Lagrangian analysis and quantum mechanics; a mathematical structure related to asymptotic expansions and the maslov index

Lagrangian Analysis and Quantum MechanicsA Mathematical Structure Related to Asymptotic Expansions and the Maslov Index Jean Leray English translation by Carolyn Schroeder The MIT Press

Lagrangian Analysis and Quantum Mechanics

A Mathematical Structure Related to

Asymptotic Expansions and the Maslov Index

Jean Leray

English translation by Carolyn Schroeder

The MIT Press

Cambridge, Massachusetts

London, England

Copyright C) 1981 by The Massachusetts Institute of Technology

All rights reserved No part of this book may be reproduced in any form or by any means, elcctronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.

This book was set in Monophoto Times Roman by Asco Trade Typesetting Ltd., Hong Kong, and printed and bound by Murray Printing Company in the United States

1 Differential equations, Partial-Asymptotic theory 2 Lagrangian functions.

3 Maslov index 4 Quantum theory I Title.

QA377.L4141982 515.3'53 81-18581

ISBN 0-262-12087-9 AACR2

To Hans Lewy

Preface xi

Index of Symbols xiii

Index of Concepts xvii

I The Fourier Transform and Symplectic Group

3 Differential Operators with Polynomial Coefficients 20

§2 Maslov Indices; Indices of Inertia; Lagrangian Manifolds andTheir Orientations 25

0 Introduction 25

1 Choice of Hermitian Structures on Z(1) 26

2 The Lagrangian Grassmannian A(l) of Z(1) 27

3 The Covering Groups of Sp(l) and the Covering Spaces of A(l) 31

4 Indices of Inertia 37

5 The Maslov Index m on A2 (1) 42

6 The Jump of the Maslov Index m(A., A ) at a Point (ti, A.')

Where dim;, n A.' = 1 47

7 The Maslov Index on Spa (l); the Mixed Inertia 51

8 Maslov Indices on A,(/) and Sp,,(/) 53

1 The Algebra W(X) of Asymptotic Equivalence Classes 68

2 Formal Numbers; Formal Functions 73

3 Integration of Elements of FO(X) 80

4 Transformation of Formal Functions by Elements of Sp2(l) 86

5 Norm and Scalar product of Formal Functions with Compact

2 Review of E Cartan's Theory of Pfaffian Forms 125

3 Lagrangian Manifolds in the Symplectic Space Z and in Its

Hypersurfaces 129

4 Calculation of aU 135

5 Resolution of the Lagrangian Equation aU = 0 139

6 Solutions of the Lagrangian Equation aU = 0 mod(1/v2) with

Positive Lagrangian Amplitude: Maslov's Quantization 143

7 Solution of Some Lagrangian Systems in One Unknown 145

Contents ix

8 Lagrangian Distributions That Are Solutions of a HomogeneousLagrangian System 151

Conclusion 151

§4 Homogeneous Lagrangian Systems in Several Unknowns 152

1 Calculation of Em_a' U,, 152

2 Resolution of the Lagrangian System aU = 0 in Which the Zeros ofdet ao Are Simple Zeros 156

3 A Special Lagrangian System aU = 0 in Which the Zeros of det aoAre Multiple Zeros 159

III Schrodinger and Klein-Gordon Equations for One-Electron Atoms in a Magnetic Field

Introduction 163

§1 A Hamiltonian H to Which Theorem 7.1 (Chapter II, §3)

Applies Easily; the Energy Levels of One-Electron Atoms with theZeeman Effect 166

1 Four Functions Whose Pairs Are All in Involution on E3 Q+ E3Except for One 166

2 Choice of a Hamiltonian H 170

3 The Quantized Tori T(l, m, n) Characterizing Solutions, Definedmod(1/v) on Compact Manifolds, of the Lagrangian System

aU = (aL2 - L2)U = (am - Mo)U = 0 mod (1/v2) 174

4 Examples: The Schrodinger and Klein-Gordon Operators 179

§2 The Lagrangian Equestion aU = 0 mod(1/v2) (a Associated to H,

U Having Lagrangian Amplitude >, 0 Defined on a Compact V) 184

0 Introduction 184

1 Solutions of the Equation aU = 0 mod(1/v2) with LagrangianAmplitude >,0 Defined on the Tori V[L0, M0] 185

2 Compact Lagrangian Manifolds V, Other Than the Tori V[L0,M0],

on Which Solutions of the Equation aU = 0 mod(1/v2) with

Lagrangian Amplitude 30 Exist 190

3 Example: The Schrodinger-Klein-Gordon Operator 204

x Contents

§3 The Lagrangian System

a U = (am - const.) U = (aL2 - const.) U = 0

When a Is the Schrodinger-Klein-Gordon Operator 207

1 Study of Problem (0.1) without Assumption (0.4) 231

2 The Schrodinger-Klein-Gordon Case 234

IV Dirac Equation with the Zeeman Effect

Introduction 238

§1 A Lagrangian Problem in Two Unknowns 238

1 Choice of Operators Commuting mod(1!v3) 238

2 Resolution of a Lagrangian Problem in Two Unknowns 240

§2 The Dirac Equation 248

1 Reduction of the Dirac Equation in Lagrangian Analysis 248

2 The Reduced Dirac Equation for a One-Electron Atom in a

Constant Magnetic Field 254

3 The Energy Levels 258

4 Crude Interpretation of the Spin in Lagrangian Analysis 262

5 The Probability of the Presence of the Electron 264

Conclusion 266

Bibliography 269

Only in the simplest cases do physicists use exact solutions, u(x), of

problems involving temporally evolving'systems Usually they use totic solutions of the type

asymp-u(v, x) = a(v, x)evw(x),

where

the phase (p is a real-valued function of x E X = R';

the amplitude a is a formal series in 1/v,

w 1

a(v, x)

=a V

whose coefficients a, are complex-valued functions of x;

the frequency v is purely imaginary

The differential equation governing the evolution,

The phase q has to satisfy a first-order differential equation that is linear if the operator a is not of first order

non-The amplitude a is computed by integrations along the characteristics

of the first-order equation that defines cp

In quantum mechanics, for example, computations are first made as if

where h is Planck's constant,

were a parameter tending to ioo; afterwards v receives its numerical valuevv.

Physicists use asymptotic solutions to deal with problems involvingequilibrium and periodicity conditions, for example, to replace problems

of wave optics with problems of geometrical optics But cp has a jump and

a has singularities on the envelope of characteristics that define cp: forexample, in geometrical optics, a has singularities on the caustics, which

xii Preface

are the images of the sources of light; nevertheless geometrical opticsholds beyond the caustics

V P Maslov introduced an index (whose definition was clarified by

I V Arnold) that described these phase jumps, and he showed by a venient use of the Fourier transform that these amplitude singularities areonly apparent singularities But he had to impose some "quantum con-ditions." These assume that v has some purely imaginary numerical value

con-vo, in contradiction with the previous assumption about v, namely, that

v is a parameter tending to i oo The assumption that v tends to i oo isnecessary for the Fourier transform to be pointwise, which is essential forMaslov's treatment A procedure, avoiding that contradiction and guided

by purely mathematical motivations, that makes use of the Fourier form, expressions of the type (1), Maslov's quantum conditions, and thedatum of a number vo does exist, but no longer tends to define a function

trans-or a class of functions by its asymptotic expansion It leads to a new

mathematical structure, lagrangian analysis, which requires the datum of aconstant vo and is based on symplectic geometry Its interest can appearonly a posteriori and could be quantum mechanics Indeed this structureallows a new interpretation of the Schrodinger, Klein-Gordon, and Diracequations provided

vo = ii = 2h1, where h is Planck's constant

Therefore the real number 2ni/vo whose choice defines this new matical structure can be called Planck's constant

mathe-The introductions, summaries, and conclusions of the chapters andparts constitute an abstract of the exposition

Historical note In Moscow in 1967 I V Arnold asked me my thoughts

on Maslov's work [10, 11] The present book is an answer to that question

It has benefited greatly from the invaluable knowledge of J Lascoux

It introduces vo for defining lagrangian functions on V (chapter II, §2,section 3) in the same manner as Planck introduced h for describing thespectrum of the blackbody Thus the book could be entitled

The Introduction of Planck's Constant into Mathematics

January 1978

College de France

Index of Symbols

N set of the natural numbers (i.e., integers > 0)

atomic energy level: III; IV

the function in III,§1,(4.23)

function: III; IV

Inert index of inertia: I,§1,2; I,§2,4; I,§2,definition 7.2

* Each chapter (1, II, III, IV) is divided into parts (§l, §2, §3, ), which in turn are divided

into sections (0, I, 2, ) References to elements of sections (for example, theorems, equations, definitions) in the same chapter, part, and section are by one or two numbers:

in the latter case the first number refers to the section and is followed by a period References

to elements of sections in another chapter, part, and/or section are by a string of numbers

separated by commas For example, a reference in chapter !, §2, section 3 to the one theorem

in this chapter, part, and section is simply theorem 3; to the one theorem in this chapter and part but section 4 is theorem 4; to the one theorem in this chapter but §3 of section 4

is §3,theorem 4; and to the one theorem in chapter II, §3, section 4 is II,§3,theorem 4 Similarly, a reference in chapter !, §2, section 3 to the first definition (of more than one)

in chapter II, §3, section 4 is II,§3,definition 4.1.

xiv Index of Symbols

matrices: II,§4,3; IV,§1,1characteristic curve: II,§3,definition 3.1; III,§1,(2.14)function: III,§1,4

element of Sp2(1)torus: III,§ 1,3lagrangian function on V: II,§2,3formal functions on V: II,§2,2unitary group: I,§2,2

lagrangian manifold: I,§2,9; I,§3,1hypersurface of Z

subset of U(1): I,§2,lemma 2.1spaces: I,§1,1; I,§3,1

differential, formal or lagrangian operator:

I,§1,definition 3.1; II,§l,definition 6.2; II,§2,definition 1.1

argumentdifferentiationLebesgue measuredeterminant2.71828 neutral element of a group: I

dimension of X : 1; II (dimension of X = 3 in III,

J-1interior product: I1,§3,2quantum number: IV,§1,example 2

IV)

element of Sp(1): I; IIfunction: III; IVfunction: II,§3,(3.10) ; 11,§3,(3.13); 111,§ 1,(2.6)element of U (1) : I

transpose of u: I,§2,2formal number or function: II,§1,2element of W(1)

elements of Xelements of Z

I I,§ 1,1space of formal or lagrangian functions ordistribution: II,§1,2; II,§1,7; II,§2,2; II,§2,3; II,§2,5Hilbert space: I,§1,1

Lie derivative: II,§3,definition 3.2neighborhood

Schwartz spaces: I,§1,1

arc: 1,§2curve: III,§1,(2.5)laplacian (A0 is the spherical laplacian):

lagrangian amplitude: II,§2,theorem 2.2arc or homotopy class

invariant measure of V: II,§3,definition 3.2characteristic vector: II,§3,definition 3.1element of A: I; II

functions : III,§1,(2.10)element of I: II,§1,1i/h = 2ni/h (h e R+): II,§2,3; II,§3,63.14159

jth homotopy group: I,§2,3Pauli matrices: IV,§1,(1.6); IV,§1,(1.7)

ri/d `x

phase: I,§2,9; I,§3,2; II,§1,2lagrangian phase: I,§3,1pfaffian forms

III,§1,(1.7)Atomic Symbols: III; IV; passim (see III,§1,4, Notations)

Q

E

p

energyspeed of lightPlanck's constantpotential vectormagnetic field1/137

Bohr magnetoncharge

mass

formal number, functions u, UR

I The Fourier Transform and Symplectic Group

A Well [18] studied it on an arbitrary field in order to extend C Siegel'swork in number theory

Summary We take up the study of the metaplectic group in order to specify its action on '(R'), *'(R'), and 9'(R') (see theorem 2) and its

action on differential operators (see theorem 3.1)

1 The Metaplectic Group Mp(1)

Let X be the vector space R' (1 > 1) provided with Lebesgue measure d'x.Let X * be its dual, and let < p, x> be the value obtained by acting p e X

IfI9,r = Sup Ix°(c'xlrf(x)I < 00

X

The topology of\Y(X) is defined by a countable fundamental system of

neighborhoods of 0, each depending on a pair of 1-indices (q, r) and arational number E > 0 as follows:

4t(q, r, E) = {f I IfI,,r < E}

The bounded sets B of 9'(X) are thus all subsets of bounded sets of '(X)

of the following form:

B({be,.}) = {fI If I,., < ba.rdq,r}, q,rEN', bq1 re1 +

The Schwartz space 59''(X) is the dual of Y (X) [13]; its elements arethe tempered distributions: such an element f' is a continuous linear

functional

.9'(X)-,C.

The value of f' on f will be denoted by fx f'(x) f (x) d'x, although thevalue of f' at x is not in general defined The bound of f' on a boundedset B in ,9'(X) is denoted by

If'IB = Sup I ff'(x) f(x) d'xI

x

The continuity of f' is equivalent to the condition that f' is bounded:

I f' Ie < oc dB The topology of 59''(X) is defined by a fundamental system

of neighborhoods of 0, each depending on a bounded set B of 99'(X) and

Y (X) is dense in Y'(X )

For the proof of the last theorem, see L Schwartz [13] : chapter VII, §4,the commentary on theorem IV, and chapter III, §3, theorem XV; alter-natively, see chapter VI, §4, theorem IV, theorem XI and its commentary.Differential operators associated with elements of Z(l) = X (D X* Let

v be an imaginary number with argument n/2: v/i > 0

Let a° be a linear function, a°: Z(1) ) R Let a°(z) = a°(x, p) be itsvalue at z = x + p [z e Z(1), x e X, p c X*] The operator

a = a° (x, l\\ -v ox-(

is a self-adjoint endomorphism of 9"(X ): the adjoint of a, which is anendomorphism of So(X ), is the restriction of a to So(X ) The operators aand the functions a° are, respectively, elements of two vector spaces sy and.sad° These spaces are both of dimension 21 and are naturally isomorphic:

We say that a is the differential operator associated to a° E d° By (1.2),sl°, which is the dual of Z(1), will be identified with Z(1)

The commutator of a and b e sl is

where z = x + p, z' = x' +p',xand x'cX,and pand p'eX*.

Each function a° e sl° is defined by a unique element a' in Z(l) such that

[a, b] = I [a', b'

V

where the right-hand side is defined by the symplectic structure

An automorphism S of '(X) transforms each a e d into an operator

b = SaS-', defined by the condition

Therefore s is an automorphism of the symplectic space Z(1)

The group of automorphisms of the symplectic space Z(1) is called thesymplectic group and is denoted Sp(l):

S E SP (I).

By (1.1),

[sa', z] = [a', s-'z] = (a° o s-')(z)

In summary:

LEMMA 1 1. Under the natural isomorphisms of sat, Z(1), and a7°, theautomorphism

a SaS-'

of si, which is defined for all S E G(1), becomes

an automorphism s of Z(1), s : a' f ' sa', s e Sp(l),

an automorphism of sl' given by a° i > a° o s-1

The function S f ' s is a natural morphism

By integration of this system of differential equations,

Se_v<P,x> = c(p)e-'(P,'), where c:X* - C

Taking the derivative with respect to p, we see that the gradient of c, cp,exists and satisfies

- vS[xev<P.x>I = -vxSe-''<P.x> + Cpe-v(P.x).

equivalently, since S and multiplication by x commute,

cp = 0

c(p) is independent of p and will be denoted c Let F be the Fourier form and let g = F-1 f e "(X) By the definition of F,

i the finite group generated by the Fourier transforms in one of the

coordinates (some base of the vector space X having been fixed);

ii the group consisting of automorphisms of 9''(X) of the form

f - e°Qf,

where Q is a real quadratic form mapping X - R;

iii the group consisting of automorphisms of 9''(X) of the form

f' -* f, where f (x) = det T f'(Tx), T an automorphism of X

Each of these groups has a restriction to 9'(X) that gives a group ofautomorphisms of 9'(X) and a restriction to *'(X) that gives a group ofunitary (that is, isometric and invertible) transformations of i*'(X) Thefollowing definition uses these properties

Definition 1.2 Let A be the collection of elements A each consisting of1°) a quadratic form X Q+ X R, whose value at (x, x') e X Q+ X is

A (x, x') = Z <Px, x> - <Lx, x'> + Z <Qx', x'), (1.9)where, if `P denotes the transpose of P,

P = `P:X - X*, L:X,- X*, Q = `Q:X -+ X*,

det L A 0;

2°) a choice of arg det L = nm(A), m(A) e Z, which allows us to defineA(A) = det L by arg A(A) = (rz/2)m(A)

1,§ 1,1

Remark det L is calculated using coordinates in X* dual to the ordinates in X and is independent of coordinates chosen such that dx' n A dx' = d'x.

co-Remark m(A) will be identified with the Maslov index by 2,(2.15) and

Proof of (1.11) Let f' e Y (X) a(SAf')/(3x and SA(df'/dx) are calculated

by differentiation of (1.10) and integration by parts; the result of thesecalculations gives the following relations among differential operators of

termined by the equation

We assume s 0 Esp Then x and x' are independent on the above

21-dimensional plane On this plane we define

A(x, x') = i <p, x> - 1 <p', x'> (1.12)

We therefore have

dA = <p, dx> - <p', dx'>, that is,

p = As, p = -As

x and Ax have to be independent Hence det;k(AX X ') i4 0 Therefore

s = 5A, which completes the proof

The sA clearly generate Sp(1) Thus:

LEMMA 1.3. The natural morphism G(l) - Sp(l) is an epimorphism

By lemma 1.2, G(1) is a Lie group and

by an expression of the form (1.10).

3°) The restriction of every S e Mp(l) to So(X) is an automorphism of,°(X)

Proof of 1°): (1.13) and (1.14); S' is identified with a subgroup of Mp(l)

.Proof of 2°) Let S e Mp(l)\EMp Then the image of S in Sp(l) is someelement sA, A e A; SSA' e S' by (1.15)

Proof of 3°) By 2°), S = cSA, SAC Now the restrictions of c,

SA,, , SA, to °(X) are automorphisms of, °(X ).

2 The Subgroup Sp2(l)Of MP(l)

Definition 2.1 We denote by Spz(l) the subgroup of Mp(l) that is

generated by the SA

Definition 2.2 - Given A E A, we define A* e A as follows:

A*(x, x') = -A(x', x), 1(A*) = i'A(A),

m(A*) = l - m(A)

LEMMA 2.1. SA 1 = SA*; thus sA1 = SA+

Proof This amounts to proving the equivalence of the following twoconditions for any f and f' EY(X):

f(x) - _ (±)"2A(A) evA(x.x') f'(x')d`x',

x rz

f (x) = (Iv2rzi) 0(A)

x

Using the expression for A given by (1.9), this is the same as the equivalence

of the following two conditions:

from the Fourier inversion formula; the

To compute compositions of the SA, we will find an explicit expressionfor SA(e°`° ), where (p' is a second-degree polynomial This is made possible

by the following definition

Definition 2.3 Choose linear coordinates in X such that d'x =dx' A A dx' and choose the dual coordinates in X* The followingnotions are independent of this choice

Let cp be a real function, twice differentiable:

cp : X -+ R.

Hess,,((p) denotes the hessian of cp, the determinant of its second derivatives.Alternatively this is the determinant of the quadratic form

X-3

Inertz((p) denotes the index of inertia of this form It is defined") whenHess((p) 0 Clearly

Inert(-q') = I - Inert(q),

arg Hess((p) = nInert(g) mod 2n

This formula makes possible the definition

Thus, for example,

If op is a real quadratic form,

rp: X a x i I<Rx, x>, where R = 1R: X - X*,

then Hess((p) and Inert((p) will be denoted Hess(R) and Inert(R) Hess(R) isthe determinant of the symmetric matrix R Inert(R) is the number ofnegative eigenvalues of R Clearly

Inert(R) = Inert(R-1), [Hess(R)]1J2[Hess(-R-1)]112 = it.

(2.3)

LEMMA 2.2. Let 9' be a real second-degree polynomial Let A e A be suchthat Hessx.(cp'(x') + A(x, x')) A 0 Denote by 9(x) the critical value of thepolynomial

Xax't * A (x,x') + V (x');

rp is a second-degree polynomial We have

SA(e'11)

Remark 2.1 This lemma assumes v/i > 0 Up to this point, it was

sufficient to assume v/i real and nonzero

Proof We know that

'It is the number of negative eigenvalues of the linear symmetric operator dx F dcp_

We then have, for I = 1, I arg p I < n1/2,

Fe°`° = v I v I evv; VF, = e"'4

In other words, when I = 1, the following result holds: Let p': X -' R be

a real second-degree polynomial such that Hess p' 0; let p(p) be thecritical value of the polynomial

xicp'(x) - CP,x>,

we have

Let us show that, since relation (2.6) holds for I = 1, it holds for all I >, 1

It suffices to choosethe coordinates x' in X such that

is equivalent to the condition

Hess,, [A(x, x') + A'(x', x")] : 0 (the Hessian is constant) (2.8)2°) This condition is equivalent by lemma 2.1 to the existence of A" E Asuch that

A" is defined by the condition that the critical value of the polynomialx' + A(x, x') + A'(x', x") + A"(x", x)

be zero

3°) Just as (1.9) defines A by P, Q, L, let A' and A" be defined by P, Q', L'and P", Q", L" The condition (2.8) for the existence of A" is expressed as1' + Q is invertible

A" can be defined by the formulas

P" + Q' = L'(P' + Q)-"L', P + Q = 1L(P' + Q)-1L,

(2.10)

Remark 2.2.

we have

Writing A + A' + A" for A(x, x') + A'(x', x") + A"(x", x),

Inertx(A + A' + A") = A' + A")

leaves x and x" independent

For any x and x", there exists an x' satisfying this relation

Now in (1.9), det L A 0 Therefore (2.7) is equivalent to (2.8)

Proof of 2°) Assumption (2.9) means that any two of the followingthree relations implies the third:

(x, P) = SA(x', P'), (x', P) = SA,(X", p"), (x", P") = P)

Then by (1.11), each of the next three relations implies the other two:

(A+A'+A")x=0, (A+A'+A")x.,=0,

(2.13)

where

A + A' + A" = A(x, x') + A'(x', x") + A"(x", x)

Now by Euler's formula, these three relations imply

A+A'+A"=0.

Therefore

(A + A' + A")s, = 0, that is, (A + A')x = 0, implies A + A' + A", = 0.Proof of 30) We have

Hessx.(A + A' + A") = Hess(P' + Q),

which gives the first statement For the other, the three pairwise equivalentrelations (2.13) can be written

(P+Q")x-`Lx' -L'x"=0,

-Lx + (P' + Q)x' -`L'x" = 0,

`L"x-L'x'+(P"+Q')x"=0.

(210) clearly expresses the equivalence of these three relations

Proof of Remark 2.2 By (2.10), the symmctric matrices

16 I,§1,2

Inert(SA, SA,, SA,.) = Inert(sA, SA,, SA,.)

Moreover, we define the Maslov index of SA, m(SA) E Z4, by

§2,8 will connect this with the index that V I Maslov actually introduced.Lemma 2.1 and (2.15) have these obvious consequences:

Inert(sA,l, sA.', sA 1) = I - Inert(sA, SA., SA ), (2.16)m(SA 1) = 1 - m(SA), m(-SA) = m(SA) + 2 mod 4

We can at last study compositions of the SA

LEMMA 2.4. Consider a triple A, A', A" of elements of A such that

Remark Condition (2.17), which is equivalent to (2.18), implies (2.20)mod 2

Proof Let Y E X Formula (1.10) holds if f' is replaced by the Diracmeasure with support y, given by

6'(x) = 6(x - y)

We obtain

q2 (SA.(>)(x)

= (21ri 0(A )eA (s y)

from which follows, by lemmas 2.2 and 2.3,2°),

(;)u2

(SASAb'x) = iA(A)A(A'){Hessx.[A(x, x')

+ A'(x', y)]}-1f2e-,A"(y.x)

Multiplying this by f'(y)d'y, where f' c- Y (X), and integrating, we get

SASA' f' = A(A(AO(j)[Hessx,(A + A' +

which gives, by lemma 2.1 and formula (2.12),

SASA.SA = ±E

Now specify the sign By definition 2.4,

arg[Hessx.(A + A' + A")] 1j2 = Z Inert(SA, SA., SA ) mod 2n

By definition 1.2, (2.16), and lemma 2.1,

arg A(A) = 2m(SA), arg A(A') = 2m(SA,) = 2 [l

Recall that Sp2 (1) denotes the group generated by the SA

LEMMA 2.5. Every element of Sp2(1) is a product of two of the SA.Proof By lemma 2.1, every element of Sp2(1) is a product of the SA.

It then suffices to prove that given U, V, W c A, there exist B and C in Asuch that

Now, by lemmas 2.3,1°) and 2.4, for every W e A and every T a genericelement of A, SWST belongs to {SA} and is generic Therefore, for Tgeneric,

SVST E {SA}, SUSVST E {SA}, ST'SW E {SA},

which gives (2.21) with

Sg = SUSVST E {SA}, SC = ST1Sk E {SA}

The restriction to Sp2(0 of the natural morphism Mp(l) - Sp(1) isclearly a natural morphism:

Proof By the preceding lemma, the kernel of this morphism is the

collection of the SASA.(A, A' c- A) such that sASA = e From this, by lemma2.1,

SA, _ ±SA.; therefore SASH- = ±E

LEMMA 2.7. The group Sp2(1) is connected

Proof Given k e Z4 (additive group of integers mod 4), let Dk be thecollection of SA such that

m(A) = k, or equivalently, i-kA(A) > 0

The collection of quadratic forms A satisfying A2(A) > 0 [or A2(A) < 0]

is connected Each Dk is thus a connected set in Sp2(1)

Given k e Z4, let SA and SA, be such that

m(SA) - -k mod 4;

p' + Q has one eigenvalue equal to zero and I - 1 eigenvalues > 0.

Let B and B'be elements of A near A and A' and such that

Hess,,,(B + B') 0.

Inerts,(B + B') takes the values 0 and 1 Since m is locally constant,

M(SB) = m(SA), m(Sa.l) = m(SA1).

We define B" E A by E By (2.20), takes the values k and

k + 1 in any neighborhood of the element (SASA.)-1

of Sp2(1) Thiselement thus belongs to Dk n Dk+1:

pk n Dk+ 1 zA 0,

which gives the lemma

The above lemmas prove the following theorem Part 1 of the theoremreduces the study of Mp(l) to that of Spz(I) Its equivalent can be found inthe work of D Shale and A Weil, but the proof we have given has es-tablished various other results that will be indispensible to us One ofthese is part 3 of the theorem This will be used in §2,8

THEOREM 2. 1°) The elements SA of Mp(l) that are defined by (1.10) generate

a subgroup Sp2(l) of Mp(l) Sp2(l) is a covering group (see Steenrod [17],1.6, 14.1) of the group Sp(l) of order 2 It is a group of automorphisms ofE/(X) that extend to unitary automorphisms of,Y(X) and to automorphisms

Of 99'(X)

2°) The formulas (2.11) and (2.14) define the inertia of every triple s, s', s"

of elements of Sp(l)\Esp such that

m(S-') = 1 - m(S), m(-S) = m(S) + 2 mod4,

Inert(S, S', S") = m(S) - m(S'-1) + m(S") mod 4

Remark 2.3 We shall see later that m is characterized by the last formulaand the property of being locally constant

Remark 2.4 Sp2(1) contains the three subgroups of G(l) defined in section

1 by (i) Fourier transformation, (ii) quadratic forms, and (iii) phisms of X

automor-Proof Let S be an element of one of the three subgroups It is easy tofind A E A such that

SSA = S,,., where A' E A

Remark 2.5 It can be shown that every S E Sp2(!) is of the form

S = S1S2S3S4,

where S3 E (i), that is, S3 is a Fourier transformation in at most

I coordinates; S, and S4 E (ii), that is, they are of the form f' i-+ e"4f',where Q is a real quadratic form; and S2 E (iii), that is, S2 has the formf' i det T f' o T, where T is an automorphism of X

3 Differential Operators with Polynomial Coefficients

By definition 1.1, the elements of Sp2(1) transform differential operatorswith polynomial coefficients into operators of the same type Section 3describes this transformation more explicitly

Let a+ and a- be two polynomials in 1/v, x, and p:

a+(v,

x, p) _ a (v, x)pa, a (v, p, x) _ >paaa (v, x)

exp ax, ap/ = k=°k!ax,op/ Y.

Proof Relation (3.3) defines a bijection a- i a+ such that, for all pEX*,

22 I,§1,3

_ [exp!( ap)] [P(P)f (x)]

The bijection a- r-+ a+ can then be defined by the relation

(v, x, p) [ exp-v ax' ap )]a-(v, p, x)

This is what the lemma asserts

Definition 3.1 Let a be a differential operator that can be expressed as

in (3.1) and (3.2) It is defined by the polynomial a° in (1/v, x, p) thatsatisfies (3.4) We say that a is the differential operator associated to thepolynomial a°

Theorem 3.1 will describe the transform SaS-1 of a by Sc Sp2(1);Lemma 1.1 has already dealt with the case in which a° is linear in (x, p).The proof of this theorem will use the following properties

LEMMA 3.2. If a and b are the operators associated to the polynomialsa° and b°, then the operator

c=ab

is associated to the polynomial co, where

c°(v, x, p) _ }[exp 2v \ay' ap)

2vCax' aqM

e-nProof If b°(v, x, p) only depends on p, then the polynomial co associated

to c = ab is

.c°(v, x, P) = {[ex-(-a , a )][a°(v, x,P)b°(Y)]Lx

Thus if b+(x,p) = b'(x)b"(p), then the polynomial associated to c = ab is

c°(v, x, P) = [exp - 2vax aq)]{Lexp2v(P' a-)]

[exp - 2v (ax' 4)] [b b°(Y, q)

This implies lemma 3.2, which has the following obvious consequence:

LEMMA 3.3. The operator

a

aq)]

y=z[a°(v, x, P)b°(v, y,q)]lq=p

If b is linear in (y, q), then

Oy, ap

-(,)]2[ao(v,x,p)bo(v,y,q)]

= 0,

from which follows

cosh[ ] a°b° = a°b°

therefore we have the following lemma

LEMMA 3.4. If b is linear in (x, p), then the operator associated to a° b° is(ab + ba)