![Contents
Preface page ix
List of abbreviations xi
1 Introduction 1
1.1 Historical remarks 1
1.2 Linear Physics 3
1.3 Ordered exponentials 13
Notes 16
2 Elementary functional methods 17
2.1 Functional differentiation 17
2.2 Linear translation 18
2.3 Quadratic (Gaussian) translation 20
2.4 Functional integration 23
2.5 Examples drawn from quantum field theory 27
2.6 Cluster decomposition 30
Notes 32
3 Schwinger–Fradkin methods 33
3.1 Proper-time representations of Schwinger and Fradkin 33
3.2 Fradkin representations for QED and QCD 37
3.3 Gauge structure in QED and QCD 40
3.4 Soluble examples: quadratic forms and perturbative
approximations 43
3.5 Pair production in generalized electric fields 45
Notes 50
4 Lasers and crossed lasers 51
4.1 Classical charged-particle propagation in a laser
(epw) field 51
4.2 The “scalar” laser solution for Gc[A] 54
vii](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-13-320.jpg)
![viii Contents
4.3 The QED laser solutions for Gc[A] and L[A] 56
4.4 Pair production via crossed lasers 62
Notes 72
5 Special variants of the Fradkin representation 75
5.1 Exact representations for scalar interactions 75
5.2 Finite-quadrature approximations 82
5.3 Exact and approximate vectorial interactions 87
5.4 The Stojkov variation 90
Notes 92
6 Quantum chaos and vectorial interactions 93
6.1 First-quantization chaos 93
6.2 Chaos suppression in second quantization 98
6.3 Fluctuation-induced chaos suppression 101
Notes 106
7 Infrared approximations 107
7.1 The Bloch–Nordsieck approximation 108
7.2 IR damping at large momentum transfers 110
7.3 Eikonal scattering amplitudes in particle physics 114
7.4 IR approximations and rescaling corrections
to non-linear ODEs 119
Notes 123
8 Models of high-energy, non-Abelian scattering 125
8.1 An Abelian separation 126
8.2 The quasi-Abelian limit 128
8.3 Loop, ladder and crossed-ladder approximations 133
8.4 Summing all the eikonal graphs 142
Notes 147
9 Unitary ordered exponentials 149
9.1 Algebraic and differential structure 149
9.2 The SU(2) adiabatic limit 150
9.3 The stochastic limit 154
9.4 Functional integration over the stochastic limit 163
Notes 167
Index 168](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-14-320.jpg)
![1
Introduction
1.1 Historical remarks
It is difficult to fix the precise beginning of the vast and disparate subject matter
which now exists under the name of “Green’s Functions”, but the origins of the
method may certainly be associated with the original and ingenious work of
George Green (1793–1841).1
That application, now called Green’s Theorem,
of Gauss’ Theorem applied to electrostatics, in modern language makes use of
the differential statement
∇2
|r − r
|−1
= −4πδ(r − r
). (1.1)
Before the advent of the Dirac delta-function, the content of (1.1) had to be ex-
pressed in a somewhat circuitous way,2
which is how Green treated the problem.
Every modern text on potential theory begins with the statement of Gauss’
Theorem,
dS · F =
d3
r
∇
· F(r
), where F(r
) is a continuous and differ-
entiable vector function, whose divergence is to be integrated over a volume
bounded by the surface
dS. Green noted that the choice F = V ∇U − U∇V
generates, for arbitrary U, V ,
dS
· [V ∇
U − U∇
V ] = −
d3
r
[U(r
)∇2
V (r
) − V (r
)∇2
U(r
)],
(1.2)
which, in vector notation, is the statement of Green’s theorem. If the further
choice U(r
) = −(4π|r − r
|)−1
is made, where r denotes the radius vector
(drawn from an origin of arbitrary location) of a point inside the integration
volume, then (1.1) and (1.2) yield
V (r) = −
1
4π
d3
r 1
|r − r|
∇2
V (r
)
+
1
4π
dS
·
1
|r − r|
∇
V (r
) − V (r
)∇ 1
|r − r|
. (1.3)
1](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-19-320.jpg)
![4 1 Introduction
inserting damping and, for example, replacing ω2
by ω2
(x2
− 1) to produce
the Duffing equation, with its manifest nonlinear behavior. What shall be done
in this section is to generate solutions to the linear problems using standard
GF methods, and then to compare the results with an alternative and equivalent
phase-space method of solution. No OEs appear in the linear analysis, but the
standard questions of retarded or advanced, causal or anticausal solutions must
be answered.
Adding spatial derivatives to (1.9) in a Lorentz-symmetric way generates the
forms of non-interacting field theory,
(µ2
− ∂2
)A(x) = j(x), (1.10)
where causality will be demanded in the sense that A(r, t) cannot be different
from zero until a signal from the source j(r, t) (traveling at the speed of light
when µ = 0) can reach the point x = (r, t); here, ∂2
denotes the d’Alembertian
operator, ∇2
− 1
c2
∂2
∂t2 , and units will be chosen in what follows such that c, the
velocity of light, is unity.
In contrast, the diffusion equation of (relatively) low-velocity fluid motion is
non-relativistic,
∂
∂t
− ν∇2
v = f(r, t), (1.11)
where ν denotes viscosity and v(r, t) is the fluid velocity; appropriate initial con-
ditions here will again demand causality. Because it has but one time derivative,
there exist but two GF solutions for this problem, one “retarded” (subscript R)
and the other “advanced” (subscript A); and it is simplest to begin the detailed
construction of these GF s with this example.
(i) Non-relativistic diffusion: The requirement of causality will select the GF
GR as the physically relevant solution of the inhomogeneous
∂
∂t
− ν∇2
GR(r − r
; t − t
) = δ(r − r
)δ(t − t
), (1.12)
which, as written in (1.12), turns out to be a function of coordinate differences.
If one knows the general solutions to the corresponding homogeneous DE,
the solution to (1.12) may be expressed as a summation over all eigenstates of
positive eigenvalues En, in the form
GR(r − r
; t − t
) = θ(t − t
)
n
un(r)u∗
n(r
) exp[−En(t − t
)], (1.13)
wheretheun (r)formacompleteorthonormalset,satisfying[En + ν∇2
]un = 0
and
n un(r)u∗
n(r
) = δ(r − r
). The θ-function of (1.13) expresses the](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-22-320.jpg)
![1.2 Linear Physics 5
retardedness of the GF; because θ(x) = 1, x 0, and θ(x) = 0, x 0, the
solution to (1.11),
v(r, t) =
d3
r
dt
GR(r − r
; t − t
)f(r
, t
) + v0(r)
will have no contribution to its first RHS term for t
t, so that an effect at t
cannot appear before its generation at t
. Here, v0(r) represents the initial con-
dition of this problem, the velocity field specified at all points r, and satisfying
the equation ∇2
v0(r) = 0. Mathematically speaking, θ(x) is really the limit of
a sequence of functions chosen such that θ(0) = 1/2; everywhere in this book
it may be represented by the integral
θ(x) =
1
2πi
+∞
−∞
dω(ω − i )−1
eiωx
, (1.14)
and its properties checked by straightforward contour integration, as well as by
the more conventional relation: δ(x) = dθ/dx.
It should be noted that the En are positive, and hence the summation of
(1.13) is sensible; here, the un(r) are just plane-wave exponentials of wave-
vector kn = 2πn/L, where n is a vector each of whose components are integers,
L3
is an appropriate normalization volume, and En = νk2
n. Were the viscosity
continued to an imaginary value, with a change of normalization, one would
be dealing with the non-relativistic Schrödinger wave equation (SWE), as in
(ii) below.
The simplest method of construction for any such GF of a linear problem is
to employ a Fourier representation,
GR(r − r
; t − t
) = (2π)−4
d3
k
dωG̃R(k, ω)eik·(r−r
)−iω(t−t
)
, (1.15)
where G̃R isdeterminedbysubstituting(1.15)into(1.12),andisclearlygivenby
G̃R = i(ω + iνk2
)−1
.Inthecomplexω-plane,theintegrandof(1.15)hasbutone
singularity, a pole at ω = −iνk2
, which multiplies the factor exp[−iω(t − t
)].
Evaluating the ω-integral by contour integration, one is forced for t − t
0 to
close the contour in the upper half ω-plane, which yields zero, while the choice
t − t
0 requires closing the contour in the lower half plane, which yields
GR(r − r
; t − t
) = θ(t − t
)(2π)−3
d3
k exp[ik · (r − r
) − νk2
(t − t
)].
(1.16)
Because one finds a non-zero value only for t − t
0, as expressed by the
θ-function of (1.16), this is the retarded GF; GA would have been obtained by
reversing the sign of the −iω(t − t
) phase of (1.15). Note that in the limit of](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-23-320.jpg)
![6 1 Introduction
zero viscosity, (1.16) reduces to θ(t − t
)δ(r − r
), so that the solution to (1.13)
is simply
v(r, t) =
t
0
dt
f(r, t
) + v0(r).
As a Gaussian integral, (1.16) may be evaluated immediately,
GR(r − r
; t − t
) = θ(t − t
)[4πν(t − t
)]−3/2
exp[−(r − r
)2
/4ν(t − t
)].
(1.17)
For small differences t − t
, most of the contribution to the r
integral over
(1.17) comes from the points r
close to r; but as the time difference increases,
more and more of the source dependence at other values of r
is available to
influence the behavior of the solution at r.
If the source dependence is chosen to be that of a delta-function in space and
time, f(r, t) = f0δ(r)δ(t − )| →0, and if the initial field v0 = 0, the integrations
over (1.17) yield the simple result
v(r, t) = θ(t)f0(4πνt)−3/2
exp[−r2
/4νt]. (1.18)
This has the form of an initial (at t = 0) and localized (at r = 0) disturbance
diffusing away to all spatial points at later times, while maintaining a constant
infinite-spatial integral,
d3
rv(r, t) = θ(t)f0.
It is worth emphasizing the physically intuitive role of such a localized source
in “generating” a solution to the homogeneous equation ( ∂
∂t
− ν∇2
)v = 0. One
imagines for t 0 a perfectly quiescent situation with v(r, t) = 0, suddenly
disturbed by a source f0 · δ(r)δ(t) rapidly “turning on and off” at t = 0, and
generating for subsequent, positive t, a solution to the homogeneous equation.
This use of a highly-localized source has long appeared in QFT, as a way
of representing the appropriate wave functions of particles “produced” at a
particular space–time point. It is an artifice, but a convenient one, and has been
used in certain fluid/vortex problems.4
(ii) Non-relativistic Schrödinger wave equation: The GFs of interest here are
typically “ingoing” or “outgoing”, with the latter chosen to represent the
“scattered wave” of probability amplitude needed to describe the physical scat-
tering of two particles which interact with each other by means of a poten-
tial V (r1 − r2). Equivalently, by a simple transformation to the CM coordi-
nates of these particles, one may treat the scattering of a particle of reduced
mass µ = m1m2/(m1 + m2) in the field of a fictitious potential field V (r),
where r = r1 − r2. In such scattering problems, the energy E of the system is](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-24-320.jpg)
![1.2 Linear Physics 7
conserved and specified in advance, so that the original SWE
ih̄
∂
∂t
ψ = −
h̄2
2µ
∇2
+ V (r) ψ (1.19)
is replaced by
Eu(r) = −
h̄2
2µ
∇2
+ V (r) u(r), (1.20)
uponusingthesubstitutionψ(r, t) = u(r) exp[−iEt/h̄];thecorrespondingGFs
of this problem are to satisfy
E +
h̄2
2µ
∇2
− V (r) G(r, r
|V ) = δ(r − r
), (1.21)
and scattering amplitudes u(r) are given by
u(r) = u0(r) +
d3
r
G(r, r
|V )V (r
)u0(r
), (1.22)
where u0(r) represents the “asymptotic input” function exp[ik · r], correspond-
ing to an incident particle of energy E = h̄2
k2
/2µ moving in the k direction.
A formal solution for the GF of this problem may be written as
G(r, r
|V ) =
n
un(r)u∗
n(r
)(E − En)−1
, (1.23)
where the summation is over all states of the complete, orthonormal set un(r)
which satisfy the time-independent SWE,
Enun = −
h̄2
2µ
∇2
+ V (r) un(r).
TheRHSof(1.21)isreproduced,uponsubstitutionof(1.23)into(1.21),because
the un are again assumed complete, satisfying
n
un(r)u∗
n(r
) = δ(r − r
).
Typically, the states |n form a continuum of positive-energy scattering states,
plus discreet bound states of negative energy. Note from (1.23) that if the scat-
tering energy E is continued to negative values, one may expect to find a pole
of the scattering amplitude when E approaches one of the isolated, negative,
bound-state energies. Note also that, because of the spatial dependence of V (r),
this GF is no longer a function of the difference of its coordinates. It will become
clear below that the distinction between the solutions Gout and Gin can be made
explicit by appending an infinitesimal ±i to the denominator E of (1.23).](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-25-320.jpg)
![THE MAN AND THE BEAST.
THE MAN AND THE BEAST.[2]
OBO, the wild man of Borneo, sat in his iron-barred cage
reading the morning paper, while he pulled vigorously at
a short, black clay pipe.
It was nearly time for the show to begin, so he could
only glance hurriedly at the stock report; for Bobo was interested in
copper.
On Mondays, Wednesdays, and Fridays there was on exhibition in
the side-show connected with Poole Brothers’ Royal Roman
Hippodrome and Three-Ring Circus what was widely advertised as
the only real wild man in captivity.
On alternate days—that is, on Tuesdays, Thursdays, and Saturdays—
the cage of Bobo was closed by a gaudily painted cover; and visitors
on those days were told that the wild man was sick.
Notwithstanding this report, there could be found on Tuesdays,
Thursdays, and Saturdays, out in one of the New York suburbs, a
middle-aged Irishman named Patsy McLockin. The connection may
not at first be evident.
Patsy wasn’t nice looking, even when he was dressed in his best
black suit; for, as the people on Blenden Street remarked, he was
too hairy.
He used to wear gloves when on the street, even in the hottest
weather; but he couldn’t very well wear gloves on his face, though if](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-33-320.jpg)
![THE SPACE ANNIHILATOR.
THE SPACE ANNIHILATOR.[3]
N the afternoon of Saturday, August 18, 1900, as I was
looking over the daily paper after my return from the
Blendheim Electric Works, where I am employed, I
noticed in the advertising department the following:
IMPORTANT NOTICE TO ENGINEERS AND SCIENTIFIC MEN.
Ten thousand dollars will be paid to the man or woman
duplicating an instrument now in the possession of this
company——
That was as far as I read. Some cheap advertising scheme, I
thought, and immediately forgot all about the paragraph.
When, however, towards the last of the month, I received the
regular issue of my pet scientific paper, I saw on the first page the
same glaring announcement. The fact of the notice being in that
paper was guarantee that the offer was bona fide, and I looked the
article over carefully.
In addition to the foregoing, the advertisement went on to state that
one of a pair of seismaphones, an invention with patent pending and
not yet in the market, had been lost. The inventor was dead, and no
one had as yet been able to construct an instrument similar to the
one now in the company’s possession.
Further particulars would be sent to any one satisfying the company
that his request for the same was not prompted by idle curiosity, but
by a desire to aid science in replacing the lost instrument.](https://image.slidesharecdn.com/30152-250405222741-9f8c1770/85/Green-s-Functions-and-Ordered-Exponentials-1st-Edition-H-M-Fried-46-320.jpg)
Green s Functions and Ordered Exponentials 1st Edition H. M. Fried
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- 5. Green s Functions and Ordered Exponentials 1st Edition H. M. Fried Digital Instant Download Author(s): H. M. Fried ISBN(s): 9780521443906, 0521443903 Edition: 1 File Details: PDF, 1.08 MB Year: 2002 Language: english
- 7. GREEN’S FUNCTIONS AND ORDERED EXPONENTIALS This book presents a functional approach to the construction, use and approxi- mation of Green’s functions and their associated ordered exponentials. After a brief historical introduction, the author discusses new solutions to problems involving particle production in crossed laser fields and non-constant electric fields. Applications to problems in potential theory and quantum field theory are covered, along with approximations for the treatment of color fluctuations in high-energy QCD scattering, and a model for summing classes of eikonal graphs in high-energy scattering problems. The book also presents a variant of the Fradkin representation which suggests a new non-perturbative approxi- mation scheme, and provides a qualitative measure of the error involved in each such approximation. In addition, it deals with adiabatic and stochastic approximations to unitary ordered exponentials. Covering the basics as well as more advanced applications, this book is suit- able for graduate students and researchers in a wide range of fields, including quantum field theory, fluid dynamics and applied mathematics. h. m. fried received his PhD from Stanford University in 1957. He spent a post-doctoral year at the Ecole Normale Supérieure in Paris and then three years teaching physics at UCLA. This was followed by a year as a visiting member of the Institute of Advanced Study in Princeton, and two years as a visiting physicist at the Courant Institute at NYU, before joining the Physics Department at Brown University. Professor Fried has lectured and performed research in university departments and institutes throughout the world, prin- cipally in Paris, Marseille and Nice, and is a Director of the Workshops on Non-Perturbative QCD, which alternate between the American University of Paris and La Citadelle, Villefranche-sur-Mer. He is now Professor Emeritus of Physics at Brown University and continues to do occasional teaching there, as well as maintaining a research program in theoretical aspects of quantum field theory.
- 8. This page intentionally left blank
- 9. GREEN’S FUNCTIONS AND ORDERED EXPONENTIALS H. M. FRIED
- 10. PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia http://www.cambridge.org © Cambridge University Press 2002 This edition © Cambridge University Press (Virtual Publishing) 2003 First published in printed format 2002 A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 44390 3 hardback ISBN 0 511 02062 7 virtual (netLibrary Edition)
- 11. This book is dedicated to the memory of three extraordinary Physi- cists and Human Beings, men who died during the ten-year period of the writing of this book. From these Scholars and Gentlemen the author was privileged to learn a little of both Physics and the Humanity which can coexist in even the greatest of scientists: Profs. Donald Yennie, Antoine Visconti, and Julian Schwinger.
- 12. This page intentionally left blank
- 13. Contents Preface page ix List of abbreviations xi 1 Introduction 1 1.1 Historical remarks 1 1.2 Linear Physics 3 1.3 Ordered exponentials 13 Notes 16 2 Elementary functional methods 17 2.1 Functional differentiation 17 2.2 Linear translation 18 2.3 Quadratic (Gaussian) translation 20 2.4 Functional integration 23 2.5 Examples drawn from quantum field theory 27 2.6 Cluster decomposition 30 Notes 32 3 Schwinger–Fradkin methods 33 3.1 Proper-time representations of Schwinger and Fradkin 33 3.2 Fradkin representations for QED and QCD 37 3.3 Gauge structure in QED and QCD 40 3.4 Soluble examples: quadratic forms and perturbative approximations 43 3.5 Pair production in generalized electric fields 45 Notes 50 4 Lasers and crossed lasers 51 4.1 Classical charged-particle propagation in a laser (epw) field 51 4.2 The “scalar” laser solution for Gc[A] 54 vii
- 14. viii Contents 4.3 The QED laser solutions for Gc[A] and L[A] 56 4.4 Pair production via crossed lasers 62 Notes 72 5 Special variants of the Fradkin representation 75 5.1 Exact representations for scalar interactions 75 5.2 Finite-quadrature approximations 82 5.3 Exact and approximate vectorial interactions 87 5.4 The Stojkov variation 90 Notes 92 6 Quantum chaos and vectorial interactions 93 6.1 First-quantization chaos 93 6.2 Chaos suppression in second quantization 98 6.3 Fluctuation-induced chaos suppression 101 Notes 106 7 Infrared approximations 107 7.1 The Bloch–Nordsieck approximation 108 7.2 IR damping at large momentum transfers 110 7.3 Eikonal scattering amplitudes in particle physics 114 7.4 IR approximations and rescaling corrections to non-linear ODEs 119 Notes 123 8 Models of high-energy, non-Abelian scattering 125 8.1 An Abelian separation 126 8.2 The quasi-Abelian limit 128 8.3 Loop, ladder and crossed-ladder approximations 133 8.4 Summing all the eikonal graphs 142 Notes 147 9 Unitary ordered exponentials 149 9.1 Algebraic and differential structure 149 9.2 The SU(2) adiabatic limit 150 9.3 The stochastic limit 154 9.4 Functional integration over the stochastic limit 163 Notes 167 Index 168
- 15. Preface Physics, and indeed all of Science and rational Life, is a causal affair. Events occur in a well-defined way; and even though nonlinear effects may mask a pre- cise understanding of an underlying mechanism, there can be no rational doubt that cause preceeds effect. The mathematical expression of this truth is couched in the language of Green’s functions (GFs), originally invented to provide solu- tions to electrostatic problems, and subsequently generalized to give compact expression to the causality which appears in time-dependent situations. At the same time, it has become at least partially clear that when a very large number of iterations of an interaction are associated with the nonlinear, or strong-coupling description of a system, it is not always possible to link specific causes with observed effects. Thus the transition to chaos observed first in the multiple repetition of simple maps, and then in the fractal behavior of physical fluids as they approach fully developed turbulence; thus the realization that strongly coupled gluons and quarks of QCD need not propagate in the causal manner expected from perturbative approximations. Causality is clearly and explicitly true in weakly coupled systems, even though this property can be masked when essential nonlinear dynamics prevent the identification of a specific effect as due to a specific cause. In recent years, utilization of GF techniques has grown to encompass an immense number of disparate subjects, including application to the large-scale structure of nonlinear systems. Whether one is treating classical or quantum mechanics, Navier–Stokes fluids or ordinary nonlinear differential equations, there is a corps of analogous problems which can advantageously be treated by these methods. In the general representation and construction of such GFs, encountered across a wide variety of fields, one meets and must deal with ordered expo- nentials (OEs); and it is for this reason that the latter subject forms an indis- pensable part of this book. OEs are interesting functions in their own right, ix
- 16. x Preface but very little is known about their non-perturbative approximations; what is presented here is intuitive, physically motivated, and with a certain connection with low-frequency approximations to nonlinear problems. Other applications, such as the use of OEs to obtain formal solutions to Euler and Navier–Stokes equations, have been left for another occasion. It must be stated clearly that some of the results stated in this book, for which the author is in part responsible, are without rigorous mathematical foundation. To a physicist, intuition has its own value, which too often becomes its only justification; but it is from this point of view that much of the material of the latter chapters should be understood. At the very least, mathematicians will find in this slim volume a number of intuitively based statements which are in need of rigorous proof, or disproof. Some of the fundamental topics presented here – such as basic functional methods, and the Schwinger/Fradkin formalism for causal GFs – follow quite closely material appearing in a previous book by the author,1 called “HMF#2”, while some references have been made to material in an even older book2 by the author, hereinafter called “HMF#1”. Including the last Section of Chapter 3, and with the exception of Chapter 7, essentially all of the remaining material presented is new, dating from the past decade. The level of the present work is again such that graduate students and pro- fessionals in mathematical science should find its material and concepts quite familiar. Dirac delta-functions, for example, are used without hesitation; and whereallreadersmaynothaveaworkingacquaintancewithfunctionalmethods, a brief introductory sketch is given, sufficient for the purpose at hand. But the techniques presented are surely applicable to a wide variety of subjects; and each reader, it is hoped, will find a significant measure of success when applying them to his or her own pressing, nonlinear problems. This book was begun during the academic year 1991–92, when the author was a Visiting Professor at the Université de Nice; and completed slowly over the following nine years at Brown University. To friends and colleagues of both institutions are due the warmest thanks and acknowledgement of many kindnesses. Comme avant, je leur remercie de tout. Brown University H. M. Fried Notes 1 Functional Methods and Eikonal Models, Éditions Frontières, Gif-sur-Yvette, France (1990), hereinafter referred to as HMF#2 2 Functional Methods and Models in Quantum Field Theory, The MIT Press, Cambridge, MA (1972), hereinafter referred to as HMF#1
- 17. Abbreviations CM Center of Mass DE Differential Equation FFT Functional Fourier Transform FI Functional Integral ĜF Generating Functional GF Green’s Function IR Infrared LHP Lower Half Plane LHS Left-Hand-Side MSA Mass-Shell Amputation ODE Ordinary Differential Equation OE Ordered Exponential QCD Quantum Chromodynamics QED Quantum Electrodynamics QFT Quantum Field Theory RHS Right-Hand-Side SC Strong Coupling SWE Schrödinger Wave Equation UHP Upper Half Plane UOE Unitary Ordered Exponential xi
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- 19. 1 Introduction 1.1 Historical remarks It is difficult to fix the precise beginning of the vast and disparate subject matter which now exists under the name of “Green’s Functions”, but the origins of the method may certainly be associated with the original and ingenious work of George Green (1793–1841).1 That application, now called Green’s Theorem, of Gauss’ Theorem applied to electrostatics, in modern language makes use of the differential statement ∇2 |r − r |−1 = −4πδ(r − r ). (1.1) Before the advent of the Dirac delta-function, the content of (1.1) had to be ex- pressed in a somewhat circuitous way,2 which is how Green treated the problem. Every modern text on potential theory begins with the statement of Gauss’ Theorem, dS · F = d3 r ∇ · F(r ), where F(r ) is a continuous and differ- entiable vector function, whose divergence is to be integrated over a volume bounded by the surface dS. Green noted that the choice F = V ∇U − U∇V generates, for arbitrary U, V , dS · [V ∇ U − U∇ V ] = − d3 r [U(r )∇2 V (r ) − V (r )∇2 U(r )], (1.2) which, in vector notation, is the statement of Green’s theorem. If the further choice U(r ) = −(4π|r − r |)−1 is made, where r denotes the radius vector (drawn from an origin of arbitrary location) of a point inside the integration volume, then (1.1) and (1.2) yield V (r) = − 1 4π d3 r 1 |r − r| ∇2 V (r ) + 1 4π dS · 1 |r − r| ∇ V (r ) − V (r )∇ 1 |r − r| . (1.3) 1
- 20. 2 1 Introduction If V (r) now refers to the electrostatic potential due to a specified charge distri- bution at points within the surface, ∇2 V (r) = −4πρ(r), (1.4) then (1.3) provides an expression for V (r) given in terms of quadratures over the “Green’s function” G(r − r ) = U(r − r ) multiplied by the charge density, to which must be added the contributions of the surface integrals of (1.3) over values of V and/or ∇V that are specified as boundary conditions. In other words, the solution to (1.4) may be written as V (r) = −4π d3 r G(r − r )ρ(r ), (1.5) to which must be added the RHS surface terms of (1.3). As long as r does not lie on dS, these surface terms satisfy the homogeneous equation of Laplace, while the volume integral of (1.5) generates a solution to the inhomogenous equation (1.4) of Poisson. This structure, of (1.5) plus appropriate solutions of the homogeneous equa- tion, has over the intervening two centuries been generalized from the relatively straightforward elliptic (1.4) to hyperbolic and partial differential equations (DEs), and to nonlinear problems such as those of Navier–Stokes fluids and quantum field theory (QFT). In each case, the solution of an inhomogeneous DE in n spatial dimensions, Dφ(r, t) = j(r, t), D = D ∂ ∂t , ∇; A(r, t) , (1.6) specified by some collection of differential operators and (in the nonlinear case) associated fields A(r, t), is given by φ(r, t) = dn r +∞ −∞ dt G(r, r ; t, t |A) j(r , t ) + S(r, t), (1.7) where the S(r, t) specify needed boundary and/or initial conditions of the prob- lem, and are solutions of the homogeneous relation DS = 0. The Green’s func- tion (GF) of the problem, G(r, r ; t, t |A) = r, t|D−1 |r , t , is a solution of the relevant generalization of the inhomogeneous (1.1), DG = δ(r − r )δ(t − t ). (1.8) In this way, Green’s original formulation of general solutions to electrostatic problemshasfoundanaturalgeneralizationtovirtuallyallfieldswhoseessential Physics is described by an inhomogeneous DE. In subsequent sections, specific forms for D−1 will be given for problems of interest in fluid motion and diffusion, whose underlying symmetry is Galilean;
- 21. 1.2 Linear Physics 3 and for the propagators of QFT, of Lorentzian symmetry. Attention will be focused mainly on hyperbolic DEs, requiring time-dependent initial conditions; and simple constructions illustrating the method of enforcing different initial conditions will be described. These relatively simple computations are associ- ated with solutions of a linear problem, and such techniques can provide only formal descriptions of nonlinear, or interacting systems, where D is a function of fields A that are to be coupled (by means of other equations) to the desired solution φ. A more explicit construction of D−1 in the presence of external interactions will also be given in terms of the exact, and most useful representation of Fradkin.3 Special variants of the Fradkin representation generate a new, non- perturbative method for exact and approximate representations of these GFs; and in these approximations, one has at least a qualitative idea of their error. For vectorial interactions, one learns in Chapter 6 of possible chaos appearing in the realization of such non-perturbative approximations; and one sees just how such chaos is naturally removed in QFT, which process suggests application to methods of chaos suppression for classical systems. One learns, in the context of any Fradkin representation, the intimate connection between such GFs and ordered exponentials (OEs), which leads, in Chapter 9, into a discussion of unitary OEs. A brief discussion of known methods of extracting the infrared, or low-frequency structure of relevant GFs is given in Chapter 7, while a solution for the “scalar laser” problem of Chapter 4 is used in Chapter 8 to construct a model GF which can be used to estimate the total cross section for particle production in a “modified multiperipheral model” at extreme, relativistic en- ergies. A new solution for pair production in the presence of a non-constant electric field is described in Chapter 3, while estimates are given in Chapter 4 for the same process in the overlapping fields of two high-intensity lasers. Some of these results are old, and some are new; but all can be given a succinct description in terms of GFs and OEs. 1.2 Linear Physics In this section will be described the simplest linear prototypes of propagator found in four distinct fields: non-relativistic fluid motion, the non-relativistic Schrödinger equation, ordinary DEs, and QFT. Motion associated with a simple harmonic oscillator driven by an arbitrary source g(t) is the simplest ODE imaginable, d2 x dt2 + ω2 x = g(t), (1.9) to be solved, for definiteness, under the initial conditions x(0) = D, dx(0)/dt = 0. More complicated problems of current interest are obtained by
- 22. 4 1 Introduction inserting damping and, for example, replacing ω2 by ω2 (x2 − 1) to produce the Duffing equation, with its manifest nonlinear behavior. What shall be done in this section is to generate solutions to the linear problems using standard GF methods, and then to compare the results with an alternative and equivalent phase-space method of solution. No OEs appear in the linear analysis, but the standard questions of retarded or advanced, causal or anticausal solutions must be answered. Adding spatial derivatives to (1.9) in a Lorentz-symmetric way generates the forms of non-interacting field theory, (µ2 − ∂2 )A(x) = j(x), (1.10) where causality will be demanded in the sense that A(r, t) cannot be different from zero until a signal from the source j(r, t) (traveling at the speed of light when µ = 0) can reach the point x = (r, t); here, ∂2 denotes the d’Alembertian operator, ∇2 − 1 c2 ∂2 ∂t2 , and units will be chosen in what follows such that c, the velocity of light, is unity. In contrast, the diffusion equation of (relatively) low-velocity fluid motion is non-relativistic, ∂ ∂t − ν∇2 v = f(r, t), (1.11) where ν denotes viscosity and v(r, t) is the fluid velocity; appropriate initial con- ditions here will again demand causality. Because it has but one time derivative, there exist but two GF solutions for this problem, one “retarded” (subscript R) and the other “advanced” (subscript A); and it is simplest to begin the detailed construction of these GF s with this example. (i) Non-relativistic diffusion: The requirement of causality will select the GF GR as the physically relevant solution of the inhomogeneous ∂ ∂t − ν∇2 GR(r − r ; t − t ) = δ(r − r )δ(t − t ), (1.12) which, as written in (1.12), turns out to be a function of coordinate differences. If one knows the general solutions to the corresponding homogeneous DE, the solution to (1.12) may be expressed as a summation over all eigenstates of positive eigenvalues En, in the form GR(r − r ; t − t ) = θ(t − t ) n un(r)u∗ n(r ) exp[−En(t − t )], (1.13) wheretheun (r)formacompleteorthonormalset,satisfying[En + ν∇2 ]un = 0 and n un(r)u∗ n(r ) = δ(r − r ). The θ-function of (1.13) expresses the
- 23. 1.2 Linear Physics 5 retardedness of the GF; because θ(x) = 1, x 0, and θ(x) = 0, x 0, the solution to (1.11), v(r, t) = d3 r dt GR(r − r ; t − t )f(r , t ) + v0(r) will have no contribution to its first RHS term for t t, so that an effect at t cannot appear before its generation at t . Here, v0(r) represents the initial con- dition of this problem, the velocity field specified at all points r, and satisfying the equation ∇2 v0(r) = 0. Mathematically speaking, θ(x) is really the limit of a sequence of functions chosen such that θ(0) = 1/2; everywhere in this book it may be represented by the integral θ(x) = 1 2πi +∞ −∞ dω(ω − i )−1 eiωx , (1.14) and its properties checked by straightforward contour integration, as well as by the more conventional relation: δ(x) = dθ/dx. It should be noted that the En are positive, and hence the summation of (1.13) is sensible; here, the un(r) are just plane-wave exponentials of wave- vector kn = 2πn/L, where n is a vector each of whose components are integers, L3 is an appropriate normalization volume, and En = νk2 n. Were the viscosity continued to an imaginary value, with a change of normalization, one would be dealing with the non-relativistic Schrödinger wave equation (SWE), as in (ii) below. The simplest method of construction for any such GF of a linear problem is to employ a Fourier representation, GR(r − r ; t − t ) = (2π)−4 d3 k dωG̃R(k, ω)eik·(r−r )−iω(t−t ) , (1.15) where G̃R isdeterminedbysubstituting(1.15)into(1.12),andisclearlygivenby G̃R = i(ω + iνk2 )−1 .Inthecomplexω-plane,theintegrandof(1.15)hasbutone singularity, a pole at ω = −iνk2 , which multiplies the factor exp[−iω(t − t )]. Evaluating the ω-integral by contour integration, one is forced for t − t 0 to close the contour in the upper half ω-plane, which yields zero, while the choice t − t 0 requires closing the contour in the lower half plane, which yields GR(r − r ; t − t ) = θ(t − t )(2π)−3 d3 k exp[ik · (r − r ) − νk2 (t − t )]. (1.16) Because one finds a non-zero value only for t − t 0, as expressed by the θ-function of (1.16), this is the retarded GF; GA would have been obtained by reversing the sign of the −iω(t − t ) phase of (1.15). Note that in the limit of
- 24. 6 1 Introduction zero viscosity, (1.16) reduces to θ(t − t )δ(r − r ), so that the solution to (1.13) is simply v(r, t) = t 0 dt f(r, t ) + v0(r). As a Gaussian integral, (1.16) may be evaluated immediately, GR(r − r ; t − t ) = θ(t − t )[4πν(t − t )]−3/2 exp[−(r − r )2 /4ν(t − t )]. (1.17) For small differences t − t , most of the contribution to the r integral over (1.17) comes from the points r close to r; but as the time difference increases, more and more of the source dependence at other values of r is available to influence the behavior of the solution at r. If the source dependence is chosen to be that of a delta-function in space and time, f(r, t) = f0δ(r)δ(t − )| →0, and if the initial field v0 = 0, the integrations over (1.17) yield the simple result v(r, t) = θ(t)f0(4πνt)−3/2 exp[−r2 /4νt]. (1.18) This has the form of an initial (at t = 0) and localized (at r = 0) disturbance diffusing away to all spatial points at later times, while maintaining a constant infinite-spatial integral, d3 rv(r, t) = θ(t)f0. It is worth emphasizing the physically intuitive role of such a localized source in “generating” a solution to the homogeneous equation ( ∂ ∂t − ν∇2 )v = 0. One imagines for t 0 a perfectly quiescent situation with v(r, t) = 0, suddenly disturbed by a source f0 · δ(r)δ(t) rapidly “turning on and off” at t = 0, and generating for subsequent, positive t, a solution to the homogeneous equation. This use of a highly-localized source has long appeared in QFT, as a way of representing the appropriate wave functions of particles “produced” at a particular space–time point. It is an artifice, but a convenient one, and has been used in certain fluid/vortex problems.4 (ii) Non-relativistic Schrödinger wave equation: The GFs of interest here are typically “ingoing” or “outgoing”, with the latter chosen to represent the “scattered wave” of probability amplitude needed to describe the physical scat- tering of two particles which interact with each other by means of a poten- tial V (r1 − r2). Equivalently, by a simple transformation to the CM coordi- nates of these particles, one may treat the scattering of a particle of reduced mass µ = m1m2/(m1 + m2) in the field of a fictitious potential field V (r), where r = r1 − r2. In such scattering problems, the energy E of the system is
- 25. 1.2 Linear Physics 7 conserved and specified in advance, so that the original SWE ih̄ ∂ ∂t ψ = − h̄2 2µ ∇2 + V (r) ψ (1.19) is replaced by Eu(r) = − h̄2 2µ ∇2 + V (r) u(r), (1.20) uponusingthesubstitutionψ(r, t) = u(r) exp[−iEt/h̄];thecorrespondingGFs of this problem are to satisfy E + h̄2 2µ ∇2 − V (r) G(r, r |V ) = δ(r − r ), (1.21) and scattering amplitudes u(r) are given by u(r) = u0(r) + d3 r G(r, r |V )V (r )u0(r ), (1.22) where u0(r) represents the “asymptotic input” function exp[ik · r], correspond- ing to an incident particle of energy E = h̄2 k2 /2µ moving in the k direction. A formal solution for the GF of this problem may be written as G(r, r |V ) = n un(r)u∗ n(r )(E − En)−1 , (1.23) where the summation is over all states of the complete, orthonormal set un(r) which satisfy the time-independent SWE, Enun = − h̄2 2µ ∇2 + V (r) un(r). TheRHSof(1.21)isreproduced,uponsubstitutionof(1.23)into(1.21),because the un are again assumed complete, satisfying n un(r)u∗ n(r ) = δ(r − r ). Typically, the states |n form a continuum of positive-energy scattering states, plus discreet bound states of negative energy. Note from (1.23) that if the scat- tering energy E is continued to negative values, one may expect to find a pole of the scattering amplitude when E approaches one of the isolated, negative, bound-state energies. Note also that, because of the spatial dependence of V (r), this GF is no longer a function of the difference of its coordinates. It will become clear below that the distinction between the solutions Gout and Gin can be made explicit by appending an infinitesimal ±i to the denominator E of (1.23).
- 26. Discovering Diverse Content Through Random Scribd Documents
- 27. “We held our rifles a little closer, and crawled through the aperture, pausing to look about us. We both nearly dropped our guns in our excitement; for, crouched in the farther corner, was a great white, hairy creature, watching us with red, flaming eyes. Then, even before we could recover ourselves, the thing gave a kind of guttural cry of anger, and started toward us. As it rose to its feet, I swear to you I turned sick as a woman. The beast was over eight feet tall, and was covered with a thick growth of hair which was snow white. Its arms were once and a half the length of those of a common man, and its head was set low on its shoulders like that of an ape or a monkey; but the skin beneath the hair was as white as yours or mine. “I heard the Lieutenant’s gun go off, but the Thing never stopped. I raised my four-bore and let drive with the left barrel; then, overcome with a nameless fear of that great white beast, I called wildly to Arthur to follow me, and plunged through the opening and ran with all my strength toward the upper passage. It was not until I felt the fresh air on my face that I stopped to take breath, and I was so weak I could scarcely stand. Then, if you can, imagine my horror to find that I was alone. The Lieutenant was nowhere in sight. I called down the passage, and I could hear my voice echoing down the dismal place, but there was no answer. “Think what you may; but I tell you it took more courage for me to force myself down into that vault again than it would to have walked up the steps to the scaffold. I crept fearfully along the passage, calling weakly every few minutes, and dreading what I should find; but—there was nothing to find.” The Colonel paused, putting his hand over his eyes, and I could see by the moonlight that his face was white and drawn. “And did you not find him in the lower passage?” I asked, when the silence had become oppressive. “No, I did not find the Lieutenant,” he answered; “but when I came to the little square opening before the vault, there were some bloody
- 28. little pieces scattered about the floor, and the place was all slippery, but there was no Lieutenant. You know it takes four horses to pull a man apart, and you can judge of the strength of that white beast when I tell you that there was not left of Arthur Stebbins a piece as big as your two hands. “As I looked at that floor with the ghastly things which covered it, a wild rage took possession of me. I knew that the creature was in the room beyond, for I could hear a crunching as a dog makes with a bone. I rushed through the opening, straight toward the corner where it was crouching. It saw me coming, and leaped to its feet. Again that sickening fear that I had felt before came over me; but I stood my ground and waited till it nearly reached me. Then, with the muzzle of my gun almost against it, I fired both barrels full into its breast. “I must have fainted or gone off my head after that, for the next thing I knew I was lying in a native’s hut on the Durbo road. Zur Khan, the man who owned the bungalow, said that he had found me four days before, wandering about on the plains, stark mad, and had taken me home.” “And the Thing in the passage?” I asked breathlessly. “Did you never go back?” “Yes; when I had recovered a little, I went back to the Mubapur Temple,” answered the Colonel; but he was silent for some minutes before he answered the first part of my question. “In my report to the Government I said that Lieutenant Arthur Stebbins was torn to pieces in the lower passage of a Mubapur Temple by an immense white ape,—but I lied,” he added quietly.
- 29. THE FOOL AND HIS JOKE. THE FOOL AND HIS JOKE. ILLIAM WATERS was not in any way what you would call a braggart, yet upon two things did he pride himself. These two things were: first, an earnest and sincere contempt for all things supernatural; and, secondly, a marksmanship with a Colt’s No. 4 revolver which bordered on the marvelous. He had on several occasions proved his bravery by such feats as sleeping alone an entire night in a house said to be haunted, and by visiting a country graveyard at midnight, and digging up a corpse. He had likewise won numerous bets by pumping six bullets into an inch and a half bull’s eye at a distance of sixteen paces, and being a healthy and vigorous animal his pride was perhaps more or less excusable. In the house in which Waters had his rooms there also lived a Fool. His particular brand of folly was practical joking, which is universally recognized by intelligent men as a particularly acute and dangerous kind of idiocy. As a child The Fool had soaked a neighbor’s cat in kerosene and then applied a match. Since then he had performed many other equally humorous feats. After much planning The Fool devised a joke, the victim of which was to be The Man Who Knew Not Fear, as the jester sneeringly called Waters. The prologue to this joke was the substitution of blanks in each of the six chambers of the No. 4 Colt’s, which hung over the headboard in William Waters’s sleeping-room, not as a weapon of defense, but as a glittering little possession dear to the heart of its owner.
- 30. The Fool had once, in the presence of all the people at the dinner table, asked Waters what he would do should he wake up at night and find a ghost in the room. “Fire a bullet straight at his heart, so be sure and wear a breastplate,” Waters answered promptly, and the laugh had been on the joker. After removing the cartridges from the revolver, The Fool withdrew the bullets from each, and placed them in his pocket. He had that day also laid in a supply of phosphorescent paint and several yards of white muslin. Waters never locked his door at night, for he was as free from fear of all things physical as from those supernatural. This of course made the program which The Fool had arranged easy to carry out, though he would not have hesitated at a little thing like stealing the key and having an impression made. He was a very thorough practical joker. That night as the French clock in the hall outside Waters’s room was striking twelve The Man Who Knew Not Fear was awakened by a rattling of chains and a dismal moaning. As he opened his eyes he saw standing in the darkest corner of the room a white-robed figure, which glistened with phosphorescent lights as it waved its arms to and fro. Without a moment’s hesitation, Waters reached for his revolver, and leveling it at the moaning figure, fired full at its breast. The Fool, chuckling to himself behind the sheet, thrust his hand upon his heart, and apparently plucking something from the folds of cloth, he tossed back toward the bed a bullet. The Man Who Knew Not Fear reached for the heavy little object that he had felt strike the bed-clothes, and his hand touching the bit of lead, he picked it up curiously, then realizing what it was that he held, he sat up stiffly in bed, and tried to raise his arm again. But his muscles refused to obey. The thought that his revolver had been tampered with never entered his head. For the first time in his life a
- 31. fear, sickening and unmanning as it was new, came over him. He recognized in that little piece of lead a bullet from the gun which had never before failed him. What was that moaning Thing upon which powder and lead had no effect? Three times he tried to raise his arm, and each time it fell back upon the bed. Meanwhile the rattling of chains began once more, and with eyes starting from his head because of his fear, Waters saw the fearsome shape advancing upon him. By a supreme effort he raised his arm, and emptied the remaining five chambers of his revolver at the approaching figure. The Fool, who had never ceased moaning while the shots were being fired, executed a rapid movement with his hands as if catching the bullets, and then slowly tossed them back, one after the other. The man in bed reached for the little balls of lead mechanically, then straightened back against the pillow, and remained perfectly motionless, staring at the Thing, which had now stopped again and was groaning dismally. For five minutes neither man moved, then The Fool, thinking that the joke was once more on him, for Waters still refused to speak, gathered his glittering robe about him, and slunk out. Back once more in his own room he undressed hurriedly, and slipped into bed. He was disappointed. He had expected that Waters would be terribly frightened, and that he could joke him unmercifully at the table for the next week. Then, too, the obstinate silence of the man puzzled him. About five o’clock in the morning he woke up vaguely alarmed. He did not know what the matter was, but he could not sleep. He could not get out of his mind that strange silence of the man down-stairs. Then, suddenly, a terrible suspicion came over him. “Not that, my God, not that!” he cried. Jumping from the bed he threw on a few clothes, and crept fearfully down to the scene of his midnight joke.
- 32. He opened the door cautiously, and, feeling for the button, turned on the electric light. Then he gave a hysterical cry, half laugh, half moan, and, rushing from the room, he fled down the hall out into the street. For this is what he had seen: in the bed propped up stiffly against the pillow, and staring with dull, unseeing eyes into the corner, sat “The Man Who Knew not Fear.” Not a muscle had he moved since The Fool had left him six hours before. One hand still held the silver-mounted revolver, while in the other were tightly clasped—six little leaden balls.
- 33. THE MAN AND THE BEAST. THE MAN AND THE BEAST.[2] OBO, the wild man of Borneo, sat in his iron-barred cage reading the morning paper, while he pulled vigorously at a short, black clay pipe. It was nearly time for the show to begin, so he could only glance hurriedly at the stock report; for Bobo was interested in copper. On Mondays, Wednesdays, and Fridays there was on exhibition in the side-show connected with Poole Brothers’ Royal Roman Hippodrome and Three-Ring Circus what was widely advertised as the only real wild man in captivity. On alternate days—that is, on Tuesdays, Thursdays, and Saturdays— the cage of Bobo was closed by a gaudily painted cover; and visitors on those days were told that the wild man was sick. Notwithstanding this report, there could be found on Tuesdays, Thursdays, and Saturdays, out in one of the New York suburbs, a middle-aged Irishman named Patsy McLockin. The connection may not at first be evident. Patsy wasn’t nice looking, even when he was dressed in his best black suit; for, as the people on Blenden Street remarked, he was too hairy. He used to wear gloves when on the street, even in the hottest weather; but he couldn’t very well wear gloves on his face, though if
- 34. he could it would have saved the small children of the neighborhood cases of fright both serious and lasting. Poole Brothers’ Royal Roman Hippodrome and Three Ring Circus was playing a winter engagement in New York City, and had been very successful. The show was to start in about two weeks for a trip through New England; and since Mrs. Patsy McLockin had consented to remain in the city till the circus came back in the fall, Bobo agreed to exhibit himself every day while on the trip. When Stetson, manager of the freaks in the side-show, had spoken to Bobo of the necessity of appearing every day while traveling, he had also mentioned a material raise in the wild man’s salary. Every two weeks during the winter Stetson had written a check for seventy-five dollars in acknowledgment of services rendered. In the event of Bobo’s agreeing to make his appearance on each of the one-day stands, Stetson was authorized by the powers above to draw these fortnightly checks for one hundred and fifty dollars, and, after much discussion in the Blenden Street home, Stetson’s offer had been accepted. On this morning Bobo was trying to decide whether to sell out his twenty-three shares of Isle Royal while that stock was at eighty-one, or to hang on to it for a while, hoping for a rise. He fully intended to sell out some time during the next two weeks, for he did not want to be bothered with the stock while on the Eastern trip. “Get together there, you freaks,” called Stetson; “the whistle has just blown, and the yaps will begin coming in soon.” Bobo tucked his paper into a little wooden box in the back of the cage, knocked the ashes out of his pipe, and curled up on the straw, pretending to go to sleep. He never worked over time, did Bobo; and up to the time when Stetson brought him his piece of meat, and began telling the people
- 35. of the terrible struggle which had taken place in the swamps of Borneo, when the wild man was captured, Bobo always pretended to be asleep. When, however, the manager reached a certain point in his narration, the nearest of the onlookers were usually startled by a savage growl, and the wild man from Borneo got up on all fours. Some hysterical woman generally screamed at this juncture, for, with the help of his make-up box, Bobo certainly did look the part. For clothes, he wore merely a ragged breech-cloth about his loins, while the rest of his body was bare, save for a tawny growth of red hair. His skin was stained a dark brown, and in several places there were great raw-looking spots, where the manager said Bobo had bitten himself. But the wild man’s face was what caused the alarm on the part of the women and children. His nose was a snout-like protuberance with great cavernous holes for nostrils, while his eyes, peeping out from under bristling brows, were small and wicked. All over his face and neck, and extending down to his breast, was a coarse growth of stiff red hair. The manager finished his harangue over Herman, the Ossified Man, pictures of whom a small boy began offering to the crowd for the sum of ten cents each. “Next, and last, I call your attention, ladies and gentlemen, to Bobo, the wild man from Borneo,” began the exhibitor. He was always glad when he came to Bobo, partly because he was the last freak to describe, and partly because the wild man always acted his part so well. The crowd rushed from in front of the platform on which the Ossified Man had been exhibited. “Don’t get so near there, boy,” shouted one of the attendants to a venturesome youth; “the wild man is liable to grab you. He killed a
- 36. man that way last week.” Stetson began his lurid tale of the fierce struggle which had taken place when the wild man was captured, and the crowd of country people listened open mouthed. “Throwing this net about his head and shoulders, we succeeded in getting the creature to the ground,” droned Stetson in a sing-song voice. This was Bobo’s cue. He yawned, exposing a set of yellow fangs, at the sight of which the small boy in the front row turned a little pale, and tried to work his way back into the crowd. Then Bobo growled. Bobo was proud of that growl. It had taken him weeks to acquire it. Beginning with a kind of guttural rumbling in his throat, he worked himself up gradually, and ended with a ferocious howl. “The wild man is hungry, you see,” said Stetson; and taking a piece of raw meat from under the wagon, he held it up to view.
- 37. “The wild man ran to the bars of the cage and shook them furiously.” (See page 35.) Bobo immediately sprang at the bars of his cage, and rattled them loudly, chattering fiercely meanwhile. The crowd fell back, leaving a clear space in front of the cage; and the wild man, reaching a hairy arm out between the iron bars, seized the meat, and crawling to a corner, buried his teeth in the bloody shank.
- 38. “This concludes the entertainment,” shouted Stetson, and the crowd reluctantly began to file out of the tent. Two months later, while Poole Brothers’ amalgamated shows were exhibiting in Vermont, Murphy, one of the side-show attendants, came to Stetson, and informed him mysteriously that Bobo was acting queer. “He don’t get out of his cage after the show’s over in the afternoon like he used to, but stays there till the evening performance.” “Nothing queer about that as I can see,” answered Stetson carelessly. “He’s been putting more life than usual into the part lately, and it probably tires him. What’s the difference whether he rests in his cage or goes over to the car? You’re probably kicking because you have to bring his supper to him.” “He used to wash the make-up off his face between the two shows,” persisted Murphy. “But now he keeps it on from ten in the morning till night.” “Well, you never take the trouble to wash the ordinary every-day dirt and grease off your face, and I don’t believe you ever would clean up if it took you the time it does Bobo,” replied Stetson irritably, and Murphy retired, muttering. But the other freaks had noticed a change in the wild man, too. Between performances Bobo used to play penny ante with the fat man and the bearded lady, both of which gentlemen now tried in vain to lure him into a game. Saturday nights, also, when the last show for the week was over, the freaks sometimes had a little “feed,” and formerly Bobo had been one of the most jovial spirits. Lately, however, he refused to attend any of these gatherings, and spent most of his time alone. As Stetson said, though, the freaks were always complaining about one another, so little attention was paid to the grumbling in the side- show tent.
- 39. The management couldn’t afford to offend Bobo, for there was no denying that the wild man was the star attraction. He was doing better work than he ever had done before. He didn’t wait for the manager to come to him to begin acting; but as soon as the crowd appeared, he was growling and tearing away at the bars of his cage. The other freaks complained; for even when the dog-faced boy was making his worst grimaces during Stetson’s description of him, most of the audience preferred standing in front of the wild man’s cage watching his antics. One Sunday night the attendant, who had been before rebuffed, again sought out Stetson with a new tale of woe. “Bobo sleeps in his cage every night now,” he declared, “and he’s been in there all day to-day.” “Perhaps he’s sick,” said Stetson, but he didn’t believe it. He himself had noticed a change of late in the wild man. The meat, which had been thrown in to him, had formerly been taken out untouched and given to the lions; but lately there hadn’t been any meat left. “He ain’t sick, neither,” declared Murphy, “but he’s too damn ugly to live. He tried to bite me when I changed the water in his dish; and yesterday, when Skoggy brought him a newspaper like he used to, to show him the stock report, Bobo tore it to pieces, and tried to hit Skoggy with that bar that’s loose in his cage. Stetson consulted with Poole Brothers, and that night the three men went to the side-show tent, which was up in readiness for the Monday’s performance. They found Bobo lying asleep in his cage. He still had on his make- up, but some way he didn’t look natural to the Poole Brothers. They didn’t go to the side-show tent very often, and it had been over two months since they had seen the wild man. The hair on his arms and breast was thicker than it used to be, and his teeth seemed longer and yellower.
- 40. Stetson opened the door of the cage and called, “Wake up, Pat. It’s time for supper.” The wild man opened his eyes quickly, and snarled like a dog which has been roused suddenly. “It’s time for supper,” repeated Stetson, stepping back and clasping his cane a little tighter. Bobo seized the little iron dish in which they brought him water, and started to hurl it at the speaker; but noticing suddenly who it was, he only growled, “Don’t want no more supper; just had mine.” The younger Poole brother looked at a half gnawed bone lying on the bottom of the cage, and muttered something which nobody heard. “Well, you’re not going to stay here all night, are you?” persisted Stetson. Bobo ran to the door of his cage and seized the bars, shaking them as he did when the show was on. “Why in hell can’t you leave me alone?” he screamed. “What do you care where I sleep? Don’t I do my work? And don’t I earn my pay? Then what you kicking about? Git along, and leave me alone; I’m sleepy.” Stetson looked at the two Poole brothers, one of whom made a sign, and the three men withdrew. “Looks as if we’d have the genuine article, instead of a fake, in a week or two more,” observed the elder Poole to the manager. He had been in the show business for some years, and wasn’t easily shocked. During the next few weeks the freaks had many causes for complaint. The Bearded Lady claimed that Bobo had spit at him when he went by the cage. But the Bearded Lady was a man of sensitive disposition, and easily offended.
- 41. There were other things more serious, however. Mlle. Mille, one of the albinos, showed Stetson a black and blue spot on her arm where the wild man had struck her when she was putting on her wig, and the snake charmer threatened to leave the show if Bobo was not locked in his cage. One night, therefore, when the wild man was asleep, three of the attendants stole into the tent and snapped a couple of strong padlocks through the staple in the door. It was a good thing that they did; for the next day Bobo had a crazy fit before the show opened up, during which he tried to tear his cage to pieces. It proved a great attraction, though; for the country people outside heard him raving, and the tent was soon packed. He stopped speaking to any one after that, and refused to answer when spoken to. He stayed in his cage all the time, sleeping there nights, and never touching the cooked food sent him from the kitchen, but there was never any meat left over for the lions. The Royal Roman Hippodrome and Three Ring Circus played to remarkably good business all summer, and finally brought up at the old winter quarters in New York. One of the first visitors upon their arrival there was Mrs. Patsy McLockin, who came to see what in the world had happened to her husband, for she hadn’t heard a word from him for over two months. Stetson took her into the room where workmen were getting every thing in order; for the show was to begin its winter indoor engagement next day. In his cage in one corner, gnawing a bloody shank of meat, crouched Bobo. Stetson took the woman over to the cage; and Mrs. McLockin, after looking at the wild man for a few seconds, broke out sobbing. “You’ve gone and made him crazy, you have,” she wailed. “Patsy, dear, don’t you know your old woman?”
- 42. But Bobo, the wild man, continued crunching his bone, and paid no attention to the woman in front of his cage. The manager stole out of the room softly, and left them together. There was nothing he could do. Each week he had gone to Bobo’s cage, and tried to talk to the wild man, telling him that he had better give up the business and settle down somewhere. But the wild man never paid any attention to him; and when one day Poole Brothers tried to take him out of his cage by force, one man was killed and Stetson himself seriously injured, so that had to be given up. All that winter the side show connected with Poole Brothers’ Royal Roman Hippodrome and Three Ring Circus played to packed houses; and probably no one paid any particular attention to a sad-faced Irish woman of middle age who spent most of the time standing in front of the cage of Bobo, the wild man, weeping silently.
- 43. AT THE END OF THE ROAD. AT THE END OF THE ROAD. T first the road was smooth and level; there were no hills, and The Man had many companions. They laughed with him and made merry, and there was no thought of care. “’Tis a pleasant life,” murmured The Man; but even as he said the words he wondered half fearfully if it could last, if the country through which they passed would always be as pleasant. Gradually the way became harder. Quite often The Man was compelled to pause for breath, for there were difficult places to get over; and when he turned for assistance to the companions who had laughed and jested with him but a little while before, he found that they had passed just beyond calling distance. At least they seemed not to hear him, for they did not stop. But the way was not all hilly; and when he came to the smoother places The Man hurried on faster than before, and, catching up with his companions, was welcomed by them, and they all made merry once more. The smoother places became rarer, however, and The Man found himself alone many times, till one day he was joined by a new companion. “He will be like the others,” said The Man bitterly: “he will not stay with me.” But the other heard him. “Do not fear,” he answered, “I will stay with you to the journey’s end. I will never leave you.”
- 44. Nevertheless, The Man did not like his new companion. He was not like the others. He never jested and made merry, and after that first time he did not speak again. He was gaunt and thin, and was clothed in rags; but he stayed with The Man when the others ran on ahead or lagged behind. One day when The Man was weary, for there was no longer any one to cheer him, and the way had become very hard, he plucked up courage to speak to his silent companion again. “’Tis true you do not leave me like the rest,” he said; “they all deserted me when we left the pleasant country; but I do not know you yet. If we must travel together we should get better acquainted.” “Mine is not a pleasant name, and few care to know me better than necessity compels,” answered the Silent One; “but had you waited a little longer you would not have needed to ask. I am known by many names, but those who know me best call me Poverty.” The Man picked himself up from where he had thrown himself to rest, and hurried on, trying to leave his companion behind. But the one in rags followed close, and when The Man stumbled and fell, exhausted by his exertions, the other was just at his heels. And about this time The Man noticed that a third wayfarer had joined them. He could not see the new comer’s face, however, for he always kept a little way behind; and there seemed to be a kind of shroud-like hood over his head. There were no longer any easy stretches in the road, and The Man moved slowly. Many times he stumbled and fell, and each time it was longer before he rose again. He wondered, but dared not ask the name of the new arrival who had moved nearer, and was now but a few steps behind. At last The Man came to a part of the way more difficult than any before; and he lay down for a few minutes to rest. After a time he tried to go on, but could not. He was too weak, and his two companions seemed to be conspiring to hold him back. He
- 45. summoned all his strength, and made one last effort to go on. At first he seemed to advance a little, but the hand of The Ragged One thrust him back. He stumbled, fell, rose again, and staggered on a few steps, then fell once more and could not rise. “This is the end,” he heard the Silent One saying; “and I have kept my word; I am still with you.” There was a sound of footsteps approaching stealthily, and The Man opened his eyes with an effort. The companion who had always lagged behind was advancing swiftly, and the black hood was drawn away from his face. Painfully The Man raised himself on his elbow and looked at the figure for a second, then fell back. “How strange that I did not know you before,” he muttered faintly, for he had seen the other’s face, and recognized that it was Death.
- 46. THE SPACE ANNIHILATOR. THE SPACE ANNIHILATOR.[3] N the afternoon of Saturday, August 18, 1900, as I was looking over the daily paper after my return from the Blendheim Electric Works, where I am employed, I noticed in the advertising department the following: IMPORTANT NOTICE TO ENGINEERS AND SCIENTIFIC MEN. Ten thousand dollars will be paid to the man or woman duplicating an instrument now in the possession of this company—— That was as far as I read. Some cheap advertising scheme, I thought, and immediately forgot all about the paragraph. When, however, towards the last of the month, I received the regular issue of my pet scientific paper, I saw on the first page the same glaring announcement. The fact of the notice being in that paper was guarantee that the offer was bona fide, and I looked the article over carefully. In addition to the foregoing, the advertisement went on to state that one of a pair of seismaphones, an invention with patent pending and not yet in the market, had been lost. The inventor was dead, and no one had as yet been able to construct an instrument similar to the one now in the company’s possession. Further particulars would be sent to any one satisfying the company that his request for the same was not prompted by idle curiosity, but by a desire to aid science in replacing the lost instrument.
- 47. Then came the greatest surprise of all; for, signed at the bottom of this interesting statement, as the man representing the company, was the name of Randolph R. Churchill, Patent Office, Washington, D. C. Now Ranny Churchill and I had been roommates at college, and I had had many a pleasant visit in his comfortable home on Fourteenth Street. He had graduated from a technical school, taken a course in patent law, and soon after secured a position as one of the governmental inspectors of patents in Washington. My annual vacation was to begin the next week, so I planned a brief trip to Washington to see the wonderful invention which no one had apparently been able to duplicate. I did not write to Churchill, but dropped in on him unexpectedly Saturday night, September 1. I had seen him two years before down on the Cape; and I could scarcely believe that the tired, careworn man who greeted me on my arrival at the Fourteenth Street house was the same merry, light- hearted Randolph Churchill I had hunted and fished with only a couple of summers ago. He seemed like a man living in constant expectation of something terrible about to happen, and, even before our first greetings were over, I noticed that he paused two or three times and listened intently. “I think I can guess to what I owe this visit,” he said as he went up- stairs with me to my room, “and I would to God I thought you would be able to accomplish what has so far proved impossible.” I told him that it was owing to his advertisement that my present trip had been undertaken, and begged him to tell me more about the wonderful invention. “Wait till after dinner,” he said, “for it is a long story. We will go to my room, and I will tell you then a tale as strange as it is true.” That dinner was the most dismal affair I ever attended. Churchill sat like a man in a trance, completely absorbed in his meditations; and
- 48. twice, after listening as I had seen him on my first arrival, he excused himself and left the table abruptly. “You and Rannie are such old friends, you mustn’t mind him to- night,” Mrs. Churchill said to me apologetically, while he was out of the room; “this terrible affair of the seismaphone has upset us both completely.” That was the only mention of the subject during dinner; but after we had sat in the library a little while discussing trivial topics, such as Robert’s progress in school and the new furnishings of the house since my last visit, Churchill and I excused ourselves and went to his private room. “I may as well start at the very beginning,” he said as he threw himself down languidly in an easy chair, after drawing out from under the table a long, narrow box, which he placed in his lap. “On the night of the tenth of last June the maid brought me the card of a man who was waiting down-stairs, and who said he wanted to see me on very important private business. I glanced at the name scrawled in red ink on the bit of card-board,‘Martin M. Bradley,’ and wondered vaguely who the man could be, as I did not remember ever having heard of him before. “I told the maid to show him up here to the den, and a few minutes later she ushered into this room the man who has been the cause of these gray hairs. “He was short and sallow, about thirty-five years of age, as I afterwards found out, though care and privations had marked him so harshly that he looked to be nearly fifty. He carried in his hand this black, leather-covered box which you see in my lap; and, after seating himself at my invitation, began: “‘You are no doubt surprised, Mr. Churchill, to have a visit from me, for you probably don’t remember ever having heard of me before; but I’ve come to you because I know you are in the patent office, and used to be a friend of mine back in the seventies, and because,
- 49. too, I’ve got something so valuable here that I don’t dare to send it up to the office in the usual way.’ “He unstrapped, as he spoke, the box, which he had not let out of his hands since he entered, and took from it two black, galvanized rubber instruments, one of which you see here.” Churchill lifted from the case a thing which resembled more than anything else the receiver of a telephone, except that both ends were turned out like the one you put to the ear. He unscrewed this outer cap and handed both parts to me to examine. About two inches in from the bell-shaped end of the cylinder was a diaphragm of peculiar looking metal, which from appearance I judged to be an alloy of copper and zinc, with something else included. Immediately over this, and tightly stretched across at unequal distances apart, were some twenty fine German silver wires. “Bradley opened one of the instruments, as I have just done,” continued Churchill, “and proceeded to explain to me its construction. “‘These two instruments,’ said he, ‘which together I call the Martin Bradley Seismaphone, are to the telephone what telegraphy without wires is to the ordinary method of sending messages. Both light and sound, as you know, travel by waves which produce sensation; one by striking against the retina of the eye, the other by striking on the drum of the ear. “‘The light wave travels with a velocity of something over 185,000 miles a second, while the sound wave moves much slower. This difference, however, is overcome by the mechanical device in the tube-like section in the middle part of the instrument. “‘As you have seen the sun’s rays collected and focused to one small spot by a reading glass, and the power intensified so that combustion takes place, so in a similar way does the seismaphone collect the sound waves, intensify and bring them to a focus here,’ and he indicated with his finger a point back of the metal diaphragm.
- 50. “‘By speaking into one of these instruments the sound passes through the wires, and strikes against the metal disk. This sets in motion a series of waves, which, traveling with the enormous velocity of which I have spoken, produce such rapid vibrations that the ear, unaided, cannot perceive the sound, but by means of the other half of the seismaphone these sound waves are collected and so transformed by the corresponding wires and diaphragm that the voice is reproduced by one instrument in exactly the tone spoken. “‘By means of the seismaphones, you and I, though separated by thousands of miles, can converse as easily as though we were in the same city, connected by an ordinary metallic current.’ “In a fifteen years’ experience with patent seekers, I have met many inventive freaks, and probably something of what I was thinking of his seismaphone showed in my face, for he stopped describing it abruptly, and handing me one of the instruments, said,— “‘I see you don’t believe a word I’ve told you, and you probably think I’m crazy; so, before I tell you anything more about the construction or possibilities of my invention, I want to ask you to take this half of the seismaphone, and go up to the top of your house. When you are ready to make the test, put the end marked “voice” to your mouth, and say in a distinct tone, “Ready, Bradley.” Then, when you see this little hammer striking against the bell, and hear a sharp tinkling inside the cylinder, put the other end to your ear and listen. Oh, you may lock me in as you go out, if you are afraid I may remove any of the bric-à-brac,’ he added, as I seemed to hesitate. “I don’t know why it was, for I am not over credulous, but something told me the man was speaking the truth. And when you stop to think of it, what was there so very improbable about it? “Who would have believed one hundred years ago that we would ever be able to communicate instantaneously with the inhabitants of another continent by any means whatever? Or, to come nearer to our own time, twenty years ago we would have scoffed at the idea
- 51. of telegraphing without wires. Why, then, was it so impossible to transmit the tones of the human voice without them? It would be only another step in the march of progress. “I took the instrument and climbed to the garret without a word. Placing the end he had indicated to my lips, I said loudly, ‘Ready, Bradley.’ Without any special expectation I then put the other end to my ear, and at the result nearly fell over backwards; for, as distinctly as if the man I had left down-stairs had been standing beside me, I heard him say,— “‘Don’t speak so loud. I can hear you at this distance if you merely whisper. Now press the little button at the end marked “ear,” and wait for the megaphone attachment.’ I did as he said, and again I jumped and nearly dropped the instrument, for the room was filled with a voice which sounded louder than a peal of thunder. “‘By pressing that button you do for the seismaphone what by putting on the horns you do for the phonograph or graphophone,’ the stentorian voice said. ‘You had better press the button in the other end, for my voice with this attachment is probably too loud for pleasure.’ “I pressed the button obediently as directed, and walked back down- stairs filled with wonder. “We shall not get to bed any earlier than Martin Bradley and I did that night, if I stop to tell you all of our conversation. I found that he was a man I had known slightly some years ago when I was trying for the patent office position. “He had in his youth been through a technical school and received a good education; but had been unable to settle down to any steady employment, preferring to devote himself to some great invention. Eight years ago he began working on this instrument, and had been developing and perfecting it ever since. “The proposition he made me was that I should go into partnership with him to get the seismaphone patented and before the public, he furnishing the device, and I the money and backing.
- 52. “We sat and talked for hours, and the morning sun found us still in our chairs discussing the immense possibilities of the invention. “It would supersede the mails. Speaking-tubes, telephones, telegraphs, and cables would give way to it. In short, the inventor of such an instrument would win for himself a name greater than a Morse or an Edison, and the fortune he could amass would exceed that of all the Vanderbilts, Goulds, and Rockefellers in the country. “Martin Bradley remained at my house all that week, and had the best of everything that money could buy. I secured a two weeks’ vacation from the patent office, and he and I worked together every hour of that time. “One day as a test he took one-half of the seismaphone and went down the Potomac a hundred and forty miles to Point Lookout, while I stayed at home with the other instrument. He had by use of the long-distance telephone hired a man down there to keep watch for the arrival of the boat he was coming on, and given him instructions to telephone me when it first hove in sight. “I sent Nellie and the children out to Chevy Chase for the day, and sat all the afternoon in front of the telephone, with the seismaphone on my knee. Several times I called to Bradley, but he did not answer. “About three o’clock, however, the ’phone rang; and, just as I had got connection, and began talking with the man down at the Point, I saw the little hammer of the seismaphone vibrating, and, putting the instrument to my ear, heard Martin Bradley say distinctly: ‘Have just sighted the lighthouse, so get down to the telephone for a message.’ “I turned to the telephone, and, sure enough, the man at the other end of the wire was telling me that the Petrel was in sight. As the boat neared the shore, Bradley kept up a running comment on events that took place. “‘We’re just pulling up the flag and firing a salute,’ he called; and scarcely did I catch his words when from the telephone at my ear, as if in echo, came the message, ‘They have just run up a flag and are firing a salute.’
- 53. “During the next week we tried every kind of test imaginable with the seismaphone, and there was not a flaw in its workings. “I was perfectly satisfied, and had started proceedings to secure a patent, when the first news of the recent trouble in China came; and then, for two weeks, as you know, the various legations were regularly slaughtered one day and reported safe on the following. “Martin Bradley was so excited that he nearly forgot his seismaphone. In the course of his wanderings he had lived for two years in Northern China, and could talk the lingo like a native, and was wild to go out there as a newspaper correspondent. “One day he came rushing to my room with a copy of a morning paper in his hand. “‘See that,’ he cried excitedly, ‘this paper says that Minister Conger was butchered in cold blood June 24, and all the others of the legation tortured to death by those yellow devils. To-morrow if you buy a paper you will read that they are safe and well. I tell you, I am going to China to find out for myself the truth of this matter, and when I do the world shall know what is true and what is false. They can put restrictions on the press, the telegraph, and the cables, but they can’t restrict Martin Bradley’s seismaphone. “‘Just think of the advertisement for the invention, too,’ he continued, getting more and more excited. ‘Every reading person in the world will know that the truth was finally obtained through Martin Bradley, by means of his greatest of all inventions, the seismaphone.’ “I tried to dissuade him, telling him of the terrible risk he would run, but he would not listen. He had lived in Peking for two years, he said, and knew the city perfectly and the customs and language of the people. “He scraped together three hundred dollars some way, the Lord only knows how, engaged a berth for San Francisco, and inside three days had made all preparations for the trip. When I found that
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