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

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 www.pdfgrip.com 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, 1906Lagrangian analysis and quantum mechanics Bibliography: p Includes indexes Differential equations, Partial-Asymptotic theory Lagrangian functions Maslov index Quantum theory I Title 81-18581 QA377.L4141982 515.3'53 ISBN 0-262-12087-9 AACR2 www.pdfgrip.com To Hans Lewy www.pdfgrip.com www.pdfgrip.com Contents xi Preface Index of Symbols Index of Concepts xiii xvii I The Fourier Transform and Symplectic Group Introduction §1 Differential Operators, The Metaplectic and Symplectic Groups Introduction 1 The Metaplectic Group Mp(l) The Subgroup Spz(l) of MP(l) Differential Operators with Polynomial Coefficients 20 §2 Maslov Indices; Indices of Inertia; Lagrangian Manifolds and 25 Their Orientations Introduction 25 26 Choice of Hermitian Structures on Z(1) The Lagrangian Grassmannian A(l) of Z(1) 27 The Covering Groups of Sp(l) and the Covering Spaces of A(l) Indices of Inertia 37 42 The Maslov Index m on A2 (1) The Jump of the Maslov Index m(A., A ) at a Point (ti, A.') 47 Where dim;, n A.' = The Maslov Index on Spa (l); the Mixed Inertia 51 Maslov Indices on A,(/) and Sp,,(/) 53 Lagrangian Manifolds 55 10 q-Orientation (q = 1, 2, 3, , cc) 56 §3 Symplectic Spaces 58 Introduction 58 Symplectic Space Z 58 The Frames of Z 60 The q-Frames of Z 61 q-Symplectic Geometries Conclusion 65 65 www.pdfgrip.com 31 Contents viii II Lagrangian Functions; Lagrangian Differential Operators Introduction 67 §1 Formal Analysis 68 68 Summary The Algebra W(X) of Asymptotic Equivalence Classes Formal Numbers; Formal Functions 68 73 Integration of Elements of FO(X) 80 Transformation of Formal Functions by Elements of Sp2(l) 86 Norm and Scalar product of Formal Functions with Compact Support 91 97 Formal Differential Operators Formal Distributions 102 §2 Lagrangian Analysis 104 104 Summary 105 Lagrangian Operators 109 Lagrangian Functions on V 115 Lagrangian Functions on V 123 The Group Sp2(Z) 123 Lagrangian Distributions §3 Homogeneous Lagrangian Systems in One Unknown 124 124 Summary Lagrangian Manifolds on Which Lagrangian Solutions of aU = 124 Are Defined 125 Review of E Cartan's Theory of Pfaffian Forms Lagrangian Manifolds in the Symplectic Space Z and in Its 129 Hypersurfaces 135 Calculation of aU 139 Resolution of the Lagrangian Equation aU = Solutions of the Lagrangian Equation aU = mod(1/v2) with 143 Positive Lagrangian Amplitude: Maslov's Quantization Solution of Some Lagrangian Systems in One Unknown 145 www.pdfgrip.com Contents ix Lagrangian Distributions That Are Solutions of a Homogeneous 151 Lagrangian System Conclusion 151 §4 Homogeneous Lagrangian Systems in Several Unknowns 152 152 Calculation of Em_a' U,, Resolution of the Lagrangian System aU = in Which the Zeros of 156 det ao Are Simple Zeros A Special Lagrangian System aU = in Which the Zeros of det ao 159 Are Multiple Zeros 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 166 Zeeman Effect Four Functions Whose Pairs Are All in Involution on E3 Q+ E3 Except for One 166 Choice of a Hamiltonian H 170 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 = mod (1/v2) 174 Examples: The Schrodinger and Klein-Gordon Operators 179 §2 The Lagrangian Equestion aU = mod(1/v2) (a Associated to H, U Having Lagrangian Amplitude >, Defined on a Compact V) 184 Introduction 184 Solutions of the Equation aU = mod(1/v2) with Lagrangian Amplitude >,0 Defined on the Tori V[L0, M0] 185 Compact Lagrangian Manifolds V, Other Than the Tori V[L0, M0], on Which Solutions of the Equation aU = mod(1/v2) with Lagrangian Amplitude 30 Exist 190 Example: The Schrodinger-Klein-Gordon Operator 204 Conclusion 207 www.pdfgrip.com Contents x §3 The Lagrangian System a U = (am - const.) U = (aL2 - const.) U = When a Is the Schrodinger-Klein-Gordon Operator 207 Introduction 207 Commutivity-of the Operators a, aL=, and am Associated to the 207 Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1) 210 Case of an Operator a Commuting with aL2 and am A Special Case 221 The Schrodinger-Klein-Gordon Case 230 Conclusion 226 §4 The Schrodinger-Klein-Gordon Equation 230 230 Introduction Study of Problem (0.1) without Assumption (0.4) 234 The Schrodinger-Klein-Gordon Case 237 Conclusion 231 IV Dirac Equation with the Zeeman Effect Introduction 238 §1 A Lagrangian Problem in Two Unknowns 238 238 Choice of Operators Commuting mod(1!v3) Resolution of a Lagrangian Problem in Two Unknowns §2 The Dirac Equation 240 248 248 Summary 248 Reduction of the Dirac Equation in Lagrangian Analysis The Reduced Dirac Equation for a One-Electron Atom in a 254 Constant Magnetic Field The Energy Levels 258 262 Crude Interpretation of the Spin in Lagrangian Analysis 264 The Probability of the Presence of the Electron Conclusion Bibliography 266 269 www.pdfgrip.com Preface Only in the simplest cases physicists use exact solutions, u(x), of problems involving temporally evolving'systems Usually they use asymptotic solutions of the type u(v, x) = a(v, x)evw(x), (1) 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 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 a lu(v,x) = 0, (2) vaxf) 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 nonlinear 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 www.pdfgrip.com 257 IV,§2,2 the last formula and formula (2.4)1 of §1, where f = b/2 by the choice (2.2) of J, give fo = - iNm (2.7) By the choice (2.2) of J and the definition §1, (2.4)2 of go, _ _Lo g0 AR (2.8) rR(A+C)dt; now by (2.14) of III,§1 and (2.5), we have, on F, dt = > 0, RHQ HQ = R Z th us dt = R QdR > 0; (2 9) hence relations (2.8) and (2.6)1 may be written = 90 L AR Jr Q A C' n NL L = - Jr Q R (2 10) Let us calculate an approximate value forgo, namely, its value for b = By (2.5), in that case the equation of I, is F:Q2 = A'(R)A"(R) - Lo, where A'(R) = RA(R) + CR, A"(R) = RA(R) - CR; (2.11) A' and A" are affine functions; by remark 2, A' is increasing and A'(R) > for R > 0; it follows by an easy calculation using residues that (' L° AR r Q A' dR = 2n; (2.12) now by (2.11)1, AR1+ AR A' R A+C' then the relations (2.10) and (2.12) prove that go = -R(1 + NL) (2.13) Formulas (2.7) and (2.13) and formula (2.4)3 of §1 prove the lemma www.pdfgrip.com IV,§2,2-IV,§2,3 258 The assumptions made in §1,2, including those of §1, example 2, are satisfied, as it is proved by the following lemma The values of the physical quantities defining A, B, and C (remark 2) are such that for the energy levels E used in section 3, LEMMA 2.2 fo = -7rN,H, go = -7r(1 + NL), NL', NLM, NM2 are small in comparison with Proof N[-,,] is expressed by (4.8) of III,§1, where Ao, Bo, and Co are defined by identifying formulas (4.6) and (4.20) in III,§1: N [Lo, Mo] = aZrUC2(1c2 L - [(L)2 - a2Z2]' Mo/h) - 1-in J E2fi (2.14) where Z, c, p, e, E, and ,Y are the physical quantities defined by remark 2, z he ~ 13 is the dimensionless fine-structure constant (2.15) ( means of the order of magnitude of), and (2.16) is the Bohr magneton Zµ hh In section 3, we choose < (µc2 /E) - ^_, a2Z2 /(2n2), where n is an integer, the magnetic energy /3,Y to be very small in comparison with µc2, and 2Mo/h to have integer values which are not large, Lo/h > The lemma follows The Energy Levels Notation Formulas (4.23) and (4.25) of III,§1 have already defined and used the function F given by F(n, k) (3.1) Z (-n ZZ ) www.pdfgrip.com 259 IV,§2,3 Note the sign of its derivatives: F,, > 0, Fk > Using F, it is possible to give the relation hk + N [hk, hm] = hn, (3.2) where N is expressed by (2.14), the form E2 = Uc2[pc2 + 2/3.,Ym]F2(n, k) In other words, to the degree of approximation used here, and assuming E > 0, E = Uc2F(n, k) + f3.3f°m (3.3) In these formulas, a is the fine-structure constant (2.15) and /3 is the Bohr magneton (2.16) The energy levels E for which system (2.3) (where l' and m' are real and have negligible squares) has admissible lagrangian solutions (definition of §1) on a compact lagrangian manifold are defined by the quadruples of quantum numbers THEOREM j±Z, m, n (3.4) ?, and n are integers (3.5) 0