![viii PREFACE
tained in one assignment of the indeterminates, but also in the original
assignment the rich physical interpretation in terms of boson operators
is regained.
But much more emerges. The group multiplication property of rep-
resentations is a consequence of a new class of identities among multi-
nomial coefficients, which themselves have a combinatorial origin and
proof, and which hold for arbitrary interpretations of the n2 indetermi-
nates, including even singular matrices of order n. The structure is fully
combinatorial. The study of these polynomials is thus brought under
the purview of combinatorics and special functions, extended to many
variables. These polynomials may be regarded as generalizations of the
functions that arise in the study of the symmetric group, with its as-
sociated catalog of symmetric functions, such as the Schur functions,
etc. Even more unexpected is that the famous MacMahon [129] Master
Theorem, a classical result in combinatorics, is the basis for Schwinger’s
[160] famous generating function approach to angular momentum theory.
Indeed, it is the MacMahon Master Theorem that unifies the angular mo-
mentum properties of composite systems in the binary build-up of such
systems from more elementary constituents.
This monograph consists, essentially, of three distinct, but interre-
lated parts: Chapters 1-4, Chapters 5-9, and Chapters 10-11. The last
two chapters are compendiums which define, develop, and summarize
concepts used in the first nine chapters.
Chapters 1-4 deal with basic angular momentum theory and the
properties of the famous Wigner D−functions, now extended to poly-
nomial forms over four commuting indeterminates, and with the prop-
erties of arbitrary many multiple Kronecker products of these extended
D−polynomials. These four chapters may be regarded as a summary of
results that subsume all of standard angular momentum theory with a
focus on the combinatorial underpinnings of these polynomials, as cap-
tured by the concept of SU(2) solid harmonics. As examples, the famous
Wigner-Clebsch-Gordan coefficients are shown to be objects that com-
binatorially come under the purview of the umbral calculus, while the
binary coupling theory of angular momentum is intrinsically an applica-
tion of the theory of graphs, specifically, binary trees, Cayley trivalent
trees, and cubic graphs. This leads to a number of combinatorial in-
terpretations of the well-known Racah sum rule and Biedenharn-Elliott
identity, and the fundamental role of Racah coefficients in the binary
recoupling theory of angular momenta.
Chapters 5-9 deal with the generalization of the solid harmonics to
polynomials called Dλ−polynomials, where λ is a partition, and these
polynomials are defined over n2 commuting indeterminates, which when
specialized to the elements of a complex matrix of order n give the in-
tegral irreducible representations of the general linear group of complex](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-13-320.jpg)
![PREFACE ix
matrices of order n, and, in particular, all inequivalent irreducible repre-
sentations of the general unitary group of matrices of order n. Again, the
focus is on the combinatorial properties of the general polynomials them-
selves, such as their unique generation by shift operator actions, which
involve diagraphs, Sylvester’s identity, Schur functions, skew Schur func-
tions, Kostka numbers, and Littlewood-Richardson numbers, all combi-
natorial concepts underlying modern treatments of the symmetric group
Sn. It is the labeling of these polynomials by Gelfand-Tsetlin patterns,
which are one-to-one with the semistandard Young-Weyl tableau, that
underlies the relationship to the symmetric group. The reduction of
the single Kronecker product Dµ ⊗ Dν =
λ cλ
µν Dλ of two such ir-
reducible polynomials into a direct sum of irreducible polynomials is
extraordinarily rich in combinatorial structures.The Dλ−polynomials
subsume many of the properties of classical Schur functions, and the
matrix Dλ(Z) might well be called a matrix Schur function.The com-
plexity of these polynomials, although elegant in their structure, allows
us to deal comprehensibly only with the Kronecker product of a pair
of such polynomials. Multiple Kronecker products and the associated
concepts of Racah coefficients, etc., and the relationship to graph theory
is beyond our reach. New viewpoints of tensor operators as operator-
valued Dλ−polynomials emerge. A comprehensive theory of (general-
ized) Racah coefficients must await further developments.
The Littlewood-Richardson numbers cλ
µν that occur in the reduction
of the Kronecker product is so pervasive that we give a great deal of
attention to their properties (Compendium B). These numbers express
the number of repetitions of a given Dλ−polynomial in the Kronecker
product reduction. They give the generalization to the general unitary
group U(n) of the familiar addition rule
j = j1 + j2, j1 + j2 − 1, . . . , |j1 − j2|
of two interacting quantum-mechanical constituents with separate an-
gular momenta j1 and j2, constituting a composite system of angular
momentum j; the Littlewood-Richardson number is 0 or 1.
Three (at least) nontrivial combinatorial objects enter into the combi-
natorial interpretation of the Littlewood-Richardson numbers: Gelfand-
Tsetlin patterns, semistandard skew tableaux, and the lattice permu-
tations associated with these entities. The intricacies of such counting
methods would appear to be a rather high price for obtaining the rule for
the addition of two angular momenta, which was deduced by physicists
from experimental spectroscopy and subsequently from algebraic tech-
niques (see Condon and Shortley [45]) that involved neither Lie algebras
nor combinatorics. But the new insights gained are well worth the effort.
These techniques underlie the development of the properties of the
Dλ−polynomials over arbitrary commuting indeterminates. One of the](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-14-320.jpg)
![PREFACE xi
(v). Extension of combinatorial foundations to other groups.
(vi). Applications to physical problems.
The very detailed Table of Contents serves as a summary of topics
covered. The readership is intended to be graduate students and re-
searchers interested in learning of the relation between symmetry and
combinatorics and of challenging unsolved problems. The many exam-
ples serve partially as exercises. It is hoped that the topics presented
promote further and more rigorous developments.
We mention some unconventional matters of style. We present signif-
icant result in italics, but do not grade and stylize them as lemmas and
theorems. Such italicized statements serve as summaries of results, and
often do not merit the title as theorems. Diagrams and figures are inte-
grated into the text, and not set aside on nearby pages, so as to have a
smooth flow of ideas. Our informality of presentation, including proofs,
does not attain the status of rigor demanded in more formal approaches,
but our purpose is better served, and our objectives met, by focusing on
algorithmic, constructive methods, as illustrated by many examples. It
is particularly encouraging to read in Andrews, Askey, and Roy [3] about
the usefulness of algorithmic based, complex, mathematical relationships
in today’s computer oriented approach. Such relations encode informa-
tion amenable to computer processing; perhaps, not to extent envisioned
by Wolfram [187], but nonetheless naturally and innovatively.
This monograph is not democratically assembled. The enormous lit-
erature on physical applications of unitary symmetry are not amenable
to a synthesis of technique, except in the broadest sense of Lie algebra
and group representations. Moreover, the subject has received little at-
tention from the combinatorial orientation presented here. Accordingly,
the monograph is heavily biased toward the understanding I have been
able to acquire over a fifteen year period of presenting lectures on these
subjects at small conferences in Poland organized by Tadeusz and Bar-
bara Lulek on Symmetry and Structural Properties of Condensed Matter,
and also at Nankai University, PR China, at the invitation of William
Y. C. Chen, Director, The Center for Combinatorics. The opportunity
to address a sizeable number of students has been particularly reward-
ing. Important special contributions to the subject have come from my
colleagues Bill Chen, Harold Galbraith, and Miguel Méndez. General
encouragement from George Andrews, Bill Chen, Gordan Drake, Harold
Galbraith, Brian Judd, Ron King, Tadeusz and Barbara Lulek, Steven
Milne, Peter Paule, Gian-Carl Rota, and in earlier years, Larry Bieden-
harn, and in later years, my son Tom and wife Marge; all have helped
to sustain the effort. I have also been inspired by the many lectures
of Gian-Carlo Rota, the comprehensive book by Stanley [163], and the
terse, but scholarly book by Macdonald [126].
James D. Louck](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-16-320.jpg)
![Chapter 1
Quantum Theory of
Angular Momentum:
Introduction
1.1 Background and Viewpoint
1.1.1 Euclidean and Cartesian 3-space
It is a basic tenet of Newtonian physics that physical phenomena take
place in Euclidean 3-space, denoted E3, which is a collection of enti-
ties called “points,” a point being a primitive undefined entity. Such
points are used in Euclidean geometry to construct other objects such
as lines, planes, triangles, etc., from a set of axioms that constitute a
deductive structure called Euclidean geometry. Euclidean geometry in
2-space and 3-space is the formalized mathematical method for deduc-
ing “facts” about the intuitive space of our surroundings and the objects
perceived to be part of those surroundings. Lines, planes, and other ob-
jects are constructed as subsets of points in E3. But the primitive object
“point” has no properties on its own; it is usually thought of as having no
“size;” a point cannot be taken apart in terms of still other points. Eu-
clidean geometry does not use the concept of “distance between points”
to characterize its objects, but rather the notion of congruence. Mac
Lane [128, p.127]) argues, however, that the notion of geometric magni-
tude is present in Euclid’s geometry and that together with congruence
and geometric ratios constitutes a geometrical description of the modern
concept of the real-number line.
Euclid’s geometry allows its objects to be “moved” around in the
E3 “space” in the modern sense of rigid body motions that includes
1](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-28-320.jpg)
![2 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
translations and rotations. But no explanation for the “cause” of such
motions is given; they are intrinsic properties of the “ambient” E3-space
( Mac Lane [128, p.76]).
Euclidean E3-space contains, as subsets, collections of three lines that
are mutually perpendicular and have a common unique point of inter-
section. The operation of perpendicular projection of a point p ∈ E3 to a
point L(p) on a given line L ⊂ E3 is defined for all points and all lines. It
is these notions of three perpendicular lines intersecting at a point, and
the operation of perpendicular projection onto these lines that is used to
describe physical phenomena in a concrete manner, based on the use of
the real numbers, and the notion of Cartesian 3-space, denoted R3.
Cartesian 3-space R3 is the real linear vector space whose elements
consist of the collection of all sequences x = (x1, x2, x3), xi ∈ R, i =
1, 2, 3, where R is the set of real numbers, where, of course, all the usual
operations of addition, multiplication by a scalar, and the distributive
rules that define a vector space are applied to such sequences. We refer
to a sequence of three real numbers (x1, x2, x3) ∈ R3 as a coordinate
point, and often as coordinates of a point in R3. Cartesian 3-space R3
comes equipped naturally with two functions defined on all pairs of points
x = (x1, x2, x3) and x = (x
1, x
2, x
3) belonging to R3, namely, a distance
function, denoted d(x, x), and a scalar product denoted, x · x, given by
d(x, x
) =
(x1 − x
1)2 + (x2 − x
2)2 + (x3 − x
3)2, (1.1)
x · x
= x1x
1 + x2x
2 + x3x
3. (1.2)
The space R3 contains subsets called lines, for example, the set of co-
ordinate points {x1, a2, a3) | x1 ∈ R}, for fixed real numbers a1 and a2,
defines a line. In particular, the subsets of R3 defined by
L1 = {(x1, 0, 0)|x1 ∈ R},
L2 = {(0, x2, 0)|x2 ∈ R}, (1.3)
L3 = {(0, 0, x3)|x3 ∈ R},
determine three mutually perpendicular lines in R3, where perpendicular
means the scalar product of points belonging to the separate lines is 0.
To relate the geometrical methods of Euclidean E3−space geometry
over points to the vector space methods of the analytical geometry of
Cartesian R3−space over the real numbers, it is convenient make the
strong assumption (from Mac Lane’s [128] observations) that each line
L ⊂ E3 is identified with the real-number line L(R) = {p(x) | x ∈ R},
where it is the custom to identity the point p(x) on the real-number line](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-29-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 3
L(R) with the real number x itself: p(x) → x. Thus, let q ∈ E3, and
let the three perpendicular lines L1(q), L2(q), L3(q) ⊂ E3, respectively,
have q as their (unique) intersection point. Identify each of these three
perpendicular geometrical lines with the real-number line L(R) = {x|x ∈
R}, so that the geometrical intersection point q is identified with a point
q → (q1, q2, q3) ∈ R3
= {(x1, x2, x3)|each xi ∈ R}. (1.4)
Thus, the three perpendicular lines L1(q), L2(q), L3(q) in E3 are identi-
fied with the three perpendicular lines in R3 given by
L1(q) → L1(R3
) = {(x1(q), q2, q3)|x1(q) ∈ R},
L2(q) → L2(R3
) = {(q1, x2(q), q3)|x2(q) ∈ R}, (1.5)
L3(q) → L3(R3
) = {(q1, q2, x3(q))|x3(q) ∈ R}.
A geometrical point p ∈ E3, which is obtained by geometric projec-
tion onto the three perpendicular lines L1(q), L2(q), L3(q) ⊂ E3 is now
mapped to a general coordinate point (x1(p), x2(p), x3(p)) ∈ R3 given by
p → (x1(p), x2(p), x3(p)) = (x1(q) + q1, x2(q) + q2, x3(q) + q3). (1.6)
The set of three perpendicular axes L1(R3), L2(R3), L3(R3) ⊂ R3 is
called a set-of-axes or a reference frame for the Cartesian R3 vector space.
The point q → (q1, q2, q3) ∈ R3 is called the origin of the Cartesian ref-
erence frame. Thus, the arithmetic description of points in Euclidean
space E3 is effected by two related objects in Cartesian space R3 : a ref-
erence frame and a set of coordinates relative to the reference frame. It
is convenient to denote this description of E3 by R3(q) : The underlying
ambient Euclidean 3-space E3 is unchanged by different choices of q ∈ E3;
the Cartesian 3-space R3(q), with origin located at q → (q1, q2, q3) is a
redescription of the (unchanged) points of E3. Equivalently, the coordi-
nate points corresponding to a coordinate frame located at any origin
(a1, a2, a3) are a redescription of the coordinate points corresponding to
a coordinate frame located at any other origin (a
1, a
2, a
3), each being a
different coordinate presentation of the same collection of points in the
underlying Euclidean space E3.
Discussions of the axioms of Euclidean geometry that suit our needs
and give a more complete picture than the brief discussion given here
can be found in Weyl [176, 177] and Mac Lane [128]. A discussion of
the origin and foundations of these concepts from the point of view of
cognitive scientists can be found in Lakoff and Núñez [100].
We complete our description of the relation between the spaces E3
and R3 by introducing the concept of a geometric vector as defined by](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-30-320.jpg)
![4 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
physicists in terms of directed line segments (see, for example, Page
[139]). This notion is well-suited to Euclidean 3-space with its point
entities. A geometrical vector is a directed line segment in E3 (a line
segment with one point called the origin and a second point called the
terminal point of the vector). Two geometrical vectors are defined as
equal if they lie along parallel lines and have the same directional sense
and length (geometrical magnitude); addition is defined in terms of the
familiar parallelogram rule; multiplication by a scalar (real numbers) is
defined as a scaling of geometrical magnitude, including the possibility
(negative number) of the reversal of direction; and the vector 0, called the
zero vector, is the vector in which the origin coincides with the terminal
point. The set of geometrical vectors satisfy the following two rules for
all points a, b ∈ E3 : (i) x(a → a) = 0; (ii) x(a → b)+x(b → c) = x(a →
c). The second property is called the transitive property of addition.
Together with the first rule, it implies that x(a → b) + x(b → a) = 0,
which gives the negative geometrical vector, x(b → a) = −x(a → b).
These definitions, together with the concept of parallel transport,
which allows for the “motion” of equal geometrical vectors necessary
to bring equal vectors into coincidence so that all geometrical vectors
can be added, determines a linear vector space of geometrical vectors
that offers an alternative means of implementing the ideas set forth by
Mac Lane [128, p.75]).
The linear space of geometric vectors with origin at point q ∈ E3
is denoted V3(q) and is isomorphic to the vector space R3(q). For q
identified as the origin (0, 0, 0), geometrical vectors are described in terms
of the corresponding Cartesian 3-space as follows: Unit vectors e1, e2, e3
are defined to be the directed line segments of unit length going from
the origin (0, 0, 0) to the respective points (1, 0, 0), (0, 1, 0), (0, 0, 1). The
general vector x = x1e1 + x2e2 + x3e3 is then the directed line segment
going from the origin (0, 0, 0) to the coordinate point (x1, x2, x3), and its
length is |x| =
x2
1 + x2
2 + x2
3.
It is convenient to think of the isomorphic vector spaces R3(q) and
V3(q) with origin at q → (q1, q2, q3) as superposed over the point q ∈ E3.
This merger of “pictures” of vector spaces makes quite clear the one-to-
one relation between vectors and coordinate points. In this sense, we
speak of a triad of unit vectors along the coordinate axes as a unit refer-
ence frame , or simply as a reference frame. Thus, we may characterize
the Euclidean 3-space corresponding to each point q ∈ E3 as consist-
ing of the two sets of objects: a reference frame (e1(q), e2(q)), e3(q))
located at the origin q → (q1, q2, q3), and the set of coordinate points
(x1(q), x2(q), x3(q)) projected onto the corresponding coordinate axes
with directional and orientational sense defined by these unit geomet-
rical vectors. This picture allows us to speak of the Cartesian 3-space
located at the Euclidean point q and having an arbitrary orientation of its](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-31-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 5
reference frame in Euclidean space, thus simplifying the description. We
do not, however, identify all the isomorphic vector spaces R3(q), q ∈ E3,
since the notion of the redescription of Euclidean 3−space by different
reference frames and coordinate points has important implications for
the description of the behavior of physical systems. (The same state-
ment applies to V3(q).)
The family of isomorphic Cartesian 3-spaces R3(q), q ∈ E3 can be
transformed into one another by geometrical congruence and other oper-
ations that leave the geometrical objects in E3 “unchanged.” For physics,
the meaning of unchanged is to be decided by experimental observation.
Physical theory must in some sense be invariant under such transforma-
tions, since the different Cartesian 3-spaces R3(q) are just a redescription
of the properties of the objects in E3 constituting a physical system. But
how a specified choice of reference frame is used to “measure” the prop-
erties of a given physical system, and how such measurements might
influence these properties, are questions that, as yet, have no universal
answers: the answers depend on additional assumptions. For example,
in Newtonian physics, it is assumed that “equations of motion” gov-
ern the motion of an object against the background Euclidean space,
and that the observation of an object and its motion have negligible (or
accountable) influence on the object and no influence on the space. In
nonrelativistic quantum theory, it is assumed that “equations of motion”
still govern the behavior of an object against the background Euclidean
space, but the new equations of motion are such that the concept of a
“property” of an object and the “measurement” of that property, such as
“point” particle and measurement of the location of the point, become
interwoven in ways that dramatically negate their classical (Newtonian)
meaning. While the equations of motion themselves are deterministic in
their time evolution, the transformation of the “properties” of the system
during the measurement process are subject to various interpretations
that need not be continuous against the Euclidean background. Special
and general relativity “correct” Newtonian relativity in a different con-
text by recognizing that objects and space do not have mutually exclusive
properties, but relate to one another in ways that require abandonment
of the Euclidean space and its assigned universal Newtonian time. De-
terministic motions are maintained in a broader measurement process
with rods and clocks, but all this requires a completely new formulation
of “equations of motion,” which are at odds with the quantum equations
of motion. A general theory that explains all the observed properties of
microscopic objects, using the successful quantum theory, and all those
of macroscopic objects, using Einstein’s successful relativity theory, re-
mains elusive. For a recent discussion, we refer to Leggett [103].
Our concern in Chapters 1-4 of this monograph is with the properties
of the angular momentum of many-particle systems in nonrelativistic
quantum mechanics, and, especially, the role of combinatorial concepts.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-32-320.jpg)
![8 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
frames such that the triad of unit vectors defining the axes are right-
handed triads, as defined by the familiar right-handed screw rule. Left-
handed frames can be included as well by the operation of inversion.
Mathematicians at the end of the nineteenth century realized that
the operations of translation and rotation can be effected by the action
of exponential differential operators (see Crofton [46], Glaisher [64] ),
which in vector notation are expressed by
ea · ∇F(x1, x2, x3) = F(x1 + a1, x2 + a2, x3 + a3), (1.15)
e−φn·R
F(x1, x2, x3) = F(x
1, x
2, x
3). (1.16)
The operator a · ∇, where ∇ = e1∂/∂x1 + e2∂/∂x2 + e3∂/∂x3 is called
the generator of the translation a = a1e1 + a2e2 + a3e3. In the second
relation, the components of R = R1e1 + R2e2 + R3e3 are the differential
operators defined by
R1 = x2∂/∂x3 − x3∂/∂x2,
R2 = x3∂/∂x1 − x1∂/∂x3, (1.17)
R3 = x1∂/∂x2 − x2∂/∂x1.
The coordinates (x
1, x
2, x
3) are the redescription of a point p ∈ E3 in
terms of the coordinates (x1, x2, x3) of the same point p as described by
the rotation of the frame (e1, e2, e3) → (e
1, e
2, e
3) in the positive sense
about the direction corresponding to the unit vector n = n1e1 + n2e2 +
n3e3, n · n = 1, where the matrix elements Rij = Rij(φ, n) in (1.12) and
(1.14) are those of
R(φ, n) = eφN
= I3 + N sin φ + N2
(1 − cos φ),
(1.18)
N =
0 −n3 n2
n3 0 −n1
−n2 n1 0
, N3
= −N.
The proper, real, orthogonal matrix R has here been parametrized by
the components of the unit vector n and the angle 0 ≤ φ 2π. The
gradient operator ∇ and the rotation operator R = R1e1 + R2e2 + R3e3
both satisfy the form invariant relation (1.13). The operator −φn · R is
called the generator of the rotation about direction n by angle φ.
We anticipate some results from the brief discussion of quantum the-
ory given below, and observe that relations (1.15)-(1.17) are precursors
to similar transformations that were to occur later in quantum theory
under the association of classical position x and linear momentum p to
operators by the rule x → x and p/ → −i∇, which, when applied to
the classical angular momentum L = x × p of a single particle located](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-35-320.jpg)
![12 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
composite whole—the original isolated system and the measuring device.
This new system then has its own collection of state functions governing
the properties of the composite whole. How all of this is to be sorted out
to obtain information about the original system is the problem of mea-
surement (see Leggett [103]). There is no generally agreed on resolution
of the problem of measurement.
The early standard book on the foundations of quantum mechanics
is by Von Neumann [174] with a more modern exposition by Mackey
[127]. Bohm [27] challenged the standard interpretation by an ingenious
example, and Bell’s [9] critical analysis reopened the entire subject of
measurement. The large book by Wheeler and Zurek [179] reviews the
history through 1980-81, with many informative comments on the Prob-
lem of Measurement and a superb list of references. Recent comments
by Griffiths [71] illustrate a popular viewpoint. The possibility of quan-
tum computers has intensified the interest in these problems and new
experiments are confirming the reality of the counterintuitive quantum
world. These important problems are, however, not the subject of this
monograph, which makes its presentations in the standard interpretation
with a focus on combinatorial aspects.
There are, fortunately, general invariance properties of the Schrödinger
equation and its solution state functions for a complex composite phys-
ical systems that can be used to classify the quantum states of physical
systems into substates available to the system.
Our focus here is on the properties of the total angular momentum
of a physical system, which is a quantity L that has a vector expression
L = L1e1 + L2e2 + L3e3 in the right-handed frame (e1, e2, e3) and the
expression L = L
1e
1 + L
2e
2 + L
3e
3 in a second rotated right-handed
frame (e
1, e
2, e
3). At a given instant of time, necessarily L = L, since
these quantities are just redescriptions of the total angular momentum of
the system at a given time. The total angular momentum is a conserved
quantity; that is, dL/dt = 0, for all time t, and this property makes
the total angular momentum an important quantity for the study of the
behavior of complex physical systems. For a system of n point particles,
the total angular momentum relative to the origin of the reference frame
(e1, e2, e3) is obtained by vector addition of that of the individual parti-
cles by L =
n
i=1 Li, where Li is expressed by the vector cross product
Li = xi×pi in terms of the vector position xi = x1ie1 +x2ie2 +x3ie3 and
the vector linear momentum pi = p1ie1 + p2ie2 + p3ie3 of the particle
labeled i. While angular momentum can be exchanged between interact-
ing particles, the total angular momentum remains constant in time for
an isolated physical system of n particles. The quantum-mechanical op-
erator interpretation of such classical physical quantities is obtained by
Schrödinger’s rule pi → −i∇i, = h/2π, where h is Planck’s constant.
The reference frame vectors (e1, e2, e3) remain intact.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-39-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 13
The viewpoints of Newtonian physics and quantum physics may be
contrasted in many ways. From the viewpoint of angular momentum,
for example, the classical angular momentum L = x × p is associated
with the actual motion of a particle, while in quantum theory, as noted
above, it becomes the generator of rotations of the state vector that
describes the properties of the object called “particle” (active viewpoint),
or, in the viewpoint we have advanced, the generator of rotations of the
coordinate frame (passive interpretation) that gives a redescription of
the state vector that describes the properties of the particle. Neither the
active nor passive viewpoint implies an actual “motion;” it is more in
accord with the concept of congruence in Euclidean geometry. An even
more contrasting feature is that the object called the electron is regarded
as a “point particle,” and whereas points in Euclidean space have no
intrinsic properties, this point object, the electron, has an “internal”
angular-momentum-like property called “spin,” as we next describe.
Internal degrees of freedom
The discovery of objects having internal degrees of freedom going be-
yond motions in Newtonian space-time led to the introduction of the
concept of the spin of a object (see Pauli [140]), even though the object
may exhibit no measurable spatial extension. For example, the electron,
the carrier of electrical conduction in metals under the influence of an
electric field, was already recognized in 1897 (see Ref. [167]) to be one of
the principal constituents of atoms, requires an intrinsic property called
spin, to account for the observed properties of the emission and absorp-
tion of radiation by atoms. A single electron is characterized not only as
being a point structure with a fixed charge and mass, but also by having
a fixed spin 1
2
. As a structure at a single point, the electron is assigned
a position x = (x1, x2, x3) ∈ R3 at time t, which in quantum theory
become the domain of definition of its space-time wavefunction ψ(x, t);
as a structure with spin, it is assigned a “state vector” |1
2
belonging to
a Hilbert space H1
2
of dimension 2, which is assumed to be spanned by a
pair of orthonormal basis vectors that are written |1
2
, 1
2
and |1
2
, −1
2
in the
Dirac bra-ket notation, with orthonormality relations 1
2
, µ | 1
2
, ν = δµ,ν,
the inner product being unspecified beyond these properties. The gen-
eral state vector of the electron in the spin space H1
2
is given by the
ket-vector |1
2
= α|1
2
, 1
2
+ β|1
2
, −1
2
, where α and β are complex coeffi-
cients. The corresponding bra-vector in the dual space is then given by
1
2
| = α∗1
2
, 1
2
| + β∗1
2
, −1
2
| with inner product 1
2
|1
2
= |α|2 + |β|2. In this
conceptualization of a point particle with intrinsic spin 1
2
, its full state
vector belongs to the tensor product space H ⊗H1
2
, which has state vec-
tors of the form Ψ1
2
= ψ ⊗ |1
2
, where ψ is a wavefunction in the usual
sense, depending only on the spatial coordinates. The action of a frame](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-40-320.jpg)
![14 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
rotation R : (e1, e2, e3) → (e
1, e
2, e
3), where e
i =
3
j=1 Rijej, i = 1, 2, 3,
is not only to effect the transformation to new spatial coordinates of the
particles given by (x1, x2, x3) → (x
1, x
2, x
3), where x
i =
3
j=1 Rijxj, i =
1, 2, 3, but also to effect the transformation of the spin-space ket-vector
basis of H1
2
by
|1
2
, 1
2
→ |1
2
, 1
2
= u11(R)|1
2
, 1
2
+ u21(R)|1
2
, −1
2
, (1.22)
|1
2
, −1
2
→ |1
2
, −1
2
= u12(R)|1
2
, 1
2
+ u22(R)|1
2
, −1
2
.
The action of the frame rotation R on the components (α, β) of the
ket-vector |1
2
= α|1
2
, 1
2
+ β|1
2
, −1
2
giving a general spin state is given by
α → α
= u∗
11(R)α + u∗
21(R)β, (1.23)
β → β
= u∗
12(R)α + u∗
22(R)β,
where
U(R) =
u11(R) u12(R)
u21(R) u22(R)
∈ SU(2) (1.24)
is an element of the group of 2 × 2 unitary unimodular matrices SU(2)
that depends on the frame rotation matrix R ∈ SO(3, R). This pair
of transformations of basis vectors and components leaves invariant the
relation
α|1
2
, 1
2
+ β|1
2
, −1
2
= α
|1
2
, 1
2
+ β
|1
2
, 1
2
, (1.25)
as required for a redescription of the state vector of the spin by a rotation
of the reference frame. (For a history of the discovery of the electron and
its properties, see Whittaker [180] and The Physical Review: The First
Hundred Years [167].
Generally, in the one-particle case above, we have in mind a single
particle with an internal spin that is a constituent of a larger isolated
physical system, with all constituents described in R3, that provides
the interactions that act on the “marked” single particle. We often
use Newtonian metaphors to identify this object, but its properties are
to be inferred from the those of its quantum-mechanical state vector
Ψ = ψ⊗|1
2
. Newtonian space-time still occurs in the domain of definition
of the wavefunction ψ. It is also possible to assign “internal coordinates”
to the new degrees of freedom of the internal space that describes the
spin, as discussed below in (1.58).
It is somewhat remarkable that the symmetry group SU(2) actually
can be used to unify the action of frame rotations R ∈ SO(3, R) on
both points in R3 and on points used to describe the internal degrees of
freedom associated with spin. From this viewpoint, the group SU(2) is
the basic group for the study of angular momentum of complex systems
with spin. We show this by using the method of Cartan [35].](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-41-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 15
The method of Cartan
In the Cartan method (see also Wigner [181] and Biedenharn and Louck
[21]), a point x = (x1, x2, x3) ∈ R3 is presented as the entries in a 2 × 2
traceless, Hermitian matrix of the form:
H(x) =
x3 x1 − ix2
x1 + ix2 −x3
=
3
i=1
xiσi, (1.26)
where the σi, i = 1, 2, 3, are the Pauli matrices:
σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
, σ3 =
1 0
0 −1
. (1.27)
Thus, each point x ∈ R3 is mapped to a traceless Hermitian matrix: x →
H(x). Conversely, given a traceless Hermitian matrix H, it is mapped to
a point x ∈ R3 by the rule xi = 1
2Tr(σiH), where TrA denotes the trace of
a matrix A. In obtaining this result, we use the following multiplication
rules of the Pauli matrices:
σ2
i = I2 = σ0, i = 1, 2, 3; σ1σ2 = iσ3, σ2σ3 = iσ1, σ3σ1 = iσ2. (1.28)
Thus, with these conventions for the placement of the components xi
into matrices, the set of points R3 is bijective with the set of matrices:
H = {H | H is a 2 × 2 traceless Hermitian matrix} . (1.29)
Traceless Hermitian matrices are mapped into traceless Hermitian
matrices by unitary similarity transformations. Moreover, the deter-
minant of a Hermitian matrix is invariant under such transformations.
Accordingly, it must be the case that, for each U ∈ SU(2), the transfor-
mation x → x defined by
U
x3 x1 − ix2
x1 + ix2 −x3
U†
=
x
3 x
1 − ix
2
x
1 + ix
2 −x
3
, (1.30)
is a real orthogonal transformation, since the determinant of this trans-
formation is −(x2
1 + x2
2 + x2
3) = −(x
2
1 + x
2
2 + x
2
3 ). The explicit transfor-
mation may be worked out to be
x
i =
3
j=1
Rij(U)xj, Rij(U) =
1
2
Tr(σiUσjU†
). (1.31)
The proof that det R(U) = 1 takes more effort, but is correct. We
also note the easily proved property (R(U))T = R(U†) = R(U−1) =](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-42-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 21
for m = j, j − 1, . . . . − j. In this formulation, under the unitary frame
rotation given by (1.41), the underlying spin-space coordinates (z1, z2)
undergo the redescription (z1, z2) → (z
1, z
2) given by
z
1 = u11z1 + u12z2,
z
2 = u21z1 + u22z2,
U = (uij)1≤i≤j≤2 ∈ SU(2), (1.59)
while the basis polynomials undergo the redescription given by
(SU Pj m)(z1, z2) = Pj m(z
1, z
2) =
m
Dj
m m(U)Pj m (z1, z2), (1.60)
in which the new coordinates are given in column matrix form by
col(z
1, z
2) = UT col(z1, z2). (Compare with (1.44) and (1.46).)
The transformation coefficients in relation (1.60) are given by (see
van der Waerden [171] and Ref. [21]):
Dj
m m(U) =
(j + m)!(j − m)!(j + m)!(j − m)! (1.61)
×
s
(u11)j+m−s(u12)m
−m+s(u21)s(u22)j−m
−s
(j + m − s)!(m − m + s)!s!(j − m − s)!
,
where the summation is over all nonnegative values of s for which all
factorials in the denominator are nonnegative.
The column matrix formulation is less specific about the charac-
ter of internal space, which is not directly accessible to measurement,
and is, perhaps, to be preferred. Mathematically, the two finite vec-
tor spaces of polynomials and unit vectors of length 2j + 1 described
above are isomorphic, and it is no restriction to use the spinor polyno-
mials Pj m(z1, z2), m = j, j − 1, . . . , −j, as basis vectors. Indeed, these
polynomials have an inner product ( , ) such that (Pj m , Pj m) = δm m,
one-to-one with that of the unit vectors sj m, m = j, j − 1, . . . , −j, as
discussed below in Sect. 1.3.1. There are many advantages to following
this approach, especially from the vantage point of combinatorics. More-
over, it is well known that the transformation properties of the spinor
basis functions already gives all inequivalent irreducible representations
of SU(2), as given above in relation (1.61), a result that we will prove
later. It is important to recognize that the use of either of these special
realizations of spin space in no way restricts the general theory of spin
space, since they do not introduce constraining conditions.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-48-320.jpg)
![1.1. BACKGROUND AND VIEWPOINT 25
Here we use the notation
(TΨ)(q) = T Ψ(q), (1.82)
where Ψ is an element of some function space F on which an operator
T acts, T : Ψ → Ψ = TΨ; q is a point in the domain of definition Q
of the functions in F; and T is the differential operator, acting in the
space of values of the functions in F, that defines the values of (TΨ)(q).
In the context at hand, such a relation applies to the operator pairs
(Ji, Ji), (Li, Li), (Si, Si), where these are the generator pairs correspond-
ing to n = ei, i = 1, 2, 3. Similar relations apply to linear combinations of
these linear operator. In particular, we have that the matrix generator
n · σ/2 of the unitary matrix representation exp (−iφn · σ/2) of order
2 of SU(2) is realized in the space of functions by
(TU(φ,n)Ψ)(X, Z) =
e−iφn·J
Ψ
(X, Z) = e−iφn·J Ψ(X, Z). (1.83)
Although the differential operators Li and Si are quite different in
structural form, they satisfy identical commutation relations, where the
commutator [A, B] of two operator A and B acting in a Hilbert space is
defined by [A, B] = AB − BA :
[L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2, (1.84)
[S1, S2] = iS3, [S2, S3] = iS1, [S3, S1] = iS2, (1.85)
[Li, Sj] = 0, i, j = 1, 2, 3. (1.86)
The operators Li and Si, with action in the space of state vectors, satisfy,
of course, these same commutation relations. The important observation
is: There can be many realizations of these commutation relations by
quite different operators, but as we shall see in Sect. 1.2 below, under
certain assumptions, they all give rise to the same matrix realizations.
Let us next relate the mathematical quantities given above to the
physical quantity called angular momentum. The classical angular mo-
mentum L of a point particle relative to a frame (e1, e2, e3) is defined
by the cross product L = x×p, where p is the linear momentum of the
particle. The quantum angular momentum L is obtained from the clas-
sical quantity by making the replacement p → −i∇, where is the
Planck constant h/2π, and ∇ is the Laplace operator with components
(∂/∂x1, ∂/∂x2, ∂/∂x3). Thus, the components of the dimensionless an-
gular momentum operators L/ = −ix×∇ are exactly the generators of
rotations given by relations (1.80). There is, of course, no such rule for
obtaining the spin operators (1.81). As remarked earlier, spin angular
momentum can either be incorporated into the quantum theory in terms
of an unspecified abstract finite-dimensional Hilbert space or by using
realizations of this space that do not restrict the mathematical content
of the theory, as we have done above.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-52-320.jpg)
![26 CHAPTER 1. QUANTUM ANGULAR MOMENTUM
The generalizations of the above one-particle results to n particles is
immediate: One simply copies relations (1.79)-(1.81) n times, adjoining
an extra index k to designate the k−th particle with spin j. The in-
finitesimal operators associated with each particle are clearly additive,
so we have that the generator of the transformation associated with the
redescription of the state vector resulting from a unitary frame rotation
by angle φ about direction n is given by n · J and have the properties
summarized by the following relations:
J = L + S, (1.87)
J =
n
k=1
J (k), (1.88)
J (k) = L(k) + S(k); (1.89)
L(k) = −ix(k)
× ∇(k)
, x(k)
=
3
i=1
xik ei ; (1.90)
S
(k)
1 = (z1k∂/∂z2k + z2k∂/∂z1k)/2,
S
(k)
2 = −i(z1k∂/∂z2k − z2k∂/∂z1k)/2, (1.91)
S
(k)
3 = (z1k∂/∂z1k − z2k∂/∂z2k)/2.
These operators are just n replicas of relations (1.79)-(1.81) with com-
mutation relations that are k replicas of relations (1.84)-(1.86) with su-
perscript (k) adjoined, where now operators having distinct values of
particle index k all mutually commute. Relation (1.78) also holds, of
course, in the interpretation as operators acting in the state space:
(n · J(k)
)Ψ
(X, Z) = (n · J (k))Ψ(X, Z), (1.92)
with similar relations for n · L(k)
and n · S(k)
. In particular, we still have
the commutation relations for the total angular momentum:
[J1, J2] = iJ3, [J1, J2] = iJ3, [J1, J2] = iJ3 (1.93)
and the associated transformations of the state vectors:
(TU(φ,n)Ψ)(X, Z) =
e−iφn·J
Ψ
(X, Z) = e−iφn· J Ψ(X, Z). (1.94)
This relatively simple n−particle physical system embodies the full con-
tent of angular momentum theory. The simplicity of these additive re-
lations does not reveal the intricacies of the theory.
Remark. Each U ∈ SU(2) effects a rotation R(U) of the reference
frame of a physical system in R3 as given by relations (1.65)-(1.67).](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-53-320.jpg)
![[234]
and see how soon the mechanic thinks he will have it
ready.”
She returned in a couple of minutes and found the
others already seated in the constable’s sedan. Mary
Louise was glad to find that the officer had put Margaret
Detweiler in front with him, not beside the tough young
man with his huge guardian in the rear seat. She
squeezed in next to Margaret, and the car started.
“The mechanic is going to drive my car to your place in
about half an hour,” announced Mary Louise. “And then
we’ll start for Philadelphia.”
“Fine!” exclaimed the constable. “That’ll give you girls a
chance to get warm. And maybe have a cup of coffee.”
“It’s marvelous coffee,” commented Mary Louise. “It just
about saved my life.”
Not another word was said about the crimes or the
secret band. Margaret Detweiler was introduced to Mrs.
Hodge as a friend of Mary Louise’s from Riverside, and
the two girls spent a pleasant half hour in the
constable’s home, sipping their freshly made coffee and
looking at the children’s Christmas toys.
The constable, who had taken the young thug away,
returned just as Mary Louise’s hired car drove up to the
door.
Mary Louise jumped up and reached for her coat.
“Wait a minute!” cautioned the constable. “Company’s
comin’ here to see you, Miss Gay! I just met somebody
askin’ for you at the hotel.... So don’t be in too much of
a rush!”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-56-320.jpg)
![[235]
From the obvious twinkle in the man’s eyes, Mary Louise
believed that Max Miller must have driven down to
Philadelphia again and, missing her there, had naturally
traced her to Center Square. But at that same moment
a yellow taxi stopped at the constable’s gate, thereby
dispelling any such illusion. Max would never ride in a
taxicab on his limited allowance!
The door of the cab opened, and a tall, handsome man
stepped out, paid the driver, and dismissed the cab. It
was Mary Louise’s father.
Flinging open the door, the girl shouted at him in
delight, so loud that Mr. Gay heard her in spite of the
noise of the departing cab. In another moment he
entered the open door of the house and held Mary
Louise tightly in his arms.
“Mary Lou!” he cried in delight. “Are you sure you’re all
right?”
“I’m fine,” she replied, ushering him into the constable’s
house. “Merry Christmas, Daddy!”
“The same to you, dear.” He gazed at her fondly. “I
believe it will be—now. You certainly look happy,
Daughter.”
“I am, Daddy. These people have treated me royally!”
She turned around and introduced her father to Mrs.
Hodge and the children, for he had already met the
constable. “And, oh, Dad, here is Margaret Detweiler,”
she added. “You remember her, don’t you?”
“I certainly do,” replied Mr. Gay, extending his hand
cordially. “My, but your grandparents are going to be
glad to see you, Margaret!”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-57-320.jpg)
![[236]
[237]
The girl blushed and looked down at the floor in
embarrassment. Wisely, Mr. Gay asked no questions.
“I have all the stolen valuables, Dad,” continued Mary
Louise. “Every single thing that was taken from
Stoddard House, and even the money!”
Mr. Gay gazed at his daughter in speechless admiration:
she had excelled his fondest hopes!
“Mary Lou, that’s—wonderful!” he said after a
moment.... “I have good news too. I caught your
thieves. Seven of ’em. They are in a Baltimore jail now.”
Both girls exclaimed aloud in amazement and delight.
Margaret Detweiler started forward and clutched the
detective’s arm.
“It’s really true, Mr. Gay?” she demanded breathlessly.
“Mrs. Ferguson—is she in jail too?”
“Locked up without any chance of getting out on bail!”
he said authoritatively.
“Oh, I’m so glad!” murmured the girl thankfully.
“Now we’ll be able to take the valuables right back to
their owners at Stoddard House, Constable Hodge,”
announced Mary Louise. “I’m not afraid to carry them,
with Dad beside me.”
Mrs. Hodge brought the jewelry and the money from its
hiding place and gave it all to Detective Gay. Both he
and Mary Louise tried to thank the Hodges for their help
and their hospitality; Mr. Gay wanted to give the
constable some sort of recompense, but the good man
refused. Only after a great deal of persuasion would he](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-58-320.jpg)
![[238]
accept a five-dollar bill as a Christmas present for his
children.
“Ready, Daddy?” inquired Mary Louise as she slipped on
her coat.
“Just a minute,” replied her father. “I want to telephone
to Mrs. Hilliard to let her know that you are safe. She’s
been terribly worried, Mary Lou.... And shall I tell her
that we’ll eat Christmas dinner with her at Stoddard
House?”
“Oh, yes! I’ve heard about the menu. There won’t be a
sweller dinner anywhere in Philadelphia than at
Stoddard House. But shall we be in time?”
Mr. Gay consulted his watch. “It’s only a little after
eleven,” he said. “We ought to make it by one o’clock.”
As soon as the telephone call was completed, the three
people got into the little car. Mary Louise herself took
the wheel, for, as she explained, she was familiar with it
by this time.
“Now tell me about your experiences, Mary Lou,” urged
her father, as soon as they were well under way.
Mary Louise explained, for Margaret’s benefit as well as
for her father’s, about deciphering the code letter and
coming up to Center Square and breaking into the
empty house in search of the valuables. But she made
light of the coldness and desolation of the dark house
and of her own hunger. She concluded with the
statement that Margaret had come that morning and let
her out with a key.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-59-320.jpg)
![[239]
“But how did you happen to have the key, Margaret?”
demanded Mr. Gay.
“I will have to tell you my whole story from the
beginning,” answered the girl. There was a tragic note
in her voice, which drew out her listeners’ sympathy, but
neither made any comment.
“Then you can decide what to do with me,” she
continued. “I guess I deserve to go to prison, but when
I assure you that I have never done anything wrong
except under compulsion, maybe you will not be so
angry with me.”
“We’re not angry with you, Margaret,” Mary Louise told
her. “Only terribly sorry. So please tell us everything.
You remember that your grandparents have never heard
anything from you since last Christmas.... So begin your
story there.”
“All right.... Let me see—I was working in that
department store in Philadelphia, and doing pretty well,
for I got commissions besides my salary on everything I
sold. I started in the cheap jewelry department and was
promoted to the expensive kind. Christmas brought me
in a lot of business, but I guess I overworked, for I got
sick the week before and had to stay home and have
the doctor. I’d already spent a good deal of money on
presents, and when my doctor’s bill was paid I found my
salary was all gone. So I went back to the store before I
should—on the twenty-third of December, I remember.”
“The twenty-third of December!” repeated Mary Louise.
“That was the day Mrs. Ferguson registered at the
Benjamin Franklin Hotel.”
“How did you know, Mary Lou?” demanded Margaret.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-60-320.jpg)
![[240]
“I went to the hotel and looked through the old
register,” she explained. “But go on, Margaret. What
happened then?”
“I found that a ring, an expensive diamond ring, had
been stolen from our department,” continued the girl.
“They insisted that it was taken before I was away, but
they couldn’t prove anything. Just the same, I know the
store detective had his eye on me.... Well, that very day
something else disappeared: a link bracelet. This time
they accused me immediately.”
“But why?”
“I don’t know, except that I was the newest salesgirl in
the department—in fact, the only girl. The store
detective stepped behind my counter and leaned down
to the floor. And he picked that bracelet right out of my
shoe!”
“How dreadful!” cried Mary Louise. “Somebody had
‘planted’ it there?”
“Of course. Mrs. Ferguson had, as I later learned. But at
the time I hadn’t a suspicion. She was standing right
near the counter, examining some rings. When she
heard me accused and told to leave the store, she
stepped forward, saying that she was sorry for me. She
asked me whether I had any family, and I told her they
were too far away for me to go to, without any money.
“‘But you’ll have trouble getting a job without a
reference,’ she said. ‘So perhaps I had better help you.’”
“The sly cat!” cried Mary Louise.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-61-320.jpg)
![[241]
[242]
Margaret nodded. “But I didn’t know it then. I simply
asked her whether she could get me a job, and she told
me to come to the Benjamin Franklin Hotel that
afternoon and ask for Mrs. Ferguson.
“Of course, I went—I had nothing else to do. She
engaged me at once as her secretary. We went out to
Center Square for a few days, and I met a lot of other
girls. Two daughters, two nieces, and a couple of
friends. We had a good time, but I didn’t do any work,
for she had two servants and a chauffeur, and I felt as if
I didn’t earn my pay.”
“Did she give you a salary?” asked Mary Louise.
“Yes,” replied Margaret. “For the first couple of weeks.
But I had to send it to my landlady in Philadelphia. After
that, Mrs. Ferguson bought my clothes and paid my
hotel bills, but she never gave me any cash.”
“So you couldn’t get away!” observed Mr. Gay.
“Exactly. Gradually I began to suspect that there was
something crooked about this bunch, and then one day
I found the diamond ring which had been stolen from
the store: among Mrs. Ferguson’s stuff at Center
Square!”
“What did you do?” demanded Mary Louise.
“I showed it to her and said I was going to take it right
back to the store, and she stood there and laughed at
me. She said it would only prove my own guilt!
“The next day we all went to Washington and stayed in
different hotels. Mrs. Ferguson kept me with her, but I
soon saw through her tricks. Her girls were all skilled](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-62-320.jpg)
![[243]
hotel thieves. She tried to teach me the business, as
she called it, but I refused to learn. So she made me
take charge of the stuff they stole. The girls would bring
their loot to her, and she’d send me with it to Center
Square. Every once in a while she would dispose of it all
to a crooked dealer who asked no questions.”
“Were you out at Center Square last Sunday, Margaret?”
interrupted Mary Louise.
“Yes. Mrs. Ferguson and I both went. We had intended
to get the place ready to spend Christmas there, but for
some reason, Mrs. Ferguson got scared. She said that
Mary Green talked too much, and she thought we ought
to clear out. She made plans to dispose of everything in
Baltimore, and then we were all going to sail to
Bermuda.... But why did you ask that, Mary Lou?”
“Because I was in that car that drove up to the house
then. I saw you and then Mrs. Ferguson. I wouldn’t
have thought of its being you, only Mary Green
admitted that she knew you. That made me suspicious.”
“You disappeared pretty quickly!”
“Rather,” laughed Mary Louise, and she told the story of
being hit over the head by a rock and of catching the
young man and having him arrested that very morning.
“That was clever!” approved her father. “Who was he,
Margaret?”
“A neighborhood bum that Mrs. Ferguson employs to
watch the place and keep the people away,” replied the
girl.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-63-320.jpg)
![[244]
“But I’m afraid I interrupted you, Margaret,” apologized
Mary Louise. “Please go on with your story.”
“There isn’t much left to tell. I was too far away from
home to run away, without any money, and I hadn’t a
single friend I could go to. All the store people thought I
was a thief, so I knew there was no use asking their
help. I just kept on, from day to day, not knowing how
it would ever end and never expecting to see my
grandparents or my Riverside friends again. Oh, you
can’t imagine how unhappy I have been!”
She stopped talking, for emotion had overcome her;
tears were rolling down her cheeks. Mary Louise laid her
hand over Margaret’s reassuringly.
“It’s all right now, isn’t it, Daddy?” she said. “We’ll take
you home to your grandparents.”
“But I can’t go back to them!” protested the other girl.
“How can I tell them what has happened? They’d be
disgraced for life.”
“You can tell them you have been working for a queer
woman who wouldn’t allow you to write home,” said Mr.
Gay. “A woman whose mind was affected, for that is the
truth. There is no doubt that Mrs. Ferguson is the victim
of a diseased mind.”
“Wouldn’t you ever tell on me?” questioned Margaret.
“No, of course not. It was in no way your fault, child....
And now try to be happy. I think I can find you a job in
Herman’s Hardware store, right in Riverside. And you
can live with your grandparents. They need you.”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-64-320.jpg)
![[245]
[246]
“It seems almost too good to be true,” breathed the
grateful girl.
Mary Louise turned to her father.
“Now for your story, Dad,” she begged. “About capturing
the thieves.”
“I think that had better be kept till dinner time,” replied
Mr. Gay. “This traffic we’re approaching will require all
your attention, Mary Lou. And besides, Mrs. Hilliard will
want to hear it too.”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-65-320.jpg)
![[247]
CHAPTER XVIII
Conclusion
Mary Louise brought the car to a stop at Stoddard
House at a quarter to one. Carrying the money and the
jewels in her father’s briefcase, and the other articles in
the basket, she and Margaret went into the hotel to get
ready for dinner while Mr. Gay returned the hired car to
the garage.
“I’ll notify the police that you’re found, Mary Lou,” he
said. “Then I’ll call your mother. I think it will be best if
she goes over to your grandparents, Margaret, and tells
them about you herself. They haven’t a telephone, and I
don’t like to frighten elderly people with telegrams.”
Both girls nodded their approval to these suggestions
and hurried into the hotel. Mrs. Hilliard was waiting for
Mary Louise with open arms; she loved the young
detective like a daughter.
“Now, run along, girls, and get ready for dinner,” she
said finally. “We are going to have one big table, instead
of all the little ones in the dining room. With a tree in
the center, and place cards, just like a jolly family party.”
“That’s swell!” exclaimed Mary Louise. “It’ll be real
Christmas after all.”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-66-320.jpg)
![[248]
“And thank you so much for the lovely handkerchiefs,
dear,” added the manager. “It was sweet of you to think
of me.... That reminds me, you haven’t had your
presents yet.”
“Put them at my place at the table,” suggested Mary
Louise. “And I’ll have presents for some of the guests,”
she added, with a significant glance at the briefcase and
basket.
When the girls returned to the first floor, after washing
their faces and powdering their noses, they found Mr.
Gay waiting for them. For a moment he did not see
them, so intent was he in the newspaper he was
reading.
“Want to see the gang’s picture?” he asked when Mary
Louise came to his side.
“Oh yes! Please!”
In spite of the fact that it was Christmas Day, a large
photograph of Mrs. Ferguson and her six accomplices
occupied much of the front page of this Philadelphia
paper. In an inset above the picture of the crooks was
Mary Louise’s smiling face!
“Daddy!” cried the girl in amazement. “Are you
responsible for this?”
“I am,” replied her father proudly. “I want everybody to
know that the credit belongs to you, Daughter.”
Other guests, who had not yet read their newspapers,
crowded about Mr. Gay eager for the exciting news.
They all remembered Pauline Brooks, and Mary Green;](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-67-320.jpg)
![[249]
several of them identified the two transients who had
stolen the other things from Stoddard House.
A loud gong sounded from the dining room, and Mrs.
Hilliard threw open the doors. The room was beautifully
decorated with greens and holly; a long table stretched
out before them, covered with a lovely lace cloth and
bearing a small Christmas tree as its centerpiece. Bright
red ribbons had been stretched from the tree to each
guest’s place, adding brilliancy to the spectacle.
“Hello, Mary Louise!” said a voice behind the young
detective, and, turning around, Mary Louise saw Mrs.
Weinberger behind her.
“Merry Christmas, Mrs. Weinberger!” she replied. “It’s
nice to see you back here.”
“I’ve come back to stay,” announced the older woman.
“I got lonely at the Bellevue. And Mrs. Macgregor is
here too, for Christmas dinner.”
It was a happy group who finally found their places
around the beautiful table and sat down. Mrs. Hilliard
was at one end, and Miss Stoddard was honored with
the seat at the other end. Mr. Gay was the only man
present, but he did not seem in the least embarrassed.
Mary Louise found her pile of presents at her place, and
Margaret Detweiler discovered a bunch of violets and a
box of candy at hers. Even in his haste, Mr. Gay had
remembered the lonely girl.
The guests ate their oyster cocktails and their
mushroom soup before any formal announcement
concerning the valuables was made. Then Mrs. Hilliard
rose from her chair.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-68-320.jpg)
![[250]
“As you all know from the papers, our criminals have
been caught by Mary Louise Gay and her father, and are
now in prison. But even better news than that is
coming. I’ll introduce Mr. Gay, whom some of you know
already, and he’ll tell you more about it.”
Everybody clapped as the famous detective stood up.
“I’m not going to make a speech,” he said, “and keep
you waiting for the turkey we’re all looking forward to. I
just thought that maybe some of you would enjoy this
wonderful dinner even more if you knew that you are
going to get everything back again which was stolen.
My daughter found all the valuables and the money this
morning in Mrs. Ferguson’s house at Center Square, and
she will now return them to their rightful owners.”
As the newspaper had not mentioned anything about
the stolen goods, the guests were not prepared for this
pleasant surprise. A loud burst of applause greeted
Mary Louise as she smilingly rose to her feet and
opened the briefcase and drew out the basket from
under the table where she had hidden it.
“I’ll begin at the beginning,” she said. “With the vase
and the silverware belonging to Stoddard House.” She
carried these articles to Mrs. Hilliard, amid appreciative
hand-clapping.
“Next, Miss Granger’s picture and her fifty dollars,” she
continued.
Tears actually came to the artist’s eyes as she took the
painting from Mary Louise’s hands.
“You keep the fifty dollars, Miss Gay,” she said. “My
picture is what I care for most.”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-69-320.jpg)
![[251]
[252]
“No, Miss Granger, no, thank you,” replied the girl
solemnly. “I am being paid a salary for my work by Mrs.
Hilliard, but I can’t accept rewards for doing my duty.”
She picked up the watches next: Mrs. Weinberger’s and
Mrs. Hilliard’s. The Walder girls would get theirs when
they returned from their holidays.
“And, last of all, Mrs. Macgregor’s diamond earrings and
her five hundred dollars,” she concluded, restoring the
jewelry and the bills to the delighted woman. “I believe
that is all, for I am wearing my own wrist-watch, and I
have my purse with its five dollars contents.”
Loud cheering accompanied the applause which
followed. When it had at last quieted down, both Mrs.
Weinberger and Mrs. Macgregor tried in vain to give
Mary Louise a reward, but she remained firm in her
refusal. Then the turkeys were brought to the dining
room, and everything else was temporarily forgotten in
the enjoyment of Christmas dinner.
When it was all over, Mr. Gay told Mary Louise to pack
her clothing and her presents while he returned the
remaining valuables to the Ritz and to the police. “For I
hope we can make the three-thirty train,” he explained.
“But with that change at the Junction, we’d have to wait
all night, shouldn’t we, Daddy?” inquired Mary Louise.
Anxious as she was to get back to Riverside, she had no
desire to spend the night in a cheerless railway station.
“No,” replied her father. “Because there’s going to be a
surprise waiting for you at the Junction.”
“Max and Norman?” guessed Mary Louise instantly. “You
mean that they’ll drive down for us?”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-70-320.jpg)
![[253]
Mr. Gay nodded. “That isn’t all,” he said.
Mary Louise did not guess the rest of the answer until
the train pulled into the Junction shortly after eight
o’clock that night. Then a war whoop that could come
from no one else but her small brother greeted her
ears, and she knew that her mother must be there too.
Yes, and there was her chum, Jane Patterson, grinning
at her from the boys’ car! And her little dog, Silky!
In another minute Mary Louise was clasping her arms
around Mrs. Gay and hugging Freckles and Jane and
Silky all at once. Max, at her side, had to be content
with pressing her arm affectionately.
Questions, Christmas greetings, words of joy and
congratulation poured so fast upon Mary Louise’s ears
that she could scarcely understand them.
“You’re home to stay, darling?” This from her mother.
“You’ll go to the senior prom with me?” demanded Max.
“You’re the most famous girl detective in the world!”
shouted Norman Wilder.
“You were a lemon to duck my party, but I’ll give
another one just in your honor,” promised Jane.
“Did you get your salary—your twenty-five bucks?”
asked Freckles.
Mary Louise nodded, smiling, to everything. Then she
got into Max’s car beside him, with Jane and Norman in
the rumble seat. Mr. Gay took the wheel of his sedan,
with his wife beside him; Margaret Detweiler, who was
quietly watching everything, sat behind with Freckles.](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-71-320.jpg)
![[254]
The drivers of the two cars did not stop for any food on
the way; they sped along as fast as they dared towards
Riverside. Old Mr. and Mrs. Detweiler were waiting up
for their precious granddaughter, their lost Margaret.
A little before midnight the cars pulled up in front of the
old couple’s home, and everybody in the party went
inside for a moment. The greeting between Margaret
and her grandparents was touching to see. Even
Norman Wilder, who prided himself on being “hard-
boiled,” admitted afterwards that the tears came to his
eyes.
Mrs. Gay discreetly drew her own party away, back to
her home, where a feast was waiting for the travelers.
This, Mary Louise felt, was her real Christmas
celebration—with her family and her three dearest
friends. Now she could tell her story and listen to the
praises which meant so much to her.
“But the best part of it all,” she concluded, “is that I’m a
real professional detective at last!”](https://image.slidesharecdn.com/571595-250529002603-7404b43f/85/Unitary-Symmetry-And-Combinatorics-James-D-Louck-72-320.jpg)
Unitary Symmetry And Combinatorics James D Louck
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- 7. This page intentionally left blank This page intentionally left blank
- 8. NE W JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI World Scientific James D. Louck UNITARY SYMMETRY AND COMBINATORICS { } Los Alamos National Laboratory Fellow Santa Fe, New Mexico, USA
- 9. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-281-472-2 ISBN-10 981-281-472-8 All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Printed in Singapore. UNITARY SYMMETRY AND COMBINATORICS ZhangJi - Unitary Symmetry.pmd 7/15/2008, 3:49 PM 1
- 10. Dedicated to the memory of my sons Samuel Victor Louck Joseph Patrick Louck And to the courage of my son and wife Thomas James Louck Margaret Marsh Louck
- 11. This page intentionally left blank This page intentionally left blank
- 12. Preface This monograph is an outgrowth of the books “Angular Momentum in Quantum Physics” and “The Racah-Wigner Algebra in Quantum The- ory,” by L. C. Biedenharn and myself, published in 1981, originally by Addison-Wesley in the Gian-Carlo Rota series “Encyclopedia of Mathe- matics and Its Applications,” and subsequently by Cambridge University Press. Biedenharn and I planned to extend the results for SU(2), which is the quantum mechanical rotation group of 2 x 2 unitary unimodular matrices, to the general unitary group U(n), based on our research over thirty years of collaboration. The plan was to use the methods of the boson calculus because of its close relationship to the creation and an- nihilation operators associated with physical processes and the natural invariance of this calculus to unitary transformations. The broad outline of such a monograph on unitary symmetry based on the boson calculus was laid out some fifteen years ago, but was never implemented. Bieden- harn became very interested in quantum groups and q-tensor operator theory, while I, under the influence of Gian-Carlo Rota and his student, William Y. C. Chen, became interested in the combinatorial basis of group representation and tensor operator theory. Biedenharn’s death in 1996 ended any possibility of a rejoining of efforts, but our earlier collaborations have had a heavy bearing on the present work. The role of combinatorics in the representation theory of groups is more encompassing than possibly could have been forseen. The funda- mental role developed here evolved from research with William Y. C. Chen and Harold W. Galbraith, postdoctoral student of mine, and col- laborator on a number of articles on symmetry in physics, all of which was tempered by Rota’s global viewpoint of the pervasiveness of com- binatorics. This monograph is about the discoveries made, as described by a algorithmic approach to enhance the computability of the complex objects encountered. It is against this background that the viewpoints advanced in this monograph emerged. Boson polynomials are homogeneous polynomials defined over a col- lection of n2 commuting boson creation operators. These polynomials give all the irreducible unitary representations of the general unitary group U(n) by the simple device of replacing the boson operators by the n2 elements of a unitary matrix. The multiplication property of these matrix group respresentations of U(n) is preserved even by the boson polynomials. This suggests that the boson operators should be taken to be commuting indeterminates, and that the properties of these homoge- neous polynomials should be developed in this context. The polynomials are themselves the basic objects, independent of any interpretation of the indeterminates over which they are defined. Then, not only are the irreducible representations of U(n) (and the general linear group) ob- vii
- 13. viii PREFACE tained in one assignment of the indeterminates, but also in the original assignment the rich physical interpretation in terms of boson operators is regained. But much more emerges. The group multiplication property of rep- resentations is a consequence of a new class of identities among multi- nomial coefficients, which themselves have a combinatorial origin and proof, and which hold for arbitrary interpretations of the n2 indetermi- nates, including even singular matrices of order n. The structure is fully combinatorial. The study of these polynomials is thus brought under the purview of combinatorics and special functions, extended to many variables. These polynomials may be regarded as generalizations of the functions that arise in the study of the symmetric group, with its as- sociated catalog of symmetric functions, such as the Schur functions, etc. Even more unexpected is that the famous MacMahon [129] Master Theorem, a classical result in combinatorics, is the basis for Schwinger’s [160] famous generating function approach to angular momentum theory. Indeed, it is the MacMahon Master Theorem that unifies the angular mo- mentum properties of composite systems in the binary build-up of such systems from more elementary constituents. This monograph consists, essentially, of three distinct, but interre- lated parts: Chapters 1-4, Chapters 5-9, and Chapters 10-11. The last two chapters are compendiums which define, develop, and summarize concepts used in the first nine chapters. Chapters 1-4 deal with basic angular momentum theory and the properties of the famous Wigner D−functions, now extended to poly- nomial forms over four commuting indeterminates, and with the prop- erties of arbitrary many multiple Kronecker products of these extended D−polynomials. These four chapters may be regarded as a summary of results that subsume all of standard angular momentum theory with a focus on the combinatorial underpinnings of these polynomials, as cap- tured by the concept of SU(2) solid harmonics. As examples, the famous Wigner-Clebsch-Gordan coefficients are shown to be objects that com- binatorially come under the purview of the umbral calculus, while the binary coupling theory of angular momentum is intrinsically an applica- tion of the theory of graphs, specifically, binary trees, Cayley trivalent trees, and cubic graphs. This leads to a number of combinatorial in- terpretations of the well-known Racah sum rule and Biedenharn-Elliott identity, and the fundamental role of Racah coefficients in the binary recoupling theory of angular momenta. Chapters 5-9 deal with the generalization of the solid harmonics to polynomials called Dλ−polynomials, where λ is a partition, and these polynomials are defined over n2 commuting indeterminates, which when specialized to the elements of a complex matrix of order n give the in- tegral irreducible representations of the general linear group of complex
- 14. PREFACE ix matrices of order n, and, in particular, all inequivalent irreducible repre- sentations of the general unitary group of matrices of order n. Again, the focus is on the combinatorial properties of the general polynomials them- selves, such as their unique generation by shift operator actions, which involve diagraphs, Sylvester’s identity, Schur functions, skew Schur func- tions, Kostka numbers, and Littlewood-Richardson numbers, all combi- natorial concepts underlying modern treatments of the symmetric group Sn. It is the labeling of these polynomials by Gelfand-Tsetlin patterns, which are one-to-one with the semistandard Young-Weyl tableau, that underlies the relationship to the symmetric group. The reduction of the single Kronecker product Dµ ⊗ Dν = λ cλ µν Dλ of two such ir- reducible polynomials into a direct sum of irreducible polynomials is extraordinarily rich in combinatorial structures.The Dλ−polynomials subsume many of the properties of classical Schur functions, and the matrix Dλ(Z) might well be called a matrix Schur function.The com- plexity of these polynomials, although elegant in their structure, allows us to deal comprehensibly only with the Kronecker product of a pair of such polynomials. Multiple Kronecker products and the associated concepts of Racah coefficients, etc., and the relationship to graph theory is beyond our reach. New viewpoints of tensor operators as operator- valued Dλ−polynomials emerge. A comprehensive theory of (general- ized) Racah coefficients must await further developments. The Littlewood-Richardson numbers cλ µν that occur in the reduction of the Kronecker product is so pervasive that we give a great deal of attention to their properties (Compendium B). These numbers express the number of repetitions of a given Dλ−polynomial in the Kronecker product reduction. They give the generalization to the general unitary group U(n) of the familiar addition rule j = j1 + j2, j1 + j2 − 1, . . . , |j1 − j2| of two interacting quantum-mechanical constituents with separate an- gular momenta j1 and j2, constituting a composite system of angular momentum j; the Littlewood-Richardson number is 0 or 1. Three (at least) nontrivial combinatorial objects enter into the combi- natorial interpretation of the Littlewood-Richardson numbers: Gelfand- Tsetlin patterns, semistandard skew tableaux, and the lattice permu- tations associated with these entities. The intricacies of such counting methods would appear to be a rather high price for obtaining the rule for the addition of two angular momenta, which was deduced by physicists from experimental spectroscopy and subsequently from algebraic tech- niques (see Condon and Shortley [45]) that involved neither Lie algebras nor combinatorics. But the new insights gained are well worth the effort. These techniques underlie the development of the properties of the Dλ−polynomials over arbitrary commuting indeterminates. One of the
- 15. x PREFACE principal purposes of this monograph is to demonstrate, by construction, the details and inter-relations of these concepts. Chapters 10-11 comprise the third part of this monograph. They consist of two extensive Compendiums A and B of results from algebra, analysis, and combinatorics that relate to the first two parts. They have been included so as to be able to refer and use the results in the main parts of the monograph without having to interrupt the flow of presentation with technical asides. The presentation of the material in the Compendiums is very uneven: some is given in great detail and some is very brief, depending on their role in the main text. There are a number of unsolved problems and unaddressed topics. Unsolved problems include the following, where further details can be found in the referenced sections: 1. Counting formula for the Clebsch-Gordon numbers that give the multiplicity of a given state of total angular momentum in the cou- pling of n angular momenta (Sect. 2.2). 2. The enumeration of the nonisomorphic unlabeled cubic graphs on 2n points that correspond to the coupling of n angular momenta (Sects. 3.3, 3.4, 4.5, 4.6). 3. Extension of the step-function formulas for Kostka numbers and Littlewood-Richardson numbers to n ≥ 4 with a geometrical inter- pretation (Sects. 9.4.3, 9.6, 11.3.7, 11.3.8). 4. The geometrical meaning of operator patterns (Sects. 9.4, 9.6, 9.7.2). 5. A comprehensive theory of multiple Kronecker products of the Dλ−polynomials and of the associated recoupling matrices; that is, the generalization of 3n − j coefficients of SU(2) and of the geometry of cubic graphs (p. 446). Inadequately addressed and nonaddressed topics include the following: (i). Full development of the properties of the skew-symmetric matrix associated with a standard labeled binary tree corresponding to the addition of angular momenta (Sect. 4.2 ). (ii). Path formulation of recoupling matrices (Sect. 2.2.10). (iii). Relation of Dλ−polynomials to special functions, such as a theory of multivariable Hermite polynomials (Sect. 11.9.4). (iv). Formulation of a comprehensive umbral calculus and invariant the- ory approach to the Dλ−polynomials (Sect. 11.9.3).
- 16. PREFACE xi (v). Extension of combinatorial foundations to other groups. (vi). Applications to physical problems. The very detailed Table of Contents serves as a summary of topics covered. The readership is intended to be graduate students and re- searchers interested in learning of the relation between symmetry and combinatorics and of challenging unsolved problems. The many exam- ples serve partially as exercises. It is hoped that the topics presented promote further and more rigorous developments. We mention some unconventional matters of style. We present signif- icant result in italics, but do not grade and stylize them as lemmas and theorems. Such italicized statements serve as summaries of results, and often do not merit the title as theorems. Diagrams and figures are inte- grated into the text, and not set aside on nearby pages, so as to have a smooth flow of ideas. Our informality of presentation, including proofs, does not attain the status of rigor demanded in more formal approaches, but our purpose is better served, and our objectives met, by focusing on algorithmic, constructive methods, as illustrated by many examples. It is particularly encouraging to read in Andrews, Askey, and Roy [3] about the usefulness of algorithmic based, complex, mathematical relationships in today’s computer oriented approach. Such relations encode informa- tion amenable to computer processing; perhaps, not to extent envisioned by Wolfram [187], but nonetheless naturally and innovatively. This monograph is not democratically assembled. The enormous lit- erature on physical applications of unitary symmetry are not amenable to a synthesis of technique, except in the broadest sense of Lie algebra and group representations. Moreover, the subject has received little at- tention from the combinatorial orientation presented here. Accordingly, the monograph is heavily biased toward the understanding I have been able to acquire over a fifteen year period of presenting lectures on these subjects at small conferences in Poland organized by Tadeusz and Bar- bara Lulek on Symmetry and Structural Properties of Condensed Matter, and also at Nankai University, PR China, at the invitation of William Y. C. Chen, Director, The Center for Combinatorics. The opportunity to address a sizeable number of students has been particularly reward- ing. Important special contributions to the subject have come from my colleagues Bill Chen, Harold Galbraith, and Miguel Méndez. General encouragement from George Andrews, Bill Chen, Gordan Drake, Harold Galbraith, Brian Judd, Ron King, Tadeusz and Barbara Lulek, Steven Milne, Peter Paule, Gian-Carl Rota, and in earlier years, Larry Bieden- harn, and in later years, my son Tom and wife Marge; all have helped to sustain the effort. I have also been inspired by the many lectures of Gian-Carlo Rota, the comprehensive book by Stanley [163], and the terse, but scholarly book by Macdonald [126]. James D. Louck
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- 18. Contents Preface vii Notation xxi 1 Quantum Angular Momentum 1 1.1 Background and Viewpoint . . . . . . . . . . . . . . . . . 1 1.1.1 Euclidean and Cartesian 3-space . . . . . . . . . . 1 1.1.2 Newtonian physics . . . . . . . . . . . . . . . . . . 9 1.1.3 Nonrelativistic quantum physics . . . . . . . . . . 10 1.1.4 Unitary frame rotations . . . . . . . . . . . . . . . 17 1.2 Abstract Angular Momentum . . . . . . . . . . . . . . . . 28 1.2.1 Brief background and history . . . . . . . . . . . . 28 1.2.2 One angular momentum . . . . . . . . . . . . . . . 30 1.2.3 Two angular momenta . . . . . . . . . . . . . . . . 35 1.3 SO(3, R) and SU(2) Solid Harmonics . . . . . . . . . . . 43 1.3.1 Inner products . . . . . . . . . . . . . . . . . . . . 49 1.4 Combinatorial Features . . . . . . . . . . . . . . . . . . . 51 1.4.1 Combinatorial definition of Wigner-Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . 51 1.4.2 Magic square realization . . . . . . . . . . . . . . . 59 1.5 Kronecker Product of Solid Harmonics . . . . . . . . . . . 61 1.6 SU(n) Solid Harmonics . . . . . . . . . . . . . . . . . . . 64 xiii
- 19. xiv CONTENTS 1.6.1 Definition and properties of SU(n) solid harmonics 64 1.6.2 MacMahon and Schwinger master theorems . . . . 66 1.6.3 Combinatorial proof of the multiplication property 67 1.6.4 Maclaurin monomials . . . . . . . . . . . . . . . . 71 1.6.5 Summary of relations . . . . . . . . . . . . . . . . 74 1.7 Generalization to U(2) . . . . . . . . . . . . . . . . . . . . 75 1.7.1 Definition of U(2) solid harmonics . . . . . . . . . 75 1.7.2 Basic multiplication properties . . . . . . . . . . . 78 1.7.3 Indeterminate and derivative actions on the U(2) solid harmonics . . . . . . . . . . . . . . . . . . . . 81 2 Composite Systems 83 2.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . 83 2.1.1 Angular momentum state vectors of a composite system . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.1.2 Group actions in a composite system . . . . . . . . 89 2.1.3 Standard form of the Kronecker direct sum . . . . 90 2.1.4 Reduction of Kronecker products . . . . . . . . . . 93 2.1.5 Recoupling matrices . . . . . . . . . . . . . . . . . 94 2.2 Binary Coupling Theory . . . . . . . . . . . . . . . . . . . 97 2.2.1 Binary trees . . . . . . . . . . . . . . . . . . . . . . 100 2.2.2 Standard labeling of binary trees . . . . . . . . . . 107 2.2.3 Generalized WCG coefficients defined in terms of binary trees . . . . . . . . . . . . . . . . . . . . . . 109 2.2.4 Binary coupled state vectors . . . . . . . . . . . . 112 2.2.5 Binary reduction of Kronecker products . . . . . . 114 2.2.6 Binary recoupling matrices . . . . . . . . . . . . . 115 2.2.7 Triangle patterns and triangle coefficients . . . . . 119 2.2.8 Racah coefficients . . . . . . . . . . . . . . . . . . 131 2.2.9 Recoupling matrices for n = 3 . . . . . . . . . . . . 137 2.2.10 Recoupling matrices for n = 4 . . . . . . . . . . . . 140
- 20. CONTENTS xv 2.2.11 Structure of general triangle coefficients and recoupling matrices . . . . . . . . . . . . . . . . . . 152 2.3 Classification of Recoupling Matrices . . . . . . . . . . . . 156 3 Graphs and Adjacency Diagrams 163 3.1 Binary Trees and Trivalent Trees . . . . . . . . . . . . . . 163 3.2 Nonisomorphic Trivalent Trees . . . . . . . . . . . . . . . 172 3.2.1 Shape labels of binary trees . . . . . . . . . . . . . 184 3.3 Cubic Graphs and Trivalent Trees . . . . . . . . . . . . . 189 3.4 Cubic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.4.1 Factoring properties of cubic graphs . . . . . . . . 200 3.4.2 Join properties of cubic graphs . . . . . . . . . . . 213 3.4.3 Cubic graph matrices . . . . . . . . . . . . . . . . 221 3.4.4 Labeled cubic graphs . . . . . . . . . . . . . . . . . 226 3.4.5 Summary and unsolved problems . . . . . . . . . . 228 4 Generating Functions 229 4.1 Pfaffians and Double Pfaffians . . . . . . . . . . . . . . . . 230 4.2 Skew-Symmetric Matrix . . . . . . . . . . . . . . . . . . . 232 4.3 Triangle Monomials . . . . . . . . . . . . . . . . . . . . . 238 4.4 Coupled Wave Functions . . . . . . . . . . . . . . . . . . . 239 4.5 Recoupling Coefficients . . . . . . . . . . . . . . . . . . . . 241 4.6 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.6.1 Cubic graph geometry of the 6 − j and 9 − j coefficients . . . . . . . . . . . . . . . . . . . . . . 255 4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 259 5 The Dλ−Polynomials: Form 261 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.2 Defining Relations . . . . . . . . . . . . . . . . . . . . . . 268 5.2.1 Another proof of the multiplication rule . . . . . . 277
- 21. xvi CONTENTS 5.3 Restriction to Fewer Variables . . . . . . . . . . . . . . . . 278 5.4 Vector Space Aspects . . . . . . . . . . . . . . . . . . . . . 280 5.5 Fundamental Structural Relations . . . . . . . . . . . . . 283 5.5.1 Structural relations . . . . . . . . . . . . . . . . . . 283 5.5.2 Principal objectives and tasks . . . . . . . . . . . . 284 6 Operator Actions in Hilbert Space 285 6.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . 285 6.2 Action of Fundamental Shift Operators . . . . . . . . . . 286 6.3 Digraph Interpretation . . . . . . . . . . . . . . . . . . . . 295 6.4 Algebra of Shift Operators . . . . . . . . . . . . . . . . . . 301 6.5 Hilbert Space and Dλ−Polynomials . . . . . . . . . . . . 304 6.6 Shift Operator Polynomials . . . . . . . . . . . . . . . . . 308 6.7 Kronecker Product Reduction . . . . . . . . . . . . . . . . 312 6.8 More on Explicit Operator Actions . . . . . . . . . . . . . 318 7 The Dλ−Polynomials: Structure 325 7.1 The Cλ(A) Matrices . . . . . . . . . . . . . . . . . . . . . 325 7.2 Reduction of Dp(Z) ⊗ Dλ(Z) . . . . . . . . . . . . . . . . 329 7.3 Binary Tree Structure: Cλ−Coefficients . . . . . . . . . . 335 7.3.1 The Dλ−polynomials for partitions with two nonzero parts . . . . . . . . . . . . . . . 342 7.3.2 Recurrence relation for the Dλ−polynomials . . . 351 8 The General Linear and Unitary Groups 355 8.1 Background and Review . . . . . . . . . . . . . . . . . . . 355 8.2 GL(n, C) and its Unitary Subgroup U(n) . . . . . . . . . 358 8.3 Commuting Hermitian Observables . . . . . . . . . . . . . 368 8.3.1 The Gelfand invariants . . . . . . . . . . . . . . . . 368 8.4 Differential Operator Actions . . . . . . . . . . . . . . . . 370
- 22. CONTENTS xvii 8.5 Eigenvalues of the Gelfand Invariants . . . . . . . . . . . . 371 8.5.1 A new class of symmetric functions . . . . . . . . 373 8.5.2 The general eigenvalues of the Gelfand invariants 375 9 Tensor Operator Theory 379 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 379 9.1.1 A basis of irreducible tensor operators . . . . . . . 382 9.2 Unit Tensor Operators . . . . . . . . . . . . . . . . . . . . 386 9.2.1 Summary of properties of unit tensor operators . . 389 9.2.2 Explicit unit tensor operators . . . . . . . . . . . . 392 9.3 Canonical Tensor Operators . . . . . . . . . . . . . . . . . 394 9.3.1 Application of the factorization lemma to canoni- cal tensor operators . . . . . . . . . . . . . . . . . 399 9.3.2 Subgroup conditions and reduced matrix elements 403 9.4 Properties of Reduced Matrix Elements . . . . . . . . . . 408 9.4.1 Unit projective operators . . . . . . . . . . . . . . 410 9.4.2 The pattern calculus . . . . . . . . . . . . . . . . . 415 9.4.3 Shift invariance of Kostka and Littlewood-Richardson numbers . . . . . . . . . . . 421 9.5 The Unitary Group U(3) . . . . . . . . . . . . . . . . . . . 422 9.5.1 The U(3) canonical tensor operators . . . . . . . . 422 9.6 The U(3) Characteristic Null Spaces . . . . . . . . . . . . 425 9.7 The U(3) : U(2) Unit Projective Operators . . . . . . . . 432 9.7.1 Coupling rules and Racah coefficients . . . . . . . 433 9.7.2 Limit relations . . . . . . . . . . . . . . . . . . . . 440 10 Compendium A. Basic Algebraic Objects 447 10.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.1.1 Group actions . . . . . . . . . . . . . . . . . . . . . 449 10.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 10.2.1 Rings of polynomials . . . . . . . . . . . . . . . . . 452
- 23. xviii CONTENTS 10.2.2 Vector spaces of polynomials . . . . . . . . . . . . 455 10.3 Abstract Hilbert Spaces . . . . . . . . . . . . . . . . . . . 456 10.3.1 Inner product spaces . . . . . . . . . . . . . . . . . 456 10.3.2 Linear operators . . . . . . . . . . . . . . . . . . . 458 10.3.3 Inner products and linear operators . . . . . . . . 460 10.3.4 Orthonormalization methods . . . . . . . . . . . . 461 10.3.5 Matrix representations of linear operators . . . . . 464 10.4 Properties of Matrices . . . . . . . . . . . . . . . . . . . . 465 10.4.1 Properties of normal matrices . . . . . . . . . . . . 467 10.4.2 Inner product on the space of complex matrices . . 468 10.4.3 Exponentiated matrices . . . . . . . . . . . . . . . 469 10.4.4 Lie bracket polynomials . . . . . . . . . . . . . . . 476 10.5 Tensor Product Spaces . . . . . . . . . . . . . . . . . . . . 486 10.6 Vector Spaces of Polynomials . . . . . . . . . . . . . . . . 488 10.7 Group Representations . . . . . . . . . . . . . . . . . . . . 494 10.7.1 Irreducible representations of groups . . . . . . . . 499 10.7.2 Schur’s lemmas . . . . . . . . . . . . . . . . . . . . 501 11 Compendium B: Combinatorial Objects 505 11.1 Partitions and Tableaux . . . . . . . . . . . . . . . . . . . 506 11.1.1 Restricted compositions . . . . . . . . . . . . . . . 512 11.1.2 Linear ordering of sequences and partitions . . . . 513 11.2 Young Frames and Tableaux . . . . . . . . . . . . . . . . . 514 11.2.1 Skew tableaux . . . . . . . . . . . . . . . . . . . . 520 11.3 Gelfand-Tsetlin Patterns . . . . . . . . . . . . . . . . . . . 523 11.3.1 Combinatorial origin of Gelfand-Tsetlin patterns . 523 11.3.2 Group theoretical origin of Gelfand-Tsetlin patterns . . . . . . . . . . . . . . . . . . . . . . . . 528 11.3.3 Skew Gelfand-Tsetlin patterns . . . . . . . . . . . 528 11.3.4 Notations . . . . . . . . . . . . . . . . . . . . . . . 532
- 24. CONTENTS xix 11.3.5 Triangular and skew Gelfand-Tsetlin patterns . . . 533 11.3.6 Words and lattice permutations . . . . . . . . . . . 535 11.3.7 Lattice permutations and Littlewood-Richardson numbers . . . . . . . . . . . . . . . . . . . . . . . 538 11.3.8 Kostka and Littlewood-Richardson numbers . . . . 550 11.4 Generating Functions and Relations . . . . . . . . . . . . 556 11.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . 557 11.4.2 Counting relations . . . . . . . . . . . . . . . . . . 558 11.4.3 Generating relations of functions . . . . . . . . . . 559 11.4.4 Operator generated functions . . . . . . . . . . . . 561 11.5 Multivariable Special Functions . . . . . . . . . . . . . . 563 11.5.1 Solid harmonics . . . . . . . . . . . . . . . . . . . . 564 11.5.2 Double harmonic functions in R3 . . . . . . . . . . 565 11.5.3 Hypergeometric functions . . . . . . . . . . . . . . 565 11.5.4 MacMahon’s master theorem . . . . . . . . . . . . 568 11.5.5 Power of a determinant . . . . . . . . . . . . . . . 568 11.6 Symmetric Functions . . . . . . . . . . . . . . . . . . . . . 570 11.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 570 11.6.2 Four basic symmetric functions . . . . . . . . . . . 571 11.6.3 Vandermonde determinants . . . . . . . . . . . . . 572 11.6.4 Schur functions . . . . . . . . . . . . . . . . . . . . 573 11.6.5 Dual bases for symmetric functions . . . . . . . . . 578 11.7 Sylvester’s Identity . . . . . . . . . . . . . . . . . . . . . . 580 11.8 Derivation of Weyl’s Dimension Formula . . . . . . . . . . 582 11.8.1 Vandermonde determinants, Bernoulli polynomi- als, and Weyl’s formula . . . . . . . . . . . . . . . 582 11.9 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . 585 11.9.1 Alternating sign matrices . . . . . . . . . . . . . . 585 11.9.2 Binary trees, graphs, and digraphs . . . . . . . . . 595 11.9.3 Umbral calculus and double tableau calculus . . . 595
- 25. xx CONTENTS 11.9.4 Special functions . . . . . . . . . . . . . . . . . . . 596 11.9.5 Other generalizations . . . . . . . . . . . . . . . . 596 Bibliography 597 Index 611
- 26. Notation General symbols R real numbers C complex numbers P positive numbers Z integers N nonnegative integers Rn Cartesian n−space Cn complex n−space En Euclidean n−space O(n, R) group of real orthogonal matrices of order n SO(n, R) group of real, proper orthogonal matrices of order n U(n) group of unitary matrices of order n SU(n) group of unimodular unitary matrices of order n GL(n, C) group of complex nonsingular matrices of order n O(n, R) group of real orthogonal matrices of order n Mp n×n(α, α) set of n × n matrix arrays with nonnegative elements with row-sum α and column-sum α × ordinary multiplication in split product, direct product ⊕ direct sum of matrices ⊗ tensor product, Kronecker product of matrices Parn set of partitions having n parts, including 0 as a part δi,j the Kronecker delta for integers i, j δA,B, δ(A, B) the Kroneker delta for sets A and B K(λ, α) the Kostka numbers cλ µ ν the Littlewood-Richardson numbers Specialized symbols Lists of specialized symbols are found on pp. 532–533, 538, 557; others are introduced as needed in the text. xxi
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- 28. Chapter 1 Quantum Theory of Angular Momentum: Introduction 1.1 Background and Viewpoint 1.1.1 Euclidean and Cartesian 3-space It is a basic tenet of Newtonian physics that physical phenomena take place in Euclidean 3-space, denoted E3, which is a collection of enti- ties called “points,” a point being a primitive undefined entity. Such points are used in Euclidean geometry to construct other objects such as lines, planes, triangles, etc., from a set of axioms that constitute a deductive structure called Euclidean geometry. Euclidean geometry in 2-space and 3-space is the formalized mathematical method for deduc- ing “facts” about the intuitive space of our surroundings and the objects perceived to be part of those surroundings. Lines, planes, and other ob- jects are constructed as subsets of points in E3. But the primitive object “point” has no properties on its own; it is usually thought of as having no “size;” a point cannot be taken apart in terms of still other points. Eu- clidean geometry does not use the concept of “distance between points” to characterize its objects, but rather the notion of congruence. Mac Lane [128, p.127]) argues, however, that the notion of geometric magni- tude is present in Euclid’s geometry and that together with congruence and geometric ratios constitutes a geometrical description of the modern concept of the real-number line. Euclid’s geometry allows its objects to be “moved” around in the E3 “space” in the modern sense of rigid body motions that includes 1
- 29. 2 CHAPTER 1. QUANTUM ANGULAR MOMENTUM translations and rotations. But no explanation for the “cause” of such motions is given; they are intrinsic properties of the “ambient” E3-space ( Mac Lane [128, p.76]). Euclidean E3-space contains, as subsets, collections of three lines that are mutually perpendicular and have a common unique point of inter- section. The operation of perpendicular projection of a point p ∈ E3 to a point L(p) on a given line L ⊂ E3 is defined for all points and all lines. It is these notions of three perpendicular lines intersecting at a point, and the operation of perpendicular projection onto these lines that is used to describe physical phenomena in a concrete manner, based on the use of the real numbers, and the notion of Cartesian 3-space, denoted R3. Cartesian 3-space R3 is the real linear vector space whose elements consist of the collection of all sequences x = (x1, x2, x3), xi ∈ R, i = 1, 2, 3, where R is the set of real numbers, where, of course, all the usual operations of addition, multiplication by a scalar, and the distributive rules that define a vector space are applied to such sequences. We refer to a sequence of three real numbers (x1, x2, x3) ∈ R3 as a coordinate point, and often as coordinates of a point in R3. Cartesian 3-space R3 comes equipped naturally with two functions defined on all pairs of points x = (x1, x2, x3) and x = (x 1, x 2, x 3) belonging to R3, namely, a distance function, denoted d(x, x), and a scalar product denoted, x · x, given by d(x, x ) = (x1 − x 1)2 + (x2 − x 2)2 + (x3 − x 3)2, (1.1) x · x = x1x 1 + x2x 2 + x3x 3. (1.2) The space R3 contains subsets called lines, for example, the set of co- ordinate points {x1, a2, a3) | x1 ∈ R}, for fixed real numbers a1 and a2, defines a line. In particular, the subsets of R3 defined by L1 = {(x1, 0, 0)|x1 ∈ R}, L2 = {(0, x2, 0)|x2 ∈ R}, (1.3) L3 = {(0, 0, x3)|x3 ∈ R}, determine three mutually perpendicular lines in R3, where perpendicular means the scalar product of points belonging to the separate lines is 0. To relate the geometrical methods of Euclidean E3−space geometry over points to the vector space methods of the analytical geometry of Cartesian R3−space over the real numbers, it is convenient make the strong assumption (from Mac Lane’s [128] observations) that each line L ⊂ E3 is identified with the real-number line L(R) = {p(x) | x ∈ R}, where it is the custom to identity the point p(x) on the real-number line
- 30. 1.1. BACKGROUND AND VIEWPOINT 3 L(R) with the real number x itself: p(x) → x. Thus, let q ∈ E3, and let the three perpendicular lines L1(q), L2(q), L3(q) ⊂ E3, respectively, have q as their (unique) intersection point. Identify each of these three perpendicular geometrical lines with the real-number line L(R) = {x|x ∈ R}, so that the geometrical intersection point q is identified with a point q → (q1, q2, q3) ∈ R3 = {(x1, x2, x3)|each xi ∈ R}. (1.4) Thus, the three perpendicular lines L1(q), L2(q), L3(q) in E3 are identi- fied with the three perpendicular lines in R3 given by L1(q) → L1(R3 ) = {(x1(q), q2, q3)|x1(q) ∈ R}, L2(q) → L2(R3 ) = {(q1, x2(q), q3)|x2(q) ∈ R}, (1.5) L3(q) → L3(R3 ) = {(q1, q2, x3(q))|x3(q) ∈ R}. A geometrical point p ∈ E3, which is obtained by geometric projec- tion onto the three perpendicular lines L1(q), L2(q), L3(q) ⊂ E3 is now mapped to a general coordinate point (x1(p), x2(p), x3(p)) ∈ R3 given by p → (x1(p), x2(p), x3(p)) = (x1(q) + q1, x2(q) + q2, x3(q) + q3). (1.6) The set of three perpendicular axes L1(R3), L2(R3), L3(R3) ⊂ R3 is called a set-of-axes or a reference frame for the Cartesian R3 vector space. The point q → (q1, q2, q3) ∈ R3 is called the origin of the Cartesian ref- erence frame. Thus, the arithmetic description of points in Euclidean space E3 is effected by two related objects in Cartesian space R3 : a ref- erence frame and a set of coordinates relative to the reference frame. It is convenient to denote this description of E3 by R3(q) : The underlying ambient Euclidean 3-space E3 is unchanged by different choices of q ∈ E3; the Cartesian 3-space R3(q), with origin located at q → (q1, q2, q3) is a redescription of the (unchanged) points of E3. Equivalently, the coordi- nate points corresponding to a coordinate frame located at any origin (a1, a2, a3) are a redescription of the coordinate points corresponding to a coordinate frame located at any other origin (a 1, a 2, a 3), each being a different coordinate presentation of the same collection of points in the underlying Euclidean space E3. Discussions of the axioms of Euclidean geometry that suit our needs and give a more complete picture than the brief discussion given here can be found in Weyl [176, 177] and Mac Lane [128]. A discussion of the origin and foundations of these concepts from the point of view of cognitive scientists can be found in Lakoff and Núñez [100]. We complete our description of the relation between the spaces E3 and R3 by introducing the concept of a geometric vector as defined by
- 31. 4 CHAPTER 1. QUANTUM ANGULAR MOMENTUM physicists in terms of directed line segments (see, for example, Page [139]). This notion is well-suited to Euclidean 3-space with its point entities. A geometrical vector is a directed line segment in E3 (a line segment with one point called the origin and a second point called the terminal point of the vector). Two geometrical vectors are defined as equal if they lie along parallel lines and have the same directional sense and length (geometrical magnitude); addition is defined in terms of the familiar parallelogram rule; multiplication by a scalar (real numbers) is defined as a scaling of geometrical magnitude, including the possibility (negative number) of the reversal of direction; and the vector 0, called the zero vector, is the vector in which the origin coincides with the terminal point. The set of geometrical vectors satisfy the following two rules for all points a, b ∈ E3 : (i) x(a → a) = 0; (ii) x(a → b)+x(b → c) = x(a → c). The second property is called the transitive property of addition. Together with the first rule, it implies that x(a → b) + x(b → a) = 0, which gives the negative geometrical vector, x(b → a) = −x(a → b). These definitions, together with the concept of parallel transport, which allows for the “motion” of equal geometrical vectors necessary to bring equal vectors into coincidence so that all geometrical vectors can be added, determines a linear vector space of geometrical vectors that offers an alternative means of implementing the ideas set forth by Mac Lane [128, p.75]). The linear space of geometric vectors with origin at point q ∈ E3 is denoted V3(q) and is isomorphic to the vector space R3(q). For q identified as the origin (0, 0, 0), geometrical vectors are described in terms of the corresponding Cartesian 3-space as follows: Unit vectors e1, e2, e3 are defined to be the directed line segments of unit length going from the origin (0, 0, 0) to the respective points (1, 0, 0), (0, 1, 0), (0, 0, 1). The general vector x = x1e1 + x2e2 + x3e3 is then the directed line segment going from the origin (0, 0, 0) to the coordinate point (x1, x2, x3), and its length is |x| = x2 1 + x2 2 + x2 3. It is convenient to think of the isomorphic vector spaces R3(q) and V3(q) with origin at q → (q1, q2, q3) as superposed over the point q ∈ E3. This merger of “pictures” of vector spaces makes quite clear the one-to- one relation between vectors and coordinate points. In this sense, we speak of a triad of unit vectors along the coordinate axes as a unit refer- ence frame , or simply as a reference frame. Thus, we may characterize the Euclidean 3-space corresponding to each point q ∈ E3 as consist- ing of the two sets of objects: a reference frame (e1(q), e2(q)), e3(q)) located at the origin q → (q1, q2, q3), and the set of coordinate points (x1(q), x2(q), x3(q)) projected onto the corresponding coordinate axes with directional and orientational sense defined by these unit geomet- rical vectors. This picture allows us to speak of the Cartesian 3-space located at the Euclidean point q and having an arbitrary orientation of its
- 32. 1.1. BACKGROUND AND VIEWPOINT 5 reference frame in Euclidean space, thus simplifying the description. We do not, however, identify all the isomorphic vector spaces R3(q), q ∈ E3, since the notion of the redescription of Euclidean 3−space by different reference frames and coordinate points has important implications for the description of the behavior of physical systems. (The same state- ment applies to V3(q).) The family of isomorphic Cartesian 3-spaces R3(q), q ∈ E3 can be transformed into one another by geometrical congruence and other oper- ations that leave the geometrical objects in E3 “unchanged.” For physics, the meaning of unchanged is to be decided by experimental observation. Physical theory must in some sense be invariant under such transforma- tions, since the different Cartesian 3-spaces R3(q) are just a redescription of the properties of the objects in E3 constituting a physical system. But how a specified choice of reference frame is used to “measure” the prop- erties of a given physical system, and how such measurements might influence these properties, are questions that, as yet, have no universal answers: the answers depend on additional assumptions. For example, in Newtonian physics, it is assumed that “equations of motion” gov- ern the motion of an object against the background Euclidean space, and that the observation of an object and its motion have negligible (or accountable) influence on the object and no influence on the space. In nonrelativistic quantum theory, it is assumed that “equations of motion” still govern the behavior of an object against the background Euclidean space, but the new equations of motion are such that the concept of a “property” of an object and the “measurement” of that property, such as “point” particle and measurement of the location of the point, become interwoven in ways that dramatically negate their classical (Newtonian) meaning. While the equations of motion themselves are deterministic in their time evolution, the transformation of the “properties” of the system during the measurement process are subject to various interpretations that need not be continuous against the Euclidean background. Special and general relativity “correct” Newtonian relativity in a different con- text by recognizing that objects and space do not have mutually exclusive properties, but relate to one another in ways that require abandonment of the Euclidean space and its assigned universal Newtonian time. De- terministic motions are maintained in a broader measurement process with rods and clocks, but all this requires a completely new formulation of “equations of motion,” which are at odds with the quantum equations of motion. A general theory that explains all the observed properties of microscopic objects, using the successful quantum theory, and all those of macroscopic objects, using Einstein’s successful relativity theory, re- mains elusive. For a recent discussion, we refer to Leggett [103]. Our concern in Chapters 1-4 of this monograph is with the properties of the angular momentum of many-particle systems in nonrelativistic quantum mechanics, and, especially, the role of combinatorial concepts.
- 33. 6 CHAPTER 1. QUANTUM ANGULAR MOMENTUM We do not concern ourselves with the puzzles of the interpretation of quantum mechanics, but rather with the development of the properties of angular momentum from the viewpoints described above. The prop- erties of angular momentum from the perspective of coordinate changes attributed to a redescription of a physical system in Euclidean 3-space E3 by the use of different Cartesian 3-spaces R3(q) leads to important properties of the quantal physical system. We concern ourselves with three types of redescriptions: translations, inversions, and rotations. The first kind of redescription is a consequence of what is called a translation of reference frames: The reference frame (e1, e2, e3) located at point q ∈ E3, which is the origin of the Cartesian 3-space R3(q), is moved by parallel transport to the point q ∈ E3 and becomes the reference frame (e 1, e 2, e 3) at the origin of the Cartesian 3-space R3(q). The description of one and the same point p ∈ E3 in terms of this pair of reference frames is best expressed in terms of the three geometric vectors x(q → p), x(q → q), and x(q → p), which satisfy the general transitive rule of addition: x(q → p) = x(q → q ) + x(q → p). (1.7) Since we can also write x(q → q) = −x(0 → q) + x(0 → q), relation (1.7) can also be written in the invariant form: x(q → p) + x(0 → q) = x(q → p) + x(0 → q ). (1.8) Here 0 ∈ E3 is an arbitrary point 0 → (0, 0, 0) ∈ R3(0). Relations (1.7) and (1.8) hold for arbitrary geometric vectors. It is the positioning of reference frames by parallel transport at the points q, q ∈ E3 that gives the operation of translation, which, in turn, gives the relations between the coordinates (x1, x2, x3) relative to the reference frame (e1, e2, e3) and the coordinates (x 1, x 2, x 3) relative to the reference frame (e 1, e 2, e 3) of the same point p ∈ E3 as follows: xi + qi = x i + q i, i = 1, 2, 3. (1.9) Thus, the translation (parallel transport) of reference frames from a point q ∈ E3 to a point q ∈ E3 gives the redescription of coordinates of a point p ∈ E3 given by (1.8); this relation is called the translation of coordinates effected by a translation of coordinate frame. The second kind of redescription is a consequence of what is called an inversion of the reference frame (e1, e2, e3) located at the point q ∈ E3. An inversion of the frame is defined by the transformation (e1, e2, e3) → (−e1, −e2, −e3). (1.10)
- 34. 1.1. BACKGROUND AND VIEWPOINT 7 The coordinates (x1, x2, x3) of each point p ∈ E3 as assigned in the Cartesian 3-space R3(q) correspondingly undergo the redescription (x1, x2, x3) → (−x1, −x2, −x3). (1.11) Reflections through planes may also defined. For example, the reflection of the coordinate frame at point q ∈ E3 through the plane containing the unit vectors (e1, e2) is defined by (e1, e2, e3) → (e1, e2, −e3) and the corresponding redescription of the coordinates (x1, x2, x3) of each point p ∈ E3 as assigned in the Cartesian 3-space R3(q) is given by (x1, x2, x3) → (x1, x2, −x3). The third kind of redescription is a consequence of what is called a rotation of the reference frame (e1, e2, e3) located at the point q ∈ E3. A rotation of the frame is defined by the linear transformation ei → e i = 3 j=1 Rij ej, i = 1, 2, 3, (1.12) where the matrix R = (Rij)1≤i,j≤3 with element Rij in row i and column j is a real, proper, orthogonal matrix; that is, a matrix with real elements such that RT R = RRT = I3, where T denotes matrix transposition, I3 the unit matrix of order 3, and det R = 1. The necessary and sufficient conditions that a general point p ∈ E3 with coordinates (x1, x2, x3) ∈ R3(q) is the redescription of the same point p with coordinates (x 1, x 2, x 3) with respect to the rotated reference frame (e 1, e 2, e 3) is expressed by the invariance of form given by x1e1 + x2e2 + x3e3 = x 1e 1 + x 2e 2 + x 3e 3. (1.13) The rotation (1.12) of the reference frame (e1, e2, e3) to the reference frame (e 1, e 2, e 3) now gives the redescription of one and the same point p ∈ E3 given by xi → x i = 3 j=1 Rij xj, i = 1, 2, 3, (1.14) where (x1, x2, x3) are the coordinates of p referred to the coordinate frame (e1, e2, e3) and (x 1, x 2, x 3) are the coordinates of p referred to the coordinate frame (e 1, e 2, e 3), where both coordinate frames are located at q and used for the description of coordinate points of the Cartesian 3−space R3(q). Relations (1.14) are necessary and sufficient conditions that the redescription of the point p is effected by a rotation of frames located at q. We will, unless otherwise noted, always choose coordinate
- 35. 8 CHAPTER 1. QUANTUM ANGULAR MOMENTUM frames such that the triad of unit vectors defining the axes are right- handed triads, as defined by the familiar right-handed screw rule. Left- handed frames can be included as well by the operation of inversion. Mathematicians at the end of the nineteenth century realized that the operations of translation and rotation can be effected by the action of exponential differential operators (see Crofton [46], Glaisher [64] ), which in vector notation are expressed by ea · ∇F(x1, x2, x3) = F(x1 + a1, x2 + a2, x3 + a3), (1.15) e−φn·R F(x1, x2, x3) = F(x 1, x 2, x 3). (1.16) The operator a · ∇, where ∇ = e1∂/∂x1 + e2∂/∂x2 + e3∂/∂x3 is called the generator of the translation a = a1e1 + a2e2 + a3e3. In the second relation, the components of R = R1e1 + R2e2 + R3e3 are the differential operators defined by R1 = x2∂/∂x3 − x3∂/∂x2, R2 = x3∂/∂x1 − x1∂/∂x3, (1.17) R3 = x1∂/∂x2 − x2∂/∂x1. The coordinates (x 1, x 2, x 3) are the redescription of a point p ∈ E3 in terms of the coordinates (x1, x2, x3) of the same point p as described by the rotation of the frame (e1, e2, e3) → (e 1, e 2, e 3) in the positive sense about the direction corresponding to the unit vector n = n1e1 + n2e2 + n3e3, n · n = 1, where the matrix elements Rij = Rij(φ, n) in (1.12) and (1.14) are those of R(φ, n) = eφN = I3 + N sin φ + N2 (1 − cos φ), (1.18) N = 0 −n3 n2 n3 0 −n1 −n2 n1 0 , N3 = −N. The proper, real, orthogonal matrix R has here been parametrized by the components of the unit vector n and the angle 0 ≤ φ 2π. The gradient operator ∇ and the rotation operator R = R1e1 + R2e2 + R3e3 both satisfy the form invariant relation (1.13). The operator −φn · R is called the generator of the rotation about direction n by angle φ. We anticipate some results from the brief discussion of quantum the- ory given below, and observe that relations (1.15)-(1.17) are precursors to similar transformations that were to occur later in quantum theory under the association of classical position x and linear momentum p to operators by the rule x → x and p/ → −i∇, which, when applied to the classical angular momentum L = x × p of a single particle located
- 36. 1.1. BACKGROUND AND VIEWPOINT 9 at position x with linear momentum p relative to the origin of a coor- dinate frame (e1, e2, e3), gives L/ = −iR = −ix×∇. Classical linear momentum and angular momentum of a point particle are identified in their quantum mechanical interpretation as the generators of translations and rotations, as described above. The founders of quantum mechanics rediscovered relations (1.15) and (1.16) in the guise of the unitary oper- ators exp (ia · p/) and exp (−iφn · L)/ acting in the Hilbert space of states of a physical system. From the definition of the quantum angu- lar momentum, L/ = −ix×∇, the operations of translation, inversion, and rotation effect the following transformations of the quantal angular momentum: translation : L → L − ia×∇, inversion : L → L, (1.19) rotation : L → L. 1.1.2 Newtonian physics A physical system is said to be isolated if it has no influence on its surroundings, and conversely. Such ideal systems do not exist in nature, but much of the progress in physics can be attributed to the approximate validity of the concept of an isolated physical system. In Newtonian physics, the Euclidean 3-space E3 is taken as the back- ground against which the changes in an isolated physical system take place: the space is considered to be void of (isolated from) all other physical objects, homogeneous (sameness at every point), and isotropic (sameness in every direction). The physical system occupies a collection (subset) of points in E3, which may be a single point, but this collection of points can change relative to the fixed Euclidean background and the fixed reference frames used to assign coordinates to the points of E3. The measure of this change requires a new concept, that of time, which itself is assumed to have any value on the real-line (ignoring units). The change of the configuration of points defining the physical system with the time parameter is called motion, where it is assumed that motion is described by increasing values of time. The change in time itself is measured by clocks, which themselves occupy points of the background Euclidean space E3, but which advance their readings in a uniform man- ner, independently of their own motion, as measured by still other clocks: There exists a universal time t for measuring all motions of physical sys- tems against the fixed Euclidean background of space; this motion is governed by Newton’s equations of motion, which determine how the coordinate points of the physical system change with time. Thus, en- ters the concept of velocity, and the recognition that the equations of motions must be form invariant with respect to the class of reference
- 37. 10 CHAPTER 1. QUANTUM ANGULAR MOMENTUM frames moving with constant velocity relative to one other. Such classes of frames are called inertial frames, and the invariance of the equations of motion is called the principle of Newtonian or Galilean relativity. The detailed mathematical description of all this is effected though the use of the Cartesian space R3 and its collection of reference frames and asso- ciated coordinate points. Thus, the reference frame (e1, e2, e3) located at some point in E3 is now taken as a Newtonian reference frame that has the dual role of assigning coordinates to all the points of R3 and of assigning coordinates to the subset of points occupied by the phys- ical system at each time t. Newton’s equation of motion are identical for all reference frames moving with constant velocity v with respect to this chosen frame. The collection of all reference frames as parametrized by v have an important role in identifying the Galilean group as the invariance group of Newton’s equations of motion. There is another class of reference frames called accelerated frames. Such frames are often attached to part of the physical system or to other moving points. Accelerating reference frames are called noniner- tial frames; they are often used to simplify the description of the internal motions of a complex physical system, but such frames are on a differ- ent footing—they provide convenient transformations of coordinates that modify the form of Newton’s equations and simplify the descriptions of the motions, as adapted to special situations. 1.1.3 Nonrelativistic quantum physics In nonrelativistic quantum physics, we retain the Newtonian notion of an isolated physical system, as well as the Cartesian space R3 and the set of Newtonian inertial frames, and the relationship between frames and points in R3, just as described above. Now, however, the coordi- nates ascribed to the points occupied by the physical system do not enjoy a deterministic motion against the space R3 as is the case for the Newtonian equations of motion of its points. The classical dual role of reference frames is lost. The role of frames as a means of assigning co- ordinates to the points of R3 against which the properties of the system are to be measured is retained, but the equation of motion in quantum mechanics, the Schrödinger equation, does not allow the classical con- cept of point particle to propagate in time: the concept of point particle must be modified. The role of coordinates of the points occupied by the classical physical system becomes that of parameters that belong to the domain of definition of a function Ψ with complex values. Time is still Newton’s universal time, which also now becomes a parameter belonging to the domain of definition of Ψ. The function Ψ with complex values Ψ(X, t), X = (x(1), x(2), . . . , x(n)), for a classical system consisting of n point particles with geometric position vectors x(1), x(2), . . . , x(n), is
- 38. 1.1. BACKGROUND AND VIEWPOINT 11 called the wavefunction or the state function of the system. The state function of the physical system, together with a set of rules that give the possible outcomes of the measurements on the system, determine the observable properties of the system. The Schrödinger equation is such that if the state function Ψ(X0, t0) of a physical system is known for some position configuration X0 at time t0, it evolves in a unitary deterministic fashion to the state function Ψ(X, t) of the physical sys- tem at time t. The meaning of coordinates and time, however, are to be inferred from the rules of measurement: The mathematical properties of the equations of motion play back into the very meaning of the prop- erties ascribed to the objects themselves; the classical concept of point particle is lost. Indeed, the quantum object called a point particle in its classical description exhibits in its quantum description the property of being “partially present” at every point in R3, in such a way that a measurement of its position at any arbitrary point x finds the object in its entirety at that point with some probability, as determined by its state function, and the object always possesses its mass, charge, and spin with certainty. Despite the holistic aspect of its position, the object can be counted in a measurement and appears as a single entity. One of the most significant mathematical properties of the equations of motions is their linear property, as expressed by the superposition of state functions: If Ψ(X, t) and Φ(X, t) are two solutions at time t, then so also is the linear combination αΨ(X, t) + βΦ(X, t), (1.20) where α and β are arbitrary complex numbers. It is this superposition property, together with the property of complex numbers given by |αΨ(X, t) + βΦ(X, t)|2 = |α|2 |Ψ(X, t)2 + |β|2 |Φ(X, t)|2 +αβ∗ Ψ(X, t)Φ∗ (X, t) + α∗ βΨ∗ (X, t)Φ(X, t), (1.21) that underlies many nonintuitive behaviors of quantum systems. The quantum-mechanical description corresponding to the classical system of many point particles, ascribes a holistic existence to the system in which each part seems to relate to every other part, even when the parts are noninteracting at the time of measurement, provided they were in- teractive in the past, and provided the system has not been previously measured. The “properties” of the system becomes entangled in intri- cate ways with those of a measuring apparatus that is introduced into the space R3 in an interactive mode with the system to determine the “value” of a given quantity associated with the originally isolated sys- tem. This, in turn, leads to fundamental questions about how the state function of the originally isolated physical system and that of the ap- paratus determine the measured properties of the original system. The properties of the measuring apparatus are presumably subject to the rules of quantum theory, which, in turn, now apply to the interacting
- 39. 12 CHAPTER 1. QUANTUM ANGULAR MOMENTUM composite whole—the original isolated system and the measuring device. This new system then has its own collection of state functions governing the properties of the composite whole. How all of this is to be sorted out to obtain information about the original system is the problem of mea- surement (see Leggett [103]). There is no generally agreed on resolution of the problem of measurement. The early standard book on the foundations of quantum mechanics is by Von Neumann [174] with a more modern exposition by Mackey [127]. Bohm [27] challenged the standard interpretation by an ingenious example, and Bell’s [9] critical analysis reopened the entire subject of measurement. The large book by Wheeler and Zurek [179] reviews the history through 1980-81, with many informative comments on the Prob- lem of Measurement and a superb list of references. Recent comments by Griffiths [71] illustrate a popular viewpoint. The possibility of quan- tum computers has intensified the interest in these problems and new experiments are confirming the reality of the counterintuitive quantum world. These important problems are, however, not the subject of this monograph, which makes its presentations in the standard interpretation with a focus on combinatorial aspects. There are, fortunately, general invariance properties of the Schrödinger equation and its solution state functions for a complex composite phys- ical systems that can be used to classify the quantum states of physical systems into substates available to the system. Our focus here is on the properties of the total angular momentum of a physical system, which is a quantity L that has a vector expression L = L1e1 + L2e2 + L3e3 in the right-handed frame (e1, e2, e3) and the expression L = L 1e 1 + L 2e 2 + L 3e 3 in a second rotated right-handed frame (e 1, e 2, e 3). At a given instant of time, necessarily L = L, since these quantities are just redescriptions of the total angular momentum of the system at a given time. The total angular momentum is a conserved quantity; that is, dL/dt = 0, for all time t, and this property makes the total angular momentum an important quantity for the study of the behavior of complex physical systems. For a system of n point particles, the total angular momentum relative to the origin of the reference frame (e1, e2, e3) is obtained by vector addition of that of the individual parti- cles by L = n i=1 Li, where Li is expressed by the vector cross product Li = xi×pi in terms of the vector position xi = x1ie1 +x2ie2 +x3ie3 and the vector linear momentum pi = p1ie1 + p2ie2 + p3ie3 of the particle labeled i. While angular momentum can be exchanged between interact- ing particles, the total angular momentum remains constant in time for an isolated physical system of n particles. The quantum-mechanical op- erator interpretation of such classical physical quantities is obtained by Schrödinger’s rule pi → −i∇i, = h/2π, where h is Planck’s constant. The reference frame vectors (e1, e2, e3) remain intact.
- 40. 1.1. BACKGROUND AND VIEWPOINT 13 The viewpoints of Newtonian physics and quantum physics may be contrasted in many ways. From the viewpoint of angular momentum, for example, the classical angular momentum L = x × p is associated with the actual motion of a particle, while in quantum theory, as noted above, it becomes the generator of rotations of the state vector that describes the properties of the object called “particle” (active viewpoint), or, in the viewpoint we have advanced, the generator of rotations of the coordinate frame (passive interpretation) that gives a redescription of the state vector that describes the properties of the particle. Neither the active nor passive viewpoint implies an actual “motion;” it is more in accord with the concept of congruence in Euclidean geometry. An even more contrasting feature is that the object called the electron is regarded as a “point particle,” and whereas points in Euclidean space have no intrinsic properties, this point object, the electron, has an “internal” angular-momentum-like property called “spin,” as we next describe. Internal degrees of freedom The discovery of objects having internal degrees of freedom going be- yond motions in Newtonian space-time led to the introduction of the concept of the spin of a object (see Pauli [140]), even though the object may exhibit no measurable spatial extension. For example, the electron, the carrier of electrical conduction in metals under the influence of an electric field, was already recognized in 1897 (see Ref. [167]) to be one of the principal constituents of atoms, requires an intrinsic property called spin, to account for the observed properties of the emission and absorp- tion of radiation by atoms. A single electron is characterized not only as being a point structure with a fixed charge and mass, but also by having a fixed spin 1 2 . As a structure at a single point, the electron is assigned a position x = (x1, x2, x3) ∈ R3 at time t, which in quantum theory become the domain of definition of its space-time wavefunction ψ(x, t); as a structure with spin, it is assigned a “state vector” |1 2 belonging to a Hilbert space H1 2 of dimension 2, which is assumed to be spanned by a pair of orthonormal basis vectors that are written |1 2 , 1 2 and |1 2 , −1 2 in the Dirac bra-ket notation, with orthonormality relations 1 2 , µ | 1 2 , ν = δµ,ν, the inner product being unspecified beyond these properties. The gen- eral state vector of the electron in the spin space H1 2 is given by the ket-vector |1 2 = α|1 2 , 1 2 + β|1 2 , −1 2 , where α and β are complex coeffi- cients. The corresponding bra-vector in the dual space is then given by 1 2 | = α∗1 2 , 1 2 | + β∗1 2 , −1 2 | with inner product 1 2 |1 2 = |α|2 + |β|2. In this conceptualization of a point particle with intrinsic spin 1 2 , its full state vector belongs to the tensor product space H ⊗H1 2 , which has state vec- tors of the form Ψ1 2 = ψ ⊗ |1 2 , where ψ is a wavefunction in the usual sense, depending only on the spatial coordinates. The action of a frame
- 41. 14 CHAPTER 1. QUANTUM ANGULAR MOMENTUM rotation R : (e1, e2, e3) → (e 1, e 2, e 3), where e i = 3 j=1 Rijej, i = 1, 2, 3, is not only to effect the transformation to new spatial coordinates of the particles given by (x1, x2, x3) → (x 1, x 2, x 3), where x i = 3 j=1 Rijxj, i = 1, 2, 3, but also to effect the transformation of the spin-space ket-vector basis of H1 2 by |1 2 , 1 2 → |1 2 , 1 2 = u11(R)|1 2 , 1 2 + u21(R)|1 2 , −1 2 , (1.22) |1 2 , −1 2 → |1 2 , −1 2 = u12(R)|1 2 , 1 2 + u22(R)|1 2 , −1 2 . The action of the frame rotation R on the components (α, β) of the ket-vector |1 2 = α|1 2 , 1 2 + β|1 2 , −1 2 giving a general spin state is given by α → α = u∗ 11(R)α + u∗ 21(R)β, (1.23) β → β = u∗ 12(R)α + u∗ 22(R)β, where U(R) = u11(R) u12(R) u21(R) u22(R) ∈ SU(2) (1.24) is an element of the group of 2 × 2 unitary unimodular matrices SU(2) that depends on the frame rotation matrix R ∈ SO(3, R). This pair of transformations of basis vectors and components leaves invariant the relation α|1 2 , 1 2 + β|1 2 , −1 2 = α |1 2 , 1 2 + β |1 2 , 1 2 , (1.25) as required for a redescription of the state vector of the spin by a rotation of the reference frame. (For a history of the discovery of the electron and its properties, see Whittaker [180] and The Physical Review: The First Hundred Years [167]. Generally, in the one-particle case above, we have in mind a single particle with an internal spin that is a constituent of a larger isolated physical system, with all constituents described in R3, that provides the interactions that act on the “marked” single particle. We often use Newtonian metaphors to identify this object, but its properties are to be inferred from the those of its quantum-mechanical state vector Ψ = ψ⊗|1 2 . Newtonian space-time still occurs in the domain of definition of the wavefunction ψ. It is also possible to assign “internal coordinates” to the new degrees of freedom of the internal space that describes the spin, as discussed below in (1.58). It is somewhat remarkable that the symmetry group SU(2) actually can be used to unify the action of frame rotations R ∈ SO(3, R) on both points in R3 and on points used to describe the internal degrees of freedom associated with spin. From this viewpoint, the group SU(2) is the basic group for the study of angular momentum of complex systems with spin. We show this by using the method of Cartan [35].
- 42. 1.1. BACKGROUND AND VIEWPOINT 15 The method of Cartan In the Cartan method (see also Wigner [181] and Biedenharn and Louck [21]), a point x = (x1, x2, x3) ∈ R3 is presented as the entries in a 2 × 2 traceless, Hermitian matrix of the form: H(x) = x3 x1 − ix2 x1 + ix2 −x3 = 3 i=1 xiσi, (1.26) where the σi, i = 1, 2, 3, are the Pauli matrices: σ1 = 0 1 1 0 , σ2 = 0 −i i 0 , σ3 = 1 0 0 −1 . (1.27) Thus, each point x ∈ R3 is mapped to a traceless Hermitian matrix: x → H(x). Conversely, given a traceless Hermitian matrix H, it is mapped to a point x ∈ R3 by the rule xi = 1 2Tr(σiH), where TrA denotes the trace of a matrix A. In obtaining this result, we use the following multiplication rules of the Pauli matrices: σ2 i = I2 = σ0, i = 1, 2, 3; σ1σ2 = iσ3, σ2σ3 = iσ1, σ3σ1 = iσ2. (1.28) Thus, with these conventions for the placement of the components xi into matrices, the set of points R3 is bijective with the set of matrices: H = {H | H is a 2 × 2 traceless Hermitian matrix} . (1.29) Traceless Hermitian matrices are mapped into traceless Hermitian matrices by unitary similarity transformations. Moreover, the deter- minant of a Hermitian matrix is invariant under such transformations. Accordingly, it must be the case that, for each U ∈ SU(2), the transfor- mation x → x defined by U x3 x1 − ix2 x1 + ix2 −x3 U† = x 3 x 1 − ix 2 x 1 + ix 2 −x 3 , (1.30) is a real orthogonal transformation, since the determinant of this trans- formation is −(x2 1 + x2 2 + x2 3) = −(x 2 1 + x 2 2 + x 2 3 ). The explicit transfor- mation may be worked out to be x i = 3 j=1 Rij(U)xj, Rij(U) = 1 2 Tr(σiUσjU† ). (1.31) The proof that det R(U) = 1 takes more effort, but is correct. We also note the easily proved property (R(U))T = R(U†) = R(U−1) =
- 43. 16 CHAPTER 1. QUANTUM ANGULAR MOMENTUM (R(U))−1. Also, because we have a homomorphism of groups, the mul- tiplication property R(U)R(U) = R(UU) holds for all pairs U, U ∈ SU(2). Given R ∈ SO(3, R), there are exactly two solutions U(R) and −U(R) that solve relation (1.30). The solution U(R) is given by U(R) = α0 − iα3 −iα1 − α2 −iα1 + α2 α0 + iα3 = α0σ0 − iα · σ, (1.32) where the parameters (α0, α) = (α0, α1, α2, α3) are given in terms of the elements Rij of R by α0 = (1 + TrR)/d, α1 = (R32 − R23)/d, α2 = (R13 − R31)/d, α3 = (R21 − R12)/d, d = 2 √ 1 + TrR, for TrR = −1; (1.33) α0 = 0, α1 = (1 + R11)/2, α2 = s2 (1 + R22)/2, α3 = s3 (1 + R33)/2, for TrR = −1, (1.34) where, for TrR = −1, we have the following definitions: s2 = sign(R12), s3 = sign(R13), where sign(x) denotes the sign of a real number x with sign(0) = +. The condition that det R = 1 requires that the parame- ters (α0, α) constitute a point on the unit sphere S3 ⊂ R4 in Cartesian 4−space; that is, α2 0 + α2 1 + α2 2 + α2 3 = 1. Relation (1.32) then gives det U(R) = 1; that is, U(R) is unimodular for all R ∈ SO(3, R). A real orthogonal matrix R with TrR = −1 is also a symmetric matrix, hence, R2 = I3, and the unitary unimodular matrix U is traceless and antihermitian; that is, TrU = 0 and U† = −U. It is also useful to give the expression for R(U) for U = U(α0, α), where (α0, α) ∈ S3 : R(α0, α) = (1.35) α2 0 + α2 1 − α2 2 − α2 3 2α1α2 − 2α0α3 2α1α3 + 2α0α2 2α1α2 + 2α0α3 α2 0 + α2 2 − α2 3 − α2 1 2α2α3 − 2α0α1 2α1α3 − 2α0α2 2α2α3 + 2α0α1 α2 0 + α2 3 − α2 1 − α2 2 . The parametrizations (1.32) and (1.35) of matrices U ∈ SU(2) and R ∈ SO(3, R) in terms of points on the unit sphere S3 are very useful for obtaining other parametrizations of these groups simply by parametriz- ing the points on the unit sphere S3, as we give below.
- 44. 1.1. BACKGROUND AND VIEWPOINT 17 The multiplication of two unitary unimodular matrices in terms of these parameters is expressed by the quaternionic multiplication rule U(α0, α)U(α 0, α ) = U(α 0, α ), (1.36) (α0, α)(α 0, α ) = (α 0, α ) = α0α 0 − α · α , α0α + α 0α + α × α . (1.37) These same relations hold, of course, upon replacing U by R. We also note the following results for unitary matrices. The group U(2) of unitary matrices is given in terms of the group of unitary uni- modular matrices SU(2) by U(2) = Uφ = eiφ U | U ∈ SU(2), 0 ≤ φ 2π . (1.38) The 2−to−1 homomorphism (1.31) of SU(2) to SO(3, R) now defines an ∞−to−1 homomorphism of U(2) to SO(3, R) : Every unitary matrix Uφ = eiφU, 0 ≤ φ 2π, corresponds to the same proper orthogonal matrix R(U). 1.1.4 Unitary frame rotations Using the 2−to−1 homomorphism Rij(U) = 1 2Tr(σiUσjU†) between the groups SU(2) and SO(3, R), we can now describe in greater detail the redescription of a particle with spin 1/2 that is effected by a frame rota- tion, where we note that (R(U))T = R(U†). Using Cartan’s method, the entire process can be described in terms of the unitary unimodular group SU(2) and its action on the various sets that enter into the description of the particle and its quantum-mechanical states, which we summarize as follows: 1. Rotation action of SU(2) on the set F of right-handed frames: (e1, e2, e3) → (e 1, e 2, e 3), (1.39) e 3 e 1 − ie 2 e 1 + ie 2 −e 3 = U e3 e1 − ie2 e1 + ie2 −e3 U† , (1.40) e i = 3 j=1 Rij(U)ej, i = 1, 2, 3. (1.41)
- 45. 18 CHAPTER 1. QUANTUM ANGULAR MOMENTUM 2. Redescription of coordinates effected by a frame rotation under the action of SU(2) : (x1, x2, x3) → (x 1, x 2, x 3), (1.42) x 3 x 1 − ix 2 x 1 + ix 2 −x 3 = U x3 x1 − ix2 x1 + ix2 −x3 U† , (1.43) x i = 3 j=1 Rij(U)xj, i = 1, 2, 3. (1.44) 3. Action of each U ∈ SU(2) on the Hilbert space H of spatial wave- functions in the tensor product space H ⊗ H1 2 : TU ψ = ψ , (TU ψ)(x) = ψ (x ), each ψ ∈ H, (1.45) x 1 x 2 x 3 = (R(U))T x1 x2 x3 . (1.46) The coordinate transformation (1.44) between column matrices may be expressed in terms of group action as (x 1, x 2, x 3) = AU (x1, x2, x3), where AU xi = j Rij(U)xj, each U ∈ SU(2). The group action (1.45) in function space is given by (1.45)-(1.46) in which the co- ordinate transformation uses (R(U))T . This is not an error; it is dictated by the requirement that the action Ag, g ∈ G, of a group G on a set Y is to satisfy, by definition, for each pair of group el- ements g, g ∈ G, the product rule Ag (Agy) = Aggy, each y ∈ Y. This rule may be verified directly for AU acting in the set of all coordinate points in accordance with (1.44), and for TU acting in the set of all functions in the Hilbert space H in accordance with (1.45)-(1.46). 4. Action of each U ∈ SU(2) on the basis of the spin space H1 2 : |1 2 , 1 2 → |1 2 , 1 2 = SU |1 2 , 1 2 = u11|1 2 , 1 2 + u21|1 2 , −1 2 , (1.47) |1 2 , −1 2 → |1 2 , −1 2 = SU |1 2 , −1 2 = u12|1 2 , 1 2 + u22|1 2 , −1 2 . (1.48) Again, the product action rule TU (TU |1 2 , ±1 2 ) = TUU |1 2 , ±1 2 can be verified.
- 46. 1.1. BACKGROUND AND VIEWPOINT 19 5. Action of each U ∈ SU(2) on a state vector Ψ ∈ H ⊗ H1 2 : TU = TU ⊗ SU : H ⊗ H1 2 → H ⊗ H1 2 , TU Ψ = TU ψ ⊗ SU |1 2 , (1.49) |1 2 = α|1 2 , 1 2 + β|1 2 , −1 2 . 6. Action of each U ∈ SU(2) on a state vector Ψ ∈ H ⊗ Hj : TU = TU ⊗ SU : H ⊗ Hj → H ⊗ Hj, TU Ψ = TU ψ ⊗ SU |j, (1.50) |j = m αjm|jm, αjm ∈ C, SU |j m = m Dj m m(U)|j m . (1.51) In order to summarize relations (1.50)-(1.51) in one place, we have an- ticipated from Sect. 1.2 below the following notation for an orthonormal basis of the spin space Hj of an object of internal spin j given by the standard Dirac ket notation: |j m, m = j, j − 1, . . . , −j; j ∈ {0, 1/2, 1, 3/2, . . . , }. (1.52) Under the action of SU these basis vectors undergo the transformation (1.51), where the functions Dj m m(U) give a unitary matrix represen- tation of order 2j + 1 of SU(2). The notation Dj m m(U) in this trans- formation denotes that these functions are defined over the elements uij, 1 ≤ i, j ≤ 2, of U ∈ SU(2), and not over matrices U. These func- tions are arranged into a matrix of order 2j + 1 by the convention of enumerating the rows and columns in the order m = j, j − 1, . . . , −j, as read across the columns from left-to-right and m = j, j − 1, . . . , −j, as read down the rows from top-to-bottom. The matrices Dj(U) are then a unitary matrix representation of SU(2); that is, they satisfy Dj (U )Dj (U) = Dj (U U) and Dj (U)(Dj (U))† = I2j+1, (1.53) for all pairs U, U ∈ SU(2). The group action (1.40) of SU(2) on reference frames in Cartesian space R3 assigns the unitary group SU(2) the primary role in the redescription of quantum states under the redescription of R3 by SU(2) frame rotations, which we henceforth call simply unitary frame rotations.
- 47. 20 CHAPTER 1. QUANTUM ANGULAR MOMENTUM Realizations of spin space We give two explicit realizations of spin space. In the first realization, Hj is replaced by unit column vectors C2j+1 of complex numbers (row vectors could also be used). Thus, we simply make the replacement of abstract basis vectors by unit column vectors of length 2j + 1 as given by |j m → sj m = col(0 . . . , 0, 1, 0, . . . , 0), (1.54) where the single 1 appears in row j − m + 1, m = j, j − 1, . . . , −j. Thus, we have that the tensor product space is realized by H⊗C2j+1 with state vectors given by Ψj(x) = ψj,j(x) ψj,j−1(x) . . . ψj,−j(x) = m ψj m(x)sj m. (1.55) Under the unitary rotations of frame given by (1.41), this tensor product space undergoes the transformation TU ψj,j(x) ψj,j−1(x) . . . ψj,−j(x) = Dj (U) (TU ψj,j)(x) (TU ψj,j−1)(x) . . . (TU ψj,−j)(x) = Dj (U) ψj,j(x) ψj,j−1(x) . . . ψj, −j(x) , (1.56) where the coordinate transformation (x1, x2, x2) → (x 1, x 2, x 2) is still given by x i = 3 k=1 Rik(U)xk, i = 1, 2, 3. (1.57) In the second realization of the abstract spin space, Hj is replaced by the polynomial space Pj of polynomials of degree 2j defined over two complex variables (z1, z2), and the abstract basis vector |j m is taken to be the polynomial with values given by z1, z2|j m → Pj m(z1, z2) = zj+m 1 zj−m 2 (j + m)!(j − m)! , (1.58)
- 48. 1.1. BACKGROUND AND VIEWPOINT 21 for m = j, j − 1, . . . . − j. In this formulation, under the unitary frame rotation given by (1.41), the underlying spin-space coordinates (z1, z2) undergo the redescription (z1, z2) → (z 1, z 2) given by z 1 = u11z1 + u12z2, z 2 = u21z1 + u22z2, U = (uij)1≤i≤j≤2 ∈ SU(2), (1.59) while the basis polynomials undergo the redescription given by (SU Pj m)(z1, z2) = Pj m(z 1, z 2) = m Dj m m(U)Pj m (z1, z2), (1.60) in which the new coordinates are given in column matrix form by col(z 1, z 2) = UT col(z1, z2). (Compare with (1.44) and (1.46).) The transformation coefficients in relation (1.60) are given by (see van der Waerden [171] and Ref. [21]): Dj m m(U) = (j + m)!(j − m)!(j + m)!(j − m)! (1.61) × s (u11)j+m−s(u12)m −m+s(u21)s(u22)j−m −s (j + m − s)!(m − m + s)!s!(j − m − s)! , where the summation is over all nonnegative values of s for which all factorials in the denominator are nonnegative. The column matrix formulation is less specific about the charac- ter of internal space, which is not directly accessible to measurement, and is, perhaps, to be preferred. Mathematically, the two finite vec- tor spaces of polynomials and unit vectors of length 2j + 1 described above are isomorphic, and it is no restriction to use the spinor polyno- mials Pj m(z1, z2), m = j, j − 1, . . . , −j, as basis vectors. Indeed, these polynomials have an inner product ( , ) such that (Pj m , Pj m) = δm m, one-to-one with that of the unit vectors sj m, m = j, j − 1, . . . , −j, as discussed below in Sect. 1.3.1. There are many advantages to following this approach, especially from the vantage point of combinatorics. More- over, it is well known that the transformation properties of the spinor basis functions already gives all inequivalent irreducible representations of SU(2), as given above in relation (1.61), a result that we will prove later. It is important to recognize that the use of either of these special realizations of spin space in no way restricts the general theory of spin space, since they do not introduce constraining conditions.
- 49. 22 CHAPTER 1. QUANTUM ANGULAR MOMENTUM Many-particle systems The above description of a single particle in R3 of spin j identified as a constituent of an isolated physical system is easily generalized to a physical systems S in which n identical particles in R3, each of spin j, are constituents, all of which may be interacting. The k−th particle is assigned the points x(k) = (x1k, x2k, x3k) ∈ R3, k = 1, 2, . . . , n relative to the Newtonian frame (e1, e2, e3) used for the description of a single particle above. We use spinor basis functions for the description of the spin states. It is convenient now to present the spatial coordinates and spin coordinates as the following 3 × n matrix X and the 2 × n matrix Z defined by X = x11 x12 · · · x1n x21 x22 · · · x2n x31 x32 · · · x3n , (1.62) Z = z11 z12 · · · z1n z21 z22 · · · z2n , (1.63) in which column k of these matrices gives the spatial coordinates x(k) = (x1k, x2k, x3k) and the spin coordinates z(k) = (z1k, z2k) of the k−th particle. These coordinates then become the domain of definition of the state vector Ψ ∈ H of the physical system with values given by Ψ(X, Z). The Hilbert space H is the n−fold tensor product space of the tensor product spaces H(k) ⊗ S (k) j of each of the particles of spin j, which we present in the form H = (H(1) ⊗ H(2) ⊗ · · · ⊗ H(n) ) ⊗ (S (1) j ⊗ S (2) j ⊗ · · · ⊗ S (n) j ). (1.64) The value of each Ψ ∈ H is denoted Ψ(X, Z) for each point X ∈ R3 ⊗ R3 ⊗ · · · ⊗ R3 and each point Z ∈ C2 ⊗ C2 ⊗ · · · ⊗ C2. The properties of this n−particle system under the action of SU(2) are summarized as follows: 1. Action of each U ∈ SU(2) on the reference frame: This action is still given by (e1, e2, e3) → (e 1, e 2, e 3), (1.65) e 3 e 1 − ie 2 e 1 + ie 2 −e 3 = U e3 e1 − ie2 e1 + ie2 −e3 U† (1.66)
- 50. 1.1. BACKGROUND AND VIEWPOINT 23 e i = 3 l=1 Ril(U)el, i = 1, 2, 3. (1.67) 2. Action of each U ∈ SU(2) on the state vector space H : The state vector Ψ ∈ H undergoes the transformation Ψ → TU Ψ given by (TU Ψ)(X, Z) = Ψ (R(U))T X, UT Z . (1.68) This formulation not only makes transparent the left action of the group SU(2), but also clearly invites the possibility of further transformations of the state vector by using right transformations X → XY and Z → ZY by an arbitrary matrix Y of order n. Thus, for example, if we choose Y = Pπ to be a permutation matrix, such transformations permute the spatial and spin coordinates of the particles. Here Pπ is any one of the n! matrices obtained by a permutation of the columns of the identity matrix In (permutation matrices). Moreover, there is also the possibility of doing right transformations by choosing Y ∈ U(n), the group of n×n unitary matrices: The occurrence of the general unitary group U(n) in the classification of the state vectors of n−particle systems is always implicit. The theory of angular momentum arises naturally from the above presentation of unitary frame rotations. Let us first show this for the case of a single particle with spin j for which we have X = x1 x2 x3 , Z = z1 z2 . (1.69) The generators of a group are defined in terms of abelian subgroups. For the group SU(2), the generator of the subgroup SU(2; φ, n) ⊂ SU(2) defined by SU(2; φ, n) = {U(φ, n) = exp(−iφn · σ/2) | 0 ≤ φ 2π} (1.70) of a frame rotation about a fixed direction n is defined by 1 2 n · σ = i d dφ U(φ, n) φ=0 . (1.71) The generator n · J = n1J1 + n2J2 + n3J3 of the transformation of the state vector Ψ → TU(φ,n) Ψ corresponding to the redescription of the state vector under the unitary frame transformation (1.67) is defined in analogy to (1.71) by ((n · J)Ψ)(X, Z) = i d dφ (TU(φ,n)Ψ)(X, Z) φ=0
- 51. 24 CHAPTER 1. QUANTUM ANGULAR MOMENTUM = i d dφ Ψ(X(φ, n), Z(φ, n)) φ=0 , (1.72) X(φ, n) = (R(φ, n))T X, Z(φ, n) = (U(φ, n))T Z. (1.73) The unitary unimodular matrix U(φ, n) is given explicitly by U(φ, n) = exp(−iφn · σ/2) (1.74) = cos(φ/2) − in3 sin(φ/2) (−in1 − n2) sin(φ/2) (−in1 + n2) sin(φ/2) cos(φ/2) + in3 sin(φ/2) . The real, proper, orthogonal matrix R(φ, n) = R(U(φ, n)) is obtained by setting α0 = cos(φ/2) and α = n sin(φ/2) in relation (1.32). We now carry out the differentiation d/dφ in (1.72), using the chain rule from calculus, and the relations i d dφ (U(φ, n))T |φ=0 = 1 2 n3 n1 + in2 n1 − in2 −n3 , (1.75) i d dφ (R(φ, n))T |φ=0 = i 0 n3 −n2 −n3 0 n1 n2 −n1 0 . (1.76) We thus obtain the following results: The generator n · J = n1J1 + n2J2 + n3J3, (1.77) which acts on functions Ψ ∈ H, is given in terms of the differential operator n· J , which acts in the set of values {Ψ(X, Z) | X ∈ R3; Z ∈ C2} by the following relations: ((n · J)Ψ)(X, Z) = (n · J )Ψ(X, Z), (1.78) J = L + S, J = L + S, (1.79) where L and S are the differential operators defined by L1 = −i(x2∂/∂x3 − x3∂/∂x2), L2 = −i(x3∂/∂x1 − x1∂/∂x3), (1.80) L3 = −i(x1∂/∂x2 − x2∂/∂x2); S1 = (z1∂/∂z2 + z2∂/∂z1)/2, S2 = −i(z1∂/∂z2 − z2∂/∂z1)/2, (1.81) S1 = (z1∂/∂z1 − z2∂/∂z2)/2.
- 52. 1.1. BACKGROUND AND VIEWPOINT 25 Here we use the notation (TΨ)(q) = T Ψ(q), (1.82) where Ψ is an element of some function space F on which an operator T acts, T : Ψ → Ψ = TΨ; q is a point in the domain of definition Q of the functions in F; and T is the differential operator, acting in the space of values of the functions in F, that defines the values of (TΨ)(q). In the context at hand, such a relation applies to the operator pairs (Ji, Ji), (Li, Li), (Si, Si), where these are the generator pairs correspond- ing to n = ei, i = 1, 2, 3. Similar relations apply to linear combinations of these linear operator. In particular, we have that the matrix generator n · σ/2 of the unitary matrix representation exp (−iφn · σ/2) of order 2 of SU(2) is realized in the space of functions by (TU(φ,n)Ψ)(X, Z) = e−iφn·J Ψ (X, Z) = e−iφn·J Ψ(X, Z). (1.83) Although the differential operators Li and Si are quite different in structural form, they satisfy identical commutation relations, where the commutator [A, B] of two operator A and B acting in a Hilbert space is defined by [A, B] = AB − BA : [L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2, (1.84) [S1, S2] = iS3, [S2, S3] = iS1, [S3, S1] = iS2, (1.85) [Li, Sj] = 0, i, j = 1, 2, 3. (1.86) The operators Li and Si, with action in the space of state vectors, satisfy, of course, these same commutation relations. The important observation is: There can be many realizations of these commutation relations by quite different operators, but as we shall see in Sect. 1.2 below, under certain assumptions, they all give rise to the same matrix realizations. Let us next relate the mathematical quantities given above to the physical quantity called angular momentum. The classical angular mo- mentum L of a point particle relative to a frame (e1, e2, e3) is defined by the cross product L = x×p, where p is the linear momentum of the particle. The quantum angular momentum L is obtained from the clas- sical quantity by making the replacement p → −i∇, where is the Planck constant h/2π, and ∇ is the Laplace operator with components (∂/∂x1, ∂/∂x2, ∂/∂x3). Thus, the components of the dimensionless an- gular momentum operators L/ = −ix×∇ are exactly the generators of rotations given by relations (1.80). There is, of course, no such rule for obtaining the spin operators (1.81). As remarked earlier, spin angular momentum can either be incorporated into the quantum theory in terms of an unspecified abstract finite-dimensional Hilbert space or by using realizations of this space that do not restrict the mathematical content of the theory, as we have done above.
- 53. 26 CHAPTER 1. QUANTUM ANGULAR MOMENTUM The generalizations of the above one-particle results to n particles is immediate: One simply copies relations (1.79)-(1.81) n times, adjoining an extra index k to designate the k−th particle with spin j. The in- finitesimal operators associated with each particle are clearly additive, so we have that the generator of the transformation associated with the redescription of the state vector resulting from a unitary frame rotation by angle φ about direction n is given by n · J and have the properties summarized by the following relations: J = L + S, (1.87) J = n k=1 J (k), (1.88) J (k) = L(k) + S(k); (1.89) L(k) = −ix(k) × ∇(k) , x(k) = 3 i=1 xik ei ; (1.90) S (k) 1 = (z1k∂/∂z2k + z2k∂/∂z1k)/2, S (k) 2 = −i(z1k∂/∂z2k − z2k∂/∂z1k)/2, (1.91) S (k) 3 = (z1k∂/∂z1k − z2k∂/∂z2k)/2. These operators are just n replicas of relations (1.79)-(1.81) with com- mutation relations that are k replicas of relations (1.84)-(1.86) with su- perscript (k) adjoined, where now operators having distinct values of particle index k all mutually commute. Relation (1.78) also holds, of course, in the interpretation as operators acting in the state space: (n · J(k) )Ψ (X, Z) = (n · J (k))Ψ(X, Z), (1.92) with similar relations for n · L(k) and n · S(k) . In particular, we still have the commutation relations for the total angular momentum: [J1, J2] = iJ3, [J1, J2] = iJ3, [J1, J2] = iJ3 (1.93) and the associated transformations of the state vectors: (TU(φ,n)Ψ)(X, Z) = e−iφn·J Ψ (X, Z) = e−iφn· J Ψ(X, Z). (1.94) This relatively simple n−particle physical system embodies the full con- tent of angular momentum theory. The simplicity of these additive re- lations does not reveal the intricacies of the theory. Remark. Each U ∈ SU(2) effects a rotation R(U) of the reference frame of a physical system in R3 as given by relations (1.65)-(1.67).
- 54. 1.1. BACKGROUND AND VIEWPOINT 27 It may seem mysterious that a unitary rotation of the reference frame should effect also a unitary transformation of the internal coordinates that describe spin. This is so because a redescription of the quantum state corresponding to the reassignment of coordinates X = R(U)X, Z = UZ, (1.95) as effected by a unitary frame rotation, treats the spin coordinates on a par with the spatial coordinates. It is this relation that defines the action of each U ∈ SU(2) to be that of an operator TU acting in the Hilbert space H of states by the rule (TU Ψ)(X, Z) = Ψ (R(U))T X, UT Z . We note that we have the option of redefining the redescription transforma- tion (1.95) by replacing U by U∗, so that the action of the operator TU on state vectors would be (TU Ψ)(X, Z) = Ψ (R(U∗))T X, U†Z . But this would lead to replacing the matrices Dj(U) defined in relation (1.16) by Dj(U∗). This is an acceptable procedure, but we have elected to use the redescription given by (1.95). The system described above leads us below to investigate abstractly all possible realizations of operators satisfying the commutation rela- tions (1.93), subject to certain conditions, together with the problem of adding two or more angular momenta, each of which has three compo- nents that satisfy these commutation relations, and the components of the separate angular momenta mutually commute. This is the problem of constructing representations of two or more copies of the Lie algebra of a group. The state vector Ψ ∈ H of every physical system in the space R3, with or without spin, possesses SU(2) unitary symmetry; that is, for each U ∈ SU(2), we have for each Ψ ∈ H that TU Ψ = Ψ ∈ H, since this transformation in the space of states available to the system is just the redescription of H corresponding to the redescription of the physical system induced by a unitary frame rotation. Ideally, the state vector space is a separable Hilbert space, and TU , each U ∈ SU(2), is a unitary operator with respect to the inner product on H. This is equivalent to requiring that the three generators Ji, i = 1, 2, 3, corresponding to rota- tions about the three basis vector ei, i = 1, 2, 3, are Hermitian operators. Such a separable Hilbert space H is said to possess SU(2) symmetry. A separable Hilbert space having SU(2) symmetry can always be decomposed into a direct sum of various subspaces HSU(2) on which the action of TU , each U ∈ SU(2), is irreducible, which means that TU : HSU(2) → HSU(2), each U ∈ SU(2), and there exists no subspace of HSU(2) with this property. The invariance of such subspaces HSU(2) under the unitary action of each TU is, however, insufficient, in general, to determine the decomposition of H into its irreducible SU(2) subspaces. For an n−particle system in R3, there are 3n − 3 degrees of spatial
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- 56. [234] and see how soon the mechanic thinks he will have it ready.” She returned in a couple of minutes and found the others already seated in the constable’s sedan. Mary Louise was glad to find that the officer had put Margaret Detweiler in front with him, not beside the tough young man with his huge guardian in the rear seat. She squeezed in next to Margaret, and the car started. “The mechanic is going to drive my car to your place in about half an hour,” announced Mary Louise. “And then we’ll start for Philadelphia.” “Fine!” exclaimed the constable. “That’ll give you girls a chance to get warm. And maybe have a cup of coffee.” “It’s marvelous coffee,” commented Mary Louise. “It just about saved my life.” Not another word was said about the crimes or the secret band. Margaret Detweiler was introduced to Mrs. Hodge as a friend of Mary Louise’s from Riverside, and the two girls spent a pleasant half hour in the constable’s home, sipping their freshly made coffee and looking at the children’s Christmas toys. The constable, who had taken the young thug away, returned just as Mary Louise’s hired car drove up to the door. Mary Louise jumped up and reached for her coat. “Wait a minute!” cautioned the constable. “Company’s comin’ here to see you, Miss Gay! I just met somebody askin’ for you at the hotel.... So don’t be in too much of a rush!”
- 57. [235] From the obvious twinkle in the man’s eyes, Mary Louise believed that Max Miller must have driven down to Philadelphia again and, missing her there, had naturally traced her to Center Square. But at that same moment a yellow taxi stopped at the constable’s gate, thereby dispelling any such illusion. Max would never ride in a taxicab on his limited allowance! The door of the cab opened, and a tall, handsome man stepped out, paid the driver, and dismissed the cab. It was Mary Louise’s father. Flinging open the door, the girl shouted at him in delight, so loud that Mr. Gay heard her in spite of the noise of the departing cab. In another moment he entered the open door of the house and held Mary Louise tightly in his arms. “Mary Lou!” he cried in delight. “Are you sure you’re all right?” “I’m fine,” she replied, ushering him into the constable’s house. “Merry Christmas, Daddy!” “The same to you, dear.” He gazed at her fondly. “I believe it will be—now. You certainly look happy, Daughter.” “I am, Daddy. These people have treated me royally!” She turned around and introduced her father to Mrs. Hodge and the children, for he had already met the constable. “And, oh, Dad, here is Margaret Detweiler,” she added. “You remember her, don’t you?” “I certainly do,” replied Mr. Gay, extending his hand cordially. “My, but your grandparents are going to be glad to see you, Margaret!”
- 58. [236] [237] The girl blushed and looked down at the floor in embarrassment. Wisely, Mr. Gay asked no questions. “I have all the stolen valuables, Dad,” continued Mary Louise. “Every single thing that was taken from Stoddard House, and even the money!” Mr. Gay gazed at his daughter in speechless admiration: she had excelled his fondest hopes! “Mary Lou, that’s—wonderful!” he said after a moment.... “I have good news too. I caught your thieves. Seven of ’em. They are in a Baltimore jail now.” Both girls exclaimed aloud in amazement and delight. Margaret Detweiler started forward and clutched the detective’s arm. “It’s really true, Mr. Gay?” she demanded breathlessly. “Mrs. Ferguson—is she in jail too?” “Locked up without any chance of getting out on bail!” he said authoritatively. “Oh, I’m so glad!” murmured the girl thankfully. “Now we’ll be able to take the valuables right back to their owners at Stoddard House, Constable Hodge,” announced Mary Louise. “I’m not afraid to carry them, with Dad beside me.” Mrs. Hodge brought the jewelry and the money from its hiding place and gave it all to Detective Gay. Both he and Mary Louise tried to thank the Hodges for their help and their hospitality; Mr. Gay wanted to give the constable some sort of recompense, but the good man refused. Only after a great deal of persuasion would he
- 59. [238] accept a five-dollar bill as a Christmas present for his children. “Ready, Daddy?” inquired Mary Louise as she slipped on her coat. “Just a minute,” replied her father. “I want to telephone to Mrs. Hilliard to let her know that you are safe. She’s been terribly worried, Mary Lou.... And shall I tell her that we’ll eat Christmas dinner with her at Stoddard House?” “Oh, yes! I’ve heard about the menu. There won’t be a sweller dinner anywhere in Philadelphia than at Stoddard House. But shall we be in time?” Mr. Gay consulted his watch. “It’s only a little after eleven,” he said. “We ought to make it by one o’clock.” As soon as the telephone call was completed, the three people got into the little car. Mary Louise herself took the wheel, for, as she explained, she was familiar with it by this time. “Now tell me about your experiences, Mary Lou,” urged her father, as soon as they were well under way. Mary Louise explained, for Margaret’s benefit as well as for her father’s, about deciphering the code letter and coming up to Center Square and breaking into the empty house in search of the valuables. But she made light of the coldness and desolation of the dark house and of her own hunger. She concluded with the statement that Margaret had come that morning and let her out with a key.
- 60. [239] “But how did you happen to have the key, Margaret?” demanded Mr. Gay. “I will have to tell you my whole story from the beginning,” answered the girl. There was a tragic note in her voice, which drew out her listeners’ sympathy, but neither made any comment. “Then you can decide what to do with me,” she continued. “I guess I deserve to go to prison, but when I assure you that I have never done anything wrong except under compulsion, maybe you will not be so angry with me.” “We’re not angry with you, Margaret,” Mary Louise told her. “Only terribly sorry. So please tell us everything. You remember that your grandparents have never heard anything from you since last Christmas.... So begin your story there.” “All right.... Let me see—I was working in that department store in Philadelphia, and doing pretty well, for I got commissions besides my salary on everything I sold. I started in the cheap jewelry department and was promoted to the expensive kind. Christmas brought me in a lot of business, but I guess I overworked, for I got sick the week before and had to stay home and have the doctor. I’d already spent a good deal of money on presents, and when my doctor’s bill was paid I found my salary was all gone. So I went back to the store before I should—on the twenty-third of December, I remember.” “The twenty-third of December!” repeated Mary Louise. “That was the day Mrs. Ferguson registered at the Benjamin Franklin Hotel.” “How did you know, Mary Lou?” demanded Margaret.
- 61. [240] “I went to the hotel and looked through the old register,” she explained. “But go on, Margaret. What happened then?” “I found that a ring, an expensive diamond ring, had been stolen from our department,” continued the girl. “They insisted that it was taken before I was away, but they couldn’t prove anything. Just the same, I know the store detective had his eye on me.... Well, that very day something else disappeared: a link bracelet. This time they accused me immediately.” “But why?” “I don’t know, except that I was the newest salesgirl in the department—in fact, the only girl. The store detective stepped behind my counter and leaned down to the floor. And he picked that bracelet right out of my shoe!” “How dreadful!” cried Mary Louise. “Somebody had ‘planted’ it there?” “Of course. Mrs. Ferguson had, as I later learned. But at the time I hadn’t a suspicion. She was standing right near the counter, examining some rings. When she heard me accused and told to leave the store, she stepped forward, saying that she was sorry for me. She asked me whether I had any family, and I told her they were too far away for me to go to, without any money. “‘But you’ll have trouble getting a job without a reference,’ she said. ‘So perhaps I had better help you.’” “The sly cat!” cried Mary Louise.
- 62. [241] [242] Margaret nodded. “But I didn’t know it then. I simply asked her whether she could get me a job, and she told me to come to the Benjamin Franklin Hotel that afternoon and ask for Mrs. Ferguson. “Of course, I went—I had nothing else to do. She engaged me at once as her secretary. We went out to Center Square for a few days, and I met a lot of other girls. Two daughters, two nieces, and a couple of friends. We had a good time, but I didn’t do any work, for she had two servants and a chauffeur, and I felt as if I didn’t earn my pay.” “Did she give you a salary?” asked Mary Louise. “Yes,” replied Margaret. “For the first couple of weeks. But I had to send it to my landlady in Philadelphia. After that, Mrs. Ferguson bought my clothes and paid my hotel bills, but she never gave me any cash.” “So you couldn’t get away!” observed Mr. Gay. “Exactly. Gradually I began to suspect that there was something crooked about this bunch, and then one day I found the diamond ring which had been stolen from the store: among Mrs. Ferguson’s stuff at Center Square!” “What did you do?” demanded Mary Louise. “I showed it to her and said I was going to take it right back to the store, and she stood there and laughed at me. She said it would only prove my own guilt! “The next day we all went to Washington and stayed in different hotels. Mrs. Ferguson kept me with her, but I soon saw through her tricks. Her girls were all skilled
- 63. [243] hotel thieves. She tried to teach me the business, as she called it, but I refused to learn. So she made me take charge of the stuff they stole. The girls would bring their loot to her, and she’d send me with it to Center Square. Every once in a while she would dispose of it all to a crooked dealer who asked no questions.” “Were you out at Center Square last Sunday, Margaret?” interrupted Mary Louise. “Yes. Mrs. Ferguson and I both went. We had intended to get the place ready to spend Christmas there, but for some reason, Mrs. Ferguson got scared. She said that Mary Green talked too much, and she thought we ought to clear out. She made plans to dispose of everything in Baltimore, and then we were all going to sail to Bermuda.... But why did you ask that, Mary Lou?” “Because I was in that car that drove up to the house then. I saw you and then Mrs. Ferguson. I wouldn’t have thought of its being you, only Mary Green admitted that she knew you. That made me suspicious.” “You disappeared pretty quickly!” “Rather,” laughed Mary Louise, and she told the story of being hit over the head by a rock and of catching the young man and having him arrested that very morning. “That was clever!” approved her father. “Who was he, Margaret?” “A neighborhood bum that Mrs. Ferguson employs to watch the place and keep the people away,” replied the girl.
- 64. [244] “But I’m afraid I interrupted you, Margaret,” apologized Mary Louise. “Please go on with your story.” “There isn’t much left to tell. I was too far away from home to run away, without any money, and I hadn’t a single friend I could go to. All the store people thought I was a thief, so I knew there was no use asking their help. I just kept on, from day to day, not knowing how it would ever end and never expecting to see my grandparents or my Riverside friends again. Oh, you can’t imagine how unhappy I have been!” She stopped talking, for emotion had overcome her; tears were rolling down her cheeks. Mary Louise laid her hand over Margaret’s reassuringly. “It’s all right now, isn’t it, Daddy?” she said. “We’ll take you home to your grandparents.” “But I can’t go back to them!” protested the other girl. “How can I tell them what has happened? They’d be disgraced for life.” “You can tell them you have been working for a queer woman who wouldn’t allow you to write home,” said Mr. Gay. “A woman whose mind was affected, for that is the truth. There is no doubt that Mrs. Ferguson is the victim of a diseased mind.” “Wouldn’t you ever tell on me?” questioned Margaret. “No, of course not. It was in no way your fault, child.... And now try to be happy. I think I can find you a job in Herman’s Hardware store, right in Riverside. And you can live with your grandparents. They need you.”
- 65. [245] [246] “It seems almost too good to be true,” breathed the grateful girl. Mary Louise turned to her father. “Now for your story, Dad,” she begged. “About capturing the thieves.” “I think that had better be kept till dinner time,” replied Mr. Gay. “This traffic we’re approaching will require all your attention, Mary Lou. And besides, Mrs. Hilliard will want to hear it too.”
- 66. [247] CHAPTER XVIII Conclusion Mary Louise brought the car to a stop at Stoddard House at a quarter to one. Carrying the money and the jewels in her father’s briefcase, and the other articles in the basket, she and Margaret went into the hotel to get ready for dinner while Mr. Gay returned the hired car to the garage. “I’ll notify the police that you’re found, Mary Lou,” he said. “Then I’ll call your mother. I think it will be best if she goes over to your grandparents, Margaret, and tells them about you herself. They haven’t a telephone, and I don’t like to frighten elderly people with telegrams.” Both girls nodded their approval to these suggestions and hurried into the hotel. Mrs. Hilliard was waiting for Mary Louise with open arms; she loved the young detective like a daughter. “Now, run along, girls, and get ready for dinner,” she said finally. “We are going to have one big table, instead of all the little ones in the dining room. With a tree in the center, and place cards, just like a jolly family party.” “That’s swell!” exclaimed Mary Louise. “It’ll be real Christmas after all.”
- 67. [248] “And thank you so much for the lovely handkerchiefs, dear,” added the manager. “It was sweet of you to think of me.... That reminds me, you haven’t had your presents yet.” “Put them at my place at the table,” suggested Mary Louise. “And I’ll have presents for some of the guests,” she added, with a significant glance at the briefcase and basket. When the girls returned to the first floor, after washing their faces and powdering their noses, they found Mr. Gay waiting for them. For a moment he did not see them, so intent was he in the newspaper he was reading. “Want to see the gang’s picture?” he asked when Mary Louise came to his side. “Oh yes! Please!” In spite of the fact that it was Christmas Day, a large photograph of Mrs. Ferguson and her six accomplices occupied much of the front page of this Philadelphia paper. In an inset above the picture of the crooks was Mary Louise’s smiling face! “Daddy!” cried the girl in amazement. “Are you responsible for this?” “I am,” replied her father proudly. “I want everybody to know that the credit belongs to you, Daughter.” Other guests, who had not yet read their newspapers, crowded about Mr. Gay eager for the exciting news. They all remembered Pauline Brooks, and Mary Green;
- 68. [249] several of them identified the two transients who had stolen the other things from Stoddard House. A loud gong sounded from the dining room, and Mrs. Hilliard threw open the doors. The room was beautifully decorated with greens and holly; a long table stretched out before them, covered with a lovely lace cloth and bearing a small Christmas tree as its centerpiece. Bright red ribbons had been stretched from the tree to each guest’s place, adding brilliancy to the spectacle. “Hello, Mary Louise!” said a voice behind the young detective, and, turning around, Mary Louise saw Mrs. Weinberger behind her. “Merry Christmas, Mrs. Weinberger!” she replied. “It’s nice to see you back here.” “I’ve come back to stay,” announced the older woman. “I got lonely at the Bellevue. And Mrs. Macgregor is here too, for Christmas dinner.” It was a happy group who finally found their places around the beautiful table and sat down. Mrs. Hilliard was at one end, and Miss Stoddard was honored with the seat at the other end. Mr. Gay was the only man present, but he did not seem in the least embarrassed. Mary Louise found her pile of presents at her place, and Margaret Detweiler discovered a bunch of violets and a box of candy at hers. Even in his haste, Mr. Gay had remembered the lonely girl. The guests ate their oyster cocktails and their mushroom soup before any formal announcement concerning the valuables was made. Then Mrs. Hilliard rose from her chair.
- 69. [250] “As you all know from the papers, our criminals have been caught by Mary Louise Gay and her father, and are now in prison. But even better news than that is coming. I’ll introduce Mr. Gay, whom some of you know already, and he’ll tell you more about it.” Everybody clapped as the famous detective stood up. “I’m not going to make a speech,” he said, “and keep you waiting for the turkey we’re all looking forward to. I just thought that maybe some of you would enjoy this wonderful dinner even more if you knew that you are going to get everything back again which was stolen. My daughter found all the valuables and the money this morning in Mrs. Ferguson’s house at Center Square, and she will now return them to their rightful owners.” As the newspaper had not mentioned anything about the stolen goods, the guests were not prepared for this pleasant surprise. A loud burst of applause greeted Mary Louise as she smilingly rose to her feet and opened the briefcase and drew out the basket from under the table where she had hidden it. “I’ll begin at the beginning,” she said. “With the vase and the silverware belonging to Stoddard House.” She carried these articles to Mrs. Hilliard, amid appreciative hand-clapping. “Next, Miss Granger’s picture and her fifty dollars,” she continued. Tears actually came to the artist’s eyes as she took the painting from Mary Louise’s hands. “You keep the fifty dollars, Miss Gay,” she said. “My picture is what I care for most.”
- 70. [251] [252] “No, Miss Granger, no, thank you,” replied the girl solemnly. “I am being paid a salary for my work by Mrs. Hilliard, but I can’t accept rewards for doing my duty.” She picked up the watches next: Mrs. Weinberger’s and Mrs. Hilliard’s. The Walder girls would get theirs when they returned from their holidays. “And, last of all, Mrs. Macgregor’s diamond earrings and her five hundred dollars,” she concluded, restoring the jewelry and the bills to the delighted woman. “I believe that is all, for I am wearing my own wrist-watch, and I have my purse with its five dollars contents.” Loud cheering accompanied the applause which followed. When it had at last quieted down, both Mrs. Weinberger and Mrs. Macgregor tried in vain to give Mary Louise a reward, but she remained firm in her refusal. Then the turkeys were brought to the dining room, and everything else was temporarily forgotten in the enjoyment of Christmas dinner. When it was all over, Mr. Gay told Mary Louise to pack her clothing and her presents while he returned the remaining valuables to the Ritz and to the police. “For I hope we can make the three-thirty train,” he explained. “But with that change at the Junction, we’d have to wait all night, shouldn’t we, Daddy?” inquired Mary Louise. Anxious as she was to get back to Riverside, she had no desire to spend the night in a cheerless railway station. “No,” replied her father. “Because there’s going to be a surprise waiting for you at the Junction.” “Max and Norman?” guessed Mary Louise instantly. “You mean that they’ll drive down for us?”
- 71. [253] Mr. Gay nodded. “That isn’t all,” he said. Mary Louise did not guess the rest of the answer until the train pulled into the Junction shortly after eight o’clock that night. Then a war whoop that could come from no one else but her small brother greeted her ears, and she knew that her mother must be there too. Yes, and there was her chum, Jane Patterson, grinning at her from the boys’ car! And her little dog, Silky! In another minute Mary Louise was clasping her arms around Mrs. Gay and hugging Freckles and Jane and Silky all at once. Max, at her side, had to be content with pressing her arm affectionately. Questions, Christmas greetings, words of joy and congratulation poured so fast upon Mary Louise’s ears that she could scarcely understand them. “You’re home to stay, darling?” This from her mother. “You’ll go to the senior prom with me?” demanded Max. “You’re the most famous girl detective in the world!” shouted Norman Wilder. “You were a lemon to duck my party, but I’ll give another one just in your honor,” promised Jane. “Did you get your salary—your twenty-five bucks?” asked Freckles. Mary Louise nodded, smiling, to everything. Then she got into Max’s car beside him, with Jane and Norman in the rumble seat. Mr. Gay took the wheel of his sedan, with his wife beside him; Margaret Detweiler, who was quietly watching everything, sat behind with Freckles.
- 72. [254] The drivers of the two cars did not stop for any food on the way; they sped along as fast as they dared towards Riverside. Old Mr. and Mrs. Detweiler were waiting up for their precious granddaughter, their lost Margaret. A little before midnight the cars pulled up in front of the old couple’s home, and everybody in the party went inside for a moment. The greeting between Margaret and her grandparents was touching to see. Even Norman Wilder, who prided himself on being “hard- boiled,” admitted afterwards that the tears came to his eyes. Mrs. Gay discreetly drew her own party away, back to her home, where a feast was waiting for the travelers. This, Mary Louise felt, was her real Christmas celebration—with her family and her three dearest friends. Now she could tell her story and listen to the praises which meant so much to her. “But the best part of it all,” she concluded, “is that I’m a real professional detective at last!”
- 73. Transcriber’s Note Retained publication and copyright information from the printed exemplar (this book is public-domain in the U.S.). Obvious typographical errors were corrected without comment. Possibly intentional spelling variations were not changed. A Table of Contents and a list of the series books were prepared for the convenience of the reader. The table of characters was reformatted to accomodate variable-width devices.
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