Table Of ContentINTERDISCIPLINARY MATHEMATICS
BY ROBERT HERMANN
1. General Algebraic Ideas
2. Linear and Tensor Algebra
3. Algebraic Topics in Systems Theory
4. Energy Momentun Tensors
5. Topics in General Relativity
6. Topics in the Mathematics of Quantum Mechanics
7. Spinors, Clifford and Cayley Algebras
8. Linear Systems Theory and Introductory Algebraic Geometry
9. Geometric Structure of Systems-Control Theory and Physics, Part A
10. Gauge Fields and Cartan-Ehresmann Connections, Part A
11. Geometric Structure of Systems-Control Theory, Part B
12. Geometric Theory of Non-Linear Differential Equations, Backlund
Transformations, and Solitons, Part A
13. Algebra-Geometric and Lie Theoretic Techniques in Systems Theory,
Part A by R. Hermann and C. Martin
LIE GROUPS: HrsTORY, FRONTIERS
AND APPLICATIONS
VOLUME V
SYMPLECTIC GEOMETRY AND FOURIER ANALYSIS
NOLAN R. WALLACH
Department of Mathematics
Rutgers University
New Brunswick, New Jersey 08903
WIT-H AN APPENDIX ON QUANTUM MECHANICS
BY ROBERT HERMANN
MA.TH SCI PRESS
53 JORDAN ROAD
BROOKLINE, MASSACHUSETTS 02146
LIE GROUPS: HISTORY, FRONTIERS AND APPLICATIONS
VOLUME V
SYMPLECTIC GEOMETRY AND FOURIER ANALYSIS
Copyright@ 1977 by Nolan R. Wallach
All rights reserved
Library of Congress Catalog Card Number: 76-43503
ISBN: 0-915692-15-5
MATH SCI PRESS
53 JORDAN ROAD
BROOKLINE, MASSACHUSETTS 02146
Printed in the United States of America
PREFACE
Most mathematicians (myself included) are awed and somewhat
mystified by what physicists call quantum mechanics. Most recent
books by physicists on quantum mechanics begin with a mathemati
cal framework (generally Hilbert space theory). They develop
standard results from functional analysis using very suggestive
terminology. They then set up some framework that leads to the
Schrodinger equation. These equations are usually studied by
separation of variables. At some point in this development the
mathematics undergoes a "transformation into physics". The
mathematical rigor dilutes and the concepts become progressively
more foreign to mathematicians. There is no classical analogue
of the mathematical softness of quantum mechanics. Classical
mechanics begins with some variant of Newton's laws. The neces
sary mathematics consists of differential equations. In general,
for concrete problems, the differential equations are hard to
solve. A variety of variational and geometric alternative inter
pretations are made. The mathematics in these schemes becomes
progressively more sophisticated. When classical mechanics
leaves the realm of mathematical rigor, it does so for purely
pragmatic reasons: The mathematical theorems are not sufficient
to handle the problem. This route to physics is (in my mind) in
no way mysterious to a mathematician.
iii
iv PREFACE
Recently, many mathematicians have attempted to make the
route to the Schrodinger equation more satisfactory to mathemati
cians. The mathematical motivation for this work comes from the
great success of Kirillov's technique for studying harmonic
analysis on nilpotent Lie groups, and Auslander-Kostant's sweep
ing extension of the theory to solvable groups. These mathemati
cal theories begin with the Stone-Von Neumann theorem, which in
a sense makes rigorous the simplest example of a route to the
Schrodinger equation: the Heisenberg quantization rules for a
free particle.
The work of Auslander-Kostant is an outgrowth of Kirillov's
theory and Kostant's theory of quantization (this theory has
been studied most significantly by Kostant, Blattner [3],
Sternberg and Guilliman). The Kostant quantization is mathe
matically rigorous. However, it replaces the "physical mystery"
by a very disheartening mathematical fact: It is not always
possible to carry out the quantization rules. Also, even when
it is formally clear that the quantization rules can be carried
out, the actual process leads to (for example) divergent (singular)
integrals which must be interpreted in some renormalized sense
(i.e., through analytic continuation). An interesting example
of this type comes from replacing the Heisenberg group by
SL(2, lR). Then the Kostant quantization involves the Kunze-
Stein intertwining operators.
PREFACE v
However, Kostant's theory has an important redeeming feature.
There are two steps in the procedure of quantization. The first
step is pre-quantization. Pre-quantization is always possible.
It also leads to interesting mathematics. Quantization becomes
a method of "cutting down" the number of variables involved in
pre-quantization. For example, in the Heisenberg quantization
rules there are two steps. Let us look at this example in a bit
of detail. In classical mechanics one labels a particle at a
time t by its position q(t) and its momentum mq'(t) = p(t).
Thus one looks at a particle as a curve in :rn.3 x :rn.3. Newton's
laws (if the force law is conservative) in Hamiltonian form
become
dpi aH
~ - aqi (q(t), p(t))
where H is the Hamiltonian of the system. If f is a function
on :rn.3 x :rn.3 (say of class C~) then
ad t f(q(t), p(t)) ~~- L ~~
Lap. aq. aq. ap.
l. l. l. l.
{H,f}(q(t), p(t))
vi PREFACE
{H,f} is given by the above formula and is called the Poisson
bracket of H with f. Now, if we set Ut(q,p) = (p(t), q(t)),
where q(O)
by uniqueness. This gives
{H,f}(Ut(q,p))
Thus we can think of the trajectories of the particles in the
Hamiltonian field as the trajectories of the one parameter group
generated by the vector field f + {H,f}. We have therefore
reinterpreted the Hamiltonian H as an operator on the Cm
functions on lR.3 x lR.3. The trajectories of the particles have
become a one-parameter group of transformations of Cm (lR.3 x lR.3).
That is, (Ut f) (x) f(Utx). The equations of motion are
ut {H,f} .
The first step in pre-quantization of this problem is to
add a complex variable. That is, look at f as "its function
value". In other words, x + (x, f(x)) becomes a function from
lR.3 x lR.3 ...... lR.3 x lR.3 x a;. The next step is to interpret
w = ~dpi A dqi as a curvature. There are many ways of accomp
lishing this (the fact that they will give the same result is
the independence of quantization in this special case); one is
to write 8 = ~pidqi' then de= w. If <P e: Cm(lR.3 x lR.3) then
we define
PREFACE vii
(o(<P)·f(x) ({cj>,f} (x) - 211i8(xcj>)f(x) + 211icj>(x)f(x))
Here is the vector field given by is
called pre-quantization. One finds by a simple computation that
(See Section 2.6.)
If cj>(q,p) = l:aiqi + l:bipi + t, (that is, a polynomial
of degree at most one). Then
For these functions we can "cut down" the number of variables in
the domain of o(cj>) by the obvious trick. Let
{f
e: c" °c JR 3 x JR 3)1 !L = 0' i = 1, z' 3 J
aqi
If f e: H and cj> is as above, then
00
O(cj>) f
This is the Heisenberg quantization of linear functions. The
"ad hoc" rules qi>-+ a/api and pi>-+ multiplication by -211i pi
have been incorporated in a formalism. The Stone-Von Neumann
viii PREFACE
theorem now says that subject to certain regularity conditions
(essential self adjointness) this is the only way we could have
cut our variables down from six to three.
In the above example two properties become apparant. The
first is that we can "pre-quantize" any function. To quantize
one must use very special functions. We will see in the discus
sion in Chapters 5 and 6 of the metaplectic representation that
the Heisenberg quantization can be extended to polynomials of
degree at most two. This gives a the quantization of a free
particle in the absence of forces. Once forces are introduced
the theory becomes murkier; unless they are of special type,
the physicist must go back to his old ways.
This monograph is the lightly edited notes for a course
that I gave at Rutgers University during the spring semester
of 1975 on symplectic geometry and Fourier analysis. The partici
pants in the course (graduate students and some professors) had
mixed backgrounds. For this reason I tried to develop the
material of the course from the most elementary perspectiv.e.
That is, the prerequisites of the course were the basic material
in a normal first year graduate program, including a decent
course in differential manifolds and elementary functional
analysis. In a course of this type the lecturer is constantly
in the situation of making compromises in the degree of generality
of the results proved. For example, in Chapter 1 a more satis
factory (from the modern point of view) development of the subject
