Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index
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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
of America.
Library of Congress Cataloging In Publication Data
Leray, Jean, 1906-
Lagrangian analysis and quantum mechanics.
Bibliography: p.
Includes indexes.
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
Contents
Preface
xi
Index of Symbols
xiii
Index of Concepts
xvii
I. The Fourier Transform and Symplectic Group
Introduction
1
§1. Differential Operators, The Metaplectic and Symplectic Groups
1
0. Introduction
1
1. The Metaplectic Group Mp(l)
1
2. The Subgroup Spz(l) of MP(l)
9
3. Differential Operators with Polynomial Coefficients
20
§2. Maslov Indices; Indices of Inertia; Lagrangian Manifolds and
Their 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
9. Lagrangian Manifolds
10. q-Orientation (q
= 1, 2,
55
3,
...,
cc)
56
§3. Symplectic Spaces
58
0. Introduction
58
1. Symplectic Space Z
58
2. The Frames of Z
60
3. The q-Frames of Z
61
4. q-Symplectic Geometries
65
Conclusion
65
viii
Contents
II. Lagrangian Functions; Lagrangian Differential Operators
Introduction 67
§1. Formal Analysis
68
0. Summary
68
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
Support 91
6. Formal Differential Operators
97
7. Formal Distributions 102
§2. Lagrangian Analysis
104
0. Summary
104
1. Lagrangian Operators
105
2. Lagrangian Functions on V
109
3. Lagrangian Functions on V
115
4. The Group Sp2(Z)
123
5. Lagrangian Distributions
123
§3. Homogeneous Lagrangian Systems in One Unknown 124
0. Summary
124
1. Lagrangian Manifolds on Which Lagrangian Solutions of aU = 0
Are Defined
124
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 Homogeneous
Lagrangian 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 of
det ao Are Simple Zeros
156
3. A Special Lagrangian System aU = 0 in Which the Zeros of det ao
Are 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 the
Zeeman Effect
166
1. Four Functions Whose Pairs Are All in Involution on E3 Q+ E3
Except for One
166
2. Choice of a Hamiltonian H
170
3. The Quantized Tori T(l, m, n) Characterizing Solutions, Defined
mod(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 Lagrangian
Amplitude >,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
Conclusion
207
x
Contents
§3. The Lagrangian System
a U = (am - const.) U = (aL2 - const.) U = 0
When a Is the Schrodinger-Klein-Gordon Operator
207
0. Introduction 207
1. Commutivity-of the Operators a, aL=, and am Associated to the
Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1)
2. Case of an Operator a Commuting with
aL2
and am
207
210
3. A Special Case
221
4. The Schrodinger-Klein-Gordon Case 226
Conclusion
230
§4. The Schrodinger-Klein-Gordon Equation
230
0. Introduction
230
1. Study of Problem (0.1) without Assumption (0.4)
231
2. The Schrodinger-Klein-Gordon Case
234
Conclusion
237
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
0. Summary
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
Preface
Only in the simplest cases do physicists use exact solutions, u(x), of
problems involving temporally evolving'systems. Usually they use asymp-
totic solutions of the type
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,
a(v,x 1 a lu(v,x)
= 0,
vaxf)
(1)
(2)
is satisfied in the sense that the left-hand side reduces to the product of
e'V and a formal series in l /v whose first terms or all of whose terms vanish.
The construction of these asymptotic solutions is well known and called
the WKB method:
The phase q has to satisfy a first-order differential equation that is non-
linear if the operator a is not of first order.
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 value
vv.
Physicists use asymptotic solutions to deal with problems involving
equilibrium 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: for
example, in geometrical optics, a has singularities on the caustics, which