![Preface and Prelude
We have titled this monograph “Applications of Unitary Symmetry and
Combinatorics” because it uses methods developed in the earlier volume
“Unitary Symmetry and Combinatorics,” World Scientific, 2008 (here-
after Ref. [46] is referred to as [L]). These applications are highly topical,
and come in three classes: (i) Those still fully mathematical in con-
tent that synthesize the common structure of doubly stochastic, magic
square, and alternating sign matrices by their common expansions as
linear combinations of permutation matrices; (ii) those with an associ-
ated physical significance such as the role of doubly stochastic matrices
and complete sets of commuting Hermitian operators in the probabilistic
interpretation of nonrelativistic quantum mechanics, the role of magic
squares in a generalization of the Regge magic square realization of the
domains of definition of the quantum numbers of angular momenta and
their counting formulas (Chapter 6), and the relation between alternat-
ing sign matrices and a class of Gelfand-Tsetlin patterns familiar from
the representation of irreducible representations of the unitary groups
(Chapter 7) and their counting formulas; and (iii) a physical applica-
tion to the diagonalization of the Heisenberg magnetic ring Hamiltonian,
viewed as a composite system in which the total angular momentum is
conserved.
A uniform viewpoint of rotations is adopted at the outset from [L],
based on the method of Cartan [15] (see also [6]), where it is fully defined
and discussed. It is not often made explicit in the present volume:
A unitary rotation of a composite system is its redescription under an
SU(2) unitary group frame rotation of a right-handed triad of perpendic-
ular unit vectors (e1, e2, e3) that serves as common reference system for
the description of all the constituent parts of the system.
This Preface serves three purposes: A prelude and synthesis of things
to come based on results obtained in [L], now focused strongly on the
basic structural elements and their role in bringing unity to the under-
standing of the angular momentum properties of complex systems viewed
as composite wholes; a summary of the contents by topics; and the usual
elements of style, readership, acknowledgments, etc.
vii](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-11-320.jpg)
![viii PREFACE AND PRELUDE
OVERVIEW AND SYNTHESIS OF BINARY COUPLING
THEORY
The theory of the binary coupling of n angular momenta is about the
pairwise addition of n angular momenta associated with n constituent
parts of a composite physical system and the construction of the asso-
ciated state vectors of the composite system from the SU(2) irreducible
angular momentum multiplets of the parts. Each such possible way of
effecting the addition is called a binary coupling scheme.
We set forth in the following paragraphs the underlying conceptual
basis of such binary coupling scheme. Each binary coupling scheme of
order n may be described in terms of a sequence having two types of
parts: n points ◦ ◦ · · · ◦ and n − 1 parenthesis pairs ( ), ( ), . . . , ( ).
A parenthesis pair ( ) constitutes a single part. Thus, the number of
parts in the full sequence is 2n − 1. By definition, the binary bracketing
of order 1 is ◦ itself, the binary bracketing of order 2 is (◦ ◦), the two
binary bracketings of order 3 are
(◦ ◦)
and
◦ (◦ ◦)
, . . . . In general,
we have the definition:
A binary bracketing Bn of order n ≥ 2 is any sequence in the n points ◦
and the n−1 parenthesis pairs ( ) that satisfies the two conditions: (i) It
contains a binary bracketing of order 2, and (ii) the mapping (◦ ◦) 7→ ◦
gives a binary bracketing of order n − 1.
Then, since the mapping (◦ ◦) 7→ ◦ again gives a binary bracketing for
n ≥ 3, the new binary bracketing of order n − 1 again contains a binary
bracketing of order 2. This implies that this mapping property can be
used repeatedly to reduce every binary bracketing of arbitrary order to
◦, the binary bracketing of order 1.
The appropriate mathematical concept for diagramming all such bi-
nary bracketings of order n is that of a binary tree of order n. We have
described in [L] a “bifurcation of points” build-up principle for construct-
ing the set Tn of all binary trees of order n in terms of levels (see [L,
Sect. 2.2]). This is a standard procedure found in many books on combi-
natorics. It can also be described in terms of an assembly of four basic
objects called forks that come in four types, as enumerated by
◦ ◦
• ,
@
@
(1)
◦ •
• ,
@
@
(2)
• ◦
• ,
@
@
(3)
• •
•
@
@
(4)
The • point at the bottom of these diagrams is called the root of the
fork, and the other two point are called the endpoints of the fork. The
assembly rule for forks into a binary tree of order n can be formulated
in term of the “pasting” together of forks.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-12-320.jpg)
![PREFACE AND PRELUDE xi
3. Select a single graph from the full set generated at Step 2 and
repeat the pasting process. putting aside all those having only ◦
endpoints, including no repetitions. Repeat this process for the
next set of graphs, etc. At Step h of this pasting process of basic
forks to the full collection generated at Step h−1, there is obtained
a huge multiset of graphs that includes all binary trees of order
2, 3, . . . , h + 2 — all those with ◦ endpoints. A very large number
of graphs with 1 • endpoint, 2 • endpoints, . . . is also obtained,
these being needed for the next step of the pasting process. All
binary trees of order n are included in the set of graphs generated
by the pasting process at step h = n − 2, these being the ones put
aside.
The pasting process is a very inefficient method for obtaining the
set of binary trees of order n. A more efficient method is to generate
this subset recursively by the following procedure: Suppose the set of
binary trees of order n has already been obtained, and that it contains
an members. Then, consider the new set of nan graphs obtained by
replacing a single ◦ endpoint by a • point in each of the an binary tree
graphs. In the next step, paste the • root of basic fork 1 over each
of these • endpoints, thus obtaining nan binary trees of order n + 1.
This multiset of cubic graphs of order n + 1 then contains the set of
an+1 binary trees of order n + 1. Thus, starting with fork 1 from the
basic set of four, we generate all binary trees of arbitrary order by this
pasting procedure. (This procedure works because every graph having
one • endpoint is included in the huge set generate in Items 1, 2, 3
above.)
Only binary trees having ◦ endpoints enter into the binary coupling
theory of angular momenta because each such coupling scheme must
correspond to a binary bracketing. The set of binary bracketings of order
n is one-to-one with the set of binary trees of order n, and the bijection
rule between the two sets can be formulated exactly. The cardinality of
the set Tn is the Catalan number an. We repeat, for convenience, many
examples of such binary bracketings and corresponding binary trees from
[L], and also give in this volume many more, including examples for
arbitrary n. As in [L], we call the binary bracketing corresponding to a
given binary tree the shape of the binary tree:
The shape ShT of a binary tree T ∈ Tn is defined to be the binary
bracketing of order n corresponding to T.
The concept of shape transformation is so basic that we discuss it
here in the Preface in its simplest realization: There are two binary
trees of order three, as given above. We now label the endpoints of
these binary trees by a permutation of some arbitrary set of elements
x1, x2, x3 :](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-15-320.jpg)
![PREFACE AND PRELUDE xiii
Then, there exists a shape transformation w(A, C) such that the labeled
shape ShT (xπ) is transformed to the labeled shape ShT′ (xπ′ ), for each
T ∈ Tn, π ∈ Sn, T′ ∈ Tn, π′ ∈ Sn; that is,
ShT (xπ)
w(A,C)
−→ ShT′ (xπ′ ),
where the shape transformation w(A, C) is a word in the two letters A
and C.
The content of this result will be amplified by many explicit examples.
It is a basic unifying result for the binary theory of the coupling of n
angular momenta.
The pairwise addition of n angular momenta J(i) = J1(i)e1+J2(i)e2+
J3(i)e3, i = 1, 2, . . . , n, with components referred to a common right-
handed inertial reference frame (e1, e2, e3), to a total angular momen-
tum J = J1e1 + J2e2 + J3e3 = J(1) + J(2) + · · · + J(n) is realized in
all possible ways by the standard labeling of each binary tree T ∈ Tn
as given in terms of its shape by ShT (J(π1), J(π2), . . . , J(πn)), where
π = (π1, π2, . . . , πn) is any permutation π ∈ Sn. The pairwise addition
encoded in a given binary tree for any given permutation π ∈ Sn is called
a coupling scheme. For example, the two coupling schemes encoded by
the shapes of the binary trees of order 3 given above are:
J(1) + J(2)
+ J(3)
,
J(3) +
J(2) + J(1)
.
For general n, there are n!an distinct coupling schemes for the pair-
wise addition of n angular momenta. The rule whereby we assign the n
angular momenta (J(1), J(2), . . . , J(n)) of the n constituents of a com-
posite system to the endpoints of a binary tree T ∈ Tn by the left-to-right
assignment to the corresponding points in the shape ShT of T is called
the standard rule labeling of the ◦ points (all endpoints) of the binary
tree T. In many instances, we need also to apply this standard rule label-
ing for the assignment of a permutation (J(π1), J(π2), . . . , J(πn)) of the
angular momenta of the constituent parts, as illustrated in the example
above. Caution must be exercised in the interpretation of all these pair-
wise additions of angular momenta in terms of the tensor product space
in which these various angular momenta act. This is reviewed from [L] in
Chapters 1 and 5 in the context of the applications made here. A stan-
dard labeling rule is required to define unambiguously objects such as
Wigner-Clebsch-Gordan coefficients and triangle coefficients to labeled
binary trees, objects that given numerical content to the theory.
To summarize: In the context of the binary coupling theory of angu-
lar momenta, we deal with standard labeled binary trees, their shapes,
and the transformations between shapes. Out these few simple under-
lying structural elements there emerges a theory of almost unlimited,](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-17-320.jpg)
![xiv PREFACE AND PRELUDE
but manageable, complexity: Mathematically, it is theory of relations
between 3(n − 1) − j coefficients and Racah coefficients and their cu-
bic graphs; physically, it is a theory of all possible ways to compound,
pairwise, the individual angular momenta of the n constituent parts of a
complex composite system to the total angular momentum of the system.
The implementation of the binary coupling theory of angular mo-
menta leads directly to the Dirac concept [24] of characterizing the
Hilbert space state vectors of each coupling scheme in terms of com-
plete sets of commuting Hermitian operators. This characterization is
described in detail in [L], and reviewed in the present volume in Chapters
1 and 5; it is described broadly as follows. The complete set of 2n mu-
tually commuting Hermitian operators for each coupling scheme T ∈ Tn
is given by
J2
(1), J2
(2), . . . , J2
(n), J2
, J3;
K2
T (1), K2
T (2), . . . , K2
T (n − 2).
The first line of operators consists of the n total angular momentum
squared of each of the constituent systems, together with the total angu-
lar momentum squared of the composite system and its 3−component.
The second line of operators consists of the squares of the n−2 so-called
intermediate angular momenta, KT (i) = (KT,1(i), KT,2(i), KT,3(i)), i =
1, 2, . . . , n − 2. There is a distinct set of such intermediate angular mo-
menta associated with each binary tree T ∈ Tn, where each KT (i) is a
0 − 1 linear combination of the constituent angular momenta J(i), i =
1, 2, . . . , n, in which the 0 and 1 coefficients are uniquely determined by
the shape of the binary tree T ∈ Tn. Thus, each standard labeled bi-
nary coupling scheme has associated with it a unique complete set of 2n
mutually commuting Hermitian operators, as given generally above. For
example, the complete sets of mutually commuting Hermitian operators
associated with the labeled binary trees T and T′ of order 3 given above
are the following, respectively:
scheme T : J2
(1), J2
(2), J2
(3), J2
, J(3); K2
T (1) =
J(1) + J(2)
2
.
scheme T′ : J2
(1), J2
(2), J2
(3), J2
, J(3); K2
T′ (1) =
J(2) + J(1)
2
.
There is a set of simultaneous eigenvectors associated with each com-
plete set of mutually commuting Hermitian operators defined above for
each binary tree T ∈ Tn of labeled shape ShT (J(π1), J(π2), . . . , J(πn)).
It is convenient now to denote this shape by ShT (jπ), where we use the
angular momentum quantum number ji in place of the angular momen-
tum operator J(i) in the labeled shape. These are the quantum numbers
ji associated with the eigenvalue ji(ji +1) of the squared “total” angular
momentum J2(i), each i = 1, 2, . . . , n, of each of the n constituents of the](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-18-320.jpg)
![xvi PREFACE AND PRELUDE
the components of each of the angular momentum operators J(i) has
the standard action on Hji
. These basic relations are presented in great
detail in Ref. [6], in [L], and reviewed in Chapter 1 of this volume. The
important point for the present work is:
Each simultaneous eigenvector |T(jπ k)j mi of the set of 2n mutually com-
muting Hermitian operators corresponding to each binary tree T ∈ Tn
of shape ShT (jπ) is a real orthogonal transformation of the eigenvec-
tors |j mi ∈ Bj. The coefficients in each such transformation are gen-
eralized Wigner-Clebsch-Gordan (WCG) coefficients, which themselves
are a product of known ordinary WCG coefficients, where the product is
uniquely determined by the shape of the labeled binary tree ShT (jπ).
This is, of course, just the expression of the property that we have con-
structed n!an uniquely defined coupled orthonormal basis sets of the
space Hj from the uncoupled basis of the same space Hj.
There is a class of subspaces of Hj of particular interest for the
present work. This class of subspaces is obtained from the basis vec-
tors BT (jπ) of Hj given above by selecting from the orthonormal basis
vectors |T(jπ k)j mi those that have a prescribed total angular momen-
tum quantum number j ∈ {jmin, jmin + 1, . . . , jmax} and, for each such j,
a prescribed projection quantum number m ∈ {j, j − 1, . . . , −j}. Thus,
the orthonormal basis vectors in this set are given by
BT (jπ, j, m) = {|T(jπ k)j mi | each k ∈ K
(j)
T (jπ)}.
We further specialize this basis set to the case π = identity permutation:
BT (j, j, m) = {|T(j k)j mi | each k ∈ K
(j)
T (j)}.
This basis set of orthonormal vectors then defines a subspace H(j, j, m) ⊂
Hj, which is the same vector space for each T ∈ Tn and for every per-
mutation jπ of j; that is, the following direct sum decomposition of the
tensor product space Hj holds:
Hj =
jmax
X
j=jmin
j
X
m=−j
⊕H(j, j, m).
We repeat: The important structural result for this vector space decom-
position is:
Each basis set BT (jπ, j, m), T ∈ Tn, π ∈ Sn, is an orthonormal basis of
one and the same space H(j, j, m).
The dimension of the tensor product space H(j, j, m) ⊂ Hj is Nj(j),
the Clebsch-Gordan (CG) number. This important number is the num-
ber of times that a given j ∈ {jmin, jmin + 1, . . . , jmax} is repeated for](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-20-320.jpg)
![PREFACE AND PRELUDE xvii
specified j. They can be calculated explicity by repeated application of
the elementary rule for the addition of two angular momenta, as de-
scribed in detail in [L]. The CG number is shape independent; that is,
Nj(j) counts the number of orthonormal basis vectors in the basis set
{|T(jπ k)j mi |for specified j m} of the space H(j, j, m) for each binary
tree T ∈ Tn and each jπ, π ∈ Sn.
It may seem highly redundant to introduce such a variety of or-
thonormal basis sets of vectors that span the same space Hj, but it
is within these vector space structures that resides the entire theory of
3(n − 1) − j coefficients. This aspect of the theory is realized through
shape transformations applied to the binary trees whose standard la-
bels appear in the binary coupled state vectors in the various basis sets
HT (jπ, j, m), T ∈ Tn, π ∈ Sn. The implementation of such shape trans-
formations into numerical-valued transformations between such state
vectors uses the notion of a recoupling matrix.
The matrix with elements that give the real orthogonal transfor-
mation matrix between distinct sets {|T(jπ k)j mi | k ∈ K
(j)
T (jπ)} and
{|T(jπ′ k′)j mi | k ∈ K
(j)
T (jπ′ )} of simultaneous basis eigenvectors, each of
which is an orthonormal basis of the vector space H(j, j, m), is called a
recoupling matrix. Thus, we have that
|T′
(jπ′ k′
)j mi =
X
k∈K
(j)
T (jπ )
RS; S′
k,j; k′,j
|T(jπ k)j mi,
where the recoupling matrix is the real orthogonal matrix, denoted RS; S′
,
with elements given by the inner product of state vectors:
RS; S′
k,j; k′,j
=
D
T(jπ k)jm T′
(jπ′ k
′
)jm
E
.
Here we have written the labeled shapes in the abbreviated forms:
S = ShT (jπ), S′
= ShT′ (jπ′ ).
As this notation indicates, these transformations are independent of m;
that is, they are invariants under SU(2) frame transformations. The
matrix elements are fully determined in each coupling scheme in terms
of known generalized WCG coefficients, since each coupled state vector in
the inner product is expressed as a linear combination of the orthonormal
basis vectors in the set Bj of the tensor product space Hj with coefficients
that are generalized WCG coefficients. Each recoupling matrix is a fully
known real orthogonal matrix of order Nj(j) in terms of its elements,
the generalized WCG coefficients. Since the inner product is real, it is
always the case that the recoupling matrix satisfies the relation
RS; S′
=
RS′
; S
tr
,](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-21-320.jpg)
![xxii PREFACE AND PRELUDE
for binary coupled angular momentum states corresponding to complete
sets of commuting Hermitian observables; it is so formulated in Chap-
ters 1 and 5, using the property that there is a unique doubly stochastic
matrix of order Nj(j) associated with each binary coupled scheme 1 and
each binary coupled scheme 2. The probability of a prepared coupled
scheme 1 state being in a measured scheme 2 coupled state is just the
(row, column) entry in the doubly stochastic matrix corresponding to
these respective coupled states. This is the answer given in the context
of conventional nonrelativistic quantum theory. We do not speculate on
the meaning of this answer to the holistic aspects of complex (or sim-
ple) quantal systems. Rather, throughout this volume, we focus on the
detailed development of topics and concepts that relate to the binary
coupling theory of angular momenta, as developed in [L], [6], and in this
volume, leaving their full interpretation for the future.
In many ways, this portion of the monograph can be considered
as a mopping-up operation for an accounting of the binary theory of
the coupling of arbitrarily many angular momentum systems within
the paradigm of conventional nonrelativistic quantum mechanics, in the
sense of Kuhn [35, p. 24]. Yet, there is the apt comparison with Com-
plexity Theory as advanced by the Santa Fe Institute — a few simple,
algorithmic-like rules that generate an almost unlimited scope of patterns
of high informational content. Moreover, binary trees viewed as branch-
ing diagrams, are omnipotent as classification schemes for objects of all
sorts — their shapes and labelings have many applications going beyond
angular momentum systems. The closely related graphs — Cayley’s [19]
trivalent trees, which originate from a single labeled binary tree — and
their joining for pairs of such binary trees define cubic graphs (see [L]).
These cubic graphs determine the classification of all binary coupling
schemes in angular momentum theory. But they surely extend beyond
the context of angular momentum systems in sorting out the diverse
patterns of regularity in nature, as discussed briefly in Appendix C.
TOPICAL CONTENTS
We summarize next the principal topics that constitute the present
monograph, and the relevant chapters.
Chapter 1, Chapter 5, Chapter 8. Total angular momentum states
(reviewed from [L]). The total angular momentum of a physical system
is a collective property. The addition of two angular momenta, a prob-
lem already solved in the seminal papers in quantum mechanics, is the
simplest example, especially for intrinsic internal quantum spaces such
as spin space. Here we review the earlier work, emphasizing that the
tensor product space in which the angular momentum operators act has
the property of all such tensor product spaces: It contains vectors that
cannot be obtained as simple products of vectors of the individual parts
of the system — tensor product spaces by their very nature are holistic;](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-26-320.jpg)
![xxiv PREFACE AND PRELUDE
Chapters 2-7. Permutation matrices and related topics. There is, per-
haps, no symmetry group more important for all of quantum physics
than the group of permutations of n objects — the permutation or sym-
metric group Sn. Permutation matrices of order n are the simplest matrix
realization of the group Sn by matrices containing a single 1 in each row
and column: They consist of the n! rearrangements of the n columns of
the identity matrix of order n. But in this work the symmetric group
makes its direct appearance in a different context than the Pauli princi-
ple; namely, through Birkhoff’s [8] theorem that proves the existence of
a subset of the set of all n! permutation matrices of order n such that
every doubly stochastic matrix of order n can be expanded with positive
real coefficients in terms of the subset of permutations matrices. We
not only present this aspect of doubly stochastic matrices, but develop a
more general theory in Chapter 3 of matrices having the same fixed line-
sum for all rows and columns. Such matrices include doubly stochastic
matrices, magic squares, and alternating sign matrices, which are all of
interest in physical theory, as discussed in [L]. Here, additional results
of interest are obtained, with Chapters 5, 6, 7 being dedicated to each
topic, respectively. We comment further on Chapter 5.
Chapter 5. Doubly stochastic matrices. These matrices are intro-
duced in Chapter 1, and the development of further properties continued
here. Recoupling matrices introduced in Chapter 1 are doubly stochas-
tic matrices. Such matrices have a probabilistic interpretation in terms
of preparation of states corresponding to complete sets of commuting
Hermitian operators. The matrix elements in a given (row, column) of
a doubly stochastic matrix is the probability of a prepared eigenstate
(labeled by the row) of a complete set of mutually commuting Hermitian
operators being the measured eigenstate (labeled by the column) of a
second (possibly the same) complete set of mutually commuting Hermi-
tian operators. Here, these eigenstates are taken to be the coupled states
corresponding to standard labeled binary trees of order n. This aspect of
doubly stochastic matrices is illustrated numerous times. To my knowl-
edge, doubly stochastic matrices were introduced into quantum theory
by Alfred Landé [39], the great atomic spectroscopist from whom I had
the privilege of hearing first-hand, while a graduate student at The Ohio
State University, his thesis that they are fundamental objects underlying
the meaning of quantum mechanics.
Chapter 8. Heisenberg’s magnetic ring. This physical problem, ad-
dressed in depth by Bethe [5] in a famous paper preceding the basic
work by Wigner [82] on angular momentum theory and Clebsch-Gordan
coefficients is a very nice application of binary coupling theory. The
Hamiltonian comes under the full purview of composite systems: Its
state vectors can be classified as eigenvectors of the square of the total
angular momentum with the usual standard action of the total angu-
lar momentum. The exact solutions are given for composite systems](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-28-320.jpg)
![PREFACE AND PRELUDE xxv
containing n = 2, 3, 4 constituents, each part with an arbitrary angular
momentum. Remarkably, this seems not to have been noticed.
For n ≥ 5, the magnetic ring problem can be reduced to the cal-
culation of the eigenstates originating from the diagonalization by a
real orthogonal matrix of a real symmetric matrix of order equal to the
Clebsch-Gordan number Nj(j), which gives the number of occurrences
(multiplicity) of a given total angular momentum state in terms of the
angular momenta of the individual constituents. This Hamiltonian ma-
trix of order Nj(j) is a real symmetric matrix uniquely determined by
certain recoupling matrices originating from binary couplings of the con-
stituent angular momenta. It is an exquisitely complex implementation
of the uniform reduction procedure for the calculation of all 3(n − 1) − j
coefficients that arise; this procedure is itself based on paths, shape trans-
formations, recoupling matrices, and their reduction properties. This
approach to the Heisenberg ring problem gives a complete and, perhaps,
very different viewpoint not present in the Bethe approach, especially,
since no complex numbers whatsoever are involved in obtaining the en-
ergy spectrum, nor need be in obtaining a complete set of orthonormal
eigenvectors. The problem is fully solved in the sense that the rules for
computing all 3(n − 1) − j coefficients that enters into the calculation of
the Hamiltonian matrix can be formulated explicitly. Unfortunately, it
is almost certain that the real orthogonal matrix required to diagonalize
the symmetric Hamilton matrix, with its complicated 3(n − 1) − j type
structure, cannot be determined algebraically. It may also be the case
that numerical computations of the elements of the symmetric Hamilto-
nian matrix are beyond reach, except for simple special cases. Of course,
complex phase factors do enter into the classification of the eigenvectors
by the cyclic invariance group Cn of the Hamiltonian, but this may not
be necessary for many applications.
Finally, there are three Appendices A, B, and C that deal with issues
raised in the main text. Appendix C, however, presents natural gener-
alizations of binary tree classifications to other problems, especially, to
composite systems where the basic constituents have U(n) symmetry.
MATTERS OF STYLE, READERSHIP, AND RECOGNITION
On matters of readership and style, we repeat portions of the first
volume, since they still prevail.
The very detailed Table of Contents serves as a summary of topics
covered. The readership is intended to be advanced graduate students
and researchers interested in learning of the relation between symmetry
and combinatorics and of challenging unsolved problems. The many ex-
amples serve partially as exercises, but this monograph is not a textbook.
It is hoped that the topics presented promote further and more rigorous
developments. While we are unable to attain the eloquence of Mac Lane
[51], his book has served as a model.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-29-320.jpg)
![xxvi PREFACE AND PRELUDE
We mention again, as in [L], some unconventional matters of style.
We present significant results 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 integrated 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.
As with the earlier volume, this continuing work is heavily indebted to
the two volumes on the quantum theory of angular momentum published
with the late Lawrence C. Biedenharn [6]. The present volume is much
more limited in scope, addressing special topics left unattended in the
earlier work, as well as new problems.
Our motivation and inspiration for working out many details of bi-
nary coupling theory originates from the great learning experiences, be-
ginning in the early 1990’s, as acquired in combinatorial research papers
with William Y. C. Chen, which were strongly encouraged by the late
Gian-Carlo Rota, and supplemented by his many informative conver-
sations and inspirational lectures at Los Alamos. We do not reference
directly many of the seminal papers by Gian-Carlo Rota, his colleagues,
and students in this volume, but these publications were foundational in
shaping the earlier volume, and are ever-present here. We acknowledge
a few of these again: Désarménien, et al. [23]; Kung and Rota [36]; Kung
[37]; Roman and Rota [68]; Rota [70]-[71]; Rota, et al. [72]; as well as
the Handbook by Graham, et al. [28]. This knowledge acquisition has
continued under the many invitations by William Y. C. Chen, Director,
The Center for Combinatorics, Nankai University, PR China, to give
lectures on these subjects to students and to participate in small con-
ferences. These opportunities expanded at about the same time in yet
another direction through similar activities, organized by Tadeusz and
Barbara Lulek, Rzeszòw University of Technology, Poland, on Symme-
try and Structural Properties of Condensed Matter, under the purview
over the years by Adam Mickiewics University, Poznań, University of
Rzeszów, and Rzeszów University of Technology. The interaction with
Chinese and Polish students and colleagues has been particularly reward-
ing. Finally, the constant encouragement by my wife Marge and son Tom
provided the friendly environment for bringing both the first and second
volumes to completion.
Editors Zhang Ji and Lai Fun Kwong deserve special mention and
thanks for their encouragement and support of this project.
James D. Louck](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-30-320.jpg)
![Notation
General symbols
, comma separator; used non-uniformly
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
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 sum of matrices
⊗ tensor product of vector spaces, Kronecker (direct)
product of matrices
δi,j the Kronecker delta for integers i, j
δA,B the Kroneker delta for sets A and B
K(λ, α) the Kostka numbers
cλ
µ ν the Littlewood-Richardson numbers
Parn set of partitions having n parts, including 0 as a part
λ, µ, ν partitions in the set Parn
|A| cardinality of a set A
[n] set of integers {1, 2, . . . , n}
Specialized symbols are introduced as needed in the text;
xxxiii](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-37-320.jpg)
![xxxiv NOTATION
the list below contains a few of the more general ones:
J(i) angular momentem of constituent
i ∈ [n] of a composite system
K(i) intermediate angular momentem
i ∈ [n − 2] of a composite system
J total angular momentem of all constituents
of a composite system
j sequence (j1, j2, . . . , jn) of quantum numbers
of the constituents of a composite system
Bn set of binary bracketings of order n
Tn set of binary trees of order n
ShT shape of a binary tree T ∈ Tn
ShT (j) shape of a standard labeled binary tree T ∈ Tn
w(A, C) word in the letters A and C
|T(j k)j mi simultaneous eigenvector of a complete set of 2n ang-
ular momentum operators J2(i), i = 1, 2, . . . , n; J2, J3;
K2(i), i = 1, 2, . . . , n − 2 — also called binary
coupled state vectors
hT(j k)j m | T′
(j′
k′
)j mi
inner product of two binary coupled state vectors
Hji
Hilbert vector space of dimension 2ji + 1 that is
irreducible under the action of SU(2)
Hj = Hj1
⊗ Hj2
⊗ · · · ⊗ Hjn
: tensor productof the spaces Hji
of dimension (2j1 + 1) · · · (2jn + 1) = N(j)
Nj(j) Clebsch-Gordan number
H(j, j, m) subspace of Hj of order Nj(j)
U†V Landé form of a doubly stochastic matrix
RS; S′
recoupling matrix for a pair of standard labeled
binary trees related by arbitrary shapes S and S′
RSh; Sh+1
recoupling matrix for a pair of standard labeled
binary trees of shapes Sh and Sh+1 related by
an elementary shape transformation
∆T (j k)j | ∆T′ (j′
k′
)j
triangle coefficient that is a 3 × (n − 1)
matrix array that encodes the structure of the
labeled forks of a pair of standard labeled binary trees](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-38-320.jpg)
![Chapter 1
Composite Quantum
Systems
1.1 Introduction
The group and angular momentum theory of composite quantum sys-
tems was initiated by Weyl [80] and Wigner [82]. It is an intricate, but
well-developed subject, as reviewed in Biedenharn and van Dam [7], and
documented by the many references in [6]. It is synthesized further by
the so-called binary coupling theory developed in great detail in [L]. It
was not realized at the time that recoupling matrices, the objects that
encode the full prescription for relating one coupling scheme to another,
are doubly stochastic matrices. This volume develops this aspect of the
theory and related topics. We review in this first chapter some of the
relevant aspects of the coupling theory of angular momenta for ease of
reference. Curiously, these developments relate to the symmetric group
Sn, which is a finite subgroup of the general unitary group and which
is also considered in considerable detail in the previous volume. But
here the symmetric group makes its appearance in the form of one of its
simplest matrix (reducible) representations, the so-called permutation
matrices. The symmetric group is one of the most important groups in
physics (Wybourne [87]), as well as mathematics (Robinson [67]). In
physics, this is partly because of the Pauli exclusion principle, which ex-
presses a collective property of the many entities that constitute a com-
posite system; in mathematics, it is partly because every finite group is
isomorphic to a symmetric group of some order. While the symmetric
group is one of the most studied of all groups, many of its properties
that relate to doubly stochastic matrices, and other matrices of physi-
cal importance, seem not to have been developed. This review chapter
provides the background and motivation for this continued study.
1](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-40-320.jpg)
![2 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS
A comprehensive definition of a composite quantal system that is
sufficiently broad in scope to capture all possible physical systems is
difficult: it will not be attempted here. Instead, we consider some general
aspects of complex systems and then restrict our attention to a definition
that is sufficient for our needs.
In some instances, a composite quantal system can be built-up by
bringing together a collection of known independent quantal systems,
initially thought of as being noninteractive, but as parts of a composite
system, the subsystems are allowed to be mutually interactive. We call
this the build-up principle for composite systems. We assume that such a
built-up composite system can also be taken apart in the sense that, if the
mutual interactions between the known parts are ignored, the subsystems
are each described independently and have their separate identities. This
is a classical intuitive notion; no attempt is made to place the “putting-
together and breaking-apart” process itself in a mathematical framework.
The mathematical model for describing a built-up composite system
utilizes the concept of a tensor product of vector spaces. The state space
of the i−th constituent of such a composite system is given by an inner
product vector space Hi, which we take to be a bra-ket vector space in
the sense of Dirac [24], and, for definiteness, it is taken to be a separable
Hilbert space. Each such Hilbert space then has an orthonormal basis
given by
Bi = {|i, kii | ki = 1, 2, . . .} , i = 1, 2, . . . , n. (1.1)
The state space of a composite system, built-up from n such independent
systems is then the tensor product space H defined by
H = H1 ⊗ H2 ⊗ · · · ⊗ Hn. (1.2)
The orthonormal basis of H is given in terms of the individual orthonor-
mal bases Bi of Hi by
B = B1 ⊗ B2 ⊗ · · · ⊗ Bn. (1.3)
A general vector in the linear vector space H is of the form
| general statei (1.4)
=
X
k1≥1
X
k2≥1
· · ·
X
kn≥1
ak1,k2,...,kn
| 1, k1i⊗ | 2, k2i ⊗ · · · ⊗ | n, kni,
where the coefficients ak1,k2,...,kn
are arbitrary complex numbers. Since,
in general, ak1,k2,...,kn
6= a1,k1
a2,k2
· · · an,kn
, it is an evident (and well-
known) that a general superposition of state vectors given by (1.4) does
not have the form
X
k1≥1
a1,k1
| 1, k1i
⊗
X
k2≥1
a1,k1
| 2, k2i
⊗ · · · ⊗
X
kn≥1
an,kn
| n, kni
.
(1.5)](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-41-320.jpg)
![1.1. INTRODUCTION 3
Thus, there are vectors in a tensor product space that cannot be written
as the tensor product of n vectors, each of which belongs to a constituent
subspace Hi. This mathematical property already foretells that compos-
ite systems have properties that are not consequences of the properties
of the individual constituents.
The definition of composite quantal systems and their interactions
with measuring devices are fundamental to the interpretation of quan-
tum mechanics. Schrödinger [73] coined the term entanglement to de-
scribe the property that quantal systems described by the superposition
of states in tensor product space are not all realizable as the tensor prod-
uct of states belonging to the constituent subsystems. Entanglement, in
its conceptual basis, is not a mysterious property. It is a natural prop-
erty of a linear theory based on vector spaces and standard methods for
building new vector spaces out of given vector spaces. But the meaning
and breadth of such mathematical constructions for the explanation of
physical processes can be profound.
We make also the following brief remarks on the methodology of ten-
sor product spaces introduced above. These observations have been made
and addressed by many authors; we make no attempt (see Wheeler and
Zurek [81]) to cite the literature, our purpose here being simply to place
results presented in this volume in the broader context:
Remarks.
1. The build-up principle for composite systems stated above is al-
ready to narrow in scope to capture the properties of many phys-
ical systems. It does not, for example, include immediately an
electron with spin: the spin property cannot be removed from the
electron; it is one of its intrinsic properties, along with it mass and
charge. Nonetheless, the energy spectrum of a single (nonrelativis-
tic) electron with spin in the presence of a central attractive poten-
tial can be described in terms of a tensor product of vector spaces
Hψ⊗Hl⊗H1/2, where Hψ, Hl, and H1/2, are, respectively, the space
of solutions of the Schrödinger radial equation, the finite-Hilbert
space of dimension 2l + 1 of orbital angular momentum states, and
the finite-Hilbert space of dimension 2 of spin states (see Ref. [6]).
This is indicative of the fact that mathematical techniques devel-
oped in specific contexts often have a validity that extends beyond
their original intent.
2. It is sometimes helpful to give concrete realizations of tensor prod-
uct spaces in terms of functions over the real or complex numbers.
This is usually possible in ordinary quantum theory, even for spin
and other internal symmetries. The abstract tensor product space
relation (1.1) is then formulated as](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-42-320.jpg)
![1.2. ANGULAR MOMENTUM STATE VECTORS 5
i = 1, 2, . . . , n. The entire composite system is described in terms of a
common right-handed inertial reference frame (e1, e2, e3) in Cartesian
3−space R3, where redescriptions of the system are effected by uni-
tary unimodular SU(2) group transformations of the reference frame
as described in detail in [L]. The angular momentum J(i) of the i−th
subsystem is given in terms of its three components relative to the refer-
ence frame by J(i) = J1(i)e1 + J2(i)e2 + J3(i)e3, where the components
satisfy the commutation relations [J1(i), J2(i)] = iJ3(i), [J2(i), J3(i)] =
iJ1(i), [J3(i), J1(i)] = iJ2(i), where the i in the commutator relation is
i =
√
−1, and not the subsystem index. The components Jk(i), k =
1, 2, 3, of J(i) have the standard action on each subspace of the states of
the subsystem, as characterized by
J2
(i)|ji mii = ji(ji + 1)|ji mii,
J3(i)|ji mii = mi|ji mii,
J+(i)|ji mii =
p
(ji − mi)(ji + mi + 1) |ji mi + 1i, (1.7)
J−(i)|ji mii =
p
(ji + mi)(ji − mi + 1) |ji mi − 1i.
The notation Hji
denotes the finite-dimensional Hilbert space of dimen-
sion dim Hji
= 2ji + 1 with orthonormal basis given by
Bji
= {|ji mii | mi = ji, ji − 1, . . . , −ji},
(1.8)
hji mi | ji m′
ii = δmi,m′
i
, each pair mi, m′
i ∈ {ji, ji − 1, . . . , −ji}.
The usual assumptions underlying the derivation of the standard rela-
tions (1.7) for the action of the angular momenta components are made;
namely, that the linear vector space Hi over the complex numbers is
equipped with an inner product with respect to which the components
Jk(i) are Hermitian operators that act linearly on the space Hi to ef-
fect a transformation to a new vector in the space. The operators J+(i)
and J−(i) are the usual Hermitian conjugate shift operators defined by
J+(i) = J1(i) + iJ2(i) and J−(i) = J1(i) − iJ2(i), where the nonindexing
i is the complex number i =
√
−1.
The angular momentum components Jk(i), k = 1, 2, 3, for distinct
subsystems i all mutually commute. It is allowed that each of the vector
spaces Hi, i = 1, 2, . . . , n can have the same or distinct definitions of inner
product, it only being required that the angular momentum components
in each subspace are Hermitian with respect to the inner product for
that subspace. In the notation for the orthonormal basis vectors Bi of
Hji
in (1.8), we have suppressed all the extra quantum labels that may
be necessary to define a basis for the full space Hi. In applications to
specific problems, such labels are to be supplied. It is the properties of](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-44-320.jpg)
![1.2. ANGULAR MOMENTUM STATE VECTORS 7
operators act in all other parts of the tensor product space. We often
use the simplified notation
J = J(1) + J(2) + · · · + J(n) (1.12)
for the sum of various angular momentum operators acting in the tensor
product space, but such expressions are always to be interpreted in the
sense of a direct sum of tensor products of operators of the tensor product
form (1.11). (See Sect. 10.5, Compendium A of [L] for a summary of the
properties of tensor product spaces in terms of the present notations.)
We introduce the following compact notations to describe the ket-
vectors of the tensor product space:
j = (j1, j2, . . . , jn), each ji ∈ {0, 1/2, 1, 3/2, . . .}, i = 1, 2, . . . , n,
m = (m1, m2, . . . , mn), each mi ∈ {ji, ji − 1, . . . , −ji},
i = 1, 2, . . . , n,
Hj = Hj1
⊗ Hj2
⊗ · · · ⊗ Hjn
, (1.13)
|j mi = |j1 m1i ⊗ |j2 m2i ⊗ · · · ⊗ |jn mni,
C(j) = {m | mi = ji, ji − 1, . . . , −ji; i = 1, 2, . . . , n}.
The set of 2n mutually commuting Hermitian operators
J2
(1), J3(1), J2
(2), J3(2), . . . , J2
(n), J3(n) (1.14)
is a complete set of operators in the tensor product space Hj, in that
the set of simultaneous eigenvectors |j mi, m ∈ C(j) is an orthonormal
basis; that is, there is no degeneracy left over. The action of the angular
momentum operators J(i), i = 1, 2, . . . , n, is the standard action given
by
J2
(i)|j mi = ji(ji + 1)|j mi,
J3(i)|j mi = mi|j mi,
J+(i)|j mi =
p
(ji − mi)(ji + mi + 1) |j m+1(i)i, (1.15)
J−(i)|j mi =
p
(ji + mi)(ji − mi + 1) |j m−1(i)i,
m±1(i) = (m1, . . . , mi ± 1, · · · , mn).
The orthonormality of the basis functions is expressed by
hj m | j m′
i = δm,m′ , each pair m, m′
∈ C(j). (1.16)
Since the collection of 2n commuting Hermitian operators (1.14) refers
to the angular momenta of the individual constituents of a physical sys-
tem, and the action of the angular momentum operators is on the basis](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-46-320.jpg)
![10 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS
The cardinality of the sets R(j) and C(j) are related by
| R(j) | =
jmax
X
j=jmin
(2j + 1)Nj(j) = N(j) =
n
Y
i=1
(2ji + 1) = | C(j) |, (1.23)
where we have defined Nj(j) = |R(j, j)|. These positive numbers are
called Clebsch-Gordan (CG) numbers. They can be generated recur-
sively as discussed in Sect. 2.1.1 of [L].
The orthonormal bases (1.16) and (1.20) of the space Hj must be
related by a unitary transformation A(j) of order N(j) =
Qn
i=1(2ji + 1)
with (row; column) indices enumerated by (m ∈ C(j); α, j, m ∈ R(j))
(see (1.35) below). Thus, we must have the invertible relations:
|(j α)j mi =
X
m∈C(j)
A(j)tr
α, j, m; m
|j mi, each α, j, m ∈ R(j), (1.24)
|j mi =
X
α,j,m∈R(j)
A(j)†
α, j, m; m
|(j α)j mi, each m ∈ C(j). (1.25)
Note. We have reversed the role of row and column indices here from
that used in [L] (see pp. 87, 90, 91, 94, 95), so that the notation accords
with that used later in Chapter 5 for coupling schemes associated with
binary trees, and the general structure set forth in Sect. 5.1.
The transformation to a coupled basis (1.20) as given by (1.24) effects
the full reduction of the n−fold Kronecker product
Dj
(U) = Dj1
(U) ⊗ Dj2
(U) ⊗ · · · ⊗ Djn
(U), U ∈ SU(2), (1.26)
of SU(2) unitary irreducible matrix representations. The matrix Dj(U),
U ∈ SU(2), is a reducible unitary representation of SU(2) of dimension
N(j), and the transformation (1.25) effects the transformation to a direct
sum of irreducible unitary representations Dj(U) (Wigner D−matrices).
We next summarize the transformation properties of the coupled and
ucoupled bases (1.25) under SU(2) frame rotations.
1.2.1 Group Actions in a Composite System
Under the action of an SU(2) frame rotation of the common frame
(e1, e2, e3) used to describe the n constituents of a physical system
in Cartesian space R3, where system i has angular momentum J(i) =
J1(i)e1 + J2(i)e2 + J3(i))e3, the orthonormal basis of the subspace
Hji
= {|ji mii | mi = ji, ji − 1, . . . , −ji} (1.27)](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-49-320.jpg)
![1.3. STANDARD FORM OF THE KRONECKER DIRECT SUM 11
of system i undergoes the standard unitary transformation
TU |ji m′
ii =
X
mi
Dji
mi m′
i
(U)|ji mii, each U ∈ SU(2). (1.28)
The uncoupled basis Hj1
⊗ Hj2
⊗ · · · ⊗ Hjn
of the angular momentum
space Hj of the collection of systems undergoes the reducible unitary
transformation given by
(TU ⊗ TU ⊗ · · · ⊗ TU ) (|j1 m′
1i ⊗ |j2 m′
2i ⊗ · · · ⊗ |jn m′
ni
=
X
m
Dj1
(U) ⊗ Dj2
(U) ⊗ · · · ⊗ Djn
(U)
m m′
×(|j1 m1i ⊗ |j2 m2i ⊗ · · · ⊗ |jn mni), (1.29)
where m = (m1, m2, . . . , mn), m′ = (m′
1, m′
2, . . . , m′
n). This relation is
described in the abbreviated notations (1.13) and (1.15)-(1.16) by
TU |j m′
i =
X
m
Dj
m m′ (U) |j mi, Dj
(U) = Dj1
(U)⊗· · ·⊗Djn
(U), (1.30)
for each U ∈ SU(2). Similarly, the coupled basis (1.20) of Hj undergoes
the irreducible unitary transformation:
TU |(j α)j m′ i =
X
m
Dj
m m′ (U)|(j α)j mi, each U ∈ SU(2). (1.31)
1.3 Standard Form of the Kronecker Direct Sum
Schur’s lemma (see Sect. 10.7.2, Compendium A in [L]) implies that the
reducible unitary Kronecker product representation Dj(U) of SU(2) de-
fined by (1.26) is reducible into a direct sum of irreducible unitary rep-
resentations Dj(U) by a unitary matrix similarity transformation U(j) of
order N(j) =
Qn
i=1(2ji + 1) :
U(j)†
Dj
(U)U(j)
= Dj
(U)
=
Djmin
(U) 0 0 · · · 0
0 Djmin+1(U) 0 . . . 0
.
.
.
.
.
.
.
.
. · · ·
.
.
.
0 0 0 · · · Djmax
(U)
, (1.32)](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-50-320.jpg)
![[263]
The white-faced figure ceased dancing. The wind in the
trees sang on. The figure, appearing to see the dragon,
drew back in trembling fright.
He approached the fiery curtain, yet his back was ever
toward it. There was yet a space between the two
sections of the curtain. The figure, darting toward this
gap, was caught in the flames.
“Oh!” Jeanne breathed. “He will die in flames!”
Marjory Dean pressed her hand hard.
Of a sudden the floor beneath the white figure opened
and swallowed him up.
Jeanne looked for the dragon. It was gone. The fiery
red of the curtain was turning to an orange glow.
“Come. You have seen.” It was Hop Long Lee who
spoke. Once again his marble-cold hand touched
Jeanne’s hand.
Ten minutes later the four figures were once more in
the street.
“Midnight in an Oriental garden,” Angelo breathed.
“That,” breathed Marjory Dean, “is drama, Oriental
drama. Give it a human touch and it could be made
supreme.”
“You—you think it could be made into a thing of
beauty?”
“Surely. Most certainly, my child. Nothing could be more
unique.”](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-56-320.jpg)
![[264]
[265]
“Come,” whispered Jeanne happily. “Come with me. The
night is young. The day is for sleep. Come. We will have
coffee by my fire. Then we will talk, talk of all this. We
will create an opera in a night. Is it not so?”
And it was so.
A weird bit of opera it was that they produced that
night. Even the atmosphere in which they worked was
fantastic. Candle light, a flickering fire that now and
then leaped into sudden conflagration, mellow-toned
gongs provided by the little lady of the cameo; such
were the elements that added to the fantastic reality of
the unreal.
In this one-act drama the giant paper dragon remained.
The flaming curtain, the setting for some weird Buddhist
ceremony, was to furnish the motif. A flesh and blood
person, whose part was to be played by Marjory Dean,
replaced the thing of white cloth and paper that had
danced a weird dance, and became entangled in the
fiery curtain. Oriental mystery, the deep hatred of some
types of yellow men for the white race, these entered
into the story.
In the plot the hero (Marjory Dean), a white boy, son of
a rich trader, caught by the lure of mystery, adventure
and tales of the magic curtain, volunteers to take the
place of a rich Chinese youth who is to endure the trial
by fire.
A very ugly old Chinaman, who holds the white boy in
high regard, learning of his plans and realizing his peril,
prepares the trap-door in the floor beneath the magic
curtain.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-57-320.jpg)
![[266]
When the hour comes for the trial by fire, the white boy,
being ignorant of the secrets that will save him, appears
doomed as the flames of the curtain surround him,
consuming the very mask from his face and leaving him
there, his identity revealed in stark reality.
Then as the rich Chinaman, who has planned the trial,
realizes the catastrophe that must befall his people if
the rich youth is burned to death, prepares to cast
himself into the flames, the floor opens to swallow the
boy up, and the curtain fades.
There is not space here to tell of the motives of love,
hate, pride and patriotism that lay back of this bit of
drama. Enough that when it was done Marjory Dean
pronounced it the most perfect bit of opera yet
produced in America.
“And you will be our diva?” Jeanne was all eagerness.
“I shall be proud to.”
“Then,” Angelo’s eyes shone, “then we are indeed rich
once more.”
“Yes. Your beautiful rugs, your desk, your ancient friend
the piano, they shall all come back to you.” In her joy
Jeanne could have embraced him. As it was she wrung
his hand in parting, and thanked him over and over for
his part in this bit of work and adventure.
“The music,” she whispered to Swen, “you will do it?”
“It is as well as done. The wind whispering in the
graveyard pines at midnight. This is done by reeds and
strings. And there are the gongs, the deep melodious
gongs of China. What more could one ask?”](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-58-320.jpg)
![[267]
[268]
What more, indeed?
“And now,” said Florence, after she had, some hours
later, listened to Jeanne’s recital of that night’s affairs,
“now that it is all over, what is there in it all for you?”
“For me?” Jeanne spread her hands wide. “Nothing.
Nothing at all.”
“Then why—?”
“Only this,” Jeanne interrupted her, “you said once that
one found the best joy in life by helping others. Well
then,” she laughed a little laugh, “I have helped a little.
“And you shall see, my time will come.”
Was she right? Does one sometimes serve himself best
by serving others? We shall see.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-59-320.jpg)
![[269]
CHAPTER XXX
A SURPRISE PARTY
Time marched on, as time has a way of doing. A week
passed, another and yet another. Each night of opera
found Jeanne, still masquerading as Pierre, at her post
among the boxes. Never forgetting that a priceless
necklace had been stolen from those boxes and that she
had run away, ever conscious of the searching eyes of
Jaeger and of the inscrutable shadow that was the lady
in black, Jeanne performed her tasks as one who walks
beneath a shadow that in a moment may be turned into
impenetrable darkness.
For all this, she still thrilled to the color, the music, the
drama, which is Grand Opera.
“Some day,” she had a way of whispering to herself,
“some happy day!” Yet that day seemed indistinct and
far away.
The dark-faced menace to her happiness, he of the evil
eye, appeared to have vanished. Perhaps he was in jail.
Who could tell?](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-60-320.jpg)
![[270]
The little Frenchman with the message, too, had
vanished. Why had he never returned to ask Pierre, the
usher in the boxes, the correct address of Petite
Jeanne? Beyond doubt he believed himself the victim of
a practical joke. “This boy Pierre knows nothing
regarding the whereabouts of that person named Petite
Jeanne.” Thus he must have reasoned. At any rate the
message was not delivered. If Jeanne had lost a relative
by death, if she had inherited a fortune or was wanted
for some misdemeanor committed in France, she
remained blissfully ignorant of it all.
Three times Rosemary Robinson had invited her to visit
her at her home. Three times, as Pierre, politely but
firmly, she had refused. “This affair,” she told herself,
“has gone far enough. Before our friendship ripens or is
blighted altogether, I must reveal to her my identity.
And that I am not yet willing to do. It might rob me of
my place in this great palace of art.”
Thanks to Marjory Dean, the little French girl’s training
in Grand Opera proceeded day by day. Without
assigning a definite reason for it, the prima donna had
insisted upon giving her hours of training each week in
the role of the juggler.
More than this, she had all but compelled Jeanne to
become her understudy in the forthcoming one-act
opera to be known as “The Magic Curtain.”
At an opportune moment Marjory Dean had introduced
the manager of the opera to all the fantastic witchery of
this new opera. He had been taken by it.
At once he had agreed that when the “Juggler” was
played, this new opera should be presented to the](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-61-320.jpg)
![[271]
public.
So Jeanne lived in a world of dreams, dreams that she
felt could never come true. “But I am learning,” she
would whisper to herself, “learning of art and life. What
more could one ask?”
Then came a curious invitation. She was to visit the
studios of Fernando Tiffin. The invitation came through
Marjory Dean. Strangest of all, she was to appear as
Pierre.
“Why Pierre?” she pondered.
“Yes, why?” Florence echoed. “But, after all, such an
invitation! Fernando Tiffin is the greatest sculptor in
America. Have you seen the fountain by the Art
Museum?”
“Where the pigeons are always bathing?”
“Yes.”
“It is beautiful.”
“He created that statue, and many others.”
“That reminds me,” Jeanne sought out her dress suit
and began searching its pockets, “an artist, an
interesting man with a beard, gave me his card. He told
me to visit his studio. He was going to tell me more
about lights and shadows.”
“Lights and shadows?”
“Yes. How they are like life. But now I have lost his
card.”](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-62-320.jpg)
![[272]
[273]
* * * * * * * *
Florence returned to the island. There she sat long in
the sunshine by the rocky shore, talking with Aunt
Bobby. She found the good lady greatly perplexed.
“They’ve served notice,” Aunt Bobby sighed, “the park
folks have. All that is to come down.” She waved an arm
toward the cottonwood thicket and the “Cathedral.” “A
big building is going up. Steam shovels are working over
on the west side now. Any day, now, we’ll have to pack
up, Meg and me.
“And where’ll we go? Back to the ships, I suppose. I
hate it for Meg. She ought to have more schoolin’. But
poor folks can’t pick and choose.”
“There will be a way out,” Florence consoled her. But
would there? Who could tell?
She hunted up Meg and advised her to look into that
mysterious package. “It may be a bomb.”
“If it is, it won’t go off by itself.”
“It may be a gun.”
“Don’t need a gun. Got two of ’em. Good ones.”
“It may be stolen treasure.”
“Well, I didn’t steal it!” Meg turned flashing eyes upon
her. And there for a time the matter ended.
* * * * * * * *
Jeanne attended the great sculptor’s party. Since she
had not been invited to accompany Marjory Dean, she](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-63-320.jpg)
![[274]
went alone. What did it matter? Miss Dean was to be
there. That was enough.
She arrived at three o’clock in the afternoon. A servant
answered the bell. She was ushered at once into a vast
place with a very high ceiling. All about her were
statues and plaster-of-paris reproductions of
masterpieces.
Scarcely had she time to glance about her when she
heard a voice, saw a face and knew she had found an
old friend—the artist who had spoken so interestingly of
life, he of the beard, was before her.
“So this is where you work?” She was overjoyed. “And
does the great Fernando Tiffin do his work here, too?”
“I am Fernando Tiffin.”
“Oh!” Jeanne swayed a little.
“You see,” the other smiled, putting out a hand to
steady her, “I, too, like to study life among those who
do not know me; to masquerade a little.”
“Masquerade!” Jeanne started. Did he, then, see
through her own pretenses? She flushed.
“But no!” She fortified herself. “How could he know?”
“You promised to tell me more about life.” She hurried
to change the subject.
“Ah, yes. How fine! There is yet time.
“You see.” He threw a switch. The place was flooded
with light. “The thing that stands before you, the ‘Fairy](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-64-320.jpg)
![[275]
and the Child,’ it is called. It is a reproduction of a great
masterpiece: a perfect reproduction, yet in this light it is
nothing; a blare of white, that is all.
“But see!” He touched one button, then another, and,
behold, the statue stood before them a thing of
exquisite beauty!
“You see?” he smiled. “Now there are shadows, perfect
shadows, just enough, and just enough light.
“Life is like that. There must be shadows. Without
shadows we could not be conscious of light. But when
the lights are too bright, the shadows too deep, then all
is wrong.
“Your bright lights of life at the Opera House, the sable
coats, the silks and jewels, they are a form of life. But
there the lights are too strong. They blind the eyes, hide
the true beauty that may be beneath it all.
“But out there on that vacant lot, in the cold and dark—
you have not forgotten?”
“I shall never forget.” Jeanne’s voice was low.
“There the shadows were too deep. It was like this.” He
touched still another button. The beauty of the statue
was once more lost, this time in a maze of shadows too
deep and strong.
“You see.” His voice was gentle.
“I see.”
“But here are more guests arriving. You may not be
aware of it, but this is to be an afternoon of opera, not](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-65-320.jpg)
![[276]
[277]
of art.”
Soon enough Jeanne was to know this, for, little as she
had dreamed it, hers on that occasion was to be the
stellar role.
It was Marjory Dean who had entered. With her was the
entire cast of “The Magic Curtain.”
“He has asked that we conduct a dress rehearsal here
for the benefit of a few choice friends,” Miss Dean
whispered in Jeanne’s ear, as soon as she could draw
her aside.
“A strange request, I’ll grant you,” she answered
Jeanne’s puzzled look. “Not half so strange as this,
however. He wishes you to take the stellar role.”
“But, Miss Dean!”
“It is his party. His word is law in many places. You will
do your best for me.” She pressed Jeanne’s hand hard.
Jeanne did her best. And undoubtedly, despite the lack
of a truly magic curtain, despite the limitations of the
improvised stage, the audience was visibly impressed.
At the end, as Jeanne sank from sight beneath the
stage, the great sculptor leaned over to whisper in
Marjory Dean’s ear:
“She will do it!”
“What did I tell you? To be sure she will!”
The operatic portion of the program at an end, the
guests were treated to a brief lecture on the art of](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-66-320.jpg)
![[278]
sculpture. Tea was served. The guests departed.
Through it all Jeanne walked about in a daze. “It is as if
I had been invited to my own wedding and did not so
much as know I was married,” she said to Florence,
later in the day.
Florence smiled and made no reply. There was more to
come, much more. Florence believed that. But Jeanne
had not so much as guessed.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-67-320.jpg)
![[279]
CHAPTER XXXI
FLORENCE MEETS THE LADY IN
BLACK
The great hour came at last. “To-night,” Jeanne had
whispered, “‘The Magic Curtain’ will unfold before
thousands! Will it be a success?”
The very thought that it might prove a failure turned her
cold. The happiness of her good friends, Angelo, Swen
and Marjory Dean was at stake. And to Jeanne the
happiness of those she respected and loved was more
dear than her own.
Night came quite suddenly on that eventful day. Great
dark clouds, sweeping in from the lake, drew the curtain
of night.
Jeanne found herself at her place among the boxes a
full hour before the time required. This was not of her
own planning. There was a mystery about this; a voice
had called her on the telephone requesting her to arrive
early.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-68-320.jpg)
![[280]
“Now I am here,” she murmured, “and the place is half
dark. Who can have requested it? What could have
been the reason?”
Still another mystery. Florence was with her. And she
was to remain. A place had been provided for her in the
box usually occupied by Rosemary Robinson and her
family.
“Of course,” she had said to Florence, “they know that
we had something to do with the discovery of the magic
curtain. It is, perhaps, because of this that you are
here.”
Florence had smiled, but had made no reply.
At this hour the great auditorium was silent, deserted.
Only from behind the drawn stage curtain came a faint
murmur, telling of last minute preparations.
“‘The Magic Curtain.’” Jeanne whispered. The words still
thrilled her. “It will be witnessed to-night by thousands.
What will be the verdict? To-morrow Angelo and Swen,
my friends of our ‘Golden Circle,’ will be rich or very,
very poor.”
“The Magic Curtain.” Surely it had been given a
generous amount of publicity. Catching a note of the
unusual, the mysterious, the uncanny in this production,
the reporters had made the most of it. An entire page of
the Sunday supplement had been devoted to it. A crude
drawing of the curtains, pictures of Hop Long Lee, of
Angelo, Swen, Marjory Dean, and even Jeanne were
there. And with these a most lurid story purporting to
be the history of this curtain of fire as it had existed
through the ages in some little known Buddhist temple.
The very names of those who, wrapped in its consuming](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-69-320.jpg)
![[281]
folds, had perished, were given in detail. Jeanne had
read, had shuddered, then had tried to laugh it off as a
reporter’s tale. In this she did not quite succeed. For her
the magic curtain contained more than a suggestion of
terror.
She was thinking of all this when an attendant, hurrying
up the orchestra aisle, paused beneath her and called
her name, the only name by which she was known at
the Opera House:
“Pierre! Oh, Pierre!”
“Here. Here I am.”
Without knowing why, she thrilled to her very finger
tips. “Is it for this that I am here?” she asked herself.
“Hurry down!” came from below. “The director wishes to
speak to you.”
“The director!” The blood froze in her veins. So this was
the end! Her masquerade had been discovered. She was
to be thrown out of the Opera House.
“And on this night of all nights!” She was ready to weep.
It was a very meek Pierre who at last stood before the
great director.
“Are you Pierre?” His tone was not harsh. She began to
hope a little.
“I am Pierre.”
“This man—” The director turned to one in the
shadows. Jeanne caught her breath. It was the great](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-70-320.jpg)
![[282]
[283]
sculptor, Fernando Tiffin.
“This man,” the director repeated, after she had
recovered from her surprise, “tells me that you know
the score of this new opera, ‘The Magic Curtain.’”
“Y-yes. Yes, I do.” What was this? Her heart throbbed
painfully.
“And that of the ‘Juggler of Notre Dame.’”
“I—I do.” This time more boldly.
“Surely this can be no crime,” she told herself.
“This has happened,” the director spoke out abruptly,
“Miss Dean is at the Robinson home. She has fallen
from a horse. She will not be able to appear to-night.
Fernando Tiffin tells me that you are prepared to
assume the leading role in these two short operas. I say
it is quite impossible. You are to be the judge.”
Staggered by this load that had been so suddenly cast
upon her slender shoulders, the little French girl seemed
about to sink to the floor. Fortunately at that instant her
eyes caught the calm, reassuring gaze of the great
sculptor. “I have said you are able.” She read this
meaning there.
“Yes.” Her shoulders were square now. “I am able.”
“Then,” said the director, “you shall try.”
Ninety minutes later by the clock, she found herself
waiting her cue, the cue that was to bid her come
dancing forth upon a great stage, the greatest in the](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-71-320.jpg)
![[284]
world. And looking down upon her, quick to applaud or
to blame, were the city’s thousands.
In the meantime, in her seat among the boxes, Florence
had met with an unusual experience. A mysterious
figure had suddenly revealed herself as one of Petite
Jeanne’s old friends. At the same time she had half
unfolded some month-old mysteries.
Petite Jeanne had hardly disappeared through the door
leading to the stage when two whispered words came
from behind Florence’s back:
“Remember me?”
With a start, the girl turned about to find herself looking
into the face of a tall woman garbed in black.
Reading uncertainty in her eyes, the woman whispered:
“Cedar Point. Gamblers’ Island. Three rubies.”
“The ‘lady cop’!” Florence sprang to her feet. She was
looking at an old friend. Many of her most thrilling
adventures had been encountered in the presence of
this lady of the police.
“So it was you!” she exclaimed in a low whisper. “You
are Jeanne’s lady in black?”
“I am the lady in black.”
“And she never recognized you?”
“I arranged it so she would not. She never saw my face.
I have been a guardian of her trail on many an
occasion.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-72-320.jpg)
![[285]
“And now!” Her figure grew tense, like that of a
springing tiger. “Now I am about to come to the end of
a great mystery. You can help me. That is why I
arranged that you should be here.”
“I?” Florence showed her astonishment.
“Sit down.”
The girl obeyed.
“Some weeks ago a priceless necklace was stolen from
this very box. You recall that?”
“How could I forget?” Florence sat up, all attention.
“Of course. Petite Jeanne, she is your best friend.
“She cast suspicion upon herself by deserting her post
here; running away. Had it not been for me, she would
have gone to jail. I had seen through her masquerade
at once. ‘This,’ I said to myself, ‘is Petite Jeanne. She
would not steal a dime.’ I convinced others. They spared
her.
“Then,” she paused for a space of seconds, “it was up to
me to find the pearls and the thief. I think I have
accomplished this; at least I have found the pearls. As I
said, you can help me. You know the people living on
that curious man-made island?”
“I—” Florence was thunderstruck.
Aunt Bobby! Meg! How could they be implicated? All this
she said to herself and was fearful.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-73-320.jpg)
![[286]
[287]
Then, like a bolt from the blue came a picture of Meg’s
birthday package.
“You know those people?” the “lady cop” insisted.
“I—why, yes, I do.”
“You will go there with me after the opera?”
“At night?”
“There is need for haste. We will go in Robinson’s big
car. Jaeger will go, and Rosemary. Perhaps Jeanne, too.
You will be ready? That is all for now.
“Only this: I think Jeanne is to have the stellar role to-
night.”
“Jeanne! The stellar role? How could that be?”
“I think it has been arranged.”
“Arranged?”
There came no answer. The lady in black was gone.](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-74-320.jpg)
![[288]
CHAPTER XXXII
SPARKLING TREASURE
The strangest moment in the little French girl’s career
was that in which, as the juggler, she tripped out upon
the Opera House stage. More than three thousand
people had assembled in this great auditorium to see
and hear their favorite, the city’s darling, Marjory Dean,
perform in her most famous role. She was not here.
They would know this at once. What would the answer
be?
The answer, after perfunctory applause, was a deep
hush of silence. It was as if the audience had said:
“Marjory Dean is not here. Ah, well, let us see what this
child can do.”
Only her tireless work under Miss Dean’s direction saved
Jeanne from utter collapse. Used as she was to the
smiling faces and boisterous applause of the good old
light opera days, this silence seemed appalling. As it
was, she played her part with a perfection that was art,
devoid of buoyancy. This, at first. But as the act
progressed she took a tight grip on herself and throwing
herself into the part, seemed to shout at the dead](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-75-320.jpg)
![[289]
audience: “You shall look! You shall hear! You must
applaud!”
For all this, when the curtain was run down upon the
scene, the applause, as before, lacked enthusiasm. She
answered but one curtain call, then crept away alone to
clench her small hands hard in an endeavor to keep
back the tears and to pray as she had never prayed
before, that Marjory Dean might arrive prepared to play
her part before the curtain went up on the second act.
But now a strange thing was happening. From one
corner of the house there came a low whisper and a
murmur. It grew and grew; it spread and spread until,
like a fire sweeping the dead grass of the prairies, it had
passed to the darkest nook of the vast auditorium.
Curiously enough, a name was on every lip;
“Petite Jeanne!”
Someone, a fan of other days, had penetrated the girl’s
mask and had seen there the light opera favorite of a
year before. A thousand people in that audience had
known and loved her in those good dead days that were
gone.
When Jeanne, having waited and hoped in vain for the
appearance of her friend and benefactor, summoned all
the courage she possessed, and once more stepped
upon the stage, she was greeted by such a round of
applause as she had never before experienced—not
even in the good old days of yesteryear.
This vast audience had suddenly taken her to its heart.
How had this come about? Ah, well, what did it matter?](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-76-320.jpg)
![[290]
They were hers, hers for one short hour. She must make
the most of this golden opportunity.
That which followed, the completing of the “Juggler,”
the opening of “The Magic Curtain,” the complete
triumph of this new American opera, will always remain
to Jeanne a beautiful dream. She walked and danced,
she sang and bowed as one in a dream.
The great moment of all came when, after answering
the fifth curtain call with her name, “Petite Jeanne!
Petite Jeanne!” echoing to the vaulted ceiling, she left
the stage to walk square into the arms of Marjory Dean.
“Why, I thought—” She paused, too astounded for
words.
“You thought I had fallen from a horse. So I did—a
leather horse with iron legs. It was in a gymnasium.
Rosemary pushed me off. Truly it did not hurt at all.”
“A frame-up!” Jeanne stared.
“Yes, a frame-up for a good cause. ‘The Magic Curtain’
was yours, not mine. You discovered it. It was through
your effort that this little opera was perfected. It was
yours, not mine. Your golden hour.”
“My golden hour!” the little French girl repeated
dreamily. “But not ever again. Not until I have sung and
sung, and studied and studied shall I appear again on
such a stage!”
“Child, you have the wisdom of the gods.”
“But the director!” Jeanne’s mood changed. “Does he
not hate you?”](https://image.slidesharecdn.com/1316236-250601160031-f4732dc9/85/Applications-Of-Unitary-Symmetry-And-Combinatorics-James-D-Louck-77-320.jpg)
Applications Of Unitary Symmetry And Combinatorics James D Louck
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- 5. Applications of Unitary Symmetry and Combinatorics 8161 tp.indd 1 4/15/11 10:17 AM
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- 7. NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI World Scientific Applications of Unitary Symmetry and Combinatorics James D Louck Los Alamos National Laboratory Fellow Santa Fe, New Mexico, USA 8161 tp.indd 2 4/15/11 10:17 AM
- 8. 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-13 978-981-4350-71-6 ISBN-10 981-4350-71-0 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 © 2011 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. APPLICATIONS OF UNITARY SYMMETRY AND COMBINATORICS LaiFun - Applications of Unitary Symmetry.pmd 2/1/2011, 4:26 PM 1
- 9. In recognition of contributions to the generation and spread of knowledge William Y. C. Chen Tadeusz and Barbara Lulek And to the memory of Lawrence C. Biedenharn Gian-Carlo Rota
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- 11. Preface and Prelude We have titled this monograph “Applications of Unitary Symmetry and Combinatorics” because it uses methods developed in the earlier volume “Unitary Symmetry and Combinatorics,” World Scientific, 2008 (here- after Ref. [46] is referred to as [L]). These applications are highly topical, and come in three classes: (i) Those still fully mathematical in con- tent that synthesize the common structure of doubly stochastic, magic square, and alternating sign matrices by their common expansions as linear combinations of permutation matrices; (ii) those with an associ- ated physical significance such as the role of doubly stochastic matrices and complete sets of commuting Hermitian operators in the probabilistic interpretation of nonrelativistic quantum mechanics, the role of magic squares in a generalization of the Regge magic square realization of the domains of definition of the quantum numbers of angular momenta and their counting formulas (Chapter 6), and the relation between alternat- ing sign matrices and a class of Gelfand-Tsetlin patterns familiar from the representation of irreducible representations of the unitary groups (Chapter 7) and their counting formulas; and (iii) a physical applica- tion to the diagonalization of the Heisenberg magnetic ring Hamiltonian, viewed as a composite system in which the total angular momentum is conserved. A uniform viewpoint of rotations is adopted at the outset from [L], based on the method of Cartan [15] (see also [6]), where it is fully defined and discussed. It is not often made explicit in the present volume: A unitary rotation of a composite system is its redescription under an SU(2) unitary group frame rotation of a right-handed triad of perpendic- ular unit vectors (e1, e2, e3) that serves as common reference system for the description of all the constituent parts of the system. This Preface serves three purposes: A prelude and synthesis of things to come based on results obtained in [L], now focused strongly on the basic structural elements and their role in bringing unity to the under- standing of the angular momentum properties of complex systems viewed as composite wholes; a summary of the contents by topics; and the usual elements of style, readership, acknowledgments, etc. vii
- 12. viii PREFACE AND PRELUDE OVERVIEW AND SYNTHESIS OF BINARY COUPLING THEORY The theory of the binary coupling of n angular momenta is about the pairwise addition of n angular momenta associated with n constituent parts of a composite physical system and the construction of the asso- ciated state vectors of the composite system from the SU(2) irreducible angular momentum multiplets of the parts. Each such possible way of effecting the addition is called a binary coupling scheme. We set forth in the following paragraphs the underlying conceptual basis of such binary coupling scheme. Each binary coupling scheme of order n may be described in terms of a sequence having two types of parts: n points ◦ ◦ · · · ◦ and n − 1 parenthesis pairs ( ), ( ), . . . , ( ). A parenthesis pair ( ) constitutes a single part. Thus, the number of parts in the full sequence is 2n − 1. By definition, the binary bracketing of order 1 is ◦ itself, the binary bracketing of order 2 is (◦ ◦), the two binary bracketings of order 3 are (◦ ◦) and ◦ (◦ ◦) , . . . . In general, we have the definition: A binary bracketing Bn of order n ≥ 2 is any sequence in the n points ◦ and the n−1 parenthesis pairs ( ) that satisfies the two conditions: (i) It contains a binary bracketing of order 2, and (ii) the mapping (◦ ◦) 7→ ◦ gives a binary bracketing of order n − 1. Then, since the mapping (◦ ◦) 7→ ◦ again gives a binary bracketing for n ≥ 3, the new binary bracketing of order n − 1 again contains a binary bracketing of order 2. This implies that this mapping property can be used repeatedly to reduce every binary bracketing of arbitrary order to ◦, the binary bracketing of order 1. The appropriate mathematical concept for diagramming all such bi- nary bracketings of order n is that of a binary tree of order n. We have described in [L] a “bifurcation of points” build-up principle for construct- ing the set Tn of all binary trees of order n in terms of levels (see [L, Sect. 2.2]). This is a standard procedure found in many books on combi- natorics. It can also be described in terms of an assembly of four basic objects called forks that come in four types, as enumerated by ◦ ◦ • , @ @ (1) ◦ • • , @ @ (2) • ◦ • , @ @ (3) • • • @ @ (4) The • point at the bottom of these diagrams is called the root of the fork, and the other two point are called the endpoints of the fork. The assembly rule for forks into a binary tree of order n can be formulated in term of the “pasting” together of forks.
- 13. PREFACE AND PRELUDE ix Our interest in viewing a binary tree as being composed of a collection of pasted forks of four basic types is because the configuration of forks that appears in the binary tree encode exactly how the pairwise addition of angular momenta of the constituents of a composite system is to be effected. Each such labeled fork has associated with it the elementary rule of addition of the two angular momenta, as well as the Wigner- Clebsch-Gordan (WCG) coefficients Cj1 j2 k m1 m2 µ that effect the coupling of the state vectors of two subsystems of a composite system having angu- lar momentum J(1) and J(2), respectively, to an intermediate angular momentum J(1) + J(2) = K, as depicted by the labeled fork: ◦ ◦ • : @ @ j1 j2 k k = j1 + j2, j1 + j2 − 1, . . . , |j1 − j2|. Similarly, the labeled basic fork 2 given by ◦ • • : @ @ ji k k′ k′ = ji + k, ji + k − 1, . . . , |ji − k| encodes the addition J(i) + K = K′ of an angular momentum J(i) of the constituent system and an intermediate angular momentum K to a “total” intermediate angular momentum K′, as well as the attended WCG coefficients Cji k k′ mi µ µ′ that effect the coupling. Labeled forks 3 and 4 have a similar interpretation. These labeled forks of a standard labeled binary tree of order n encode the constituent angular momenta that enter into the description of the basic SU(2) irreducible state vectors of a composite system. A build-up rule for the pasting of forks can be described as follows: 1. Select a fork from the set of four forks above and place the • root point over any • endpoint of the four forks, merging the two • points to a single • point. Repeat this pasting process for each basic fork. This step gives a set of seventeen distinct graphs, where we include the basic fork containing the two ◦ points in the collection:
- 14. x PREFACE AND PRELUDE ◦ ◦ • @ @ ◦ ◦ • ◦ • @ @ @ @ ◦ • • ◦ • @ @ @ @ • ◦ • ◦ • @ @ @ @ • • • ◦ • @ @ @ @ ◦ ◦ • ◦ • @ @ @ ◦ • • ◦ • @ @ @ • ◦ • ◦ • @ @ @ • • • ◦ • @ @ @ ◦ ◦ • • • @ @ @ ◦ ◦ • • • @ @ @ @ ◦ • • • • @ @ @ ◦ • • • • @ @ @ @ • ◦ • • • @ @ @ • ◦ • • • @ @ @ @ • • • • • @ @ @ • • • • • @ @ @ @ 2. Select a single graph from the set of seventeen graphs generated at Step 1 and repeat the pasting process with each of the four basic forks. This gives back the three graphs from the collection above having only endpoints of type ◦, which includes all binary tree graphs of order 2 and order 3, and, in addition, ninety-six more graphs as follows: (1)(4)(6) from the six graphs above hav- ing one • endpoints; (2)(4)(6) from the six graphs having two • endpoints, and (3)(4)(2) from the three graphs having three • end- points. From this large collection, put aside those having only ◦ endpoints, including no repetitions. This gives the set of all binary trees or order 2, 3, 4: ◦ ◦ • @ @ ◦ ◦ • ◦ • @ @ @ ◦ ◦ • ◦ • @ @ @ @ ◦ ◦ • ◦ • ◦ • @ @ @ @ @ ◦ ◦ • ◦ ◦ • • @ @ @ @ @ ◦ ◦ ◦ ◦ • • •, , , , e e e e @ @ ◦ ◦ • ◦ ◦ • • @ @ @ @ @ @ ◦ ◦ • ◦ • ◦ • @ @ @ @ @ @
- 15. PREFACE AND PRELUDE xi 3. Select a single graph from the full set generated at Step 2 and repeat the pasting process. putting aside all those having only ◦ endpoints, including no repetitions. Repeat this process for the next set of graphs, etc. At Step h of this pasting process of basic forks to the full collection generated at Step h−1, there is obtained a huge multiset of graphs that includes all binary trees of order 2, 3, . . . , h + 2 — all those with ◦ endpoints. A very large number of graphs with 1 • endpoint, 2 • endpoints, . . . is also obtained, these being needed for the next step of the pasting process. All binary trees of order n are included in the set of graphs generated by the pasting process at step h = n − 2, these being the ones put aside. The pasting process is a very inefficient method for obtaining the set of binary trees of order n. A more efficient method is to generate this subset recursively by the following procedure: Suppose the set of binary trees of order n has already been obtained, and that it contains an members. Then, consider the new set of nan graphs obtained by replacing a single ◦ endpoint by a • point in each of the an binary tree graphs. In the next step, paste the • root of basic fork 1 over each of these • endpoints, thus obtaining nan binary trees of order n + 1. This multiset of cubic graphs of order n + 1 then contains the set of an+1 binary trees of order n + 1. Thus, starting with fork 1 from the basic set of four, we generate all binary trees of arbitrary order by this pasting procedure. (This procedure works because every graph having one • endpoint is included in the huge set generate in Items 1, 2, 3 above.) Only binary trees having ◦ endpoints enter into the binary coupling theory of angular momenta because each such coupling scheme must correspond to a binary bracketing. The set of binary bracketings of order n is one-to-one with the set of binary trees of order n, and the bijection rule between the two sets can be formulated exactly. The cardinality of the set Tn is the Catalan number an. We repeat, for convenience, many examples of such binary bracketings and corresponding binary trees from [L], and also give in this volume many more, including examples for arbitrary n. As in [L], we call the binary bracketing corresponding to a given binary tree the shape of the binary tree: The shape ShT of a binary tree T ∈ Tn is defined to be the binary bracketing of order n corresponding to T. The concept of shape transformation is so basic that we discuss it here in the Preface in its simplest realization: There are two binary trees of order three, as given above. We now label the endpoints of these binary trees by a permutation of some arbitrary set of elements x1, x2, x3 :
- 16. xii PREFACE AND PRELUDE ◦ ◦ • ◦ • , @ @ @ x1 x2 x3 T = ◦ ◦ • ◦ • @ @ @ @ x2 x1 x3 T′ = We initially ignore the labels. The first graph corresponds to the binary bracketing ((◦ ◦) ◦), and the second one to (◦ (◦ ◦)). The binary bracket- ing, now called the shape of the binary tree, is defined in the obvious way by reading left-to-right across the binary tree, and inserting a parenthesis pair for each • point: ShT = ((◦ ◦) ◦), ShT′ = (◦ (◦ ◦)). These shapes are called unlabeled shapes. If the ◦ endpoints are labeled by a permutation of distinct symbols such as x1, x2, x3, the ◦ points of the shape are labeled by the corresponding symbols to obtain the labeled shapes ShT (x1, x2, x3) = ((x1 x2) x3), ShT′ (x2, x1, x3) = (x2 (x1 x3)). What is interesting now is that the first labeled shape can be transformed to the second labeled shape by the elementary operations C and A of commutation and association: ((x1 x2) x3) C −→ ((x2 x1) x3) A −→ (x2 (x1 x3)), where the action of commutation C and that of association A have their usual meaning: C : (x y) 7→ (y x), A : (x y) z 7→ x (y z). Thus, we have the shape transformation: ShT (x1, x2, x3) AC −→ ShT′ (x2, x1, x3), where the convention for the action of commutation and association is that C acts first on the shape ShT (x1, x2, x3) to effect the transformation to the shape ShT (x2, x1, x3) followed by the action of A on ShT (x2, x1, x3) to give the shape ShT′ (x2, x1, x3). We state at the outset the generalization of this result: The set Tn of binary trees of order n is unambiguously enumerated by its set of shapes ShT , T ∈ Tn, the number of which is given by the Catalan numbers an. Let x1, x2, . . . , xn be a collection of n arbi- trary distinct objects and xπ = (xπ1 , xπ2 , . . . , xπn ) an arbitrary permu- tation of the xi, where π = (π1, π2, . . . , πn) is an arbitrary permuta- tion in the group Sn of all permutations of the reference set 1, 2, . . . , n.
- 17. PREFACE AND PRELUDE xiii Then, there exists a shape transformation w(A, C) such that the labeled shape ShT (xπ) is transformed to the labeled shape ShT′ (xπ′ ), for each T ∈ Tn, π ∈ Sn, T′ ∈ Tn, π′ ∈ Sn; that is, ShT (xπ) w(A,C) −→ ShT′ (xπ′ ), where the shape transformation w(A, C) is a word in the two letters A and C. The content of this result will be amplified by many explicit examples. It is a basic unifying result for the binary theory of the coupling of n angular momenta. The pairwise addition of n angular momenta J(i) = J1(i)e1+J2(i)e2+ J3(i)e3, i = 1, 2, . . . , n, with components referred to a common right- handed inertial reference frame (e1, e2, e3), to a total angular momen- tum J = J1e1 + J2e2 + J3e3 = J(1) + J(2) + · · · + J(n) is realized in all possible ways by the standard labeling of each binary tree T ∈ Tn as given in terms of its shape by ShT (J(π1), J(π2), . . . , J(πn)), where π = (π1, π2, . . . , πn) is any permutation π ∈ Sn. The pairwise addition encoded in a given binary tree for any given permutation π ∈ Sn is called a coupling scheme. For example, the two coupling schemes encoded by the shapes of the binary trees of order 3 given above are: J(1) + J(2) + J(3) , J(3) + J(2) + J(1) . For general n, there are n!an distinct coupling schemes for the pair- wise addition of n angular momenta. The rule whereby we assign the n angular momenta (J(1), J(2), . . . , J(n)) of the n constituents of a com- posite system to the endpoints of a binary tree T ∈ Tn by the left-to-right assignment to the corresponding points in the shape ShT of T is called the standard rule labeling of the ◦ points (all endpoints) of the binary tree T. In many instances, we need also to apply this standard rule label- ing for the assignment of a permutation (J(π1), J(π2), . . . , J(πn)) of the angular momenta of the constituent parts, as illustrated in the example above. Caution must be exercised in the interpretation of all these pair- wise additions of angular momenta in terms of the tensor product space in which these various angular momenta act. This is reviewed from [L] in Chapters 1 and 5 in the context of the applications made here. A stan- dard labeling rule is required to define unambiguously objects such as Wigner-Clebsch-Gordan coefficients and triangle coefficients to labeled binary trees, objects that given numerical content to the theory. To summarize: In the context of the binary coupling theory of angu- lar momenta, we deal with standard labeled binary trees, their shapes, and the transformations between shapes. Out these few simple under- lying structural elements there emerges a theory of almost unlimited,
- 18. xiv PREFACE AND PRELUDE but manageable, complexity: Mathematically, it is theory of relations between 3(n − 1) − j coefficients and Racah coefficients and their cu- bic graphs; physically, it is a theory of all possible ways to compound, pairwise, the individual angular momenta of the n constituent parts of a complex composite system to the total angular momentum of the system. The implementation of the binary coupling theory of angular mo- menta leads directly to the Dirac concept [24] of characterizing the Hilbert space state vectors of each coupling scheme in terms of com- plete sets of commuting Hermitian operators. This characterization is described in detail in [L], and reviewed in the present volume in Chapters 1 and 5; it is described broadly as follows. The complete set of 2n mu- tually commuting Hermitian operators for each coupling scheme T ∈ Tn is given by J2 (1), J2 (2), . . . , J2 (n), J2 , J3; K2 T (1), K2 T (2), . . . , K2 T (n − 2). The first line of operators consists of the n total angular momentum squared of each of the constituent systems, together with the total angu- lar momentum squared of the composite system and its 3−component. The second line of operators consists of the squares of the n−2 so-called intermediate angular momenta, KT (i) = (KT,1(i), KT,2(i), KT,3(i)), i = 1, 2, . . . , n − 2. There is a distinct set of such intermediate angular mo- menta associated with each binary tree T ∈ Tn, where each KT (i) is a 0 − 1 linear combination of the constituent angular momenta J(i), i = 1, 2, . . . , n, in which the 0 and 1 coefficients are uniquely determined by the shape of the binary tree T ∈ Tn. Thus, each standard labeled bi- nary coupling scheme has associated with it a unique complete set of 2n mutually commuting Hermitian operators, as given generally above. For example, the complete sets of mutually commuting Hermitian operators associated with the labeled binary trees T and T′ of order 3 given above are the following, respectively: scheme T : J2 (1), J2 (2), J2 (3), J2 , J(3); K2 T (1) = J(1) + J(2) 2 . scheme T′ : J2 (1), J2 (2), J2 (3), J2 , J(3); K2 T′ (1) = J(2) + J(1) 2 . There is a set of simultaneous eigenvectors associated with each com- plete set of mutually commuting Hermitian operators defined above for each binary tree T ∈ Tn of labeled shape ShT (J(π1), J(π2), . . . , J(πn)). It is convenient now to denote this shape by ShT (jπ), where we use the angular momentum quantum number ji in place of the angular momen- tum operator J(i) in the labeled shape. These are the quantum numbers ji associated with the eigenvalue ji(ji +1) of the squared “total” angular momentum J2(i), each i = 1, 2, . . . , n, of each of the n constituents of the
- 19. PREFACE AND PRELUDE xv composite physical system in question. The simultaneous eigenvectors in this set are denoted in the Dirac ket-vector notation by |T(jπ k)j mi, where jπ = (jπ1 , jπ2 , . . . , jπn ), and where j denotes the total angular momentum quantum number of the eigenvalue j(j + 1) of the squared total angular momentum J2 and m the eigenvalue of the 3−component J3. We generally are interested in the finite set of vectors enumerated, for specified j = (j1, j2, . . . , jn), each ji ∈ {0, 1/2, 1, 3/2, 2, . . .}, by the range of values of the total angular momentum quantum numbers j m; and by the range of values of all the intermediate quantum numbers k = (k1, k2, . . . , kn−2). The latter are associated with the eigenvalues ki(ki + 1) of the squared intermediate angular momenta for given T ∈ Tn. This gives the set of eigenvectors associated with the labeled shape ShT (jπ), and defines an orthonormal basis of a finite-dimensional tensor product Hilbert space denoted by Hj = Hj1 ⊗Hj2 ⊗· · ·⊗Hjn of dimension equal to (2j1 + 1)(2j2 + 1) · · · (2jn + 1). The domain of definition of j m for each coupled state vector |T(jπ k)j mi corresponding to the binary tree T of shape ShT (jπ) is j ∈ {jmin, jmin + 1, . . . , jmax}, where jmin is the least nonnegative integer or positive half-odd integer among the 2n sums of the form ±j1 ± j2 ± · · · ± jn, and jmax = j1 + j2 + · · · + jn. The domain of definition of the intermediate quantum number ki depends on the labeled shape of the binary tree T; it belongs to a uniquely defined set as given by k ∈ K (j) T (jπ), the details of which are not important here. Thus, an orthonormal basis of the space Hj is given, for each T ∈ Tn and each labeled shape ShT (jπ), by BT (jπ) = n |T(jπ k)j mi j ∈ {jmin, jmin + 1, . . . , jmax}; and for each j, k ∈ K (j) T (jπ); m ∈ {j, j − 1, . . . , −j o . In all, we have n!an sets of coupled orthonormal basis vectors, each set BT (jπ) giving a basis of the same tensor product space Hj. We digress a moment to recall that the basic origin of the tensor prod- uct Hilbert space Hj is just the vector space formed from the tensor prod- uct of the individual Hilbert spaces Hji of dimension 2ji +1 on which the angular momentum J(i) has the standard action, with the commuting Hermitian operators J2(i), J3(i) being diagonal with eigenvalues ji(ji+1) and mi. The orthonormal basis of the space Hj is now the uncoupled ba- sis |j mi = |j1 m1i ⊗ |j2 m2i ⊗ · · · ⊗ |jn mni, in which we can have ji ∈ {0, 1/2, 1, 3/2, 2, . . .}, and, for each selected ji, the so-called projection quantum number mi assumes all values mi = ji, ji −1, . . . , −ji. Thus, the uncoupled orthonormal basis Bj = {|j mi | each mi = ji, ji − 1, . . . , −ji} of Hj is the space of simultaneous eigenvectors of the complete mutu- ally commuting Hermitian operators J2(i), J3(i), i = 1, 2, . . . , n, where
- 20. xvi PREFACE AND PRELUDE the components of each of the angular momentum operators J(i) has the standard action on Hji . These basic relations are presented in great detail in Ref. [6], in [L], and reviewed in Chapter 1 of this volume. The important point for the present work is: Each simultaneous eigenvector |T(jπ k)j mi of the set of 2n mutually com- muting Hermitian operators corresponding to each binary tree T ∈ Tn of shape ShT (jπ) is a real orthogonal transformation of the eigenvec- tors |j mi ∈ Bj. The coefficients in each such transformation are gen- eralized Wigner-Clebsch-Gordan (WCG) coefficients, which themselves are a product of known ordinary WCG coefficients, where the product is uniquely determined by the shape of the labeled binary tree ShT (jπ). This is, of course, just the expression of the property that we have con- structed n!an uniquely defined coupled orthonormal basis sets of the space Hj from the uncoupled basis of the same space Hj. There is a class of subspaces of Hj of particular interest for the present work. This class of subspaces is obtained from the basis vec- tors BT (jπ) of Hj given above by selecting from the orthonormal basis vectors |T(jπ k)j mi those that have a prescribed total angular momen- tum quantum number j ∈ {jmin, jmin + 1, . . . , jmax} and, for each such j, a prescribed projection quantum number m ∈ {j, j − 1, . . . , −j}. Thus, the orthonormal basis vectors in this set are given by BT (jπ, j, m) = {|T(jπ k)j mi | each k ∈ K (j) T (jπ)}. We further specialize this basis set to the case π = identity permutation: BT (j, j, m) = {|T(j k)j mi | each k ∈ K (j) T (j)}. This basis set of orthonormal vectors then defines a subspace H(j, j, m) ⊂ Hj, which is the same vector space for each T ∈ Tn and for every per- mutation jπ of j; that is, the following direct sum decomposition of the tensor product space Hj holds: Hj = jmax X j=jmin j X m=−j ⊕H(j, j, m). We repeat: The important structural result for this vector space decom- position is: Each basis set BT (jπ, j, m), T ∈ Tn, π ∈ Sn, is an orthonormal basis of one and the same space H(j, j, m). The dimension of the tensor product space H(j, j, m) ⊂ Hj is Nj(j), the Clebsch-Gordan (CG) number. This important number is the num- ber of times that a given j ∈ {jmin, jmin + 1, . . . , jmax} is repeated for
- 21. PREFACE AND PRELUDE xvii specified j. They can be calculated explicity by repeated application of the elementary rule for the addition of two angular momenta, as de- scribed in detail in [L]. The CG number is shape independent; that is, Nj(j) counts the number of orthonormal basis vectors in the basis set {|T(jπ k)j mi |for specified j m} of the space H(j, j, m) for each binary tree T ∈ Tn and each jπ, π ∈ Sn. It may seem highly redundant to introduce such a variety of or- thonormal basis sets of vectors that span the same space Hj, but it is within these vector space structures that resides the entire theory of 3(n − 1) − j coefficients. This aspect of the theory is realized through shape transformations applied to the binary trees whose standard la- bels appear in the binary coupled state vectors in the various basis sets HT (jπ, j, m), T ∈ Tn, π ∈ Sn. The implementation of such shape trans- formations into numerical-valued transformations between such state vectors uses the notion of a recoupling matrix. The matrix with elements that give the real orthogonal transfor- mation matrix between distinct sets {|T(jπ k)j mi | k ∈ K (j) T (jπ)} and {|T(jπ′ k′)j mi | k ∈ K (j) T (jπ′ )} of simultaneous basis eigenvectors, each of which is an orthonormal basis of the vector space H(j, j, m), is called a recoupling matrix. Thus, we have that |T′ (jπ′ k′ )j mi = X k∈K (j) T (jπ ) RS; S′ k,j; k′,j |T(jπ k)j mi, where the recoupling matrix is the real orthogonal matrix, denoted RS; S′ , with elements given by the inner product of state vectors: RS; S′ k,j; k′,j = D T(jπ k)jm T′ (jπ′ k ′ )jm E . Here we have written the labeled shapes in the abbreviated forms: S = ShT (jπ), S′ = ShT′ (jπ′ ). As this notation indicates, these transformations are independent of m; that is, they are invariants under SU(2) frame transformations. The matrix elements are fully determined in each coupling scheme in terms of known generalized WCG coefficients, since each coupled state vector in the inner product is expressed as a linear combination of the orthonormal basis vectors in the set Bj of the tensor product space Hj with coefficients that are generalized WCG coefficients. Each recoupling matrix is a fully known real orthogonal matrix of order Nj(j) in terms of its elements, the generalized WCG coefficients. Since the inner product is real, it is always the case that the recoupling matrix satisfies the relation RS; S′ = RS′ ; S tr ,
- 22. xviii PREFACE AND PRELUDE where tr denotes the transpose of the matrix. But the most significant property of a recoupling matrix that originates from the completeness of the mutually commuting Hermitian operators that define each binary coupled state is the multiplication property: RS1; S3 = RS1; S2 RS2; S3 , which holds for arbitrary binary trees T1, T2, T3 ∈ Tn and for all possi- ble labeled shapes S1 = ShT1 (jπ(1) ), S2 = ShT2 (jπ(2) ), S3 = ShT3 (jπ(3) ), where each permutation π(i) ∈ Sn. It is this elementary multiplication rule that accounts fully for what are known as the Racah sum-rule be- tween Racah coefficients and the Biedenharn-Elliott identity between Racah coefficients. Indeed, when iterated, this multiplication rule gener- ates infinite classes of relations between 3(n − 1) − j coefficients, n ≥ 3. We can now bring together the notion of general shape transforma- tions as realized in terms of words in the elementary association actions A and commutation actions C to arrive at the fundamental relation un- derlying the properties of the set of binary coupled angular momenta state vectors: Let w(A, C) = L1L2 · · · Lr, where each Lh is either an elementary association operation A or a ele- mentary commutation operation C, give a word w(A, C) that effects the shape transformation given by S1 L1 −→ S2 L2 −→ S3 L3 −→ · · · Lr −→ Sr+1. The abbreviated shapes are defined by Sh = ShTh (jπ(h) ), h = 1, 2, . . . , r, with corresponding elementary shape transformations given by Sh Lh −→ Sh+1, h = 1, 2, . . . , r. Thus, the transformation from the initial shape S1 to the final shape Sr+1 is effected by a succession of elementary shape transformations via S2, S3, . . . , Sr. The matrix elements of the product of recoupling matrices given by RS1; Sr+1 = RS1; S2 RS1; S2 RS2;S3 · · · RSr; Sr+1 is equal to the inner product of binary coupled state vectors given by RS1; Sr+1 k(1),j; k(r+1),j = D T1(jπ(1) k(1) )j m Tr+1(jπ(r+1) k(r+1) )j m E . Then, the main result is: There exists such a shape transformation by elementary shape operations between every pair of specified initial and final shapes, S1 and Sr+1.
- 23. PREFACE AND PRELUDE xix We introduce yet another very useful notation and nomenclature for the matrix elements of a general recoupling matrix RS; S′ , which are given by the inner product of state vectors above. This is the concept of a triangle coefficient, which encodes the detailed coupling instructions of its labeled forks discussed above. A triangle coefficient has a left- triangle pattern and a right-triangle pattern. The left-triangle pattern is a 3×(n−1) matrix array whose 1×3 columns, n−1 in number, are the quantum numbers that encode the elementary addition of two angular momenta of the n−1 labeled forks that constitute the fully labeled binary tree T(jπ k)j. For example, the column corresponding to the labeled fork 1 given earlier is col(j1 j2 k) with a similar rule for labeled forks of the other three types. The left-triangle pattern of a triangle coefficient is this collection of n − 1 columns, read off the fully labeled binary tree, and assembled into a 3×(n−1) matrix pattern by a standard rule. The right- triangle pattern is the 3×(n−1) triangular array constructed in the same manner from the fully labeled binary tree T′(jπ′ k ′ )j. These two triangle patterns denoted, respectively, by ∆T (jπ k)j and ∆T′ (jπ′ k ′ )j, are now used to define the triangle coefficient of order 2(n − 1) : n ∆T (jπ k)j ∆T′ (jπ′ k ′ )j o = RS; S′ k,j; k′,j = D T(jπ k)jm T′ (jπ′ k ′ )jm E . The main result for the present discussion is: The triangle coeffi- cients (matrix elements of recoupling matrices) for the matrix elements of the elementary commutation operation C and association operation A in the product of recoupling matrices is a basic phase factor of the form (−1)a+b−c for a C−transformation, and a definite numerical object of the form (−1)a+b−c p (2k + 1)(2k′ + 1) W for an A−transformation, where W is a Racah coefficient. It follows that the matrix elements RS1; Sr+1 k(1),j; k(r+1),j of the recoupling matrix is always a summation over a number of Racah coefficients equal to the number of associations A that occur in the word w(A, C), where the details of the multiple sum- mations depend strongly on the shapes of the underlying pair of binary trees related by the A−transformations. Each word w(A, C) = L1L2 · · · Lr that effects a shape transformation between binary coupling schemes corresponding to a shape S1 and Sr+1 has associated with it a path, which is defined to be path = S1 L1 −→ S2 L2 −→ · · · Lr −→ Sr+1. But there are many distinct words w1(A, C), w2(A, C), . . . that effect the same transformation S1 wi(A,C) −→ Sr+1, i = 1, 2, . . .
- 24. xx PREFACE AND PRELUDE via different intermediate shapes. Hence, there are correspondingly many paths of the same or different lengths between the same given pair of shapes, where the length of a path is defined to be the number of asso- ciations A in the path. There is, of course, always a path of minimum length. It is this many-fold structure of paths that gives rise to different expressions of one and the same 3(n − 1) − j coefficient, as well as to a myriad of relations between such coefficients. An arbitrary triangle coefficient of order 2(n−1) is always expressible as a product of recoupling matrices related by elementary shape trans- formations that give either simple phase transformations or a triangle coefficient of order four, since irreducible triangle coefficients of order four (those not equal to a phase factor or zero) are always of the form (−1)a+b−c p (2k + 1)(2k′ + 1) W. Thus, triangle coefficients of order four may be taken as the fundamental objects out of which are built all tri- angle coefficients. Triangle coefficients provide a universal notation for capturing the structure of all recoupling matrices. They possess some general simplifying structural properties that are inherited from the pair of standard labeled binary trees whose labeled fork structure they en- code. For the description of this structure, we introduce the notion of a common fork: Two standard labeled binary trees are said to have a common fork if each binary tree contains a labeled fork having endpoints with the same pair of labels, disregarding order.Then, the left and right patterns of the triangle coefficient corresponding to a pair of standard labeled binary tree with a common fork has a column in its left pattern and one in its right pattern for which the entries in the first two rows are either the same or the reversal of one another: If the order of the labels in the two columns is the same, then the triangle coefficient is equal to the reduced triangle coefficient obtained by removal of the column from each pattern and multiplying the reduced pattern by a Kronecker delta factor in the intermediate quantum labels in row three of the common fork column. A similar reduction occurs should the quantum labels of the common fork be reversed, except now a basic phase factor multiplies the reduced triangle coefficient. Of course, if the resulting reduced trian- gle coefficient contains still a pair of columns corresponding to a common fork, then a further reduction takes place. This continues until a trian- gle coefficient of order 2(n − 1) containing s columns corresponding to s common forks is reduced to a product of basic phase factors times an irreducible triangle coefficient of order 2(n − s). An irreducible triangle coefficient is one for which the corresponding pair of labeled binary trees has no common fork — the triangle coefficient has no common columns. But the structure of irreducible triangle coefficients does not stop here. Each irreducible triangle coefficient defines a cubic graph. In particular, irreducible triangle coefficients of order 2(n − 1) enumerate all possible “types” of cubic graphs of order 2(n − 1) that can occur in the binary theory of the coupling of angular momenta. The cubic graph
- 25. PREFACE AND PRELUDE xxi C∗ 2(n−1) of an irreducible triangle coefficient of order 2(n − 1) is obtained by the very simple rule: Label 2(n − 1) points by the 2(n − 1) triplets (triangle) of quantum labels constituting the columns of the triangle coefficient, and draw a line between each pair of points that is labeled by a triangle containing a common symbol. This defines a graph with 2(n − 1) points, 3(n − 1) lines, with three lines incident on each point, which is the definition of a cubic graph C∗ 2(n−1) of “angular momentum type” of order 2(n − 1). While cubic graphs do not enter directly into such calculations, they are the objects that are used to classify a given collection of 3(n − 1) − j coefficients into types. In summary: We have crafted above a conceptual and graphical frame- work that gives a uniform procedure for computing all 3(n − 1) − j co- efficients, based in the final step on computing the matrix elements of a recoupling matrix expanded into a product of recoupling matrices corre- sponding to a path of elementary shape transformations. The reduction process, applied to the matrix elements in this product, then automatically reduces in consequence of common forks to give the desired expression for the matrix elements of an arbitrary recoupling matrix (triangle coef- ficient),which is equal to the inner product of state vectors fully labeled by the simultaneous eigenvectors of the respective pair of complete sets of 2n commuting Hermitian operators. Moreover, properties of recoupling matrices can be used to generate arbitrarily many relations among irre- ducible triangle matrices of mixed orders, and expressions for one and the same 3(n − 1) − j in terms of different coupling schemes. The uni- fication of the binary coupling theory of angular momenta is achieved. There remain, however, unresolved problems such as: A procedure for obtaining paths of minimum length, a counting formula for the number of cubic graphs of order 2(n − 1) of “angular momentum type,” and the physical meaning of the existence of many paths for expressing the same 3(n−1)−j coefficient. Chapters 1, 5, and 8 provide more comprehensive details of results presented in this overview. We have taken the unusual step of presenting this overview here in the Preface so as to have in one place a reasonable statement of the coherence brought to the subject of angular momentum coupling theory by the methods outlined above, unblurred by the intricate steps needed in its implementation. The binary coupling theory of angular momenta has relevance to quantum measurement theory. Measurements of the properties of com- posite systems is only at the present time, in sophisticated experimental set-ups, revealing the behavior of systems prepared in an initial defi- nite state that remains unmeasured (undisturbed) until some later time, when a second measurement is made on the system. It is the prediction of what such a second measurement will yield that is at issue. This prob- lem can be phrased very precisely in terms of doubly stochastic matrices
- 26. xxii PREFACE AND PRELUDE for binary coupled angular momentum states corresponding to complete sets of commuting Hermitian observables; it is so formulated in Chap- ters 1 and 5, using the property that there is a unique doubly stochastic matrix of order Nj(j) associated with each binary coupled scheme 1 and each binary coupled scheme 2. The probability of a prepared coupled scheme 1 state being in a measured scheme 2 coupled state is just the (row, column) entry in the doubly stochastic matrix corresponding to these respective coupled states. This is the answer given in the context of conventional nonrelativistic quantum theory. We do not speculate on the meaning of this answer to the holistic aspects of complex (or sim- ple) quantal systems. Rather, throughout this volume, we focus on the detailed development of topics and concepts that relate to the binary coupling theory of angular momenta, as developed in [L], [6], and in this volume, leaving their full interpretation for the future. In many ways, this portion of the monograph can be considered as a mopping-up operation for an accounting of the binary theory of the coupling of arbitrarily many angular momentum systems within the paradigm of conventional nonrelativistic quantum mechanics, in the sense of Kuhn [35, p. 24]. Yet, there is the apt comparison with Com- plexity Theory as advanced by the Santa Fe Institute — a few simple, algorithmic-like rules that generate an almost unlimited scope of patterns of high informational content. Moreover, binary trees viewed as branch- ing diagrams, are omnipotent as classification schemes for objects of all sorts — their shapes and labelings have many applications going beyond angular momentum systems. The closely related graphs — Cayley’s [19] trivalent trees, which originate from a single labeled binary tree — and their joining for pairs of such binary trees define cubic graphs (see [L]). These cubic graphs determine the classification of all binary coupling schemes in angular momentum theory. But they surely extend beyond the context of angular momentum systems in sorting out the diverse patterns of regularity in nature, as discussed briefly in Appendix C. TOPICAL CONTENTS We summarize next the principal topics that constitute the present monograph, and the relevant chapters. Chapter 1, Chapter 5, Chapter 8. Total angular momentum states (reviewed from [L]). The total angular momentum of a physical system is a collective property. The addition of two angular momenta, a prob- lem already solved in the seminal papers in quantum mechanics, is the simplest example, especially for intrinsic internal quantum spaces such as spin space. Here we review the earlier work, emphasizing that the tensor product space in which the angular momentum operators act has the property of all such tensor product spaces: It contains vectors that cannot be obtained as simple products of vectors of the individual parts of the system — tensor product spaces by their very nature are holistic;
- 27. PREFACE AND PRELUDE xxiii that is, are superpositions of the tensor product of the constituent sys- tem state vectors. For the most part, our discussions in Chapters 1 and 5 are a summary of material from the first volume needed for this mono- graph, but now focused more strongly on the properties of the unitary matrices Zj(U(j), V (j)), called recoupling matrices, where U(j) and V (j) are unitary matrices that given the transformation coefficients from the uncoupled basis to the coupled basis that determines the composite sys- tem state vectors. These recoupling matrices satisfy the very important multiplication rule: Zj (U(j) , V (j) ) Zj (V (j) , W(j) ) = Zj (U(j) , W(j) ). In this relation, each of the unitary matrices U(j), V (j), W(j) gives the transformation coefficients of a complete coupled set of state vectors that are simultaneous eigenvectors of the mutually commuting set of Hermi- tian operators given by the squares of the constituent angular momenta J2(i), i = 1, 2, . . . , n, the squared total angular momentum J2 and its 3−component J3, and an additional set of operators (distinct for each of the three cases), or sets of parameter spaces, that complete the set of state vectors. Each set of coupled state vectors then spans the same ten- sor product space Hj, and the elements of the recoupling matrices give the transformation coefficients from one coupled basis set to the other, either for the full tensor product space Hj, or well-defined subspaces. It is also the case that each of the three recoupling matrices is a dou- bly stochastic matrix, each of which has a probabilistic interpretation in exactly the same sense as that for von Neumann’s density matrices. Thus, the product rule has implications for measurements carried out on systems described by the state vectors corresponding to complete sets of mutually commuting observables. The above vector space structures are more comprehensive than the notation indicates. This is because we have suppressed the labels in the SU(2) irreducible multiplet ket vector notation |ji mii ∈ Hji . More generally, these ket vectors are given by |αi, ji, mi i and constitute a complete set of eigenvectors for the i−th system Si of the full system S; it is the quantum labels in the sequence αi that originate from the eigen- values of a complete set of mutually commuting Hermitian operators (or other complete labeling schemes) that includes J2(i) and J3(i) that give a complete set of eigenvectors of system Si, which itself could be a com- posite system with repeated values of the angular momentum quantum numbers ji, as controlled by the labels αi. The basic multiplication prop- erty of the recoupling matrix still holds under an appropriate adaptation of the notations. The key concept is always completeness, first in the set of mutually commuting observables, and then of the simultaneous eigen- vectors. Thus, many complex quantal systems come under the purview of the angular momentum structure of composite systems, as we have outlined above.
- 28. xxiv PREFACE AND PRELUDE Chapters 2-7. Permutation matrices and related topics. There is, per- haps, no symmetry group more important for all of quantum physics than the group of permutations of n objects — the permutation or sym- metric group Sn. Permutation matrices of order n are the simplest matrix realization of the group Sn by matrices containing a single 1 in each row and column: They consist of the n! rearrangements of the n columns of the identity matrix of order n. But in this work the symmetric group makes its direct appearance in a different context than the Pauli princi- ple; namely, through Birkhoff’s [8] theorem that proves the existence of a subset of the set of all n! permutation matrices of order n such that every doubly stochastic matrix of order n can be expanded with positive real coefficients in terms of the subset of permutations matrices. We not only present this aspect of doubly stochastic matrices, but develop a more general theory in Chapter 3 of matrices having the same fixed line- sum for all rows and columns. Such matrices include doubly stochastic matrices, magic squares, and alternating sign matrices, which are all of interest in physical theory, as discussed in [L]. Here, additional results of interest are obtained, with Chapters 5, 6, 7 being dedicated to each topic, respectively. We comment further on Chapter 5. Chapter 5. Doubly stochastic matrices. These matrices are intro- duced in Chapter 1, and the development of further properties continued here. Recoupling matrices introduced in Chapter 1 are doubly stochas- tic matrices. Such matrices have a probabilistic interpretation in terms of preparation of states corresponding to complete sets of commuting Hermitian operators. The matrix elements in a given (row, column) of a doubly stochastic matrix is the probability of a prepared eigenstate (labeled by the row) of a complete set of mutually commuting Hermitian operators being the measured eigenstate (labeled by the column) of a second (possibly the same) complete set of mutually commuting Hermi- tian operators. Here, these eigenstates are taken to be the coupled states corresponding to standard labeled binary trees of order n. This aspect of doubly stochastic matrices is illustrated numerous times. To my knowl- edge, doubly stochastic matrices were introduced into quantum theory by Alfred Landé [39], the great atomic spectroscopist from whom I had the privilege of hearing first-hand, while a graduate student at The Ohio State University, his thesis that they are fundamental objects underlying the meaning of quantum mechanics. Chapter 8. Heisenberg’s magnetic ring. This physical problem, ad- dressed in depth by Bethe [5] in a famous paper preceding the basic work by Wigner [82] on angular momentum theory and Clebsch-Gordan coefficients is a very nice application of binary coupling theory. The Hamiltonian comes under the full purview of composite systems: Its state vectors can be classified as eigenvectors of the square of the total angular momentum with the usual standard action of the total angu- lar momentum. The exact solutions are given for composite systems
- 29. PREFACE AND PRELUDE xxv containing n = 2, 3, 4 constituents, each part with an arbitrary angular momentum. Remarkably, this seems not to have been noticed. For n ≥ 5, the magnetic ring problem can be reduced to the cal- culation of the eigenstates originating from the diagonalization by a real orthogonal matrix of a real symmetric matrix of order equal to the Clebsch-Gordan number Nj(j), which gives the number of occurrences (multiplicity) of a given total angular momentum state in terms of the angular momenta of the individual constituents. This Hamiltonian ma- trix of order Nj(j) is a real symmetric matrix uniquely determined by certain recoupling matrices originating from binary couplings of the con- stituent angular momenta. It is an exquisitely complex implementation of the uniform reduction procedure for the calculation of all 3(n − 1) − j coefficients that arise; this procedure is itself based on paths, shape trans- formations, recoupling matrices, and their reduction properties. This approach to the Heisenberg ring problem gives a complete and, perhaps, very different viewpoint not present in the Bethe approach, especially, since no complex numbers whatsoever are involved in obtaining the en- ergy spectrum, nor need be in obtaining a complete set of orthonormal eigenvectors. The problem is fully solved in the sense that the rules for computing all 3(n − 1) − j coefficients that enters into the calculation of the Hamiltonian matrix can be formulated explicitly. Unfortunately, it is almost certain that the real orthogonal matrix required to diagonalize the symmetric Hamilton matrix, with its complicated 3(n − 1) − j type structure, cannot be determined algebraically. It may also be the case that numerical computations of the elements of the symmetric Hamilto- nian matrix are beyond reach, except for simple special cases. Of course, complex phase factors do enter into the classification of the eigenvectors by the cyclic invariance group Cn of the Hamiltonian, but this may not be necessary for many applications. Finally, there are three Appendices A, B, and C that deal with issues raised in the main text. Appendix C, however, presents natural gener- alizations of binary tree classifications to other problems, especially, to composite systems where the basic constituents have U(n) symmetry. MATTERS OF STYLE, READERSHIP, AND RECOGNITION On matters of readership and style, we repeat portions of the first volume, since they still prevail. The very detailed Table of Contents serves as a summary of topics covered. The readership is intended to be advanced graduate students and researchers interested in learning of the relation between symmetry and combinatorics and of challenging unsolved problems. The many ex- amples serve partially as exercises, but this monograph is not a textbook. It is hoped that the topics presented promote further and more rigorous developments. While we are unable to attain the eloquence of Mac Lane [51], his book has served as a model.
- 30. xxvi PREFACE AND PRELUDE We mention again, as in [L], some unconventional matters of style. We present significant results 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 integrated 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. As with the earlier volume, this continuing work is heavily indebted to the two volumes on the quantum theory of angular momentum published with the late Lawrence C. Biedenharn [6]. The present volume is much more limited in scope, addressing special topics left unattended in the earlier work, as well as new problems. Our motivation and inspiration for working out many details of bi- nary coupling theory originates from the great learning experiences, be- ginning in the early 1990’s, as acquired in combinatorial research papers with William Y. C. Chen, which were strongly encouraged by the late Gian-Carlo Rota, and supplemented by his many informative conver- sations and inspirational lectures at Los Alamos. We do not reference directly many of the seminal papers by Gian-Carlo Rota, his colleagues, and students in this volume, but these publications were foundational in shaping the earlier volume, and are ever-present here. We acknowledge a few of these again: Désarménien, et al. [23]; Kung and Rota [36]; Kung [37]; Roman and Rota [68]; Rota [70]-[71]; Rota, et al. [72]; as well as the Handbook by Graham, et al. [28]. This knowledge acquisition has continued under the many invitations by William Y. C. Chen, Director, The Center for Combinatorics, Nankai University, PR China, to give lectures on these subjects to students and to participate in small con- ferences. These opportunities expanded at about the same time in yet another direction through similar activities, organized by Tadeusz and Barbara Lulek, Rzeszòw University of Technology, Poland, on Symme- try and Structural Properties of Condensed Matter, under the purview over the years by Adam Mickiewics University, Poznań, University of Rzeszów, and Rzeszów University of Technology. The interaction with Chinese and Polish students and colleagues has been particularly reward- ing. Finally, the constant encouragement by my wife Marge and son Tom provided the friendly environment for bringing both the first and second volumes to completion. Editors Zhang Ji and Lai Fun Kwong deserve special mention and thanks for their encouragement and support of this project. James D. Louck
- 31. Contents Preface and Prelude vii Notation xxxiii 1 Composite Quantum Systems 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Angular Momentum State Vectors of a Composite System 4 1.2.1 Group Actions in a Composite System . . . . . . . 10 1.3 Standard Form of the Kronecker Direct Sum . . . . . . . 11 1.3.1 Reduction of Kronecker Products . . . . . . . . . . 12 1.4 Recoupling Matrices . . . . . . . . . . . . . . . . . . . . . 14 1.5 Preliminary Results on Doubly Stochastic Matrices and Permutation Matrices . . . . . . . . . . . . . . . . . . . . 19 1.6 Relationship between Doubly Stochastic Matrices and Density Matrices in Angular Momentum Theory . . . . . 22 2 Algebra of Permutation Matrices 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Basis Sets of Permutation Matrices . . . . . . . . . . . . . 31 2.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . 41 3 Coordinates of A in Basis PΣn(e,p) 43 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 xxvii
- 32. xxviii CONTENTS 3.2 The A-Expansion Rule in the Basis PΣn(e,p) . . . . . . . . 45 3.3 Dual Matrices in the Basis Set Σn(e, p) . . . . . . . . . . . 47 3.3.1 Dual Matrices for Σ3(e, p) . . . . . . . . . . . . . . 48 3.3.2 Dual Matrices for Σ4(e, p) . . . . . . . . . . . . . . 50 3.4 The General Dual Matrices in the Basis Σn(e, p) . . . . . 53 3.4.1 Relation between the A-Expansion and Dual Matrices . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Further Applications of Permutation Matrices 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 An Algebra of Young Operators . . . . . . . . . . . . . . . 60 4.3 Matrix Schur Functions . . . . . . . . . . . . . . . . . . . 63 4.4 Real Orthogonal Irreducible Representations of Sn . . . . 67 4.4.1 Matrix Schur Function Real Orthogonal Irreducible Representations . . . . . . . . . . . . . . . . . . . . 67 4.4.2 Jucys-Murphy Real Orthogonal Representations . 69 4.5 Left and Right Regular Representations of Finite Groups 72 5 Doubly Stochastic Matrices in Angular Momentum Theory 81 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Abstractions and Interpretations . . . . . . . . . . . . . . 89 5.3 Permutation Matrices as Doubly Stochastic . . . . . . . . 91 5.4 The Doubly Stochastic Matrix for a Single System with Angular Momentum J . . . . . . . . . . . . . . . . . . . . 92 5.4.1 Spin-1/2 System . . . . . . . . . . . . . . . . . . . 92 5.4.2 Angular Momentum−j System . . . . . . . . . . . 94 5.5 Doubly Stochastic Matrices for Composite Angular Momentum Systems . . . . . . . . . . . . . . . . . . . . . 97 5.5.1 Pair of Spin-1/2 Systems . . . . . . . . . . . . . . 97 5.5.2 Pair of Spin-1/2 Systems as a Composite System . 99 5.6 Binary Coupling of Angular Momenta . . . . . . . . . . . 104
- 33. CONTENTS xxix 5.6.1 Complete Sets of Commuting Hermitian Observables . . . . . . . . . . . . . . . . . . . . . . 104 5.6.2 Domain of Definition RT (j) . . . . . . . . . . . . . 106 5.6.3 Binary Bracketings, Shapes, and Binary Trees . . . 109 5.7 State Vectors: Uncoupled and Coupled . . . . . . . . . . . 115 5.8 General Binary Tree Couplings and Doubly Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 140 5.8.2 Uncoupled States . . . . . . . . . . . . . . . . . . . 142 5.8.3 Generalized WCG Coefficients . . . . . . . . . . . 143 5.8.4 Binary Tree Coupled State Vectors . . . . . . . . . 145 5.8.5 Racah Sum-Rule and Biedenharn-Elliott Identity as Transition Probability Amplitude Relations . . 153 5.8.6 Symmetries of the 6 − j and 9 − j Coefficients . . 165 5.8.7 General Binary Tree Shape Transformations . . . . 167 5.8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . 172 5.8.9 Expansion of Doubly Stochastic Matrices into Permutation Matrices . . . . . . . . . . . . . . . . 174 6 Magic Squares 177 6.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2 Magic Squares and Addition of Angular Momenta . . . . 180 6.3 Rational Generating Function of Hn(r) . . . . . . . . . . . 186 7 Alternating Sign Matrices 195 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.2 Standard Gelfand-Tsetlin Patterns . . . . . . . . . . . . . 197 7.2.1 A-Matrix Arrays . . . . . . . . . . . . . . . . . . . 199 7.2.2 Strict Gelfand-Tsetlin Patterns . . . . . . . . . . . 202 7.3 Strict Gelfand-Tsetlin Patterns for λ = (n n − 1 · · · 2 1) . 202 7.3.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . 204
- 34. xxx CONTENTS 7.4 Sign-Reversal-Shift Invariant Polynomials . . . . . . . . . 206 7.5 The Requirement of Zeros . . . . . . . . . . . . . . . . . . 211 7.6 The Incidence Matrix Formulation . . . . . . . . . . . . . 219 8 The Heisenberg Magnetic Ring 223 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.2 Matrix Elements of H in the Uncoupled and Coupled Bases . . . . . . . . . . . . . . . . . . . . . . . . 226 8.3 Exact Solution of the Heisenberg Ring Magnet for n = 2, 3, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.4 The Heisenberg Ring Hamiltonian: Even n . . . . . . . . 235 8.4.1 Summary of Properties of Recoupling Matrices . . 240 8.4.2 Maximal Angular Momentum Eigenvalues . . . . . 242 8.4.3 Shapes and Paths for Coupling Schemes I and II . 243 8.4.4 Determination of the Shape Transformations . . . 245 8.4.5 The Transformation Method for n = 4 . . . . . . . 249 8.4.6 The General 3(2f − 1) − j Coefficients . . . . . . . 253 8.4.7 The General 3(2f − 1) − j Coefficients Continued . 255 8.5 The Heisenberg Ring Hamiltonian: Odd n . . . . . . . . . 261 8.5.1 Matrix Representations of H . . . . . . . . . . . . 266 8.5.2 Matrix Elements of Rj2;j1 : The 6f − j Coefficients 269 8.5.3 Matrix Elements of Rj3; j1 : The 3(f + 1) − j Coefficients . . . . . . . . . . . . . . . . . . . . . . 276 8.5.4 Properties of Normal Matrices . . . . . . . . . . . 287 8.6 Recount, Synthesis, and Critique . . . . . . . . . . . . . . 289 8.7 Action of the Cyclic Group . . . . . . . . . . . . . . . . . 292 8.7.1 Representations of the Cyclic Group . . . . . . . . 295 8.7.2 The Action of the Cyclic Group on Coupled State Vectors . . . . . . . . . . . . . . . . . . . . . . . . 299 8.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 304
- 35. CONTENTS xxxi A Counting Formulas for Compositions and Partitions 305 A.1 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 B No Single Coupling Scheme for n ≥ 5 313 B.1 No Single Coupling Scheme Diagonalizing H for n ≥ 5 . . 313 C Generalization of Binary Coupling Schemes 317 C.1 Generalized Systems . . . . . . . . . . . . . . . . . . . . . 317 C.2 The Composite U(n) System Problem . . . . . . . . . . . 321 Bibliography 327 Index 335 Errata and Related Notes 343
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- 37. Notation General symbols , comma separator; used non-uniformly 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 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 sum of matrices ⊗ tensor product of vector spaces, Kronecker (direct) product of matrices δi,j the Kronecker delta for integers i, j δA,B the Kroneker delta for sets A and B K(λ, α) the Kostka numbers cλ µ ν the Littlewood-Richardson numbers Parn set of partitions having n parts, including 0 as a part λ, µ, ν partitions in the set Parn |A| cardinality of a set A [n] set of integers {1, 2, . . . , n} Specialized symbols are introduced as needed in the text; xxxiii
- 38. xxxiv NOTATION the list below contains a few of the more general ones: J(i) angular momentem of constituent i ∈ [n] of a composite system K(i) intermediate angular momentem i ∈ [n − 2] of a composite system J total angular momentem of all constituents of a composite system j sequence (j1, j2, . . . , jn) of quantum numbers of the constituents of a composite system Bn set of binary bracketings of order n Tn set of binary trees of order n ShT shape of a binary tree T ∈ Tn ShT (j) shape of a standard labeled binary tree T ∈ Tn w(A, C) word in the letters A and C |T(j k)j mi simultaneous eigenvector of a complete set of 2n ang- ular momentum operators J2(i), i = 1, 2, . . . , n; J2, J3; K2(i), i = 1, 2, . . . , n − 2 — also called binary coupled state vectors hT(j k)j m | T′ (j′ k′ )j mi inner product of two binary coupled state vectors Hji Hilbert vector space of dimension 2ji + 1 that is irreducible under the action of SU(2) Hj = Hj1 ⊗ Hj2 ⊗ · · · ⊗ Hjn : tensor productof the spaces Hji of dimension (2j1 + 1) · · · (2jn + 1) = N(j) Nj(j) Clebsch-Gordan number H(j, j, m) subspace of Hj of order Nj(j) U†V Landé form of a doubly stochastic matrix RS; S′ recoupling matrix for a pair of standard labeled binary trees related by arbitrary shapes S and S′ RSh; Sh+1 recoupling matrix for a pair of standard labeled binary trees of shapes Sh and Sh+1 related by an elementary shape transformation ∆T (j k)j | ∆T′ (j′ k′ )j triangle coefficient that is a 3 × (n − 1) matrix array that encodes the structure of the labeled forks of a pair of standard labeled binary trees
- 39. NOTATION xxxv Gλ Gelfand-Tsetlin (GT) pattern of shape λ λ m member of Gλ E(x) linear matrix form eij(x) element of E(x) Pn vector space of linear forms PΣn basis set Σn of permutation matrices PΣn(e) basis set Σn(e) of permutation matrices PΣn(e,p) basis set Σn(e, p) of permutation matrices An set of doubly stochastic matrices of order n Mn(r) set of magic squares of order n and line-sum r ASn set of alternating sign matrices of order n lA line-sum of a matrix A of fixed line-sum
- 40. Chapter 1 Composite Quantum Systems 1.1 Introduction The group and angular momentum theory of composite quantum sys- tems was initiated by Weyl [80] and Wigner [82]. It is an intricate, but well-developed subject, as reviewed in Biedenharn and van Dam [7], and documented by the many references in [6]. It is synthesized further by the so-called binary coupling theory developed in great detail in [L]. It was not realized at the time that recoupling matrices, the objects that encode the full prescription for relating one coupling scheme to another, are doubly stochastic matrices. This volume develops this aspect of the theory and related topics. We review in this first chapter some of the relevant aspects of the coupling theory of angular momenta for ease of reference. Curiously, these developments relate to the symmetric group Sn, which is a finite subgroup of the general unitary group and which is also considered in considerable detail in the previous volume. But here the symmetric group makes its appearance in the form of one of its simplest matrix (reducible) representations, the so-called permutation matrices. The symmetric group is one of the most important groups in physics (Wybourne [87]), as well as mathematics (Robinson [67]). In physics, this is partly because of the Pauli exclusion principle, which ex- presses a collective property of the many entities that constitute a com- posite system; in mathematics, it is partly because every finite group is isomorphic to a symmetric group of some order. While the symmetric group is one of the most studied of all groups, many of its properties that relate to doubly stochastic matrices, and other matrices of physi- cal importance, seem not to have been developed. This review chapter provides the background and motivation for this continued study. 1
- 41. 2 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS A comprehensive definition of a composite quantal system that is sufficiently broad in scope to capture all possible physical systems is difficult: it will not be attempted here. Instead, we consider some general aspects of complex systems and then restrict our attention to a definition that is sufficient for our needs. In some instances, a composite quantal system can be built-up by bringing together a collection of known independent quantal systems, initially thought of as being noninteractive, but as parts of a composite system, the subsystems are allowed to be mutually interactive. We call this the build-up principle for composite systems. We assume that such a built-up composite system can also be taken apart in the sense that, if the mutual interactions between the known parts are ignored, the subsystems are each described independently and have their separate identities. This is a classical intuitive notion; no attempt is made to place the “putting- together and breaking-apart” process itself in a mathematical framework. The mathematical model for describing a built-up composite system utilizes the concept of a tensor product of vector spaces. The state space of the i−th constituent of such a composite system is given by an inner product vector space Hi, which we take to be a bra-ket vector space in the sense of Dirac [24], and, for definiteness, it is taken to be a separable Hilbert space. Each such Hilbert space then has an orthonormal basis given by Bi = {|i, kii | ki = 1, 2, . . .} , i = 1, 2, . . . , n. (1.1) The state space of a composite system, built-up from n such independent systems is then the tensor product space H defined by H = H1 ⊗ H2 ⊗ · · · ⊗ Hn. (1.2) The orthonormal basis of H is given in terms of the individual orthonor- mal bases Bi of Hi by B = B1 ⊗ B2 ⊗ · · · ⊗ Bn. (1.3) A general vector in the linear vector space H is of the form | general statei (1.4) = X k1≥1 X k2≥1 · · · X kn≥1 ak1,k2,...,kn | 1, k1i⊗ | 2, k2i ⊗ · · · ⊗ | n, kni, where the coefficients ak1,k2,...,kn are arbitrary complex numbers. Since, in general, ak1,k2,...,kn 6= a1,k1 a2,k2 · · · an,kn , it is an evident (and well- known) that a general superposition of state vectors given by (1.4) does not have the form X k1≥1 a1,k1 | 1, k1i ⊗ X k2≥1 a1,k1 | 2, k2i ⊗ · · · ⊗ X kn≥1 an,kn | n, kni . (1.5)
- 42. 1.1. INTRODUCTION 3 Thus, there are vectors in a tensor product space that cannot be written as the tensor product of n vectors, each of which belongs to a constituent subspace Hi. This mathematical property already foretells that compos- ite systems have properties that are not consequences of the properties of the individual constituents. The definition of composite quantal systems and their interactions with measuring devices are fundamental to the interpretation of quan- tum mechanics. Schrödinger [73] coined the term entanglement to de- scribe the property that quantal systems described by the superposition of states in tensor product space are not all realizable as the tensor prod- uct of states belonging to the constituent subsystems. Entanglement, in its conceptual basis, is not a mysterious property. It is a natural prop- erty of a linear theory based on vector spaces and standard methods for building new vector spaces out of given vector spaces. But the meaning and breadth of such mathematical constructions for the explanation of physical processes can be profound. We make also the following brief remarks on the methodology of ten- sor product spaces introduced above. These observations have been made and addressed by many authors; we make no attempt (see Wheeler and Zurek [81]) to cite the literature, our purpose here being simply to place results presented in this volume in the broader context: Remarks. 1. The build-up principle for composite systems stated above is al- ready to narrow in scope to capture the properties of many phys- ical systems. It does not, for example, include immediately an electron with spin: the spin property cannot be removed from the electron; it is one of its intrinsic properties, along with it mass and charge. Nonetheless, the energy spectrum of a single (nonrelativis- tic) electron with spin in the presence of a central attractive poten- tial can be described in terms of a tensor product of vector spaces Hψ⊗Hl⊗H1/2, where Hψ, Hl, and H1/2, are, respectively, the space of solutions of the Schrödinger radial equation, the finite-Hilbert space of dimension 2l + 1 of orbital angular momentum states, and the finite-Hilbert space of dimension 2 of spin states (see Ref. [6]). This is indicative of the fact that mathematical techniques devel- oped in specific contexts often have a validity that extends beyond their original intent. 2. It is sometimes helpful to give concrete realizations of tensor prod- uct spaces in terms of functions over the real or complex numbers. This is usually possible in ordinary quantum theory, even for spin and other internal symmetries. The abstract tensor product space relation (1.1) is then formulated as
- 43. 4 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS ψ(z1, z2, . . . , zn) (1.6) = X k1≥1,k2≥1,...,kn≥1 ak1,k2,...,kn ψ (1) k1 (z1)ψ (2) k2 (z2) · · · ψ (n) kn (zn), where ψ (i) ki ∈ Hi, i = 1, 2, . . . , n, with values ψ (i) ki (zi), where zi is a set of real or complex numbers appropriate to the description of the desired property of the i−th part of the system. In the sense of presentation of state vectors in the form (1.6), the tensor product property of state vectors can be referred to as the factorization assumption for composite systems. 3. The converse of the build-up principle and its extensions is more difficult to formulate as a general principle, where the first naive question is: Can a composite system be taken apart to reveal its basic constituents? This question is regressive, since it can again be asked of the basic constituents. It does not have an answer without further qualifications. 4. There are fundamental issues associated with the very notion of an isolated quantal system: How does Newton’s third law tran- scribe to quantal systems? Specifically, how does a quantal system interact with its environment such as instruments, classical and quantal, designed to measure certain of its properties. The impli- cation of a measurement performed on a subsystem of a composite system are particularly intriguing, since the subsystems remain as entangled parts of the composite system independent of separation distance, if the whole system is left undisturbed between the time of its preparation and the time of the measurement. 5. Many properties of composite physical systems can be presented ex- actly by focusing on the properties of tensor products of subspaces of the general state space that can described exactly in terms of separable Hilbert spaces and their tensor products. We continue now by describing the general setting for composite phys- ical systems from the viewpoint of their angular momentum subspaces. 1.2 Angular Momentum State Vectors of a Composite System We consider those composite quantal systems such that the state space Hi of the i–th part of the system contains at least one subspace characterized by the angular momentum J(i) of the subsystem, each
- 44. 1.2. ANGULAR MOMENTUM STATE VECTORS 5 i = 1, 2, . . . , n. The entire composite system is described in terms of a common right-handed inertial reference frame (e1, e2, e3) in Cartesian 3−space R3, where redescriptions of the system are effected by uni- tary unimodular SU(2) group transformations of the reference frame as described in detail in [L]. The angular momentum J(i) of the i−th subsystem is given in terms of its three components relative to the refer- ence frame by J(i) = J1(i)e1 + J2(i)e2 + J3(i)e3, where the components satisfy the commutation relations [J1(i), J2(i)] = iJ3(i), [J2(i), J3(i)] = iJ1(i), [J3(i), J1(i)] = iJ2(i), where the i in the commutator relation is i = √ −1, and not the subsystem index. The components Jk(i), k = 1, 2, 3, of J(i) have the standard action on each subspace of the states of the subsystem, as characterized by J2 (i)|ji mii = ji(ji + 1)|ji mii, J3(i)|ji mii = mi|ji mii, J+(i)|ji mii = p (ji − mi)(ji + mi + 1) |ji mi + 1i, (1.7) J−(i)|ji mii = p (ji + mi)(ji − mi + 1) |ji mi − 1i. The notation Hji denotes the finite-dimensional Hilbert space of dimen- sion dim Hji = 2ji + 1 with orthonormal basis given by Bji = {|ji mii | mi = ji, ji − 1, . . . , −ji}, (1.8) hji mi | ji m′ ii = δmi,m′ i , each pair mi, m′ i ∈ {ji, ji − 1, . . . , −ji}. The usual assumptions underlying the derivation of the standard rela- tions (1.7) for the action of the angular momenta components are made; namely, that the linear vector space Hi over the complex numbers is equipped with an inner product with respect to which the components Jk(i) are Hermitian operators that act linearly on the space Hi to ef- fect a transformation to a new vector in the space. The operators J+(i) and J−(i) are the usual Hermitian conjugate shift operators defined by J+(i) = J1(i) + iJ2(i) and J−(i) = J1(i) − iJ2(i), where the nonindexing i is the complex number i = √ −1. The angular momentum components Jk(i), k = 1, 2, 3, for distinct subsystems i all mutually commute. It is allowed that each of the vector spaces Hi, i = 1, 2, . . . , n can have the same or distinct definitions of inner product, it only being required that the angular momentum components in each subspace are Hermitian with respect to the inner product for that subspace. In the notation for the orthonormal basis vectors Bi of Hji in (1.8), we have suppressed all the extra quantum labels that may be necessary to define a basis for the full space Hi. In applications to specific problems, such labels are to be supplied. It is the properties of
- 45. 6 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS the fundamental standard angular momentum multiplets defined by (1.7) and (1.8) in the tensor product space Hj1 ⊗ Hj2 ⊗ · · · ⊗ Hjn ⊂ H1 ⊗ H2 ⊗ · · · ⊗ Hn (1.9) that are the subject of interest here. The analysis concerns only finite- dimensional Hilbert spaces and is fully rigorous. We point out that while we use the term angular momentum to describe the operators with the action (1.7) on the basis (1.8), it would, perhaps, be more appropriate to refer to the space Hji as an irreducible SU(2)-multiplet, since it is not necessary that such operators be interpreted physically as angular momenta. For example, the analysis can be applied to Gell-Mann’s eight- fold way, since the irreducible SU(3)−multiplet is realized in terms of the eight-dimensional Hilbert space H(2,1,0) can be presented as the direct sum of the angular momentum vector spaces as given by H(2,1,0) = H(2,1)⊕H(2,0)⊕H(1,0)⊕H(1,1) = H1/2⊕H1⊕H′ 1/2⊕H0. (1.10) The mapping from the subspace H(a,b) to the angular momentum sub- space Hj is given by j = (a − b)/2, which conceals the fact that the two spaces H(2,1) and H(1,0), each of which has j = 1/2, are, in fact, per- pendicular. Thus, while the SU(2)−multiplet content is the same, the physical content is quite different, since the two multiplets correspond to particles with different properties within the context of the unitary group SU(3). The space H(2,1,0) can clearly be incorporated with the framework of relation (1.9) by taking direct sums and paying careful attention to notations. The concept of the tensor product space of SU(2)−multiplets is suffi- ciently rich in structure to accommodate rather diverse applications and illustrate properties of composite systems. A principal property always to be kept in mind is that this tensor product space is a linear vec- tor space; hence, arbitrary linear combinations of vectors belonging to the space are allowed, and such superpositions show interference in the probabilistic interpretation of measurements. The total angular momentum operator for the composite system (1.9) is defined by J = n X i=1 ⊕ (Ij1 ⊗ · · · ⊗ J(i) ⊗ · · · ⊗ Ijn ) , (1.11) where in the direct sum the identity operators Ij1 , Ij2 , . . . , Ijn appear in the corresponding positions 1, 2, . . . , n, except in position i, where J(i) stands. This notation and the positioning of J(i) signify that the angular momentum operator J(i) acts in the Hilbert space Hji , and that the unit
- 46. 1.2. ANGULAR MOMENTUM STATE VECTORS 7 operators act in all other parts of the tensor product space. We often use the simplified notation J = J(1) + J(2) + · · · + J(n) (1.12) for the sum of various angular momentum operators acting in the tensor product space, but such expressions are always to be interpreted in the sense of a direct sum of tensor products of operators of the tensor product form (1.11). (See Sect. 10.5, Compendium A of [L] for a summary of the properties of tensor product spaces in terms of the present notations.) We introduce the following compact notations to describe the ket- vectors of the tensor product space: j = (j1, j2, . . . , jn), each ji ∈ {0, 1/2, 1, 3/2, . . .}, i = 1, 2, . . . , n, m = (m1, m2, . . . , mn), each mi ∈ {ji, ji − 1, . . . , −ji}, i = 1, 2, . . . , n, Hj = Hj1 ⊗ Hj2 ⊗ · · · ⊗ Hjn , (1.13) |j mi = |j1 m1i ⊗ |j2 m2i ⊗ · · · ⊗ |jn mni, C(j) = {m | mi = ji, ji − 1, . . . , −ji; i = 1, 2, . . . , n}. The set of 2n mutually commuting Hermitian operators J2 (1), J3(1), J2 (2), J3(2), . . . , J2 (n), J3(n) (1.14) is a complete set of operators in the tensor product space Hj, in that the set of simultaneous eigenvectors |j mi, m ∈ C(j) is an orthonormal basis; that is, there is no degeneracy left over. The action of the angular momentum operators J(i), i = 1, 2, . . . , n, is the standard action given by J2 (i)|j mi = ji(ji + 1)|j mi, J3(i)|j mi = mi|j mi, J+(i)|j mi = p (ji − mi)(ji + mi + 1) |j m+1(i)i, (1.15) J−(i)|j mi = p (ji + mi)(ji − mi + 1) |j m−1(i)i, m±1(i) = (m1, . . . , mi ± 1, · · · , mn). The orthonormality of the basis functions is expressed by hj m | j m′ i = δm,m′ , each pair m, m′ ∈ C(j). (1.16) Since the collection of 2n commuting Hermitian operators (1.14) refers to the angular momenta of the individual constituents of a physical sys- tem, and the action of the angular momentum operators is on the basis
- 47. 8 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS vectors of each separate space, the basis |j mi, m ∈ C(j), is referred to as the uncoupled basis of the space Hj. One of the most important observables for a composite system is the total angular momentum defined by (1.11). A set of n + 2 mutually commuting Hermitian operators, which includes the square of the total angular momentum J and J3 is the following: J2 (1), J2 (2), . . . , J2 (n), J2 , J3. (1.17) This set of n + 2 commuting Hermitian operators is an incomplete set with respect to the construction of the states of total angular momentum; that is, the simultaneous state vectors of the n+2 operators (1.17) do not determine a basis of the space Hj. There are many ways to complete such an incomplete basis. For example, an additional set of n−2 independent SU(2) invariant Hermitian operators, commuting among themselves, as well as with each operator in the set (1.17), can serve this purpose. Other methods of labeling can also be used. For the present discussion, we make the following assumptions: Assumptions. The incomplete set of simultaneous eigenvectors of the n + 2 angular momentum operators (1.17) has been extended to a basis of the space Hj with properties as follows: A basis set of vectors can be enumerated in terms of an indexing set R(j) of the form R(j) = α = (α1, α2, . . . , αn−2), j, m j ∈ D(j); α ∈ A(j)(j); m = j, j − 1, . . . , −j , (1.18) where the domains of definition D(j) of j and A(j)(j) of α have the properties as follows. These domains of definition are to be such that for given quantum numbers j the cardinality of the set R(j) is given by | R(j) | = | C(j) | = n Y i=1 (2j1 + 1). (1.19) Moreover, these labels are to be such that the space Hj has the orthonor- mal basis given by the ket-vectors |(j α)j mi, α, j, m ∈ R(j), h(j α)j m | (j α′ )j′ m′ i = δj,j′ δm,m′ δα,α′ , (1.20) α, j, m ∈ R(j); α′, j′ , m′ ∈ R(j). It is always the case that D(j) is independent of how the extension to a basis through the parameters α is effected and that, for given j, the domain of m is m = j, j − 1, . . . , −j. The sequence of quantum labels α also belongs to some domain of definition A(j)(j) that can depend on j.
- 48. 1.2. ANGULAR MOMENTUM STATE VECTORS 9 The actions of the commuting angular momentum operators (1.17) and the total angular momentum J on the orthonormal basis set (1.20) are given by J2 (i)|(j α)j mi = ji(ji + 1)|(j α)j mi, i = 1, 2, . . . , n, J2 |(j α)j mi = j(j + 1)|(j α)j mi, J3|(j α)j mi = m|(j α)j m+1i, (1.21) J+|(j α)j mi = p (j − m)(j + m + 1)|(j α)j m+1i, J−|(j α)j mi = p (j + m)(j − m + 1)|(j α)j m−1i. The notation for the ket-vectors in (1.20) and (1.21) places the to- tal angular momentum quantum number j and its projection m in the subscript position to accentuate their special role. The set R(j) enumer- ates an alternative unique orthonormal basis (1.20) of the space Hj that contains the total angular momentum quantum numbers j, m. Any basis set with the properties (1.20)-(1.21) is called a coupled basis of Hj. For n = 2, the uncoupled basis set is {|j1 m1i⊗|j2 m2i | (m1, m2) ∈ C(j1, j2)}, where C(j1, j2) = {(m1, m2)|mi = ji, ji − 1, . . . , −ji, i = 1, 2}; and the coupled basis set is {|(j1 j2)j mi | (j, m) ∈ R(j1, j2)}, where R(j1, j2) = {(j, m)|j = j1 + j2, j1 + j2 − 1, . . . , |j1 − j2|; m = j, j − 1, . . . , −j}. No extra α labels are required. For n = 3, one extra label α1 is required, and at this point in our discussions, we leave the domain of definition of α1 unspecified. Angular momentum coupling theory of composite systems is about the various ways of providing the extra set of α labels and their domains of definition, together with the values of the total angular momentum quantum number j, such that the space Hj is spanned by the vectors |(j α)j mi. It turns out, as shown below, that the set of values that the total angular momentum quantum number j can assume is independent of the αi; the values of j being j = jmin, jmin + 1, . . . , jmax, for well- defined minimum and maximum values of j that are expressed in terms of j1, j2, . . . , jn. Thus, the burden of completing any basis is placed on assigning the labels αi in the set R(j, j) = {α = (α1, α2, . . . , αn−2) | α ∈ A(j) (j)}. (1.22) Such an assignment is called an α−coupling scheme. Since there are many ways of completing an incomplete basis of a finite vector space, there are also many coupling schemes. In this sense, the structure of the coupling scheme set R(j, j) is the key object in angular momentum coupling theory; all the details of defining the coupling scheme are to be provided by the domain of definition α ∈ A(j)(j).
- 49. 10 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS The cardinality of the sets R(j) and C(j) are related by | R(j) | = jmax X j=jmin (2j + 1)Nj(j) = N(j) = n Y i=1 (2ji + 1) = | C(j) |, (1.23) where we have defined Nj(j) = |R(j, j)|. These positive numbers are called Clebsch-Gordan (CG) numbers. They can be generated recur- sively as discussed in Sect. 2.1.1 of [L]. The orthonormal bases (1.16) and (1.20) of the space Hj must be related by a unitary transformation A(j) of order N(j) = Qn i=1(2ji + 1) with (row; column) indices enumerated by (m ∈ C(j); α, j, m ∈ R(j)) (see (1.35) below). Thus, we must have the invertible relations: |(j α)j mi = X m∈C(j) A(j)tr α, j, m; m |j mi, each α, j, m ∈ R(j), (1.24) |j mi = X α,j,m∈R(j) A(j)† α, j, m; m |(j α)j mi, each m ∈ C(j). (1.25) Note. We have reversed the role of row and column indices here from that used in [L] (see pp. 87, 90, 91, 94, 95), so that the notation accords with that used later in Chapter 5 for coupling schemes associated with binary trees, and the general structure set forth in Sect. 5.1. The transformation to a coupled basis (1.20) as given by (1.24) effects the full reduction of the n−fold Kronecker product Dj (U) = Dj1 (U) ⊗ Dj2 (U) ⊗ · · · ⊗ Djn (U), U ∈ SU(2), (1.26) of SU(2) unitary irreducible matrix representations. The matrix Dj(U), U ∈ SU(2), is a reducible unitary representation of SU(2) of dimension N(j), and the transformation (1.25) effects the transformation to a direct sum of irreducible unitary representations Dj(U) (Wigner D−matrices). We next summarize the transformation properties of the coupled and ucoupled bases (1.25) under SU(2) frame rotations. 1.2.1 Group Actions in a Composite System Under the action of an SU(2) frame rotation of the common frame (e1, e2, e3) used to describe the n constituents of a physical system in Cartesian space R3, where system i has angular momentum J(i) = J1(i)e1 + J2(i)e2 + J3(i))e3, the orthonormal basis of the subspace Hji = {|ji mii | mi = ji, ji − 1, . . . , −ji} (1.27)
- 50. 1.3. STANDARD FORM OF THE KRONECKER DIRECT SUM 11 of system i undergoes the standard unitary transformation TU |ji m′ ii = X mi Dji mi m′ i (U)|ji mii, each U ∈ SU(2). (1.28) The uncoupled basis Hj1 ⊗ Hj2 ⊗ · · · ⊗ Hjn of the angular momentum space Hj of the collection of systems undergoes the reducible unitary transformation given by (TU ⊗ TU ⊗ · · · ⊗ TU ) (|j1 m′ 1i ⊗ |j2 m′ 2i ⊗ · · · ⊗ |jn m′ ni = X m Dj1 (U) ⊗ Dj2 (U) ⊗ · · · ⊗ Djn (U) m m′ ×(|j1 m1i ⊗ |j2 m2i ⊗ · · · ⊗ |jn mni), (1.29) where m = (m1, m2, . . . , mn), m′ = (m′ 1, m′ 2, . . . , m′ n). This relation is described in the abbreviated notations (1.13) and (1.15)-(1.16) by TU |j m′ i = X m Dj m m′ (U) |j mi, Dj (U) = Dj1 (U)⊗· · ·⊗Djn (U), (1.30) for each U ∈ SU(2). Similarly, the coupled basis (1.20) of Hj undergoes the irreducible unitary transformation: TU |(j α)j m′ i = X m Dj m m′ (U)|(j α)j mi, each U ∈ SU(2). (1.31) 1.3 Standard Form of the Kronecker Direct Sum Schur’s lemma (see Sect. 10.7.2, Compendium A in [L]) implies that the reducible unitary Kronecker product representation Dj(U) of SU(2) de- fined by (1.26) is reducible into a direct sum of irreducible unitary rep- resentations Dj(U) by a unitary matrix similarity transformation U(j) of order N(j) = Qn i=1(2ji + 1) : U(j)† Dj (U)U(j) = Dj (U) = Djmin (U) 0 0 · · · 0 0 Djmin+1(U) 0 . . . 0 . . . . . . . . . · · · . . . 0 0 0 · · · Djmax (U) , (1.32)
- 51. 12 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS each U ∈ SU(2), where the block form on the right defines the matrix Dj(U) for each j = (j1, j2, . . . , jn), which is also of order N(j). Each ma- trix Dj(U), j = jmin, jmin + 1, . . . , jmax, is itself a matrix direct sum of block form consisting of the same standard irreducible matrix represen- tation Dj(U) of order 2j + 1 of SU(2) repeated Nj(j) times, as given by the Kronecker product Dj (U) = INj(j) ⊗ Dj (U) = Dj(U) 0 0 · · · 0 0 Dj(U) 0 . . . 0 . . . . . . . . . · · · . . . 0 0 0 · · · Dj(U) . (1.33) In this relation, INj (j) is the unit matrix of order Nj(j), the Clebsch- Gordan number. The reason for adopting a standard form for the Kro- necker direct sum, as given explicitly by (1.32)-(1.33), is so that we can be very specific about the structure of the unitary matrix U(j) that effects the reduction. 1.3.1 Reduction of Kronecker Products The reduction of the Kronecker product Dj(U) into the standard form of the Kronecker direct sum by the unitary matrix similarity transfor- mation in (1.32) is not unique. There are nondenumerably infinitely many unitary matrices U(j) of order N(j) = Qn i=1(2ji +1) that effect the transformation U(j)† Dj (U)U(j) = jmax X j=jmin ⊕Dj (U). (1.34) The rows and columns of U(j)† are labeled by the indexing sets R(j) and C(j) as given by (1.18) and (1.13), respectively: U(j)† α, j, m; m = h(j α)j m | j mi = A(j)† α, j, m; m , (1.35) where we note that these matrix elements are also the transformation coefficients between the coupled and uncoupled basis vectors given by (1.25). The rows and columns can always be ordered such that U(j) effects the standard reduction given by (1.32)-(1.33); that is, given any coupling scheme, the transformation of the Kronecker product to the standard Kronecker direct sum can always be realized.
- 52. 1.3. STANDARD FORM OF THE KRONECKER DIRECT SUM 13 There is an intrinsic non-uniqueness in the transformation (1.32) due to the multiplicity structure (1.33) of any standard reduction. Thus, define the matrix W(j,j) of order N(j) to be the direct product given by W(j,j) = W(j,j) ⊗ I2j+1, (1.36) where W(j,j) is an arbitrary complex matrix of order Nj(j), the CG number. Then, the matrix W(j,j) commutes with the direct sum matrix Dj defined by (1.33): W(j,j) Dj (U) = Dj (U)W(j,j) , each U ∈ SU(2). (1.37) We may choose W(j,j) in (1.36) to be an arbitrary unitary matrix of order Nj(j); that is, W(j,j) ∈ U(Nj(j)), the group of unitary matrices of order Nj(j). Then, the direct sum matrix W(j) = jmax X j=jmin ⊕W(j,j) = jmax X j=jmin ⊕ W(j,j) ⊗ I2j+1 , W(j,j) ∈ U(Nj(j)), (1.38) is a unitary matrix belonging to the unitary group U(N(j)); it has the commuting property given by W(j) Dj (U) = Dj (U)W(j) , each U ∈ SU(2). (1.39) Thus, if we define the unitary matrix V (j) by V (j) = W(j) U(j), hence, V (j) U(j)† = W(j) , (1.40) then V (j) also effects, for each U ∈ SU(2), the transformation: V (j)† Dj (U)V (j) = U(j)† Dj (U)U(j) = jmax X j=jmin ⊕Dj (U). (1.41) Each unitary matrix V (j) effects exactly the same reduction of the Kro- necker product representation Dj(U) of SU(2) into standard Kronecker direct sum form as does U(j). We call all unitary similarity transforma- tions with the property (1.41) standard reductions. Summary: Define the subgroup H(N(j)) of the unitary group U(N(j)) of order N(j) = Qn i=1(2ji + 1) by H(N(j)) = jmax X j=jmin ⊕ W(j,j) ⊗ I2j+1 W(j,j) ∈ U(Nj(j)) . (1.42)
- 53. 14 CHAPTER 1. COMPOSITE QUANTUM SYSTEMS Then, if the unitary matrix element U(j) effects the standard reduction, so does every unitary matrix V (j) such that V (j) U(j)† ∈ H(N(j)). (1.43) 1.4 Recoupling Matrices Let U(j) and V (j) be unitary matrices of order N(j)) = Qn i=1(2ji + 1) that effect the standard reduction (1.34). The unitary matrix U(j)† cor- responds to an α−coupling scheme and has its rows enumerated by the elements of a set R(j) of the form: R(j) = α ∈ R(j, j), j, m j = jmin, jmin + 1, . . . , jmax; m = j, j − 1, . . . , −j . (1.44) The domain of definition R(j, j) of each αi quantum number in the se- quence α is itself a set of the form: R(j, j) = {α = (α1, α2, . . . , αn−2) | αi ∈ Ai(j, j)} , (1.45) where each set Ai(j, j) is uniquely defined in terms of the given angular momenta j = (j1, j2, . . . , jn) and j in accordance with the prescribed α−coupling scheme. Similarly, the unitary matrix V (j)† corresponds to a β−coupling scheme and has its rows enumerated by the elements of a set S(j) of the form: S(j) = β ∈ S(j, j), j, m j = jmin, jmin + 1, . . . , jmax; m = j, j − 1, . . . , −j , (1.46) S(j, j) = {β = (β1, β2, . . . , βn−2) | βi ∈ Bi(j, j)} . (1.47) The column indexing set for each of U(j) and V (j) is the same set of projection quantum numbers C(j). There is a set of coupled state vectors associated with each of the unitary matrices U(j) and V (j) given by |(j α)j mi = X m∈C(j) U(j)tr α, j, m; m |j mi, α, j, m ∈ R(j), (1.48) |(j β)j mi = X m∈C(j) V (j)tr β, j, m; m |j mi, β, j, m ∈ S(j).
- 54. 1.4. RECOUPLING MATRICES 15 The unitary matrices U(j) and V (j) in these transformations are matrices of order N(j) = Qn i=1(2ji+1) in consequence of the equality of cardinality of the sets that enumerate the rows and columns: |R(j)| = |S(j)| = |C(j)| = N(j). (1.49) Both the α−coupled basis and β−coupled basis are orthonormal basis sets of the same tensor product space Hj and satisfy all of the standard relations (1.41)-(1.43). Since these orthonormal basis sets span the same vector space, they are related by a unitary transformation of the form: |(j β)j mi = X α∈R(j,j) Zj α; β (U(j) , V (j) ) |(j α)j mi, (1.50) Zj α; β (U(j) , V (j) ) = h(j α)j m | (j β)j mi = U(j)† V (j) α, j, m; β, j, m , α ∈ R(j, j), β ∈ S(j, j); |R(j, j)| = |S(j, j)| = Nj(j). It is the same value of j and m that appear in both sides of the first rela- tion because the vectors in each basis set are eigenvectors of J2 and J3. Moreover, the transformation coefficients Zj α; β (U(j), V (j)) are indepen- dent of the value m = j, j−1, . . . , −j of the projection quantum number, as the notation indicates. This is true because the general relation (1.50) can be generated from |(j β)j ji = X α∈R(j,j) Zj α; β (U(j) , V (j) ) |(j α)j ji (1.51) by the standard action of the lowering operator J−, which does not affect the transformation coefficients. The unitary transformation coefficients Zj α; β (U(j), V (j)) are called recoupling coefficients because they effect the transformation from one set of coupled state vectors to a second set, which here is from the α−coupling scheme to the β−coupling scheme, as given by (1.50). The matrix Zj(U(j), V (j)) with rows elements enumerated by α ∈ R(j, j) and column elements enumerated by β ∈ S(j, j) is defined by Zj (U(j) , V (j) ) α; β = Zj α; β (U(j) , V (j) ) (1.52) is called a recoupling matrix. It is assumed that a total order relation can be imposed on the sequences α ∈ R(j, j) and β ∈ S(j, j); hence, the
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- 56. [263] The white-faced figure ceased dancing. The wind in the trees sang on. The figure, appearing to see the dragon, drew back in trembling fright. He approached the fiery curtain, yet his back was ever toward it. There was yet a space between the two sections of the curtain. The figure, darting toward this gap, was caught in the flames. “Oh!” Jeanne breathed. “He will die in flames!” Marjory Dean pressed her hand hard. Of a sudden the floor beneath the white figure opened and swallowed him up. Jeanne looked for the dragon. It was gone. The fiery red of the curtain was turning to an orange glow. “Come. You have seen.” It was Hop Long Lee who spoke. Once again his marble-cold hand touched Jeanne’s hand. Ten minutes later the four figures were once more in the street. “Midnight in an Oriental garden,” Angelo breathed. “That,” breathed Marjory Dean, “is drama, Oriental drama. Give it a human touch and it could be made supreme.” “You—you think it could be made into a thing of beauty?” “Surely. Most certainly, my child. Nothing could be more unique.”
- 57. [264] [265] “Come,” whispered Jeanne happily. “Come with me. The night is young. The day is for sleep. Come. We will have coffee by my fire. Then we will talk, talk of all this. We will create an opera in a night. Is it not so?” And it was so. A weird bit of opera it was that they produced that night. Even the atmosphere in which they worked was fantastic. Candle light, a flickering fire that now and then leaped into sudden conflagration, mellow-toned gongs provided by the little lady of the cameo; such were the elements that added to the fantastic reality of the unreal. In this one-act drama the giant paper dragon remained. The flaming curtain, the setting for some weird Buddhist ceremony, was to furnish the motif. A flesh and blood person, whose part was to be played by Marjory Dean, replaced the thing of white cloth and paper that had danced a weird dance, and became entangled in the fiery curtain. Oriental mystery, the deep hatred of some types of yellow men for the white race, these entered into the story. In the plot the hero (Marjory Dean), a white boy, son of a rich trader, caught by the lure of mystery, adventure and tales of the magic curtain, volunteers to take the place of a rich Chinese youth who is to endure the trial by fire. A very ugly old Chinaman, who holds the white boy in high regard, learning of his plans and realizing his peril, prepares the trap-door in the floor beneath the magic curtain.
- 58. [266] When the hour comes for the trial by fire, the white boy, being ignorant of the secrets that will save him, appears doomed as the flames of the curtain surround him, consuming the very mask from his face and leaving him there, his identity revealed in stark reality. Then as the rich Chinaman, who has planned the trial, realizes the catastrophe that must befall his people if the rich youth is burned to death, prepares to cast himself into the flames, the floor opens to swallow the boy up, and the curtain fades. There is not space here to tell of the motives of love, hate, pride and patriotism that lay back of this bit of drama. Enough that when it was done Marjory Dean pronounced it the most perfect bit of opera yet produced in America. “And you will be our diva?” Jeanne was all eagerness. “I shall be proud to.” “Then,” Angelo’s eyes shone, “then we are indeed rich once more.” “Yes. Your beautiful rugs, your desk, your ancient friend the piano, they shall all come back to you.” In her joy Jeanne could have embraced him. As it was she wrung his hand in parting, and thanked him over and over for his part in this bit of work and adventure. “The music,” she whispered to Swen, “you will do it?” “It is as well as done. The wind whispering in the graveyard pines at midnight. This is done by reeds and strings. And there are the gongs, the deep melodious gongs of China. What more could one ask?”
- 59. [267] [268] What more, indeed? “And now,” said Florence, after she had, some hours later, listened to Jeanne’s recital of that night’s affairs, “now that it is all over, what is there in it all for you?” “For me?” Jeanne spread her hands wide. “Nothing. Nothing at all.” “Then why—?” “Only this,” Jeanne interrupted her, “you said once that one found the best joy in life by helping others. Well then,” she laughed a little laugh, “I have helped a little. “And you shall see, my time will come.” Was she right? Does one sometimes serve himself best by serving others? We shall see.
- 60. [269] CHAPTER XXX A SURPRISE PARTY Time marched on, as time has a way of doing. A week passed, another and yet another. Each night of opera found Jeanne, still masquerading as Pierre, at her post among the boxes. Never forgetting that a priceless necklace had been stolen from those boxes and that she had run away, ever conscious of the searching eyes of Jaeger and of the inscrutable shadow that was the lady in black, Jeanne performed her tasks as one who walks beneath a shadow that in a moment may be turned into impenetrable darkness. For all this, she still thrilled to the color, the music, the drama, which is Grand Opera. “Some day,” she had a way of whispering to herself, “some happy day!” Yet that day seemed indistinct and far away. The dark-faced menace to her happiness, he of the evil eye, appeared to have vanished. Perhaps he was in jail. Who could tell?
- 61. [270] The little Frenchman with the message, too, had vanished. Why had he never returned to ask Pierre, the usher in the boxes, the correct address of Petite Jeanne? Beyond doubt he believed himself the victim of a practical joke. “This boy Pierre knows nothing regarding the whereabouts of that person named Petite Jeanne.” Thus he must have reasoned. At any rate the message was not delivered. If Jeanne had lost a relative by death, if she had inherited a fortune or was wanted for some misdemeanor committed in France, she remained blissfully ignorant of it all. Three times Rosemary Robinson had invited her to visit her at her home. Three times, as Pierre, politely but firmly, she had refused. “This affair,” she told herself, “has gone far enough. Before our friendship ripens or is blighted altogether, I must reveal to her my identity. And that I am not yet willing to do. It might rob me of my place in this great palace of art.” Thanks to Marjory Dean, the little French girl’s training in Grand Opera proceeded day by day. Without assigning a definite reason for it, the prima donna had insisted upon giving her hours of training each week in the role of the juggler. More than this, she had all but compelled Jeanne to become her understudy in the forthcoming one-act opera to be known as “The Magic Curtain.” At an opportune moment Marjory Dean had introduced the manager of the opera to all the fantastic witchery of this new opera. He had been taken by it. At once he had agreed that when the “Juggler” was played, this new opera should be presented to the
- 62. [271] public. So Jeanne lived in a world of dreams, dreams that she felt could never come true. “But I am learning,” she would whisper to herself, “learning of art and life. What more could one ask?” Then came a curious invitation. She was to visit the studios of Fernando Tiffin. The invitation came through Marjory Dean. Strangest of all, she was to appear as Pierre. “Why Pierre?” she pondered. “Yes, why?” Florence echoed. “But, after all, such an invitation! Fernando Tiffin is the greatest sculptor in America. Have you seen the fountain by the Art Museum?” “Where the pigeons are always bathing?” “Yes.” “It is beautiful.” “He created that statue, and many others.” “That reminds me,” Jeanne sought out her dress suit and began searching its pockets, “an artist, an interesting man with a beard, gave me his card. He told me to visit his studio. He was going to tell me more about lights and shadows.” “Lights and shadows?” “Yes. How they are like life. But now I have lost his card.”
- 63. [272] [273] * * * * * * * * Florence returned to the island. There she sat long in the sunshine by the rocky shore, talking with Aunt Bobby. She found the good lady greatly perplexed. “They’ve served notice,” Aunt Bobby sighed, “the park folks have. All that is to come down.” She waved an arm toward the cottonwood thicket and the “Cathedral.” “A big building is going up. Steam shovels are working over on the west side now. Any day, now, we’ll have to pack up, Meg and me. “And where’ll we go? Back to the ships, I suppose. I hate it for Meg. She ought to have more schoolin’. But poor folks can’t pick and choose.” “There will be a way out,” Florence consoled her. But would there? Who could tell? She hunted up Meg and advised her to look into that mysterious package. “It may be a bomb.” “If it is, it won’t go off by itself.” “It may be a gun.” “Don’t need a gun. Got two of ’em. Good ones.” “It may be stolen treasure.” “Well, I didn’t steal it!” Meg turned flashing eyes upon her. And there for a time the matter ended. * * * * * * * * Jeanne attended the great sculptor’s party. Since she had not been invited to accompany Marjory Dean, she
- 64. [274] went alone. What did it matter? Miss Dean was to be there. That was enough. She arrived at three o’clock in the afternoon. A servant answered the bell. She was ushered at once into a vast place with a very high ceiling. All about her were statues and plaster-of-paris reproductions of masterpieces. Scarcely had she time to glance about her when she heard a voice, saw a face and knew she had found an old friend—the artist who had spoken so interestingly of life, he of the beard, was before her. “So this is where you work?” She was overjoyed. “And does the great Fernando Tiffin do his work here, too?” “I am Fernando Tiffin.” “Oh!” Jeanne swayed a little. “You see,” the other smiled, putting out a hand to steady her, “I, too, like to study life among those who do not know me; to masquerade a little.” “Masquerade!” Jeanne started. Did he, then, see through her own pretenses? She flushed. “But no!” She fortified herself. “How could he know?” “You promised to tell me more about life.” She hurried to change the subject. “Ah, yes. How fine! There is yet time. “You see.” He threw a switch. The place was flooded with light. “The thing that stands before you, the ‘Fairy
- 65. [275] and the Child,’ it is called. It is a reproduction of a great masterpiece: a perfect reproduction, yet in this light it is nothing; a blare of white, that is all. “But see!” He touched one button, then another, and, behold, the statue stood before them a thing of exquisite beauty! “You see?” he smiled. “Now there are shadows, perfect shadows, just enough, and just enough light. “Life is like that. There must be shadows. Without shadows we could not be conscious of light. But when the lights are too bright, the shadows too deep, then all is wrong. “Your bright lights of life at the Opera House, the sable coats, the silks and jewels, they are a form of life. But there the lights are too strong. They blind the eyes, hide the true beauty that may be beneath it all. “But out there on that vacant lot, in the cold and dark— you have not forgotten?” “I shall never forget.” Jeanne’s voice was low. “There the shadows were too deep. It was like this.” He touched still another button. The beauty of the statue was once more lost, this time in a maze of shadows too deep and strong. “You see.” His voice was gentle. “I see.” “But here are more guests arriving. You may not be aware of it, but this is to be an afternoon of opera, not
- 66. [276] [277] of art.” Soon enough Jeanne was to know this, for, little as she had dreamed it, hers on that occasion was to be the stellar role. It was Marjory Dean who had entered. With her was the entire cast of “The Magic Curtain.” “He has asked that we conduct a dress rehearsal here for the benefit of a few choice friends,” Miss Dean whispered in Jeanne’s ear, as soon as she could draw her aside. “A strange request, I’ll grant you,” she answered Jeanne’s puzzled look. “Not half so strange as this, however. He wishes you to take the stellar role.” “But, Miss Dean!” “It is his party. His word is law in many places. You will do your best for me.” She pressed Jeanne’s hand hard. Jeanne did her best. And undoubtedly, despite the lack of a truly magic curtain, despite the limitations of the improvised stage, the audience was visibly impressed. At the end, as Jeanne sank from sight beneath the stage, the great sculptor leaned over to whisper in Marjory Dean’s ear: “She will do it!” “What did I tell you? To be sure she will!” The operatic portion of the program at an end, the guests were treated to a brief lecture on the art of
- 67. [278] sculpture. Tea was served. The guests departed. Through it all Jeanne walked about in a daze. “It is as if I had been invited to my own wedding and did not so much as know I was married,” she said to Florence, later in the day. Florence smiled and made no reply. There was more to come, much more. Florence believed that. But Jeanne had not so much as guessed.
- 68. [279] CHAPTER XXXI FLORENCE MEETS THE LADY IN BLACK The great hour came at last. “To-night,” Jeanne had whispered, “‘The Magic Curtain’ will unfold before thousands! Will it be a success?” The very thought that it might prove a failure turned her cold. The happiness of her good friends, Angelo, Swen and Marjory Dean was at stake. And to Jeanne the happiness of those she respected and loved was more dear than her own. Night came quite suddenly on that eventful day. Great dark clouds, sweeping in from the lake, drew the curtain of night. Jeanne found herself at her place among the boxes a full hour before the time required. This was not of her own planning. There was a mystery about this; a voice had called her on the telephone requesting her to arrive early.
- 69. [280] “Now I am here,” she murmured, “and the place is half dark. Who can have requested it? What could have been the reason?” Still another mystery. Florence was with her. And she was to remain. A place had been provided for her in the box usually occupied by Rosemary Robinson and her family. “Of course,” she had said to Florence, “they know that we had something to do with the discovery of the magic curtain. It is, perhaps, because of this that you are here.” Florence had smiled, but had made no reply. At this hour the great auditorium was silent, deserted. Only from behind the drawn stage curtain came a faint murmur, telling of last minute preparations. “‘The Magic Curtain.’” Jeanne whispered. The words still thrilled her. “It will be witnessed to-night by thousands. What will be the verdict? To-morrow Angelo and Swen, my friends of our ‘Golden Circle,’ will be rich or very, very poor.” “The Magic Curtain.” Surely it had been given a generous amount of publicity. Catching a note of the unusual, the mysterious, the uncanny in this production, the reporters had made the most of it. An entire page of the Sunday supplement had been devoted to it. A crude drawing of the curtains, pictures of Hop Long Lee, of Angelo, Swen, Marjory Dean, and even Jeanne were there. And with these a most lurid story purporting to be the history of this curtain of fire as it had existed through the ages in some little known Buddhist temple. The very names of those who, wrapped in its consuming
- 70. [281] folds, had perished, were given in detail. Jeanne had read, had shuddered, then had tried to laugh it off as a reporter’s tale. In this she did not quite succeed. For her the magic curtain contained more than a suggestion of terror. She was thinking of all this when an attendant, hurrying up the orchestra aisle, paused beneath her and called her name, the only name by which she was known at the Opera House: “Pierre! Oh, Pierre!” “Here. Here I am.” Without knowing why, she thrilled to her very finger tips. “Is it for this that I am here?” she asked herself. “Hurry down!” came from below. “The director wishes to speak to you.” “The director!” The blood froze in her veins. So this was the end! Her masquerade had been discovered. She was to be thrown out of the Opera House. “And on this night of all nights!” She was ready to weep. It was a very meek Pierre who at last stood before the great director. “Are you Pierre?” His tone was not harsh. She began to hope a little. “I am Pierre.” “This man—” The director turned to one in the shadows. Jeanne caught her breath. It was the great
- 71. [282] [283] sculptor, Fernando Tiffin. “This man,” the director repeated, after she had recovered from her surprise, “tells me that you know the score of this new opera, ‘The Magic Curtain.’” “Y-yes. Yes, I do.” What was this? Her heart throbbed painfully. “And that of the ‘Juggler of Notre Dame.’” “I—I do.” This time more boldly. “Surely this can be no crime,” she told herself. “This has happened,” the director spoke out abruptly, “Miss Dean is at the Robinson home. She has fallen from a horse. She will not be able to appear to-night. Fernando Tiffin tells me that you are prepared to assume the leading role in these two short operas. I say it is quite impossible. You are to be the judge.” Staggered by this load that had been so suddenly cast upon her slender shoulders, the little French girl seemed about to sink to the floor. Fortunately at that instant her eyes caught the calm, reassuring gaze of the great sculptor. “I have said you are able.” She read this meaning there. “Yes.” Her shoulders were square now. “I am able.” “Then,” said the director, “you shall try.” Ninety minutes later by the clock, she found herself waiting her cue, the cue that was to bid her come dancing forth upon a great stage, the greatest in the
- 72. [284] world. And looking down upon her, quick to applaud or to blame, were the city’s thousands. In the meantime, in her seat among the boxes, Florence had met with an unusual experience. A mysterious figure had suddenly revealed herself as one of Petite Jeanne’s old friends. At the same time she had half unfolded some month-old mysteries. Petite Jeanne had hardly disappeared through the door leading to the stage when two whispered words came from behind Florence’s back: “Remember me?” With a start, the girl turned about to find herself looking into the face of a tall woman garbed in black. Reading uncertainty in her eyes, the woman whispered: “Cedar Point. Gamblers’ Island. Three rubies.” “The ‘lady cop’!” Florence sprang to her feet. She was looking at an old friend. Many of her most thrilling adventures had been encountered in the presence of this lady of the police. “So it was you!” she exclaimed in a low whisper. “You are Jeanne’s lady in black?” “I am the lady in black.” “And she never recognized you?” “I arranged it so she would not. She never saw my face. I have been a guardian of her trail on many an occasion.
- 73. [285] “And now!” Her figure grew tense, like that of a springing tiger. “Now I am about to come to the end of a great mystery. You can help me. That is why I arranged that you should be here.” “I?” Florence showed her astonishment. “Sit down.” The girl obeyed. “Some weeks ago a priceless necklace was stolen from this very box. You recall that?” “How could I forget?” Florence sat up, all attention. “Of course. Petite Jeanne, she is your best friend. “She cast suspicion upon herself by deserting her post here; running away. Had it not been for me, she would have gone to jail. I had seen through her masquerade at once. ‘This,’ I said to myself, ‘is Petite Jeanne. She would not steal a dime.’ I convinced others. They spared her. “Then,” she paused for a space of seconds, “it was up to me to find the pearls and the thief. I think I have accomplished this; at least I have found the pearls. As I said, you can help me. You know the people living on that curious man-made island?” “I—” Florence was thunderstruck. Aunt Bobby! Meg! How could they be implicated? All this she said to herself and was fearful.
- 74. [286] [287] Then, like a bolt from the blue came a picture of Meg’s birthday package. “You know those people?” the “lady cop” insisted. “I—why, yes, I do.” “You will go there with me after the opera?” “At night?” “There is need for haste. We will go in Robinson’s big car. Jaeger will go, and Rosemary. Perhaps Jeanne, too. You will be ready? That is all for now. “Only this: I think Jeanne is to have the stellar role to- night.” “Jeanne! The stellar role? How could that be?” “I think it has been arranged.” “Arranged?” There came no answer. The lady in black was gone.
- 75. [288] CHAPTER XXXII SPARKLING TREASURE The strangest moment in the little French girl’s career was that in which, as the juggler, she tripped out upon the Opera House stage. More than three thousand people had assembled in this great auditorium to see and hear their favorite, the city’s darling, Marjory Dean, perform in her most famous role. She was not here. They would know this at once. What would the answer be? The answer, after perfunctory applause, was a deep hush of silence. It was as if the audience had said: “Marjory Dean is not here. Ah, well, let us see what this child can do.” Only her tireless work under Miss Dean’s direction saved Jeanne from utter collapse. Used as she was to the smiling faces and boisterous applause of the good old light opera days, this silence seemed appalling. As it was, she played her part with a perfection that was art, devoid of buoyancy. This, at first. But as the act progressed she took a tight grip on herself and throwing herself into the part, seemed to shout at the dead
- 76. [289] audience: “You shall look! You shall hear! You must applaud!” For all this, when the curtain was run down upon the scene, the applause, as before, lacked enthusiasm. She answered but one curtain call, then crept away alone to clench her small hands hard in an endeavor to keep back the tears and to pray as she had never prayed before, that Marjory Dean might arrive prepared to play her part before the curtain went up on the second act. But now a strange thing was happening. From one corner of the house there came a low whisper and a murmur. It grew and grew; it spread and spread until, like a fire sweeping the dead grass of the prairies, it had passed to the darkest nook of the vast auditorium. Curiously enough, a name was on every lip; “Petite Jeanne!” Someone, a fan of other days, had penetrated the girl’s mask and had seen there the light opera favorite of a year before. A thousand people in that audience had known and loved her in those good dead days that were gone. When Jeanne, having waited and hoped in vain for the appearance of her friend and benefactor, summoned all the courage she possessed, and once more stepped upon the stage, she was greeted by such a round of applause as she had never before experienced—not even in the good old days of yesteryear. This vast audience had suddenly taken her to its heart. How had this come about? Ah, well, what did it matter?
- 77. [290] They were hers, hers for one short hour. She must make the most of this golden opportunity. That which followed, the completing of the “Juggler,” the opening of “The Magic Curtain,” the complete triumph of this new American opera, will always remain to Jeanne a beautiful dream. She walked and danced, she sang and bowed as one in a dream. The great moment of all came when, after answering the fifth curtain call with her name, “Petite Jeanne! Petite Jeanne!” echoing to the vaulted ceiling, she left the stage to walk square into the arms of Marjory Dean. “Why, I thought—” She paused, too astounded for words. “You thought I had fallen from a horse. So I did—a leather horse with iron legs. It was in a gymnasium. Rosemary pushed me off. Truly it did not hurt at all.” “A frame-up!” Jeanne stared. “Yes, a frame-up for a good cause. ‘The Magic Curtain’ was yours, not mine. You discovered it. It was through your effort that this little opera was perfected. It was yours, not mine. Your golden hour.” “My golden hour!” the little French girl repeated dreamily. “But not ever again. Not until I have sung and sung, and studied and studied shall I appear again on such a stage!” “Child, you have the wisdom of the gods.” “But the director!” Jeanne’s mood changed. “Does he not hate you?”
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