Lagrangian Analysis And Quantum Mechanics A Mathematical Structure Related To Asymptotic Expansions And The Maslov Index Jean Leray

Lagrangian Analysis And Quantum Mechanics A
Mathematical Structure Related To Asymptotic
Expansions And The Maslov Index Jean Leray
download
https://ebookbell.com/product/lagrangian-analysis-and-quantum-
mechanics-a-mathematical-structure-related-to-asymptotic-
expansions-and-the-maslov-index-jean-leray-56388524
Explore and download more ebooks at ebookbell.com

Here are some recommended products that we believe you will be
interested in. You can click the link to download.
Random Fields And Stochastic Lagrangian Models Analysis And
Applications In Turbulence And Porous Media Karl K Sabelfeld Nikolai A
Simonov
https://ebookbell.com/product/random-fields-and-stochastic-lagrangian-
models-analysis-and-applications-in-turbulence-and-porous-media-karl-
k-sabelfeld-nikolai-a-simonov-51109598
Global Formulations Of Lagrangian And Hamiltonian Dynamics On
Manifolds A Geometric Approach To Modeling And Analysis Lee
https://ebookbell.com/product/global-formulations-of-lagrangian-and-
hamiltonian-dynamics-on-manifolds-a-geometric-approach-to-modeling-
and-analysis-lee-6751620
Lagrangian Shadows And Triangulated Categories Paul Biran Octav Cornea
https://ebookbell.com/product/lagrangian-shadows-and-triangulated-
categories-paul-biran-octav-cornea-52181364
Lagrangian And Hamiltonian Mechanics A Modern Approach With Core
Principles And Underlying Topics 1st Edition Jos Rachid Mohallem
https://ebookbell.com/product/lagrangian-and-hamiltonian-mechanics-a-
modern-approach-with-core-principles-and-underlying-topics-1st-
edition-jos-rachid-mohallem-57159256

Lagrangian Fluid Dynamics Bennett A
https://ebookbell.com/product/lagrangian-fluid-dynamics-
bennett-a-2040990
Lagrangian Probability Distributions Consul Pc Famoye F
https://ebookbell.com/product/lagrangian-probability-distributions-
consul-pc-famoye-f-2042084
Lagrangian And Hamiltonian Methods For Nonlinear Control 2006
Proceedings From The 3rd Ifac Workshop Nagoya Japan July 2006 1st
Edition Francesco Bullo
https://ebookbell.com/product/lagrangian-and-hamiltonian-methods-for-
nonlinear-control-2006-proceedings-from-the-3rd-ifac-workshop-nagoya-
japan-july-2006-1st-edition-francesco-bullo-2159370
Lagrangian Modeling Of The Atmosphere John Lin Dominik Brunner
https://ebookbell.com/product/lagrangian-modeling-of-the-atmosphere-
john-lin-dominik-brunner-4307116
Lagrangian Optics 1st Edition Vasudevan Lakshminarayanan Ajoy K Ghatak
https://ebookbell.com/product/lagrangian-optics-1st-edition-vasudevan-
lakshminarayanan-ajoy-k-ghatak-4594140

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

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

xii Preface
are the images of the sources of light; nevertheless geometrical optics
holds 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 con-
venient use of the Fourier transform that these amplitude singularities are
only apparent singularities. But he had to impose some "quantum con-
ditions." These assume that v has some purely imaginary numerical value
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 is
necessary for the Fourier transform to be pointwise, which is essential for
Maslov's treatment. A procedure, avoiding that contradiction and guided
by purely mathematical motivations, that makes use of the Fourier trans-
form, expressions of the type (1), Maslov's quantum conditions, and the
datum of a number vo does exist, but no longer tends to define a function
or a class of functions by its asymptotic expansion. It leads to a new
mathematical structure, lagrangian analysis, which requires the datum of a
constant vo and is based on symplectic geometry. Its interest can appear
only a posteriori and could be quantum mechanics. Indeed this structure
allows a new interpretation of the Schrodinger, Klein-Gordon, and Dirac
equations provided
vo = ii = 2h1, where h is Planck's constant.
Therefore the real number 2ni/vo whose choice defines this new mathe-
matical structure can be called Planck's constant.
The introductions, summaries, and conclusions of the chapters and
parts 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 the
spectrum of the blackbody. Thus the book could be entitled
The Introduction of Planck's Constant into Mathematics.
January 1978
College de France
Paris 05

Index of Symbols
A I,§1,definition 1.2*
C field of the complex numbers; = C{0}
E3 3-dimensional euclidean space
I II,§1,1
N set of the natural numbers (i.e., integers > 0)
R field of the real numbers
R+ set of the real positive numbers
Sn n-dimensional sphere
Z ring of the integers
A element of A: I; II
B bounded set: I; II
A, B, C functions of M, coefficients of the
Schrodinger-Klein-Gordon
operator: III,§1,example 4
E neutral element of a group: I; II
atomic energy level: III; IV
F any function: II
the function in III,§1,(4.23)
G group: I; II
function: III; IV
H hamiltonian: II,§3,1; II,§3,definition 6.1
Hess hessian: I,§1,definition 2.3
Ik,Jk elements of E3: III,§1,1
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,§1,1
characteristic curve: II,§3,definition 3.1; III,§1,(2.14)
function: III,§1,4
L, M, P, Q, R functions: 1II,§ 1,1
N function of (L, M): III,§1,(2.9); NL and N,M are its
derivatives
R (I, II), Ro (III, IV) frame in symplectic geometry: I,§3,2; I,§3,3
SP(1)
SPz(1)
S
T
U
UR
U(1)
V
W
W(1)
X' Z
a
arg
d
d'x
det
e
symplectic group: I,§1,definition 1.1
the covering group, of order 2, of the symplectic
group: I,§l,definition 2.1
element of Sp2(1)
torus: III,§ 1,3
lagrangian function on V: II,§2,3
formal functions on V: II,§2,2
unitary group: I,§2,2
lagrangian manifold: I,§2,9; I,§3,1
hypersurface of Z
subset of U(1): I,§2,lemma 2.1
spaces: 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
argument
differentiation
Lebesgue measure
determinant
2.71828.. .
neutral element of a group: I
dimension of X : 1; II (dimension of X = 3 in III,
J-1
interior product: I1,§3,2
quantum number: IV,§1,example 2
IV)

Index of Symbols
1, m, n
m, MR
S
r
A
xv
quantum numbers: III; IV
Maslov index: I,§1,definition 1.2; I,§1,(2.15);
I,§2,definition 5.3; I,§2,theorem 5; I,§3,theorem 1;
Q3,3
element of Sp(1): I; II
function: III; IV
function: II,§3,(3.10) ; 11,§3,(3.13); 111,§ 1,(2.6)
element of U (1) : I
transpose of u: I,§2,2
formal number or function: II,§1,2
element of W(1)
elements of X
elements of Z
I I,§ 1,1
space of formal or lagrangian functions or
distribution: II,§1,2; II,§1,7; II,§2,2; II,§2,3; II,§2,5
Hilbert space: I,§1,1
Lie derivative: II,§3,definition 3.2
neighborhood
Schwartz spaces: I,§1,1
arc: 1,§2
curve: III,§1,(2.5)
laplacian (A0 is the spherical laplacian):
III,§3,(2.4)
A lagrangian grassmannian: I,§2,2; I,§3,1
A exterior product
rI projection : II,§1,theorem' 2.1
apparent contour, EsP: I,§1,definition 1.3
Y,,: I,§2,9
ER : I,§3,2
b,'I',0 Euler angles: III,§1,(1.12)
C) open set in Z: II,§1,6; II,§2,1

xvi
#0
Y
11, 11 v
K
A
A,p
V
Vp
it
ni
a[ ],Ori
X
V
m,m
toi
Index of Symbols
function : III,§1,(2.8)
open set in E3 Q E3: III,§1,1
amplitude: II,§1,2
lagrangian amplitude: II,§2,theorem 2.2
arc or homotopy class
invariant measure of V: II,§3,definition 3.2
characteristic vector: II,§3,definition 3.1
element of A: I; II
functions : III,§1,(2.10)
element of I: II,§1,1
i/h = 2ni/h (h e R+): II,§2,3; II,§3,6
3.14159...
jth homotopy group: I,§2,3
Pauli matrices: IV,§1,(1.6); IV,§1,(1.7)
ri/d `x
phase: I,§2,9; I,§3,2; II,§1,2
lagrangian phase: I,§3,1
pfaffian forms
III,§1,(1.7)
Atomic Symbols: III; IV; passim (see III,§1,4, Notations)
Q
E
p
energy
speed of light
Planck's constant
potential vector
magnetic field
1/137
Bohr magneton
charge
mass

Index of Concepts
amplitude
asymptotic class
characteristic curve K
characteristic vector K
energy E
Euler angles (1),`I',0
formal number, functions u, UR
frames R;(1,,12,13);(JI;J2;J3)
groups Sp(1); U(1)
hamiltonian H
hessian Hess
homotopy 1j
index of inertia Inert
interior product i4
lagrangian amplitude #0
lagrangian function U
lagrangian manifold V
lagrangian operator a
lagrangian phase
Lie derivative Sf
Maslov index m
matrix J(k) Q
operator a
Planck's constant VO = i/h
quantum numbers 1, m,n,j
spaces X, Z, ', A r, ,9", ,9,
symplectic space z

I The Fourier Transform and Symplectic Group
Introduction
Chapter I explains the connection between two very classical notions: the
Fourier transform and the symplectic group.
It will make possible the study of asymptotic solutions of partial differen-
tial equations in chapter II.
§1. Differential Operators, the Metaplectic and Symplectic Groups
0. Introduction
Historical account. The metaplectic group was defined by I. Segal [14];
his study was taken up by D. Shale [15]. V C. Buslaev [3, 11] showed
that it made Maslov's theory independent of the choice of coordinates.
A. Well [18] studied it on an arbitrary field in order to extend C. Siegel's
work 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
onxEX.
Spaces of functions and distributions on X. The Hilbert space. °(X) con-
sists of functions f : X - C satisfying
If (x) I'
d'x1/2
< ao.
Ifi = (tf2dIx)
The Schwartz space 9'(X) [13] consists of infinitely differentiable,
rapidly decreasing functions f : X - C. That is, for all pairs of 1-indices
(q, r)
IfI9,r = Sup Ix°(c'xlrf(x)I < 00.
X
The topology ofY(X) is defined by a countable fundamental system of

2 I,§1,1
neighborhoods of 0, each depending on a pair of 1-indices (q, r) and a
rational 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 are
the 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 the
value of f' at x is not in general defined. The bound of f' on a bounded
set 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' I e < 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
a number e > 0, as follows:
"(B, E) _{ f I I f' B<_ E}.
Unlike the above, this topology cannot be given by a countable fundamen-
tal system of neighborhoods of zero.
Let us recall the following theorems.. °(X) can be identified with a sub-
space of .So'(X ):
5(X)ca/l'(X)c.9''(X).
The Fourier transform is a continuous automorphism of ,9''(X) whose
restrictions to ,Y(X) and 5o(X) are, respectively, a unitary automorphism
and a continuous automorphism.
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 its
value 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 an
endomorphism of So(X ), is the restriction of a to So(X ). The operators a
and 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
[a, b] = ab - ba E C;
c c- C denotes the endomorphism of 9"(X):
c : f H c f df E Y' (X ).
In order to study this commutator, we give Z(l) the symplectic structure
] defined by
[z, z'] = (P, x'> - < P x>,
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°(z) = [a', z]. (1.1)
This gives a natural isomorphism
Z(1) E) a' H a° e °. (1.2)
The commutator of a and b c- d is clearly

[a, b] =
1
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
bSf = Saf Vf c 9"(X).
b 0 0 if a 0 0. In general, b 0 d.
Definition 1.1. G(1) is the group of continuous automorphisms S of 9"(X)
that transform sad into itself in the sense that
SaS ' e..4 Va e si.
G(1) is clearly a semigroup. If S E G(1),
a i--+ SaS
-'
is clearly an automorphism of d. Therefore S-' E G(1), and G(l) is a group.
Under the natural isomorphism Z(1) - sad, the automorphism (1.5) of
d becomes an automorphism of the vector space Z(1):
s : a
1
F--' sa 1. (1.6)
Since S commutes with the automorphisms of .9'(X) given by c e C, and
since [a, b] E C, we have
[SaS-', SbS-1] = [a, b],
or, considering (1.3) and the equivalence of (1.5) and (1.6),
[sa', sb'] = [a', b']
Therefore s is an automorphism of the symplectic space Z(1).
The group of automorphisms of the symplectic space Z(1) is called the
symplectic 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°, the
automorphism
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
G(1) - Sp(1). (1.7)
LEMMA 1.2. The kernel of the morphism (1.7) is a subgroup of G(1) con-
sisting of automorphisms of 9'(X) of the form
f - cf, where f c .9'(X) and c e IC (complex plane minus the origin).
Remark. This subgroup will be written as t.
Proof. All c c t commute with all a e sad and thus belong to the kernel.
Conversely, let S be an element of the kernel. Therefore S is an auto-
morphism of 9"(X) commuting with all a E sad. Let p e X*. We have
-y<P.x)
vax+ple =0.
Therefore, since S and (1/v)(0/8x) + p commute,
1 a
+ p)Se-'<P-'> = 0.
vc?x
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 trans-
form and let g = F-1 f e ."(X). By the definition of F,
(iL)'I2
f(x) = 2ni
e-'<P.X>g(p)d'p. (1.8)
fX
Since Se-v<o.=>
= ce-v<P X>,
we obtain
Sf = Cf Vf E .9'(X).
Now 9'(X) is dense in .9''(X). Therefore S = c e C. This proves the lemma.
Some other subgroups of G(l) will be needed in proving that the map
G(l) - Sp(l) is an epimorphism. They are
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 of
automorphisms of .9'(X) and a restriction to .*'(X) that gives a group of
unitary (that is, isometric and invertible) transformations of .i*'(X). The
following definition uses these properties.
Definition 1.2. Let A be the collection of elements A each consisting of
1°) 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 define
A(A) = det L by arg A(A) = (rz/2)m(A).

1,§ 1,1 7
Remark. det L is calculated using coordinates in X* dual to the co-
ordinates in X and is independent of coordinates chosen such that
dx' n . A dx' = d'x.
Remark. m(A) will be identified with the Maslov index by 2,(2.15) and
§2,8,(8.6).
To each A we associate SA, an endomorphism of Y(X) defined by
rzi]`I2 A(A) I
(SAf)(x) [IV,,
where f' E ,9(X), arg[i]t'2 = nl/4. (1.10)
Clearly SA is a product of elements belonging to the groups (i), (ii), and
(iii). Therefore SA is an automorphism of Y (X) that extends by continuity
to a unitary automorphism of A (X) and to an automorphism of 9"(X).
These three automorphisms will be denoted SA; SA e G(l).
The image sA of SA in Sp(l) is characterized as follows (where Ax is the
gradient of A with respect to x):
(x, p) = sA(x', p') is equivalent to
p = Ax(x, x'), p' = -Ax.(x, x').
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 these
calculations gives the following relations among differential operators of
V 8x -
Px = - SA('Lx)SA',
SA (v Ox
+ Qx}SA' = Lx;
writing
(X, P) = SA(x', P'),
these relations mean
P - Px = -'Lx', p' + Qx' = Lx bx' E X, p' e X
This is proposition (1.11).
Definition 1.3. We shall write Esp for the set of s e Sp(l) such that x and
x' are not independent on the 21-dimensional plane in Z(l) Q Z(l) de-

termined by the equation
(x, P) = s(x', p').
Let us recall the well-known theorem that the set of sA characterized by
(1.11) is Sp(l)ESp.
Proof. Clearly sA 0 Esp. Conversely, let s e Sp(l). On the 21-dimensional
plane in Z(l) Q Z(1) determined by the equation
(x, P) = s(x', p')
we have, since s is symplectic,
<p, dx> - <dp, x> = <p', dx'> - <dp', x'>.
Therefore
? d [<P, x> - <p', x'>] _ <p, dx> - <p', dx'>.
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
G(1)/t = Sp(l ). (1.13)
[C is the center of G(l) because the center of Sp(l) is just the identity
element.]
Definition 1.4. The metaplectic group Mp(l) is the subgroup of G(l)

I,§1,1-I,§1,2 9
consisting of those elements whose restriction to ff (X) is a unitary auto-
morphism of ,*°(X).
We have SA e Mp(l) VA. Now the SA generate Sp(l), so the natural
morphism
Mp(l) - Sp(1)
is an epimorphism. By (1.13), all elements of G(l) can be written uniquely
in the form
cS, where S e Mp(l), c > 0.
Writing R+ for the multiplicative group of real numbers > 0, we obtain
G(l) = R+ x Mp(l). (1.14)
The study of G(l) therefore reduces to that of Mp(l), which has the follow-
ing properties:
THEOREM 1. Mp(l) is a group of automorphisms of S°'(X) whose restric-
tions to.W'(X) are unitary automorphisms.
1°) Let S' be the multiplicative group of complex numbers of modulus 1.
Then
Mp(1)/S' = Sp(l). (1.15)
2°) Let EMp be the hypersurface of Mp(l) that projects onto Esp. Every
element of Mp(l)EMp can be written as cSA, where c e S' and SA is given
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 some
element 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.

10 I,§1,2
The purpose of this section is to prove that Sp2(l) is a covering group
of Sp(l) of order 2.
In order to prove this, we calculate inverses and compositions of the
elements 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 two
conditions for any f and f' EY(X):
f(x) - _ (±)"2A(A) evA(x.x') f'(x')d`x',
x
rz
f (x) =
(Iv2rz
i
) 0(A)
x
Using the expression for A given by (1.9), this is the same as the equivalence
of the following two conditions:
f(x) = J
X x
f'(x') = (jj)'detLI jev<1x'>f(x)dIx.
The equivalence is deduced
lemma follows.
from the Fourier inversion formula; the
To compute compositions of the SA, we will find an explicit expression
for 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 following
notions are independent of this choice.
Let cp be a real function, twice differentiable:
cp : X -+ R.

1,§1,2 11
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") when
Hess((p) 0. Clearly
Inert(-q') = I - Inert(q),
arg Hess((p) = n Inert(g) mod 2n.
This formula makes possible the definition
arg Hess((p) = n Inert(q). (2.1)
Thus, for example,
[Hess (9)]112 = IHess((p)I112itnert(N).
(2.2)
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) is
the determinant of the symmetric matrix R. Inert(R) is the number of
negative 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 such
that Hessx.(cp'(x') + A(x, x')) A 0. Denote by 9(x) the critical value of the
polynomial
Xax't--* A (x,x') + V (x');
rp is a second-degree polynomial. We have
SA(e'11)
= A(A)[Hessx.(cp' + A)]-1l2e"°. (2.4)
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_

12
exp [ - 2] dx = 2n.
L J
Therefore if c e C and I arg p I< n1/2,
x + c)2l dx = Iarg f < .
JexP[-1(X J J VP 4
We then have, for any p e C,
fex[_vx - 2(x + c)zl dx
J exp - 2u [vp + µ(x + c)] )} dx = e (P
= ev(O
'C
I,§1,2
where (p is the critical value of the function
x i- p'(x) - px, where gyp' p (x + c)2.
2v
The Fourier transform F is the automorphism of 9 defined by
1/2
(Ff')(p) = 2nli
Je<>f(x)dlxdf' E.5(X) (2.5)
We then have, for I = 1, I arg p I < n1/2,
Fe°`° = v I v I evv; = e"'4
VF,
11P 1/71-
Since F is a continuous automorphism of 5''(X), the preceding formula
remains valid for p = -Ev, r e $; then
f 1'/E if r > 0
iJIEI if r < 0.
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 the
critical value of the polynomial
xicp'(x) - CP,x>,
we have

1,§1,2 13
Fe°`° =
[Hess p']-1ne"*. (2.6)
Let us show that, since relation (2.6) holds for I = 1, it holds for all I >, 1.
It suffices to choose the coordinates x' in X such that
(P'(x) = i (pi(x').
j=1
Now using the definitions (1.9) of A, (1.10) of SA, and (2.5) of F, we have
in the case P = Q = 0,
(Snf') (x) = A(A) (Ff ') (Lx).
Then (2.6) establishes (2.4) in this case. From the definitions of A and SA,
the general case is clearly equivalent to this one.
Before taking compositions of the SA,we consider compositions of
the sA:
LEMMA 2.3. 1°) Let A and A' E A. The condition
$ASA' 0 E5p (2.7)
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 A
such that
SASA.sA.. = e [identity element of Sp(l)]. (2.9)
A" is defined by the condition that the critical value of the polynomial
x' + 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 as
1' + Q is invertible.
A" can be defined by the formulas
P" + Q' = L'(P' + Q)-"L', P + Q = 1L(P' + Q)-1L,
(2.10)

14 I,§1,2
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")
= Inert,-.(A + A' + A"). (2.11)
.(A + A' + A") =
A2(A)A2(A')
Hess (2.12)
x A2(A'*)
Proof of I°). By (1.11), the relations
(x, P) = SA(x', P'), (x', p') = p")
may be written
p = Ax(x, x'), p' = -As (x, x') = A'x,(x', x"),
p" -A'x.,(x' x").
It results from the elimination of p' and x' in these relations that
(x, P) = SASA'(x", p")
The condition (2.7) that SASA' 0 Esp is then equivalent to each of the
following conditions:
The elimination of p' and x' in the preceding step leaves x and x"
independent.
The relation
A, (x, x') + A'x-(x', x") = 0
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 following
three 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)

I,§1,2 is
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 equivalent
relations (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
P" + Q', (P' + Q)-1, P + Q"
can be transformed one into the other. They therefore have the same
inertia. This is (2.11).
By (2.10)3,
Hess(P' + Q) = (det L) (det L')/(-1)' det L".
By definition 2.2, this is (2.12).
Definition 2.4. Given
3A, SA SA' SA" = e,
we define
Inert(sA, 5A,, sA.-) = Inertx(A + A' + A") [see (2.11)]. (2.14)
We define

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
m(SA) = m(A) mod 4. (2.15)
§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
SASA, SA,. = e. (2.17)
Then
SASA'SA.. = ±E [E is the identity element of Mp(1)]. (2.18)
We have
(2.19)
if and only if
Inert(SA, SA,, SA,.) = m(SA) - m(SA,.) mod4. (2.20)
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 Dirac
measure 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°),

I,§1,2
(;)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 -
n 1 n
argA(A"*)
=
2
[l - m(SA..)] mod 2n.
Therefore
17
arg(± 1) =
n
2
[Inert(SA, SA., SA..) - m(SA) + m(SA.1) - M(SA")] mod 271,
which proves the lemma.
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 A
such that
SUSvSW = SBSS. (2.21)
Now, by lemmas 2.3,1°) and 2.4, for every W e A and every T a generic
element of A, SWST belongs to {SA} and is generic. Therefore, for T
generic,

18
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) is
clearly a natural morphism:
SP2(1) - SP(1)-
LEMMA 2.6. The kernel of this morphism is the subgroup
S° = {E, -E}.
Therefore
Sp2(1)/S° = Sp(1).
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 lemma
2.1,
Therefore, by (1.11),
A'(x, x') = A*(x, x') Vx, x' c- X.
Consequently by definition 1.2.
A(A') = ±1 (A*),
and
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 the
collection 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

1,§1,2 19
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). This
element 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 theorem
reduces the study of Mp(l) to that of Spz(I). Its equivalent can be found in
the 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 of
these 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 of
E/(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
ss's" = e [identity element of Sp(l)].
The inertia is a locally constant function (discontinuous on Esp) with values
in {0, 1, ... , 1} satisfying
Inert(s, s', s") = Inert(s", s, s') _
= I - Inert(s"-1
s'-1, s-1).
Let Esp2 be the hypersurface of Sp2(l) that is mapped onto Esp in Sp(l)
under the natural projection. The elements SA defined by (1.10) are the
elements of Sp2(l)Esp= . Let S, S', S" be a triple of such elements satisfying

20 I,§1,2-I,§1,3
SS'S" = E [identity element of Sp2(1)].
Let s, s', s" be the images of these elements under the natural projection onto
Sp(l). We define
Inert(S, S', S") = Inert(s, s', s").
3°) Formula (2.15) and definition 1.2 define the Maslov index m on
Sp2(1)Y-SD,. It is a locally constant function (discontinuous on 1sP2) with
values in Z4. It satisfies
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 formula
and 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) automor-
phisms of X.
Proof. Let S be an element of one of the three subgroups. It is easy to
find 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 form
f' 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 operators
with polynomial coefficients into operators of the same type. Section 3
describes 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)

I,§1,3 21
(a a multi-index). We consider the two differential operators
a+ (v, x,
v a x) :f a, (v' v e x
x)af( )'
a (v,
ax
, x : f
vex
[aa (v, .)f( )]-
LEMMA 3.1. These two operators are identical, that is,
(v
x1 a
)
l = a v1 a 1
-
a+
,vax 'vax,
xJ
'
if and only if there exists a polynomial a° in 1/v, x, and p such that
+
a (v, x, P) _ [exp 1
2v
aax, aa
/] a '(v, x, p)-
_ 1 a a
a (v, p, x) exp - - -, - a°(v, x, p).
[ 2v ax ap
The notation is the following:
2
ax
,
ap
a = a
(xi and pi dual coordinates in X and X*);
ij
1 ax api
a s °° 1
,(
k/ a a k
exp ax, ap/ = k=°k!ax,
op/
Y.
Proof. Relation (3.3) defines a bijection a- i-- a+ such that, for all
pEX*,
a+(v, x, p) = e v<P,x>a+
(v
x l a )Y<P,x>
V ax
e
v(P,x>a-
(v'
1 a
'
x) ev<P,x>
I vex
- 1a1
v Ox, a
p )]a (v, p, x),
P + vaxJ aQ (v, x)
-[eXp

since, by Taylor's formula, for every polynomial P: X/ C and every
function f : X - C,
P
P (p
+ v ax)f (x) Y P. P(P) (vex) f (x)

22 I,§1,3
_ [exp!(. ap)] [P(P)f (x)]
The bijection a- r-+ a+ can then be defined by the relation
a+ _ 1 a a
(v, x, p) exp- )]a-(v, p, x)
[ v ax' ap
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) that
satisfies (3.4). We say that a is the differential operator associated to the
polynomial 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 polynomials
a° and b°, then the operator
c=ab
is associated to the polynomial co, where
c°(v, x, p) _ }[exp 2v ay' ap)
2v Cax' aqM
[a°(v, x, P)b°(v, y, q)] Y='- (3.5)
e-n
Proof. If b°(v, x, p) only depends on p, then the polynomial co associated
to c = ab is
c°(v, x, P) _ [exp - 2v ax' ap
)][a4 (v, x, P)b°(P)]
{[exp
2v (ax p + aq)] [a+(v, x, P)b°(q)]}
9-P
_ 1 a a
{[exp
2v ax ap)] [a°(v, x, P)b°(q)}q=r
Similarly, if h°(v, x, p) only depends on x, then the polynomial associated
to c = ab is

I,§1,3 23
.
c°(v, x, P) = {[ex-(-
a , a )][a°(v, x, P)b°(Y)]Lx
Y P
Thus if b+(x,p) = b'(x)b"(p), then the polynomial associated to c = ab is
c°(v, x, P) = [exp - 2v ax aq)] {Lexp 2v (P' a-)]
[a°(v, x, p)b'(y)]
1)y=x
- {[exp
2v ax' aq) 2v (ay aq) + 2v (ay' P)]
[a°(v, x, }y_}-
4=p
This is (3.5) since, by (3.4),
[exp
-2v (ax' 4)] [b b°(Y, q)
This implies lemma 3.2, which has the following obvious consequence:
LEMMA 3.3. The operator
c = 2(ab + ba)
is associated to the polynomial
1 o a
x, p) _ cosh
[2V(8Y'2
1
2v
o
ax
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).

24 I,§1,3
This lemma enables us to prove the following theorem.
THEOREM 3.1. The transform SaS-1 of a by S is the differential operator
associated to the polynomial a° o s -I [S E Sp2(l); s is the image of S in Sp(l)].
Proof. Let b be a differential operator associated to a polynomial b°
that is linear or affine in (x, p); lemma 1.1 shows that theorem 3.1 holds
for b. To prove the theorem by induction on the degree of a° in (x, p),
it suffices to prove that, if the theorem holds for a°, then it holds for a°b°.
Since the theorem holds for a° and b°, the operators associated to the
polynomials
a°b° and (a°b°) o s-1 = (a° o s-1)(b° o s-1)
are, respectively, by lemma 3.4,
4(ab + ba);
2(SaS-1SbS-1 + SbS-1SaS-1) = 4S(ab + ba)S-1.
The theorem thus holds for a°b°, which completes the proof.
We supplement this by a theorem about adjoint operators.
Definition 3.2. Recall that.,Y(X) has a scalar product:
(f 19) = jf(x)d1x bf, 9 E Af (X),
x
where g(x) is the complex conjugate of g(x). Two differential operators a
and b are said to be adjoint if
(af I 9) = (f I b9) df, 9 E -*'(X). (3.6)
THEOREM 3.2. Two differential operators a and b associated to two poly-
nomials a° and b° are adjoint if and only if
b°(v,x, p) = a°(v,x, p) VvCiR, xcX, pEX*. (3.7)
Proof. It is clear that (3.6) is equivalent to
b (v, p, x) = a+(v, x, p),
that is to say, since v is pure imaginary, to

I,§1,3-1,§2,0 25
l r 1 a
ax, a
)]QVx, p),

)Jb°(v, x, P) exp -2v (-a
-2v (ax, P / P
and hence to (3.7).
Theorems 3.1 and 3.2 obviously have the following corollary.
COROLLARY 3.1. If a* is the adjoint of a, then VS E Sp2(l), Sa*S-1 is the
adjoint of SaS -1.
Theorem 3.2 clearly has the following corollary, which will be important
later.
COROLLARY 3.2. The operator a associated to a polynomial a° is self-
adjoint if and only if the polynomial a° is real valued Vv e iR, x e X, p e X*.
§2. Maslov Indices; Indices of Inertia; Lagrangian Manifolds and
Their Orientations
0. Introduction
Historical account. Following V. C. Buslaev [3], [11], §1 has defined a
Maslov index mod4 on Sp2(l) by (2.15) and has connected it by (2.20) to
an index of inertia that is a function of a pair of elements of Sp(l).
On the other hand, V. 1. Arnold [1], [11] defined another Maslov index
on the covering space of the lagrangian grassmannian A(l) of Z(l); this
index is connected to the preceding one and to a second index of inertia
that is a function of a triple of points of A(l). J. M. Souriau [16] has given
a variant of the definition of the Maslov index that is considered in this
section.
Summary. Chapter I, §3. and chapter II use these two Maslov indices and
a third index of inertia, which is a function of an element of Sp(l) and a
point of A(l).
We review and modify the various definitions of these indices (Arnold's,
section 5; Maslov's, section 6; Buslaev's, section 7) so as to clarify their
properties (sections 4-8). In §3 those properties that will be used in
chapter II are set forth.
First of all we must recall and supplement the topological properties
of Sp(l) and A(l) (theorem 3). To study these properties we follow Arnold
in employing a hermitian structure on Z(l) (sections 1 and 2).

26 I,§2,1
1. Choice of Hermitian Structures on Z(1)
Let (. I ) be the scalar product defining a hermitian structure on Z(1);
clearly
Im(zIz')= -Im(z'Iz)
is a symplectic structure on Z(1). Now in §1,1, a symplectic structure
[ , - ] was defined on Z(1).
LEMMA I.I. Restriction to X defines a homeomorphism between the set
of hermitian structures ( I -) on Z(1) such that
Im(z I z') = [z, z'], ix = X*, (1.1)
and the set of euclidean structures on X.
Proof. (i) The restriction to X of a hermitian structure on Z(1) satisfying
(1.1) is euclidean since
[x, X] = 0 Vx, X' E X.
Observe that, by (1.1),
(z I z') = [iz, z] + i[z, z'],
and in particular
(XIX')= [ix, X] Vx, X, C- X.
Hence
z
ix =
lax
Ox
E X.
2
a
(ii) A given (- I ) on X defines
by (1.4), the restriction of i to X,
i,: X -> X*;
the restriction of i to X
i2: X* -+ X,
because i2 = -il' since i2 = -1;
hence the automorphism i of Z(1),

1,§2,1-1,§2,2 27
i(x, P) = (i2P, ilx); (1.5)
finally, by (1.2), the hermitian structure on Z(1).
The restriction to X of hermitian structures on Z(1) satisfying (1.1) is
thus an injective mapping of the set of such structures into the set of
euclidean structures on X.
(iii) It is bijective. Indeed, the automorphism i of Z(1) defined by the
given ( I ) on X, that is, by (1.5), satisfies
i2 = -1, [iz, z'] = [iz', z],
since `i1 = i1, `i2 = i2; the function ( I
), which (1.2) defines on Z(1),
is clearly linear in z e C', and
satisfies (z', z) = (z', z),
hence is sesquilinear,
satisfies Ix + iyI2 = IxI2 + IyJ2,
and indeed defines a hermitian structure.
Lemma 1.1 has as the following corollary.
LEMMA 1.2. The set of hermitian structures on Z(1) satisfying (1.1) is an
open convex cone. It is therefore connected.
Remark 1. We choose arbitrarily one of these hermitian structures on
Z(1), which we shall use to define topological notions (the Maslov indices).
By the preceding lemma, these notions will not depend on this choice.
2. The Lagrangian Grassmannian A(1) of Z(1)
Definition 2.1 A subspace of Z(1) is called isotropic when the restriction of
[ , ] to this subspace is identically zero, that is, by (1.1), when the res-
triction of the hermitian structure on Z(1) is a euclidean structure on this
subspace.
Every orthonormal frame of an isotropic subspace of dimension k is thus
composed of vectors orthogonal in Z(1); hence k < 1.
Definition 2.2. The isotropic subspaces of maximal dimension I are
called lagrangian subspaces; the collection of lagrangian subspaces A(1) is
called the lagrangian grassmannian:
XandX*EA(1).

28 I,§2,2
Let A E A(l) and let r be an orthonormal frame of A. It is a frame of Z(1):
the elements of Z(1) (respectively A) are linear combinations with complex
(respectively real) coefficients of the vectors that make up r.
Let U(1) denote the group of unitary automorphisms u of Z(l) (that is,
uu* = e, where u* = 'u, and e is the identity). By (1.1),
U(1) C Sp(1).
Further let A' E A(l) and let r' be an orthonormal frame of A'. There is a
unique element u in U(l) such that
r = ur'
from which follows
A = ua.'.
The group U(1) thus acts transitively on A(l). The same holds a fortiori
for Sp(l); whence 1°) of the lemma below, where St(l) [respectively, 0(l)]
denotes the stabilizer of X* in Sp(l) [respectively, U(1)], that is, the
subgroup of s such that sX * = X *.
Now 0(l) is clearly the orthogonal group. Lemma 2.3 characterizes
St(l); part 2 shows why the stabilizer of X* interests us more than that of
X.
LEMMA 2.1. 1 ° We have
A(l) = Sp(l)/St(l) = U(1)/0(1). (2.1)
2°) Let W(l) be the set of symmetric elements w in U(1), that is, the set
of elements w such that 'w = w; thus w c- W(1) means w = 'w = w-'. The
diagram
U(l) a u - U'u = w E W(l)
U(1)/0(1) = A(l) a2 = UX*
defines a natural homeomorphism
Then
z c- a. is equivalent to z + w(A)z = 0.

I,§2,2 29
Let
z = x + iy, where x and y c X.
Assume
10 sp(w(A),
where sp(w) is the spectrum of w, a 0-chain of the unit circle S'. Then
z e 2 is equivalent toy = i
e + w(A)x,
(2.5)
e - w(it)
where i(e + w(A)]/[e - w(.)] is a real symmetric matrix (that is, equal to
its transpose).
3°) dim(.1 n A') is the multiplicity of 1 in sp(w(,)w-1(.1')).
Remark 2.1. Part 3 is preparation for the topological definition of the
Maslov index (section 5).
The proof of lemma 2.1 is based on the following lemma. Writing
u E U(l) in terms of its eigenvectors and eigenvalues, the proof of lemma 2.2
is clear.
LEMMA 2.2. 1°) Let u c- U(l). A necessary and sufficient condition for
u e W(l) is that all of its eigenvectors can be chosen to be real.
2°) Every surjective mapping
F : S1 - S' (S' is the unit circle in C)
defines a surjective mapping
W(l) a w i-+ F(w) E W(l).
Proof of lemma 2.1,2°). The diagram (2.2) defines a mapping (2.3) since, if
u and uv c- U(l) have the same image in A(l) = U(l)/O(l), then v c- 0(l), and
so
uv `(uv) = uv `v `u = U 'U.
By lemma 2.2,2°), given w c- W(l), there exists some u c- W(l) such that
w = u2. Then w = u 'u, and so the map (2.3) is surjective.
Since a. = uX *, where u c- U(l), the condition z c- ) means u-'z c- X*, or
Re(u-lz) = 0, or u-1z + u-12 = 0, or z + w2 = 0. The map (2.3) is
therefore injective.

30 I,§2,2
Proof of lemma 2.1,3°). Let w = w(,1), w' = w(A'). Then A n A' is given
by the equations
z+w2=0, z+w'2=0;
that is,
r) A': w-'z =w'-'z, z= WE.
Let T be the analytic subspace of Z(l) given by the equation
T: w-'z = w'-'z.
Then dim, T = k, where k is the multiplicity of 1 in sp(ww'-'). The equation
of An A' in Tjs
z+wz=0.
By lemma 2.2,2°), there exists a u c- W(l) such that
-w=U2 =u-'u.
Thus the equation of A n A' in T may be written
uz=iii.
The isomorphism
T=)
therefore maps A n A' onto the real part Rk of Ck, and so
dim, A n A' = k.
LEMMA 2.3. The stabilizer St(l) of X * in Sp(l) has the following properties:
1°) The elements s of St(l) are characterized as follows:
s(x', P) = (x, P)
is equivalent to
x = slx', P = 'Si'(P' + s2x'), (2.6)
where sl is an arbitrary automorphism of X and s2 = 's2 is an arbitrary
symmetric morphism X -+ X*.
2°) An element s of St(l) is the projection of two elements S of Sp2(l)
defined by

I,§2,2-I,§2,3 31
(Sf)(W ) _ ,/d et s1 1 [e("12)Cx',s=x'>f(x')]z
=sI,X (2.7)
Remark 2.2. We denote by St2(1) the subgroup of Sp2(1) whose projection
onto Sp(1) is St(1). By Remark 2.5 in §1, St2(1) is the set of S E Sp2(1) that
act pointwise on .°(X): the value of Sf at a point x of X depends only on
the behavior of fat a point x' of X (in fact on the value of f at x').
Proof of 1 °). The elements of the stabilizer of X* in the group of auto-
rnorphisms of the vector space Z(1) are the mappings (x', p') i-- (x, p)
defined by
x = six', p = s*(p + s2x'),
where sl and s* are automorphisms of X and X* and s2 is a morphism
X -a X*. These elements belong to Sp(1) when
t -1 t
s = S1 S2 = S2.
Proof of 2°). Formula (2.7) defines an automorphism S of Y'(X) that
belongs to Sp2(1) by Remark 2.4 in §1. Clearly
1
x-(Sf) = S[f -s1x'], I ax(Sf) = S[tsl1 (-'V ax'
+ S2x'J f].
Hence, for any a in .sd (§1,1),
la la
a°(X,- )(Sf) = Sao(s1x', ts11 + S2x' f,
vax v8x
that is, by (2.6),
S -1 aS is associated to a° o s,
and so s is the natural image in Sp(1) of ± S E Sp2(1).
3. The Covering Groups of Sp(1) and the Covering Spaces of A(1)
The properties of these covering groups and spaces (2) follow from prop-
erties of n1 [Sp(1)] and n1 [A(1)], which are obtained by studying n1 [U(1)].
Here ik denotes the kth homotopy group, (see Steenrod [17]; we note that
N. Steenrod uses the expression symplectic groups in a different sense than
we do.)
2See Steenrod [17], 1.6, 14.1.

32 I,§2,3
LEMMA 3.1. 1°) The inclusion O(l) c St(l) induces an isomorphism
rzk[O(1)] ' nk[St(l)] t1k E N.
2°) The inclusion U(l) c Sp(l) induces an isomorphism
lrk[U(1)] it,k[Sp(l)] Vk E N.
3°) The morphism
nl [U(1)]
3 Y
--r
1 d(det u)
E Z (3.1)
27ri det u
is a natural isomorphism:
ir,[U(1)] Z.
Proof of 1°). The elements s of St(l) are characterized by (2.6); those for
which s2 = 0 form a subgroup GL(l) of St(l). The inclusions
O(l) c GL(l) c St(l)
induce natural morphisms
ltk[O(1)] itk[GL(l)] 7rk[St(l)].
The second morphism is an isomorphism, since
St(l) = GL(l) x R", where n = 1(1 + 1)/2.
It has to be shown that i is an isomorphism. Now GL(l) acts transitively
on the set Q+ of positive definite quadratic forms on X, and O(l) is the
stabilizer of one of them. Hence
GL(l)/O(l) = Q+, where Q+ is convex.
The exactness of the homotopy sequence of this fibration (see Steenrod
[17], 17.3, 17.4) proves that i is indeed an isomorphism.
Proof of 2°). The inclusions
U(l) c Sp(l), St(l) n U(l) = 0(1) c St(l)
define a mapping (see Steenrod [17], 17.5) of the fibration
U(l)/O(l) = A(l) into Sp(l)/St(l) = A(1);
its restriction to A(l) is the identity. This mapping induces a morphism of

I,§2,3
33
the homotopy sequences of these two fibrations (see Steenrod [17], 17.3,
17.11, 17.5):
nk+I[A(l)] -4 nk[O(l)] 4 nk[U(l)] p' nk[A(l)] ... no[o(l)]
l i° 1 io Ii, i i° i i°
7rk+i[A(l)] 4 n,[St(l)] - nk[Sp(l)] P rzk[A(l)] ... no[St(l)]
This diagram, in which the lines are exact, is thus commutative. Since the
mappings io are isomorphisms, it follows that the mappings it are neces-
sarily isomorphisms.
Proof of 3°). (Steenrod [17], 25.2, proves part of this by other means.) We
denote by det (that is, determinant) the epimorphism
U(l) 3 u E- det u c S' (3.2)
and by SU(l) its kernel. Since u c SU(l) when det u = 1, we have
U(l)/SU(l) = S';
(3.2) is the natural projection of U(l) onto S'. The homotopy sequence of
this fibration contains the following, which is thus exact:
it1[SU(l)] -4 ni[U(l)] -°' rrl[S1] * no[SU(l)]; (3.3)
here p is induced by the morphism (3.2). Since SU(l) is connected,
iro[SU(l)] is trivial. Let us compute n1[SU(l)].
SU(l) acts transitively on the sphere
521-1:Izl = 1.
The stabilizer of the vector (1, 0, ... , 0) in C' is SU(l - 1); thus
SU(l)/SU(l - 1) = S21-1
The homotopy sequence of this fibration contains the following, which is
thus exact:
it2 [S21-1]
nl [SU(l - 1)] '' 7r1 [SU(l)] _v
nl[Su i]
where 7x1[521-'] and n2[S21-'] are trivial for l 3 2 (see Steenrod [17],
21.2). Thus i' is an isomorphism. Now 7r, [SU(1)] is trivial, since SU(1) is
trivial, and so
it1[SU(l)] is trivial. (3.4)

34 I,§2,3
Since 1r1 [S U(1)] and no [S U(1)] are trivial in the exact sequence (3.3), p is
an isomorphism. Now
9G1[S1]3FH-
If d- EZ
r
is an isomorphism.
The composition of p, which is induced by (3.2), with this isomorphism
is an isomorphism n1 [U(1)] -+ Z, which clearly is defined by (3.1).
LEMMA 3.2. 1 °) The composition of the natural isomorphism n 1
[A(1)]
n1 [W(1)] [cf. (2.3)] and the morphism
nl[j'1'(l)]-3
f ddetw)EZ (3.5)
7ri
is a natural isomorphism (Arnold [1]):
[A(1)] ^' Z.
2°) The fibration Sp(l)/St(l) = A(1) defines a monomorphism
p : Z '=' it1 [Sp(1)] - nl (A(1)] = Z, (3.6)
which is multiplication by 2 on Z.
Proof of 1 °). The homeomorphism (2.3) allows us to define
det ). = det w e S 1; (3.7)
the mapping
A(l)a2"detA c- S' (3.8)
is clearly an epimorphism. By (2.2) we have
det A = detz
u, if i = uX *.
Hence for all u e U(1)
det(uA) = det 2 u det A. '(3.9)
The mapping (3.8) thus defines a fibration on which U(1) permutes the
fibers. The fibration is
A(1)/SA(l) = S 1,

Random documents with unrelated
content Scribd suggests to you:

Sighing softly to the river
Comes the loving breeze,
Setting nature all a-quiver,
Rustling through the trees!
And the brook in rippling measure
Laughs for very love,
While the poplars, in their pleasure,
Wave their arms above!
River, river, little river,
May thy loving prosper ever.
Heaven speed thee, poplar tree.
May thy wooing happy be!
Yet, the breeze is but a rover,
When he wings away,
Brook and poplar mourn a lover!
Sighing well-a-day!
Ah, the doing and undoing
That the rogue could tell!
When the breeze is out a-wooing,
Who can woo so well?
Pretty brook, thy dream is over
For thy love is but a rover!
Sad the lot of poplar trees,
Courted by the fickle breeze!

Good children, list, if you're inclined,
And wicked children too—
This pretty ballad is designed
Especially for you.
Two ogres dwelt in Wickham Wold—
Each traits distinctive had:
The younger was as good as gold,
The elder was as bad.
A wicked, disobedient son
Was James M'Alpine, and
A contrast to the elder one,
Good Applebody Bland.
M'Alpine—brutes like him are few—
In greediness delights,
A melancholy victim to
Unchastened appetites.
Good, well-bred children every day
He ravenously ate,—
All boys were fish who found their way
Into M'Alpine's net:
Boys whose good breeding is innate,
Whose sums are always right;
And boys who don't expostulate
When sent to bed at night,
And kindly boys who never search
The nests of birds of song;
And serious boys for whom, in church,
No sermon is too long.
Contrast with James's greedy haste
A d h i h d

And comprehensive hand,
The nice discriminating taste
Of Applebody Bland.
Bland only eats bad boys, who swear—
Who can behave, but don't—
Disgraceful lads who say "don't care,"
And "shan't," and "can't," and "won't."
Who wet their shoes and learn to box,
And say what isn't true,
Who bite their nails and jam their frocks,
And make long noses too;
Who kick a nurse's aged shin,
And sit in sulky mopes;
And boys who twirl poor kittens in
Distracting zoëtropes.
But James, when he was quite a youth,
Had often been to school,
And though so bad, to tell the truth,
He wasn't quite a fool.

At logic few with him could vie;
To his peculiar sect
He could propose a fallacy
With singular effect.
So, when his Mentors said, "Expound—
Why eat good children—why?"
Upon his Mentors he would round
With this absurd reply:
"I have been taught to love the good—
The pure—the unalloyed—
And wicked boys, I've understood,
I always should avoid.
"Why do I eat good children—why?
Because I love them so!"
(But this was empty sophistry,
As your Papa can show.)
Now, though the learning of his friends
Was truly not immense,
They had a way of fitting ends
By rule of common sense.
"Away, away!" his Mentors cried,
"Thou uncongenial pest!
A quirk's a thing we can't abide,
A quibble we detest!
"A fallacy in your reply
Our intellect descries,
Although we don't pretend to spy
Exactly where it lies.
"In misery and penal woes
Must end a glutton's joys;

Must end a glutton s joys;
And learn how ogres punish those
Who dare to eat good boys.
"Secured by fetter, cramp, and chain,
And gagged securely—so—
You shall be placed in Drury Lane,
Where only good lads go.
"Surrounded there by virtuous boys,
You'll suffer torture wus
Than that which constantly annoys
Disgraceful Tantalus.
("If you would learn the woes that vex
Poor Tantalus, down there,
Pray borrow of Papa an ex-
Purgated Lempriere.)
"But as for Bland who, as it seems,
Eats only naughty boys,
We've planned a recompense that teems
With gastronomic joys.
"Where wicked youths in crowds are stowed
He shall unquestioned rule,
And have the run of Hackney Road
Reformatory School!"

When I was a lad I served a term
As office boy to an Attorney's firm;
I cleaned the windows and I swept the floor,
And I polished up the handle of the big front door.
I polished up that handle so successfullee,
That now I am the Ruler of the Queen's Navee!
As office boy I made such a mark
That they gave me the post of a junior clerk;
I served the writs with a smile so bland,
And I copied all the letters in a big round hand.
I copied all the letters in a hand so free,
In serving writs I made such a name
That an articled clerk I soon became;
I wore clean collars and a brand-new suit
For the Pass Examination at the Institute:
And that Pass Examination did so well for me,
Of legal knowledge I acquired such a grip
That they took me into the partnership,
And that junior partnership, I ween,
Was the only ship that I ever had seen:
But that kind of ship so suited me,
I grew so rich that I was sent
By a pocket borough into Parliament;
I always voted at my Party's call,
And I never thought of thinking for myself at all.
I thought so little, they rewarded me,
By making me the Ruler of the Queen's Navee!
Now, landsmen all, whoever you may be,

If you want to rise to the top of the tree—
If your soul isn't fettered to an office stool,
Be careful to be guided by this golden rule—
Stick close to your desks and never go to sea,
And you all may be Rulers of the Queen's Navee!

Earl Joyce he was a kind old party
Whom nothing ever could put out,
Though eighty-two, he still was hearty,
Excepting as regarded gout.
He had one unexampled daughter,
The Lady Minnie-haha Joyce,
Fair Minnie-haha, "Laughing Water,"
So called from her melodious voice.
By Nature planned for lover-capture,
Her beauty every heart assailed;
The good old nobleman with rapture
Observed how widely she prevailed.
Aloof from all the lordly flockings
Of titled swells who worshipped her,
There stood, in pumps and cotton stockings,
One humble lover—Oliver.
He was no peer by Fortune petted,
His name recalled no bygone age;
He was no lordling coronetted—
Alas! he was a simple page!
With vain appeals he never bored her,
But stood in silent sorrow by—
He knew how fondly he adored her,
And knew, alas! how hopelessly!

Well grounded by a village tutor
In languages alive and past,
He'd say unto himself, "Knee-suitor,
Oh, do not go beyond your last!"
But though his name could boast no handle,
He could not every hope resign;
As moths will hover round a candle,
So hovered he about her shrine.
The brilliant candle dazed the moth well:
One day she sang to her Papa
The air that Marie sings with Bothwell
In Niedermeyer's opera.
(Therein a stable boy, it's stated,
Devoutly loved a noble dame,
Who ardently reciprocated
His rather injudicious flame.)
And then, before the piano closing
(He listened coyly at the door),
She sang a song of her composing—
I give one verse from half a score:

Ballad
Why, pretty page, art ever sighing?
Is sorrow in thy heartlet lying?
Come, set a-ringing
Thy laugh entrancing,
And ever singing
And ever dancing.
Ever singing, Tra! la! la!
Ever dancing, Tra! la! la!
Ever singing, ever dancing,
Ever singing, Tra! la! la!
He skipped for joy like little muttons,
He danced like Esmeralda's kid.
(She did not mean a boy in buttons,
Although he fancied that she did.)
Poor lad! convinced he thus would win her,
He wore out many pairs of soles;
He danced when taking down the dinner—
He danced when bringing up the coals.
He danced and sang (however laden)
With his incessant "Tra! la! la!"
Which much surprised the noble maiden,
And puzzled even her Papa.
He nourished now his flame and fanned it,
He even danced at work below.
The upper servants wouldn't stand it,
And Bowles the butler told him so.

At length on impulse acting blindly,
His love he laid completely bare;
The gentle Earl received him kindly
And told the lad to take a chair.
"Oh, sir," the suitor uttered sadly,
"Don't give your indignation vent;
I fear you think I'm acting madly,
Perhaps you think me insolent?"
The kindly Earl repelled the notion;
His noble bosom heaved a sigh,
His fingers trembled with emotion,
A tear stood in his mild blue eye:
For, oh! the scene recalled too plainly
The half-forgotten time when he,
A boy of nine, had worshipped vainly
A governess of forty-three!
"My boy," he said, in tone consoling,
"Give up this idle fancy—do—
The song you heard my daughter trolling
Did not, indeed, refer to you.
"I feel for you, poor boy, acutely;
I would not wish to give you pain;
Your pangs I estimate minutely,—
I, too, have loved, and loved in vain.

, , ,
"But still your humble rank and station
For Minnie surely are not meet"—
He said much more in conversation
Which it were needless to repeat.
Now I'm prepared to bet a guinea,
Were this a mere dramatic case,
The page would have eloped with Minnie.
But, no—he only left his place.
The simple Truth is my detective,
With me Sensation can't abide;
The Likely beats the mere Effective,
And Nature is my only guide.

Oh, listen to the tale of Mister William, if you please,
Whom naughty, naughty judges sent away beyond the seas.
He forged a party's will, which caused anxiety and strife,
Resulting in his getting penal servitude for life.
He was a kindly goodly man, and naturally prone,
Instead of taking others' gold, to give away his own.
But he had heard of Vice, and longed for only once to strike—
To plan one little wickedness—to see what it was like.
He argued with himself, and said, "A spotless man am I;
I can't be more respectable, however hard I try;
For six and thirty years I've always been as good as gold,
And now for half-an-hour I'll deal in infamy untold!
"A baby who is wicked at the early age of one,
And then reforms—and dies at thirty-six a spotless son,
Is never, never saddled with his babyhood's defect,
But earns from worthy men consideration and respect.
"So one who never revelled in discreditable tricks
Until he reached the comfortable age of thirty-six,
Is free for half-an-hour to perpetrate a deed of shame,
Without incurring permanent disgrace, or even blame.
"That babies don't commit such crimes as forgery is true,
But little sins develop, if you leave 'em to accrue;
And he who shuns all vices as successive seasons roll,
Should reap at length the benefit of so much self-control.
"The common sin of babyhood—objecting to be drest—
If you leave it to accumulate at compound interest,
For anything you know, may represent, if you're alive,
A burglary or murder at the age of thirty-five.
"Still, I wouldn't take advantage of this fact, but be content
With d bl f ll it' i t

With some pardonable folly—it's a mere experiment.
The greater the temptation to go wrong, the less the sin;
So with something that's particularly tempting I'll begin.
"I would not steal a penny, for my income's very fair—
I do not want a penny—I have pennies and to spare—
And if I stole a penny from a money-bag or till,
The sin would be enormous—the temptation being nil.
"But if I broke asunder all such pettifogging bounds,
And forged a party's Will for (say) Five Hundred Thousand Pounds,
With such an irresistible temptation to a haul,
Of course the sin must be infinitesimally small.
"There's Wilson who is dying—he has wealth from Stock and rent—
If I divert his riches from their natural descent,
I'm placed in a position to indulge each little whim."
So he diverted them—and they, in turn, diverted him.
Unfortunately, though, by some unpardonable flaw,
Temptation isn't recognised by Britain's Common Law;
Men found him out by some peculiarity of touch,
And William got a "lifer," which annoyed him very much.
For ah! he never reconciled himself to life in gaol,
He fretted and he pined, and grew dispirited and pale;
He was numbered like a cabman, too, which told upon him so,

He was numbered like a cabman, too, which told upon him so,
That his spirits, once so buoyant, grew uncomfortably low.
And sympathetic gaolers would remark, "It's very true,
He ain't been brought up common, like the likes of me and you."
So they took him into hospital, and gave him mutton chops,
And chocolate, and arrowroot, and buns, and malt and hops.
Kind clergymen, besides, grew interested in his fate,
Affected by the details of his pitiable state.
They waited on the Secretary, somewhere in Whitehall,
Who said he would receive them any day they liked to call.
"Consider, sir, the hardship of this interesting case:
A prison life brings with it something very like disgrace;
It's telling on young William, who's reduced to skin and bone—
Remember he's a gentleman, with money of his own.
"He had an ample income, and of course he stands in need
Of sherry with his dinner, and his customary weed;
No delicacies now can pass his gentlemanly lips—
He misses his sea-bathing and his continental trips.
"He says the other prisoners are commonplace and rude;
He says he cannot relish the disgusting prison food,
For when a boy they taught him to distinguish Good from Bad,
And other educational advantages he's had.
"A b l tt i d d thi f

"A burglar or garrotter, or, indeed, a common thief
Is very glad to batten on potatoes and on beef,
Or anything, in short, that prison kitchens can afford,—
A cut above the diet in a common workhouse ward.
"But beef and mutton-broth don't seem to suit our William's whim,
A boon to other prisoners—a punishment to him:
It never was intended that the discipline of gaol
Should dash a convict's spirits, sir, or make him thin or pale."
"Good Gracious Me!" that sympathetic Secretary cried,
"Suppose in prison fetters Mister William should have died!
Dear me, of course! Imprisonment for Life his sentence saith:
I'm very glad you mentioned it—it might have been For Death!
"Release him with a ticket—he'll be better then, no doubt,
And tell him I apologise." So Mister William's out.
I hope he will be careful in his manuscripts, I'm sure,
And not begin experimentalising any more.

Would you know the kind of maid
Sets my heart a flame-a?
Eyes must be downcast and staid,
Cheeks must flush for shame-a!
She may neither dance nor sing,
But, demure in everything,
Hang her head in modest way
With pouting lips that seem to say,
"Kiss me, kiss me, kiss me, kiss me,
Though I die of shame-a!"
Please you, that's the kind of maid
Sets my heart a flame-a!
When a maid is bold and gay
With a tongue goes clang-a,
Flaunting it in brave array,
Maiden may go hang-a!
Sunflower gay and hollyhock
Never shall my garden stock;
Mine the blushing rose of May,
With pouting lips that seem to say
"Oh, kiss me, kiss me, kiss me, kiss me,
Though I die for shame-a!"
Please you, that's the kind of maid
Sets my heart a flame-a!

A proud Pasha was Bailey Ben,
His wives were three, his tails were ten;
His form was dignified, but stout,
Men called him "Little Roundabout."
His Importance
Pale Pilgrims came from o'er the sea
To wait on Pasha Bailey B.,
All bearing presents in a crowd,
For B. was poor as well as proud.
His Presents
They brought him onions strung on ropes,
And cold boiled beef, and telescopes,
And balls of string, and shrimps, and guns,
And chops, and tacks, and hats, and buns.
More of them
They brought him white kid gloves, and pails,
And candlesticks, and potted quails,
And capstan-bars, and scales and weights,
And ornaments for empty grates.
Why I mention these
My tale is not of these—oh no!
I only mention them to show
The divers gifts that divers men
Brought o'er the sea to Bailey Ben.
His Confidant
A confidant had Bailey B.,
A gay Mongolian dog was he;
I am not good at Turkish names

I am not good at Turkish names,
And so I call him Simple James.
His Confidant's Countenance
A dreadful legend you might trace
In Simple James's honest face,
For there you read, in Nature's print,
"A Scoundrel of the Deepest Tint."
His Character
A deed of blood, or fire, or flames,
Was meat and drink to Simple James:
To hide his guilt he did not plan,
But owned himself a bad young man.
The Author to his Reader

And why on earth good Bailey Ben
(The wisest, noblest, best of men)
Made Simple James his right-hand man
Is quite beyond my mental span.
The same, continued
But there—enough of gruesome deeds!
My heart, in thinking of them, bleeds;
And so let Simple James take wing,—
'Tis not of him I'm going to sing.
The Pasha's Clerk
Good Pasha Bailey kept a clerk
(For Bailey only made his mark),
His name was Matthew Wycombe Coo,
A man of nearly forty-two.
His Accomplishments
No person that I ever knew
Could "yödel" half as well as Coo,
And Highlanders exclaimed, "Eh, weel!"
When Coo began to dance a reel.
His Kindness to the Pasha's Wives

He used to dance and sing and play
In such an unaffected way,
He cheered the unexciting lives
Of Pasha Bailey's lovely wives.
But why should I encumber you
With histories of Matthew Coo?
Let Matthew Coo at once take wing.—
'Tis not of Coo I'm going to sing.
The Author's Muse
Let me recall my wandering Muse
She shall be steady if I choose—
She roves, instead of helping me
To tell the deeds of Bailey B.
The Pasha's Visitor
One morning knocked, at half-past eight,
A tall Red Indian at his gate.
In Turkey, as you're p'raps aware,
Red Indians are extremely rare.
The Visitor's Outfit
Mocassins decked his graceful legs,
His eyes were black, and round as eggs,
And on his neck, instead of beads,
Hung several Catawampous seeds.
What the Visitor said
"Ho, ho!" he said, "thou pale-faced one,
Poor offspring of an Eastern sun,
You've never seen the Red Man skip
Upon the banks of Mississip!"

Upon the banks of Mississip!"
The Author's Moderation
To say that Bailey oped his eyes
Would feebly paint his great surprise—
To say it almost made him die
Would be to paint it much too high.
But why should I ransack my head
To tell you all that Indian said;
We'll let the Indian man take wing,—
'Tis not of him I'm going to sing.
The Reader to the Author
Come, come, I say, that's quite enough
Of this absurd disjointed stuff;
Now let's get on to that affair
About Lieutenant-Colonel Flare.

Welcome to our website – the perfect destination for book lovers and
knowledge seekers. We believe that every book holds a new world,
offering opportunities for learning, discovery, and personal growth.
That’s why we are dedicated to bringing you a diverse collection of
books, ranging from classic literature and specialized publications to
self-development guides and children's books.
More than just a book-buying platform, we strive to be a bridge
connecting you with timeless cultural and intellectual values. With an
elegant, user-friendly interface and a smart search system, you can
quickly find the books that best suit your interests. Additionally,
our special promotions and home delivery services help you save time
and fully enjoy the joy of reading.
Join us on a journey of knowledge exploration, passion nurturing, and
personal growth every day!
ebookbell.com

Lagrangian Analysis And Quantum Mechanics A Mathematical Structure Related To Asymptotic Expansions And The Maslov Index Jean Leray

More Related Content

Similar to Lagrangian Analysis And Quantum Mechanics A Mathematical Structure Related To Asymptotic Expansions And The Maslov Index Jean Leray (20)

Recently uploaded (20)

Lagrangian Analysis And Quantum Mechanics A Mathematical Structure Related To Asymptotic Expansions And The Maslov Index Jean Leray