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BOOK REVIEWS

5

It may note be amiss here to remark on the price of this out

standing book. At approxim ately $8 0 the price comes to roughly 40

cents per page Ou ch But, do n't leave; there are roughly 30 lines

per page compared to the 40 lines per page of Van der Waer-

den's Algebra (Viertel Auflage, Springer-Verlag (1959)), which is

of comparable size. Thus, set in the denser Springer-Verlag mode,

this book would shrink to 3 /4's the present num ber of

215

pages, to

^ 1 5 5 pages, which comes to 55 cents per page while Springer

books average < 20 cents per page

I sym path ize with an au tho r's plight: H e does not set prices Re

gardless, I recommend this excellent text for those who can afford

it .

I found nary a typo, and the treatm ent of the selected top

ics is not only lucid but im peccable. It brou ght the sam e delight

that I experienced reading Kaplansky's non-pareil Commutative

Rings and Lambek's Lectures on Rings and Modules, which is to

say that the author's love and command of the subject shines on

every page.

C A R L F A I T H

R U T GE R S, T HE ST AT E U NI VE R SIT Y

BULLETIN (New Series) OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 22, Number 2, April 1990

1990 American Mathematical Society

0273-0979/90 $1 .00 + $.25 per page

Harmonic analysis in phase space, by Gerald Folland. Princeton

University Press, Princeton, NJ, 1989, $17.50 (paper), $55.00

(cloth). ISBN 0-691-08528-5

"The phrase harmonic analysis in phase space is a concise if

somewhat inadequate name for the area of analysis R" that in

volves the Heisenberg group, quantization, the Weyl operational

calculus, the metaplectic representation, wave packets, and related

concepts: It is meant to suggest analysis

on

the configuration space

R" done by working in the phase space R" x R" . The ideas that

fall under this rubric have originated in several fieldsFourier

analysis, partial differential equations, mathematical physics, rep

resentation theory, and number theory, among others. As a result,

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6

BOOK REVIEWS

althoug h these ideas are individu ally well know n to workers in such

fields, their close kinship and the cross-fertilization they can pro

vide have often been insufficiently appreciated. One

of

the prin

cipal objectives

of

this mono graph

is to

give

a

coherent account

of this material, comprising not justan efficient tourof the major

avenues but alsoanexplorationofsome p icturesque byways."

The paragraph above is taken from thepreface to Folland's

splendidly written bookandis a very good summaryofitscon

tents. To putthese co ntents into perspectiveletmeremind you

what the representation theoryofthe Heisenb erg grou p looks like

to group theorists. The salient factsareeasyto summarize; they

consist

of

the following four statem ents:

1. Theirreducible infinite-dimensional representationsofH

n

are in on e-o ne correspondence with thenonzero real num bers.

Given anynonz ero real num ber, h, there existsa unique irre

ducible representation

(*) p

h

:H

n

-+U V)

with the property that the restriction

ofp

h

to

the center

R ofH

n

is the representation

(**) t-+cxp 2niht)I

v

.

This resultisknownasthe Stone-Von N eum ann theorem.

2. LetAbeanelementofthe group Sp )andletx

A

be the

automorphismofH

n

associated with A. By comp osing *)w ith

T

A

we getano ther representation

ofH

n

withtheproperty **).

Therefore,bythe Stone-Vo n Ne um ann theorem, this representa

tionhasto betheStone-Vo n N eum ann representation in anew

guise: i.e., there exists

a

unitary operator

T

A

V

V

such that

Ph

T

A

X

)

T

l

pk

X

)

T

A

for allx GH

n

. Moreover, T

A

is unique uptoaconstant mu ltiple

of modulus one;sothe ma p

***

A

-

T

A

is

a

projective representation

of

Sp(w)

onV.

3.

Let

Mp(fl) be the double cover

of

Sp(w). Then, by adjusting

the multipliers infront ofthe T

A

sone canmake ***)intoan

honest unitary representation of M p ( ). This representationis

known as themetaplectic oro scillator, orharmonic , orSegal-

Shale-Weil) representation.

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B O O K R E V IE WS

7

4. The m etaplectic representation is not irreducible; however, it

splits into two components, an

even

component and an

odd

com

ponent; and both of these are irreducible.

The subject of Folland's book is the remarkable world of math

ematical reality hiding behind the facade of the simple theory that

I 've outlined above. The representation p

h

was, in some sense,

"discovered" in the twenties by the physicists when they noticed

that the Heisenberg canonical commutation relations could be re

alized by the scheme:

Q

t

multiplication by 2nx

i

Since then we have become accustomed to thinking of the un

derlying Hubert space V of this representation as being L (R

w

) .

However, as Folland points out, this representation occurs in na

ture in many other guises. For instance, it occurs as holomorphic

functions on C

n

(Fock space), as sections of a certain line bund le

over T

n

(the lattice desc ription of p

h

), as holom orphic func

tions on the generalized Siegel upper-half space, as sections of a

certain line bundle over the Lagrangian Grassmannian, and (last

but not least) as the symplectic analogue of the spin representation

of the double cover of SO 2n).

In each of these guises the representation theory of the Heisen

berg group conn ects with other areas of m athem atics: In its L R

n

)

guise it connects with classical quantum physics and with the the

ory of pseudo differential op erato rs. As Fock space it conn ects with

qu antu m field theory (vacuu m states, creation and annihilation o p

erato rs, etc.) an d with complex variables. As a line bu nd le over

T it connects with analytic num ber theory (in particular, the

theory of the theta function) and with phenomena in solid-state

physics (such as the de Ha as -v an Alphen effect). As the Siegel

upper half-plane it connects again with analytic number theory

(transformation properties of the theta function) and, finally, in

its last guise, with such esoteric matters of modern elementary

particle physics as BRS cohomo logy. All these marvelous in ter

connections are beautifully described in Folland's book. The only

place I know of where the picture is laid out as well is in Mackey's

book [2]. However, the overlap between these two books is not

great. In particular, in Folland's book the focus of interest is on

the implications of this whole picture for analysis. One has to go

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