapter 2: The Differential Structure of the Euclidean Space • Chapter 3: Changing Observers and Tensor Analysis. Part II: Interactions and Gauge Theory • Chapter 4: Basic Principles in Physics • Chapter 5: Observers, Physical Fields, and Bundles • Chapter 6: Symmetry Breaking in Physics • Chapter 7: Gauss’ Surface Theory. Part III: Fundamental Gauge Theories in Physics • Chapter 8: Einstein’s Theory of Special Relativity • Chapter 9: Maxwell’s Theory of Electromagnetism • Chapter 10: Dirac’s Relativistic Electron • Chapter 11: The Standard Model in Particle Physics • Chapter 12: Einstein’s Theory of General Relativity and Cosmology • Chapter 13: A Glance at String Theory and the Graviton • Chapter 14: The Sigma Model. Interestingly enough, it turns out that the Standard Model in particle physics is related to many deep questions in both mathematics and physics. We will see that the question about the structure of the fundamental forces in nature has influenced implicitly or explicitly the development of a large part of mathematics. One of our heros will be Carl Friedrich Gauss (1777–1855), one of the greatest mathematicians of all time. We will encounter his highly influential work again and again. In the German Museum in Munich, one can read the following inscription under Gauss’ impressive portrait:
- 37. 10 Prologue His spirit lifted the deepest secrets of numbers, space, and nature; he mea- sured the orbits of the planets, the form and the forces of the earth; in his mind he carried the mathematical science of a coming century. On the occasion of Gauss’ death, Sartorius von Waltershausen wrote the following in 1855: From time to time in the past, certain brilliant, unusually gifted person- alities have arisen from their environment, who by virtue of the creative power of their thoughts and the energy of their actions have had such an overall positive influence on the intellectual development of mankind, that they at the same time stand tall as markers between the centuries. . . Such epoch-making mental giants in the history of mathematics and the natural sciences are Archimedes of Syracuse in ancient times, Newton toward the end of the dark ages and Gauss in our present day, whose shining, glorious career has come to an end after the cold hand of death touched his at one time deeply-thinking head on February 23 of this year. Another hero will be Bernhard Riemann (1826–1866) – a pupil of Gauss. Riemann’s legacy influenced strongly mathematics and physics of the 20th century, as we will show in this treatise.18 The two Standard Models in modern physics concerning cosmology and elementary particles are closely related to modern differential geometry. This will be thoroughly studied in Volume III. We will show that both Einstein’s general theory of relativity and the Standard Model in particle physics are gauge theories. From the mathematical point of view, the fundamental forces in nature are curvatures of appropriate fiber bundles. Historically, math- ematicians have tried to understand the curvature of geometric objects. At the very beginning, there was Gauss’ theorema egregium19 telling us that cur- vature is an intrinsic property of a surface. On the other side, in the history of physics, physicists have tried to understand the forces in nature. Nowadays we know that both mathematicians and physicists have approached the same goal coming from different sides. We can summarize this by saying briefly that force = curvature. For the convenience of the reader, we will also discuss in Volume II that many of the mathematical concepts arising in quantum field theory are rooted in the geometry of the Euclidean space (e.g., Lie groups and Lie algebras, operator algebras, Grassmann algebras, Clifford algebras, differential forms and coho- mology, Hodge duality, projective structures, symplectic structures, contact structures, conformal structures, Riemann surfaces, and supersymmetry). Volume IV. Quantum physics differs from classical relativistic field the- ories by adding the process of quantization. From the physical point of view, 18 We also recommend the beautiful monograph written by Krzysztof Maurin, Rie- mann’s Legacy, Kluwer, Dordrecht, 1997. 19 The Latin expression theorema egregium means the beautiful theorem.
- 38. Prologue 11 there appear additional quantum effects based on random quantum fluctu- ations. From the mathematical point of view, one has to deform classical theories in an appropriate way. Volume IV is devoted to the mathematical and physical methods of quantization. For this, we coin the term Quantum Mathematics. Volume IV represents the first systematic textbook on Quan- tum Mathematics. This volume will be divided into the following parts. Part I: Finite Quantum Mathematics • Chapter 1: Many-Particle Systems, Probability, and Information • Chapter 2: Quantum Systems and Hilbert Spaces • Chapter 3: Quantum Information. Part II: Symmetry and Quantization • Chapter 4: Finite Groups and the Paradigm of Symmetric Functions • Chapter 5: Compact Lie Groups • Chapter 6: The Poincaré Group • Chapter 7: Applications to Analytic S-Matrix Theory • Chapter 8: The Yang–Baxter Equation, Hopf Algebras, and Quantum Groups. Part III: Operators Algebras and Quantization • Chapter 9: States and Observables • Chapter 10: Local Operator Algebras and Causality. Part IV: Topology and Quantization • Chapter 11: Basic Ideas • Chapter 12: Cohomology and Homology in Physics • Chapter 13: The Atiyah–Singer Index Theorem and Spectral Geome- try. Part V : Interactions between Mathematics and Physics • Chapter 14: Geometric Quantization • Chapter 15: Stochastic Quantization • Chapter 16: Progress in Mathematics by Using Ideas Originated in Quantum Physics • Chapter 17: Mathematics – a Cosmic Eye of Humanity. Typically, quantum fields are interacting physical systems with an infinite number of degrees of freedom and very strong singularities. In mathematics, • interactions lead to nonlinear terms, and • infinite-dimensional systems are described in terms of functional analysis. Therefore, the right mathematical setting for quantum field theory is nonlin- ear functional analysis. This branch of mathematics has been very successful in the rigorous treatment of nonlinear partial differential equations concern- ing elasticity and plasticity theory, hydrodynamics, and the theory of general relativity. But the actual state of the art of nonlinear functional analysis does not yet allow for the rigorous investigation of realistic models in quan- tum field theory, like the Standard Model in particle physics. Physicists say, we cannot wait until mathematics is ready. Therefore, we have to develop our
- 39. 12 Prologue own non-rigorous methods, and we have to check the success of our methods by comparing them with experimental data. In order to help mathematicians to enter the world of physicists, we will proceed as follows. (i) Rigorous methods: We first develop quantum mathematics in finite- dimensional spaces. In this case, we can use rigorous methods based on the theory of Hilbert spaces, operator algebras, and discrete functional integrals. (ii) Formal methods. The formulas from (i) can be generalized in a straight- forward, but formal way to infinite-dimensional systems. This way, the mathematician should learn where the formulas of the physicists come from and how to handle these formulas in order to compute physical effects. What remains is to solve the open problem of rigorous justification. The point will be the investigation of limits and pseudo-limits if the number of particles goes to infinity. By a pseudo-limit, we understand the extraction of maximal information from an ill-defined object, as in the method of renormalization. The experi- ence of physicists and mathematicians shows that we cannot expect the limits or pseudo-limits to exist for all possible quantities. The rule of thumb is as follows: concentrate on quantities which can be measured in physical experi- ments. This seriously complicates the subject. We will frequently encounter the Feynman functional integral. From the mnemonic point of view, this is a marvellous tool. But it lacks mathematical rigor. We will follow the advise given by Evariste Galois (1811–1832): Unfortunately what is little recognized is that the most worthwhile scien- tific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties. Volume V. The mathematician should notice that it is the ultimate goal of a physicist to compute real numbers which can be measured in physical experiments. For reaching this goal, the physicist mixes rigorous arguments with heuristic ones in an ingenious way. In order to make mathematicians familiar with this method of doing science, in Volume V we will study the physics of the Standard Model in particle physics. In particular, we will show how to compute a number of physical effects. In this respect, symmetries will play an important role. For example, this will concern the representation the- ory of compact Lie groups (e.g., gauge groups in gauge theory), noncompact Lie groups (the Poincaré group and its universal covering group in relativis- tic physics), infinite-dimensional Lie algebras (e.g., the Virasoro algebra in string theory), and supersymmetric generalizations. Volume VI. The last volume will be devoted to combining the Standard Model in particle physics with gravitation. We will study several possible approaches to this fascinating, but still completely open problem. The leading candidate is string theory. In connection with the string theory of physicists,
- 40. Prologue 13 a completely new way of thinking has emerged which we will call physical mathematics, a term already used in Kishore Marathe’s nice survey article on the role of knot theory in modern mathematics, physics, and biology.20 Distinguish the following: • By mathematical physics, we traditionally understand a branch of mathe- matics which answers questions coming from physics by applying rigorous mathematical methods. The heart of mathematical physics are mathemat- ical proofs (e.g., existence proofs for solutions of partial differential equa- tions or operator equations). • By physical mathematics, we understand a branch of physics which is mo- tivated by the question about the fundamental forces in nature. Using physical pictures, physicists are able to conjecture deep mathematical re- sults (e.g., the existence and the properties of new topological invariants for manifolds and knots). The heart of physical mathematics is physical intuition, but not the mathematical proof. The hero of physical mathematics is the physicist Edward Witten (born 1951) from the Institute for Advanced Study in Princeton. At the International Congress of Mathematicians in Kyoto (Japan) in 1990, Witten was awarded the Fields medal. In the last 15 years, physical mathematics was very suc- cessful in feeding fascinating new ideas into mathematics. The main method of physical mathematics goes like this: • start with a model in quantum field theory based on an appropriate La- grangian; • quantize this model by means of the corresponding Feynman functional integral; • extract essential information from the functional integral by using the method of stationary phase. The point is that this method yields beautiful mathematical conjectures, but it is not able to give rigorous proofs. Unfortunately, for getting proofs, math- ematicians have to follow quite different sophisticated routes. It is a challenge to mathematicians to understand better the magic weapon of physical math- ematics. The magic weapon of physical mathematics will be called the Witten functor. This functor translates physical structures into mathematical structures. With respect to the Witten functor, one observes the following general evo- lution principle in mathematics. 20 K. Marathe, A chapter in physical mathematics: theory of knots in the sciences, pp. 873–888. In: Mathematics Unlimited – 2001 and Beyond edited by B. En- gquist and W. Schmid, Springer, Berlin, 2001.
- 41. 14 Prologue (i) From quantity to quality: In the 1920s, modern algebra was founded by passing from concrete mathematical objects like numbers to abstract mathematical structures like groups, rings, fields, and algebras. Here, one only considers the relations between the objects, but not the individual structure of the objects. For example, Emmy Noether emphasized in the 1920s that, in the setting of algebraic topology created by Poincaré at the end of the 19th century, it is very useful to pass from Betti numbers to homology groups. In turn, it was discovered in the 1930s that cohomology groups are in fact richer in structure than homology groups. The point is that cohomology groups possess a natural multiplicative structure which generates the cohomology ring of topological spaces. For example, the product S2 ×S4 of a 2-dimensional sphere with a 4-dimensional sphere has the same homology and cohomology groups as the 3-dimensional complex projective space P3 C. However, these two manifolds are not topologically equivalent, since their cohomology rings are different. (ii) Combining abstract structures with each other: For example, Lie groups are obtained by combining the notion of manifold with the notion of group. In turn, fiber bundles occur by combining manifolds with Lie groups. (iii) Functors between abstract structures: In the late 1940s, the theory of categories emerged in the context of algebraic topology. For example, the Galois functor simplifies the study of field extensions by mapping fields to groups. The Lie functor simplifies the investigation of Lie groups by mapping Lie groups to Lie algebras. Moreover, the homology functor sim- plifies the structural analysis of topological spaces (geometric objects) by mapping topological spaces to groups called homology groups. Combin- ing the homology functor with the general concept of duality, we arrive at the cohomology functor which maps topological spaces to cohomology groups. Cohomology plays a fundamental role in modern physics. (iv) Statistics of abstract structures: In physical mathematics, one considers the statistics of physical states in terms of functional integrals. The point is that the states are equivalence classes of mathematical structures. In the language of mathematics, the physical state spaces are moduli spaces. For example, in string theory the states of strings are Riemann surfaces modulo conformal equivalence. Thus, the state space of all those strings which possess a fixed genus g is nothing other than Riemann’s famous moduli space Mg which can be described by a universal covering space of Mg called the Teichmüller space Tg. Mathematicians know that the theory of moduli spaces is a challenge in algebraic geometry, since such objects carry singularities, as a rule. Physicists expect that those singu- larities are responsible for essential physical effects. Another typical feature of physical mathematics is the description of many- particle systems by partition functions which encode essential information. As we will show, the Feynman functional integral is nothing other than a
- 42. Prologue 15 partition function which encodes the essential properties of quantum fields. From the physical point of view, the Riemann zeta function is a partition function for the infinite system of prime numbers. The notion of partition function unifies • statistical physics, • quantum mechanics, • quantum field theory, and • number theory. Summarizing, I dare say that The most important notion of modern physics is the Feynman func- tional integral as a partition function for the states of many-particle systems. It is a challenge of mathematics to understand this notion in a better way than known today. A panorama of mathematics. For the investigation of problems in quantum field theory, we need a broad spectrum of mathematical branches. This concerns (a) algebra, algebraic geometry, and number theory, (b) analysis and functional analysis, (c) geometry and topology, (d) information theory, theory of probability, and stochastic processes, (e) scientific computing. In particular, we will deal with the following subjects: • Lie groups and symmetry, Lie algebras, Kac–Moody algebras (gauge groups, permutation groups, the Poincaré group in relativistic physics, conformal symmetry), • graded Lie algebras (supersymmetry between bosons and fermions), • calculus of variations and partial differential equations (the principle of critical action), • distributions (also called generalized functions) and partial differential equations (Green’s functions, correlation functions, propagator kernels, or resolvent kernels), • distributions and renormalization (the Epstein–Glaser approach to quan- tum field theory via the S-matrix), • geometric optics and Huygens’ principle (symplectic geometry, contact transformations, Poisson structures, Finsler geometry), • Einstein’s Brownian motion, diffusion, stochastic processes and the Wiener integral, Feynman’s functional integrals, Gaussian integrals in the theory of probability, Fresnel integrals in geometric optics, the method of stationary phase,
- 43. 16 Prologue • non-Euclidean geometry, covariant derivatives and connections on fiber bundles (Einstein’s theory of general relativity for the universe, and the Standard Model in elementary particle physics), • the geometrization of physics (Minkowski space geometry and Einstein’s theory of special relativity, pseudo-Riemannian geometry and Einstein’s theory of general relativity, Hilbert space geometry and quantum states, projective geometry and quantum states, Kähler geometry and strings, conformal geometry and strings), • spectral theory for operators in Hilbert spaces and quantum systems, • operator algebras and many-particle systems (states and observables), • quantization of classical systems (method of operator algebras, Feynman’s functional integrals, Weyl quantization, geometric quantization, deforma- tion quantization, stochastic quantization, the Riemann–Hilbert problem, Hopf algebras and renormalization), • combinatorics (Feynman diagrams, Hopf algebras), • quantum information, quantum computers, and operator algebras, • conformal quantum field theory and operator algebras, • noncommutative geometry and operator algebras, • vertex algebras (sporadic groups, monster and moonshine), • Grassmann algebras and differential forms (de Rham cohomology), • cohomology, Hilbert’s theory of syzygies, and BRST quantization of gauge field theories, • number theory and statistical physics, • topology (mapping degree, Hopf bundle, Morse theory, Lyusternik–Schni- relman theory, homology, cohomology, homotopy, characteristic classes, ho- mological algebra, K-theory), • topological quantum numbers (e.g., the Gauss–Bonnet theorem, Chern classes, and Chern numbers, Morse numbers, Floer homology), • the Riemann–Roch–Hirzebruch theorem and the Atiyah–Singer index the- orem, • analytic continuation, functions of several complex variables (sheaf theory), • string theory, conformal symmetry, moduli spaces of Riemann surfaces, and Kähler manifolds. The role of proofs. Mathematics relies on proofs based on perfect logic. The reader should note that, in this treatise, the terms • proposition, • theorem (important proposition), and • proof are used in the rigorous sense of mathematics. In addition, for helping the reader in understanding the basic ideas, we also use ‘motivations’, ‘formal proofs’, ‘heuristic arguments’ and so on, which emphasize intuition, but lack rigor. Because of the rich material to be studied, it is impossible to provide the reader with full proofs for all the different subjects. However, for missing
- 44. Prologue 17 proofs we add references to carefully selected sources. Many of the missing proofs can be found in the following monographs: • E. Zeidler, Applied Functional Analysis, Vols. 1, 2, Springer, New York. 1995. • E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. 1–4, Sprin- ger, New York, 1985–88. For getting an overview, the reader should also consult the following book:21 • E. Zeidler (Ed.), Oxford Users’ Guide to Mathematics, Oxford University Press, 2004 (1300 pages). At the end of the Oxford Users’ Guide to Mathematics, the interested reader may find a chronology of mathematics and physics from ancient to present times embedded in the cultural history of mankind. Perspectives. At the International Congress of Mathematicians in Paris in 1900, Hilbert formulated 23 open problems for the mathematics of the 20th century. Many of them have been solved.22 Hilbert said the following to the audience in 1900: Each progress in mathematics is based on the discovery of stronger tools and easier methods, which at the same time makes it easier to understand earlier methods. By making these stronger tools and easier methods his own, it is possible for the individual researcher to orientate himself in the different branches of mathematics. . . When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related prob- lems. . . The organic unity of mathematics is inherent in the nature of this sci- ence, for mathematics is the foundation of all exact knowledge of natural phenomena. For the 21th century, the open problem of quantum field theory represents a great challenge. It is completely unclear how long the solution of this prob- lem will take. In fact, there are long-term problems in mathematics. As an example, let us consider Fermat’s Last Theorem where the solution needed more than 350 years. In ancient times, Pythagoras (508–500 B.C.) knew that the equation x2 + y2 = z2 has an infinite number of integer solutions (e.g., x = 3, y = 4, z = 5). In 1637, Pierre de Fermat (1601–1665), claimed that the equation xn + yn = zn , n = 3, 4, . . . 21 The German version reads as E. Zeidler, Teubner-Taschenbuch der Mathematik, Vols. 1, 2, Teubner, Wiesbaden, 2003. The English translation of the second volume is in preparation. 22 See D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 8 (1902), 437– 479, and B. Yandell, The Honors Class: Hilbert’s Problems and Their Solvers, Natick, Massachusetts, 2001.
- 45. 18 Prologue has no nontrivial integer solution. In his copy of the Arithmetica by Diophan- tus (250 A.C.), Fermat wrote the following: It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which however the margin is not large enough to contain. The history of this problem can be found in the bestseller by Simon Singh, Fermat’s Last Theorem: The Story of a Riddle that Confounded the World’s Greatest Minds for 358 Years, Fourth Estate, London, 1997. The final proof was given by Andrew Wiles (born 1953) in Princeton in 1994.23 The proof, based on the Galois functor, is of extraordinary complexity, and it uses many sophisticated tools from number theory and algebraic geometry developed in the 19th and 20th century. However, in the sense of Hilbert’s philosophy for hard problems quoted above, let us describe the basic idea behind the solution. In this connection, it turns out that there is a beautiful geometric result of general interest behind Fermat’s Last Theorem.24 The fundamental geometric result tells us that25 (M) Each elliptic curve is modular. Roughly speaking, the proof of Fermat’s last theorem proceeds now like this: (i) Suppose that Fermat’s claim is wrong. Then, there exists a nontrivial triplet x, y, z of integers such that xn + yn = zn for some fixed natural number n ≥ 3. (ii) The triplet x, y, z can be used in order to construct a specific elliptic curve (the Frey curve), which is not modular, a contradiction to (M). It remains to sketch the meaning of the geometric principle (M). To begin with, consider the equation of the complex unit circle x2 + y2 = 1 where x and y are complex parameters. The unit circle allows a parametriza- tion either by periodic functions, x = cos ϕ, y = sin ϕ, ϕ ∈ C, or by rational functions, 23 A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 142 (1994), 443–551. 24 We refer to the beautiful lecture given by Don Zagier, Leçon inaugurale, Jeudi 17 Mai 2001, Collège de France, Paris. See also H. Darmon, A proof of the full Shimura–Taniyama–Weil conjecture is announced, Notices Amer. Math. Soc. 46 (1999), 1397–1401. Much background material can be found in the fascinating textbook by Y. Hellagouarch, Invitation to the Mathematics of Fermat–Wiles, Academic Press, New York. 25 A comprehensive survey article on modular forms can be found in Zagier (1995).
- 46. Prologue 19 x = 2 1 + t2 − 1, y = 2t 1 + t2 , t ∈ C, provided we set t := tan ϕ 2 . Recall that each compact Riemann surface of genus zero is conformally and topologically equivalent to the real two- dimensional sphere called the Riemann sphere. In particular, the complex unit circle considered above is such a Riemann surface of genus zero. More- over, compact Riemann surfaces of genus one are conformally and topologi- cally equivalent to some real two-dimensional torus. Such Riemann surfaces are also called elliptic curves. For example, given three pairwise different complex numbers e1, e2, e3, the equation y2 = 4(x − e1)(x − e2)(x − e3) with complex parameters x and y represents an elliptic curve which allows the global parametrization x = ℘(t), y = ℘ (t), t ∈ C by the Weierstrass ℘-function. This is an elliptic (i.e., double-periodic) func- tion whose two complex periods depend on e1, e2, e3. The fundamental geo- metric result reads now as follows: (i) Each compact Riemann surface of genus zero (i.e., each complex curve of circle type) allows two global parametrizations by either periodic func- tions or rational functions. (ii) Each compact Riemann surface of genus one (i.e., each elliptic curve) allows two global parametrizations by either double-periodic functions or modular functions. (iii) Each compact Riemann surface of genus g ≥ 2 can be globally parame- trized by automorphic functions.26 The global parametrization (i) of elliptic curves by elliptic functions is one of the most famous results of 19th century mathematics due to Jacobi, Riemann, and Weierstrass. The general result (ii) on the global parametrization of el- liptic curves by modular functions was only proved in 1999, i.e., it was shown that the full Shimura–Taniyama–Weil conjecture is true. Statement (iii) rep- resents the famous uniformization theorem for compact Riemann surfaces which was proved independently by Koebe and Poincaré in 1907 after strong efforts made by Poincaré and Klein. The existence of double-periodic func- tions was discovered by Gauss in 1797 while studying the geometric properties of the lemniscate introduced by Jakob Bernoulli (1654–1705). Therefore, the innocent looking three statements (i), (ii), (iii) above are the result of 200 years of intense mathematical research. Summarizing, in the sense of Hilbert, 26 Much material on Riemann surfaces, elliptic curves, zeta functions, Galois theory, and so on, can be found in the volume edited by M. Waldschmidt et al., From Number Theory to Physics, Springer, New York, 1995.
- 47. 20 Prologue the famous Fermat conjecture could finally be solved because it could be re- duced to the general idea of modular curves. In a fascinating essay on the future of mathematics, Arthur Jaffe (born 1937) from Harvard University wrote the following:27 Mathematical research should be as broad and as original as possible, with very long range-goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be dis- covered. Studying the physics and mathematics of the fundamental forces in na- ture, there arises the question about the philosophical background. Concern- ing this, let me finish with two quotations. Erich Worbs writes in his Gauss biography: Sartorius von Waltershausen reports that Gauss once said there were ques- tions of infinitely higher value than the mathematical ones, namely, those about our relation to God, our determination, and our future. Only, he con- cluded, their solutions lie far beyond our comprehension, and completely outside the field of science. In the Harnack Building of the Max-Planck Society in Berlin, one can read the following words by Johann Wolfgang von Goethe: The greatest joy of a thinking man is to have explored the explorable and just to admire the unexplorable. 27 Ordering the universe: the role of mathematics, Notices Amer. Math. Soc. 236 (1984), 589–608.
- 48. 1. Historical Introduction If we wish to foresee the future of mathematics our proper course is to study the history and present condition of the science. Henri Poincaré (1854–1912) It is not the knowledge but the learning, not the possessing, but the earn- ing, not the being there but the getting there, which gives us the greatest pleasure. Carl Friedrich Gauss (1777–1855) to his Hungarian friend Janos Bólyai For me, as a young man, Hilbert (1858–1943) became the kind of math- ematician which I admired, a man with an enormous power of abstract thought, combined with a fully developed sense for the physical reality. Norbert Wiener (1894–1964) In the fall 1926, the young John von Neumann (1903–1957) arrived in Göttingen to take up his duties as Hilbert’s assistant. These were the hec- tic years during which quantum mechanics was developing with breakneck speed, with a new idea popping up every few weeks from all over the hori- zon. The theoretical physicists Born, Dirac, Heisenberg, Jordan, Pauli, and Schrödinger who were developing the new theory were groping for adequate mathematical tools. It finally dawned upon them that their ‘observables’ had properties which made them look like Hermitean operators in Hilbert space, and that by an extraordinary coincidence, the ‘spectrum’ of Hilbert (which he had chosen around 1900 from a superficial analogy) was to be the central conception in the explanation of the ‘spectra’ of atoms. It was therefore natural that they should enlist Hilbert’s help to put some mathematical sense in their formal computations. With the assistance of Nordheim and von Neumann, Hilbert first tried integral operators in the space L2, but that needed the use of the Dirac delta function δ, a concept which was for the mathematicians of that time self-contradictory. John von Neumann therefore resolved to try another approach. Jean Dieudonné (1906–1992) History of Functional Analysis1 Stimulated by an interest in quantum mechanics, John von Neumann be- gan the work in operator theory which he was to continue as long as he lived. Most of the ideas essential for an abstract theory had already been 1 North–Holland, Amsterdam, 1981 (reprinted with permission).
- 49. 22 1. Historical Introduction developed by the Hungarian mathematician Fryges Riesz, who had estab- lished the spectral theory for bounded Hermitean operators in a form very much like as regarded now standard. Von Neumann saw the need to ex- tend Riesz’s treatment to unbounded operators and found a clue to doing this in Carleman’s highly original work on integral operators with singular kernels. . . The result was a paper von Neumann submitted for publication to the Mathematische Zeitschrift but later withdrew. The reason for this with- drawal was that in 1928 Erhard Schmidt and myself, independently, saw the role which could be played in the theory by the concept of the adjoint operator, and the importance which should be attached to self-adjoint operators. When von Neumann learned from Professor Schmidt of this ob- servation, he was able to rewrite his paper in a much more satisfactory and complete form. . . Incidentally, for permission to withdraw the paper, the publisher exacted from Professor von Neumann a promise to write a book on quantum mechanics. The book soon appeared and has become one of the classics of modern physics.2 Marshall Harvey Stone (1903–1989) 1.1 The Revolution of Physics At the beginning of the 20th century, Max Planck (1858–1947) and Albert Einstein (1879–1955) completely revolutionized physics. In 1900, Max Planck derived the universal radiation law for stars by postulating that The action in our world is quantized. Let us discuss this fundamental physical principle. The action is the most im- portant physical quantity in nature. For any process, the action is the product of energy × time for a small time interval. The total action during a fixed time interval is then given by an integral summing over small time intervals. The fundamental principle of least (or more precisely, critical) action tells us the following: A process in nature proceeds in such a way that the action becomes minimal under appropriate boundary conditions. More precisely, the action is critical. This means that the first variation of the action S vanishes, δS = 0. In 1918 Emmy Noether (1882–1935) proved a fundamental mathematical theorem. The famous Noether theorem tells us that Conservation laws in physics are caused by symmetries of physical systems. 2 J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955 (first German edition: Springer, Berlin, 1932). This quo- tation is taken from F. Browder (Ed.), Functional Analysis and Related Fields, Springer, Berlin, 1970 (reprinted with permission).
- 50. 1.1 The Revolution of Physics 23 To explain this basic principle for describing nature in terms of mathemat- ics, consider our solar system. The motion of the sun and the planets only depends on the initial positions and initial velocities. Obviously, the motion of the solar system is invariant under time translations, spatial translations, and rotations. This is responsible for conservation of energy, momentum, and angular momentum, respectively. For example, invariance under time trans- lations means the following. If a process of the physical system is possible, x = x(t), then each process is also possible which is obtained by time trans- lation, x = x(t + const). According to Planck, the smallest amount of action in nature is equal to h = 6.260 0755 · 10−34 Js (1.1) where 1 Joule = 1 kg · m2 /s2 . We also introduce := h/2π. The universal constant h is the famous Planck quantum of action (or the Planck constant). Observe that the action of typical processes in daily life has the magnitude of 1 Js. Therefore, the Planck constant is tiny. Nevertheless, the quantization of action has enormous consequences. For example, consider a mass point on the real line which moves periodically, q(t) = const · sin(ωt) where t denotes time, and ω is called the angular frequency of the harmonic oscillator. Since the sine function has period 2π, the harmonic oscillator has the time period T = 2π/ω. By definition, the frequency ν is the number of oscillations per second. Hence T = 1/ν, and ω = 2πν. If E denotes the energy of the harmonic oscillator, then the product ET is a typical action value for the oscillations of the harmonic oscillator. Therefore, according to Planck’s quantization of action, it seems to be quite natural to postulate that ET = nh for n = 0, 1, 2, . . . This yields Planck’s quantization rule for the energy of the harmonic oscillator, E = nω, n = 0, 1, 2, . . . from the year 1900. About 25 years later, the young physicist Werner Heisen- berg (1901–1976) invented the full quantization procedure of classical me- chanics. Using implicitly the commutation rule qp − pq = i (1.2) for the position q and the momentum p of a quantum particle, Heisenberg obtained the precise formula E = n + 1 2 ω, n = 0, 1, 2, . . . (1.3) for the quantized energy levels of a harmonic oscillator. Heisenberg’s quantum mechanics changed completely the paradigm of physics. In classical physics,
- 51. 24 1. Historical Introduction observables are real numbers. In Heisenberg’s approach, observables are ab- stract quantities which obey certain commutation rules. More than fifty years before Heisenberg’s discovery, the great Norwegian mathematician Sophus Lie (1842–1899) found out that commutation rules of type (1.2) play a fun- damental role when trying to study continuous symmetry groups by means of linearization. In 1934, for this kind of algebraic structure, the term “Lie algebra” was coined by Hermann Weyl (1885–1955). Lie algebras and their generalizations to infinite dimensions, like the Virasoro algebra and super- symmetric algebras in string theory and conformal quantum field theory, are crucial for modern quantum physics. The Heisenberg formula (1.3) tells us that the ground state of each harmonic oscillator has the non-vanishing en- ergy E = ω 2 . (1.4) This fact causes tremendous difficulties in quantum field theories. Since a quantum field has an infinite number of degrees of freedom, the ground state has an infinite energy. There are tricks to cure the situation a little bit, but the infinite ground state energy is the deeper reason for the appearance of nasty divergent quantities in quantum field theory. Physicists have developed the ingenious method of renormalization in order to extract finite quanti- ties that can be measured in physical experiments. Surprisingly enough, in quantum electrodynamics there is an extremely precise coincidence with the renormalized theoretical values and the values measured in particle acceler- ator experiments. No one understands this. Nowadays many physicists are convinced that this approach is not the final word. There must be a deeper theory behind. One promising candidate is string theory. At the end of his life, Albert Einstein wrote the following about his first years. Between the ages of 12–16, I familiarized myself with the elements of math- ematics. In doing so I had the good fortune of discovering books which were not too particular in their logical rigor. In 1896, at the age of 17, I entered the Swiss Institute of Technology (ETH) in Zurich. There I had excellent teachers, for example, Hurwitz (1859–1919) and Minkowski (1864–1909), so that I really could get a sound mathematical education. However, most of the time, I worked in the phys- ical laboratory, fascinated by the direct contact with experience. The rest of the time I used, in the main, to study at home the works of Kirchhoff (1824–1887), Helmholtz (1821–1894), Hertz (1857–1894), and so on. The fact that I neglected mathematics to a certain extent had its cause not merely in my stronger interest in the natural sciences than in mathemat- ics, but also in the following strange experience. I saw that mathematics was split up into numerous specialities, each of which could easily absorb the short life granted to us. Consequently, I saw myself in the position of Buridan’s ass which was unable to decide upon any specific bundle of hay. This was obviously due to the fact that my intuition was not strong enough in the field of mathematics in order to differentiate clearly that
- 52. 1.1 The Revolution of Physics 25 which was fundamentally important, and that which is really basic, from the rest of the more or less dispensable erudition, and it was not clear to me as a student that the approach to a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods. This only dawned upon me gradually after years of independent scientific work. True enough, physics was also divided into separate fields. In this field, however, I soon learned to scent out that which was able to lead to fundamentals.3 After his studies, Einstein got a position at the Swiss patent office in Bern. In 1905 Einstein published four fundamental papers on the theory of special relativity, the equivalence between mass and energy, the Brownian motion, and the light particle hypothesis in Volume 17 of the journal Annalen der Physik. The theory of special relativity completely changed our philosophy about space and time. According to Einstein, there is no absolute time, but time changes from observer to observer. This follows from the surprising fact that the velocity of light has the same value in each inertial system, which was established experimentally by Albert Michelson (1852–1931) in 1887. From his principle of relativity, Einstein deduced that a point particle of rest mass m0 and momentum vector p has a positive energy E given by E2 = m2 0c4 + c2 p2 (1.5) where c denotes the velocity of light in a vacuum. If the particle moves with sub-velocity of light, x = x(t), than it has the mass m = m0 1 − ẋ(t)2/c2 . (1.6) If the particle rests, then we get E = m0c2 . (1.7) This magic energy formula governs the energy production in our sun by helium synthesis. Thus, our life depends crucially on this formula. Unfortu- nately, the atomic bomb is based on this formula, too. Let us now discuss the historical background of Einstein’s light parti- cle hypothesis. Maxwell (1831–1879) conjectured in 1862 that light is an electromagnetic wave. In 1886 Heinrich Hertz established the existence of electromagnetic waves by a famous experiment carried out at Kiel Univer- sity (Germany). When electromagnetic radiation is incident on the surface 3 This is the English translation of Einstein’s handwritten letter copied in the following book: Albert Einstein als Philosoph und Naturforscher (Albert Ein- stein as philosopher and scientist). Edited by P. Schilpp, Kohlhammer Verlag, Stuttgart (printed with permission).
- 53. 26 1. Historical Introduction of a metal, it is observed that electrons may be ejected. This phenomenon is called the photoelectric effect. This effect was first observed by Heinrich Hertz in 1887. Fifteen years later, Philipp Lenard (1862–1947) observed that the maximum kinetic energy of the electrons does not depend on the intensity of light. In order to explain the photoelectric effect, Einstein postulated in 1905 that electromagnetic waves are quantized. That is, light consists of light particles (or light quanta) which were coined photons in 1926 by the physical chemist Gilbert Lewis. According to Einstein, a light particle (photon) has the energy E given by Planck’s quantum hypothesis, E = hν. (1.8) Here, ν is the frequency of light, which is related to the wave length λ by the dispersion relation λν = c. Hence E = hc/λ. This means that a blue photon has more energy than a red one. Since a photon moves with light speed, its rest mass must be zero. Thus, from (1.5) we obtain |p| = E/c. If we introduce the angular frequency ω = 2πν, then we obtain the final expression E = ω, p = k, |k| = ω c (1.9) for the energy E and the momentum vector p of a photon. Here, the wave vector k of length k = ω/c is parallel to the vector p. Nowadays we know that light particles are quanta, and that quantum particles are physical objects which possess a strange structure. Quanta combine features of both waves and particles. In the photoelectric effect, a photon hits an electron such that the electron leaves the metal. The energy of the electron is given by E = ω − W where the so-called work function W depends on the binding energy of the electrons in the atoms of the metal. This energy formula suggests that for small angular frequencies ω no electrons can leave the metal, since there would be E 0, a contradiction. In fact, this has been observed in experi- ments. Careful experiments were performed by Millikan (1868–1953) in 1916. He found out that a typical constant in his experiments coincided with the Planck constant, as predicted by Einstein. In 1921 Einstein was awarded the Nobel prize in physics for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect. As a curiosity let us mention, that Max Planck, while recommending Einstein enthusiastically for a membership in the Prussian Academy in Berlin, wrote the following: That sometimes, in his speculations, he went too far, such as, for example, in his hypothesis of the light quanta, should not be held too much against him.
- 54. 1.2 Quantization in a Nutshell 27 1.2 Quantization in a Nutshell In 1926 Born discovered the fundamental fact that quantum physics is intrin- sically connected with random processes. Hence the mathematical theory of probability plays a crucial role in quantum physics. Already Maxwell (1831– 1879) had emphasized: The true logic of this world lies in probability theory. Before discussing the randomness of quantum processes and the challenge of quantization, let us mention that Maxwell strongly influenced the physics of the 20th century. As we will show later on, Einstein’s theory of special relativity follows from the invariance of the Maxwell equations in electro- magnetism under Lorentz transformations. Moreover, the generalization of the Maxwell equations from the commutative gauge group U(1) to the non- commutative gauge groups SU(2) and SU(3) leads to the Standard Model in particle physics. Finally, statistical physics can be traced back to Maxwell’s statistical velocity distribution of molecules. From the physical point of view, quantum mechanics and quantum field theory are described best by the Feynman approach via Feynman diagrams, transition amplitudes, Feynman propagators (Green’s functions), and func- tional integrals. In order to make the reader familiar with the fascinating story of this approach, let us start with a quotation taken from Freeman Dyson’s book Disturbing the Universe, Harper Row, New York, 1979:4 Dick Feynman (1918–1988) was a profoundly original scientist. He refused to take anybody’s word for anything. This meant that he was forced to rediscover or reinvent for himself almost the whole physics. It took him five years of concentrated work to reinvent quantum mechanics. He said that he couldn’t understand the official version of quantum mechanics that was taught in the textbooks and so he had to begin afresh from the beginning. This was a heroic enterprise. He worked harder during those years than anybody else I ever knew. At the end he had his version of quantum mechanics that he could understand. . . The calculations that I did for Hans Bethe,5 using the orthodox method, took me several months of work and several hundred sheets of paper. Dick could get the same answer, calculating on a blackboard, in half an hour. . . In orthodox physics, it can be said: Suppose an electron is in this state at a certain time, then you calculate what it will do next by solving the Schrödinger equation introduced by Schrödinger in 1926. Instead of this, Dick simply said: 4 Reprinted by permission of Basic Books, a member of Perseus Books, L.L.C. 5 Hans Bethe (1906–2005) was awarded the 1967 Nobel prize in physics for his con- tributions to nuclear reactions, especially his discoveries concerning the energy production in stars. See H. Bethe, R. Bacher, and M. Livingstone, Basic Bethe: Seminal Articles on Nuclear Physics 1936–37, American Institute of Physics, 1986.
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. wonderful as they doubtless are, are indisputably eclipsed by the structures formed by many insects; and the regular villages of the beaver, by far the most sagacious architect amongst quadrupeds, must yield the palm to a wasp's nest. You will think me here guilty of exaggeration, and that, blinded by my attachment to a favourite pursuit, I am elevating the little objects, which I wish to recommend to your study, to a rank beyond their just claim. So far, however, am I from being conscious of any such prejudice, that I do not hesitate to go further, and assert that the pyramids of Egypt, as the work of man, are not more wonderful for their size and solidity than are the structures built by some insects. To describe the most remarkable of these is my present object: and that some method may be observed, I shall in this letter describe the habitations of insects living in a state of solitude, and built each by a single architect; and in a subsequent one, those of insects living in societies, built by the united labours of many. The former class may be conveniently subdivided into habitations built by the parent insect, not for its own use, but for the convenience of its future young; and those which are formed by the insect that inhabits them for its own accommodation. To the first I shall now call your attention. The solitary insects which construct habitations for their future young without any view to their own accommodation, chiefly belong to the order Hymenoptera, and are principally different species of wild bees. Of these the most simple are built by Colletes[758] succincta, fodiens, c. The situation which the parent bee chooses, is either the dry earth of a bank, or the vacuities of stone walls cemented with earth instead of mortar. Having excavated a cylinder about two inches in depth, running usually in a horizontal direction, the bee occupies it with three or four cells about half an inch long,
- 57. and one-sixth broad, shaped like a thimble, the end of one fitting into the mouth of another. The substance of which these cells are formed is two or three layers of a silky membrane, composed of a kind of glue secreted by the animal, resembling gold-beater's leaf, but much finer, and so thin and transparent that the colour of an included object may be seen through them. As soon as one cell is completed, the bee deposits an egg within, and nearly fills it with a paste composed of pollen and honey; which having done, she proceeds to form another cell, storing it in like manner until the whole is finished, when she carefully stops up the mouth of the orifice with earth. Our countryman Grew seems to have found a series of these nests in a singular situation—the middle of the pith of an old elder-branch—in which they were placed lengthwise one after another with a thin boundary between each[759]. Cells composed of a similar membranaceous substance, but placed in a different situation, are constructed by Anthidium manicatum[760]. This gay insect does not excavate holes for their reception, but places them in the cavities of old trees, or of any other object that suits its purpose. Sir Thomas Cullum discovered the nest of one in the inside of the lock of a garden-gate, in which I have also since twice found them. It should seem, however, that such situations would be too cold for the grubs without a coating of some non-conducting substance. The parent bee, therefore, after having constructed the cells, laid an egg in each, and filled them with a store of suitable food, plasters them with a covering of vermiform masses, apparently composed of honey and pollen; and having done this, aware, long before Count Rumford's experiments, what materials conduct heat most slowly, she attacks the woolly leaves of Stachys lanata, Agrostemma coronaria, and similar plants, and with her mandibles industriously scrapes off the wool, which with her fore legs she rolls into a little ball and carries to her nest. This wool she sticks upon the plaster that covers her cells, and thus closely envelops them with a warm coating of down impervious to every change of temperature[761].
- 58. The bee last described may be said to exercise the trade of a clothier. Another numerous family would be more properly compared to carpenters, boring with incredible labour out of the solid wood long cylindrical tubes, and dividing them into various cells. Amongst these, one of the most remarkable is Xylocopa[762] violacea, a large species, a native of Southern Europe, distinguished by beautiful wings of a deep violet colour, and found commonly in gardens, in the upright putrescent espaliers or vine-props of which, and occasionally in the garden seats, doors and window-shutters, she makes her nest. In the beginning of spring, after repeated and careful surveys, she fixes upon a piece of wood suitable for her purpose, and with her strong mandibles begins the process of boring. First proceeding obliquely downwards, she soon points her course in a direction parallel with the sides of the wood, and at length with unwearied exertion forms a cylindrical hole or tunnel not less than twelve or fifteen inches long and half an inch broad. Sometimes, where the diameter will admit of it, three or four of these pipes, nearly parallel with each other, are bored in the same piece. Herculean as this task, which is the labour of several days, appears, it is but a small part of what our industrious bee cheerfully undertakes. As yet she has completed but the shell of the destined habitation of her offspring; each of which, to the number of ten or twelve, will require a separate and distinct apartment. How, you will ask, is she to form these? With what materials can she construct the floors and ceilings? Why truly God doth instruct her to discretion and doth teach her. In excavating her tunnel she has detached a large quantity of fibres, which lie on the ground like a heap of saw-dust. This material supplies all her wants. Having deposited an egg at the bottom of the cylinder along with the requisite store of pollen and honey, she next, at the height of about three quarters of an inch, (which is the depth of each cell,) constructs of particles of the saw-dust glued together, and also to the sides of the tunnel, what may be called an annular stage or scaffolding. When this is sufficiently hardened, its interior edge affords support for a second ring of the same materials, and thus the ceiling is gradually formed of these concentric circles, till
- 59. there remains only a small orifice in its centre, which is also closed with a circular mass of agglutinated particles of saw-dust. When this partition, which serves as the ceiling of the first cell and the flooring of the second, is finished, it is about the thickness of a crown-piece, and exhibits the appearance of as many concentric circles as the animal has made pauses in her labour. One cell being finished, she proceeds to another, which she furnishes and completes in the same manner, and so on until she has divided her whole tunnel into ten or twelve apartments. Here, if you have followed me in this detail with the interest which I wish it to inspire, a query will suggest itself. It will strike you that such a laborious undertaking as the constructing and furnishing these cells, cannot be the work of one or even of two days. Considering that every cell requires a store of honey and pollen, not to be collected but with long toil, and that a considerable interval must be spent in agglutinating the floors of each, it will be very obvious to you that the last egg in the last cell must be laid many days after the first. We are certain, therefore, that the first egg will become a grub, and consequently a perfect bee, many days before the last. What then becomes of it? you will ask. It is impossible that it should make its escape through eleven superincumbent cells without destroying the immature tenants; and it seems equally impossible that it should remain patiently in confinement below them until they are all disclosed. This dilemma our heaven-taught architect has provided against. With forethought never enough to be admired she has not constructed her tunnel with one opening only, but at the further end has pierced another orifice, a kind of back-door, through which the insects produced by the first-laid eggs successively emerge into day. In fact, all the young bees, even the uppermost, go out by this road; for, by an exquisite instinct, each grub, when about to become a pupa, places itself in its cell with its head downwards, and thus is necessitated, when arrived at its last state, to pierce its cell in this direction[763].
- 60. Ceratina albilabris of Spinola, who has given an interesting account of its manners, forms its cell upon the general plan of the bee just described, but, more economical of labour, chooses a branch of briar or bramble, in the pith of which she excavates a canal about a foot long and one line, or sometimes more, in diameter, with from eight to twelve cells separated from each other by partitions of particles of pith glued together[764]. Such are the curious habitations of the carpenter bees. Next I shall introduce you to the not less interesting structures of another family which carry on the trade of masons, (Megachile muraria,) building their solid houses solely of artificial stone. The first step of the mother bee is to fix upon a proper situation for the future mansion of her offspring. For this she usually selects an angle, sheltered by any projection, on the south side of a stone wall. Her next care is to provide materials for the structure. The chief of these is sand, which she carefully selects grain by grain from such as contains some mixture of earth. These grains she glues together with her viscid saliva into masses the size of small shot, and transports by means of her jaws to the site of her castle[765]. With a number of these masses, which are the artificial stone of which her building is to be composed, united by a cement preferable to ours, she first forms the basis or foundation of the whole. Next she raises the walls of a cell, which is about an inch in length and half an inch broad, and before its orifice is closed in form resembles a thimble. This, after depositing an egg and a supply of honey and pollen, she covers in, and then proceeds to the erection of a second, which she finishes in the same manner, until the whole number, which varies from four to eight, is completed. The vacuities between the cells, which are not placed in any regular order, some being parallel to the wall, others perpendicular to it, and others inclined to it at different angles, this laborious architect fills up with the same material of which the cells are composed, and then bestows upon the whole group a common covering of coarser grains of sand. The form of the whole nest, which when finished is a solid mass of stone so hard as not to be
- 61. easily penetrated with the blade of a knife, is an irregular oblong of the same colour as the sand, and to a casual observer more resembling a splash of mud than an artificial structure. These bees sometimes are more economical of their labour, and repair old nests, for the possession of which they have very desperate combats. One would have supposed that the inhabitants of a castle so fortified might defy the attacks of every insect marauder. Yet an Ichneumon and a beetle (Clerus apiarius) both contrive to introduce their eggs into the cells, and the larvæ proceeding from them devour their inhabitants[766]. Other bees of the same family with that last described, use different materials in the construction of their nests. Some employ fine earth made into a kind of mortar with gluten. Another (Osmia[767] cærulescens), as we learn from De Geer, forms its nest of argillaceous earth mixed with chalk, upon stone walls, and sometimes probably nidificates in chalk-pits. O. bicornis selects the hollows of large stones for the site of its dwelling; while others prefer the holes in wood. The works thus far described require in general less genius than labour and patience: but it is far otherwise with the nests of the last tribe of artificers amongst wild bees, to which I shall advert—the hangers of tapestry, or upholsterers—those which line the holes excavated in the earth for the reception of their young, with an elegant coating of flowers or of leaves. Amongst the most interesting of these is Megachile[768] Papaveris, a species whose manners have been admirably described by Reaumur. This little bee, as though fascinated with the colour most attractive to our eyes, invariably chooses for the hangings of her apartments the most brilliant scarlet, selecting for its material the petals of the wild poppy, which she dexterously cuts into the proper form. Her first process is to excavate in some pathway a burrow, cylindrical at the entrance but swelled out below, to the depth of about three inches. Having polished the walls of this little apartment, she next flies to a neighbouring field, cuts out oval portions of the flowers of poppies,
- 62. seizes them between her legs and returns with them to her cell; and though separated from the wrinkled petal of a half-expanded flower, she knows how to straighten their folds, and, if too large, to fit them for her purpose by cutting off the superfluous parts. Beginning at the bottom, she overlays the walls of her mansion with this brilliant tapestry, extending it also on the surface of the ground round the margin of the orifice. The bottom is rendered warm by three or four coats, and the sides have never less than two. The little upholsterer, having completed the hangings of her apartment, next fills it with pollen and honey to the height of about half an inch; then, after committing an egg to it, she wraps over the poppy lining so that even the roof may be of this material; and lastly closes its mouth with a small hillock of earth[769]. The great depth of the cell compared with the space which the single egg and the accompanying food deposited in it occupy, deserves particular notice. This is not more than half an inch at the bottom, the remaining two inches and a half being subsequently filled with earth. —When you next favour me with a visit, I can show you the cells of this interesting insect as yet unknown to British entomologists, for which I am indebted to the kindness of M. Latreille, who first scientifically described the species[770]. Megachile centuncularis, M. Willughbiella, and other species of the same family, like the preceding, cover the walls of their cells with a coating of leaves, but are content with a more sober colour, generally selecting for their hangings the leaves of trees, especially of the rose, whence they have been known by the name of the leaf- cutter bees. They differ also from M. Papaveris in excavating longer burrows, and filling them with several thimble-shaped cells composed of portions of leaves so curiously convoluted, that, if we were ignorant in what school they have been taught to construct them, we should never credit their being the work of an insect. Their entertaining history, so long ago as 1670, attracted the attention of our countrymen Ray, Lister, Willughby, and Sir Edward King; but we
- 63. are indebted for the most complete account of their procedures to Reaumur. The mother bee first excavates a cylindrical hole eight or ten inches long, in a horizontal direction, either in the ground or in the trunk of a rotten willow-tree, or occasionally in other decaying wood. This cavity she fills with six or seven cells wholly composed of portions of leaf, of the shape of a thimble, the convex end of one closely fitting into the open end of another. Her first process is to form the exterior coating, which is composed of three or four pieces of larger dimensions than the rest, and of an oval form. The second coating is formed of portions of equal size, narrow at one end but gradually widening towards the other, where the width equals half the length. One side of these pieces is the serrate margin of the leaf from which it was taken, which, as the pieces are made to lap one over the other, is kept on the outside, and that which has been cut within. The little animal now forms a third coating of similar materials, the middle of which, as the most skilful workman would do in similar circumstances, she places over the margins of those that form the first tube, thus covering and strengthening the junctures. Repeating the same process, she gives a fourth and sometimes a fifth coating to her nest, taking care, at the closed end or narrow extremity of the cell, to bend the leaves so as to form a convex termination. Having thus finished a cell, her next business is to fill it to within half a line of the orifice, with a rose-coloured conserve composed of honey and pollen, usually collected from the flowers of thistles; and then having deposited her egg, she closes the orifice with three pieces of leaf so exactly circular, that a pair of compasses could not define their margin with more truth; and coinciding so precisely with the walls of the cell, as to be retained in their situation merely by the nicety of their adaptation. After this covering is fitted in, there remains still a concavity which receives the convex end of the succeeding cell; and in this manner the indefatigable little animal proceeds until she has completed the six or seven cells which compose her cylinder.
- 64. The process which one of these bees employs in cutting the pieces of leaf that compose her nest is worthy of attention. Nothing can be more expeditious: she is not longer about it than we should be with a pair of scissors. After hovering for some moments over a rose- bush, as if to reconnoitre the ground, the bee alights upon the leaf which she has selected, usually taking her station upon its edge so that the margin passes between her legs. With her strong mandibles she cuts without intermission in a curve line so as to detach a triangular portion. When this hangs by the last fibre, lest its weight should carry her to the ground, she balances her little wings for flight, and the very moment it parts from the leaf flies off with it in triumph; the detached portion remaining bent between her legs in a direction perpendicular to her body. Thus without rule or compasses do these diminutive creatures mete out the materials of their work into portions of an ellipse, into ovals or circles, accurately accommodating the dimensions of the several pieces of each figure to each other. What other architect could carry impressed upon the tablet of his memory the entire idea of the edifice which he has to erect, and, destitute of square or plumb-line, cut out his materials in their exact dimensions without making a single mistake? Yet this is what our little bee invariably does. So far are human art and reason excelled by the teaching of the Almighty[771]. Other insects besides bees construct habitations of different kinds for their young, as various species of burrowing wasps (Fossores), Geotrupes, c., which deposit their eggs in cylindrical excavations that become the abode of the future larvæ. In the procedures of most of these, nothing worth particularizing occurs; but one species called by Reaumur the mason-wasp, (Odynerus muraria,) referred to in a former letter, works upon so singular a plan, that it would be improper to pass it over in silence, especially as these nests may be found in this country in most sandy banks exposed to the sun. This insect bores a cylindrical cavity from two to three inches deep, in hard sand which its mandibles alone would be scarcely capable of penetrating, were it not provided with a slightly glutinous liquor which it pours out of its mouth, that, like the vinegar with which
- 65. Hannibal softened the Alps, acts upon the cement of the sand, and renders the separation of the grains easy to the double pickaxe with which our little pioneer is furnished. But the most remarkable circumstance is the mode in which it disposes of the excavated materials. Instead of throwing them at random on a heap, it carefully forms them into little oblong pellets, and arranges them round the entrance of the hole so as to form a tunnel, which, when the excavation is completed, is often not less than two or three inches in length. For the greater part of its height this tunnel is upright, but towards the top it bends into a curve, always however retaining its cylindrical form. The little masses are so attached to each other in this cylinder, as to leave numerous vacuities between them, which give it the appearance of filagree-work. You will readily divine that the excavated hole is intended for the reception of an egg, but for what purpose the external tunnel is meant is not so apparent. One use, and perhaps the most important, would seem to be to prevent the incursions of the artful Ichneumons, Chrysidæ, c. which are ever on the watch to insinuate their parasitic young into the nests of other insects: it may render their access to the nest more difficult; they may dread to enter into so long and dark a defile. I have seen, however, more than once a Chrysis come out of these tunnels. That its use is only temporary, is plain from the circumstance that the insect employs the whole fabric, when its egg is laid and store of food procured, in filling up the remaining vacuity of the hole; taking down the pellets, which are very conveniently at hand, and placing them in it until the entrance is filled[772].—Latreille informs us, that a nearly similar tunnel, but composed of grains of earth, is built at the entrance of its cell by a bee of his family of pioneers[773]. Under this head, too, may be most conveniently arranged the very singular habitations of the larvæ of the Linnæan genus Cynips, the gall-fly, though they can with no propriety be said to be constructed by the mother, who, provided with an instrument as potent as an enchanter's wand, has but to pierce the site of the foundation, and
- 66. commodious apartments, as if by magic, spring up and surround the germe of her future descendants. I allude to those vegetable excrescencies termed galls, some of which resembling beautiful berries and others apples, you must have frequently observed on the leaves of the oak, and of which one species, the Aleppo gall, as I have before noticed, is of such importance in the ingenious art de peindre la parole et de parler aux yeux[774]. All these tumours owe their origin to the deposition of an egg in the substance out of which they grow. This egg, too small almost for perception, the parent insect, a little four-winged fly, introduces into a puncture made by her curious spiral sting, and in a few hours it becomes surrounded with a fleshy chamber, which not only serves its young for shelter and defence, but also for food; the future little hermit feeding upon its interior and there undergoing its metamorphosis. Nothing can be more varied than these habitations. Some are of a globular form, a bright red colour, and smooth fleshy consistence, resembling beautiful fruits, for which indeed, as you have before been told, they are eaten in the Levant: others, beset with spines or clothed with hair, are so much like seed-vessels, that an eminent modern chemist has contended respecting the Aleppo gall, that it is actually a capsule[775]. Some are exactly round; others like little mushrooms; others resemble artichokes; while others again might be taken for flowers: in short, they are of a hundred different forms, and of all sizes from that of a pin's head to that of a walnut. Nor is their situation on the plant less diversified. Some are found upon the leaf itself; others upon the footstalks only; others upon the roots; and others upon the buds[776]. Some of them cause the branches upon which they grow to shoot out into such singular forms, that the plants producing them were esteemed by the old botanists distinct species. Of this kind is the Rose-willow, which old Gerard figures and describes as not only making a gallant shew, but also yeelding a most cooling aire in the heat of summer, being set up in houses for the decking of the same. This willow is nothing more than one of the common species, whose twigs, in consequence of the deposition of the egg of a Cynips in their summits, there shoot out into
- 67. numerous leaves totally different in shape from the other leaves of the tree, and arranged not much unlike those composing the flower of a rose, adhering to the stem even after the others fall off. Sir James Smith mentions a similar lusus on the Provence willows, which at first he took for a tufted lichen[777]. From the same cause the twigs of the common wild rose often shoot out into a beautiful tuft of numerous reddish moss-like fibres wholly dissimilar from the leaves of the plant, deemed by the old naturalists a very valuable medical substance, to which they erroneously gave the name of Bedeguar. None of these variations is accidental or common to several of the tribe, but each peculiar to the galls formed by a single and distinct species of Cynips. How the mere insertion of an egg into the substance of a leaf or twig, even if accompanied, as some imagine, by a peculiar fluid, should cause the growth of such singular protuberances around it, philosophers are as little able to explain, as why the insertion of a particle of variolous matter into a child's arm should cover it with pustules of small pox. In both cases the effects seem to proceed from some action of the foreign substance upon the secreting vessels of the animal or vegetable: but of the nature of this action we know nothing. Thus much is ascertained by the observations of Reaumur and Malpighi—that the production of the gall, which however large attains its full size in a day or two[778], is caused by the egg or some accompanying fluid: not by the larva, which does not appear until the gall is fully formed[779]; that the galls which spring from leaves almost constantly take their origin from nerves[780]; and that the egg, at the same time that it causes the growth of the gall, itself derives nourishment from the substance that surrounds it, becoming considerably larger before it is hatched than it was when first deposited[781].—When chemically analysed, galls are found to contain only the same principles as the plant from which they spring, but in a more concentrated state.
- 68. No productions of nature seem to have puzzled the ancient philosophers more than galls. The commentator on Dioscorides, Mathiolus, who agreeably to the doctrine of those days ascribed their origin to spontaneous generation, gravely informs us that weighty prognostications as to the events of the ensuing year may be deduced from ascertaining whether they contain spiders, worms, or flies. Other philosophers, who knew that except by rare accident no other animals are to be found in galls, besides grubs of different kinds which they rationally conceived to spring from eggs, were chiefly at a loss to account for the conveyance of these eggs into the middle of a substance in which they could find no external orifice. They therefore inferred that they were the eggs of insects deposited in the earth, which had been drawn up by the roots of trees along with the sap, and after passing through different vessels had stopped, some in the leaves, others in the twigs, and had there hatched and produced galls! Redi's solution of the difficulty was even more extraordinary. This philosopher, who had so triumphantly combated the absurdities of spontaneous generation, fell himself into greater. Not having been able to witness the deposition of eggs by the parent flies in the plants that produce galls, he took it for granted that the grubs which he found within them could not spring from eggs: and he was equally unwilling to admit their origin from spontaneous generation,—an admission which would have been fatal to his own most brilliant discoveries. He therefore cut the knot, by supposing that to the same vegetative soul by which fruits and plants are produced, is committed the charge of creating the larvæ found in galls[782]! An instance truly humiliating, how little we can infer from a man's just ideas on one point, that he will not be guilty of the most pitiable absurdity on another! Though by far the greater part of the vegetable excrescencies termed galls, are caused by insects of the genus Cynips, they do not always originate from this tribe. Some are produced by weevils belonging to Schüppel's genus Ceutorhynchus; as those on the roots of kedlock (Sinapis arvensis), which I have ascertained to be inhabited by the larvæ of Curculio contractus Marsh., Rhynchænus
- 69. assimilis, F. From the knob-like galls on turnips called in some places the anbury, I have bred another of these weevils, (Curculio pleurostigma, Marsh., Rhynchænus sulcicollis, Gyll.) and I have little doubt that the same insects, or species allied to them, cause the clubbing of the roots of cabbages. It seems to be a beetle of the same family that is figured by Reaumur[783], as causing the galls on the leaves of the lime-tree. Others owe their origin to moths, as those resembling a nutmeg which Reaumur received from Cyprus[784]; and others again to two-winged flies, as the woody galls of the thistle caused by Trypeta Cardui[785], and the cottony galls found on ground ivy, wild thyme, c. as well as a very singular one on the juniper resembling a flower, described by De Geer[786], all which are the work of minute gall-gnats (Cecidomyiæ, Latr.). Some of these last convert even the flowers of plants into a kind of galls, as T. Loti of De Geer[787], which inhabits the blossoms of Lotus corniculatus; and one which I have myself observed to render the flowers of Erysimum Barbarea like a hop blossom. A similar monstrous appearance is communicated to the flowers of Teucrium supinum by a little field-bug, Tingis Teucrii of Host[788], and to another plant of the same genus by one of the same tribe described by Reaumur[789]. In these two last instances, however, the habitations do not seem strictly entitled to the appellation of galls, as they originate not from the egg, but from the larva, which, in the operation of extracting the sap, in some way imparts a morbid action to the juices, causing the flower to expand unnaturally: and the same remark is applicable to the gall-like swellings formed by many Aphides, as A. Pistaciæ, which causes the leaves of different species of Pistacia to expand into red finger-like cavities; A. Abietis, which converts the buds or young shoots of the fir into a very beautiful gall, somewhat resembling a fir-cone, or a pine-apple in miniature; and A. Bursariæ, which with its brood inhabits angular utriculi on the leafstalk of the black poplar, numbers of which I have observed on those trees by the road-side from Hull to Cottingham.—The majority of galls are what entomologists have denominated monothalamous,
- 70. or consisting of only one chamber or cell; but some are polythalamous, or consisting of several. Having thus described the most remarkable of the habitations constructed by the parent insects for the accommodation of their future young, I proceed to the second kind mentioned, namely, those which are formed by the insect itself for its own use. These may be again subdivided into such as are the work of the insects in their larva state; and such as are formed by perfect insects. Many larvæ of all orders need no other habitations than the holes which they form in seeking for, or eating, the substances upon which they feed. Of this description are the majority of subterranean larvæ, and those which feed on wood, as the Bostrichi or labyrinth beetles; the Anobia which excavate the little circular holes frequently met with in ancient furniture and the wood work of old houses; and many larvæ of other orders, particularly Lepidoptera. One of these last, the larva of Cossus ligniperda differs from its congeners in fabricating for its residence during winter a habitation of pieces of wood lined with fine silk[790]. Under this division, too, come the singular habitations of the subcutaneous larvæ, so called from the circumstance of their feeding upon the parenchyma included between the upper and under cuticles of the leaves of plants, between which, though the whole leaf is often not thicker than a sheet of writing-paper, they find at once food and lodging. You must have been at some time struck by certain white zigzag or labyrinth- like lines on the leaves of the dandelion, bramble, and numerous other plants: the next-time you meet with one of them, if you hold it up to the light you will perceive that the colour of these lines is owing to the pulpy substance of the leaf having there been removed; and at the further end you will probably remark a dark- coloured speck, which, when carefully extricated from its covering,
- 71. you will find to be the little miner of the tortuous galleries which you are admiring. Some of these minute larvæ, to which the parenchyma of a leaf is a vast country, requiring several weeks to be traversed by the slow process of mining which they adopt—that of eating the excavated materials as they proceed—are transformed into beetles (Cionus Thapsi, c.); others into flies; and a still greater number into very minute moths, as Gracillaria? Wilkella, Clerkella, c. Many of these last are little miracles of nature, which has lavished on them the most splendid tints tastefully combined with gold, silver and pearl: so that, were they but formed upon a larger scale, they would far eclipse all other animals in richness of decoration. Another tribe of larvæ, not very numerous, content themselves for their habitations with simple holes, into which they retire occasionally. Many of these are merely cylindrical burrows in the ground, as those formed by the larvæ of field-crickets, Cicindelæ and Ephemeræ. But the larvæ of the very remarkable lepidopterous genus (Nycterobius of Mr. MacLeay) before alluded to[791], excavate for themselves dwellings of a more artificial construction; forming cylindrical holes in the trees of New Holland, particularly the different species of Banksia, to which they are very destructive, and defending the entrance against the attacks of the Mantes and other carnivorous insects by a sort of trap-door composed of silk interwoven with leaves and pieces of excrement, securely fastened at the upper end, but left loose at the lower for the free passage of the occupant. This abode they regularly quit at sun-set, for the purpose of laying in a store of the leaves on which they feed. These they drag by one at a time into their cell until the approach of light, when they retreat precipitately into it, and there remain closely secluded the whole day, enjoying the booty which their nocturnal range has provided. One species lifts up the loose end of its door by its tail, and enters backward, dragging after it a leaf of Banksia serrata, which it holds by the footstalk[792]. A third description of larvæ, chiefly of the two lepidopterous tribes Tortricidæ and Tineidæ, form into convenient habitations the leaves
- 72. of the plants on which they feed. Some of these merely connect together with a few silken threads several leaves so as to form an irregular packet, in the centre of which the little hermit lives. Others confine themselves to a single leaf, of which they simply fold one part over the other. A third description form and inhabit a sort of roll, by some species made cylindrical, by others conical, resembling the papers into which grocers put their sugar, and as accurately constructed, only there is an opening left at the smaller extremity for the egress of the insect in case of need. If you were to see one of these rolls, you would immediately ask by what mechanism it could possibly be made—how an insect without fingers could contrive to bend a leaf into a roll, and to keep it in that form until fastened with the silk which holds it together? The following is the operation. The little caterpillar first fixes a series of silken cables from one side of the leaf to the other. She next pulls at these cables with her feet; and when she has forced the sides to approach, she fastens them together with shorter threads of silk. If the insect finds that one of the larger nerves of the leaf is so strong as to resist her efforts, she weakens it by gnawing it here and there half through. What engineer could act more sagaciously?—To form one of the conical or horn-shaped rolls, which are not composed of a whole leaf, but of a long triangular portion cut out of the edge, some other manœuvres are requisite. Placing herself upon the leaf, the caterpillar cuts out with her jaws the piece which is to compose her roll. She does not however entirely detach it: it would then want a base. She detaches that part only which is to form the contour of the horn. This portion is a triangular strap, which she rolls as she cuts. When the body of the horn is finished, as it is intended to be fixed upon the leaf in nearly an upright position, it is necessary to elevate it. To effect this, she proceeds as we should with an inclined obelisk. She attaches threads or little cables towards the point of the pyramid, and raises it by the weight of her body[793]. A still greater degree of dexterity is manifested in fabricating the habitations of the larvæ of some other moths which feed on the leaves of the rose-tree, apple, elm, and oak, on the under-side of
- 73. which they may in summer be often found. These form an oblong cavity in the interior of a leaf by eating the parenchyma between the two membranes composing its upper and under side, which, after having detached them from the surrounding portion, it joins with silk so artfully that the seams are scarcely discoverable even with a lens, so as to compose a case or horn, cylindrical in the middle, its anterior orifice circular, its posterior triangular. Were this dwelling cylindrical in every part, the form of the two pieces that compose it would be very simple; but the different shape of the two ends renders it necessary that each side should have peculiar and dissimilar curvatures; and Reaumur assures us, that these are as complex and difficult to imitate as the contours of the pieces of cloth that compose the back of a coat. Some of this tribe, whose proceedings I had the pleasure of witnessing a short time since upon the alders in the Hull Botanic Garden, more ingenious than their brethren, and willing to save the labour of sewing up two seams in their dwelling, insinuate themselves near the edge of a leaf instead of in its middle. Here they form their excavation, mining into the very crenatures between the two surfaces of the leaf, which, being joined together at the edge, there form one seam of the case, and from their dentated figure give it a very singular appearance, not unlike that of some fishes which have fins upon their backs. The opposite side they are necessarily forced to cut and sew up, but even in this operation they show an ingenuity and contrivance worthy of admiration. The moths, which cut out their suit from the middle of the leaf, wholly detach the two surfaces that compose it before they proceed to join them together, the serrated incisions made by their teeth, which, if they do not cut as fast, in this respect are more effective than any scissors, interlacing each other so as to support the separated portions until they are properly joined. But it is obvious that this process cannot be followed by those moths which cut out their house from the edge of a leaf. If these were to detach the inner side before they had joined the two pieces together, the builder as well as his dwelling would inevitably fall. They therefore, before making any incision, prudently run (as a sempstress would call it) loosely together in distant points the two
- 74. membranes on that side. Then putting out their heads they cut the intermediate portions, carefully avoiding the larger nerves of the leaf; afterwards they sew up the detached sides more closely, and only intersect the nerves when their labour is completed[794].—The habitation made by a moth, which lives upon a species of Astragalus, is in like manner formed of the epidermis of the leaves, but in this several corrugated pieces project over each other, so as to resemble the furbelows once in fashion[795]. Other larvæ construct their habitations wholly of silk. Of this description is that of a moth, whose abode, except as to the materials which compose it, is formed on the same general plan as that just described, and the larva in like manner feeds only on the parenchyma of the leaf. In the beginning of spring, if you examine the leaves of your pear-trees, you will scarcely fail to meet with some beset on the under surface with several perpendicular downy russet-coloured projections, about a quarter of an inch high, and not much thicker than a pin, of a cylindrical shape, with a protuberance at the base, and altogether resembling at first sight so many spines growing out of the leaf. You would never suspect that these could be the habitations of insects; yet that they are is certain. Detach one of them, and give it a gentle squeeze, and you will see emerge from the lower end a minute caterpillar with a yellowish body and black head. Examine the place from which you have removed it, and you will perceive a round excavation in the cuticle and parenchyma of the leaf, the size of the end of the tube by which it was concealed. This excavation is the work of the above-mentioned caterpillar, which obtains its food by moving its little tent from one part of the leaf to the other, and eating away the space immediately under it. It touches no other part; and when these insects abound, as they often do to the great injury of pear-trees[796], you will perceive every leaf bristled with them, and covered with little withered specks, the vestiges of their former meals. The case in which the caterpillar resides, and which is quite essential to its existence, is composed of silk spun from its mouth almost as soon as it is
- 75. excluded from the egg. As it increases in size, it enlarges its habitation by slitting it in two, and introducing a strip of new materials. But the most curious circumstance in the history of this little Arab is the mode by which it retains its tent in a perpendicular posture. This it effects partly by attaching silken threads from the protuberance at the base to the surrounding surface of the leaf. But being not merely a mechanician, but a profound natural philosopher well acquainted with the properties of air, it has another resource when any extraordinary violence threatens to overturn its slender turret. It forms a vacuum in the protuberance at the base, and thus as effectually fastens it to the leaf as if an air-pump had been employed! This vacuum is caused by the insect's retreating on the least alarm up its narrow case, which its body completely fills, and thus leaving the space below free of air. In detaching one of these cases you may easily convince yourself of the fact. If you seize it suddenly while the insect is at the bottom, you will find that it is readily pulled off, the silken cords giving way to a very slight force; but if, proceeding gently, you give the insect time to retreat, the case will be held so closely to the leaf as to require a much stronger effort to loosen it. As if aware that, should the air get admission from below, and thus render a vacuum impracticable, the strongest bulwark of its fortress would be destroyed, our little philosopher carefully avoids gnawing a hole in the leaf, contenting itself with the pasturage afforded by the parenchyma above the lower epidermis; and when the produce of this area is consumed, it gnaws asunder the cords of its tent, and pitches it at a short distance as before. Having attained its full growth, it assumes the pupa state, and after a while issues out of its confinement a small brown moth, with long hind legs, the Phalæna Tinea serratella of Linné[797]. Some larvæ, which form their covering of pure silk, are not content with a single coating, but actually envelop themselves in another, open on one side and very much resembling a cloak; whence Reaumur called them Teignes à fourreau à manteau. What is very striking in the construction of this cloak, is, that the silk, instead of being woven into one uniform close texture, is formed into
- 76. numerous transparent scales over-wrapping each other, and altogether very much resembling the scales of a fish[798]. These mantle-covered cases, one of which I once had the pleasure of discovering, are inhabited by the larva of a little moth apparently first described by Dr. Zincken genannt Sommer, who calls it Tinea palliatella[799]. Various substances besides silk are fabricated into habitations by other larvæ, though usually joined together either with silk or an analogous gummy material. Thus Diurnea? Lichenum forms of pieces of lichen a dwelling resembling one of the turreted Helices, many of which I observed in June 1812 on an oak in Barham. The larvæ of another moth, which also feeds upon lichens, instead of employing these vegetables in forming its habitation, composes it of grains of stone eroded from the walls of buildings upon which its food is found, and connected by a silken cement. These insects were the subject of a paper in the Memoirs of the French Academy[800], by M. de la Voye, who, from the circumstance of their being found in great abundance on mouldering walls, attributed to them the power of eating stone, and regarded them as the authors of injuries proceeding solely from the hand of time: for the insects themselves are so minute, and the coating of grains of stone composing their cases is so trifling, that Reaumur observes they could scarcely make any perceptible impression on a wall from which they had procured materials for ages[801].—Another lepidopterous larva, but of a much larger size and different genus, the case of which is preserved in the cabinet of the President of the Linnean Society, who pointed it out to me, employs the spines apparently of some species of Mimosa, which are ranged side by side so as to form a very elegant fluted cylinder. A similar arrangement of pieces of small twigs is observable in the habitation of the females[802] of the larvæ of a moth referred by Von Scheven to Bombyx vestita, F.; which Ochsenheimer regards as synonymous with Psyche graminella, while P. Viciella of the Wiener Verzeichniss covers itself with short portions of the stems of grasses placed transversely, and united by means of silk into a five-
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