An Introduction to Symplectic Geometry
Rolf Berndt
Graduate Studies in Mathematics Volume 26
American Mathematical Society
Selected Titles in This Series 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 20(10 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000
23 Alberto Candel and Lawrence Conlon, Foliations 1, 2000 22 Gfinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000
21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andrds I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. 11: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms. 1997
16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997
15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume 1: Elementary theory, 1997
Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics. 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 14
3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994
2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1
Ethan Akin, The general topology of dynamical systems, 1993
An Introduction to Symplectic Geometry
An Introduction to Symplectic Geometry Rolf Berndt Translated by
Michael Klucznik
Graduate Studies in Mathematics Volume 26
0 ""1
American Mathematical Society Providence, Rhode Island
Editorial Board James Humphreys (Chair) David Saltman David Sattinger Ronald Stern 2000 Mathematics Subject Classification. Primary 53C15, 53Dxx, 20G20, 81S10.
Originally published in the German language by Ftiedr. Vieweg & Sohn Verlagsgesellschaft mbH, D-65189 Wiesbaden, Germany, as "Rolf Berndt: Einfnhruug in die Symplektische Geometrie. 1. Auflage (1st edition)" © by Ftiedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1998. Translated from the German by Michael Klucznik ABSTRACT. The notions of symplectic form, symplectic manifold and symplectic group appear in many different contexts in analysis, geometry, function theory and dynamical systems. This book assembles tools from different mathematical regions necessary to define these notions and to introduce their application. Among the topics treated here are symplectic and Kiihler vector spaces,
the symplectic group and Siegel's half space, symplectic and contact manifolds, the theorem of Darboux, methods of constructing symplectic manifolds: Kiihler manifolds, coadjoint orbits and symplectic reduction, Hamiltonian systems, the moment map, and a glimpse into geometric quantization (in particular the theorem of Groenewold and van Hove) leading to some rudiments of the representation theory of the Heisenberg and the Jacobi group. The goal of the book is to provide an entrance into a fascinating area linking several mathematical disciplines and parts of theoretical physics.
Library of Congress Cataloging-In-Publication Data Berndt, Rolf. ]Einfiihrung in die symplektische Geometric. English] An introduction to sympletic geometry / Rolf Berndt ; translated by Michael Klucznik.
p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 26) Includes bibliographical references and index.
ISBN 0-8218-2056-7 (alk. paper) 1. Symplectic manifolds. 2. Geometry, Differential
I. Title.
II. Series.
QA649.B47 2000 516.3'6- dc2l
00-033139
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://wv.ame.org/
10987654321
060504030201
Contents
Preface
ix
Chapter 0. Some Aspects of Theoretical Mechanics §0.1. The Lagrange equations §0.2. Hamilton's equations §0.3. The Hamilton-Jacobi equation §0.4. A symplectic interpretation §0.5. Hamilton's equations via the Poisson bracket §0.6. Towards quantization Chapter 1. Symplectic Algebra §1.1. Symplectic vector spaces §1.2. Symplectic morphisms and symplectic groups §1.3. Subspaces of symplectic vector spaces §1.4. Complex structures of real symplectic spaces Chapter 2. Symplectic Manifolds §2.1. Symplectic manifolds and their morphisms §2.2. Darboux's theorem §2.3. The cotangent bundle §2.4. Kiihler manifolds §2.5. Coadjoint orbits §2.6. Complex projective space §2.7. Symplectic invariants (a quick view) vii
viii
Contents
Chapter 3. Hamiltonian Vector Fields and the Poisson Bracket §3.1. Preliminaries §3.2. Hamiltonian systems §3.3. Poisson brackets §3.4. Contact manifolds
71
71
74
79 85
Chapter 4. The Moment Map 93 §4.1. Definitions 93 §4.2. Constructions and examples 97 §4.3. Reduction of phase spaces by the consideration of symmetry 104 Chapter 5. Quantization §5.1. Homogeneous quadratic polynomials and 912 §5.2. Polynomials of degree 1 and the Heisenberg group §5.3. Polynomials of degree 2 and the Jacobi group §5.4. The Groenewold-van Hove theorem §5.5. Towards the general case
111
Appendix A. Differentiable Manifolds and Vector Bundles §A.1. Differentiable manifolds and their tangent spaces §A.2. Vector bundles and their sections §A.3. The tangent and the cotangent bundles §A.4. Tensors and differential forms §A.5. Connections
135
Appendix B. Lie Groups and Lie Algebras §B.1. Lie algebras and vector fields §B.2. Lie groups and invariant vector fields §B.3. One-parameter subgroups and the exponent map
163
Appendix C. A Little Cohomology Theory
171
§C.1. §C.2.
§C.3.
Cohomology of groups Cohomology of Lie algebras Cohomology of manifolds
Appendix D. Representations of Groups §D.1. Linear representations §D.2. Continuous and unitary representations §D.3. On the construction of representations
111
114 120 124 128
135 144 146 150 158
163 165
167
171
173
174 177
177 179 180
Contents
ix
Bibliography
185
Index
189
Symbols
193
Preface I.e caractere propre des mcthodes de ''Analyse et de Is Gt ometrie modernes consiste daps 1'emploi d'un petit nombre de principes genttraux. independants de Is situation respective des differentes parties on des valeurs relatives des differents symboles; et les
consequences sont d'autant plus tttendues que les principes eux-memes ont plus de generalite. from G. DARBOUX: Prancipes de GEometrie Analytique
This text is written for the graduate student who has previous training in analysis and linear algebra, as for instance S. Fang's Analysis I and Linear Algebra. It is meant as an introduction to what is today an intensive area of research linking several disciplines of mathematics and physics in the sense of the Greek word ouµrrMecet.v (which means to interconnect, or to interrelate in English).' The difficulty (but also the fascination) of the area is the wide variety of mathematical machinery required. In order to introduce this interrelation, this text includes extensive appendices which include definitions and developments not usually covered in the basic training of students but which lay the groundwork for the specific constructions 11 want to thank P. Slodowy for pointing out to me that the name symptectic group, which eventually gave rise to the term sympkche geometry, was proposed by If. WEYL, [W], 1938, in his book, The Classical Groups (see footnote on p. 165). The symplectic group was also called the comples group or an Abelion linear group, this last to honor ABEL, who was the first to study them. xi
xii
Preface
needed in symplectic geometry. Furthermore, more advanced topics will continue to rely heavily on other disciplines, in particular on results from the study of differential equations. Specifically, the text tries to reach the following two goals:
To present the idea of the formalism of symplectic forms, to introduce the symplectic group, and especially to describe the symplectic manifolds. This will be accompanied by the presentation of many examples of how they come to arise; in particular the quotient manifolds of group actions will be described, and
To demonstrate the connections and interworking between mathematical objects and the formalism of theoretical mechanics; in particular, the Hamiltonian formalism, and that of the quantum formalism, namely the process of quantization. The pursuit of these goals proceeds according to the following plan. We begin in Chapter 0 with a brief introduction of a few topics from theoretical mechanics needed later in the text. The material of this chapter will already be familiar to physics students; however, for the majority of mathematics students, who have not learned the connections of their subject to physics, this material will perhaps be new. We are constrained, in the first chapter, to consider symplectic (and a little later Kdhler) vector spaces. This is followed by the introduction of the associated notion of a symplectic group Sp(V) along with its generation. We continue with the introduction of several specific and theoretically important subspaces, the isotropic, coisotropic and Lagrangian subspaces, as well as the hyperbolic planes and spaces and the radical of a symplectic space. Our first result will be to show that the symplectic subspaces of a given dimension and rank are fixed up to symplectic isomorphism. A consequence is then that the Lagrangian subspaces form a homogeneous space £(V) for
the action of the group Sp(V). The greatest effort will be devoted to the description of the spaces of positive complex structures compatible with the given symplectic structure. The second major result will be that this space is a homogeneous space, and is, for dim V = 2n, isomorphic to the Siegel
half space S) = Spn(R)/U(n). The second chapter is dedicated to the central object of the book, namely symplectic manifolds. Here the consideration of differential forms is unavoid-
able. In Appendix A their calculus will be given. The first result of this chapter is then the derivation of a theorem by Darboux that says that the symplectic manifolds are all locally equivalent. This is in sharp contrast to the situation with Riemannian manifolds, whose definition is otherwise
xiii
Preface
somewhat parallel to that of the symplectic manifolds. The chapter will then take a glance at new research by considering the assignment of invariants to symplectic manifolds; in particular. the symplectic capacities and the pseudoholomorphic curves will be given. In the course of the second chapter. we will present several examples of symplectic manifolds: First, the example which forms the origin of the theory and remains
the primary application to physics is the cotangent bundle T'Q of a given manifold Q.
Second. the general Kahler manifold.
Third, the coadjoint orbits. This description of symplectic manifolds with the operation of a Lie group G can be taken as the second major result of this chapter. We describe a theorem of Kost.ant and Souriau that says that for a given Lie group G with Lie algebra g satisfying the condition that the first two cohomology groups
vanish, that is H'(g) = H2(9) = 0. there is. up to covering. a one-to-one correspondence between the symplectic manifolds with
transitive G-action and the G-orbits in the dual space g' of g. Here we will need several facts from the theory of Lie algebras and systems of differential equations. and we will at least, cover some of
the rudiments we require. This will then offer yet another means for introducing one of the central concepts of the field. namely the moment map. This will, however. be somewhat postponed so that
In the fourth and last example. complex projective space can be presented as a symplectic manifold: this will be seen as a specific example of the third example. as well as, the second: that is. as a coadjoint orbit as well as as a Kahler manifold. As preparation for the higher level construction of symplectic manifolds.
Chapter 3 will introduce the standard concepts of a Hamiltonian vector field and a Poisson bracket. With the aid of these ideas, we can give the Hamiltonian formulation of classical mechanics and establish the following fundamental short exact sequence:
0 -. R - .F(M)
Ham Af -0,
where.F(M) is the space of smooth functions f defined on the symplectic manifold and given the structure of Lie algebra via the Poisson bracket, and Ham M is the Lie algebra of Hamiltonian vector fields on the manifold. The third chapter continues with a brief introduction to contact manifolds. A theory for these manifolds in odd dimension can be developed
Preface
xiv
which corresponds precisely to that of the symplectic manifolds. On the other hand, both may be viewed as pre-symplectic manifolds. Here the connection will be given through the example of a contact manifold as the surface of constant energy of a Hamiltonian system. The fourth and fifth chapters will be a mix of further mathematical constructions and their physical interpretations. This will begin with the description of the moment map attached to the situation of a Lie group G acting symplectically on a symplectic manifold such that every Hamiltonian vector field is global Hamiltonian. This is a certain function
4i:M-+g", g=LieG. The most important examples of the moment maps are the Ad`_equivariant ones, that is, those that satisfy a compatibility condition with respect to the coadjoint representation Ad. The first result of Chapter 4 is that for a symplectic form w = -dt9 and a G invariant 1-form t9 such an Ad`-equivariant map can be constructed. This will then be applied to the cotangent bundle T"Q, as well as to the tangent bundle TQ, where it will turn out that for a regular Lagrangian function L E F(Q) the associated moment map is an integral for the Lagrangian equation associated to L. As examples, we will discuss the linear and angular momenta in the context of the formalism of the moment map, and so make clear the reason for this choice of terminology. Next, we describe symplectic reduction. Here, we are given a symplectic
C-operation on M and an Ad'-equivariant moment map 4i; under some relatively easy-to-check conditions, for p E g', the quotient
Mµ = i-1(µ)lG, is again a symplectic manifold. This central result of Chapter 4 has many applications, including the construction of further examples of symplectic manifolds (in particular, we obtain other proofs that the projective space PI(C) as well as the coadjoint orbits are symplectic). Another application is the result of classical mechanics on the reduction of the number of variables by the application of symmetry, leading to the appearance of some integrals of the motion. In the fifth and last chapter, we consider quantization; that is, the transition from classical mechanics to quantum mechanics, which leads to many interesting mathematical questions. The first case to be considered is the simplest: M = R2" = 7'R^. In this case the important tools are the groups SL2(R), the Heisenberg group Heis2n(R), the Jacobi group Ga (R) (as a semidirect product of the Heisenberg and symplectic groups) and their associated Lie algebras. It will follow that quantization assigns to the polynomials of degree less than or equal to 2 in the variables p and q of R2., an operator on L2(R) with the help of the Schrodinger representation of
Preface
xv
the Heisenberg group and the Weil representation of the symplectic group (more precisely, its metaplectic covering). The theorem of Groenewold and van Hove then says that this quantization is maximal; that is, it cannot be extended to polynomials of higher degree. The remainder of the fifth chapter consists in laying the groundwork for the general situation, which essentially follows KIRILLOV [Ki]. Here a subalgebra p, the primary quantities, comes into play, which for the case
of M = T*Q turns out to be the arbitrary functions in q and the linear functions in p. Here yet more functional analysis and topology are required in order to demonstrate the result of Kirillov that for a symplectic manifold,
with an algebra p in F(M) of primary quantities relative to the Poisson bracket, a quantization is possible. That is, there is a map which assigns to each f E p a self-adjoint operator f on Hilbert space N satisfying the conditions
(1) the function 1 corresponds to the identity idN,
(2) the Poisson bracket of the two functions corresponds to the Lie bracket of operators, and (3) the algebra of operators operates irreducibly. There is a one-to-one correspondence between the set of equivalence classes of such representations of p and the cohomology group HI(M,C'). In the first two appendices, manifolds, vector bundles, Lie groups and algebras, vector fields, tensors, differential forms and their basic handling
are covered. In particular, the various derivation processes are covered so that one may follow the proofs in the cited literature. A quick reading of this synopsis is perhaps recommended as an entrance to the second chapter. In Chapter 2 some material about cohomology groups will also be required. The third appendix presents some of the rudiments of cohomology theory. In the final appendix, the central concept of coadjoint orbits is prepared by a consideration of the fundamental concepts and constructions of representation theory. As already mentioned, somewhat more from the theory of differential equations than is usually presented in a beginner's course on the topic, in particular Frobenius' theorem, is required to fully follow the treatment of symplectic geometry given here. Since in these cases the difficulty is not in grasping the statements, this material is left out of the appendices and simply used in the text as needed, though again without proof. It is not the intention of this text to compete with the treatment of the classical and current literature over the research in the various subtopics of symplectic geometry as can be found, for example, in the books by
xvi
Preface
ABRAHAM-MARSDEN [AM], AEBISCHER of al. [Ae], GUILLEMJN-STERNBERG [GS], HOFER-ZEHNDER [HZ], MCDUFF-SALAMON [MS], SIEGEL [Sill, SOURIAU [so], VAISMAN [V], WALLACH [W] and WOODHOUSE [WO].
Instead we have tried to introduce the reader to the material in these sources and, moreover, to follow the work contained in, for example, GROMOV [Gr] and KIRILLOV [Ki]. In the hope that this will provide each reader with a starting point into this fascinating area a few parts of chapters 1, 2, and 4 may be skipped by those whose interests lie in physics, and one may begin directly with the sections on Hamiltonian vectorfields, moment maps and quantization. This text is, with minor changes, a translation of the book "Einfiihrung in die Symplektische Geometrie" (Vieweg, 1998). The production of this text has only been possible through the help of many. U. Schmickler-Hirzebruch and G. Fischer, on the staff of Vieweg-Verlag, have made many valuable
suggestions, as has E. Dunne from the American Mathematical Society. My colleagues J. Michalilek, 0. Riemenschneider and P. Slodowy, from the Mathematische Seminar of the Universitiit Hamburg, were always, as ever, willing to discuss these topics. A. G6nther prepared one draft of this text, and I. Kowing did a newer draft and also showed great patience for my eternal desire to have something or other changed. I also had very successful technical consultation with F. Berndt, D. Nitschke and R. Schmidt. The last of these went through the German text with great attention and smoothed
out at least some of what was rough in the text. I would also thank T. Wurzbacher, W. Foerg-Rob and P. Wagner for carefully reading (parts of) the German text and finding some misprints, wrong signs and other mistakes. The translation was done by M. Klucznik, who had an enormous task in producing very fluent English (at least in my opinion) and a fine layout of my often rather involved German style. It is a great joy for me to thank each of these.
R. Berndt
Chapter 0
Some Aspects of Theoretical Mechanics
Symplectic structures arise in a natural way in theoretical mechanics, in particular during the process of quantizat.ion. that is. in the passage from classical to quantum mechanics. In order to motivate the study of symplectic geometry. we will begin with a rough sketch of the relevant physics. although
we will not cover all the concepts of this field nor give all of the relevant definitions. As references. one may consult the first chapter of VAISAIA` [V]. A more complete description of the principles of classical mechanics can be found in WOODHOUSE [Wo]. in the third chapter of ARNOLD [A] and in the third and fifth chapters of ABRAHAM-XIARSDEN [AM]. A further highly recommendable classical source is the first chapter of SIEGEL-MIOSER [SM].
For the process of quantization. we refer the reader to §15.4 of KIRILLOV [Ki]. It is the goal of this text to later return and cover the topics of this chapter in greater detail.
0.1. The Lagrange equations The purpose of theoretical mechanics is the discovery of principles which allow one to describe the time development of the state of a physical system. In classical mechanics such a state is given as a point P on an n dimensional real manifold Q (see Section A.1). Q is called the configuration space. and P is described b y the local coordinates q1, ... , q.. called position variables. The time development of the system is then described by the curve 7 = P(t),
t .-- P(t)
with
P(to) = P°. I
0. Some Aspects of Theoretical Mechanics
2
or in the local coordinates as
t'-'qi(t)
gi(to)=q0,
with
i=1,...,n.
Here physical principles must be found which allow one to give the curve as a solution to a differential equation. The starting point for this determination
is the classical mechanical principle of least action. For this it is assumed that the system has a Lagrange function L of the form L = L(q, 4, t),
which is gotten as the difference of the kinetic and the potential energies,
L=Eye - E, which is also written as
L=T-V The principle of least action now says that the change in the system proceeds so that the curve -y minimizes the path integral eJLdt. l to
The variational calculus now says (am Coua4w -HILBERT [CII], p. 170) that for the minimum curve y = q(t) the system satisfies the Euler-Lagrange equations (1)
d 8L
8L
dt 8qi
8qi
0,
This can be seen as a system of ordinary differential equations in a 2n-dimensional space TQ with local coordinates qn,
(which can be understood as the tangent bundle over the configuration space
Q (see Section A.3)). The desired curve y on Q is the projection of the solution curve ry of (1) onto TQ.
0.2. Hamilton's equations Classical mechanics now takes the following formulation: for a given Lagrange function L the coordinates position and velocity, (q, 4), are replaced by the coordinates position and momentum (q, p) made possible by the transformation 8L pi = aqt, i = 1,...,n.
0.2. Hamilton's equations
3
The basis of this concept is the Legendre transformation (see ARNOLD [A]. p. 61 f.) between tangent and cotangent bundles (see Section A.3)
TQ -* T'Q, '--'
(q, 4)
(q, p)
Then the time development described on TQ by the Lagrange function L = L(q, 4, t) (which we can and will assume to be convex in the second argument:
see, for example. ARNOLD ([A], p. 65)) is replaced by the Hamiltonian function H on phase space T'Q defined by with
H(P, q, t) :=P4-L(q,4,t)
p= aQ
where we have used the usual abbreviated symbols for the n-tuple 8L P=(Pi,...,P-),
8L
8L
aq = \aql ..., 57q-
etc.
The Lagrange equations (1) are here translated into Hamilton's equations
8H (2)
q =
ap
8H ,
p = -
.
aq
Because the total differential of H = H(p, q, t) (see Section A.4) gives
dH =
dp + aq dq +
dt
and by the definition H = p4 - L(q, 4. t). we also get
dH = 4dp -
-q dq -
dt.
8L
Comparing() 1 and p = a4 , we get
OH OH OL OH 8L -p. W=-q=-;, aq = - aq 8t .
Hamilton's equations (2) are now (when H is independent of t) a system of ordinary differential equations, which, given a particular set of initial conditions p0, q', gives a unique curve ry' in phase space T*Q whose projection ry onto the configuration space Q solves the original problem. The Hamiltonian function is also written in the form
H=H(p,q,t)=(T)+V, where V is the potential energy of the system and T is the kinetic energy given in terms of the variables q and p.
0. Some Aspects of Theoretical Mechanics
4
0.3. The Hamilton-Jacobi equation Yet another formulation of the problem passes from the solution of a system
of ordinary differential equations to the solution of a partial differential equation. The resulting partial differential equation is the Hamilton-Jacobi equation
H(q,
(3)
9S,t) +
5
=0
for the action function S. Here, giving a solution which is dependent on t, the n variables q, and the n initial parameters a, S = S(q, t, a),
is equivalent to giving a solution q = q(t), p = p(t) of (2). Here we present only the following consideration:
Let S = S(q, t, a) be a solution of (3) with (a82S
k) # 0.
det
Then the n equations as Sae
= be, I= 1,...,n,
in the q; are solvable in the q, = cpi (t, a, b) , i = 1, ... , n. This allows one to write
as
PC=
aqe
as a function of t, a. and b:
pt = 't(t, a, b). These qi, p, satisfy Hamilton's equations (2), since differentiating
(+)
H I q,
Fq s
(q, t, a), t) +
=0
with respect to at gives
k
aH 02S + apk Sae aqk
025 5a-1 -5i
= 0,
t =..., 1, n.
And differentiating as = be with respect to t gives
Ys aqk sac
qk +
a2S at aac
= 0.
0.3. The Hamilton-Jacobi equation
5
Taking the difference of the two equations yields a2S
OH
Oak Oat
OPk
- qk
t=1....,n,
= 0,
I
and since
det {
as ag Oa
)oo
we arrive at half of Hamilton's equations. Next, differentiating (+) with respect to qt gives
OH 02S
02S o OPk Oqk aqt + Oqe Ot =
OH
Oqe +
k
and then, differentiating pe =
as aqt
with respect to t, we get 025
of =
aqk aqe
qk +
O2S
at aqt
Taking the difference of these two equations yields, considering the satisfac-
tion of the relation
= q,
W
OH
Pe=-aqt
There is yet another way (which at first glance looks different) to derive the Hamilton-Jacobi equation (see ARNIOLD [A], pp. 253-5). Here the path integral SgO,tt (q, t) =
J
Ldt
ti
is taken along the curve y from (qe. te) to (q. t) that minimizes the integral, and it is shown that
dS = pdq - Hdt. Then it is immediately clear that for S the equations
as _ hold, and therefore also (3).
q, t)
and
as = P 9
0. Some Aspects of Theoretical Mechanics
6
0.4. A symplectic interpretation Here we continue from Section 0.2. The Hamiltonian function H defines a Hamiltonian vector field XH on phase space Q. Relative to the usual coordinates (q, p), this is defined by (see Section A.4)
XH
:
8H 8
8H 8
"PiNi -
NiBpi
Given a vector field X, the question immediately arises as to the existence of integral curves 'y; that is, curves whose tangent vectors ^ (t) at every point of -y(t) are equal to the given vector of the vector field at that point, which, in symbols then, is ry' (t) = XH ('y(t)). For
7(t) = (q(t), p(t)) this condition leads to Hamilton's equations (2) 8H . 8H
8p = q, 8q = -p-
With the help of a little bit more from the theory of differential forms (see Section A.4), this can be reformulated as follows: there exist a 2-form
w = > dqi n dpi E 02
on T'Q
i
and an inner product i such that, from any vector field X, the 2-form w gives us a 1-form i(X)w. Then Hamilton's equations (2) are equivalent to (4)
i(XH)w = d. H.
0.5. Hamilton's equations via the Poisson bracket On any pair of (arbitrarily often) differentiable functions f, g on phase space T'Q we may take their Poisson bracket { , }, 1 defined by the equation
r of 8g {f,9}8qi
Bpi
8g
Of 8pi8gi
This Poisson bracket endows the space of functions.F(T'Q) with the structure of a Lie algebra (see Section B.1). This will be discussed in more detail later. Here we only note that Hamilton's equations (2) can be written with the help of the Poisson bracket as (5)
¢ = {q, H}, p = {p, H).
l Warning- in the literature (for example, in [KI[) one sometimes takes for if, g} the negative of what is taken here)
0.6. Towards quantization
From this it does not take too much work to see that, generally, for the time development of an observable given by f. the above system must satisfy the condition (5')
f = (f, H).
0.6. Towards quantization The term quantization will indicate the process by which a corresponding quantum system is constructed from a given classical system. Thus we must find a transition from the point in phase space T'Q which describes the state of a classical system to an element v (more accurately. a 1-dimensional sub-
space vC) of a complex Hilbert space f with the aid of the probability distribution for the state of a quantum mechanical system. This transition should give, for the Hamilton function H and the classical observables f. corresponding self-adjoint operators H and f in fl for the quantum situation. Thus one would expect that the equation (5')
f={f.H}. giving the time course in T'Q in the classical case should pass to an analogous equation on the operators
f =c[f,H]j where [ , J is here the natural Lie bracket,
[A.B] =AB-BA. where c = - 2* (c is. up to the factor i, a factor ensuring the symmetry of the operators, a constant from physics, and h is called Planck's constant). We will later examine for which f E .F(T*M) one may define a mapping f i-, f whose images are self-adjoint operators and which satisfies
1'--.1=id7 and
{f1f } = c [fl, f2]
.
With this process considerable problems will appear, but we will finally see that, at least for the so-called primary quantities (that is. f either a polynomial of degree 2 in q and p or a linear function of the pl, ... , p and an arbitrary function of the q , . . . . ,
solutions can be found.
Chapter 1
Symplectic Algebra
The soon to be introduced symplectic manifolds can be thought of locally as symplectic vector spaces. It is therefore necessary to define and study vector spaces with additional structure. This additional structure is given by
a) a scalar product, b) a symplectic form,
c) a complex structure. To begin, we rework some old and trusted linear and inultilinear algebra. As a final result, we will describe all spaces with a given symplectic structure compatible with a complex structure as a Siegel half space. This space is important not only for geometry, but also for function theory, although as to the latter, we will only be able indicate a small part of this importance. The second chapter of VAISMAN: Symplectic Geometry and Secondary Characteristic Classes [V], as well as the third chapter of ABRAHAM--MARSDEN (AM] serve as good guides to this chapter. For additional background, E. ARTIN's Geometric Algebra [Ar) can be recommended.
1.1. Symplectic vector spaces Now we let K be an arbitrary field of characteristic 0. Later, we will restrict to K = R. Also let V be a finite-dimensional K vector space (with dim V = p). Then the basic underlying concept of symplectic space is given by the following definition. DEFINITION 1.1. A symplectie form
w:VXV,K 9
1. Symplectic Algebra
10
is an antisymmetric and nondegenerate bilinear form; that is, it satisfies
w(v,v)=0 forallvEV, and if
w(v,w)=0 forallvEV, then w = 0. A vector space V is called a symplectic vector space if it is equipped with a symplectic form. Remark 1.2. (1) ABRAHAM-MARSDEN [AMJ consider infinite-dimensional symplec-
tic vector spaces, as well. (2) In the case of K = R this definition is parallel to the definition of a Euclidean vector space; that is, to a space having a scalar product, a symmetric positive definite bilinear form which is usually denoted
bysorby(, ).
Let e = (el,...,ey)
be a basis of V. Then the bilinear form w on V in terms of a can be given in matrix form by
wc=(wj)EMp(K)
with wj=w(e:,ei)
For K = R there is a nice classification of the normal forms of symmetric and skew-symmetric bilinear forms. PROPOSITION 1.3. Let V be a p-dimensional R vector space.
i) In the case that s is a symmetric bilinear form of rank r, then is a basis a of V relative to which
with ei = ±1, i = 1, ... , r.
3g =
ii) In the case that w is an antisymmetric bilinear form of rank r, then r = 2n and then is a basis e of V relative to which En 0 0 wr = -E, 0 0 with En E Mn(R) the unit matrix. 0
0
0
1.1. Symplectic vector spaces
11
Proof. i) By a process analogous to the Gram-Schmidt orthonormalization, it can be assumed that since s is symmetric, it satisfies the polarization identity
s(v, w) = (1/4)(s(v+w,v+w) -s(v-w,v-w)). Thus for s # 0, there is an ei E V with s (ei, ei) # 0. ei can be multiplied by a scalar, giving el with el := s (el, el) = f1. Let and
Vi := Rej
V2 := {v E V, s (e, e1) = 0}.
It is then clear that VlnV2={0}, and so V1+V2=V; thenforvEV, v - Els (v, el) el E V2.
One may now continue by induction. So long as s 0 on V2 x V21 we may find an e2 # 0 in V2 with s (e2, e2) = E2 = ±1, and so on. ii) For w:# 0, there must be e1, e,,+l E V with w (el, e,,+1) # 0. After multiplying el by a scalar, it can be assumed that w (el, 1. Since w is skew-symmetric, we have w (ei, el) = w (en+i,
0,
and the matrix for w' in the plane El spanned by el and
01 0
is
.
}
Now let V2 be the w--orthogonal complement of El, so
V2:_{vEVjw(v,vl)=0 forallvlEEl}. We have El nV2={0}and V=El GV2,and for vEV v - w(v,
el) e,,+1 E V2.
Given that w # 0 on V21 one may repeat the above procedure for V2 and so that w (e2, 1. Inductively we get the obtain e2 as well as
0
claimed matrix we.
The statement ii) clearly holds as well for more general fields K # R.
Let V` denote the dual space of V and e` the dual basis to e on V', which satisfies ei (ei) = (ei, a=) = 8 j.
One of the basic results of (multi-)linear algebra (see also Section A.4) is that the space A9(V, K) of skew-symmetric q-linear functions from V9 to K is isomorphic to the qth exterior product A9 V' of V*. AQV ` has as K-basis the set e;, A ... A e,*, with it < ... < iq.
1. Symplectic Algebra
12
In particular, an antisymmetric bilinear form w with the matrix wg = (wig) relative to e can also be written as w = E wi.7 ei A e!.
iv Written this way, w works as a function by sending (v, w) E V x V to w (v, w) = E wi) (e; (v)et (w) - e; (w)ef(v)). i<j The statement ii) can be reformulated as COROLLARY 1.4. By an appropriate choice of basis e, the antisymmetric bilinear form w can be written as n
w=Ee; Ae;
.
i=1
Such a representation will be called the canonical form of w, and e a symplectic basis of V. Then n
w (v, u) = Dxiyi - xiyi), i=1
when for v E V the components relative tog are given by p-2n
n
n
v=
xiei + E yiei+n + : zie2n+i i=1
i=1
i=1
Of particular importance for symplectic geometry is naturally the non-
degenerate skew-symmetric bilinear form w; in this case p = r = 2n, so that V must have even dimension. A criterion for w E A2V' to be nondegenerate is that the nth exterior power wn = w A ... A w of w must be a nonzero multiple of the volume form
T=e;A...Ae,EAPV*. For r, we have P
r (v1, ... , v.) =det(ail)
with vi = E ail ei, i = 1, ... , p. =1
For w = Eel A en+i, we have, in the usual notation, wn = n!(-1)(n/21r, where [x] is the function that returns the largest integer < x for all x E R. In general, (_1) (n/21
n!
w
defines an orientation on V (see ABRAHAM-MARSDEN [AM], pp. 165-166).
1.1. Symplectic vector spaces
13
A symplectic form w makes possible the identification of wb : V
r
V. wb(v).
with wb(v)(v') = w (v, v') for v, v' E V. We will use the letter i for a general inner product i
: V x AqV - Aq-1V'. (v, d) .-, i(v)d,
where i(v)t7 is the (q - 1)-linear function given by i (v)99(VI..... Vq-I) = 19 (v. VI...., vq_I),
and so wb(v) = i(v)w; E V. It follows easily for w. as in Corollary 1.4. that
i(e3)w = wb(ej) = ej+n i(e1+n)w = wb(ej+n) = -e*
j = 1....,n,
J.
Although it is almost trivial, we recommend
EXERCISE 1.5. Given a bilinear form w on V with dine V = m. show that the following are equivalent.: a) w is non-degenerate. b) wb is an isomorphism.
and. in the case that m = 2n.
C) w"= )A...Aw#0. Although these statements are left unproved, we will not hesitate to use them later in the text. In parallel to the situation in Euclidian geometry, we may form the following definition in symplectic geometry.
DEFINITION 1.6. Two vectors v, w from the symplectic vector space (L; w) are called w-orthogonal, skew-orthogonal or - when there is no doubt about which w is intended - simply orthogonal, whenever w (v, w) = 0.
This is also indicated by v 1 w.
1. Symplectic Algebra
14
EXAMPLE 1.7. Ken with the symplectic form w given by w (v, v') _
(xi2 - xyi) for v = xlel + ... + xnen + glen+1 + ... +yne2n, + y'e2
v' = x11e1 +...
with the canonical basis e = (el,... ,
is the standard symplectic space.
Given Proposition 1.3, every 2n-dimensional symplectic space can be described this way. EXAMPLE 1.8. Let W be an n-dimensional K vector space and W* its dual space. Then V = W ® W* is a symplectic space with
w:VxV -+K given by
w (t1 + 71, t2 +7'2) = n(t2) - r (ti) for t1i t2 E W, 71,1 E W*. Remark 1.9. Relative to this last example, note that a symplectic space V has many decompositions V = W ® W*. Let e be a symplectic basis for
V, such that W is the span of el,... , en. Then W is isomorphic, via wb, to the subspace spanned by en+1, ... , e2n; so V = W ®W * with the form defined in the previous example.
EXAMPLE 1.10. There is for every k E N a p = (2n + k)-dimensional K of K vector space U with a skew-symmetric bilinear form w: U x U rank 2n. Then the annihilator of w,
Uo:={uEU; j(u,w)=0 for all wEU}, has dimension k, and i induces a symplectic form w on V := U/Uo. We will say that the symplectic space (V, w) is given by the reduction of (U, w).
1.2. Symplectic morphisms and symplectic groups Just as in the case of a Euclidian vector space, where the scalar product permits one to define orthogonal morphisms, we have, in symplectic geometry, a natural definition of morphism:
DEFINITION 1.11. Let (V,wi) be two symplectic vector spaces and 0 : V1 -+ V2 a linear map. Then we call 0 symplectic whenever (*)
w2 (0 (v), 0 (w)) = w1(v, w)
for all v, w E V1.
1.2. Symplectic morphisms and symplectic groups
15
Remark 1.12. A symplectic morphism is necessarily injective, since if 0(v) = 0, then (*) forces v = 0, since w, is non-degenerate. For dim Vi = dim V2 < oo, 0 must therefore be an isomorphism. Symplectic isomorphisms will be called symplectomorphisms.
When (VI1 w1) = (V2, w2) = (V, w), any symplectic map 0 must be an automorphism of (V; w). The collection of symplectic automorphisms forms a group under the usual composition, called the symplectic group of (V; w)
and denoted Sp(V). In particular, for V = Ken with the standard form, (unfortunately it is also written as this group will be written as in the literature!). are naturally matrices from G" (K). They The elements M E can be written in the following way. For the standard form, we have
=E
,
,
which, with the use of the matrix
J=J
0
-E
E 0
and column vectors v, v' with tv = (xl,... , x, yl,... ,
can also be
written as
w (v, v') = tvJv'. The matrix M leaves this form invariant, that is. W (Mv, Mv') = w (v, v'),
exactly when
(+)
tMJM = J.
Then from simple computation, we get
Remark 1.13. For A, B, C, D E M (K), the following are equivalent:
t)
M = ` C D) E Spn(K),
ii)
tAC ='CA, tBD = tDB, 'AD - 'CB = E, ,
iii)
AtB = BVA, CD = D`C, AtD - BC = E,,.
1. Symplectic Algebra
16
Special symplectic matrices are J as well as
uv
=
(0 V`)
with V* =
(IV)-t,
and
m= (0 E)
withtS=S.
PROPOSITION 1.14. These matrices generate Spn(K).
0
Proof. See EICHLER [E], p. 47.
As a consequence of this proposition, we immediately get that
det M = 1
for M E Spn(K),
since this is clearly the case for the generators. This statement can also be derived from the fact that a symplectic automorphism yfi relative to w must also fix w", the volume form. Therefore, we have
(det0)ei A...Ae2n(e1,..., e2")
e! A... AeL(0e1,
=ejA...Ae (eI,...,e2n), and thus det 0 = 1. PROPOSITION 1.15. Let M E Spn(K) and X an eigenvalue of M with multiplicity k. Then 1/A is also an eigenvalue with multiplicity k. Proof. Consider
P (t) = det(M - tE2"), the characteristic polynomial of M. Then by using (+) and the fact that det M = 1, we have P (t) = det(1M - tE2n) = det (J-' (LM - tE2,=)J)
=
det(M_1
- tE2n) = det M-1 det(E2n - tM)
= t2n det(M - (1/t)Fj ).
0 Remark 1.16. For K = R, if M E Sp,t(K) as a complex matrix has the eigenvalue A E C. then M also has the eigenvalues A, 1/A and 1/A.
These statements about eigenvalues are fundamental to the qualitative theory, including that of the stability of Hamiltonian systems. Here we give just a few comments on the topic of stability (see ARNOLD [A], p. 227):
1.3. Subspaces of symplectic vector spaces
17
DEFINITION 1.17. A morphism ¢, of V is called stable if for each e > 0 there is a i9 > 0 such that
11O vII<e, forallNEN, as soon as 11vJJ<19. EXERCISE 1.18. If 0 E Sp(V) has an eigenvalue A with JAI # 1. then 0 is not stable. EXERCISE 1.19. Whenever the eigenvalues A of 0 E End V are distinct and have norm 1, the transformation is stable. DEFINITION 1.20. 0 E Sp(V) is called strongly stable if every neighbor 41 E Sp (V) (that is, given a fixed basis for V. the matrix entries of 01 differ from those of 0 by less than a fixed e) is stable.
EXERCISE 1.21. If for 0 E Sp(V) all 2n eigenvalules are distinct and have norm 1, ¢ is strongly stable.
The symplectic group is a significant object in both function theory and algebraic geometry. For clarification the reader may wish to consult SIEGEL [Sil, Si2]. Spr(R) is a Lie group and has as associated Lie algebra spn(R) = Lie (see Section B.2). It is easy to see that with the standard form w we have op. (R) = {M E W (Mv, w) + w (v, Mw) = 0 for all v, to E liter }
_ (M E
tMJ + JM = 0}.
Here, as in the last proposition, we have. for Al E sp,,(R) that if A is an eigenvalue then so is -A, and both have the same multiplicity. Particular symplectic morphisms are the symplectic transvections r,,,,a. For W E V and A E R, r,,,,a : V - V is defined by v -* Aw (v, w) to.
EXERCISE 1.22. Show that
a) r,,,\ E Sp(V). b) Sp(V) is generated by the symplectic transvectionss (see JACOBSON [Ja], p. 373)-
1.3. Subspaces of symplectic vector spaces Let W be a k-dimensional linear subspace of the 2n-dimensional symplectic space (V, w). Then k = dimd(W) and 21 := rank wlm(w) remain unchanged under any symplectic morphism from Sp (V). We will show that these two integers, k and 21, classify the subspaces of V, in that they are the only two independent symplectic invariants for subspaces.
1. Symplectic Algebra
18
As in the proof of Proposition 1.3, let
W1:={vEV; w(v,w)=0 for allwEW}. W 1 is called the skew (or w-) orthogonal space of W (and often simply called its orthogonal).
Remark 1.23. We have
dim W -L = dim V - dim W = 2n - k. Then the dimension formula from linear algebra (see, for example, LANG [Li], P. 95), dim V = dim W + dim W ° for
(w)=0foralwEW},
W°:=IV
can be here applied to show that (for an arbitrary non-degenerate bilinear form w) W-L can be identified with W° via wb
Specific to the symplectic algebra is the appearance of the radical rad W of W of the form
radW:=WnW1. This is also given by
rad W= {w E W; i (w) w I w= 0} and satisfies
dim rad W = dim W - rank w I W = k - 21. Clearly, the structure of w-orthogonal spaces has the following properties: Remark 1.24.
a) W-CW,1for W'CW, b) (W-)1=W,
c) (W+W')1=WlnW1, d) (WnW')1=W1+W1. A third space fixed by W is defined by
°W:=W+Wl. The usual dimension formula here says that
dim °W = dim W + dim W1- dim rad W = 2n - (k - 21). From this space we can form yet more spaces, in particular Wr`d := W/rad W,
1.3. Subspaces of symplectic vector spaces
19
the symplectic space associated to W. This satisfies dim R"'d = 21, and for a subspace U C V with
Wr`d = rad W e U, (U, w1 u) is symplectic and isomorphic to Wred. U is also w--orthogonal
to rad W. Therefore, we can also write W = rad W1U. Further, U' with Wl = rad W ® U' is symplectic and isomorphic to the reduced space
(Wl)red = Wl/rad W. The most important examples of subspaces of a symplectic space (V, w) are the following. DEFINITION 1.25.
1) A subspace Q of V with ";IQ = 0 is called an isotropic subspace of (V, w).
2) A subspace W C V with w'N, non-degenerate is called a symplectic subspace of V.
3) A subspace W C V with W1 isotropic is called coisotropic.
4) A subspace L C V which is both isotropic and coisotropic (thus with Ll = L) is called a Lagrangian subspace. Before we discuss Lagrangian subspaces further, we prove the earlier statement that the dimension and rank are the only two independent symplectic invariants of a subspace. This result will be a corollary of the following otherwise useful discussion, which is here taken from AIrrIN's masterful parallel presentations of orthogonal and symplectic geometry in his Geometric Algebra ((Ar], pp. 114 f). A two-dimensional symplectic space P is, according to Proposition 1.3, of the form
P=eK+e.K
with w(e, e)=w(e., e.) =0, w(e, e.) = 1.
P is also called a hyperbolic plane and (e, e.) a hyperbolic pair. An orthogonal sum of hyperbolic planes is then called a hyperbolic space
H.,. = P11.... P,. By the same proposition, a symplectic space is always an orthogonal sum of hyperbolic planes and so is a hyperbolic space. An important step in proving our claim will be the following theorem, which says that every subspace of a symplectic space has a symplectic hull. On the way the notion of symplectic morphism will be extended to a morphism of those subspaces U which are compatible with the restriction of the symplectic form to U.
1. Symplectic Algebra
20
THEOREM 1.26. Let V be symplectic and U a subspace, written in the forIn U = rad U.LW Let e1, ... , er be a basis of rad U. Then there are elements e1., ... , e,.. E V such that P, = e,K + e,.K is hyperbolic (i = 1, ... , r) with
for i
P,1Pj
j, and P,1W for all i.
Then
V DU:=P11...1P,.1W is symplectic with U D U.
An injeetive symplectic morphism 0 from U into a symplectic space V' can be extended to a symplectic morphism : U - V'.
Proof. i) The proof of the first statement goes by induction. For r = 0 there is nothing to show. Now assume the claim is true for r' < r. Let Uo :_ (e1,.. , er-1)1W Then er is orthogonal to Uo, er 0 U0, and rad Uo = rad Uo = (ell ...,er-1).
Consequently, we have er E Uo and er ¢ rad Uo , and so there is an a E Uo f o r which w (er. a) # 0. The plane spanned by er and a is contained in Uo , and in this plane a can be changed to er. so that e, er. is a hyperbolic pair. It follows that
Pr:=erK+er.K is contained in Ua and is orthogonal to Uo, and thus Uo C P, .L. Since dim rad Uo = r - 1, the induction hypothesis may be applied to
Uo = rad Uo1W C P,l, giving the existence of hyperbolic planes in Uo
P:=(ei,ei.)C PT,
i=1,..., r-1,
which are pairwise orthogonal and each orthogonal to W. Since all P; are also orthogonal to Pr, we have that
U:= P11...1pr1W is the desired symplectic space. ii) Let 0 be a symplectic morphism from U to V' and e; W': = 0 (W). Then O (U) = (e, ... , e'r)1W'.
(e,) so that
1.3. Subspaces of symplectic vector spaces
21
From i) there are elements e=. E V so that for P' = (e,, e;,) we have
PJP for i0 j, andP'1W', i=1,...,r. The rule
(e,.) := e;. then gives the desired extension of 0.
A consequence of this theorem is a special case of a theorem of Witt. COROLLARY 1.27. Let V and V be isomorphic symplectic spaces, U C V
a subspace and 0 an injective symplectic morphism from U to V. 0 can be extended to an isornorphisrn
¢:V-.V. Proof. From the previous theorem, 0 can be extended to a morphism U -+ V. Thus we may assume that U is symplectic, and that V = U.LU-.
Let U' := 0 (U) and Ul be such that V = U'J U'1. It only remains to show that Ul and U'l are symplectically isomorphic. But this is clear, since both spaces have the same dimension and no radical, and symplectic spaces of the same dimension are isomorphic.
It is also now dear that dimension and rank of a subspace U C V are the only symplectic invariants; given two subspaces U, with the same rank and dimensional, we can as in the theorem decompose them into rad UU and W;, pair the bases of the radicals and W; against each other, and then use the corollary to extend this map to a svmplectic automorphism of V. Besides the isotropic subspaces, the Lagrangian subspaces are the most important. Let's collect here some of the theory which can now be stated, given what has already been covered. For a subspace W C V with dim W = k, we have
W isotropic W
coisotropic
W Lagrange
WCW
k< n,
aWDW1
k> n,
e' W = Wi
k = n.
A Lagrangian subspace L can thus be described as maximal isotropic. We then also have (with the identification V V' via wb as in Section 1.1)
V=LeL1=V'. Remark 1.28. Let W be a coisotropic subspace of V. Then the image of L n W under projection to W,Yd is a Lagrangian subspace. EXERCISE 1.29. Prove this last remark. (See VAISMAN [V], p. 35.)
1. Symplectic Algebra
22
DEFINITION 1.30. Denote by G (V) the collection of Lagrangian subspaces L C V.
From the previous corollary, Sp (V) acts transitively on L(V); that is, for any pair LI, L2 E G (V), there is a ¢ E Sp (V) with
0 (LI) = L2 In such a case, it is a general fact that L (V) has the structure of a homogeneous space. The concept of homogeneous space will later play a very important role (see Section 2.5). We will briefly make its acquaintance here with a special example. Denote by GL the isotropy group of L EC (V); that is,
GL :_ {O E Sp (V), 0 (L) = L}.
Let e = (e;, ej.)f=l,...,,, be an L-related symplectic basis of V; that is, (el,... , e,) = L and (el., ... , e,,,.) = L1. Then, relative to this basis, V K2R, Sp(V) ?d L Lo :_ (ei,...,en), where (et) are the canonical unit vectors inK2ia, and
GL '-
GL.
=I( A B \I ;BD ='DB,'AD =
This is because M(Lo) = Lo. and thus
\C D/\0/EL° forces C = 0, so that from Remark 1.13 about the general form of symplectic matrices we get the claimed transitive action. From this we get a bijection between C (V) and the set of cosets of GL,.
in Sp (K); thus Sp,,(K)/GLO 2L (V), M i--> M(Lo), because, from
M(Lo) = M'(Lo), it follows that
M-IM' E GLo. In a similar vein, we can describe another family of interesting subspaces of a given symplectic space. Fix T (L), for a given Lagrangian subspace L C V, to be the collection of all Lagrangian subspaces L' transverse to L, so that
T(L):= {L' EL(V), LPL'=V}. The proofs already given show that GL operates transitively on T (L). Take
(e=) to be a basis of L and (e;.) a basis of L' (so that both together form a symplectic basis of V). Then, relative to this basis, the isotropy group
1.3. Subspaces of symplectic vector spaces
23
GL,L' C GI C SpV that fixes L and L' can be identified with the group of matrices \ A A E GL,,(K), 0 f ' 0 iA-I/ C
which is naturally isomorphic to GL"(K). We then get that 7 (L) GL/GL"(K),
In VAISMAN [V], pp. 36-38, the following statement and a few of its consequences are discussed.
THEOREM 1.31. T (L) has a natural structure as an affine space over K of dimension n (n + 1)/2.
An affine space is a triple (A, V, ir), where A is a set, V is a K vector space and 7r : A x A -* V is a morphism, such that point ao E A, is a bijection and for all a, b, c E A we have
1rIfor a fixed
ir(a, b) + a(b, c) = ir(a, c).
This definition can easily be brought into agreement with the definitions which often appear in elementary textbooks. Of the proof which is contained in VAISMAN [V], we give only a sketch:
By choosing the earlier required basis, the matrix of a morphism which carries L' E T (L) to L" E T (L) has the form
(A
B
has the form
which then modulo GL(En L, = 0
E,
l
with X ='X.
This shows already thatT (L) rel/ative to a fixed basis of V is in bijection to a set of symmetric n x n matrices, and therefore can be seen as an (n (n + 1)/2) -dimensional affine subspace of K"2. This observation can be made basis-independent, in that the symmetric matrix X can be defined as a coordinate-independent quadratic form q on the dual space P. The following remarks can then be established:
Remark 1.32. For any pair (L, L') of Lagrangian subspaces of (V, w), there exists a mutually transverse Lagrangian space L". Remark 1.33. Let L, L', L" be Lagrangian subspaces of (V, w), and let L fl L' = L fl L". Then there exists a (not unique) symplectic transformation of V which fixes every vector of L and carries L' to L".
EXERCISE 1.34. The reader would benefit from supplying proofs for these remarks (which in any case can be found in VAISMAN [V]).
1. Symplectic Algebra
24
1.4. Complex structures of real symplectic spaces Up to now, we have considered only R2n equipped with
i) the canonical Euclidian structure with the form 2n
s (v, w) := (v, w) = tvw = > vow'
for V, W E R2n (as columns),
j=1
ii) the canonical symplectic structure with the form n
w (v, w) = =vJnw = E(Vi wn+i - Vn+i wi) i=1
Now, as R vector spaces, R21
v= 1
Cn.. With the identification X
the operation z .+ iz corresponds to the operation v '-. -Jnv = This operation,
Jn : R2n -- R2n
with Jn = - id,
supplies R2n with a complex structure. It is then natural ask in how many ways R2n or, indeed, an arbitrary R vector space, can be supplied with a complex structure. This question will be considered further below. For now, we give a brief survey.
The invertible linear morphisms of R2n, that is, the invertible matrices
M, that i) preserve the canonical scalar product, s, form the orthogonal group O(2n); ii) preserve the symplectic standard form, w, form the symplectic group Spn(R);
iii) preserve the complex structure J, that is, with MJ = JM, are of the form
M-
(X -Y YX
and form the general linear group GLn(C) via M i-. X + iY.
It can be shown that O(2n) n Spn(R) = Spn(R) fl GLn(C) = GLn(C) n O(2n) = U(n). The unitary group U(n) then preserves the hermitian scalar product h (v, w) = s (v, w) - i w (v, w).
1.4. Complex structures of real symplectic spaces
25
As a consequence of earlier material, we have, for the space of all Lagrangian subspaces of R2n. L (R 2")
U(n)/O(n),
and the space of all positively compatible complex structures J on R2n is in bijection with
s7n = {Z E Ill"(C): tZ = Z. Im Z > 0} = Sp"(R)/U(n). All of this will now be discussed in greater generality (following VAISMMAN
((V], pp. 40 ff.). DEFINITION 1.35. Let. V be an 1EY vector space. J E Aut V is called a complex structure on V if and only if
J2 = -id%,.
In the case that V is symplectic with the form w, then we call the complex structure J compatible with w if w(Jv, Ju') = w(v. w) for all v. w E V.
Slightly changing the notation as in the previous example. (V, J) with an arbitrary J can be made into a C vector space via V"--l V := Jv.
Further. J can be extended linearily to the complexification
6', :=VSRC. Then J has the eigenvalue fv=l, and from Jw = Aw we deduce that
-w=J2u+_AJu,_.A2u'. The eigenspace of \1.2 = f
is n-dimensional, and is given by
V+:={v-%/---I JvvEV},
V :={v+v/ I Jv. vEV}.
We then have that
V, =VC+ gV,, and
c- V - y-IJV defines a C vector space isomorphism between (l:
J) and (1' . ).
In the case that J is a complex structure compatible with the symplectic form w. we have
g(v, w) := w(v. Jw)
for v, w E V.
From the compatibility of J and w we have also g(Jv, w) = w(v, w)!
1. Symplectic Algebra
26
and from J2 = -1 and the skew-symmetry of w we also get g(v, w) = g(w, v) and
g(Jv. Jw) = g(v, w). Therefore g is a symmetric bilinear form, and, like w, is also non-degenerate. g will be called an w-compatible pseudohermitian metric. When g(v, v) > 0 for all v E V, we call g a hermitian 1 metric, J a positive compatible complex
structure and the triple (V, u;, J) a Knhler vector space. THEOREM 1.36. Every real symplectic vector space (V, w) can be given a compatible positive complex structure J and a hermitian structure g. Any two such structures Jo and Jl are homotopic in the following sense: there
is a differentiable family Jt, 0 < t < 1, of positive compatible complex structures on V defining a path from Jo with Jl. Remark 1.37. An analogous statement holds for Hilbert spaces having a skew-symmetric weakly nondegenerate bilinear form w. The following reasoning for the general case reduces the proof of ABRAHAM-MARSDEN
((AK, p. 173) to our simpler case. Proof. Let 7 be a Euclidean scalar product on V. and let A : V
V be
defined by
y(Av, w) = w(v, w) Since w is skew-symmetric, we have
for all v, w E V.
y(Av, w) = -w(w, v) = -y(v, Aw), and so
y(A2v, w) = --y(Av, Aw) = y(v, A2w); this is to say that A2 is self-adjoint and, since
y(A2v, v) = -y(Av, Av) < 0 is negative, must have negative, but not necessarily distinct, eigenvalues -A j2, Ai > 0 (j = 1,... , 2n). V then has a y-orthonormal--basis a of eigenvectors for A2, al, ... , a2,,. Let B E Aut V with Ma(B) = A2n
Then B is the unique self--adjoint positive operator with
B2=-A2. 'This notation will later be clarified: It will be shown that g can be extended to a hermitian metric on the complexificatlon of V.
1.4. Complex structures of real symplectic spaces
27
For J := AB-i we have, when we consider the eigenspaces of A and B,
J = B-1A, and thus
J2 = AB-i. B-1A = -E. This J is compatible with w, and so
w(Jv, Jw) = w(AB-iv, AB-'w) --y(AB-ivy A2B-'w) = y(A2B-lv, AB-1w) = = -w(B-iv, A2B-iw) = w(B-iv, Bw) = w(BB-iv, w) = w(v, w). for g with
g (v, w) := w (v, Jw), and we have
g(v, v) = w(v, Jv) = y(Av, Jv) = -y(v, AJv) = y(v, Bv) > 0 by the construction of B. This J satisfies the requirements of the theorem. Since it is dependent on the chosen scalar product y, we write J = J.. It can be seen that every positive compatible complex structure J arises in this manner from some such y (one need just note that J = J. with g from (*)). The last statement of the theorem is now easy. Let Jo and Ji be given; then they are of the form J.,o and Jy, , and can be carried from one to the other via the family Jy with
yt:=tya+(1-t)7i (O
V = RZ", with
w (v, w) _ `rvJnw,
Jn = and
`
-1 q),
(v, w) _ 'vw,
/
we clearly have A = -Jn. So A2 = -E, B = E, and so J = -Jn gives the positive complex structure, and g (v, w) = 'vw.
1. Symplectic Algebra
28
The scalar product described above is called the hermitian metric, since g induces a hermitian bilinear form gc on the complexification Vc of V by
9c(vCiv,w)=-9c(v, f Tw)=vi9(v,w)
for v,WEV.
Then there is the restriction of gc to V.
gc(v - v/--l Jv, w -
Jw) = 2 [9 (v, w) - viw(v, w)],
and then h (v, w) := g (v, w) - %/'--I w(v, w) for v, to E V is a usual hermitian metric on (V, J) as C vector space. This immediately gives us
Remark 1.38. The map V D v ,-+
-(v - vr1 Jv) E V+
is an isomorphism of the hermitian spaces (V, J, h) and (V+, 9c).
From linear algebra (V, J, h) has an orthonormal basis relative to h, thus a C-basis (ej ), j = 1, ... , n, with h (ej, ek) = bjk. Since h = 9 - vl'--l w and g (v, w) = 9 (Jv, Jw) = w (v, Jw), this is equivalent to g(ej, ek) = 9(Jej, Jek) = bjk, g(ej, Jek) = 0. From this we may conclude that (ej, Jej; j = 1, ... , n) is a real g orthonormal basis of V and, with ej. = Jej, the family (ej, ej.), j = 1, ... , n, is a symplectic basis of V.
In the other direction, if (ej, Jej) is a real unitary basis of (V, w, J), that is, a g orthonormal basis and at the same time a symplectic basis, then (ej) is also an h-basis, and we have a corresponding C-basis of V+ given by This is called a complex unitary basis. The automorphisms of the structure (V, w, J) which leave the symplectic form w, as well as the metric g (or, equivalently, those which commute with
J), fixed are called the unitary transformations and generate the unitary group U(V, J). These automorphisms are also characterized by carrying unitary bases into unitary bases. Other equivalent descriptions are given as the complex-linear transformations of (V, J) which fix the metric h, or (see the last remark) as the group of complex linear transformations of V+ which fix the metric gc. By fixing a basis one may derive the relationships between matrix groups as described at the beginning of this section: The choice of a unitary basis (ej; j = 1, ... , n) allows one to identify (V, J, h) as
1.4. Complex structures of real symplectic spaces
29
well as (V+, gj with C", where h in the complex coordinates (zj) relative to (ej) is the canonical hermitian metric
h(z,z')zizi. 1=1
Here we then have that
U(V, J) c U (n) = {U E GLn(C); UtU = En}. On the other hand, for the Euclidian vector space (V, g), O(V, g) is the orthogonal group of linear isomorphisms which fix g. With the help of a g orthonormal basis this group is identified with O(2n). The above discussion can be summarized in the following proposition. THEOREM 1.39. Let (V, w) be a real symplectic space and J a positive compatible complex structure. Then U(V, J) = Sp (V) fl O(V, g).
In particular, for V = R2n with
the canonical basis e = (ei, e;,) and the coordinates (
v=
Vi.
the symplectic standard form n
w(v, w) = tvJw = E(vi wi. - vi. wi), i=1
the canonical scalar product
9 (v,w) = tvw = Dviwi + vi. wi=), i=1
the complex structure defined by Jei = ei., Jei. = -ej, vi. (i = 1.... , n) satisfy U(n) = Spn(R) fl O(2n).
then the complex coordinates zi = vi +
This leads to the following alternative formulation of the description given in Section 1.3 of the space G (V) of Lagrangian subspaces L C V in the case that V has a compatible positive complex struture J. This arises from the fact that in the above introduced notation, any g orthonormal basis (ei)i=1,...,n of L (ei, Jej) gives rise to a real unitary basis of V. Each such basis is called an L-related unitary basis. For L' E L (V) with L'-related
1. Symplectic Algebra
30
basis (e!, Je,) there is a unitary transformation which takes (ei, Jei) to (e;, Je,), and so L to V. The unitary group U(V. J) operates transitively on G (V), and the isotropy group which fixes an L is the group which carries L-related bases to themselves and is thus isomorphic to the orthogonal group O(L,g). Thus we have shown THEOREM 1.40. We have that
,C(V) .,, U(V, J)/O(L, g) '=' U(n)/O(n).
Now we will fulfill the promise made at the beginning of this section; that is, we describe the collection
9=J(V,w) of all compatible positive complex structures J on real symplectic space (V, w), and so the set of possible ways (V, w) can be made into a Kahler vector space. From this it will follow that 3 can be identified with the Siegel upper half plane 15n ^-' Spn(R)/U(n)
From Remark 1.37, for a fixed J E J, the space (V, J) can be identified with a subspace V+ of the complexification of V. Here V+ is a Lagrangian subspace of V, where the symplectic structure w of V is linearly extended to V. This is because we have
w(v -
Jv, W -
Jw)
= w(v, w) - w(Jv, Jw) -
(w(Jv, w) + w(v, Jw)) = 0.
Furthermore, since g(v, w) := w(v, Jw),
-/ w(v, `v) = 2g(a, a) > 0 for 0 96 v := a -
Ja E V+,
and, additionally, Jv = -,./--l(vl - v2) for
V=V+$Vc 9v=vl+v2, Then the Lagrangian subspace F of (V, w) is called positive if
forall0&vEF. Remark 1.41. There is a natural bijection between 9 = 3(V, w) and the collection L+ = L+(V,,) of positive Lagrangian subspaces of (Va, w).
Proof. i) The mapping 3 - G+ is defined from the map J - V+ ii) For F E G+ F is also a Lagrangian subspace of V, since w is real. From the positivity of F, it follows that F fl F = {0} and so V = F (D F. Now we can define J : V -+ V by
vl,v2EF
1.4. Complex structures of real symplectic spaces
31
Then J2 = -id, w (Jv, Jw) = w (v. w). for all v, w E V, and J (V) = V, and so for v E Vc we have the bijection
vEVav=v. Finally, we have
w(v, Jv) = -2vr--1w(vi, 1Y1) > 0
for 0 34v = vl + v1 E V,
and so J E J(V, w). iii) The morphisms 3 -. G+ from i) and L+ - J from ii) are clearly inverse to one another.
A Lagrangian subspace L. of a complex symplectic space VV is called a real Lagrangian subspace, if it is the complexification of a Lagrangian subspace L C V; that is, L, = L OR C. This is satisfied precisely when Lc is carried to itself by complex conjugation. that is. when Lc = L. In V c. all real Lagrangian subspaces Lc are transversal to every F E C+. This is because for
00v=a+vr 1bELcnF witha,bEL we have
0<- /
u; (v, v)_-2w(a,b)=0.
and so also the converse. The remainder of the treatment in VAISMAN [V] is based on the description of the space T (L) of Lagrangian subspaces transverse to a fixed Lagrangian subspace as an affine n (n + 1)/2-dimension space, as given in Theorem 1.31. Since this description was not fully given here, we can only give a sketch of VAISMAN'S treatment, but enough to see the idea: Let L, be a fixed real Lagrangian space. Then, as described above.
C+ = C+(ti') C T (Lc). That T (LJ is an affine space means that for each
pair F E C+ and L' E T (La) with L' real and transversal to L, (thus Lc E3 L' = V) there is a symplectic transformation 0 which fixes every point
of L, and carries L' into F. If e = (ei, ei.) (e; E L. ei. E L') is a basis of V, respectively V. appropriate to L. then ¢ (see Theorem 1.31). relative to this basis, is written as a matrix l
,, : I
with ZEM,, (C), Z=1Z.
Then det Z 96 0, since F is /also transversal to L. The positivity of F sets yet another condition on Z. And thus qi can be written, relative to e, as.
((ei), (ei.)) (0 Z ) = ((e,). (ei.) + (ei)Z),
I. Symplectic Algebra
32
and so
n
O(ei) = ei, 0(ei.) = ei. +EejZiji, (Z;j) = Z. j=1
In fact, 0 (ej.) is a basis of F, and a small calculation shows that F is positive, and so
forOOvEF exactly when
Ire Z = 2 V--1 is a positive definite matrix, which can also be written simply as Ire Z > 0.
This shows that, with a choice of a basis e, the mapping F - Z gives a mapping from C+ into the so-called Siegel upper half plane J
, = {Z E MM(C), tZ = Z, Im Z > 0}.
With a bit more care and the help of Remark 1.41, one can show THEOREM 1.42. For a real 2n-dimensional symplectic space (V, w), the set 3 (V, w) of positive compatible complex structures on V can be identified with the Siegel upper half plane fin.
The identification of the theorem depends on the symplectic basis a of V. Should the symplectic basis be changed to e, then, since (*)
(e,e.)=(e,e.)(C
D)
with ( C D
,
there is a matrix Z associated to F relative toe with the property that F has the bases e. + eZ as well as e. + eZ. So there is a A E CLn(C) with e. + eZ = (e. + eZ)A. Applying the transformation formula (*), we get
eB+e`.D+(eA+e.C)Z=e,A+eZA, and thus
A=CZ+D and Z=(AZ+B)(CZ+D)-1. From complex function theory, the mapping
Z I. Z = (AZ + B) (CZ + D)-' =: (
C
D) (z)
is recognizable as a complex-analytic automorphism of $,,.
1.4. Complex structures of real symplectic spaces
33
We have just seen that operates on gj,,, and previously we have established the bijection t (V) = Therefore it is not so surprising that b,,, and hence ,7(V, w), can be described as a homogeneous space. It turns out that .7(V, w)
S7n ^-' Sp,(R)/U(n).
Thus the operation 0 E Sp(V) can be extended to an operation on V,; which
is also called 0. This 0 then also operates on C+, and gives a transitive operation of Sp (V) on C+. This is because the equality
h(v, w) :_
(*),
w)
for v, w E F from C..
allows one to define a hermitian metric h on F, and for an orthonormal basis of F, we have that (A, -vi74);_1,,.. is then a symplectic basis of V. Analogously, for F' from C+ with h orthonormal basis (f,), (f j',
-v -1 f i)i=1,..,,,, is a symplectic basis of V,. Then there is a d E Sp (Ve)
with
f;.
(f=) = A',
This 0 maps F to F' and commutes with complex conjugation, and therefore is in Sp (V ). Thus, the transitivity of the operation of Sp (V) on G+ is
demonstrated. The isotropy group of F in Sp(V) is, because of (*), the unitary group U(F, h). And so we have shown THEOREM 1.43.
.7 (V, w) = C+(V) ^' fn = Sp(V)/U(F, h) =' Sp.,(R)/U(n) EXERCISE 1.44. The reader is recommended to show directly (that is, independently of what has come before) that:
a) G = SL2(R) operates on bi = {r = x + iy E C, y > 0} by
(M, r) -. M(r) := ar
for M=[ a d
+d
f E SL2(R) and r E 551,
b) The following isomorphism holds:
Sp1(R)/U(1)
S51?` SL2(R)/SG(2)
c) The Poincare metric ds2 =
dx2 +dye y2
is SL2(R)-invariant.
34
1. Symplectic Algebra
The Siegel half space $ has many uses in the area of moduli problems of Abelian varieties. It can be given the structure of a complex manifold of dimension n(n + 1)/2 (see SATAKE [Sa], p. 78). The study of the geometry of these manifolds and the holomorphic, as well as the meromorphic, functions on 15 with known invariant or covariant properties under the operation of the group or its subgroups was initiated by SIEGEL (see his Symplectic Geometry [Sil] or Topics in Complex Function Theory [Si2]); for some time it was exactly these topics introduced by Siegel that formed the subject of symplectic geometry. One may find a new and particularly nice treatment of these traditional topics in MUMFORD: Tata Lectures on Theta, Vol. 2 ([Mul]). Nowadays, however, symplectic geometry refers to a much broader range of topics, which we will consider in the next chapters.
Chapter 2
Symplectic Manifolds
Symplectic geometry arises from the globalization of the symplectic algebra
considered in the previous chapter. The central concept is that of a symplectic manifold. We begin, in this chapter, by defining these, and continue by studying some of their properties and by giving several examples. A good reference for this material is the second chapter of AEBISCHER et al. ]Ae] and Section 3.2 of ABRAHAM-MARSDEN [AM].
2.1. Symplectic manifolds and their morphisms In what follows we will assume that M is a smooth manifold of dimension p, that is, a C°°-manifold in the sense of Section A.1, and assume, as well, that it is real, unless something is said to the contrary. DEFINITION 2.1. M is called a symplectic manifold, if there is defined on M a closed nondegenerate 2-form w; that is, an w E c12(M) such that
i)dw=0, ii) on each tangent space T,,,M, m E Al, if
w(X, Y) = 0 for all Y E TmM, then X = 0.
The assumptions on w say that its restriction to each m E M makes the tangent space TmM into a symplectic vector space. Thus it is already clear that the dimension p of M is even; thus p = 2n. In the next section, it will be shown that all symplectic manifolds of the same dimension are locally the same. This is in sharp contrast to the situation in Riemannian geometry, and indicates that symplectic geometry is essentially a global theory. 35
2. Symplectic Manifolds
36
However, not every even-dimensional manifold has a symplectic struoture. In AEBISCHER et al. ([Ae], p.3) is given the example of M = S4; however, this uses a cohomological result (see Appendix C), which we should not yet go into. Given two symplectic manifolds (M, w) and (M', w'), let F : M M' be a smooth map, that is, differentiable in the sense of Section A.I.
DEFINITION 2.2. The map F is called symplectic, or a morphism of symplectic manifolds, so long as
rw' = w. Given a symplectic diffeomorphism F, F-I is also symplectic, and F is called a symplectomorphism. Sp (M) denotes the group of symplectomorphisms from M to itself.
2.2. Darboux's theorem Darboux's theorem has, in its simplest form, the following formulation. To every point m of a symplectic manifold (M, w) of dimension 2n, there correspond an open neighborhood U of m and a smooth map
with rwo=wIu, where wo is the standard symplectic form on R. It follows immediately that for an appropriate choice of symplectic coordinates x = (q, p), p = (pl, ... , pn) and q = (q,, ... , qn ), w can be written on U in the form n
w = dgAdpdq,Adp;. i=1
In AEBISCHER et al. ([Ae], p. 17) a proof is given which follows that of GUILLEMIN-STERNBERG ([GS], pp. 156ff). We will use this proof here
and supply somewhat more detail. It does go back (as the proof in [AM], p. 175) to MOSER (1965) and WEINSTEIN (1977). As a preliminary we give a standard result on the behavior of parameter-dependent differential forms
under transformations, which will be useful in other places. Although the details are somewhat tedious, its proof will provide a good exercise in the calculus of differential forms described in Section A.4. LEMMA 2.3. Let M and M' be smooth manifolds and Ft : M M', t E Ht, a smooth 1-parameter family of maps. Let Xt denote the tangent field of M' along Ft; that is, Xt is the map
M - TM', m
'--i (m', Xt(m)), m' = Ft(m),
2.2. Darboux's theorem
37
where Xt(m) is the tangent vector at m' = Ft(m) E M' to the curve s F,(m). Let (at)tER be a 1-parameter family of differential forms on M. Then we have
3i (Ft at) = 1 t {( tt + i(Xt)dat) + d(FF (i(Xt)oe))
= Fi (4dt t + i(Xt)do,t + d(i(Xt)at)) so long as all the maps Ft are difeomorphisms from M to M' and, therefore, Xt gives a vector field on M'.
The derivative gi applied to a one-parameter (t) differential simply means the differentiation of the coefficients with respect to that parameter. In any case, we use the usual notation for differential forms (see Section A.4). An exception is the symbol Ft (i(Xt)at), whose meaning as a differential form on M must be clarified in part b) of the proof, since with the general preconditions Xt is not a vector field on M' and therefore the symbol i(Xt)at does not give rise to an inner product in the usual sense. Proof. a) The statement will first be proved for the special case M =
M' = N x I, where N is an n-manifold, I is an interval, Ft = vt with r(it(x, a) = (x, 8 + t) for x E N, and 8 E I. A differential form at on N x I of degree k, which depends on x, a and the parameter t, can be written as at = ds A a (x, a, t) dxk + b (x, s, t) dxk+i where the coefficients are given in the abbreviated form a(x, a, t) dxk
ai,...ik (x, 8, t) dxi, A ... A dxik. i1G..<4
With this, we clearly have
ytat =dsAa(x, s + t, t)dxk+b(x, s+t, t)dxk+i and so
(1)
d (iP at) = ds A T (x, s + t, t)dxk + -(x, s + t, t) dxk+i dt 8s as
(x,a+t,t)dxk+
+dsA
5i(x, a+t, t)dxk+i
We immediately deduce that (2)
ti ( j- J = ds A
(x, s + t, t) dxk +
(x, s + t, t) dxk+l
2. Symplectic Manifolds
38
The vector field Xt, in this special situation, can be written as Xt = $ And so we then have
a ) of = a(x, s, t) d?
i(Xt)crt = i ( and
d (i(X1)ot) = d
(a(.)(xIs
t ) dx, A ... A
(')
_
dii)
(8am (x, s, t) ds A dxt, ... A dxtk (i) s)
+
jOl...ik)
ax 7-
(x, s, t) dxj Adxt, A... AdxiR},
which can be written more briefly as d (i(Xt)ot) = as (x, s, t) ds A dxk + dsa (x, s, t)
dxk+1.
Pulling back by ;t, we get (3)
t,I
d (i(Xt) at) =
84
as
(x, s + t, t) ds A dxk + 4a (x, s + t, t)
dxk+1_
With the same notation, dot = -ds A 4a (x, s, t) dxk+1 +
8b (x, s, t) ds A dxk+l + 4b (x, s, t) dxk+2 8s
is then i (Xt) dot = -4a (x, s, t)dxk+1 + 8 (x, s, t) dxk+1 and
(4)
rJlt i (Xt) dot = -dxa (x, s + t, t) dxk+1 + 8 (x, s + t, t) dxk+1
Adding (2), (3) and (4) and equating this with (1) gives the claim for this special case.
2.2. Darboux :s theorem
39
b) The general case will now be derived from the case a). via the following decomposition:
Ft=Foi,toj with j:A1-Al xI. m,-+(m.0),
t;'"t:AMxl-Alxl. (m, s) -. (m. S + t).
F:MxI - A1'. (m, s)
F,s(m)
given by the diagram
The image of the curve in Al x I given by s (m, s + t). which for s = 0 passing through (m, t) is the curve s'- F,,+t(m) passing through Fj(m) = m' at s = 0. The tangent vector X1(m) from the tangent field along Ft at the point m' = Fi(m) = F(m, t) is then the image of the tangent vector to the curve {(m, s+ t), s E I}, which was previously given as a4; that is. we have (5)
(F) m.t) (
X1(m)
To prepare the key conclusion, we only need to trace a vector 17 = Xm ETmM
through the diagram which gives Ft as a composition. And so n is carried
by j. to (6)
(,1, 0) E T(m,o)(A1 x I),
by (0t).j to (7)
(77, 0) ET(m,t)(M x I),
2. Symplectic Manifolds
40
and finally by (Ft). = F.(0t).j. to (Ft)*+n(0) E TFe(m)M.
(8)
By Lemma 2.3, in its sharper statement that all the Ft are diffeomorphisms
from M to M', Xt can be understood as a vector field on M', and for a given (k + 1)-form at on M' it gives meaning to i (Xt) at as a k-form on M'. Under the general weaker statement, Ft (i (Xt)) at is also meaningful
T
and can be thought of in the f o l l o w i n g w a y.
define
Ft
(i(Xt)0t)m..,r/k)
:= (at)Ft(m) (Xt(m), (Ft)*m(th), ...,
and, with the presumption that m' = Ft(m) = F(m,t), this with (5) and (8) gives
Ft (i(Xt)at),n(rll,-..,t1k) (at)F(t,m )
8 (m,t)
(F*)(m.t)
I
(m,o),...,(F.)(m,t>(rlk,0)
,
Then pulling back by F to (m, t) gives
r, (i (Xt)at)m(tfl,-..,thk)
_ (rcl)(mt),...,(tik,0) (Ll(m,t)(mO) _
:
8 88
F*at
((rh, 0), ...
0))
and then pulling back by rot to (m, 0) gives, with (7),
Fi (i(Xt)at)m(m,...,nk)
l
= tyi (i (88) F*at)
((rh,0),...,(rlk,0)),
II (m.0)
and, pulling back by j to m,
Ft (i(Xt)at)m(th,...,r/k)
_ (j*tyi) (i (8s) (rh, F'at) (m,0)
,tlk)
2.2. Darboux's theorem
41
This means that
Fi (i (Xt) at) = j`C, (i (8s) F'at)
(9)
and so
Fe (i(Xt)dat) =j''ipi (i (8s) d(F'at))
(10)
.
Now, since j' and F" are independent of t, we get d Wt
(Ft at) =
d Wt
(j*
d
F'at) = je dt(tl'i F'at),
and applying part a) of the proof to F'at we get. since d j* = j'd, d . (Ft dt
at) =j 't
d(F'at) dt
+?otGt
(: (as) d(F'ot)) +j'd
(i-)
F'at))
and further, from (9) and (10),
dt(Fiat) =j'tP;F`
t +j'tjiF'(i(X1)dat)
With this the claim is shown. Under the stronger condition that all the Ft are diffeomorphisms, this can also be written more easily as
dt(Ft
at) = F1
(dol
+ i (Xt) dat + d (i (Xt)at))
.
0 Remark 2.4. This formula degenerates when M = M'. at is independnent of t and Ft is the flow arising from a vector field X on M. Then the formula for the Lie derivative of a along X is
Lxa = i(X )da + d(i(X )a).
EXERCISE 2.5. Prove the remark. The above notion of a flow Ft to a given vector field X on M will be used later. Here we will only reproduce Theorem 8.1 from STERNBERG ([St], p. 90), which says that for every mo E M there exist a neighborhood U of mo,
an e > 0 and a family of differentiable maps Ft : U M with
42
2. Symplectic Manifolds
i)
F: (-E,e)xU - M, (t, m)
Ft(m) is differentiable,
ii) for Iti, IsI, Is + tj < e and m E U with Ft(m) E U, we have Fa+t(m) = FF(Ft(m)), iii) form E U, Xm is a tangent vector at t = 0 to the curve t ,-+ Ft (m). For the further properties we will later need we refer to Section A.4 as well as to ABRAHAM-MARSDEN ([AM], pp. 61-67).
The result stated at the beginning of the section is then a direct consequence of the general theorem: THEOREM 2.6. (Darboux's Theorem) Let wo and wi be two nondegenerate and closed forms of degree 2 on a 2n-dimensional manifold M with wok,,, = wi Im for some m E M. Then there exist a neighborhood U of m and a difeomorphism F : U -. F (U) C M with F (m) = m and F`wl = wo.
Proof. The idea of the proof is to use a deformation argument to get a neighborhood U of m and a family (F()tEj, I = [0,1], of diffeomorphisms from U to Ft(U) such that Fo = id,
F1 = F,
Ft (m) = m, and Ftwt = wo, for all t E I with wt :_ (1- t)wo + twi, and so, in particular, F'wl = wo. The Ft are realized as flows to the time dependent vector field Y on Ft(U) with da t (m') = Yt(m`)
(o)
for all m' E U.
The determination of Yt and thus of Ft will proceed in several steps. a) F om the equation wo = 1 t wt it follows from Lemma 2.3 as a necessary
condition that 0=
dt
(Ft wt) = Ft
(awl + i(Yt)dwt + d(i(Yt)wt))
.
2.2. Darboux's theorem
43
Since wo and wl are closed, so are all wt; that is, dwt = 0; the necessary condition is equivalent to
dtwt = "0 - wl =: a.
d(i(Y)wt)
(*)
This will now be taken as the defining equation for Y.
b) Since a is closed, there are, by Poincare's lemma, a neighborhood U1 of m and a 1-form a on U with do = a and a(m) = 0. The defining equation (*) becomes i(}')Wt = a.
(**)
c) We have, for all t E I, that wt (m) = wo (rn) . and therefore that it is nondegenerate at rtt. It follows that this also holds in a neighborhood Up of m, contained in Ul. Therefore, there is in U0 a vector field 1 which satisfies (**) and therefore also (*). Because of the normalization a(m) = 0, Y(m) = 0, we get, from the existence and uniqueness theorem for systems of ordinary differential equations. that the family (l' )tEi of vector fields can be integrated to obtain a family (FF)tE/ of diffeomorphisms which on an open neighborhood U of m satisfy Ft(U) C U0 for all t, and satisfy (o) with the normalizations FO = id and F, (m) = m. By construction, it follows that for
tEI,
dt
(Ft"wt) = 0,
and then also the expected equality
Fiwt=Fpwo=wo. d) The 1-form a from part b). whose existence was guaranteed by Poincart's lemma, can be made more explicit with the application of a little analysis. Namely, this says that there is a neighborhood Ui of in which has a smooth retraction from Ul to {m}, that is, a family of maps c"t
: Ul ---> Ut with Vi = id, (pt(m) = m for all t and wo : U, --. {m}.
For each (Vt)tEj we have a tangent field along Oat in the sense of Lemma 2.3,
and with its help we have i
a-Soa= f d (vio,)dt 0
and
f (c (d + i(X,)d,) + 0
i(X! )a))dt.
2. Symplectic Manifolds
44
Then, since a(m) = 0, a is closed and independent of t. We have I
a = da with a =
J(i(Xt)u)dt. 0
Then by substitution into the given formulas we get from this a the desired family (Y)tEj, respectively (Ft)tEI, as in c). In GUILLEMIN-STERN BERG ([GS], p. 156) there is a further equivariant sharpening of this theorem, which we will now describe, although this deals with the situation of manifolds with group operations, which we will study in detail later. We will consider the operation of a compact group G on the symplectic manifold M. Let m E M be a fixed point under this group operation. Then wo, as well as wI, will be G-invariant symplectic forms on M. Then there exist a G-invariant neighborhood U of m and a G-equivariant
diffeomorphism F from U into M with F (m) = m and F*wl = "70. The proof of this statement requires adding just a little bit more additional work to the proof just given. The result stated at the beginning of this section is then a consequence of Darboux's theorem:
COROLLARY 2.7. For each point m on the symplectic manifold (M, w) them- exist an open neighborhood U of m and a symptectomorphism F of U onto a subset F (U) of R2" equipped with the standard symplectic form wo.
Proof. Here we will require a little more external yet routine analysis. This will assure the plausible existence of a diffeomorphism Fl : UI , U of a neighborhood UI of the origin of the tangent spaces T,,,M H22n to a neighborhood U of m in M. Then wI := Fi w is a symplectic form on U1. Therefore, after a linear transformation, it can be assumed that w1lo = wolo.
Darboux's Theorem 2.6 now guarantees that there exist a neighborhood Uo C UI of 0 and a diffeomorphism
Fo:Uo - Fo(Uo)CU1with Fo(0)=0and Fowl=wo.
F
(FI o Fo)-I, which is clearly the desired symplectomorphism.
As we have already mentioned, the coordinates given by the corollary will be called symplectic and will be written as (q, p). The fact, just proved, that all symplectic manifolds of the same dimension are locally the same immediately raises questions about finding global distinguishing features. We will take a glance at some results on these questions at the end of this chapter in Section 2.7, and explore a little of what is today a very active area of research. But first we need to give a few examples of symplectic manifolds.
2.4. Kahler manifolds
45
2.3. The cotangent bundle The most important example of a symplectic manifold for physical applica-
tions is the cotangent bundle, M = T'Q, to an n-dimensional manifold Q (see Section A.3). Q here plays the role of configuration space and M that of phase space (see Section 0.2). Such an M is necessarily a 2n-dimensional manifold. As coordinates of a neighborhood U of a point m E Al we will (which we will sometimes take, as usual, (q, p) = (q,.... , q,,, pi , ... ,
give in the natural order p, q; here we just want to make sure that it is understood that the q parametrize the configuration space Q, and that it is customary to indicate the coordinates of the base space first when speaking of a bundle). A 1-form 79 is defined on M = T'Q and is given on U by
d = pdq = > R dqj 1=1
This form is called the Liouville form. The form can also be understood, using the notation from Appendix A, as follows: t9 is defined as a 1-form on M, given by 19-(11) :=
for in E M. (Thus m = (q, µ,,), where k is a 1-form on Q, 71 E T,,,M, 7r is the
Q carrying m = (q, p,,) to q, and (7r=), canonical projection M = T'Q is the induced map T,,,M - TTQ.) For the negative of the inner derivative w := -dd, we have, in terms of the (q, p)-coordinates,
-dd=dgAdp=Jdq,Adp;. ==1
This form is clearly closed and non-degenerate, and so defines a symplect.ie
structure on M = T'Q. Each diffeomorphism F : Q - Q naturally extends to F := (F-')' = F'-1 a diffeomorphism of M = T'Q to itself, which is a symplectic morphism. EXERCISE 2.8. Prove this last comment.
2.4. Kiihler manifolds A Kuhler manifold is, roughly speaking, a complex n-manifold (thus the transformation functions between the charts are holomorphic), equipped with a Kiihler metric, that is, a hernitian metric for which the associated 2 form w is closed. This metric was introduced by KAHLER in 1932 [K], and taken up by WEIL [We] among others, and has, because of the peculiar properties of Kahler manifolds, become particularly significant. These manifolds form an important class of examples of symplectic manifolds. In
2. Symplectic Manifolds
46
order to introduce them, we need the material of Section 1.4. There, among other things, we defined: A complex structure on a 2n--dimension R vector space V is a J E
Aut V with J2 = -idv. -- A symplectic R vector space (V, w) is called Kdhler, if it has an wcvmpatible complex structure J (with J E Sp(V)) which satisfies w (v, Jv) > 0.
it follows from the discussion in [K] and [We], that for a C vector space W the following sets are canonically isomorphic: The set of hermitian forms h on W; that is, the set of h : W x W C which are sesquilinear in h(x, y) = h(y, x) for all x and y E W.
The set of symmetric R-bilinear forms g on W that are invariant under multiplication with i; that is, those R-bilinear maps g : W x W -+ R satisfying g(x, y) = g(y, x) = g(ix, iy) for all x and y E W.
The set of antisymmetric R-bilinear forms w on W that are invariant under multiplication by i; that is, those R-bilinear maps g : W x W - R satisfying w(x, y) = -w(y, x) = w(ix, iy) for all x
andyEW. The isomorphisms are given by
g=Reh, w=-Imh, h(x, y) = g(x, y) + ig(x, iy) = w(ix, y) + iw(x, y). Thus h is positive definite precisely when g is.
It is now natural to generalize the question asked in Section 1.4 of whether and in how many ways a given R vector space can be supplied with a complex structure for a given real manifold M. The answer to this question depends on the answer to the question of whether every real tanR2n can be supplied with a complex structure Jm so gent space T,,, M that these structures vary smoothly from point to point (more precisely: they satisfy an integrability condition). We will not pursue this question further here, but will assume that M comes as a complex n-manifold. Then the tangent space TmM ^ C", as an R vector space, has in a natural sense a complex structure J,,,. This will correspond to the choice of local coordinates zj = xj + iyj (j = 1, ... , n) and the concurrent identification of the basis 8
_
1
m-2
8
8 9xjlm-t
jlm
=1,...,n,
2.4. Kahler manifolds
47
of T,mM as C vector space with the basis a
_
a
as R vector space. The multiplication by i =
n,
in T,,, M as C vector
space will be given by the map J,,,, IM,
Jm (7Im)
55-1
Jn, (Im)
i:i;
8
=1,...,n.
The compatibility of these structures J,,, with the holomorphic coordinate transformation functions allows one to give a formulation of the abovementioned integrability condition. We will not go into this here (see CHERN [Ch], p. 14, for details). We continue with the definitions.
DEFINITION 2.9. A complex n-manifold M with a symplectic structure (as real 2n-manifold) is called a Kahler manifold, if for every point m E Al the l[t vector space (T,,,M, w,,,, Jm) is Kahler.
This is equivalent to the description given at the beginning of this section:
DEFINITION 2.10. Let M be a complex n-manifold with a hermitian metric g. Then M is a Kohler manifold if the skew symmetric bilinear form w (. , ) := g (J., ) is as an exterior 2--form, a closed differential form on M.
To this we add a few points of clarification. By a hermitian metric g we mean, as in AEBISCHER et al. ([Ae], p. 24), a Riemanian metric such
that for every point m E M, g,,, is a J,,,-invariant inner product on the 2n-dimensional R vector space TmM, and then J,n is compatible with gm in the sense that g,n(Jmv, Jmw) = g,n(v, w) for all v, w E TmM. As already seen in Section 1.4, g,,, is, in the usual sense, the real part of a hermitian scalar product on TmM as n--dimensional C vector space, namely
(
')h = h(. , .) = 9m(- , ) + In the case when the differential form w associated to this g,n is closed, we call it a Kahler metric. The findings from Section 1.4 on the connection between skew-symmetric bilinear forms w and symmetric ones g apply here ,
immediately to verify the equivalence of Definitions 2.9 and 2.10. In practice we find ourselves with the following procedure: we are given a complex n-manifold M on which there is a Riemannian metric g, so that
g,n and J,n are compatible at every point m E M. This means a nondegenerate 2-form w is also given. To see M as Kahler, and so also as
2. Symplectic Manifolds
48
symplectic, we must prove that dw = 0. A useful criterion in this situation is due to MUMFORD (IMu], p. 87). Let 0 be a group of diffeomorphisms acting on M, which under the operation
GxM -b M, (g, m)
O9(m) _: gm,
leave the complex structure and the metric h unchanged. For m E M denote by
Gm= (9EG;gm=m) the isotropy group of in. Then 09 induces, for each g E Gm, a map ((4g)=)m : TmM ~ TmM,
and so a representation Pm of Gm in TmM, thus a homomorphism
Pm : Gm - Auti(T,,,M). We have
THEOREM 2.11. (Mwnford's criterion) If Jm E e n(Gm) for all m E M, then dw = 0.
Proof. Since G leaves the complex structure and the metric fixed. G also leaves w and hence dw unchanged. Therefore, for all g E Gm and u, v, w E 7,n M..
dwm(p,n(9)u, 8m(g)v, gm(9)W) = dwm(u, V, w).
Here setting p (g) = .In, and applying the formula twice yields
dwm(u, v, w) _m(Jmu, Jmv, J,nu) = clw.n(Jmt+, Jmu, Jmw)
=
-v, -w) _ -dwm(u, v, w) = 0.
0 This criterion can be used to show that complex projective space P (C"+I) _ CP" is a Kahler manifold (see Section 2.6). AEBIsCHER et al. ((Ae], pp. 27 ff.) go on to prove the following criterion. The condition d.o = 0 is equivalent to
VxJ = O
for all X E r (TM).
By V, we here mean the connection corresponding to the R.iemannian metric g (see Section A.4); by X a vector field, thus a global section of the tangent bundle; and the complex structure J appears as a tensor field of type (1,1)1 thus as a global section of T(RI)M
49
2.4. Kahler manifolds
They ([Ae], pp. 28 ff.) then use this criterion to show that the unit ball in C", B" :_ {z E C", [[zj[ < 1}, with the aid of the Bergmann-metric with the kernel n!
K,,(z,tP)
1 (1-Zip)"+lz,iPE B",
a"
is equipped with a Kahler metric.
Towards a formalism of complex differential forms: the Kffhler form. In texts in which complex manifolds are the central theme (see CHERN ([Ch], p. 53), WEIL ([We], p. 41) or KAHLER [K]), the description of Kahler manifolds uses the standard formalism of the real differential forms as described in Section A.4, a = E b;,...,Qdx;, A...Adx;?
extended to the complexes. Following the Wirtinger calculus of function theory (see FIsCHER-LIES [FL], pp. 22-23), we assign to the complex coordinates zj = xj + iyj (j = 1, ... , n) the symbols dzj = dx; + idyj,
(*)
dzj = dxj - idy;,
as well as the differential operators known from the Cauchy-Riemann differential equations,
a at;
1
2
a axi
a
i ft,
,
1 a a a azj = 2 \ ax; + = ayj
and define differential forms, for example
f! = E cjkdzj A dz,t (Cjk are C-valued functions.). j.k
This is called a form of type (1,1), since it is homogeneous of degree 1 in the dzj and the dz-k. Such a form is called closed when dig = 0, where here
d=a+$
with
a&jk 0910=F azt dzt Adz; A dzk
j,k.l
and
ac dz1 A dzj A dzk.
&I = j,kj
This will now be used in an application. A complex differential manifold Al of complex dimension n is called hermitian if TM has a hermitian structure; that is, for every point m E M the n-dimensional C vector space T,mM has
a hermitian scalar product (, )," assigned in a smooth way. For a chart
2. Symplectic Manifolds
50
(cp, U) with coordinates z = (z1, ... , z,.) a positive definite hermitian matrix is defined, for each m E U, by
H = (Hjk) with Hjk =
U' k = 1, .... n). axj , azk To this matrix can then be associated the (1,1)-form S2 = (i/2) E Hjkdzj A dzk, j,k
which CHERN called the Kahter form. 51 is clearly real in the sense that we have
_ -(i/2)
Hjkdzj A dzk = (i/2) 1: Hkjdzk A dzj = R. j,k
j,k
As an easy variant (see below) of Definition 2.10, we have
DEFINITION 2.12. M is called a Kahler manifold if M is hermitian and the corresponding Kiihler form iZ is closed.
Of particular importance, especially for applications of Kiihler forms, is the fact that f) is closed exactly when D locally has a potential; more precisely,
THEOREM 2.13. Let M be hermitian with a Kahler form f1. Then M is Kahler precisely when there is locally an R-valued differentiable function f satisfying
n = i8f.
EXERCISE 2.14. Prove this. (A proof can be found in KAHI.ER [K], as well as in CHERN ((ChJ, p. 56).
The connection of Definition 2.10 with the real theory described above is that the Kiihler form via (*) can be transformed into a real differential form.
EXERCISE 2.15. By a short calculation, show that
0=-
8jk(dxj A dxk + dyj A dyk) +
ajkdxj A dyk. j,k
j
Here the (ajk) = A = ReH form a symmetric matrix, while the (Ojk) _ B = Im H form a skew-symmetric one. We have
H(z, z') = tzHz' = txAx' + tyAy' + txBy' - tyBa' + i(txBx' + tyBy' + tyAx' - txAy')
2.5. Coadjoint orbits
51
This can then be likened to the earlier real theory. There we spoke of a real compatible metric g on Al as a 2n -dimensional real manifold with a complex structure J. thus with g(e', w) = g(w, v) = g(Jv. Jw) for all v. u' E T..Al R2". When the matrix of g is replaced by
9=(C D) and
J=(-1 1
10
we arrive at the condition
g= (-B
A
with A = to and B = -tB.
Then. from tv = (tx.ty),ty' _ (tx'.ty'). tz = tx+ity and tz' = tx'+i'y', we have
H(z. z') = g(v, w) + ig(Jv, w). The written formalism of the symmetric form g gives the assignment of the skew symmetric form as ,O(v. U') = g(Jv. w).
After this is multiplied by 1/2 it has the associated differential form
ajk(dxj Adxk+dyj Ady)L) - >ajkdxj A dyk j.k
j
This is not exactly the above Kiihler form f2 for H. but rather that for H. WEIL, by the way, associated to H the (1.1) form hjkdij A dzk.
i/2
j.k
(WEIL ((We]. pp. 15 and 41)). and this is then in full agreement with the real formalism.
2.5. Coadjoint orbits This section will need to lean more heavily on the representation theory of Lie groups and algebras (see Appendix D) and on the theory of systems of differential equations on manifolds than either the preceding or the following sections will. However, courage will soon be rewarded, for we will gain a better description and the means for constructing many more manifolds; this will be needed in the discussion of the moment map and of quantization. Coadjoint orbits arise in a natural way for any given Lie group G. The group operates via a coadjoint representation Ad' on the dual space g' of the Lie algebra g of G. The orbits of this action are called coadjoint orbits and can (under known conditions) be made into symplectic manifolds. More
2. Symplectic Manifolds
52
precisely, the goal of this section will be to discuss the following theorem of Kostant and Souriau
Let G be a Lie group with H1(g) = H2(g) = {0} for g = Lie G. (Here by Hk(g) we mean the k-th cohomology group of g (see Appendix Q. Then there is (up to covering) a one-to-one correspondence between the symplectic manifolds with transitive G-operation and G-orbits in g*. The study of the coadjoint orbits was introduced by KIRILLOV, and
the reader may find in [Ki], pp. 226 if., a readable introduction to this topic. Here though, we will follow the treatment of AEBISCHER et al. ([AeJ,
pp. 32-39), where one may find a summary of the detailed treatment of GUILLEMIN-STERNBERG ([GS], pp. 172 ff.).
The coadjoint representation on f24(G)
AQg*. For what follows, we fix G to be a Lie group and g its Lie algebra, which we identify with TeG or with the space VI(M) of all left-invariant vector fields X on C (for
preliminaries, see Appendix B). Analogously, g* with TeG is identified with
the space of left-invariant differential forms of degree 1, and from this it then follows that the space f2q(G) of left-invariant q-forms can be identified with AQg*, the space of alternating q-forms on g. Here is a good time to make the following fundamental observations. Remark 2.16. Via this identification of I (G) with AQg*, the operators of exterior differentiation on 52l (G) are exactly the coboundary operators 6 on AQg* defined in Section C.2. Thus, in particular, the space Z2(g) of 2-cycles in A'g* can be identified with the space of left-invariant closed differentials on G.
Proof. i) Here, as in much of what follows, the Lie derivative acts on differential forms and vector fields of a manifold M. This is described in Sections A.4 and B.2. These actions are in fact linked according to the following formula. For a E 0Q(M) and X1, ..., XQ E )(M) we have (see ABRAHAM-MARSDEN ([AK, p. 117))
(Lxc)(X1,...,Xq) = LX(a(Xi,...,Xq)) (*)
q
i=1
a(Xi, ... , [X, Xd,... , Xq).
As described in Section A.4 and as a special case of Lemma 2.3, the Lie derivative is related to inner multiplication through the formula (**)
LXa = d(i(X)a) +i(X)da.
2.5. Coadjoint orbits
53
ii) Now let a be an element of f2' (G) and X. Y elements of LA(M). Then a(Y) is constant on G and, therefore.
Lx(a(Y)) = 0. The formula (*) for q = 1 then says that
(Lxa)(Y) + a([X, Y]) = 0. Since i(X)a = a(X) is also constant on G. it further follows from (**) that.
(i(X)da)(Y) = -a([X. Y]). This can then be formulated as
da(X, Y) = -a([X. YJ). and with this we have the formula for 5. when a is taken as an element of g. iii) Let. w be an element of f22 (G) and X. Y, Z elements of Vj(G). Then w(Y. Z) is constant, and therefore
Lxw(Y. Z) = 0. The equation (*) for q = 2 then gives
(Lxw)(Y Z) + w([X. Y]. Z) + w(Y, [X. Z]) = 0. Front (**) we arrive at
(Lxw)(Y, Z) = (i(X)dw)(Y. Z) + d(i(X)w)(Y. Z). which then. using the result of part ii) for a = i(X)w. gives (Lxw)(Y, Z) = dw(X. Y. Z) - w(X. [Y, ZJ). Putting these two statements together. we get
dw(X. Y, Z) = -w([X, YJ, Z) +w([X. ZJ. Y) - w.'(IY. Z], X), thus the formula for b for q = 2. iv) Through further and analogous iteration we get the formula for the general case. EXERCISE 2.17. Verify the formula (*) for q = 1 and 2. and fill in the details for step iv) of the proof.
The coadjoint representation on flq (G) can now be easily described: conjugation with go E G acts as an inner automorphism of G: Kg,
G --.
9 ''
G. 9og90
Kg induces a map of the tangent spaces: in particular. for the case TeG = g Ad(go) :=
g
g.
2. Symplectic Manifolds
This map gives, through a roughly similar mechanism, the homomorphism
Ad : G -+ Aut g,
go -Ad go, which is called the adjoint representation of G. This gives rise (see Section D.1) to the coadjoint representation
Ad' : G - Aut g`. For q > 1, this induces a representation
Ad':G-+ AutAQg', which is also often denoted by the symbol Ad#. In those cases where AQg' is identified with the space of left invariant q--forms w on G, we get the formula Ad'(go)w = (rigo)*w = (eyu)'w
with a the right translation on G. Unlike KnRtLLOV [KI , we do not study the coadjoint orbits in g' directly,
that is, the G-orbits in the coadjoint representation in g' G#ry :_ {Ad'(go)t9; go E G} for V E g',
but rather study the orbits in A29 . We begin with some notation. Let (M, w) be a symplectic manifold on which G operates on the left as differentiable maps
GxM
M,
(g, m) i-+ gm.
Then we may consider two maps
Og:M-+M
forallgEG
with0g(m):=gm
and
bm:G--+M
withryn(g):=gm
forallmEM.
In the situation where w is fixed by each 0g, g E G, we then have
09*w=w forallgEG. We call this given operation of G on M a symplectic operation. For the right translation p and the left translation A we clearly have Og 0 10m = 10m o ag, -Ogm = OM 0 Log.
With the help of 0,, forms can be pulled back from M to G; in particular, the closed form w on M induces a dosed form on G. More precisely, we have the following statement:
2.5. Coadjoint orbits
55
THEOREM 2.18.
a) A symplectic group operation G x M M defines a map
41 : M
Z2(9),
in
-' Winw,
with
T(gm)= Ad(9)(`I'(m))
(#)
b) 41(M) is the union of G-orbits in Z2(g).
In the case that the operation of G is transitive, the image of 4(M) consists of one orbit. Because of the commutation rule a C-morphism.
is also called
Proof. a) ty;,, w, as the image of a 2-form, is itself a 2-form on G. It is left-invariant, and so we have 'k is a G-morphism, and so we have vlpm w = (hm o Log)t w = Lo.*q
, w = P9'I'(m) = Ad' (9)41(m).
From the fact that 1(m) = ipM w is closed, we have d (rt'm w) = >Gmdw = 0.
b) The orbit through the point 41(m) has the form G* (m) := {'P(gm) = Ad`(g)4' (m); g E G} (also =: G#T(m)). Clearly,
$(M) = U G#(m), mEM
and for a transitive operation also 11(M) = G#(m).
O
Here we now immediately pose, for w E Z2(g),
QUESTION 2.19. Are there, for a given G-orbit G#w in Z2(g), a symplectic manifold M and a map 4 : M Z2(g) as above with G#w = 41(M)? This gives us cause to ask whether M can be constructed as a homogeneous space of the form M = G/H, where H is a closed Lie subgoup of G. To find such an H, the difficulty will be in showing that a symplectic form w on G/H can be defined, so that for the given closed w E Z2(g) we have
2. Symplectic Manifolds
56
pr'w = w, where pr: G - G/H is the natural projection. So let w E Z2(g) be a closed form on G, and set l)
:= {X E g; i (X) w = 0}.
Remark 2.20. hW is a subalgebra; 4w = {0} when w is non-degenerate.
Proof. The last statement follows immediately from the definition of an inner product i (X) w. To prove the first statement we need to show that for X, Y E h, we have [X, YJ E 4, After the application of the Lie derivative, this has the same appearance as the proof of the previous remark. In detail, for Z E g in terms of the formulas (*) used there, we have 0 = Ly(w(X, Z)) = (Lyw)(X, Z) + w([Y, XJ, Z) + w(X, [Y, Z)).
and in terms of (**) Lyw = i(Y)dw + d(i(Y)w) = 0, implying that w is closed. Both statements together then give, for X, Y E
w([Y.XJ,Z)=0, and so [X, YJ E lj ,.
D
One of the central theorems of the theory of Lie groups now says that, given such an l,,,, there exists, up to covering, exactly one connected sub-
group H,,, C G with Lie H. ^_- 4, In this case, this delivers the desired Hw.
THEOREM 2.21. Suppose that H., is closed. Then there is exactly one symplcctic form ;a on X := G/H, with w = pr* , where
pr:G-G/H4,=Mw is the canonical projection. Proof. Here again we will need some external help. We begin by showing
that the condition that H is closed implies that G/H;,, = M,,, is really a differentiable manifold. We note first that the orbits aH4J (which is an abbreviation for ai(H,,,)), a E G, generate the points of M, At the least, this gives a locally transparent description of M,,, in coordinates, in which
the orbit aH, of a E C in G can be seen as an integral manifold of a differential system A (also called a distribution). Here we can use a few concepts and the central statement from the theory of systems of (partial) differential equations (as can be found in the text STERNBERG ([St), p. 130), for example):
2.5. Coadjoint orbits
57
i) A c-dimensional differential system A on an n-dimensional differentiable manifold G (which does not necessarily need to be, as in this special case, a Lie group) is, for 1 < c < n, a map which assigns to each a E G a c-dimensional subspace A (a) of TUG:
GE)a-A(a)CTUG and dimA(a)=c. ii) Such a system A is called smooth if for each a E G, there exist (a) a neighborhood U(a) of a in G, and (b) smooth vector fields Z1..... ZZ on U(a), such that for every m E U(a) the vectors (Zl )m, ... , (Z,),,, form a basis of A (p) (this condition takes on a much more elegant form in the language of vector bundles). iii) A differential system A is said to be involutive if and only if A is smooth and
[X,YJEA for allX,YE.A, where XEAmeans that XaEA(a)for all aEG. iv) A submanifold N ' M of M is called an integral manifold of A, if i.(TqN) = A (i(q)) for all q E N.
This thus means that the tangent space TqN of N is, at every point q E N, isomorphic to the given subspace A (i (q)) of the tangent spaces T,igiG given by the differential system A. (It would perhaps be better to call this a maximal integral manifold, since then it would make sense to consider objects N with i.(TTN) C A (i (q)).) A criterion for the existence of these maximal integral manifolds is delivered by the following: THEOREM 2.22. (Frobenius' Theorem) Let .A be a c-dimensional smooth differential system on a differentiable n -manifold G. Then:
a) A is involutive exactly when every point of a E G lies in an integral manifold N = Na. b) Should a) be satisfied, there are local coordinates xl,... , x so that the integral manifold has the form
N = {xlx; is constant, i = 1, ... , k},
k = n - c.
This form of Frobenius' theorem, so written with the help of vector fields. has a mirror image in the world of differential forms (which historically came first). A proof of Frobenius' theorem in this form can be found in KAHLER [K1J.
2. Symplectic Manifolds
58
v) Now we return yet again to the special situation of the proof of Theorem 2.21. In particular, we want to interpret the Lie algebras as subspaces of tangent spaces in order to define a differential system D by
A (e) := f)
for the identity element e E G
and
for a E G with the left translation Aa. (a) :_ From Remark 2.20, it immediately follows that this 0 is involutive and, from iv), that every point a E G has an integral manifold N.. These are
clearly given by Na = aH,,,, and further, from Frobenius' theorem (part b), Na, at every a E G, can be locally written as
Na = {xlxi is constant , i = 1, ... , k},
k = n - dim fk,.
The tangent space TzNa then has as basis
a a OXk+l,...,. Now the given closed 2-form w on G has the form n
W = E a=j(x) dxi A dxj.
iv It follows from the construction of fj that i (X) w = 0 for X E h,,, thus, in particular, for X = -, i = k + 1, ... , n. This and the fact that dw = 0 is closed show that w, in these coordinates, can be dependent only
on xl,...,xk. Thus k
w =1:
a+S(xl,...,xk)dx;Adxf.
i<j
With this, we are essentially done. The integral manifolds Na of 0 are the fibers of the projection pr : G M,,, = G/H,,, and M,,, is described in the local coordinates (xi.... , xk). For k
0:=
aij(xl,...,xk)&, Adxj s<j
we naturally have pr`w = w. It is now routine to see that in this manner also a global form 0 on M,,, with the desired properties can be given.
0
In GUILLEMIN-STERNBERG ((GS], p. 174), Theorem 2.21 is given for a somewhat more general case. In this generality it lays the groundwork for the so-called symplectic reduction, which has the goal of projecting a
given manifold onto a (smaller) symplectic manifold by exploiting additional
2.5. Coadjoint orbits
59
symmetries. We will give a presentation of this material in Section 4.3. The proof uses the same ideas and is only a little harder.
THEoR.EM 2.23. Let M be a differentiable manifold and w a closed 2form on M. Then the vector space of vector fields X on M with
i(X)w-0
(*)
is closed with respect to the Lie bracket. In the case that the dimension of the space of those X satisfying (*) is constant at every point m E M, w defines an integrable differential system A. Should there, further, be given a manifold MO and a submersion p : M MO (that is, p is a differentiable surjection which induces surjective maps on the tangent spaces) so that the integral manifolds of A are the fibers of p, then there is exactly one symplectic form wo on Mn with p`wo = w.
The symplectic manifold of an orbit G. With this last theorem, we can now answer Question 2.19. For a given orbit G#w in Z2(g), H,,, and (since under the presumptions H, is closed) the manifold M, are fixed. Then G#w is the image of M . under the map T. Thus
'I+(M')=G#w. Proof. When mo = eH.. is interpreted as a point of M, we have, for
a E G. with the canonical projection pr : G -+ M . = G/H and in the notation introduced at the beginning of the section,
pra=aH,, =amo=V,.w(a); thus pr
When we further define T : MW -+ Z2(g) as in Theorem 2.18
by
for w as in Theorem 2.21, we get
T (mo) = pr`w = w. Since ' is a G-morphism, we conclude that T (amo) = Ad* (a) +l'(m)) = Ad* (a) w
and so
4, (M')=G#w, since C operates transitively on M,,,.
O
Now that this construction of Mme; for a given G-orbit of w E Z2(g) for closed Hw has positively answered Question 2.19, we naturally turn to the question of its uniqueness
2. Symplectic Manifolds
60
QUESTION 2.24. Is there more than one homogeneous symplectic manifold attached to a given orbit G*w?
To answer this question, we let a symplectic manifold M = G/H with symplectic form fl be given, where H in G is a closed but not necessarily connected subgroup, and with a symplectic transitive operation G x M M. Then this operation induces, for all b E G, a map bbH :
G -+ M,
a ' - ObH (a) = abH. By Theorem 2.18, there further exists a map
W:M-*Z2(9) We then set
w:=T(eH)=i,*H11, and because of the transitivity of the G-operation we have that 41 (M) = {Ad`(a) w; a E G} = G*w
consists of but a single G--orbit. The uniqueness can now be positively answered, since, by the details of the proof of Theorem 2.21, we have that M,,, is the same as the given M. To see this, as in the proof of Theorem 2.21, for g = Lie G let h,,:_ {X E g; i (X) w = 0}. Thus n (X, ') = 0}, hw = {X E g; and since Q is nondegenerate, we also have
h,o={X E9; But this shows that h := Lie H = 44 since
(WeH).Io:g=TeG - TTHM=9/h, X -X+h means that X E his equivalent to (ikeH),X = 0. As things are now arranged, H, is the connected subgroup of C associated to h,,, = h, thus the connected
component of unity in H. And so M,,, = C/H,,,, when not the same as M = G/H, is then shown to be a covering of M. With this, we have now proved THEOREM 2.25. Homogeneous sympleetic manifolds G/H are, up to cov-
ering, parametrized by the C-orbits G*w in Z2(g), if a given w gives rise to a closed subgroup H, in G. From these results the statement made at the beginning of the section quickly follows:
2.5. Coadjoint orbits
61
THEOREM 2.26. (Kostant-Souriau) For H'(g) = H2(g) = {0} there is, up to covering, a one-to-one correspondence between the symplectic manifolds for G and the G-orbits in g'. Proof. i) The cohomology groups Hk(g) are introduced in Section C.2. H2(g) = {0} says that for every w E Z2(g) there is a,3 E g' with d,3 = w. In a similar vein, H' (g) = {0} is equivalent to Z' (g) = B' (g). But B' (g) _ d(A°g') = {0} (since A°g' = R), and therefore d,3 = d8' implies /3' and so 0 E g' with d/3 = w uniquely given by w. ii) This says that there is a one-to-one correspondence between the G-orbits in Z2(g) G#w = {Ad'(a)w; a E G} and the G-orbits in g' C#(3 = {Ad* (a),3; a E G}.
Thus, a bijective map
Ad'(a),3"Ad'(a)w for all a E C is given, and it follows that d (Ad'(a)i3) = d (p.3) = en w = Ad'(a):.r.
iii) The isotropy group G,3 := {a E G; Ad* (a) 0 = 3}
is per se closed. Then H,,, is also shown to be closed as soon as it is shown that H,,, is the connected component of unity of C3. We have
gyLie G3= {X Eg; Lx,3=0}, and thus /3 = Ad'(a),3 = g,',3 is, by the definition of the Lie derivative, for a = exp X equivalent to Lx(3 = 0. This is further equivalent to i (X) w = 0. Then for X E g, Lemma 2.3 says that, in connection with the formula (#+) in the proof of Remark 2.16, Lx,3 = i(X)d/3+ d (i(X),3); thus
Lx)3 =i(X)w, since i (X) /3 is locally constant (because X and /3 both are left-invariant). From this we now get that
9.1 ={XEg; and so H, is contained in C,3 and therefore closed.
2. Symplectic Manifolds
62
A further criterion that M, = G/HW really be a manifold was discovered by CHU. It says that H,,, is closed when G is simply connected (for a proof, see GUILLEMIN-STERNBERG ([GS], p. 179)). Of greater practical significance for giving explicit symplectic forms is the following theorem, where a coadjoint orbit in g" also shows up.
THEOREM 2.27. Let G be a Lie group with H1(9) = Ha(g) = {0), 0 E
g', w = d,9, and
43 =G/H4J-G*w-G08 the associated symplectic manifold to the form D. Then it follows that (X3, 1Y.8) = -fj ([X, Y]) with
X0.= pr.X, Y,9:= for X,Y E g. Here again, pr: C - G/H;, is the canonical projection. Proof. From Theorem 2.21, w as a symplectic form on G/H,,, is uniquely determined by pr* 0 = w. On the one hand, we have
iv (Xq, Ys) = iv (pr.X, pr.Y) = (pr` w)(X, Y) = w (X, Y). On the other hand, as given in step ii) of the proof of Remark 2.16,
-0 ([X, Y]) = dQ (X, Y) = w (X, Y), which can here be recognized as the formula for the coboundary operator 6 in the cohomology theory in Section C.2. 0 We close this section with an example which should give a first feeling for the power of Theorem 2.27: one can recover the volume form wo of the two-dimensional sphere S2 in the following way. EXERCISE 2.28. Take G = SO(3) and identify S2 with an orbit G*# for 0 E g' = so(3)*, and use Theorem 2.27 to define a symplectic form rv on M,3 = SO(3)/SO(2) S2.
Hint. As indicated in B.2 and used more explicitly in Example 4.22, the elements B E g = so(3) can be identified with b E R3 in such a way that the Lie products [B, B'] correspond to the vector products b x Y. In this way, we come up with a form w = xl dxa A dx3 + x2dx3 A dx1 + x3dxj A dxa
which, if we put coordinates on S2 by
xI = coscicos X2 = cos ti sin Q5,
X3 = sin e9,
2.6. Complex projective space
63
pulls back to wo = cos t9d0 A dt9.
Remark 2.29. In the same way, one can construct symplectic manifolds
as orbits G#3 for the Galilei and the Poincare group which are of great importance for physics. One is tempted to do it here, but it would take too much space. The interested reader may consult GUILLEMIN-STERNBERG [GS], pp. 122--130, 437--445, or SouRIAU [So], pp. 144-192, and also the thesis of M.A. EL GRADECHI [EG].
2.6. Complex projective space From the treatment of coadjoint orbits, one can build the moment map as in GUILLEMIN-STERNBERG [GS]. But we will deal with this central topic
later. Now we turn our attention to another concrete example which will serve to make the previous concepts and techniques somewhat more explicit. Later, the study of the moment map will give us cause to consider further examples.
Let P" = P" (C) = P (C "+') be complex projective space, thus the space of all (complex!) lines passing through the origin in C"+1 and denote by ;r the map
C"+1 \{0} - P",
Tr :
z = (zo, , z") - Z-As in Section A.1, z. will mean the equivalence class gotten by setting z - z' whenever z; = \zi, i = 0, ... , n, for a,\ E C. P" is now an example of a real symplectic 2n--manifold. This can be shown in a variety of ways; we shall do so by demonstrating that P" is Kiihler (and therefore, from Section 2.4, symplectic). This itself can be demonstrated in a variety of ways. i) In MUMFORD ([Mu], pp. 86-87) the following metric is introduced on IP". Let V = VHerm denote the space of (n + 1)-rowed hennitian matrices,
V = {AEM"+1(C); to=A}, thought of as an R vector space, which then has dimension (n + 1)2. A differentiable map ¢ can be given by P"
z-
V,
'-
(Aj) = (ztzj) for z with IZ12 = 1.
The group U(n + 1) _ IS E GL"+I(C); £SS = E"+1} operates on V by conjugation and on P" by (S, z.,.) e--- S(z-) :_ (Sz)..
2. Symplectic Manifolds
64
0 is then equivariant for the operation: 0 (S (z-)) = SO(z,)S-1.
V has a U(n + l)-invariant positive definite symmetric standard bilinear form q, given by
q (A, B) = tr(AB). induces at every point z. = 7r (z) a map of the tangent spaces
(0.)(:) : TT(.)P"
T .,-)V = V.
By pulling back, this gives a U(n + 1)-invariant Riemannian metric on P" as real manifold. A rather brutal computation (as conceded by Mumford) shows that the associated (see Section 2.4) hermitian metric on the complex manifold, called the Ftebini-Study metric, in the coordinates
xi=z;/zo, i=1,...,n, for zo#0, has the form ds2 =
EdXi
dx,)
-
1+F,E1xj12
(1+EIxjI2)2
Here we give a few steps of the calculation for the enlightment of the interested reader. To C" x
V,
P- (C)
.
x
)- =
z- ''
1
+IIxI2 (x
with x as column, is associated the map of tangent spaces
C" = TzC" io* T;a(z)P" "' T (=)v = V, which takes the tangent vector u to the curve -y (t) = x + ut in C" to the tangent vector to the image curve (4 y)(t) at t = 0 in the space V. This vector can be transformed (with (x, u) = Exiu;) to a (u) := ¢.io.u = 1
+ Ixl2u0 ) (1' )+\x/(0'`) 1
(u'1+Ixla'u)(x)(i)).
'
Then as a bilinear form on T(=)P" we take the pullback of the bilinear form living on V; thus
g (u, u) = tr(a (u) a (u)). From this, after a little calculation involving substitutions of the form
tr((I )(1,r)()(1,ii))=(1+(x,u))2.
2.6. Complex projective space
65
we arrive at
9 (u, u') =
(u, u') + (u', u) 1+ Ix12
+ (x, u) (u', x) - (u, x) (x,(1u')+ (x12)2
This can now be recognized as the two-fold real part of the hermitian metric given above (after substitution of it = ei and u' = e1).
ii) In AEaISCHER et al. ([Ae], p. 40), this metric is carried over to the definition of Pn as the space of one-dimensional subspaces of Cn+i. Then the map Cn+1\{0}
*
Pn D Uo = {z.,,, zo # 0}
-
C°, X,
with xi = zi/zo for (i = 1, ... , n), induces the map of tangent spaces Cn+1 = TZCn+1
Since any {
T=lm
TZCn = Cn,
0 can be interpreted as a tangent vector to the curve t i-- ry (t) = z + t.
in C"+1
it can be shown that C is the tangent vector at t = 0 to the image curve t ,-+ 7o(t) = (4P0l1`)(t) in Cn
via the transformation
C = (Cl, ... I W with (i = &/zo - Cozi/zo For two such tangent vectors C and (' , we can, in the following "natural" way, find a hermitian scalar product ((, )). On Cn+1 the usual hermitian scalar product is given by the formula n
(c e) = F, &i. i=0
Denote by W the orthogonal complement in T2(Cn+1) = Cn+1 of the lines passing through X. Then
(zC)1=:W25 C", and W can be identified with TXCn = Cn since a.(C"+1) = ir.(W). So we have = cz +' for a uniquely defined c = (C, z)/(z, z) E C and n E W with (cPo r).W = W O-07) = C. Clearly,
(z, z)£ - (C, z)z
2. Symplectic Manifolds
66
and it follows that
(z, z) - (C z)(z, f')
(rl, n)
(Z, Z)
Now as hermitian scalar product for two tangent vectors in Tf(:)P" with the coordinate vectors C. respectively C', we take (z, z) Here we can substitute for , respectively i;', as well as z an (n + 1)-tuple with (;poir).(l;) = ( as well as cpon (z) = x (that is, & = (;zo, Co = 0), and then we get straightaway the form of the metric given above in i). The formalism developed in Section 2.4 can now be applied to h(
, ):=Re((,))
This then says that this is a Riemannian metric on P" which is J-invariant; that is, for the complex structure J for which JIT(z) operates on TT(.)P" as multiplication by i = v1---1 on W, we have
hp(Jpv, Jw) = hp(v, w) for p = 7r (z). To show that the exterior 2-form w associated to h is closed, one uses Mumford's criterion from Section 2.4. To this end, let G be the group of diffeomorphisms of P" which leave the complex structure and h invariant; thus G = SU(n + 1). (Here, we are exploiting the classical isomorphism
P"
SU(n + 1)/U(n).)
It is then observed that
SU(n +
z U(W)
and that the representation pp in Mumford's criterion, Theorem 2.11, is here given by U (W). B,r(z) : SU (n +
Then we clearly have JI,r(z) = iE,i E U(W), and the criterion says that (-
,)=h
is closed, thus that P" is Kiibler. Mumford proves this directly in (Mu], p. 87. In AEBiscHER et al. [Ae] this is crystalized to the criterion of Theorem 2.11. iii) Thus with a little calculation one not only arrives at the metric but also gets a good feeling as to its origin. On the other hand, one may arrive at the metric a little more quickly using the formalism of Definition 2.12, by
2.6. Complex projective space
67
giving a potential from which the Kahler form is gotten through differentiation. And this is given on U; = {(z).,,, z; # 0} C P", with the coordinates
xEC",by
I
f = log K with K(x, T) = + > xJaJ. J=1
Then, in the formalism of Section 2.4, this can be written as
r( ! f
,Oaf
tlxjAdxj
n xkdk K2
Y
K
J,k i=I and this can be interpreted as a Kahler form from a hermitian matrix, which belongs to the Fubini--Study metrics given earlier. iv) It is then pretty nice to demonstrate (AEBISCHER et al. ([Ae], pp. 41 ff.)) that P" can also be realized as a coadjoint orbit. Here again the underlying group is naturally G = SU(n + 1). It can be shown that
9=su(n+1)={AEg((n+1,C); `A=-A, trA=O} is the associated Lie algebra. On the vector space of these traceless skew
hermitian matrices, we can give a scalar product (,) by the formula
(A, B) = Re(tr AFB). This makes possible an identification
9 - 8w, A
with cpA(B)
FGA
(A. B),
which we will consider fixed in what follows. Now we consider the coadjoint orbit in g',
G#B = {Ad'(a) B; a E G} = {a-IBa; a E G}, for the special case B = B0 with
Bo_- aE_E_ _
((1/n) E 0 0
-1 )
Then we have
and this is a-'E_a
G#Bo SU(n + 1)/U(n) - P", E_ precisely for
a=( T 0t
t0
Theorem 2.27 now says that the symplectic form won P" is given by
w (X, Y) :_ (-B, IAx, AY))
2. Symplectic Manifolds
68
for AX, AY E g tangent vectors such that
pr. AX = X, pr. AY = Y for pr : SU (n + 1) - SU (n + 1)/U (n).
2.7. Symplectic invariants (a quick view) On the basis of Darboux's Theorem 2.6, all symplectic manifolds of a fixed dimension are locally isomorphic. Thus to differentiate between them we
must find attributes of the global object. Here we will give only a hint of two such attributes. A more precise and fuller account of this active area of research can be read in HOFER-ZEHNDER ([HZ], chapter 4) and in AEBISCHER et al. ([Ae], chapter 6). Both treatments rely on the earlier
work of GROMOV, who during his study of symplectic embeddings [Gr] formulated the following statement. THEOREM 2.30. (Gromov's squeezing theorem) For r, R E Rio, let
B(r) :_ {(x, y) E R2"; Ix12 + Iy12 < r2}
be the open ball in R' and Z(R) :_ {(x, y) E R2"; x + yl < R2} an open cylinder. Let 0 be a symplectic embedding of B(r) defined by
q(B(r)) C Z(R). Then
R > r.
EXERCISE 2.31. Prove this for the case that 0 is linear.
Pseudoholomorphic curves. A proof of the above theorem is given in GRoMov [Gr] and uses the concept of peeudoholomorphic curves. This concept, along with some important applications and a proof of Gromov's theorem, is given by AEBISCHER et al. ([Ae], chapter 6). Let M be an almost-complex manifold (that is, there is a J E Aut(TM) with J2 = -1 (see also Section 2.4)), and let E be a Riemannian surface, thus a onedimensional complex manifold, which is an almost-complex manifold under the map of tangent spaces induced by multiplication by i on E. DEFINITION 2.32. A pseudoholomorphic curve is a smooth map f : E M for which the induced map
Tf:=f.:TE-.TM is (i, J)-linear; that is, it satisfies f. o i = J 0 f..
2.7. Symplectic invariants (a quick view)
69
Here the pseudoholomorphic curves are particulary interesting when there is a symplectic structure w on M which is compatible with the almost complex structure. As an example we may pick out a statement appearing in AEBISCHER et at. ([Ae]. Lemma 6.3.6 on p. 128). Let f : E -. Al be a pseudoholomorphic curve, µ a metric on E in the conformal class of the coin-
plex structure i on E, and g a metric on M. The smooth map f : E Al induces the map f.: TE - TM, and so, at each point, a homomorphism TfZ = f.Z : TZE
TJ(Z)M,
ZEE,
between Euclidean vector spaces. and therefore a norm II Tf 112= tr(T f )` o T f.
The metric u on E fixes a volume form dr,,. The energy of f is defined by E(f) := f II Tf 112 d-r,, E
and the area off by
a(f) =
Jfi). E
The quoted lemma then says that for compatible (w. J. g) we have
E(f) = 2a(f) and that this is a topological invariant. Using the notation [w] for the de Rham class of w, this can also be written as ([w], f. (E)). As a consequence. for example, one may derive that a pseudoholomorphic curve is a minimal surface relative to the metric.
Symplectic Capacities. In GROMOV [Gr] the following concept is also introduced and studied. DEFINITION 2.33. The symplectic radius, rad M, of a symplectic manifold M is the supremum of all r for which there is a symplectic embedding of the ball B(r) in M.
This motif was adopted and further expanded to the notion of the symplectic capacity by EKELAND, HOFER, VITERBO, ZEHNDER and others. This
was a part of their study of the question of the existence of given closed curves, called Hamiltonian trajectories (to be defined in the next chapter), on the hypersurfaces of constant energy on symplectic manifolds. There is for this situation the following axiomatization (see AEBISCHER et al. ([Ae]. pp. 73-74), or HOFER-ZEHNDER ([HZ]. p. 51)).
70
2. Symplectic Manifolds
DEFINITION 2.34. c is called a symplectic capacity of dimension 2n when c is a map which assigns to every symplectic 2n-manifold (M, w) (eventually with boundary) an element of [0, oo] satisfying the following properties:
Cl (Monotonicity): Whenever there exists a symplectic embedding
':(M,w)- (M',
we have
c(M, w) < c(M', w'). C2 (Conformality): We have
c(M, aw) = Ialc(M, w) for all a E W. C3 (Non-triviality): We have, for the standard form wo on R2", c(B(1), wo) = c(Z(1), wo) = 7r-
The condition C3 can sometimes be weakened to
C3' (Weak non-triviality): We have 0 < c(B(1),wo) and c(Z(1),wo) < oo. That a symplectic capacity actually is a symplectic invariant follows immediately from condition Cl. These axioms, however, do not fix a uniquely defined capacity function. Remark 2.35. 1) It can be shown that the square of the symplectic radius is a capacity. 2) For n = 1, thus in the case of a two-dimensional symplectic manifold, the total volume,
c(M,w)=I
f
M
U,
is an example of a capacity function, which for (M, w) c (R2, wo) agrees with the Lebesgue measure. For n > 1, (vol M)1V" is not a capacity function,
because of C3 and the fact that the cylinder has infinite volume. EXERCISE 2.36. Show that, for an open subset U C R2" and .1 54 0 with the standard form wo, we have c(AU, wo) = A2c(U, wo),
and calculate c(B(r), wj). With the help of the symplectic capacity, Gromov's squeezing theorem can be proven. For this and other applications, as well as for a proof of the existence of capacities, the reader is referred to HOFER-ZEHNDER ([HZ], in particular, Chapters 2-4), and to AEBISCRER et al. ([Ae], Chapter 4).
Chapter 3
Hamiltonian Vector Fields and the Poisson Bracket
Already in Sections 0.4 and 0.5 it was shown how the fundamental Hamilton's equations of theoretical mechanics can be elegantly formulated (and then also studied) with the help of the formalism of symplectic geometry. This formulation will be further detailed in this chapter. Sources for this material are, among others, KIRILLOV ([Kij, pp. 231-233), GUILLEMINSTERNBERG ([GSJ, pp. 88 ff.) as well as ABRAHAM-MAP.SDEN ([AM], pp. 187-208).
3.1. Preliminaries For a given vector field X E V(M), there are, as we've already partly seen in Chapter 2, several operations on the spaces of functions, vector fields and differential forms on M. You may find a collected, systematical overview on these topics in Sections A.4, A.5 and B.I. However, the information we now need is also collected here.
i) Lie derivative of a function. For X E V (M) and f E Jr(M), we define Lx f E F (M) by Lx f (p) = dfp(Xp) for p E M; thus
L x (p)
a, (x) of (x), when X I U Ox;
_
a;
ax,
8 71
72
3. Hamiltonian Vector Fields and the Poisson Bracket
in the chart (U, cp) with the coordinates x. Then Lx is an element of
Der (.F (M)) = {D E End.F(M); D(f g) = f Dg+g Df; f,g E .F(M)}.
ii) Lie derivative of a differential form. For X E V (M) and 3 E flk(M), we define LXw E Qk(M) by Lx,3 = dt (FF,3) I tom'
where Ft denotes the associated flow of X.
iii) Lie derivative of a vector field. For X E V (M) and Y E V (M), with Ft the flow associated to X, we define LxY E V (M) by LxY = j (F_t.Y)jt=o, and it satisfies
LxY = [X, Y]. Here we are using the fact that V (M) is a Lie algebra with the Lie bracket [ , ] as product.
iv) Inner multiplication and the fundamental duality. For X E V (M) and w E 122(M), the inner product of X and w, i (X) w (also written wb(X)) E 1l'(M), is defined by (i (X) w) (Y) = w (X, Y), for Y E V (M). If w is non-degenerate, this gives rise to an isomorphism
fll(M).
V (M)
In the standard symplectic coordinates x = (q, p) of M we thus have w = =t dqq A dpi, and it follows that for n
d=
E
E fI
J=1
the vector field w* (t9) = X E V (M) corresponds to X=En
=1
(a' a -a= 8 ). , dqi
aPi
v) A collection of formulas. A few of the following statements have already been used previously. The reader may find proofs in ABRAHAMMARSDEN ([AM], pp. 109-120), but we also recommend that they be done as exercises. Let a be in f1k, X, Xo, ... , Xk, Y elements of V (M) and f an element of F (M). Then we have
3.1. Preliminaries
73
k
1. da (Xo, ... , Xk) = >(-1)'Lx, (a (Xo,... , Xi, ... , Xk)) i=0
+E(-1) +3a([Xi,Xi].Xo,.. ,J.Ci... , tj..... Xk). i
2. i (X) a is R-bilinear in both X and a, and
i(fX)a= fi(X)a=i(X)fa, i(X)i(X)a=0, i (X)(a A 0) = (i (X) a) A /3 + (-1)ka A (i (X )13)
(for /3 E &I*) .
3. Lxa = d(i (X) a) + i (X) da. 4. Lxa is R-bilinear in both X and a, and
Lx (aA/3) = Lxa A/3+aALx/3.
5. (Lxa)(Xl,...,Xk) = Lx (a(Xi,...,Xk)) k
- Ea(X., ..., [X. Xi],..., Xk). i=1
6. Lfxa = fLxa+df A (i(X)a). 7. i ([X, Y]) a = Lxi (Y) a - i (Y)Lxa.
8. Lxda = dLxa. 9. Lxi (X) a = i (X )Lxa.
10. Lix,yja = LxLya - LyLxa. Remark 3.1. These formulas are dependent on the given formula for the
wedge product ofaiEQl(M)(i=1,...,k)andXjEV(M)(j=1,...,k). which we have taken from ABRAHAM-MARSDEN. With this convention, we have the formula (see ABRAHAM-MARSDEN ([AM], p.104))
(al A ... A ak)(X1, .... Xk) = > sgn (a)al(Xo(1)) ..... ak(XC(k)). CESk
where the summation is taken over all the permutations o of 1, ... , k. However, in the normalization that appears in, for example, KIRILLOV [Ki], one must take in the above formula (1) an additional factor of k + 1 on the left side as well as applying an additional factor to the operator i(X).
3. Hamiltonian Vector Fields and the Poisson Bracket
74
3.2. Hamiltonian systems The following concept is central to the whole theory.
DEFINITION 3.2. Let (M, w) be a symplectic manifold and H E F (M). Then a vector field XH on M is called a Hamiltonian vector field with the energy function H, if for XH we have i (XH) w = dH.
(M, w, XH) is then called a Hamiltonian system. The following important statements are, after the material of the previous section, but remarks.
Remark 3.3. If (q, p) are the canonical coordinates of w, then, by Section 3.1 iv), in these coordinates 8H 8 8H 8 XH 8qi 8Qi api ,
=
since dH
8pi i
-
(Ldgj + 4dpi).
Remark 3.4. If (M, w, XH) is a Hamiltonian system, then there is an integral curve y = y(t) = (q(t), p(t)), t E I, for the vector field XH, precisely when for this curve the Hamiltonian equations hold in their classical form,
8H
8H
8Pi -
8q}
And then the theorem of the conservation of energy has the form that
H(y(t))is aconstant for all tEI. Proof. y(t) = (q(t),p(t)) is an integral curve for XH precisely when ti(t) _ (XH). (t) for all t. With XH as in the previous remark, this can be brought, with the use of Hamilton's equations, to the above form. And, when y = y(t) is an integral curve to XH, we have
dtH y (t)) = dH.r(t)(ti(t)) = w7(t)
((XH)7(t), (XH)n(t)) = 0.
0 As an example of the usefulness of this Hamiltonian formalism we will show how to handle the example of the motion of a particle of mass m and charge e in an electromagnetic field of electric field strength E = (E1, E2, E3) and magnetic field strength B = (B1, B2, B3). Physics (for the appropriate
3.2. Hamiltonian systems
75
physics we refer the reader to LANDAU-LIFSHITZ ((LL], p. 45)) then gives the equations of motion for these particles with velocity v = (vi, v2, v3) and momentum p = (pi, p2, p,3) as
dp=eE+-vxB, and so dtt = e E, + e (vj Bk - vk Bj),
c
for each of the triplets (i, j, k) E {(1, 2, 3), (2, 3, 1)j (3, 1, 2)}. In this situation these quantities satisfy the relation
P
3
M
-V /
V,
j=1
which, in the classical case of v2 « c2, has the approximation p = mv. These equations allow one to find integral curves -y of a Hamiltonian system (M, U;, XH).
EXERCISE 3.5. It is recommended that the reader determine the integral curves for the following systems (for the formulation of Maxwell's equations with differential forms see, for example, SCHOTTENLOHER ((Sch], p. 191).
a) In the classical case, for a time independent field, we have M = T'R3 with the coordinates (q, p) = (Q1, q2, q3, pl, p2, p3), 3
w = wp - (e/c)w8 with wo = E dqj A dpi, j=1
Bidgj A dqk,
WB = (i,j,k} 3
p; + 0(q) with -gradO = E.
H(q, P) = 2m 3_1
b) In the relativistic case, there is the Minkowski space R1.3 with the coordinates q = (qo, Q1i 92i q3) and the metric ds2 = dqa - dqi - dq2 - dq3,
76
3. Hamiltonian Vector Fields and the Poisson Bracket
so
M = T'R1,3 with the coordinates (q, p), w = CIDO + e WF with 3
uro=dgoAdpo-dq,Adpi, wF = wB + WE, 3
WE=>E=dq,Adgo, i=1
H(q, p) = H(q, p) (1/(2m))(po -pi -pa -p3). Here one looks for curves ^y = -y(s) _ (q(s), p(s)) satisfying
a(s) _ (XH)ry(5) Hint. Use the following formula from relativity: dt
ds
1
c
1-u/
c) A variant of b) arises for the case that
M = T'R"3, w = curd, and
H(q, p) = H(q, p - (e/c)A(q)) with A = (Ao, Al, A2, A3), 3
so that da = wF for a = > A. dq;. i=o
It is easy to see that b) and c) describe equivalent problems. In GUILLEMINSTERNBERG (EGS], pp. 143-144) a recommendation is given of how a particle with spin would be described in this formalism. Now we fill in more details of the general theory.
Remark 3.6. (Liouville's Theorem) Let Ft be the flow of XH. Then Ft is symplectic; that is, we have
lw = w, and consequently Ft preserves the volume form r,,. Proof. From Lemma 2.3, we have
d (Fiw) =F (i(XH)dw+d(i(XH)w)).
3.2. Hamiltonian systems
77
Thus, since dw = 0 and i (XH) w = dH,
(Ftw)=Fi(ddH)=0. From this, Ft *w is seen to be independent of t, and since FO = id. it must in fact be w. From Section 1.1. the volume form of the given w is fixed by
rm =
(-1)!n/21w n!
Since Ft preserves the 2-form w, it must also preserve r,,,.
0
Since the Hamiltonian vector fields have such nice properties, we are compelled to give a special symbolism for them. Thus we let Ham(M) denote the vector space of all Hamiltonian vector fields and Ham°(M) denote the local Hamiltonian vector fields X E V (M) that have the property that
to every m E M there is a neighborhood U of m with XIU E Ham(U). The question of the characterization of Hamiltonian vector fields has the following answer.
THEOREM 3.7. X is an element of Ham°(M) exactly when one of the following equivalent conditions is satisfied:
i) i (X) w is closed, or
ii) Lxw = 0 holds, or iii) the flow Ft of X consists of symplectic maps.
Proof. That a local Hamiltonian vector field satisfies the conditions i) iii) is clear from the previous remark. Poincar6's lemma says that a closed form is locally exact, and so for i (X) w there is a local function H with
dHIU =i(X)wIU. From the definition in 3.1 ii) of the Lie derivative of a differential form,
we see that LXw = 0 is equivalent to w being fixed by Ft. This, along with the proof of the first equation in Remark 3.6, shows that i (X) w is
0
closed.
Because of relation (10) in 3.1, LLX,i iw = LXLyw - LyLXw,
local Hamiltonian vector fields give rise to a Lie algebra in V(M). Hamiltonian vector fields are naturally locally Hamiltonian. For the reverse statement, a topological condition, which assures that closed 1-forms are also globally exact, is useful. This says, in effect (see section C.3), that the first
homology group, H'(M, R), is {0}, which is exactly the fact needed. In particular,
3. Hamiltonian Vector Fields and the Poisson Bracket
78
Remark 3.8. For M with H'(M, R) = {0} we have that Ham(M) = Ham°(M) is a Lie subalgebra of the Lie algebra V (M). In general the codimension of Ham in Ham° is exactly
bl(M) = dimH1(M, R), the first Betti number of M. An example of a locally Hamiltonian vector field which is not Hamiltonian will now be given. We begin with a vector field X, defined on the 2-dimensional torus T = R2/Z2, with the coordinates (z, y), by X(x,y) _ (a, b),
a, b E R with a2+0960.
Naturally, here we have
m=dandy. Then i (X) w = (i (X )dx) A dy - dx A (i (X )dy) = ady - bdx
is closed. In other words, this says that X is locally Hamiltonian. But a locally Hamiltonian vector field with no zero points on a compact symplectic
manifold cannot be Hamiltonian. Thus if X = X H, the function H must have a critical point (thus, a maximum or a minimum) on the compact manifold; but then X would have a zero point. Remark 3.9. Let i denote the identification of each real number c with the constant function on M taking the value c at each point. Let j assign to each element h E 1(M) the associated Hamiltonian vector field Xh. Then there is a sequence of R vector spaces
0- R
.F(M) '+Ham(M)-+0.
The fundamental duality w# now says that this sequence is exact; this follows
precisely because it is the constant functions which give rise to the trivial
vector field. Thus Imi = ker j. We will call this the fundamental exact sequence .
It is of great significance that this sequence is not only an exact sequence of vector spaces, but is also short exact with regard to the Lie algebra structure, which is defined on F (M) by the Poisson brackets (see the following section).
3.3. Poisson brackets
79
3.3. Poisson brackets There are many ways to introduce the Poisson brackets, which then naturally lead to the same result for the Poisson bracket of two functions in canonical coordinates (at least, up to sign. depending on whether the result is given in terms of wo = dq A dp or wo = dp A dq). Here, we will continue to follow ABRAHAM-MARSDEN ([AM], p. 191). where Poisson brackets for 1-forms are first introduced. For this we will fix the symplectic manifold (Al, W). The isomorphism given in Section 3.1 iv).
V (M) ,
w
Q'(M),
will be abbreviated to wb(X)
Xb for X E V (M),
w#(19)
t9# for d E S21(M).
respectively,
DEFINITION 3.10. For a, 3 E Q1 (M), the Poisson bracket of a and 3 is the 1-form
{a. 3} := -[a3#]b From this we get the commutative diagram
V(M) x V(M)
Id
V (M) Wb
S21(M) x S21(M)
{.}
S21(Af )
Since V (M) has a Lie algebra structure with respect to the Lie bracket [ , so does 111(M) with respect to the Poisson bracket { , }.
],
THEOREM 3.11. For a, $ E 01 (M), we have
{a, Q} _-LQ#$+Lj.a+d(i(a#)i(/3#)W) Proof. Here we will use the calculus of the Lie derivatives (see Section 3.1). The starting point is formula (1) of Section 3.1 (this is the same as material in Section C.2): (dw)(X,Y, Z) = LX(w (Y, Z)) + Ly(w (Z, X)) + Lz(w (X, Y))
-w([X,Y],Z) -w([Y,Z],X) -w([Z,X],Y).
3. Hamiltonian Vector Fields and the Poisson Bracket
80
For X = a#, Y =,3*, it follows from the observation w(a#, Z) = a (Z) that 0 = LQ* (,d (Z)) - L,3* (a (Z)) - LZ (i (a#) i ((3#) w) +{a, 01(Z) +a (LO, Z) - ,3 (LQ* Z), and from this, with an application of the standard formula LxY = [X, Y], as well as (1) and (5) from Section 3.1, we derive that 0 = (La*(3)(Z) - (Lq*a)(Z) - d (i (a#) i (f#) w)(Z) + {a, 3}(Z).
0 COROLLARY 3.12. If both or and,3 E 01(M) are closed, then {a, #} is exact.
Proof. Since for a closed 1-form y we have, from (3) in Section 3.1,
Lxy = d(i(X )y), Theorem 3.11 immediately gives the result.
0
The closed and the exact 1-forms generate Lie subalgebras of III(M). Since every function f E F (M) brings along a 1-form d f and via w# also a vector field X f? we may define Poisson brackets on functions.
DEFINITION 3.13. For f, g E F(M) we take as Poisson bracket the function
If, 9}
-i (X f) i (Xg) w,
thus
= w (X f, Xg).
In KIRILLOV ([Ki], p. 232), the statement of the following theorem is taken for the definition of the Poisson bracket. THEOREM 3.14. For f, g E .F (M) we have
if, g} = -Lxf9 = Lx9f Proof. On the basis of the definition of X9, we have
i(X9)w=dg and on the basis of the definition of the Lie derivative
Lx,g =dg(X1) = w (Xg, Xf) = i (Xf) i (X9) , = -w (Xf, Xg) = -Lxp f.
0 COROLLARY 3.15. For fo E F (M), g ,-* {fo, g} is a derivation.
3.3. Poisson brackets
81
Proof. For g, h E F (M) we clearly have
{fo, 9h} _ -Lx1o(gh) _ -d(9h)(Xf.) _ -((dg)h +gdh)(XJ0) = -(Lxiog)h - (Lx1oh)9 = {fo, 9}h + { fo, h}g.
0 COROLLARY 3.16. The following statements are equivalent:
(1) f is constant on the curves of X9,
(2) g is constant on the curves of Xj, and (3) {f, g} = 0. Proof Let Ft be the flow of the vector field X1. Then on the basis of the definition of the Lie derivative of a function (see Section 3.1 i), and Sections A.4 and A.5) we get that d -j(goFt)=Ft Lxjg.
EXERCISE 3.17. Prove this last equality.
Now from Theorem 3.14 we also have
(goFt)=-{f,9}oFt. Thus g is constant along Ft exactly when (f, g} = 0. Because of the antisymmetry of (f, g}, we also get the equivalence of ii). Remark 3.18. The relation {H, H} = 0 then says that H remains constant on the curves of the tangent vector field XH, and therefore leads to the theorem of the conservation of energy.
Finally, we can bring the Poisson brackets from Section 0.5 into the picture: COROLLARY 3.19. In canonical coordinates, i.e., with a chart in whose coordinates (q, p) the symplectic form w has the standard form wt = dq A dp,
we have, for functions f, g E F (M),
Ofa9of891l
aqi 8p;
ap; aq8
3. Hamiltonian Vector Fields and the Poisson Bracket
82
Proof. From Theorem 3.14
{f, g} = Lxof = c (X9), and from Section 3.1 iv)
-
=w#(dg)=ag a
X9
aP a9
ag a eq aP'
and so q}
dq + !M
l
ag a
Of agOf ag
aq ap
aq ap
Op 8q
0 COROLLARY 3.20. For a Hamiltonian vector field XH E Ham(M) with }low F` we have
(foFF)={foFt,H} for all f E.F(M). Proof. From Theorem 3.14 we have
If o Ft, H} = Lx,, (f o Ft), and so
{foFt,H}=d(foFf)(XH)=
(foFg).
0 The connection between the Poisson bracket of 1-forms and that of functions is now at hand.
THEOREM 3.21. For f, g E.F(M), we have d { f, g} = {ti, dg}.
Proof. From Theorem 3.11
{df, dg} = -Lxldg + Lx,df + d (i (X f) i (Xq) w)
= d(-Lxfg+Lxef +i(Xf)i(Xq)w), and from Theorem 3.14 we have
{df,dg}=d(i(Xf)i(Xg)w)=d{f,g}. 0 Since ftl (M) with the Poisson bracket is a Lie algebra, we can easily arrive at the following important fact.
3.3. Poisson brackets
83
THEOREM 3.22. F (M), as Ii vector space with the Poisson bracket, has the structure of a Lie algebra.
Proof. Since d and w# are R-linear, the map f -, X f is also R-linear and it follows that If. g} = -i (X f) i (X9) w is R-bilinear. It is then clear that If, f } = 0. It remains to verify the Jacobi identity. We begin with
{f,{g, h}} = -Lxf{g,h} = Lxf(L. Bh), {g,{h, f}} = Lx9(Lx,,f) _ -Lxg(Lxfh), {h,{f,g}} = Lxlf,g}h.
But from Theorem 3.21, we have
X{f,9} = (d{ f, g})# = {4f, dg}* = -[(df)#, (dg)#], and therefore
X{f,9} _ -[Xf, X9],
0
from which the claim follows.
The reader should pay particular attention to this last relationship. COROLLARY 3.23. For f, g E F (M), we have
X{ f.9} = -[X f, X9],
and so the Hamiltonian vector fields Ham(M) generate a Lie algebra. The
fundamental exact sequence in Remark 3.9 is, with -j(f) = -X f, of the form
0 - R ' .F (M) - Ham(M) - 0, and, therefore, also an exact sequence of Lie algebras.
In the context of this formalism, we can give a useful criterion for when a diffeomorphism of symplectic manifolds is symplectic.
THEOREM 3.24. (Jacobi, 1837) Let (M, w) and (M', w') be symplectic manifolds and F : M - M' a diffeomorphism. Then F is symplectic exactly when for all h E .F (M') we have
F.Xh.F = Xh Proof. We begin with
XA =W '# (dh) and Xhop = w# (d (h o F)). Thus, for all Y' E V (M'), w (Xh, Y') = dh (Y').
3. Hamiltonian Vector Fields and the Poisson Bracket
84
i) When F is symplectic, we also have, for all Y E V (M),
w (F.XhoF, F.Y) = w (XhoF, Y) = d (h o F)(Y). Then, with the usual transformation formalism,
w'(F.Xh.F, F.Y) = dh (F.Y) = w'(Xh, F.Y). Since w' is nondegenerate, it follows that F.XhoF = Xh is closed. ii) For h E F(M') we have
i(XhoF)w = d(h o F) = Fdh = F'(i(Xh) w). The precondition F.Xh,,F = Xh, along with the general formula
F'(i(X')o) = i((F-1).X')F'o, gives
i(XhoF)W = i(XhoF)F*w'
Since every X,,, can be taken locally to be of the form (Xh.F)m for an h E F (M), we arrive at the claim F'w' = w. 0 Symplectic maps can be defined by the property that they preserve the symplectic structure. They can also be characterized as preserving the Poisson bracket:
THEOREM 3.25. The difeomorphism F from M to M' is symplectic precisely when F preserves the Poisson bracket of functions on M or the Poisson bracket of 1-forms on M. In other words, exactly when 171:.F(MI) F(M),
f
r-. f o F,
or, respectively,
F*: ill (M')
0I (M),
F't9, are Lie algebra homomorphisms with relation to the Poisson bracket. 19
+-+
Proof. For f, g E F (M'), we have, from Theorem 3.14,
If, g} o F = L(F-t).x;(f o F), and
if as well. Thus, from Theorem 3.24, both are the same exactly when F is symplectic. We leave the proof of the corresponding statements for 1-forms as an exercise. 0 Now we are able to say whether the coordinates of a chart are canonical:
85
3.4. Contact manifolds
Remark 3.26. Let (U, gyp) be a chart with coordinates (q, p). Then this is a symplectic chart, that is, with
wo=1: dg1Adpi, we have w = V*wo exactly when
{q1, gi} _ {p:,Pj} = 0 and {q:, pi} = btj, for i, j = 1, ... , n. EXERCISE 3.27. Prove the remark.
3.4. Contact manifolds We have now seen how specifying a 2-form with particular properties on a manifold will endow this manifold with the structure of a symplectic manifold, and that it will then necessarily have even dimension. In a precisely analogous manner, specifying a 1-form will define a contact structure on a manifold, which then forces the dimension to be odd. These contact manifolds can be discussed in a theory parallel to that of symplectic manifolds; however, both can also be developed as special cases of the general theory of presymplectic manifolds. In any case, it will soon be clear that the physically most important examples of contact manifolds arise as hypersurfaces of constant energy in Hamiltonian systems or from the treatment of time-dependent Hamiltonian functions. For this reason, we will give a few highlights from the theory of contact manifolds as a finish to this chapter on Hamiltonian systems. We begin by defining the major concept of this material. DEFINITION 3.28. Let M be a differentiable manifold with dim M = 2n + k, k > 0, supplied with a 2--form w which has rank 2n everywhere. Then w is called a presymptectic form and (M, w) a presymplectic manifold For k = 0 we get the definition of symplectic manifold, and for k = 1 we get that M is a so-called weak contact manifold. Darboux's Theorem 2.6 for symplectic manifolds transfers without difficulty to the following statement about the normal form of w. THEOREM 3.29. Let M be a (2n+k) --dimensional differentiable manifold
and w a closed form of rank 2n. Then for every point m E M there is a chart (U, gyp) containing m with coordinates (ql, ... gn,P1, ... ,Pn, W1..... Wk),
so that in these coordinateswlu can be written as wIU =
dqj A dpj.
j=1
86
3. Hamiltonian Vector Fields and the Poisson Bracket
ABRAHAM-MARSDEN ([AM], p. 372) give a proof by falling back on the proof of Darboux's theorem that we gave in Section 2.2. On the other hand, STERNBERG ([St], pp. 137-140) gives a direct proof. The concept of a so-called weak contact manifold can be made sharper as in the following definition.
DEFINITION 3.30. A differentiable (2n + 1)-dimensional manifold M is called a contact manifold, if there is on M a 1-form t9 with t9 A (dt9)" # 0 everywhere on M. L9 is then called a contact form.
This notation is in agreement with that found in AEBISCHER et at. [Ae], VAISMAN [V] and BLAIR [BI]; this last is a good place to begin one's study of this theory. In BLAIR'S text one may also find (pp. 11-12) some thoughts as to the choice of vocabulary. ABRAHAM-MARSDEN [AM] call the above defined object an exact contact manifold, and use the term contact manifold for what is here called a weak contact manifold.
It is clear that with w = dt9 contact manifolds are weak contact manifolds. Corresponding to Darboux's Theorem 2.6, we have here a statement about the normal form of a contact manifold. THEOREM 3.31. Let M be a (2n+1)-dimensional differentiable manifold with a contact form t9. Then there is, for every point m E M, a chart (U, gyp) containing m with coordinates (q1, ... , qn, pl, ... , p,,, w) such that n
r9lu=dw+>pj4. i=I
Proof. Let w = -dt9. Then by Theorem 3.29 there are a chart (U, gyp) and coordinates (q1, ... , qn, pi, ... , pn, w1) such that in U
d(t9->pidb) = 0, i=1
and so locally for w = w(q1i... , w1) dw. j=1
Then, since 6 A (dt9)" ordinates.
0, (ql, .... qn, p1, ... , pn, w) are also usable as co-
0
3.4. Contact manifolds
87
The reader may find a more direct proof in AEBISCHER et al. ([Ae]. pp. 168-171).
The study of contact manifolds will be enlightened by the introduction of a particular differential system; that, is, by the introduction of a system of smoothly varying subspaces of the tangent spaces of a fixed dimension, as we have already seen in the discussion of FYobenius' theorem in Section 2.5.
Alternatively, this may be given as particular subbundles of the tangent bundle TM. DEFINITION 3.32. Let. w be an element, of S22(M). Then we call
R,,,.:_ {(m, Xm) E TMI i(Xm)wm = 0}
a characteristic bundle of w. X E V(M) is called a characteristic vector field of w if
i(X)w=0. Remark 3.33. When w E S12(M) has constant rank, then Rte, is a subbun-
dle of TM, and its sections generate a differential system (the differential system of the characteristic vector fields). When w is closed. R,,, is involutive. Proof. The proof is not hard (see ABRAHAM-MMARSDEN ([AM], p. 371)).
Since for two characteristic vector fields X. Y. [X, Y] is also characteristic, we get the result from the formulas in Section 3.1. Remark 3.34. In particular, when w defines a weak contact structure on Al. then R,,, is a vector bundle of rank 1. thus the characteristic line bundle.
In analogy to the association of a characteristic bundle to a 2-form one may also associate a bundle to a 1-form. DEFINITION 3.35. Let t9 be an element of 1l'(M) and t9m 54 0 for all m E M. Then we call Rfl :_ {(m, Xm) E TM: t9m(Xm) = 0}
the characteristic bundle of V.
Remark 3.36. (Al, t9) is a contact manifold precisely when dt9 is nondegenerate on all the fibers of R,t. Proof. Since R,9 is a vector bundle of rank 2n, t9 is thus non-degenerate
on Rd precisely when the n-th exterior power (dt9)" 34 0. And this is equivalent to saying that t9 A (dt9)" 54 0 on all of Al.
88
3. Hamiltonian Vector Fields and the Poisson Bracket
One should be careful about the word usage, which wee have here taken from ABRAHAM-MARSDEN [AM]. In BLAIR [B1], the differential system
D of a section of R, is called a contact distribution and an element X of a one -dimensional complement of D in V(M) a characteristic vector field with the contact structure given by d. Such an X is then fixed by d(X) = 1 and dd(X, Y) = 0 for all Y E V(M). The following examples should make clear what may be confusing at this first glance. EXAMPLE 3.37. Let M = lR2,+1 come with a contact structure via n
d=dw - EPidh. i=1
Then X = 8,, is, in the above sense, a characteristic vector field of d, and,
for w=dr9dq,Adp,, we have d(X) = 1 and i(X)u) = 0; that is, X also spans a one-dimensional space of characteristic vector fields to w in the sense of the first definition. Further, the contact distribution D is here spanned by
Xi:=0q,+p,8,, and
i=1,...,n,
and then, clearly, for i = 1, ... , n, d(Xi) = 19(Xn+i) = 0.
EXAMPLE 3.38. Let Se be a regular energy surface for a Hamiltonian system (M, w, H), that is. a connected component of H-1 (e) for a regular value e of H: in other words, for those in which dH,,, 0 0 for all m E H-1(e). S, is then a submanifold of M of codimension 1. And it is (Sei t'w) for t : Se --+ M, a weak contact manifold. XH IS, is a characteristic vector field of Ow and induces the characteristic line bundle of Ow . To see this is not difficult. In particular, we have i(XHIs)t'w = 0 because of the relation, defined on XH, wm((XH)m, t) = (dH)m(t) = 0 EXAMPLE 3.39. In order to produce a true contact manifold, we sharpen
the last example, and arrive at the physically particularly interesting case of
M=T'Q -Q
of the phase space of the configuration space Q with w = -ddo (see Section 2.3) and
H=K+VoaQ, where V is a real potential function on Q and K is the kinetic energy associated to the Riemannian metric. Then (Se, i'do) is a contact manifold, and
3.4. Contact manifolds
89
it can be easily shown that t90 A (dt3o)n on Se has no zeroes (see ABRAHAMMARSDEN ((AM, p. 373)).
This last example can be formulated in more abstract terms with the help of a theorem.
THEOREM 3.40. Let i : S ' lR2i+2 be the immersion of a smooth hypersurface, where no tangent space of S meets the origin of R2n+2. Then S has a contact structure. And it will be given by i' a for a = xldx2 - x2dx1 + ... + x2 +Idx2n+2 - x2n+2dx2n+I, when x1i ... , X2n+2 are the coordinates in R2n+2 (for a proof, see BLAIR (Bi], pp. 9-10)). The next example takes the form of a recommended exercise.
EXERCISE 3.41. Give the 3-dimensional torus T = R3/Z3 a contact structure as well as an associated characteristic vector field so that the contact distribution D can be made explicit. EXAMPLE 3.42. Closely related to Example 3.39 is the example, particularly interesting for the physicist, of the construction of a contact manifold arising from a time-dependent Hamiltonian function. Here, one is given a symplectic manifold (M, w) and a time direction R on R x M. Then we let
7r2:RxM - M, (t, m) '--
in,
C D:= axw, and t be the vector field on R x M given by
t(s,m} = (1,0) E TR x TmM = T(,,,,,) (R x M) for (s, m) E R x M. Then the following statements are easy to see (see ABRAHAM--MARSDEN ([AM), p. 374)).
Remark 3.43.
i) (R x M, w) is a weak contact manifold.
ii) R;, is generated from the vector field t E V(R x M). iii) if w = dt9, then for 0 = dt + 7r2W also D = dt9, and (R x M, z9) is a contact manifold.
EXAMPLE 3.44. This last example can be varied by the inclusion of a time-dependent Hamiltonian function H. For this the following formalism is useful. Let
X.RxM-+TM
3. Hamiltonian Vector Fields and the Poisson Bracket
90
be a time-dependent vector field; that is, for every fixed t E R, a fixed vector field on M is given. Then X represents, via
X:RxM -+ T(RxM), (t, M) '-'
((t, tn), (1, X(t,m)),
a vector field X E V(R x M), called the suspension of X. Tb an integral curve 7 of X through m E M (that is,
ry:I t
M, -Y (t),
with -y(t) = X(t,.,(t)) for all t E I and y(O) = m) there is associated the integral curve 7 of X through (0, m) E R x M given by
ry:I
RxM,
t '--. (t, 7(t)), y(t)) = (1, X(t,. (t))) and 1-Y(0) = (0, m). If we now have with 7(t) a symplectic manifold (M, w) along with a time-dependent Hamiltonian function H E.F(R x M), then with R, Ht:M m --r Ht(m) = H(t, m),
XH, is, as in the above, a time-dependent Hamiltonian vector field. At each 'M E M the vector is represented by (XH,) ,, and XH is the associated suspension. If we now put wH :=Ca +dHAdt, we get the following statement as a variant of Remark 3.43. Remark 3.45.
i) (R x M, wH) is a weak contact manifold. ii) XH generates the characteristic bundle RAH which satisfies
t(XH)WH = 0 and i(XH)dt = I. iii) For w = dig, tH := 1r2W + Hdt and we have wH = d19H. In the case that H+(t901r2)(XH) vanishes nowhere, we have that (R x M, 19H) is a contact manifold.
Thus here the contact manifold carries physical information, and the conservation of energy, which in the case of time-independent Hamiltonian functions H can be written as LXHH = 0, is here
LXHH=
3.4. Contact manifolds
91
The phenomena described in the examples can in large part be found in the following influential viewpoint of WEINSTEIN which forms the starting point
for further study (see AEBISCHER et al. ([Ae], p. 174)). Let (M, w) be a symplectic manifold and i : S - M a hypersurface given as the vanishing
set of f E F(M) with df Is # 0. Then all the multiples of the Hamiltonian vector field X f are called the characteristic line fields on S. and from this
GS:={cXf;CER}. S is said to be of contact type if there is 1-form fl on S with
dt9 = i'w and t9(X) 6 0 for all X E Cs\{O}. In the context of the concepts presented at the beginning of this section, every cX f is then a characteristic vector field to d in the sense of Bt.AiR as well as a characteristic vector field to d O in the sense of Definition 3.35. The contact structure on S is naturally not uniquely defined; given /3, a closed 1-form on S with (t9+13)(X) 0, then t9+,3 also defines a contact structure on S.
We close the chapter with a nice result that shows just how tightly symplectic and contact manifolds are interwoven. THEOREM 3.46. A manifold with an orientable contact structure can be realized as a hypersurface of contact type in a symplectic manifold.
A proof of this statement, as well as an entrance to this modern development, can be found in AEBISCHER et at. ([Ae], pp. 167-218).
Chapter 4
The Moment Map
A very helpful technique used in classical mechanics for the solution of complex problems consists of deriving integrals (that is, expressions which remain constant with the motion of the system) by analyzing the symmetry of the given system. The conservation of momentum and angular momentum in systems with invariance under respectively translations, rotation.-,, is the most frequent example. In this, the following beautiful and at. first glance rather abstract formalism has been crystalized from the influential work of
Souriau, Kostant, Smale and Marsden. This is linked to the discussion in Section 2.5.
4.1. Definitions For what follows, we fix (M, w) to be a symplectic manifold, on which the Lie group G operates symplectically via 0; that is, for
GxM (g, m)
M, ' -'
gm = Og(m),
where all the Qg, g E G, are symplectic diffeomorphisms with ¢e(m) = m and Ogg,(m) = tg(og,(m)). We let g denote the Lie algebra of G and g" its dual. In this situation, g can be realized as the tangent space TG to G at the element e E G. and this is in turn identified with the left--invariant vector fields VI(G) on G. In the same manner, 9' can be realized as the cotangent space T, G, which in turn is the left. invariant 1-forms on G. For X E g, we denote by XM the vector field on M which for all m E M is given by the rule
(XAff)(m)
at f(oexpix(m))It=o
for f E F (M), 93
4. The Moment Map
94
or, equivalently, by (XM)m
or, yet again, by (XM)m
for u,,,, :G - M,
(Vm)seX
g'-- V4. (g) = gm. EXERCISE 4.1. It is recommended that the reader, using the introductory material of Appendix A, verify that these three definitions coincide with one another, and also verify the formula Lx.v w = 0.
X,M will be called the infinitesimal generator of the operation on M associated to X. DEFINITION 4.2. A map 4? : M - 9' is called a moment map for the
group operation 0 if for all Y E g we have
d (Y)=i(YM)w
(ie)
with
6(Y):M -+ m
R, 4b (m)(Y).
Remark 4.3. Here we have what might at first sight appear to be confusing notation: 6 (Y)(m) = 4 i (m)(Y)
f o r m E M, Y E 9,
which says, in any case, that 4i is fixed by 4i. With the abbreviation of X f for the (Hamiltonian) vector field (*) associated to f E F (M), this is equivalent to
Xi(y) = Yu. NOTATION 4.4. (M, w, 0, 4?) is also called a Hamiltonian G-space.
Remark 4.5. Not every symplectic group operation has an associated moment map. The existence of a moment map 4' for a given operation 0 is guaranteed only when the local Hamiltonian vector fields for M are also Hamiltonian. Then Theorem 3.7 says that the symplectic diffeomorphisms 0s produce local Hamiltonian vector fields Y;/, which then, under the precondition Ham°(M) = Ham(M), are of the form YM = Xji(y).
Remark 4.6. Should 4i and 4" both be moment maps for the group operation 0, then there is a u E g' with
4?(m)-4Y(m)=u forallmEM.
4.1. Definitions
95
The meaning of the moment maps will be made a little clearer in the following statement about their behavior. THEOREM 4.7. Let fi be a moment map for the operation ¢ of G on Al, and let H E F (M) be invariant under this operation, that is,
H (m) = H (Og(m))
for all mEM and9EG.
Then 4' is an integral for the vector field XH associated to H; that is, for the flow Ft, t E I associated to XH, we have 4) (Ft(m)) = 4, (m) for all »t E Af, t E I. Proof. For all Y E g, we have
H (mexpty(m)) = H (m), since H is invariant. Here, after differentiating with respect to t and evaluating at t = 0, we get (dH)m((Y.Yt)m) = 0,
and so LYA, H = 0 for Yaf =
From Theorem 3.14, we then have
(Y)}=0, and from Corollary 3.16, we have `F (S')(Ft(m)) (Y) (m) for every Y E g. Thus we arrive at the claim.
O
In the most important examples, the moment map has yet another nice equivariance property. In order to describe this we recall the definition of the coadjoint representation from Section 2.5 or D.3. The adjoint representation Ad is given by
G x TeG - TeG, (g, Y)
Ad(g)Y = Ad9Y with Ado = (p 1 \9 and the coadjoint representation Ad' by G X (TeG)`
(TeG)' = T, *G.,
Ad`(g)a = (Ad(g''))'a =: Ady`._,a,1
(g, a) from which we define (Ad(g))*
: (TTG)`
(TAG)',
a e {Y -a (Ad(g)Y)}, 1Thia usage is common in the literature of the moment map, and we will use it here also.
4. The Moment Map
96
and so
(Ad(g)'a)(Y) = a (Ad(g)Y). DEFINITION 4.8. A moment map 4' on the group operation 0 is called Ad'-equivariant, when
(09 (m))=Ad; 4D (m) for all mEM andgEG; that is, when, for all g E.G, the following diagram commutes:
m
M
Ad'(g)= Ad,
w
g
In ABRAHAM-MARSDEN [AM] the following is introduced as the means for weakening this equivariance.
DEFINITION 4.9. Let G be a Lie group and g its Lie algebra. Then we call o a coadjoint cocycle when or is a map or : G -+ g' which satisfies the cocycle identity
a (gh) = a (g) + Adg_, a (h) for all g, h E G. Such a cocycle 6 is called a coboundary if there is a u E g' with
6(g)=u-Ad9_,µ foraUgEG. The cocycles modulo the coboundaries give a cohomology group, which is to be understood in connection with the general theory of Appendix C.
A Hamiltonian C-space (M, w, ¢, 4') can be represented by a coadjoint cocycle a (respectively, a cohomology class [a]), in that for g E C and t; E g we can take a(g)(l;) as the value of the map 1,bg,t, which is defined by
,g,E M -' m '-'
R,
--i
and which can be shown to be constant (see ABRAHAM-MARSDEN ([AM], p. 277)).
4.2. Constructions and examples
97
4.2. Constructions and examples We will now discuss only the Ad* equivariant moment maps; we will try to explain the reason for the term moment map, in that the classical momentum and the angular momentum will both appear in the formalism. Before we can manage this, however, we must make a few general observations. We proceed by taking as fixed a symplectic operation
0:GxM- MMl; that is, for
GxM3(9,m)igm=tp9(m)EM all the ¢9, g E G, are symplectomorphisms. For a 1; E g = Lie G we denote by CAt the associated infinitesimal generator, that is, the vector field on M which, as defined at the beginning of Section 4.1, is given by (CA!)rn :_ -dttlexpt'(()It=o'
We will later need several properties of this generator:
Remark 4.10. For
E 9 and g E G we have
((Ad94)A1)m =
im
in the notation of ABRAHAM-MARSDEN ([AM], p. 269).
Proof. It follows from the definition of infinitesimal generators that dl OexptAdt;(m')It=O+
and from the definition of Ad9 that d dtO9(exPg)9-,(7n)It-o'
Then, since 0 is a group operation, Wt-
(0g o Oexptt;(g-Im))
It=O+
which finally can be interpreted as the displacement of
by 0.9. at
the point g (g-Im) = m. Thus ((Ad9C)Af)rn = (O90g-1rn(CM)g-'m 11
Remark 4.11. Fore, g E g we have [Cm, i7m] _ -[C, rl]AI. EXERCISE 4.12. Prove the remark. See ABRAHAM-MARSDEN ([AM], p. 269).
4. The Moment Map
98
Remark 4.13. The notion of the infinitesimal generator is functorial. That is, when we are given two manifolds M and N with G-operations 0, respectively, ip and an equivariant map F : M - N (thus with Forty = e, 9oF
for all 9EG),then we have.,for tE9, TFoCM=CNoF, where CM and i N are the respective generators on M, respectively N. Thus we have the following commutative diagram: F
M far 11
TM
TF=F.
TN.
EXERCISE 4.14. Prove the remark. See ABRAHAM-MARSDEN ([AM, p. 270).
We recommend that readers take another opportunity to acquaint themselves with the notion of infinitesimal generators by proving the following. EXERCISE 4.15. Let Ad : G x TeG - TeG be the adjoint representation. Then for l; E TeG g the associated infinitesimal generator is &T G =: add with adt : TeG n
TTG,
[C nl
Many of the examples of moment maps stem from the following situation.
THEOREM 4.16. Let 0 be a symplectic operation of G on M. Let the symplectic form w of M be exact (that is, w = -0) and let the 1 -form V be G-invariant; thus Oyt9 = d for all g E G. Then
4i:M--+ g' with
't (m) W = (i (CM)19)(m) = Vml(C.lf)m) defines an Ad' -equivariant moment map for 0.
Proof. i) Since t9 is G-invariant, we have (compare with Section 3.1 ii))
LfMt9=0,
4.2. Constructions and examples
99
and because of (3) in Section 3.1 also 0.
d(i(EM) t9) +
Thus i(eM)w,
d
that is, (t;):=i(£M)t9 foreE9, thus satisfying the characterizing relation of the moment map b. ii) To demonstrate the Ad'-equivariance, we must show that
ii (C)(Og(m)) = $ (Adg-i)(m) for all g E G, M E M and
E 9.
But from i) this is equivalent to ((Adg-1j)af)19)(m),
(i(Cnf)t9)(O (m))
and so to and, because of Remark` 4.10, even to tt
19grn(SM)gm = t9
(that is, the property that 6g19 = i) says
that, for all Y, E TmG, we have t9gm((og.)mYm) = t9m(Ym).
)m gives, on account of
Here substituting Y,,, =
r. the claimed equality t9gm((SM)gm) = t9m (((AdgM)rn).
0 This theorem will now be used on the phase space, i.e. the cotangent bundle M = T'Q of the configuration space. Q. In this situation G operates via diffeomorphically on Q; that is, we have
GxQ (g, q)
Q, i--r
gq = cpg(q), Mpg diffeomorphic for all g E G.
In light of Section 2.3 this G-operation on Q can be carried forward to a symplectic G-operation cP (also called a canonical transformation) on M = T*Q according to T"Q,
G x T*Q (9, (q, aq))
'-'
(9q = +pg(q), V*g-,aq) =: O(q, aq),
4. The Moment Map
100
where the point m E M = T'Q is written as the pair m = (q, %) with gEQand agETQQ. THEOREM 4.17. 0 has an Ad'--equivariant moment map
4?:M=T*Q_.g'. For m = (q, aq) E M and C E g with infinitesimal generator q on Q, thus,
[
d dtVexpq(q)It_e,
(SQ)q =
4i is given by 4'
(q, aq) :=
This can also be written as
$ w = 1' where, for a vector field X E V (Q),
P (X) : T*Q
R,
(q, aq) is defined as the momentum to X.
'-' aq(Xq),
Proof. The G-operation c, on Q is extended to a C-operation 0 on M = T'Q so that the projection 7r : T'Q -. Q is G-equivariant; thus cpgo7r= 7ro0y.
The equivariance in the construction of the infinitesimal generators, as
demonstrated in Remark 4.13, implies that, for M = T*Q, N = Q and
F=7r, Qo7r=7r.oEAf
(*)
The definition of the canonical 1-form 79 on M (see Section 2.3), along with d1rm = (7r.),,, : TmM
T q Q f o r m = (q, aq) and X m E T
(q,ay)(X(q,ay)) = aq (7'(q,0q)X(q,av)) Employing this and (*), we get (i (EM) 10) (q, aq) =
says that
-
((W(q,Qa))
= aq((4Q)q)
= P (eQ)(q, aq) From this and Theorem 4.16, we can deduce that 4? is an Ad'-equivariant map.
4.2. Constructions and examples
101
Were the usual coordinates (q, p) for a point m E T'Q used, then the momentum corresponding to Xq = E (X, (q) i9q,) E TQQ would be P (X) (q, p) =
piX:(q)-
In the treatment of quantization in the next section the following relations will receive meaning. These will be relations on lifting, of functions f E F (Q) as position functions to functions f E F (M) such that f = f o n for the projection
rr:M=T`Q
Q,
(q, aq)
q.
Remark 4.18. For X, Y E V (Q) and f, g E F (Q) we have
i) {P (X), P (Y)} = -P ([X, Y]),
ii) {J}=0 , iii)
{f,P(X)}=X(f).
The proofs follow by routine computations (which appear in ABRAHAM-MARSDEN ([AM], p. 284)).
From Theorem 4.17, a moment map can be established on the tangent bundle M = TQ whenever Q is a Riemannian manifold with a scalar prod-
uct (
,
) on the tangent spaces TqQ (it is actually sufficient that Q be
pseudorieinannian) and a group C operates by isometries V. on Q such that this operation can be extended in a natural way to symplectomorphisms ;pg = Tcpg on TQ. Then from Theorem 4.17, the following statement can be deduced (or it can be proven directly in exact analogy to the proof of that theorem). THEOREM 4.19. The moment map (P associated to cp is given, for
(q,vq)ETQ with qEQ, vgETgQ, by
4 (q, vq)
4' (C) (q, vq) := (vg,
for t; E g.
ABRAHAM-MARSDEN ([AM], p. 285) show that this is a special case
of a general statement connected to Noether's theorem in the context of the Lagrangian formalism regarding TQ, which (pp. 208 ff.) is developed via the Hamiltonian formalism on T'Q. On this topic we have only enough
4. The Moment Map
102
space to say a few words. A Lagrange function L E F (TQ) can be assigned a fiber derivative FL, that is, a map
FL:TQ-.T'Q, which is defined by
(q, v4)'-' (q, d"Lq), (This where d"Lq E TTQ at a vq E TqQ has the value in Section 0.2 in the transition from the corresponds to the relation p = Lagrangian to the Hamiltonian formalism.) L is called regular when FL is a local diffeomorphism. Precisely in this case the symplectic standard form wo on TQ is pulled back via FL to a symplectic form WL on TQ: wL = (FL)'wo. Correspondingly, the Liouville form d carries over to >9L :_ (FL)`t9.
ABRAHAM-MARSDEN ([AM), pp. 285-286) now prove
THEOREM 4.20. Let the regular Lagrangian function L E F(Q) be Ginvariant, that is,
Loo.q=L foraligEG. Then we have
i) t9L is also G--invariant; that is, O;$L = tL for all g E G.
ii) For this G-operation, an Ad'-equivariant moment map 4 can be given by
6
vq) =
for E E 9-
iii) The moment map it is an integral for the L-associated Lagrange equation.
Now we offer several examples.
EXAMPLE 4.21. Let Q = R", and let C = R" operate on R" via translations
GxQ -+ (8, q)
,--+
Q,
s+q=,p.(9)
Then the infinitesimal generator associated to E R" = 9 is also t; (and therefore independent of q). From Theorem 4.17 the associated moment map lb on T*Q is given in the standard coordinates (q, p) of M by
4.2. Constructions and examples
103
p) _
Me
which is also
4, (q,P)=P. the momentum. This is to be understood in connection with the conservation statement in Theorem 4.7. For every system with a Hamiltonian function invariant
under the action of G = lR", the momentum is a conserved quantity (an observation which the physicist makes. however, without the need of the whole apparatus here constructed.) EXAMPLE 4.22. Let Q = W', and let G be a Lie subgroup of GL" (1R). The elements q E Q will be thought of as columns, and G will operate as usual through multiplication; thus
GxQ -
Q.
(A, q)
Aq = P.t(q)
' -'
The infinitesimal generator for B E g = Lie G C Af,,(R) is BQ with (BQ)q = Bq. We have further. from Theorem 4.17. that there is then an Ad'-equivariant moment map given by
4 (B)(q P) = P (Bq) where p is meant to be a row vector.
In the special case of n = 3 and G = SO(3). we have (see the end of Section B.2)
g = so (3) = {B E A13(1R). B = -1B}
)R3
with
B=
0
-b3
b2
b3
0 bl
-bt
b1
b=
b2
E 1R3.
-b2 0 b3 Were we now to realize the moment map 4 on TQ. as in Theorem 4.19, the result would be. with the standard scalar product (.) on 1R3. 6 (B) (q, v)
= (v, Bq) = (b x q, v) = det(b. q, v) = (g x v, b).
Thus, with the identification of so (3) with R3 and 1R3 with (1R3)". we have
4(qv)=qxv.
4. The Moment Map
104
which is the usual angular momentum. For instance, for the harmonic oscillator with the Hamiltonian function
H(q,4)=(1/2)(IIq 112+114112), this moment map is an integral. EXAMPLE 4.23. The Lie group G operates on itself by left translation
GxG -+
G,
(g, h) - gh _ Ag(h). Then the infinitesimal generator associated to this operation &, for f E g = Lie G is given by the right-invariant vector field that takes on the value at e; thus (£G )g = (eg, )el;, where Bg is the right translation. From this we get for the moment map on T'G 4' ( )(g, ag) _
1(Pgs)e!O
_ (pgag);
that is, 4' (g, ag) = B?ag(e)
ABRAHAM-MARSDEN [AM] discuss further examples in their exercises. Also related to the material of this section, GUILLEM[N-STERNBERG [GS]
introduce the example of the operation of the Euclidean group E (3) _ SO(3) x 1R3 on 1R3.
4.3. Reduction of phase spaces by the consideration of symmetry A classical theorem, going all the way back to Jacobi and Liouville, says that by giving k first integrals whose Poisson brackets vanish. Hamilton's equations can be reduced to a system of equations in 2k fewer variables. In a similar way, rotational invariance in an n-body problem allows the elimination of four variables. These two processes indicate how, in a general way, with the help of symplectic reduction, one can reduce from higher to lower dimensional symplectic manifolds when a symmetry group operates on the given manifold. Going back to Elie Cartan, these procedures allow one to form quotient spaces and are a completely general central theme of later constructions; they have, in fact, already been seen at the end of Section 2.5 in the discussion of construction procedures for symplectic manifolds. Here we will follow the treatment of ABRAHAM-MARSDEN ([AM], pp. 298 ff.). We assume that we are given
a symplectic manifold (M, w),
a symplectic operation 0: G x M M of a Lie group G on M,
4.3. Reduction of phase spaces by the consideration of symmetry
105
and for g = Lie G an associated Ad'-equivariant moment map 4' :
Then we denote by G,, for a u E g' the isotropy group Gµ := {g E G; Ad9_,µ = Ea}.
It is a general fact that this is a closed subgroup of G and therefore also a Lie group. Since 4i is Ad'-equivariant, the space Mr, := `y-1(1t)1GN
of GN-orbits on the fibers 4-1(p) makes sense. It is called the reduced space associated to the triple M, 4i and p, and it is this reduced phase space which
will find application in the important special case of M = TQ. We will begin by giving a few technical conditions that insure that Afµ is at least a smooth manifold.
i) Let p E g' be a regular value for 4?; that is, for all m E -D-1 (P), the map T4i,,, (also written as (4',),,,) of the respective tangent spaces TmA1 in Tg' = g is surjective (which, because of Sard's theorem, must be true for almost all p). Then, with the methods described in Appendix A (see ABRAHAM-MARSDEN [AM], p. 49), one may show that the fibers of 45-1(p)
form a submanifold of M. and that this manifold has dim 44-1 (µ) = dim Aldim G.
ii) GF, operates without fixed points and properly on 4>-1(p). Here properly means the following: if (mj) and (¢,,,m3) are convergent series in Al, then ¢(gg) has a convergent subsequence in C. This condition is, for example, automatically satisfied when G is compact. It is then a fundamental statement (see, for example, ABRAHAM-MARSDEN ([AM], p. 266)) that
the above introduced space M. = 4r1(µ)/G,, is a manifold and that the canonical projection 7rµ . 4;-1(ld) - Mµ = 4'-1(1,)/Gµ
is a submersion. That Al. in this situation is symplectic is the content of the following theorem. THEOREM 4.24. Let (M, w) be 9ymplectic with a symplectic G-operation and an Ad* -equivariant moment map satisfying the conditions given above. Then Mµ = 4t-1 (µ)/Gµ has a uniquely defined syniplectic form wµ with T,,w,,
where ir, : 4D (y) - Mµ is the canonical projection and i,, : (P -1(µ) is the inclusion.
M
4. The Moment Map
106
The proof requires the following statement.
LEMMA 4.25. Form E 4)-t(p) and Gm := {O9m; 9 E G} we have o)
Tm(Gpm) = Tm(Gm) flTm($-I{ft)},
ii) T.($ (µ)) and Tm(Gm) are w-orthogonal complements of one another. EXERCISE 4.26. Prove the lemma. In case of emergency, one may consult ABRAHAM-MARSDEN ([AM], p. 299).
Proof. Now the proof of the theorem follows in several steps. a) For v E Tm (f-I (µ)), let [v] = (ir,,).m(v) be the associated equivalence
class in Tm(f-I(p))ITm(G, m). The equality irµw,, = i;w says that w,,([v], [w]) = w (v, w) for all v, w E Tm(-6-1(µ))
Since n and (a,,), are surjective, w,, is clearly uniquely defined. b) It follows immediately from part ii) of Lemma 4.25 that w,, is welldefined. c) w is closed, and so d (ir,;w,,) = d(iµw) = i7,dw = 0.
and so also irµdw, = 0. Then, because 7r,, is surjective, we can conclude that &A',,= 0 is closed.
d) w,, is non-degenerate. Thus from
w,,([v], [w))=0
for
allwETm''1(i.c)
it follows that
w(v,w)=0 for all wETm4b-I(,u), thus v E Tm(Gm) from part ii) of the Lemma 4.25, and in this situation 0 v E Tm(Gpm) from part i); that is, [v] = 0. It should be remarked that when w = dd and t9 is G-invariant, w. need not be exact. Remark 4.27. Since it is symplectic, the manifold M. has even dimension. Because of general principles that we cannot go into here, it turns out
that dim M. = dim 4i-1(u) - dim G. = dim M - dim G - dim G,,.
4.3. Reduction of phase spaces by the consideration of symmetry
107
Remark 4.28. If p is a regular value of fi, the operation of Gm is locally free. Then the reduction described in Theorem 4.24 can be taken, at least locally, in known interesting cases; this remains true even if the global conditions are not satisfied. For the sake of completeness, we will now specialize the construction to
the important case for physical applications of M = T'Q. Here G operates on Q and then, as discussed in the previous section, also on M. The associated moment map is, as in Theorem 4.17, given by
aq) = aq((tQ)q) for e E g, q E Q, aq E TqQ Let the conditions for Theorem 4.24 be satisfied; then, moreover, G. operates without fixed points and properly on Q, so that Q,.:= Q/GN is again a manifold. THEOREM 4.29. Let am be a Gµ -equivariant 1 form on Q with values in that is, with for all. E g. Given the canonical symplectic 2 -form wo on T'Q, the form S2µ := WO - 7r'dam
is then also a symplectic form on T'Q. This then further induces a symplectic form on T*QM, and there is a symplectic embedding
x1,:MM- 7"Qto a subbundle over Qm. Ya is then a diffeomorphism of T'QM precisely when 9 = 9N = Lie GM.
Sketch of proof. For
Fm:={(q,aq)ET'Q; aq((i;Q)q)=0 for all
E 9M}
we have, using, for example, results about quotients of bundles, T*QM = Fµ/GM.
From the definition of the fiber 4we get (see Theorem 4.17) for all t; E g}.
-1(µ) = {(q, aq) E T*Q; Now define zGµ : b
(ie)
(q, aq) a symplectic map with V)µ
-
Fµ,
'-' (q, aq - (c w0 4 _t():
)q),
4. The Moment Map
108
Q
Q/Gµ=Q.
Vim is clearly an embedding, and is surjective precisely when g = gµ. We also have that am is G.--equivariant. With this pr o ioN factors through the quotients Mj, and so defines x,.. From the definition of symplectic structure on M. Xv is symplectic. 0 Remark 4.30. ABRAHAM-MARSDEN ([AM], Section 4.5) discuss more concrete systems with symmetry; these examples really bring Theorem 4.29 to life. They also construct differential forms aµ of degree 1 for this theorem.
It is worth remarking that for p = 0, one may assume that aµ = 0. When
Gisabelianwehave g=gM,andsoM,,-T'Q,,. Examples. EXAMPLE 4.31. The theorem of Jacobi and Liouville discussed at the beginning of this section now takes the following form. Let (M, w) be a symplectic manifold, and let fl, ... , fk E F (M) with { fi, ff } = 0 for all i, j. Since the associated flows Ki, respectively K,, to X f, and X f; commute (this is a consequence of Corollary 3.16), we get (locally) a symplectic operation
of G = Rk defined on M. The associated moment map is then 4t = Kl x ... x Kk. It can then be taken that the dfi are independent at every point, so that every u E Im 4; is a regular value of 0. Since G is abelian, we have Gµ = G, and it gives rise to a symplectic manifold M. = 4i-1(p)/G of dimension 2n - 2k. It then remains to show (see ABRAHAM-MARSDEN ([AM), p. 304)) that every invariant Hamiltonian system on M canonically induces a Hamiltonian
system on M,,. In the important special case n = k, the system is called completely integrable. In this sense, an example of a completely integrable system is the Hamil-
tonian system (M = T'R3, wo, Hv) belonging to a central force field, thus with Hv(q, p) = (1/2)p2 + V(q), where V (q) is an SO(3) invariant potential. Then f2=111112, f3=I3 with I=qxp fi =H, are integrals with vanishing Poisson bracket.
4.3. Reduction of phase spaces by the consideration of symmetry
109
EXERCISE 4.32. Verify this last statement.
EXAMPLE 4.33. Let XH be a Hamniltonian vector field on M. Its flow gives a symplectic operation of R on M. whose moment map 4) is then the function H itself. Thus, for a regular value E E R of H, we have a symplectic
structure on H-'(E)/R =: ME. In these quotients every orbit and every solution to the H-associated Hamilton's equations can be seen as a point: ME is therefore called the manifold for the solution of constant energy E. We have dim ME = dim M - 2. EXAMPLE 4.34. For G = SO(3). the adjoint operation of G on g = R3 can be written as in Example 4.22. For µ E R3. p 34 0. G,, = SI then corresponds to the angular velocity of the µ-spanned straight line. The = Al,, for the associated angular momentum reduction of M to 4i goes back to JACOBI. We have dim Al,, = dim M - dim G - dim G,, _ dim Al - 4 = 2. EXAMPLE 4.35. Example 4.33 can be further extended to assign symplectic structures to specially given manifolds which can be considered as quotients in this scheme. This is the case for complex projective space
IP" = P"(C) discussed in Section 2.6. and gives us a proof that this is a symplectic manifold. Let Al = R2("+I) = 7"R"+' with the canonical symplectic form wo =
E=l dqi A dpi, and let (1
n+1
H (q, P) = 1 2J
-1
(q? +P?)
be the Hamiltonian function of the harmonic oscillator. Then we have (see Remark 3.3) n+I
Xy(9 p) =
Pi
i=I
a
a l.
- 9i api aqi
and the associated flow is Ft : (q, p) s -. (q cos t + p sin t, p cos t - q sin t).
EXERCISE 4.36. Prove this last statement.
Since Ft is periodic with period 2a, Ft defines a symplectic operation of
SI on M. Because of the compactness of SI. the operation is proper and clearly fixed-point-free. The value 1/2 is a regular value for H, and we have
H-I(1/2) =
Stn+I. Theorem 4.29, considered in the case of Example 4.33,
then says that
H-I(1/2)/R = H-I(1/2)/S' = S2n+,/SI = Pn(C) is a symplectic manifold of real dimension 2n.
4. The Moment Map
110
EXAMPLE 4.37. We may also construct examples of symplectic manifolds with the help of the coadjoint orbit using the procedures discussed in Section 2.5, which can be, at least partly, adapted here. As in Example 4.23,
let G be a Lie group, and A : G x G - G the action of G on itself by left translations; thus
(9, h) -As(h). A then denotes the usual extension of this operation on M = G. Then, as shown in every example so far, the associated moment map 4i is given by
t (9, ag) = 4%. Every µ E g' is a regular value for ', and we have 4-1(µ) = {(9, a9) E T* G, a9 ((p9.)9 () = µ (e) for all E g}; that is, here a is the right invariant 1-form aµ whose value at e is p. Thus
a = a,, with (ar.), =,a. Moreover, it can be seen that
G,,={gEG, A9ap =aµ} and yet further that
-1(µ)lGv = G/G,u = G*µ E g', where the last isomorphism is given by the coadjoint representation. In this way the coadjoint orbit G#µ is seen to be a symplectic manifold. This is cited as the Kirillov-Kostant-Souriau theorem in ABRAHAM-MARSDEN ((AM], p. 302). They go on to explicitly construct the associated symplectic form w,,. The result which we gave as Theorem 2.27 then Comes out. EXERCISE 4.38. Let M = C" be considered as a real symplectic manifold
with w(x,y) := Im(t4) for x,y E C. G = S' = {( E C, I(I = 11 operates on M by multiplication:
((,x)'lb((x)=(x for (ES', xEM=C". Make explicit the above concepts in this simple situation and prove that 4C is a symplectic operation, to which, for x E C" and Y E g = Lie S' -_1fI, (x)(Y) := (1/2) 11 x 11' Y gives a moment map. Are these Ad'-equivariant? What does the associated reduced symplectic space look like?
Here we stop so that we can save room for the last chapter, which will treat quantization. The reader interested in further concrete examples is unconditionally recommended to look over the remainder of Section 4.3, as well as Sections 4.4 and 4.5, in ABRAHAM-MARSDEN [AM]. This material
should now be fairly easy to understand.
Chapter 5
Quantization
As we have already encountered in Section 0.6, the process of quantization converts the description of the time progression of physical systems on a syinplectic manifold (in particular on that of the phase space) as described in classical mechanics to a similar progression on Hilbert space in quantum mechanics. The book by WALLACH [Wa] is dedicated to a critical account of these problems and includes many historical comments. Also well worth reading is the book by WOODHOUSE [Wo]. Yet another fully detailed account of these topics can be found in ABRAHAM-MARSDEN ([AM],
Section 5.4, pp. 425 ff.). Shorter introductions occur in KIRILLOV ([Ki], Section 15.4) and in GUILLEMIN-STERNBERG ([GS], Section 34). However, these treatments require more supplementary materials from functional anal-
ysis and/or representation theory then we have covered in this treatise; a proper treatment of this preliminary material would carry us too far afield of the material of this text. The coverage of the general case in Section 5.5 is therefore, maybe, not based as soundly on the fundamentals as it could be. Fortunately the easiest conceivable situation, namely M = R2n = T'1gn can be covered without a great deal of theoretical apparatus, and yet in it we can already see much of the problems and concepts coming into play. To make possible the introduction of the Heisenberg and Jacobi groups, which are equally important for physics and mathematics, we treat this simplest case in Sections 5.1 to 5.4 in quite some detail.
5.1. Homogeneous quadratic polynomials and s(2 W e begin by letting M = R2,, with the coordinates (q,, ... , qn, pI, .... pn). (However, when we really want to do some calculations, we will restrict to the case n = 1.) The starting point will be the fundamental exact sequence 111
5. Quantization
112
for Lie algebras (see Sections 3.2 and 3.3)
0- R
0.
Here i is the embedding which identifies c E R with the constant function
f (m) = c for all m E M, and j assigns to H E F (M) the Hamiltonian vector field
" XH = E 8H 8
-1
8H 8
epjCk-av.9pj
F(M) generates an (infinite-dimensional) Lie algebra with the Poisson bracket
If,
Of 89
Of 89}
8Pj 89j j=1 aqi epj as does Ham(M) with the Lie bracket [, ], for vector fields. Thus we have
for f, 9 E F (M). Now in order to pass to the quantization, we search for an R-linear map which assigns to elements of F (M) or, at a minimum, elements f from the largest possible subset of F (M), self-adjoint operators f in a Hilbert space ?{ such that the Lie structure is preserved in the sense that [X1, X9] = -Xi1,9}
{*)
U142}= 41,f21= c(flf2 - hfl),
where c = -
(c is, up to the factor i, a factor ensuring the symmetry of the operators, a constant from physics, and h is called Planck's constant). Moreover, this map should extend (s*)
T=1:=id 7j.
In view of the fact that M can be understood as M = TR", we may take ?{ = L2(R"), which is here the smallest non-trivial Hilbert space. This search can now proceed with the following general procedure.
We look for a Poisson subalgebra F° of .F(M) which is isomorphic to the Lie algebra of a Lie group G.
a:Yo =+g=Lie G. Then (see Appendix D) an irreducible unitary representation of G
7r : C -' Aut?{,
g - A(9), has an associated infinitesimal representation of g, which for X E g is given by
dir(X )v = dt ir(exp(tX)v) Ito.
5.1. Homogeneous quadratic polynomials and S12
113
This representation lives on the subspace 7{,,_, of smooth vectors v in 7{, for which the differentiation process is practicable. There are many deep theorems from representation theory (in particular, from NELSON and HARISHCHANDRA) which deal with this space and give answers to the question of when 71 is dense in W. In the given case, 7t = S(R") is the Schwartz space, and all works well. By differentiation of the unitarity condition
(7r(exp tX)v, ir(exp tX)w) = (v. w)
) the scalar product in f, we get
for v, w E 7{x and
(da(X)v, w) + (v,dzr(X)w) = 0.
Thus the operators dzr(X) are skew hermitian. These become hermitian after multiplication by ±i, and so. because dir(a({f, g})) = [d r(a(f)), dir(a(g))), the transformation F°
f
a(f) = X
fid7r(X)
da(X)
f
satisfies the condition
if, g) =±idir(a{f,g}) = T-i[f, g]. Choosing here the factor -i, we see that the condition (*) is also satisfied; that is, once the constant is normalized to 1 by an appropriate choice of units (as is often done in the case of physical problems). It is naturally easy here instead of multiplication by -i to use multiplication by c-1. so that the relation (*) is realized with the constant c. As the most obvious candidate for 2 to which the general procedure applies, we consider the Poisson algebra .F2 of the quadratic polynomials in F. thus (on account of the simplifying assumption n = 1) .7° = F2 = (gp,P2,g2) with the relations
{g2,p2} = 4qp, {gp,P2} = 2p2,
{gp,g2} = -2q2.
.F2 is, via j. isomorphic to the subalgebra Ham2(M) of Ham(M), which for
Ht = qp,
H2 = (1/2)p2, is generated by the vector fields
XH,=qa--P op
,
H3 = (1/2)q2
XH2=Paq,
Xjf,= -q
.
o'! P
5. Quantization
114
Thus F2 is isomorphic to the Lie algebra g = S12 = Lie SL2(R), which can be written as s12=(H,F,G) with
H=(
-1), F=( 0 0 ), G=(1 0
and so
[F,G]=H, [H,F]=2F, [H, G]=-2G. EXERCISE 5.1. It is recommended that the reader fill in the simple computational details for this example.
The reader may find discussions of the representations of SL2(R) and $12 in many sources (see in particular LANG: SL2(R) [L3], KNAPP [Kn], Chapter II, or BERNDT-SLODOWY [BSI). We will also give some details in Section 5.3. EXERCISE 5.2. Let V be the vector space with basis (vjU E No}. For µ E R\{0} let 42 act by Hvj = (j + 1/2)vj,
Fvj = -(1/(21+))vj+2,
Gvj = (µl2)j(.j - 1)vj-2 Verify that this describes a representation of s[2.
It is worth bringing to the reader's attention that this representation is isomorphic to an infinitesimal representation of a unitary projective representation 7rw of the group SL2(R), the so-called Well representation, which after a little preparation will be covered in Section 5.3.
5.2. Polynomials of degree 1 and the Heisenberg group The treatment above allows (at least, in principle) the quantization of systems consisting of homogeneous quadratic Hamiltonian functions in p and q. This should lead to, in particular, a description of the harmonic oscillators, but it is not quite enough. A little more can be accomplished by considering, instead, the case 0 = .F<2(M) of polynomials of degree < 2 in p, q. .F'<2(M) is also a Lie subalgebra relative to the Poisson bracket. As preparation for considering this case, we next study the case of F1 (M), which is the homogeneous linear polynomials in p, q. From the fact that (1)
{q+, pi}=b+j, {qi,q,}=0, {pi,pj}=0,
i,j=1,...,n,
5.2. Polynomials of degree I and the Heisenberg group
115
we see that .F1(M) is not, a Lie subalgebra of 1(AI) (but .F<1(M) is). The associated Hamiltonian vector fields are XP =
a 57q
and X.
a 57p-
and so 11(Af) is bijectively carried by j to the constant vector fields Hamc(M)=lag+1R--R2n.
Ham,,(M) is, in any case., a Lie subalgebra of Ham(M). which consists of the infinitesimal translations of 1R2n. \N,,e thus have before us the following situation: the Lie algebra 91 = II22n can be identified. via the map t. with Ham (M). There is. however, a difference from the situation covered in Section 5.1, where 512 could be identified with .F2(Af) and also with a subHam(Af) algebra Ham2(.F) of Harn(F). Here there is no lifting of A : 91 to a Lie algebra homomorphism of 91 to.F(M) as in the following diagram:
0
IR
.F(M) - J Ham(M)
0
91
However, there is a Lie algebra, namely .F<1(Af ), which is the image under j of Ham,(AI) = 91: this is, however, not an isomorphism. but has a kernel Ker j = R. This situation gives us cause to look at the group 1R2n of all translations of IR2n, and to search for a group H (1R2n) and an associated projection j on 1R2n so that for the associated Lie algebras j is carried to j. Thus we are looking for H (R2") and j with
H(fl22n)
1R 2n
.F<1(M)--91=1R2n
that is, for a group satisfying the Heisenberg commutation relations (1) This task is solved by
5. Quantization
116
The construction of the Heisenberg group. Let (V, w) be a symplectic R vector space of dimension 2n. This gives rise to an associated Heisenberg group which is developed in various ways in the literature. We give a description of these. a) The set
H(V)=VxR={h=(v,K); vEV,KER} is made into the Heisenberg group by the multiplication rule h1h2 = (vl + V2, x1 + K2 + W(ul, V2)).
(3)
EXERCISE 5.3. Some simple calculations will show the following:
i) H(V) is in fact a group. We have
h-1 = (-v, -K) and
hhlh-1 = (vi, Kl +2w(v, v1)). ii) H(V) can be realized as the group of (2n + 2)-rowed matrices. In particular, for v = (A, µ) (A, p E Rn as rows), we have
1 0
h-
0
µ
K p 0 0 1 'Ca A
1
0
0
0
1
iii) The one-parameter subgroups of H(V) are of the form G,,,,, := {(tv, ts), t E R}, v E V, K E R. iv) For the/ associated Lie algebra h = b (V) we have 4 = V x R with [(ul, t9l), (u2, 192)] = (0, 2w(ul, U2)) for (ttl,'91), (U2, $2) E h
(4)
v) For the adjoint representation of H(V) on l1 we have Ad(,,,.) (u, 19) = (u, 0 + 2w(v, u)).
vi) Forn=1,l =(P,Q,R)iswith [Q,R]=[P,R]=0and[P,Q]= 2R,
P-
0
0
0
0
1
0 0
0
0 0 0
0
0 0 0 -1
0 0 0 0 0 1
' Q=
1
0
0
0
0o00 0
0 0 0
R-
0 0 0 0 0 1
0000 0 0 0 0
5.2. Polynomials of degree I and the Heisenberg group
117
And now returning to the various constructions of the Heisenberg group:
b) For the second treatment, consider, for S' = {( E C, I(I = 1}, the set
H'(V)=V xS' ={h'=(v,(),vEV,(ES1}, made into a group by hi h'2= (v1 + v2, (I(2e(w(vi, tri))),
(3')
e(u) := exp(21riu).
Clearly we have an isomorphism of groups
H(V)/Z ^- H'(V) with
h=(v,K)'-'h'=(v,e(a)) c) GUILLEMIN-STERNBERG ([GS], p. 94) consider V x S' with the
product hlh2 = (vl + v2, (1(2 exp((i/2)uj(v1, v2)))
(3*)
This is written as H*(V). d) Often (see for example the fundamental article of A. WEIL [Wel] or the book of LION-VERGNE [LV]), given an arbitrary K vector space W with
dual space W* and associated canonical bilinear form (,) : W x W' - K, one takes as the associated Heisenberg group
Heis(W) := W x W' xK={h= (q, p,tc),gEW,pEW ,KEK} with
hlh2 = (ql + q2, P1 + P2, Kl + K2 + (ql, p2)), or
Heis'(W):=WxW*xSl={h'=(q,p,(); qEW, pEW*,(ES1) with hlh2 = (q1 + q2, Pl + p2, (1(2e((gl,p2))).
Here Heis (W) can be realized as a matrix group, specifically as a subgroup of GLr+2(K), via 1
q
K
0
0
1
h -- 0 1 tp
In what follows we will most often take as our starting point the description in a), which is also the one most often seen in the modern literature. The Lie algebra h of the Heisenberg group will naturally be called the Heisenberg algebra. For the natural basis elements (n = 1)
P=(1,0,0),
Q=(0,1,0),
R=(0,0,1)
5. Quantization
118
of b we have, on account of (4), [P, Q] = 2R,
(5)
[R, P] = [R, Q] = 0 ,
and thus
o(1) = 2R, o(q) = P, r(p) = Q defines a Lie algebra isomorphism o of F
begin by stating an analog to Exercise 5.1, that for n = 1 and a E R\{0}, an irreducible representation of 4 = (P, Q, R) on the space W spanned by vj, j E No, is given by Pvt = vj+I,
(6)
Qvs = -pjvi-I
,
Rvv = (µ/2)v,.
For quantization, this is not yet particularly helpful. It is important, however, that with just a little legwork a non-trivial unitary representation of the Heisenberg group can be found. And it can be fairly easily calculated that, for h = (A, µ, rc.) E H(R2n) and each m E ]R, a unitary representation of H(R2n) on L2(R') can be given by (7),
(mss (h)f)(x) = em(K + (2x + ))tp)f (x + A),
f E L2(flt"),
which for m 36 0 is called the Schrodinger representation of index m. EXERCISE 5.4. Verify the necessary computations.
In the general theory this can be made to arise as a simple example of an induced representation in the sense of the theory of MAcKEY (see Appendix D; for more information, see, for example, KIRILLOV ([Ki], pp. 218-219), the already mentioned book by LION-VERCNE [LV] or the article Of CARTIER [Ca]). GUILLEM N-STERINBERG ([GS], p. 97) give the following
representation r of H'(R2), which there arises as a result of a discussion on geometric optics(!). It is given with h = (q, p, () E H'(R') by the expression (7')
(r(h)f)(x) =
(-I e-iy°neu`p f (x - q) for f E L2(R").
This corresponds to the representation 7rs for m = -1 given above. The fundamental theorem of Stone and van Neumann says that such a representation rrs (respectively r) of the Heisenberg group through its action on the center (that is, here through ors (0, 0, rc) = em(tc)) is, up to equivalence, uniquely defined. For a more precise formulation of this theorem and an accessible proof, the reader is referred to LION-VERGNE ([LV], pp. 19 if.)
5.2. Polynomials of degree 1 and the Heisenberg group
119
It is yet another easy exercise to calculate that from (7) it follows that (on account of simplicity we take n = 1)
7((t, 0, 0)) f (x)It _0 iS d 7r'
((0, t, 0)) f (x) It=o
fi(x), = 4irimxf (x),
emirs ((0,0,t)) f (x)It-0 = 2aimf(x). For the skew hermitian operators
das (P} =
das (Q) = 4nimx,
do (R) = 21rim
the commutation rules (5) are satisfied. After multiplication by -i (and normalization by 4am = 1) we get self-adjoint operators 4=(1/i), P = x,
1 = 1,
(7)
which is the first step in the process of quantization. Physicists may be uneasy that the position variable q represents a differential operator while p corresponds to a multiplication operator. But this follows from the particular form of the Schrodinger representation used here, which is the usual one from the standpoint of function theory. This can be easily converted to the form more familiar to physicists by considering instead a 7rs -equivalent representation. The form we have chosen will very soon lead to the usual formula for the Well representation, and so we will keep this possibly unfamiliar usage for now and make the appropriate transformation only in the larger context. From the general theory (see Appendix D) it makes sense to carry this over to the complex numbers. In particular, let
Yf := (1/2)(P±iQ), Then tic =b®RC=Y±,Zo' with [Z4,Y±]=0
(5')
and
Zo := -iR. [Y+, Y-
Zo,
as well as
(r)
s (Y) =
1 d
2 dx
21rmx
and
* (Zo) = 2irm.
EXERCISE 5.5. Verify that for p > 0 on V = (vj)jEpb an irreducible representation of ll, is given by Y+v, = vi+i,
Y vj = -jpvi-I,
Z0vj = uv,,
j = 0,1, 2, ... .
5. Quantization
120
Show that this may be realized as functions on L2(R), beginning with
th = fo,
with fo(x) = e-2Rmx2
and then letting Yf and Zo act as differential operators, as given by (7), to specify vj (j E N).
5.3. Polynomials of degree 2 and the Jacobi group We will again focus, on account of its simplicity, on the case n = 1. For the most part, the following is not difficult to generalize to n > 1. Since F
0J=sl2+h of the Jacobi group GJ(R), which can be brought to light in the following way. The group SL2(R) operates on the right on H(R) via
(M, h) F--4 hm = (vM, #c) for M E SL2(R) and h = (v, ,c) E H(R). This gives us justification to form the semidirect product
GJ(R) := SL2(R) x H(R) = {g = (M, h)IM E SL2(R), h E H(R)} with the multiplication rule
gg' = (MM', vM'+v',' EXERCISE 5.6. Verify the following:
a) We have gJ = (H, F, G, P, Q, R) with [F, G] = H,
[H, F] = 2F,
[P, Q] = 2R,
(H, Q] = Q,
[H. G] = -2G,
IH, P] = -P, [F, P] = -Q, IG, Q] = -P
and all the other Lie brackets are 0.
b) The map a: -F<- 2 -+ gJ with
a(1) = 2R, a(q) = P,
a(P) = Q.
a(qp) = H, a(q2) = -2G, a(p2) = 2F is a Lie algebra isomorphism.
5.3. Polynomials of degree 2 and the Jacobi group
121
Hint for a): GJ(R) can be realized as the subgroup of Sp2(R) in which H(1R2) is embedded as described in Section 5.2 while introducing the Heisenberg group, and SL2(R) is embedded as follows:
M=
labl c
a 0
b
0
0100
d JJ ~ c 0 d 0 0
0
0
1
Further details on this and for all that will follow in Section 5.3 can be found in BERNDT-SCHMIDT ([BeS], Chapters 1 and 2).
The operation of SL2 on H fixes, in connection with the already mentioned theorem of Stone and von Neumann, a distinguished projective representation of SL2, the so-called Weil representation. Using the Schrodinger
representation, r := 7rs on L2(lR), from Exercise 5.5, we can let every M E SL2 give a rIVI, which acts by
rM(h) := T (hM) for h E H, giving a representation of H. Since this representation has the same effect as r on the center of H, it follows from the theorem of Stone and von Neumann that they are equivalent; that is, for every Al E SL2, there exists a unitary
operator U(M) such that (8)
U(M) T(h) U(M-I) = r(hM
)
for all h E H.
This relation fixes U up to a constant factor by a general statement from representation theory (the so-called Schur's lemma, see for example, KIRILLOV ([Ki], p. 119)). So we have (9)
U(MI)U(M2)=c(MI, M2)U(MIM12).
Therefore c(MI, M2) E CI = {z E C, I z I = 11, and for these c we have. on the basis of the associativity law in SL2, the relation (see KIRILLOV ([Kl], p. 218))
c(MI, M2)c(All M2, hi3) =c(MI, M2Af3)c(M2, M3). This then says that (10)
SL23M- U(M)EAutf
defines a projective representation of SL2 on f. Remark 5.7. A projective representation in the above sense g 7r (g) of a group G into the space 7.1 induces a usual representation z in the projective space P (71) associated to R. This is defined by v,. - it (g) (v-) = (7r (g) v) _ for v. E 1P (7-l).
5. Quantization
122
This procedure accommodates well to quantum theory, since it is not the element v of the Hilbert space that. has physical meaning, but the 1-dimensional subspace spanned by any such v 0 0.
There is now a standard realization of (10), which is today most often called the Weil representation, although it also goes under the names of the Shale-Weil representation, the metaplectic representation or the oscillator
representation. Also the name of Segal appears in this connection. This representation, here denoted by U instead of by 7rw, is fixed by giving the actions of the generators of SL2. This leads to the following formulas, which first appeared in WEIL's fundamental article [Well. which should be consulted since it introduces many other important concepts as well (also see, for example, LION-VERGNE ([LV], pp. 1-63), ([Mu2], pp. 133 f.), or GUILLEMIN-STERNBERG ([GS], pp. 47 65)). For f E L2(R) we postulate
irw (d (a))f (x) = f (ax)IaI1/2
ford(a)= 7rw (n (v)) J (x)
=
a 0
a01
0
1
I, a#0,
J (x)e2amiVX2
forn(v)= I 7rw (w)f (x) = J(x) = f f
I,
vER,
(u)e2'riumrdu
for w = The derivation of these formulas needs more general considerations, but that they fix a representation can be verified by elementary means. Those who do not want to try this themselves may find the calculations in MumFORD ([Mu2l, pp. 134-135). Together with the Schrodinger representation given in the previous sec-
tion, 7rs = r, we obtain via 7rsw(M, h) = 7rs(h)7rw(M)
a (projective) representation of the Jacobi group G-'(R), which we here call the Schrodinger- Weil representation.
5.3. Polynomials of degree 2 and the Jacobi group
123
EXERCISE 5.8. It is not such an easy calculation to determine the infinitesimal representation corresponding to lrswy. For the already given operators (7) and (7') we have also
fril,(F) = 27rimx2.
11'(G) =
(11)
(d Barn
r)2.
frit!(H) = 2+xd for
F=
C0
I. G=I 1 0 I. H=(00 -1 10
respectively, for the complexification
X :E :_ (1/2)(H ± i(F + G)), Z = -i(F - G). frw(X±)
= a + 2 x 1rmx2
I6anr (d)2,
frlv(Z) = 2amx2 - Sam ( T a;)2. An extension of the last exercise allows us to obtain a realization of the infinitesimal representation frsw on V = (vj) jEry,. where
9! _ (Zo, Yf, Xf. Z) for u = 2irm. > 0 operates via
Zovj = µvj,
Zvj
Y+Vj = Vj+i,
Y-vj =
-pjvj-1.
(j + 1/2)v,,
X+vj
- 2p vj+2,
X-vj
jj(j - 1)'vj-2
The reader may find the necessary calculations in the notation used here in BERNDT-SCHMIDT ([BeS[, Chapters 1 and 2) as well as, among others, LION-VERGNE ([LVJ, p. 198) and GUILLEMIN-STERNBERG ([GS]. p.
101), where one must ever pay attention to the fact that the group law of the Heisenberg group has various formulations. In any case. we get here a representation of the Lie algebra relative to { , }: .F<2 = (1, p, q? 42, 9P, P2 )
through the differential operators (7) and (11). which operate on a dense subspace of L2(R) and after multiplication by (-i) solve the quantizat.ion task from the beginning of Section 5.1.
5. Quantization
124
5.4. The Groenewold-van Hove theorem The investigations in Sections 5.1-5.3 dealt with the topic of quantization in the case of polynomials of degree 2, leading to the specification of the operators in (7) and (11). This then raises the question of whether this can be somewhat extended. The theorem of Groenewold and van Hove says, to a first approximation, that this is not possible. This will be made more precise in a moment; but first, we prove a statement of general interest for an arbitrary field K of characteristic 0. THEOREM 5.9. The algebra of the quadratic polynomials p = K[q, PI <2 is, relative to the Poisson bracket, a maximal subalgebra of the algebra
K[q, p] of all polynomials.
Proof. i) The formulas
{(l/2)q2, p'qk} = jpy-lqk+l -kpi+lqk-',
{(1/2)p2, p1gk} =
{pq, pigk} _ (j - k)pigk show that by (possibly several-fold) left-Poisson bracket multiplication with q2, respectively p2, every monomial p1gk, j + k = 1, in the set f1 of the homogeneous polynomials of degree I can be carried to any other; even more, that every homogeneous polynomial of degree 1 generates in this manner the whole of the monomials pigk, j + k = 1.
ii) Let f E K [q, p] be a polynomial of degree > 2. Because {q, f } _
and {p, f } _ aq
-1
we can assume that, by taking left-bracket multiplication with p, respectively q, often enough, an algebra a contained in the algebra p has a polynomial with maximal homogeneous component of degree 3, and after subtraction of the quadratic part the polynomial is homogeneous of degree 3. From i), then, the whole of F3 lies in the algebra a. iii) Because {q3, p3} = 9g2p2
is a polynomial of degree 4, we have from i) that the whole of F4 is in the algebra. And now, since {q3, pn} =
(3n)g2pn-1,
this can be extended inductively to show that all polynomials of degree higher than 2 similarly lie in a. 0
5.4. The Groeneaold-van Hove theorem
125
From ('7) in Section 5.2 as well as (11) in Section 5.3. after the normalization 47rm = 1, the map a :
p2 '---+ ix2.
9P'-
\I xd +1/2
is a representation of p = .7 <2 which acts by skew hermitian operators on a dense subspace of 9{ = L2(]R), and after multiplication by (-i) can be made hermitian. This can be shown by direct calculation, without returning to the formulas for the Schrodinger-Weil representation. EXERCISE 5.10. Verify the above construction. and verify the statements in the following construction. To adapt our presentation to the one in G UILLEMIN -STERN BERG ([GS],
pp. 101-104), which goes back to CHERNOFF and will be here varied a bit, and to simultaneously manufacture the properties most familiar to physicists, we introduce a Lie algebra isomorphism q of p given by
77(1) = -1, rl(g2) = -p2.
TIM = -p, n(p2) = -q2,
rl(p) = -q, i (qp) = -qp.
Then oil is a Lie algebra isomorphism of p, and A = iaq is given by
A(1) =
1,
A(q) = x,
A(p) _ -i (12)
aj,
A(q2) = x2, A(p2) =
-
A(qp) = -i(x/ +1/2). Setting f = A f , we see that C77 satisfies the quantization conditions (* ) and (**) given at the beginning, and so
{f,g} _ -i[f,g] and 1=1.
5. Quantization
126
To simplify the notation, we now put
A(q) = x =: Q and A(p) = -i
ai
=: P
(the old meanings of the letters P and Q will no longer be needed). Then we clearly have (13)
A(q2) = Q2
and
A(p2) = P2.
In this situation, Groenewold-van Hove's theorem is contained in the following:
THEOREM 5.11. There are no linear maps A from K[q, p] into an associative K algebra that satisfy (*) and (**) as well as (13) with A(q) = Q and A(p) = P.
This means in particular that the quantization map A cannot be extended to p, since every properly contained Poisson algebra of p is by Theorem 5.9 equal to K (q, p].
Proof. i) The basic idea of the proof is that, because of the plethora of relations satisfied by the Poisson brackets, the polynomials can be represented by a Poisson bracket in a variety of ways. In our case we particularly consider (14)
q2p2 = (1/3){g2p,p2q}
and (15)
g2p2 = (i/9)(g3, p3}.
From these we will construct a contradiction which says that A assigns to the same Poisson bracket differing values. This proceeds by a few calculations and will be given in several steps, beginning with: ii) We have
A(qp) = (1/2)(QP+PQ).
(16)
In fact, from {q, p} = 1 it follows with (*) that
[Q, P] = QP - PQ = i
(17)
and from {q2, p2} = 4qp 4A(qp) = -i[Q2, P2] = -i(Q2P2
- P2Q2);
with (17), this gives
4A(qp) = -i(QPQP + iQP - P2Q2) and by the further repetitive application of (17) then
4A(qp) = -i(2i(QP + PQ)).
5.4. The Groeneuold-van Hove theorem
127
iii) We have (18)
A(q3) = Q3.
Setting A(q3) =: X. it immediately follows from {q3, q} = 0 that [X. Q] = 0,
and, from {q3, p} = 3q2 with (*) and (13).
[X. P] = 3iQ2. Trivially. in an associative algebra, we have [Q3, Q] = 0. and because of (17)
[Q3, P] = Q3P -PQ3 = Q2PQ - PQ3 + iQ2 = ... = 3iQ2. From here, it immediately follows that y = X - Q3 commutes with P and Q. and then, because [Y, PQ] = YPQ - PQY = [Y. P]Q + P[Y, Q] = 0. also with PQ, and, by entirely analogous reasoning. also with QP. This can be used in the following manner. Because {q3, qp} = 3q3 we get from (16) and (*) that
[X, (1/2)(QP + PQ)] = 3iA(q 3) = 3iX,
but since y = X - Q3 commutes with QP and PQ. we also get
[X, (1/2)(QP + PQ)] = [Q3 (1/2)(QP + PQ)] = (1/2)(Q4P+Q3PQ - QPQ3 - PQ4). and, after further repeated applications of (17). [X, (1 /2) (QP + PQ)] = -3iQ3; thus X = Q3.
iv) With an analogous calculation. it can be shown that
A(p3)=P3.
(19)
v) We have
A(g2p) = (1/2)(Q2P+PQ2). From {q3, p2} = 6q2p it follows from (*), (18) and (17) that (20)
6iA(q2p) _ [Q3 , P2] = Q3P2 - P2Q3 = Q2PQP + iQ2P - P2Q3
... = 3i(Q2P + PQ2).
5. Quantization
128
vi) And again by the same methods, we arrive at (21)
A(gp2) = (1/2)(QP2 + P2Q)
vii) On the basis of the formulas (18) and (19), we have
(1/9)A{q3, pt} = (_i/9)[Q3,P'J. From this. we further get. after a few similar calculations (or, when we don't care about the purity of our method, quickly by returning to (12)),
(1/9)A{g3.p3} = -2/3
- 2iQP+Q2P2.
And then by (20) and (21), from what. is in any case a tedious computation, we finally arrive at (1/3)A{q2p, p2q} = (-i/12)[(Q2P+PQ2), (P2Q+QP2)1
= -1/3 - 2iQP + Q2P2. From (14) and (15) we have the two equations giving the contradiction promised in part i) of the proof. EXERCISE 5.12. Supply the missing calculations in the above proof.
Remark 5.13. The operators used here operate on a dense subspace of L2(R) and cache into being with the help of the irreducible representation 7r' . We can now attempt to extend the area of applicability of quantization, by enlarging the space on which the operators of the quantization operate. For instance, we can try to avoid giving the operators as arising from an irreducible representation. ABRAHAM--A'IARSDEN ([AM[, pp. 435
439) demonstrate the impossibility of extending the quantization to p for the case of the space of functions with values in a finite dimensional vector space., in which the steps presented here are just slightly generalized.
5.5. Towards the general case Kirillov sketches ([Ki], pp. 241 f.) the following picture of the process for converting from classical mechanics to quantum mechanics. Although this has already been described at various occasions, we reproduce it yet, once more to provide it unifying view for the general problems we will now handle.
In classical mechanics, the physical quantities are real functions f E .F(A!). The symplectic manifold Al is, in the first view, the phase space, thus the cotangent bundle T'Q to a configuration space Q; but in actuality the conversion does not only concern itself with this situation. The time course of a physical system is fixed by giving it. Hamiltonian vector field X11 E V (AI), where H E .1(M) is the energy of the system. Then the
5.5. Towards the general case
129
traversal of an integral curve -y of X11 describes the time course of a quantity f E F (141) and is given by the generalized Hamiltonian equation
J' _ (f. H}. A group G is called a symmetry group of the given system, when it operates via symplectic transformations on Al. In quantum mechanics the phase space, or the symplectic manifold Al, is replaced by the projective space P (il) to an appropriate Hilbert space ? { for the system. The physical quantities f are then carried over to self -adjoint operators f . which then operate on (or, more precisely, in) f{ (on the rather subtle problem of the appropriate space of definition of these operators, we
have no room here to elucidate). The value of the quantity f or f that is the state of the system is then described by a unit vector v1 E 11 which is a random variable with the distribution function p (s) = (E9vl, v1), where E4 is the spectral projection measure of the operator f. This means that in order to describe the state of the system by vi, the quantities f must have a fixed value a, for which vi is an eigenvector with eigenvalue a.
The time course is then given by a one-parameter group of unitary operators on }{ of the form U(t) = e 12".
where h is Planck's constant. H is here a self -adjoint operator, called the energy operator. so that then the time course of a quantity f = F satisfies ih [H, F]. G is a symmetry group of the system, when G operates as unitary operators on W.
A quantization is now the process of constructing, for a given classical system, a corresponding quantum system. It is however very difficult (when it can be done at all) to give a unique interpretation to the word corresponding. Classical mechanics should, in all possible interpretations, be understandable as the limit case of quantum mechanics for h - 0. It should not be expected that the reversal of this process is per se unique. Be that as it may, KIRILLOV showed that. this process is, in many known and simple cases, independent of the choice of the quantization process. In common with the sources we here cite KLRILLOV ([Ki], p. 243), ABRAHAMhIARSDEN ([AM], pp. 433 f.), and GUILLEMIN-STERNBERG ([GS], p. 265),
and, in view of the experience gained in Sections 5.1 through 5.4. we can attempt the following scheme. We will consider in F (M) the physical quantities of a given physical system and select from them certain ones, the so-called primary quantities
5. Quantization
130
f, which relative to the Poisson bracket generate a Lie algebra p. The transformation to a quantum system should now proceed so that
fHf is an R-linear map of p into a Lie algebra of (self-adjoint) operators on a Hilbert space 7t with
{fi,f2} =cIfI,f21 for fl, f2 E p c .F(M). In Kirillov [Ki], we have c = A., where, on the other hand, the Poisson (*)
bracket has the opposite sign from the one we have here used. In ABRAHAMMARSDEN [AM] as well as in GUILLEMIN-STERNBERG [GS], c is set equal
to -i. Further, it is required that the constant functions are contained in p and that we then have (**)
1 = 1 = idx. A rule which delivers exactly this is called a prequantization. Otherwise
(see, for example, ABRAHAM-MARSDEN ([AM], p. 434)) for a so-called full
quantization the operation of the i's must satisfy an irreducibility condition in the sense that there be no proper general invariant subspace, or at least, only finitely many. For p = .F<2, the construction in Sections 5.1-5.3 gave a full quantization, since it arose as the infinitesimal representation of an irreducible representation ni, of the Jacobi group. The statement of the Groenewold-van Hove Theorem 5.11 says that the domain of this quantization cannot be extended. We can now follow a somewhat different line, which is here taken from KIRILLOV ([Ki], p. 243). And this begins by taking, for M = 7"'R", the subalgebra p of F(M) spanned by the linear functions in pi , ... , p,,, and the commutative algebra po of functions in p which depend only on q,..., q,,. This algebra is infinite-dimensional and has the property that
if, g} c po for all f E p, g E po. In agreement with the terminology of Kirillov, this algebra p will be called the algebra of primary quantities. Under known conditions, it gives rise to a representation of the Poisson .algebra F(M), which then, on restriction to p, satisfies a minimality condition for the representation space 7i, which, it turns out, corresponds to the above-mentioned condition for full quantization.
The construction of a representation of the Lie algebra of primary quantities p. i) The starting point for this construction is the map .F(M) 3 f . -, Xf E Ham(M),
5.5. Towards the general case
131
which gives the vector field Xf, after Theorem B.4. as a derivation on F (M). and therefore as a linear differential operator. This extends to
f'f= -iXf and satisfies, because X{ f.y) _ -[X f, X9] (see Corollary 3.23), the condition
(*) with c = -i. Since the image of the constant functions in Ham(M) is zero, this procedure must be altered if we wish (**) to be satisfied. Therefore Kirillov takes, with the 1-form a on M.
f:=-iXf+f +a(Xf).
(+)
In this way, (*) is satisfied so long as for a we have da = w. EXERCISE 5.14. Using the relation 1) from Section 3.1, verify that
da (XI, X2) = Lx, (a (X2)) - Lxs(a (XI)) - a ([XI. X2]).
In the special case of M = T'Q one can take -a = t9 = E pidgi. and then (+) has the form
n(Ofa_afa1
f = -i j=1
ap, a9j
99l apj J +
f_
n
of
pf gpj j=1
and so in particular a a Pi = -igaj' qj = iapj +qj
ii) If w is not exact, one can use the above construction in a generalized form. In this case, by using the fact that w is closed and thus locally exact, a cocycle h (see Section A.2) is constructed which has an associated vector bundle E of rank 1. thus a line bundle. Then the construction is generalized by assigning sections f of this bundle to the primary quantities f as in (+). This then goes through (see Theorem 1 of KIRILLOV ([Ki], p. 245)), in the case that the appropriate normalization of the integral of w
Ic over an arbitrary 2-cycle ry in M is a whole number (an integral multiple of Planck's constant h in the normalization of Kirillov). It is further shown by Kirillov that THEOREM 5.15. There exists a one-to-one correspondence between the equivalence classes of representations of p and the elements of the cohomology group HI(M. C').
5. Quantization
132
In particular for connected M, H' (M, C*) is trivial, and therefore this quantization is unique. In the general picture constructed so far, nothing has been said about the Hilbert space R. nor has it been required that the operators f should operate as self-adjoint operators, or that the choice of f is required to be restricted to p. The operators which appear in (+) are (differential) operators which
operate on F (M) or on the space r (E) of sections of the line bundle E. As already mentioned in the discussion of thequantization concept, it is desirable that the space on which the operators f corresponding to f operate be as small as possible (so that the operation is irreducible or at least of finite multiplicity). In the explicit discussion of the examples M = T`Q = R2"
in Sections 5.1-5.3, there appears, somewhat informally, the space N = L2(R"), which then is the desired space; namely, for n = I we get the quantization of g1 with the help of the Schrodinger-Weil representation agy of the Jacobi group GJ(R). In the next most general case M = T'Q we see that f = L2(Q, du), with an appropriate measure dµ, is the proper space. Now for the most general situation we will greatly simplify the discussion, but still hope to make it clear that the choice of p makes it possible to manage with this small Hilbert space as the domain for the operators f . When M is not the cotangent space to a manifold, one would expect a Hilbert space of functions in n = (1/2) dim M variables. The local separation of variables in q and p via the form w_
dq,Adpi
is perhaps not globally possible. Halving the dimension is however possible with the help of the local concepts. This is given by defining on M a Lagrange distribution L; that is, by defining at every m E M, a Lagrangian subspace L,,, in TmM (on which u4n vanishes), so that these spaces vary from point to point in a differentiable way. (This can, like the concept of differentiable vector spaces, be made more precise). Analytically this leads to the choice of a maximal commutative subalgebra Fo (U) C Y (U), relative to { , }, for all neighborhoods U C M. Then we have
Fo(U)={fEF(U)IXf=0forallXEL}, and ['0(E, U) denotes the space of sections of r (E) for which the functions f E Fo(U) are assigned operators f, in accordance with the rule in ii), that operate as multiplication operators. This leads to a subspace [o(E) of the space of sections of t (E), which can also be characterized as the space of sections which are annihilated by covariant differentiation (see Section A.4) along arbitrary X E L. The primary quantities f E p are excellent, because
5.5. Towards the general case
133
they satisfy { f,.F0(U)} C .F0(U) for all U.
This condition implies that the operators f associated to f leave r0(E) stable. In the special case of M = T*R" this translates to the result that the f are primary, that is, sums of arbitrary functions of qI,... , q" and linear functions of Pi , ... , p". 'H is then the completion of I'c(E) relative to a fitting scalar product. so that the operators are selfadjoint.
Remark 5.16. This last briefly mentioned theme is introduced in KIRILLOV ([Ki], p. 241-248) with somewhat more, but still not full, detail, and shows that at least the topics introduced in the previous chapters and in the appendices can be collected into a general theory. The general theory goes yet further. but requires yet more application of functional analysis.
I hope that the interested reader is now sufficiently prepared to study the original sources: ABRAHAM-MARSDEN [AM], GUILLEMIN-STERNBERG
[GS], KIRILLOV [Ki], WALLACH [Wa] and WOODHOUSE [Wo]. In par-
ticular, we refer the reader one last time to WALLACH [Wa], which has an appendix by R. HERMANN that is a readable introduction to quantum mechanics with many historical and critical remarks.
je coji? Yo no se si t.e coji, plums suavisima. o si coji to sombra. J.R. JIMENEL
Appendix A
Differentiable Manifolds and Vector Bundles
The following definitions are standard in function theory and differential geometry, and also pass into symplectic geometry. The reader will find this material covered in [AM], p. 31 if., or [A], pp. 77 if. and 163 if. The following presentation is also influenced by the books of H. Cartan: Formes Differentielles [C], Hollman and Rummler: Alternirmnde Differentiolformen [HR], Sternberg: Lectures on Differential Geometry [St], and Chern: Complex Manifolds without Potential Theory (Ch), as well as the article by Kt filer: Der innere Di femntialkalkiil [K2].
A.1. Differentiable manifolds and their tangent spaces The concept of differentiable manifold is fundamental to the study of symplectic (as well as Riemannian) geometry, and so we begin with the definition of a p-dimensional (p E N) differentiable manifold. Here, differentiable will mean infinitely differentiable, although in many cases this will be stronger than needed. DEFINITION A.1. A connected Hausdorff topological space M with a countable basis is called a p-manifold if every point m E M lies in an open neighborhood t1 C M holneomorphic to an open neighborhood of R.
In particular, M is locally compact, and is also referred to as a locally Euclidian space of dimension p. 135
A. Differentiable Manifolds and Vector Bundles
136
DEFINITION A.2. A chart for Al is a pair (
of the image of U under V will be given by the coordinates x(XI, =. . .
, x ,). ).
have r,=cp,(m), i = 1,...,p.
So for
DEFINITION A.3. Two charts cp : U 1R'' and gyp' : U' compatible when the transformation functions `p o `p
1 I ;p(Urnu,)
and
'p o
RP are called
-I Io,(Iin11')
are differentiable.
In general, we will abbreviate this to Al is a differentiable manifold. Analogous to this concept of real manifold is the concept of a complex differentiable manifold of dimension p. In this case each chart V gives a homeomorphism of an open neighborhood U C 141 to an open set of Cr. One then requires that the transformation function between any two charts (cp, U) and (,p', U') be a holomorphic function. There is not, enough space here to cover the many examples that. can and do arise in the literature. For this reason only those examples which show up frequently in this text will be discussed.
i) Open submanifolds. Let Al be a differentiable manifold with atlas a, and 1410 an open set in Al. Then the family (Al() fl U, cpjA
with (cp, U)
in a. is an atlas ao on A-!0. Such A10, with the given differential structure, is called an open submanifold.
ii) p-dimensional submanifolds of R". In FORSTER ([F]. p. 128) and LANG ((L5], p. 363). one finds the following definition.
A. 1. Differentiable manifolds and their tangent spaces
137
DEFINITION A.4. A subset M C R9 is called a p-dimensional differentiable submanifold of IlPQ if for every, point m E M there exist an open neigh-
IR (i = 1, ... , q - p)
borhood V C IIt9 and differentiable functions f, : V with
(i) M n V = {x E V : fl(x) _ ... = fq_p(x) = 0},
(2) rank Df(in)=q-p, where
fi 0(h....' f9-P) _ Df = 8(x1 ...,xq) 1
0XI Of
8z
is the Jacobian off = (fl..... fq_p). It can be shown that such a differentiable submanifold is a differentiable manifold in the previous sense. Among the simplest examples are the hypersurfaces, with q = p + 1. Important examples of this class are the hyperplanes HP = {x E RP+1
f(x) = 01,
where f is a linear function p+l
p+1
.f (x) = ao + F, aixi,
ai2 # 0,
ao, ... , aq,+l E lR,
i=1
i=1
as well as the p-sphere SP
r
P+1
x72
x E RP+1
1
_1 l 1
I`
iii) Complex projective space PP(C). This is PP(C) = {(x)_ : z E CP+1\{0} },
the space of all complex lines in CP+1 passing through the origin; that. is, z - z' if there is a A E C` with z; = \zi for i = 1, .... p + 1. In this case. an
atlas a, for i = 1,...,p+ 1, is given by
Vi`i:U,={(z).: zi#0} m = (z)-
-
CP, ZI
zi
zP+1
Zi
Zi
Zi
x =
(where the symbol - indicates that this entry should be omitted). Clearly the transformation functions are holomorphic; for instance, V = p(l) and
A. Differentiable Manifolds and Vector Bundles
138
P' _ P(P+I) We have
p'(m) = x' _
zp+I zI
Z2'. zI zI
zP
zP+I
zP+1
it follows that i
'P O
1
I cp(UinUU+t) (x) =
r 1 ,2I , ... , X P - 1 l xP x P
xP
is then holomorphic, since xP = zp+I/zI 0 0 in UI fl Up+I.
Morphisms of differentiable manifolds. The appropriate concept of maps for differentiable manifolds are those maps which are compatible with the differential structures. DEFINITION A.5. Let M and N be differentiable manifolds of dimensions
p and q, respectively. A map F : M -+ N is called a differentiable map or a morphism precisely when the following are satisfied:
i) F is continuous, and ii) F is given locally by differentiable maps.
This means that there exist atlases o on M and a on N which fix the underlying differential structures of the manifolds, so that, for charts (gyp, U) from o and (u(', V) from f3, on w := U fl F- I (V) the composition
V(W) -, W Fi V is differentiable.
-O(V)
A.1. Differentiable manifolds and their tangent spaces
139
Remark A.6.
(1) The condition ii) is independent of the choice of atlases for M and N. (2) The phrase F is given by differentiable maps should have the meaning that o F o ;p-I gives a q-tuple of differentiable functions in p variables in the local coordinates. (3) The composition of differentiable maps is again a differentiable map.
DEFINITION A.7. A morphism of differentiable manifolds F : M -p N is called an isomorphism or a dif feomorphism. when F is a homeomorphism
to N and F-1 : N - M is also a morphism. As an indication of just how general the concept of a submanifold of R q is, we have the following statement.
THEOREM A.8. (Whitney) A differentiable manifold of dimension p is diffeomorphic to a submanifold of R2P+1.
Differentiable functions. For N = R, we get, as a special case of the definition of a differentiable map, the concept of a differentiable function on a differentiable manifold M. For an open subset V of M, denote by F(V) the ring of differentiable functions defined on V, namely those functions
f : V - R8 such that, for every chart (ep, U) of an atlas a of M with can be given by a differentiable U r V 0, the function f o function in the p variables xl, ... , xp. The function f o p-1 for a typical chart (gyp, U) will, in agreement with
the practice found in the literature, also be denoted simply by f. Thus f (m) = f (x) for the value of the function f for which m E V in the chart (gyp, U) corresponds to the point with coordinates x. Analogously, we denote by O(V) the ring of holomorphic functions for an open set V of a complex differentiable manifold.
Tangent spaces. There are many ways in which one may assign the tangent space to a point in of a differentiable manifold M. The central point is the demonstration that it is a vector space which has the same dimension as M. The following definition formalizes the visual image that the tangent space
in m E M is the set of the tangent vectors to all curves on M which pass through the given point. DEFINITION A.9. Let m be a point in M. Then the tangent space of M
at m is T.. M := CuRM/,,,,
A. Differentiable Manifolds and Vector Bundles
140
where Cu,,,M is the class of all pairs (J, y), where J is an open interval of R containing 0 and y : J - e M is a differentiable map with y(0) = m, with an equivalence relation defined on Cu,,,M by
(J, y) - (J', -/) precisely when d( , o'>')
for a chart (:p, U) with in E U.
With the help of the chain rule, it can be shown that this definition is independent of the choice of the chart. Remark A. 10. When M C R9 is a submanifold, this definition simplifies
as follows: v E R° is a tangent vector to M at m E M (thus E TmM) precisely when there is, for a given e > 0, a map y e) - M with y(0) = m and ''y(0) = v. This remark prepares the general statement that TmM, in a natural way, has the structure of a vector space of dimension p. THEOREM A.11. For each chart (cp, U) with m E U there exists a bijection
h: TmM-+IPp given by
(d(o)
Xm
(0)) i-l+ ..,p
Now, given a second chart (gyp', U'), with m E U' and v'(m) = y, and transformation functions y = y(x) = cp' o cp-l I+p(UnU') (x), then we have for the map h' associated to gyp', on account of the chain rule,
hi(Xm) _ E axt (x) hj(Xm) j j=1
(i = 1,...,p)
It is thus clear that T,,, M has a canonical vector space structure which is independent of the choice of coordinates. It is conventional, for a chart (cp, U) containing m with the coordinates x, to take as a basis for T,,,M the dual members of the canonical basis e of RP, so 13 Balm
=
\
axl lm19 OXplm
A. I. Differentiable manifolds and their tangent spaces
131
(the elements here can only be read as symbols). Then we write P
a TrnM:) Xmaiaxilm with a,EIR
(i=1....,p).
1=1
The transformation to another chart (p', U') with coordinates y then leads to the following calculation:
Xm = E b3 j=1
8 I
yi m
with bj = E ax, (x)a, r=1
This interpretation is consistent because of how partial derivatives behave under transformation, and it makes clear the following alternative.
The interpretation of TmM as the space of derivatives. Denote by .F(m) the collection of all the differentiable functions locally defined at m E
M, thus all pairs (U, f) with U an open neighborhood of m of Al and f a differentiable function on U. Further, denote by .Fm the set of all equivalence
classes (U, f)- =: f with
(U, 1)-(U'.f') precisely when there is a neighborhood U" of m contained in U fl U' with f I p" = f'I u,,. Such an equivalence class (U, f ) - is called a germ of f . Fm can be easily given the structure of a ring, and from the value f (m) of f at m the germ (U, f)- also has a uniquely defined value 1(m) at m. DEFINITION A.12. The tangent space of Al at m is TmM = Der (.Fm, R).
Here Der (.Fm, lit) is the set of all R-linear maps
L :.Fm -+ R with
L(f g) = f(m)Lg+g(m)LJ for all j, g E .Fm. Such linear maps are called derivations and can be interpreted as directional derivatives on the basis of the following theorems, which show that Definitions A.9 and A.12 are equivalent. Note that, as the space of linear maps, the space of derivations has an R vector space structure in a natural way.
THEOREM A.M. There is an isomorphism
v : Cum Ml-
Der (.Fm, R)
given by
Xm = (J,'Y)-
LX_
with
d(f dt y) (0),
A. Differentiable Manifolds and Vector Bundles
142
where f is a representative for the germ f E .fm. If yU) is a curve with cp; o-y(i) (t) = b;?t, we get the associated derivation, here denoted by L3. Clearly L3 f _ d(f o (i)) (0) _ 8(f -1) (x), oxi which, by a slight abuse of notation, may be also written as
L,f
jmf.
And so this is the derivative of f in the direction of 7(() (the xj-axis in the chart (W, U)).
Cotangent spaces. As background for the later introduction of differential forms, the following concept will be useful. DEFINITION A.14. Let M E M. Then the cotangent space to the point
mis TmM = Horn (T .. M, R),
i.e., the dual space of the tangent space, consisting of the R-linear maps an from TmM to R. These maps will also be called cotangent vectors.
We now introduce some conventional notation. Let an E T;,,M be the map
am : T,,,M -b R, Xm -+ (Xm,am) := am(Xm) Since TmM is a p-dimensional R vector space, this is also true for Tn, M. The standard basis 1m of TmM associated to the chart cp with coordinates x corresponds to a dual basis, which will be denoted as (dx)m. Thus (dx)m denotes the p-tuple of linear forms on TmM with j)m`f
\
9 1., (
= bii-
An am E 7`,,,M with
am = > ai (dxi)m,
ai E R (i = 1, ... , p)
would in the coordinates y of another chart cp` and the correspondingly
constructed basis (dy)m be written as
am = E bi(dyi)m.
With the notation x = x(y) = cp o
Vr-1(y)
A.I. Differentiable manifolds and their tangent spaces
143
for the transformation functions between these two charts, we can evaluate the coefficients:
P
b, = E
c3xj i(y)aj
j=1
Remark A.15. Every f E F(m), and thus every differentiable function which can be defined on a neighborhood of m, gives rise to a cotangent vector (l)m, namely the linear form defined by (Xm, (df)m) =
d( dt
(0)
(here y again stands for a curve through m with the tangent vector Xm). This is in agreement with the notation dxi in the case that f = xi is the ith coordinate function of the chart cp, and with the usual concept of the total differential. Since, relative to a basis (dx)m, (df ),,, can be represented by
(df)m = E ai(dxi)m, we have that (dx)m is the dual basis to
a - \l
axj
Im, (df )m> =
j
Im,
(x) (more precisely, aj =
a(fax i
i) (gy(m)).
Thus P
(tf )m = E 8x; (x)(dxj)m j=1 /
/
Maps of tangent and cotangent spaces. Let F : M N be a morphism of differentiable manifolds, m E M and F(m) = n E N. Then in the following way we get a map
(F.)m : TmM , of the tangent spaces. Given Xm E TmM represented by the curve (J, y) in M and y(0) = m; then (J, F o y) is a curve in N with F(-y(O)) = n and its tangent vector can be taken as the image of Xm. Thus
(F.)m(Xm) = (J, F o 1'),. =: Yn E TN, and this is well defined. Dual to this stands the map Fm : 7T N -. TTM
of the cotangent spaces given by the following procedure. For 8n E T;, N we take for the image F;nj3 in T,,,M the form which for all Xm E FmM is defined by
FmFn(Xm) = Thus, in other words, (Xm, FmF fl) _ ((F.)m)(m, A,).
144
A. Differentiable Manifolds and Vector Bundles
F is called a submersion if F.,,, is surjective for all m E M.
A.2. Vector bundles and their sections The concept, which is general in physics and particularly frequently encountered in symplectic geometry, of a vector field X on a manifold M, is easy to grasp. The concept consists of a map which assigns to each point a tangent vector Xm E TmM. Somewhat more delicate to state precisely is when such a vector field should be called continuous or differentiable. This succeeds
quite easily, however, with the help of the concept of vector bundle. We will now study this concept and will follow the treatment by FORSTER in Riemannsche Fldchen ([FRF], pp. 195 ff.), where one may find the proofs which are here omitted. As a more complete coverage of this material, ABRAHAM-MARSDEN ([AM], pp. 37 f.) and the early pages of CHERN [Ch] can be recommended. Grossly stated, a vector space bundle, or, more frequently as well as more briefly, a vector bundle, is a manifold which assigns in a smooth way to each
point m of a basis manifold M a copy of a standard vector space V. Here one may imagine, for instance, the collection of all m of M = S2, the sphere in R3, with attached tangent spaces. This is then a 4-dimensional structure. We now make this both more general and more precise, and examine some of its properties. We assume for now that K = JR or C. DEFINITION A.16. Let E and it! be topological spaces and 7r: E -+ M a continuous map. Assume further that each fiber E,,, = 7r-1(m), m E M, has the structure of an n-dimensional K vector space. 7r : E - M or more briefly just E is called a K vector(space) bundle of rank n over M precisely when the following condition is satisfied. For every point m E M there are an open neighborhood U and a homeomorphism h from E11 := 7r-1(U) to U x K" with the following properties:
i) The projection map factors through h; that is, the following diagram is commutative: h
U
A.2. Vector bundles and their sections
145
ii) For every m E U, the map hIE,,, is a vector space isomorphism of
The map h : EU -' U x K" is called a linear chart of E over U. If it = (U,)jEj is an open covering of M and the hi : EU, -+ Ui x K" are linear charts, then the family 21 of the hi is called an atlas of E. A vector bundle of rank n is called trivial when there exists a global linear chart.
h:E-MxK".
Remark A.17. A vector bundle is thus defined to be locally trivial, and so, under local examination, the concept of a vector bundle delivers nothing new. It is only with a global study that they become interesting. THEOREM A.18. Let E -+ M be a vector bundle of rank n over M, and i an element from an index set I; let
iEI,
hi:EU,-4UixK",
be the linear charts of an atlas of E. Then there are uniquely defined continuous maps
tpij:UifU.-4GL"(K), such that for the maps
hij:=hiohil:(UiflUj)xK"- (U1flUj)xK" we have
hij(m, v) = (m, tli,j(m)v)
for all
(m, v) E (Ui fl Uj) x K".
On U; fl Uj fl Uk the cocycle relation holds:
thj Ojk = ''ik-
A. Differentiable Manifolds and Vector Bundles
146
Remark A.19. As a matter of notation, the maps ii j are called transformation functions, and the family (vG:I), i, j E I is called the atlas 21= (h;),El associated to the cocycle. DEFINITION A.20. Let M be a real or complex differentiable manifold,
E M an R or C vector space bundle of rank n over M, and
21=(h,:EU4-U,xK",iEI) an atlas of E. The atlas is called differentiable if the associated transformation functions tb,, are differentiable. Two differentiable atlases 21 and 2l' are called compatible if 21 U 21' is again a differentiable atlas.
It is easy to see that this compatibility is an equivalence relation. An equivalence class of compatible differentiable atlases is called a differentiable
linear structure on E. A differentiable vector bundle is a vector bundle E M equipped with a differentiable linear structure over a differentiable manifold. This definition deals with the real case. When V is a C vector space, everything that follows can be easily carried over. Remark A.21. For a differentiable vector bundle on a differentiable manifold with the projection map r : E - M, E is itself a differentiable manifold and rr is a differentiable map.
A differentiable vector bundle E - M is called differentiably trivial if the differentiable linear structure contains an atlas which consists of but a
single chart, E M x K". DEFINITION A.22. Let E _ M be a differentiable vector bundle and U C M. A differentiable section of E over U is a differentiable map
f:U-#E
with
rof=idr.
The collection of all sections will be denoted by I'(E, U) K vector space.
.
This is again a
Analogously one can, for every r E No, define the sections of class Cr. The resulting spaces are then denoted by Cr (U).
A.3. The tangent and the cotangent bundles Differentiable vector fields can now be defined as differentiable sections of the associated tangent bundle E = TM of a differentiable manifold M. In this somewhat general framework, we will consider the construction of vector bundles.
For an open subset U of the differentiable manifold M, denote by GL"(-F(U))
A.3. The tangent and the cotangent bundles
147
the group of all invertible n x n-matrices with coefficients in the space .F(U) of differentiable functions on U. If it = is an open covering of M. then denote by Z' (!t. GL,, (-F))
the set of all 1-cocycles relative to it, that is to say, the families (i'1j)i.jEl with
li',j E GL,(.F(Ui n Uj)) and
1Vij7Jjk = L''ik over Ui n Uj n Uk for all i, j, k E I.
If 21 is a differentiable atlas of a vector bundle over Al, then the family of the transformation functions of 21 generate such a 1-cocycle. In the reverse direction, from every 1-cocycle from Zl (ft. GLn (.F)) a differentiable vector bundle of rank n can be constructed.
THEOREM A.23. Let Al be a differentiable manifold, U = (UI),EI an open covering of Al and (irij) a family from Z1(I1.GLn(.F)). Then there are a differentiable vector bundle 7r : E - M of rank n and a differentiable atlas K")iEl of E whose transformation functions are the given $Yij.
Of the proof, we will only say enough to show that the bundle E - Al is given by the space
E':=UU,xK' x{i}CAfxKnxI iEI
by introducing an equivalence relation (m. V, i) - (m'. v', i') precisely when
m=m'
and
v=yii.(m)v'.
Then, with the help of a few facts from topology. one can show that E arises
as E'/ As an application we will now develop the concepts of the tangent bundle TM and the cotangent bundle T' M. To this end, let M again be covered by It = (U1)iE j and, as we did earlier, denote by Bp(i) : Ui -+ K" the coordinate
map on the ith chart. Then, for the tangent bundle TM, we can take for the transformation functions t4'ij those matrices which are the Jacobians of the transformations on the charts Bp(i) o thus
i'ij(m) =
(cp(')(m))
A. Differentiable Manifolds and Vector Bundles
148
If ;pW has the coordinates y and 1p(i) the coordinates x, this says that from
h;;=hiohjl:(U,nU;)xKP -. (U;nUj)xKP. (m, a) --+ (m,ap;i(m)a = b), we may calculate that the vector a = (aµ) in the x-coordinates has, in the with y--coordinates, the components b = P
"Y"
N=1
For the cotangent bundleT'M one correspondingly takes O>> (m) =
which for the vectors a* and b' is the transformation given by
Remark A.24. As a set, these are
TM = U Tm M, respectively, 7-M = U T,,, M; mEM
mEM
that is to say, they are the union of all tangent, respectively cotangent spaces, to all points m E M. A differentiable vector field, thus a global differentiable section, X : M - TM can be written for a chart (gyp, U) with the coordinates x = (x1,.. . , xP) (with a slight but already familiar abuse of notation) by Iu
X
P
=
a.(x) µ=1
Cxµ
with
ap E F(U).
For a chart (ip', U') with coordinates y = (yl,... , yP) we have, correspondingly, X I u, _
b,, (y)
8
with b E F(U'),
v=1
and in the overlapping set U n U' we have (so long as it's not empty) the transformation rule bi(y) =
'.(y)aµ(x(y))
(contra).
Correspondingly, a field of cotangent vectors can be considered, which we call differential forms of degree 1 (or 1-forms), and can be introduced as differentiable global sections
a:M-+T'M
A.3. The tangent and the cotangent bundles
149
of the cotangent bundle. For these, we have analogously arJU =
a,,(x)dxµ,
a;E .F(U),
respectively
aJu, = Eb (y)dy,,, b;, E .F(U), with the transformation rule b,(y)
-(x(y))aN(x(y))
(co).
DEFINITioN A.25. The differentiable global vector fields on M will be
denoted by V(M), and the 1-forms by 1'(M). Remark A.26. In the older literature, particularly in many texts from physics, one finds the following definitions: A contra, respectively covariant
vector (or one-fold contra, respectively covariant tensor) on M is a rule which assigns to a chart (w, U) of M with coordinates x a system of p differentiable functions A,, with the condition that for two charts (cp, U) and (gyp', U') with U fl U' 36 0 a relation as in (contra), respectively (co), exists between the function systems A,,, and A,,,,. In this sense the components of a vector field describe a one-fold contravariant tensor, and those from a degree 1 differential form a one-fold covariant tensor. EXAMPLE A.27. For every differentiable function on M, its total differential df can be represented by the 1--form which in a chart (gyp, U) with the coordinates x has the form P Of OXI, df = µ-1 - d x,.
Maps of vector fields and 1-forms. The description of the maps of tangent and cotangent spaces at the end of Section A.1 leads immediately to the fact that vector fields and their dual objects are mapped by a differentiable map F : M N in the following way. if F is injective, then to X E V(M) is assigned a vector field F.X on F(M) C N such that at every n = F(m) E F(M) (F.X )n = (F.)mXm is assigned. When F(M) is a submanifold, one can prove that this vector field is again differentiable. And this is the case when F is an embedding, that is, all F.m are injective and F is a homeomorphisin of F(M). In order to pull back a 1-form 13 E 01(N) on M, the injectivity of F is not needed. F*,3 E 01 (M) is defined at every m E M so that (F"J3)m := FF[3n with n = F(m). This assignment gives a differentiable section of T* M.
A. Differentiable Manifolds and Vector Bundles
150
With a little more formalism it can be seen (see, for example, ABRAHAM-
MARSDEN ([AM], p. 45)) that a differentiable map F : M N gives rise
to a differentiable map TF : TM - TN of the tangent bundles and analogously to a differentiable map 7F: T'N -' T'M of the cotangent bundles.
A.4. Tensors and differential forms With the help of some multilinear algebra (see, for example GREUB (Gb], MARCUS [Ma], or, for the complete background, BOURBAKI ([Bo], Chap-
ter III, §5)) one may carry this concept yet further. Given three (finitedimensional) vector spaces T, T' and W, and a number q E N, let L (T x T', W) be the K vector space of bilinear maps
f : T xT'-+W. The tensor product of T and T', T ®T', is then the (up to isomorphism, uniquely determined) K vector space with
L(T x T', W)
Hom (T 0 T', W) for all W.
Let Lq(T,W) be the K vector space of q-linear maps f : Tq
W, and Aq(T, W) the K vector space of all alternating q-linear maps f : Tq -+ W. Then Aq T is the qth exterior power of T, thus the (up to isomorphism, uniquely determined) K vector space with q
Aq (T, W) = Hom (AT, w) for all W. Now put Lq(T) := Lq(T, K) and Aq(T) := Aq(T, K); so that, in particular,
L1(T) = A, (T) = Hom (T, K) = T'
is the dual space of T and A.(T) = l\qP. Let t = (t1, ... , tn) be a basis of T and t'
t') a basis of T'.
Then
t 0 e_ (ti 0 t')i j=1,_,.,n tq = Iti, 19 194-q)1
is a basis of T ®T', is a basis of is a basis of
®q T,
nqT,
t' = (ti, ... , tn} with t; (t,) = (ti, tJ) = bij is a basis of T'. With these algebraic constructs we may now obtain yet more vector bundles
E on M, whose fibers E,n are, for example, ®q T, M or A12 M. In the generalization of the definitions of the previous section, an 1 -fold co and qfold contravariant tensor on M is then, for example, a differentiable section
A.4. Tensors and differential forms
151
X of this bundle E such that X is a map which assigns to each m E Al an element 9
t
11
XmE®TmM®(®TmM) every
point m from a chart o of an atlas a of M, the components X,,, relative to the standard bases and is differentiable in the following sense. For associated to w, given as described above by
Im for T..M, respectively. t` = (dx),,, for T,*,, AT,
t YX_
are differentiable. Moreover a change of charts carries this assignment via an adequate assembly of transformation rules for both (co) and (contra).
Remark A.28. Without exploiting multilinear algebra, one can, as in Remark A.26, also give a definition of these in the style of the old definition.
An I-fold co and q-fold contravariant tensor is a set of data which assigns to every chart (gyp. U) of M a system
X
1
'
s=1,... 1
of differentiable functions in p(U) with the condition that for two charts cp and cp' with U fl U' 36 0 between the systems X,, and X' a relation exists which gives for every index i a (contra) transformation formula and for every index j a (co) transformation formula. EXAMPLE A.29. As an example of a 1-fold contra and 2-fold covariant tensor, we have i9x ax (x(y)) (y) (x(?!))(A,)a;32(x) (APy,j2(y) =
E
ayj
rt,11.42
Of greater significance, especially for physics, are those tensors which moreover realize a known symmetry condition. Before introducing these, we next consider an example of one of the fundamental objects of Riemannian geometry.
The R.iemannian metric. For each m E M, let a scalar product (
, ) in TmM be given-that is, a symmetric positive definite bilinear form. In the canonical basis representation this means that for v, w E TmM
(u, w) = E vjwjg7j
ij
with
o;; (w(mll :_ _8
Ial
for i. i= 1..... v.
A. Differentiable ,Manifolds and Vector Bundles
152
The assignment of an atlas a of M for each m E M and every chart (gyp, U) allows one to assign to m' -- i (gij(V (m))) for m E U
a system of functions, defining then a twofold covariant tensor. This tensor is called a metric or a Riemannian fundamental tensor, when
a) the matrix (gi1(x)) for all x E cp(U) is symmetric and positive definite,
b) the functions gil (x) are all differentiable, and
c) between the function systems of compatible charts, there is the following relationship between the transformation functions:
9ij(y) _
a (x(y))j (x(y))9rs(x(1I)}.
Naturally, a metric tensor can be better understood as the differentiable section of a bundle E over M which has as fibers Em the second symmetric power S2T,M of the cotangent space 7 M. It is also frequently fixed with the help of a degree 2 symmetric differential form in the following way (in the standard--chart (gyp, U) with the coordinates x): ds2
= F 9ijdxi dxl.
A Riemannian fundamental tensor in this form gives rise to a Riemannian metric, which for m, m' E M and a curve y linking the two points defines the distance between the points along -y by
f Vz. With this the manifold receives yet another additional structure which is often of great help. One may derive, from a metric fundamental tensor, the Riemannian curvature tensor which is required for the formulation of the theory of general relativity. The following theorem is therefore of particular interest THEOREM A.30. On a differentiable manifold there is but one Riemannian fundamental tensor. Proof. See HOLMANN-RUMMLER ([HR], p. 108) or LANG ([L2], p.231).
0
A.4. Tensors and differential forms
153
Of great significance is also the concept of pseudo -Rieman.niann manifold. which comes from a slight weakening of the conditions on the metric fundamental tensor. In particular one weakens condition a) in the above
definition to the requirement that the matrix (gi,(x)) be symmetric and non-degenerate. From total skew-symmetric covariant tensors one gets the (exterior) qth degree differential forms: differentiable sections CY(q) of the bundle E over M whose fibers E,,, are the qth exterior powers A9 T,,, Al' of the cotangent space are called exterior qth degree differential forms or simply q-forms. In the standard chart (V. U) with the coordinates x, they have the reduced representation a(q)Ic-
=
ai,...iq dxi, A ... A dxiq,
ai1,...,iq E.F(U).
1
Using the notation I. :_ {(i1.... J J1 < i1 < ... < iq < p}, this is frequently more briefly written as a(q)
a(i)dxi, A... A dxiq = (i)Elq
a(i)dxq i) (i)Elq
The set of exterior q-forms on Al forms an R vector space, in which the value at the points m E M is declared to be the product. This space is denoted by Q 9(M), and, putting Q°(M) we have the definition: P
(DS29(1) _: QW). q=o
The calculus of differential forms. The concept of differential forms allows one to define yet other operations in addition to taking sums. Here we will only give a short overview. Begin by letting U C RP be an open set. It will later be made clear that all can be carried over to differential forms on manifolds.
1) As already noted, two differential forms
a(q) =
a(i)dx9)
d(q)
_
E 1lq(U)
can be added: a(q) + Q(q) := t'(a(i) + a(i))dxl(i).
Moreover, the multiplication of a differential form by a function f E .F(U) is well defined by for(q) :_ E fa(i)dxq(i). and therefore 119(U) can be considered as an .F(U)-module.
A. Differentiable Manifolds and Vector Bundles
154
Moreover, n(U) = ®St'(U) is itself a ring. The product is given by setting (dxi, A... A dxiq) A (dx A ... A dxx,) := X(ir)dxf, A ... A dxtq+r,
where X(a) = 0 whenever all the i and j are not different, and otherwise to
X(a) is the sign of the permutation 7r which carries
the ordered form (l1, ... , lq+r) with 1 < 11 < ... < lq+r. Linearity then permits a product to be defined by nq(U) x Str(U) (a(q), a(r))
.--y SZq+r(U), a(q) A a(r)
This product is associative, but not commutative. It is called exterior multiplication and satisfies the commutation rule a(q) A a(r) = (-1)gra(r) A a(q).
From the interpretation of the differential forms a(q) and a(r) as alternating q-, respectively r-, linear forms on TmM, we get that a(q) A a(r), through antisymmetrization of the operation of a(q) x a(r) on (TmM)q+r, defines a (q + r)-linear map. Remark A.31. Differential forms may be added coefficient by coefficient and (by writing one after the other) may be associatively multiplied under the commutation rules
dxi A dx, _ -dx, A dxi 0
for i 36 j, otherwise.
For the specification of differential forms, one can exploit the already used ordered or reduced representation, so that an element of the skew-symmetric representation is usually written as P
a(q) = E 1 qb71 dx)1 A ... A dx79 q
31,...Jq=1
and so it then holds that bil...jq
= 0,
when j1, ... , jq are not q different integers,
= X(7r)ail...iq,
when jl,... , jq are pairwise different and (i1, ... , iq) is carried by the permutation 7r
to (ii,...,jq). 2) The assignment discussed at the end of Section A.3, which assigns to every f E .F(U) = Q°(U) a 1-form df E 11(U), can be extended to an arbitrary q-form, and then delivers the exterior differentiation.
A.4. Tensors and differential forms
155
DEFINITION A.32. For a differential form a(Q) in the ordered represen-
tation a(q)
_
a(i)dxi, A ... A dxi,,,
the total differential is defined to be do(q)
da(i) A dxi, A ... A dxiq,
da(i)
ax(k) dxk.
For further calculations da(Q) is brought to the ordered or the skewsymmetric forms. The operation d is then extended linearly to the whole of 11(U).
DEFINITION A.33. A differential form a E 11(U) is called closed when
dw = 0, and exact when there is a 0 E 11(U) with a = 0. The total differential d satisfies the following easily verified calculation
rules for aES1Q(U) and0E1r(U):
d(a + 3) = da + d(3 (for q= r), = da A /3 + (-1)Qa A d0,
d(a A p)
d(da) = 0. Poincare's lemma says that for convex U every closed differential is exact. With the help of cohomology groups (see Section B.3) one may significantly
sharpen this important statement. 3) In the interpretation of the differential as a differentiable section of a bundle, the appropriate behavior under transformation from one chart to another is already built in. To make our notation compatible, we postulate the following general behavior under substitution of variables. Let U C RP be open with coordinates x = (x l, ... , xp), V C W open with coordinates
y = (yl,..., y,-), and
F:U -. V.
x -- F(x) = y,
a differentiable map with the components fi(xl,... xp), i = 1, ... , r. Every differential
E E b(r)dyi, A ... A dyiq E 11(V) q=0
(_)
on V then assigns a differential on U by F ' ( 3 = /3F =
> b(i) o Fdfi, A ... A dfsq. q=0 (i)
It is not hard to prove that this. In the special case of the earlier coordinate change with the given preconditions, the following holds.
A. Differentiable Manifolds and Vector Bundles
156
Remark A.34. The map F : U F' : O(V)
V induces a map St(U),
Q --+ F`p = OF, which obeys the rules
(,3+Q')F = j3F+p'F, (0 A 3')F = OF n YF, and therefore is a ring homomorphism. Since substitution and differentiation commute, we have
d(F'l3) = d(13F) = (d)3)F = F'(d13) for all Q E Q(V).
With this remark, it is clear that all the earlier operations defined only on U C RP, thus locally defined, are also globally defined, and so hold for differential forms on manifolds. When a vector field X E V (M) is given, we may define yet another pair of operations on the differential forms St(M) on M.
4) The inner multiplication i(X) decreases the degree of a differential form a(q) of f&(M) by one. It is defined by assigning to i(X)a(q), for X E V(M), the map i(X)a(g)(X1,---,Xq-1) := a[9)(X,X1,...,Xq_1). It is not hard to see (see ABRAHAM-MARSDEN ([AM], p. 115)) that i(X)a(q) E Q9-1(M) really holds, and that i(X) satisfies the following rules:
i) i(X) is R-linear with i(X)(a(g) A 13(r)) = i(X)a(g) A #(r) + (-1)ga(q) A i(X),3(r), ii) i (f X)a(q) = f i(X)a(q),
iii) i(X)df = Lx f for f E.F(M), and iv) F*(i(F.X)/3(q)) = i(X)F`f3(q) for F : M -. M' a diffeomorphism and 1(q) E n (M). 5) The Lie derivative Lx doesn't change the degree of a differential form. For Q(q) E f(M) the Lie derivative Lxa(q) is defined at every point m E M by (Lxa(g))m
:= d Here Fx means the flow associated to X E V(M). Grossly stated, this ((Fx(t)`a(g))mLo.
is a one-parameter family Fx(t) of diffeomorphisms in the direction X.
A.4. Tensors and differential forms
157
Somewhat more precisely, a fundamental statement, which will be heavily
used in what comes next, and which can be proven with the help of the theorem for the solution of systems of ordinary differential equations (see ABRAHAM-MARSDEN ([AM], p. 67) or STERNBERG ([St], p. 90)), says that
the integral curves ryx to a differentiable vector field X with yx(0) = X,n' for all m' = yx (0) from a neighborhood U of m form a flow F = Fx ; that is, a family (Fx(t))t of diffeomorphisms from U to F(U), where the parameter t E R is taken from an interval containing 0. Next for the complete X for which U = M can be taken, later also made more general (see ABRAHAMMARSDEN ([AM], p. 90)) and thus made independent of U, we can make the following definition:
Lxa(a) =
(Fx(t)a(a))
dt
t=o.
For these Lx we have the computation rules (see ABRAHAM-MARSDEN ([AM], pp. 113 ff.))
i) Lxda = dLxa for a E 1(M), ii) Lxa = d(i(X)a) + i(X )da, iii) Lxa is R-bilinear in X and a with Lx (aA/3) = LxaAO+a ALx,3,
iv) L fxa = f Lxa + df A i(X)a for f E .F(M),
v) Lxi(X)a = i(X)Lxa. In complete analogy, one can define the Lie derivative for a vector field Y E V(M) as
Lx Y:= dt
(Fx(-t).
Y)It_o'
This is, like the definition of Lxa(Q) above, a kind of symbolic notation expressing the derivative in the direction of the vector field fx. In principle one has to go back to the definition of maps of cotangent and tangent spaces from the end of Section A.2, and arrive at
(Lx Y).. = dt ((Fx (t). 1)n Yn) Jt-o
for
n := Fx (t)m
and
(Lx a)m = d (Fx (t) m an)It-o This will be used further in Section B.1, and more details can be found in ABRAHAM-MARSDEN (]AM], p. 90/1).
A. Differentiable Manifolds and Vector Bundles
158
There is yet another important concept of derivative, the covariant derivatives. However, these are only definable in the context of connections, which will now be introduced in a general context.
A.S. Connections In [Ch], p. 33, CHERN introduces the general concept of a connection on a differentiable vector bundle E over a manifold M. Here we only consider the case of the tangent bundle TM on M (see ARNOLD [A] and ABRAHAMMARSDEN [AM]). In what follows we give a formulation of under what conditions a differentiable vector field X E V (M), viewed as a differentiable global section of TM over M, can be understood as consisting of parallel vectors. Or - possibly easier to visualize - how a vector Xm E TmM along M passing through m can be displaced parallelly. This a curve y : J can be accomplished by making this image more precise. That is, between the tangent spaces T,,,M and Tm'M of two neighboring points m and m', which are isomorphic only through an abstract isomorphism, we look for a connection, i.e. a concrete isomorphism 'm,m' depending differentiably on (m, m') and such that 0m,m = id. One way to make this more precise consists of defining a map for a given vector field X E V(M) at every point m E M, which to every Xm E TmM and every direction Ym E TmM assigns a derivative (VyX)m of Xm in the direction Ym. Xm does not change in the direction fixed by Ym if this derivative (VyX)m is 0. This then leads to the following definition (see ABRAHAM-MARSDEN ([AM], p. 145)).
DEFINITION A.35. A connection V on a differentiable manifold M is a map
V : V(M) X V(M) (X, Y)
--f V(M),
- VyX,
satisfying
a) V is R-bilinear in X, Y E V (M),
b) VfyX = fVyX, and Vy(fX) = fVyX + (Lyf)X for all f E F(M). VyX is called a covariant derivative of X along Y (to the connection V).
Lyf means the Lie derivative of f, as covered in the last section. In a for TmM with chart (.p, U), with coordinates x and the usual basis ej = m E U, let the Christoffel symbol r?k of the connection V be introduced as v
Vefek =
i=I
rikei.
159
A.5. Connections
For X = (1)
aiei and Y = > bjej we then have
(DyX) _
ciei
with ci = Y
C78xai
j
i
bj +E r.,kakbj. j,k
Now let y : J -' M be a curve in M. Unfortunately the associated tangent vectors (t) E TmM for m = -y(t) do not form a vector field Y on M, but it still makes sense to see these as a part of such a vector field and to denote by V (t)X the covariant derivative of X along y. We get DEFINITION A.36. The vector field X E V(M) consists of parallel vectors along y when Ory(t)X = 0 for all t E J.
With the use of the frequently needed notation
X=EXiei we have, from (1). for the covariant derivative of X along y (2) (V7(t)X)i =
dt(Xi(y(t))) +E r;k(y(t))Xk(y(t))7i(t) (i =1,...,p). j,k
When now the tangent vectors y' (t) to the curve y are substituted for X, we get, from (2), as a condition for the tangent vectors being parallel in the sense of the direction fixed by the connection V, the system of differential equations (3)
yi(t) +
rj,k(I(t))y,(t)yk(t) = 0. j,k
The most important special case is when the connection is fixed and M has the structure of a Riemannian (or pseudo-Riemannian) manifold and then one is given a metric fundamental tensor g = (gik), respectively, a symmetric 2-form d32
=
gikdXidxk
Then the solution curve y of the differential equation (3) is called a geodesic. It is a fundamental fact that for such a curve the length of -y,
has an extreme value. For the given gik,respectively, tg-I
(9'k) := = (9k,)-I, we calculate the Christoffel symbol of the connections to g as
r = (1/2) E(ai9j1 + aj9ii - ai9ij)9tk 23
I
A. Differentiable Manifolds and Vector Bundles
160
Connections on vector bundles. Inspection of formula (1) shows that in the definition of connections given above, the coordinates bj in the direction Y' remain unchanged under the covariant derivative. As in CHERN ([Ch], p. 33), one can give a definition for a connection in the general situation of a differentiable vector bundle E of rank d over M with r (E) as the space of global sections over M.
DEFLNiTION A.37. A connection V on E - M is a map
V:F(E)--.I,(T'M(9 E) with
i) V( + 77) _ V£ + Vr) for all , p E r'(E),
ii) V(f) = df ®t; + fV for all t E I'(E), f E.F(M). Locally on an open set U C M with a d -frame 9i, .. , sd (that is, sections s{ E r(E, U) so that sl (m), ... , sd(m) are linearly independent for all m E U) this is written as d
Os;_F'w;s;, i=1,...,d, j=1
or in matrix notation as Vs = ws, where the
E r(E, U), thus { =
are 1-forms on U. For
fist with j E F(U),
we then have because of i) and ii)
vE _
(4)
w. +
ti+d )s
Thus (5)
dC;+1: J=1
l1jktj)dxk, k=1
i=1,...,d.
j=1
For the case E = TM, the equations (4) and (5) can be equated with equation (1), and then it can be seen that Definition A.36 is carried into Definition A.37 when for X, Y E v (m) = r (TM) one substitutes
VyX = (VX,Y), where here standard pairing
I'(T`M ® TM) x V(M) - V(M) is induced by the dx;
a axj
= 6{j.
A.5. Connections
161
The generalization of Definition A.36 of connections makes it possible to make sense of a vector field X being parallel to a curve. Thus, as a consequence of Definition A.37, a section £ E r (E) will be called horizontal when it satisfies or, locally, d
dpi +EwJtj = 0,
i = 1,...,d.
i=1
This can be understood as a system of partial differential equations. In general, they will have no solution, but carry over to a solvable system of ordinary differential equations, when one searches for a horizontal section over a given curve 7 on M. Up to this point we have considered only one chart si. Suppose now that another d-frame is given by
and a fixed d-frame
s' = As, where A is a nonsingular d x d matrix of differentiable functions on U. If w' relative to s' is given by
Vs' ='O's" we derive, as the transformation law of connections, (6)
w 'A = dA + Aw
We further introduce a d x d matrix fI of 2-forms, called the curvature matrix relative to s, according to the formula
11:=dw+wnw. Then we can deduce, after exterior derivation of (6) with the matrix Q' corresponding to s', that the transformation law is S2'A = AS2.
Here it can be seen that the vanishing of f2 is independent of the choice of the d-frame. A connection w with
52=0 is called flat or integrable. When the connection is derived from a metric, the curvature has the geometric meaning that is studied in differential geometry. In any case, one can by the exterior derivation of the defining equation of S1 arrive at the Bianchi identity
dQ+11Aw-wAS2=0.
A. Differentiable Manifolds and Vector Bundles
162
Given a connection V on the tangent bundle TM, as in (4) and (5), with the help of the Christoffel symbols I)k and the 1-forms written as
j=1
one may now give, in a natural way, definitions for covariant derivatives of differential forms, tensors and, more generally, tensors constructed from differential forms (see for example, ABRAHAM-MARSDEN [AM], p. 148) and KAHLER [K2], p. 440)). For a differential form a E f2(Q)(M) and
h = 1,... , p, the covariant derivative in the direction his as dha r, A era az := h - r with
era = Q, when a = dz,. A$ + y, where dxr does not arise in y; thus era = i(Xr)a with inner multiplication discussed in Section A.4. It is but a short step to generate a (q + 1)-form by the sum v
dxh A dhO, h=1
which then, on the basis of the symmetry of PA, agrees with the exterior differential do, also discussed in Section A.4. The covariant derivative of a p-fold contra and q-fold covariant tensor t is a (p, q + 1)-tensor Vt given by 3... iy = a (V t)i ...jqe - axh tjl...jq
+
rtj..jq hi E il...iy I
il..-l is ... + ,tjl...jgrhi ii ..iy
tl.Y2...jgl'lhj1
- ... - L.. u...: hjy' !
For a much more general treatment which avoids all these indices, the interested reader is referred to Equations Diferentieiks a Points Singuliers Reguliers by DELIGNE [D].
Appendix B
Lie Groups and Lie Algebras
As in Appendix A, we shall explicate from this large and important area of mathematics only the major results and concepts needed in symplectic geometry. As a guideline, Chapter 6 of KIRILLOV's book Elements of the Theory of Representations [Ki] and also the corresponding sections from ABRAHAM-MARSDEN [AM] will serve well. The interested reader might
also want to keep at hand a book with Lie groups or Lie algebras in the title. Particularly recommended are the classics of WIGNER [Wi], WEYL [W] and CHEVALLEY [Ce].
B.1. Lie algebras and vector fields Lie algebras made their first appearance as infinitesimal Lie groups, but quickly acquired meaning as self-standing algebraic objects. In the following, the ground field K will be assumed to be R or C. The theory of Lie algebras can, however, be considered for a completely general field.
DEFINITION B.I. A K vector space g is called a Lie algebra if a Lie product (or a Lie bracket) is defined on it; that is, a K-bilinear map [
,
]:gxg-*g
satisfying
(Antisymmetry) [X, X] = 0 for all X E g, (Jacobi identity) [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0
for allX,Y,Zeg. 163
B. Lie Groups and Lie Algebras
164
Remark B.2. If g is finite-dimensional with basis (X1..., Xn), it is clearly enough to know the values of the Lie brackets [Xi, X j] for pairs of basis elements. In other words, one must know the coefficients cikj (i, j, k =
I,-, n) from the expressions
=
[Xi,Xj]
n
EcjXk k=1
These coefficients are called the structure constants of g (relative to the given
basis (XI,... , EXAMPLE B.3.
(1) The Lie algebra g is called abelian or commutative whenever
(X,Y]=0
X,YE9.
for all
Every K vector space can be considered as a commutative Lie algebra when equipped with this trivial Lie bracket.
(2) Every associative K-algebra A can be taken to be a Lie algebra, by taking for the Lie bracket
[X,Y] := XY - YX
for
X,Y E A.
In this manner, for the space of all p x p matrices A = Mp(K) we have the Lie algebra called gt,(K).
(3) Every (not necessarily associative) K-algebra A allows the construction of a Lie algebra Der A, given by the K-linear maps called derivations; that is, maps satisfying
D : A -* A, with D(XY) = (DX )Y + XDY for all X,Y E A and [DI, D2] := DiD2 - D2DI as Lie bracket. DerA is called the algebra of derivations, or also differentiation, on A. By an easy refinement of Theorem A. 13, it can be shown (see ABRAHAMMARSDEN ([AIM, p. 83)) that:
THEOREM B.4. The space V (M) of differentiable vector fields on a differentiable manifold M is isomorphic to Der (.1 (M)) as K vector spaces.
This isomorphism is given by assigning to a vector field X on M the derivation Lx defined by Lx f (m) =
(cf )m) for
f E .T(M) and M E M.
Lx f will be called the Lie derivative of f relative to X, and has already been introduced at the end of Section A.3 as well as in the discussion of the
B.2. Lie groups and invariant vector fields
165
definition of tangent spaces in Section A.1, where it was written, because of local considerations, as Lx f (m) = Lxmf. The isomorphism of the theorem gives a Lie algebra structure on V (M), where (X, Y] for X, Y E V (M) is the uniquely defined vector field given by L(x,Yj = [Lx, LY] For a diffeomorphism F : M M'; in accordance with the conclusions from
Section A.3. F.X E V(M') is taken as the image of X E V(M). The map F. is compatible with the Lie bracket (see ABRAHAM-MARSDEN ([AM], p. 85)).
Remark B.5. F. is a Lie algebra homomorphism; that is, F. is K-linear with
F.[X,Y] = [F.X,F.Y] for X,Y E V(M).
The Lie derivative of a vector field. It is not very easy to prove the statement that the map defined in terms of flows, Fx to X E V(M), at the conclusion of Section A.4.
Lx : V(M)
V(M),
Y --i LXY :=
dt is the same map given in terms of the Lie bracket as
(Fx(-t).
Y)I
e.o,
LXY = [X, Y].
B.2. Lie groups and invariant vector fields DEFINITION B.6. A Lie group G is a finite-dimensional manifold, on which a group structure is defined, whereby the group operations are given by differentiable maps; that is, the maps G x G ,G, and G -- G, (9, h)
9h,
9'--' .9-I
are differentiable.
Here, we will focus on the case of real Lie groups; analogous constructs can be formed over the complex numbers. EXAMPLE B.7.
(1) G =R" with addition as product. (2) G = R>0 with multiplication as product; or analogously,
G=SI={z EC,jzj=1}.
B. Lie Groups and Lie Algebras
166
(3) G = GL(n, R) with matrix multiplication as product. (4) G = G1 X G2, where both GI and G2 are Lie groups.
(5) G = H(R) = {h = (A, µ, tc) E R3), the Heisenberg group of degree 1 with the product
hh' = (A+A',p+µ',K+IC +Ap.'-A'p). Every Lie group G gives rise to a Lie algebra g = Lie G, which can be thought of as the linearization of G and whose structure is fixed by G, here in a neighborhood of the unity. For this assignment there are several equivalent constructions. Here we take as the starting point a concept which also finds many other uses. NOTATION B.B. For each go E G denote by Ay,p the function defined by
g . -. ay9
909,
the so-called left translation given by go, and by py,o the right translation given by
9'-'Pa,9;=990 and ppu are diffeomorphisms of C to itself. They can also be used to shift given vector fields X on G. This makes possible the concepts of right-invariant and left-invariant vector fields on G. For every go, the maps A
DEFINITION B.9. X E V(G) is called left-invariant whenever
(a9o).X = X
for all
go E G.
The space VI(G) of left-invariant vector fields on G is clearly a vector subspace of V(G), which can then be proved to be a Lie subalgebra of the Lie algebra V(G) (and therefore is carried over to g = LieG ). Remark B.10. The tangent space TeG to G at the unity element e E G is isomorphic to V (G) as a vector space:
TG.
V ,(G)
Here API is given by
pI (X) := Xe
for
X E V (G)
and 4P2 by
(Z;) := X
with
X9 :_ (A9).t
l E T,,G.
for
Then we clearly have
t = VIV2W
///
C
= Ae). (`` = S
tt
B.3. One-parameter subgroups and the exponent map
167
Moreover, from the left-invariance of X (A9)sXe = Xg,
thus
idTG and
Remark B.11. If X, Y E V (G), then [X, Y] is also in V (G). Then, because of Remark B.2, we have [X, Y].
[(A9.)X,
And this allows us to fix a definition: DEFINITION B.12. The vector space TG with the Lie algebra structure induced by the isomorphism with VI(G) is called the Lie algebra g = Lie G of G. EXAMPLE B.13.
LieGL,,(R) = M(R), {X E
Lie
0},
E
as
as o
(3) := Lie 0(3)
R3
with
X
C
[X, YJ - C x n (the vector product in R3).
B.3. One-parameter subgroups and the exponent map As already indicated, the passage from a Lie group G to its Lie algebra g can, to a certain degree, be reversed. To make this statement more precise requires the following concepts, which can only be briefly mentioned here. To each £ E TeG denote by X its associated left-invariant vector field (with Xe = ). Then denote by
yf:R -b G, t
f--. exp t.,
the integral curve to X which for t = 0 goes through e and whose tangent vector 7f(t) at every point yy(t) is equal to X.,,(1). That such a curve exists is, as already discussed at the end of Section B.1, a consequence of the
B. Lie Groups and Lie Algebras
168
existence and uniqueness theorem for solutions of systems of ordinary differential equations (and a simple corollary). For such a curve, it can be shown (see, for example, ABRAHAM-MARSDEN ([AM], p. 255)) that
exp (t + s)£ = exp tH exp sl;;
that is,
ryf:R -+C is a (differentiable) group homomorphism. 'yf is called a 1-parameter subgroup of G. This now makes possible the next definition. DEFINITION B.14. The map
exp:TeG
G,
% (1) = exp
is called the exponential map of the Lie algebra 9 = TeG in G.
This map turns out to be differentiable and induces the identity map on the tangent space To(TeG) ^- TeG. Therefore it is a local diffeomorphism, but not a diffeomorphism on G. Remark B.15. For a differentiable homomorphism F : H --+ G between two Lie groups H and G and for the induced map
TeF :_ (F,)e : TeH -+ TeG we have the commutation rule
F(expH g) = expc(TeF)rl
for ail
n E TeH = Lie H.
Since the map ry
: R -P F(expH ti?)
is a 1-parameter subgroup of G, it is of the form
-y(t) = expc t
with
d y(t)I t o = (TTF)(rl);
that is,
F(expH,) ='r(1) = 7f(1) = exp
= expc(TeF)il. An important special case is given by conjugation: 1C9 : C --i G, h +-----+ ghg-1 = p9-,a9h
for
g E C.
This is a differentiable map, and actually an inner automorphism of G. Now denote by I a9) : TG - TeG Ad9 := 1'e K9 = Te(P9
B.3. One parameter subgroups and the exponent map
169
the adjoint map associated to g E G. As a consequence of the last remark, it follows that exp(Adgl;) = rcg expe = g (expt;)g-1 for all f E TAG and g E G. EXAMPLE B.16.
(1) G = R", with addition as product, has p = Lie G = R', and exp: R" ---' R'& is the identity.
(2) For G =
and its subgroups, the exponential map is the
generalization of the exponential function of matrices; thus
f),
exp : 00
A i--. > Am/m!
.
.=o
To every A E M,,(R) there belongs the 1-parameter subgroup -YA, given by 00
1'A(t) = exp to = E tmAm/m! M=0
From this, for C E GL,,(R), we derive the frequently used computation rule exp (CAC-1) = C(exp A)C-1.
Appendix C
A Little Cohomology Theory
Homology and cohomology groups are important tools for both describing and characterizing objects in almost every area of mathematics. They are introduced under the topics of homological algebra and/or algebraic topology. Here we will only give a few definitions from various sources (for instance, KIRILLOV [Ki], and GUILLEMIN-STERNBERG [GS]) and treat enough of their elementary properties to suffice for our study of symplectic geometry. For a systematic introduction, we recommend MACLANE'S book Homology [ML] or GODEMENT's book Topologie algebrique et theorie des faisceaux [Go], which has retained its value as a classical introduction to sheaves and their cohomology.
C.1. Cohomology of groups Let G and M be groups with M abelian, and let G operate on M from the left, that is, there is a map
GxM -+ M, (g, m) '-' gm, with
(gg')m = g(g'm),
em = m,
g(m+m') =gm+grn' 171
C. A Little Cohomology Theory
172
for all g, g' E G and m, m' E M. Then (see KIRILLOV ([Ki , p. 21)) an n-dimensional cochain c is an (n + 1)-linear map n+1
with for all
9,9o.. ,gn E G.
The collection of all n-dimensional (or, more simply, n-) cochains is a group, and will be denoted by Cn(G, M). Then a coboundary operator d
Cn+1(G,
d : C?(G, M)
'--i
c
M),
dc,
is given by n+1
dc(go,...,9n+1) _ E(-1)'c(90,...,9i,...,gn+1) i^o
If c = db,
then c E Cn(G, M) is called a coboundary of the cochain b E Cn-1(G, M); and c is called a cocycle, if
do=0. Then we get in C' the subgroup of cocycles Z' and the subgroup of coboundaries B. The following statement is central.
Remark C.I. We have
dod=0.
EXERCISE C.2. Prove the remark.
We thus have
C"(G, M) D Z"(G, M) D Bn(G, M), and we can form the factor group
H'(G, M) := Z'(G, M)/B'(G, M). This is again a group, and is called the n-th cohomology group of the group G with coefficients in M. Taking a direct sum, we obtain
H* (G, M) := ®Hn(G, M). n
This is a graded ring or a graded algebra, if M is not only a group, but a ring or an algebra as well.
C.2. Cohomology of Lie algebras
173
For practical computation. the cochain functions c can also be replaced by c:
c(hl, ... , hn) := c(e, hl, hih2,... , hi ... hn). For these c the coboundary operator is then dc(h1,...,hn+1) = hlc(h2,...,hn+1) n
+ D-1)'e(h1, ..
. hi-1, hihi+1. hi+2,
, hn+I )
i=1
EXERCISE C.3. Prove this last formula.
As a hint as to the usefulness of this concept, we inform the reader of the following example. Given two groups Go and G1 with an operation of G1 on Z(Go) = {g E Go, 990 = 909, 9o E Go}. the center of Go, one can identify H2(G1, Z(Go)) with the set of equivalence classes of central extensions 1
G1 by Go (see KIRILLOV ([Ki], p. 18)).
C.2. Cohomology of Lie algebras Let g be a Lie algebra and V a g-module (see GUILLEMIN-STERNBERG ([GS]. p. 417)). Then denote by Ck(g, V) the collection of all n-cochains, that is, the antisymmetric n-linear maps
j gx...xg-,V. Then take as coboundary operator the operator b : Ck(g, V) -,
Ck+1(g, V)
defined by k
bf(F.O..... k) :=
i
E(-'Y 1f(So,...,.i.....l;k) i=0
+
,Sj,.. .l;k)
i<j
Thus, with this arrangement, we have for k = 0
df (4) = U, for k = I
6f
f1) = W (6) - W (to) - f ([to, 6 ]),
C. A Little Cohomology Theory
174
and for k = 2 1,f2) = fof(S1,S2) - S1f(So,S2) +
EXERCISE C.4. Verify that 62 = 0.
Thus, in analogy to Section C.1, we may define the cohomology groups Hk(9, V)
Zk(9, V)/Bk(g, V).
In the special case that V is a trivial g-module (that is, l;v = 0 for all E g and v E V), the first terms of the coboundary operators do not appear, and if V is simply K, we write
Hk(9) := Hk(9, K). The vanishing of HI (g) and H2(g) allows for important reductions in symplectic geometry (see, for example, the theorem of Kostant and Souriau in GUILLEMIN-STERNBERG ([GS], p. 179), which is covered in Section 2.5). Here let us but list the following general statements from GUILLEMINSTERNBERG ([GS], pp. 418 ff.)):
(1) H'(g, V) = {0} for all V precisely when every representation of g is completely reducible.
(2) If the above is satisfied, it follows that H2(g) _ {0}. (3) If g is semisimple (that is, it has no commutative ideals), we have, as in 1), that H1(g, V) = 0 for all V.
C.3. Cohomology of manifolds It is especially important to actually compute homology and cohomology groups for topological and geometric objects, for example manifolds, varieties, schemes, etc. This has led to a whole art of technical apparati making such computations possible. Here we give a brief treatment of the following related concepts (see KIRILLOV ((Ki]. p. 9)). Let G be an abelian group, M a manifold, and U = {Ua}aEI a covering of M by open sets. A k-cochain of U with coefficients in G is then a skewsymmetric map c defined on every (k + 1)-tuple (no, ... , ak) E Ik+1 with
Ua,,n...nUQk00 to G. Thus
c(...,aiI...Iaj ....) = -c(.... a1,...,ai,...).
C.3. Cohomology of manifolds
175
Ck (it, G) denotes the group of these k-cochains. The coboundary map d:
Ck(it,
G) -i Ck+i (it, G)
is given by k+1
(-1)ic(ao,...,6i,...ak+1)
dc(ao,...,ak+1) = i=0
We then have that d2 = 0, and therefore, in complete analogy to what has come before, we define
Hk(f1, G) := Zk(it, G)/Bk(it, G) as the Cech cohomology groups of the open covering it. This shows a way to give M itself cohomology groups. Namely given a finer open covering it' of it, we get, in a natural way, homomorphisms
Tff : Hk(it, G) - Hk(it', G). This construct allows us to define inductive limits over a directed system of open coverings:
Hk(M,G) := limHc(it,G).
u
Fortunately these cohomology groups can already be computed from a single covering of it (LERAY's Theorem):
Hk(M,G) = Hk(it,G), when it = {UQ} and the sets UQ and their intersections all have trivial cohomology in the dimensions n > 1. A completely different viewpoint allows one to attach cohomology groups to a real or complex CI-manifold M with the help of differential forms (see
Section A.4). For K = R or C, denote by Zk(M. K) the closed K-valued k-forms on M; thus w E SZk(M) with dw = 0. Denote by Bk(M, K) the boundaries of w, thus those for which there is a 19 E SZk-1(M) such that dig = w. Then the de Rham cohomology groups HDR(M, K) are defined by
HDR(M, K) = Zk(M, K)/Bk(M, K). De Rham's theorem says that these cohomology groups are the same as the above-defined Cech cohomology groups:
HDR(M, K) = Hk(M, K).
Appendix D
Representations of Groups
Many of the concepts from representation theory have already played a role in the previous appendices. Here, we collect together in one place all sorts of important concepts which are needed in connection with symplectic geometry. As a guideline to this material we take §7 from KIFULLOV's book [Ki]. An introduction to the parts of the theory most relevant to us is also offered in LANG's book [L4].
D.1. Linear representations Let G be a group and V a K vector space. Here again we are mostly thinking about K = ][t or C, although much of what we will say holds also for a general field. We call T a representation of G in V when T is a homomorphism of G in Aut V, i.e. when T (glg2) = T (gl) T (g2) for g1, g2 E G.
dim V is taken as the dimension of T. Two representations T and T' of G in V and V', respectively, are called equivalent when there is a bijective intertwining operator between the two; that is, a linear isomorphism U
V - V' with UT (g) = T'(g)U for all g E G. The set of intertwining operators U of V and V' will also be designated by C (T, V). A representation T is called reducible if the representation space V properly contains a non-trivial subspace V1 invariant under all T (g) with g E G. The restriction of T (g) to Vl then defines a representation T1 of G on V1, which will be called a subrepresentation of T. There is then also 177
D. Representations of Groups
178
a natural representation of G on the quotient space V/Vi, which is then called the quotient representation (in KIRILLOV ([Ki], p. 110), this is called a factor representation of T). If the invariant subspace V1 C V allows for an invariant subspace complement T2, then the representation T on V is called decomposable and is written as T = T1 i@ T2. A representation is called (algebraically) irreducible if it contains no nontrivial subrepresentation other than itself. A representation T in V is called completely reducible if every invariant subspace of V possesses an invariant complement. This is then equivalent to V being completely decomposable into a sum of irreducible subrepresentations.
One of the major tasks of representation theory is to give a complete list of all isomorphism classes of irreducible representations, or at least of all irreducible unitary representations (see the next section). Another task is to take a given completely reducible representation T and to reduce it; that is, to write it as a direct sum of its irreducible parts T; along with their multiplicities
mult(T T) := dim C (Ti, T). A character X of G is a homomorphism
X:G-'Ci={CEC, 15I=1}. For a given representation T of G in V, T' is called the contragredient representation if T' in V' is given by
g'-' T'(9)
T (9-1)'
Under these conditions, it is then a fact that for a map of representations F : V -+ V the canonical map F' : (V')' --+ V' with
F'(v'')(v) = v'' (F (v)) again gives a map of representations. It is also but a simple exercise to show that from representations Ti of G in V (i = 1, 2) one can obtain new representations in the following two ways.
i) The direct sum T1 ® T2, as a representation of G on V1 ® V2, is given by (T1 (DT2)(9)(vr + v2) = Ti(9)vr + T2(g)v2
for v1 E V1, V2 E V2.
ii) The tensor product T1 0 T2, as a representation of G of the tensor product V1 0 V2, is given by (T1 O T2)(9)(vl ®v2) = Tl(9)vl (&T2(9)v2 for v1 E V1, v2 E V2.
D.2. Continuous and unitary representations
179
It is a standard result of representation theory that the tensor product of given irreducible representations can be decomposed into irreducible representations. Parallel to the concept of linear representations, we discuss the projective representations. These are maps T : G -. Ant V satisfying T (9192) = c(91,92)T (91)T(92) for all 91,92 E G,
where c : G x G - K is a function satisfying the functional equation c(9i,92)c(9192,93) =c(g1,92g3)c(92,93) for 91,92,93 E G, which then insures the equality T (9192)T (93) = T (9i)T (9293)
Such a projective representation of G on V then induces a representation on the projective space, P(V) associated to V, which is defined by
T (9)(v..) = (T (9)v),
,
for v- E P(V).
D.2. Continuous and unitary representations In the applications in this text, we will most often encounter G which are Lie groups, and so we have a differentiable and topological structure. Thus, we will need all types of restrictions on the concept of representations of G. Here we will assume the following data is fixed: T is a continuous representation, thus a representation of a topological group G in a topological vector space V (mostly a Hilbert space) with
GxV V, (g, v) '--- T (g)v is continuous. For such continuous representations, the definitions of the previous section are extended in the appropriate sense. For example, in this sense, T1 would be a subrepresentation of T on V1 C V when V1 is a closed invariant subspace of V. The extension of the other definitions is entirely analogous.
Unitary representations. A representation 7r of a (not necessarily topological) group G in the space V is called unitary if V is a Hilbert space and the operators T (g) for all g E G are unitary. Unitary representations are the most important in applications to physics. They are particularly tractable, since every unitary representation is completely reducible. The collection of isomorphism classes of irreducible unitary representations of G is called the unitary dual and is denoted by G. G can be supplied with a topological structure (see KIRILLOV ((Ki], p. 113)).
D. Representations of Groups
180
EXAMPLE D.1. Elementary examples are the following:
i) For an abelian group G, every 7r E G is one-dimensional. ii) For a compact group G, every 7r E G is finite-dimensional.
iii) For G = IIt" we have G = W", and for G = SI we have d = Z. Schur's lemma says LEMMA D.2.
a) If Tl and T2 are (algebraically) irreducible representations, then every intertwining operator A E C (TI,T2) is either 0 or invertible. b) A unitary representation 7r is irreducible precisely when dimC(7r,7r) = 1.
D.3. On the construction of representations There are several ways to approach the representations of a group, in particular for the unitary dual G. Here we give a few hints.
Reduction of the regular representation. For a locally compact topological group, denote by V = L2(G) the Hilbert space of quadratic integrable functions on V with respect to a suitable measure, dA (a Haar measure, see, for example, KIRILLOV ([Ki], p. 130)). Denote by g the representation ¢ E L2(G) given by g (go) 0 (g) = 0 (ggo) for g, go E G
the so-called right regular representation. Correspondingly, the left regular representation A is given by ,\ (go) 0 (g) _ O W, g) -
An intertwining operator U between g and \ is given by
044 ¢ with
q (g) := O
(g_1).
In the general case, the question arises as to whether all 7r E G are subrepresentations of g respectively A. For compact G, this is the case (this is a consequence of the Peter-Weyl theorem).
D.3. On the construction of representations
181
A variant. We now treat a variant of this. We work in the following situation. G operates contiuously on a differentiable manifold M; that is, there is given a map
GxM (g, m)
M,
'-' gm = co9(m),
where all the spy, for g E G, are continuous, and both P9192 = `P93 V92
for all 91, 92 E G
as well as V, = idM
hold. Then gyp, given by
0 (9) 0 (m) _ o (Vy-, (m)) for Q, E F (M),
is a representation of G on the space F (M) of all arbitrarily often differentiable functions on M. This theme can now be varied in a myriad of ways. To show the simplest case, let j be an automorphic factor, that is, a function
j:GxM (9, m)
liY',
(9, m),
satisfying the functional equation j (g1g2, m) = j (91, 92m) j (92, m)
for all gl, 92 E G, m E M.
Then from i with
Pj(9) f (m) = 0 (9-' (m)) j (9-1, m) we get another representation of G on F (M). The standard example in this connection is the operation of G = SL2(R) on 1R2 given by multiplication of the matrix g E G with the column vector m E R2. Then 0 with
,p (9) f (m) = f (9-1 m) for f E V = R[q, p]
is a representation of G on the polynomial ring R[q, p] in two variables. Clearly the subspaces of homogeneous polynomials of degree l > 0 remain invariant and give irreducible representations.
Eigenspace representations. As in the previous example, G operates contiuously on a differentiable manifold M. Let D be a C-invariant differential operator on F (M), and let E (D. A) be the eigenspace of D with eigenvalue A. Then gyp, defined as in the previous example, is a representation on E (D, A), called an eigenspace representation. These representations have been particularly propagated by HELGASON.
D. Representations of Groups
182
The infinitesimal method (for Lie groups). Let it be a continuous representation of a real Lie group G in a topological vector space V. Then denote by V,, the space of analytic vectors in V, that is, those v E V for which
g - 7r (9) v is a real analytic map. From results of HARISH-CHANDRA and NELSON, V,,
is dense in V. it assigns, as an infinitesimal representation d7r, a representation of the Lie algebra g of G on V, given by d7r (X) v := Wt 7r (exp tX )vl t_e
for X E g, v E
(A representation of a Lie algebra in a vector space W is a homomorphism of Lie algebra-, g - . End V, where End V is a Lie algebra as explained in Section B.1. That d7r gives such a representation can be shown as an exercise.)
The infinitesimal method consists of these steps, but followed in the reverse order:
a) Determination of the irreducible representations it of g, or of the complexification g, = g ® C. (This is somewhat easy to do for the Heisenberg algebra g = b and for g = s12. The elements for these were provided in Sections 5.1 and 5.2; the rest is a recommended exercise.)
b) Examination of which of these a can be integrated to unitary representations 7r; that is, the determination of those representations it of G with d7r = F.
In important cases it turns out that it is uniquely determined by dlr. If G = H (iit) is the Heisenberg group, one may take for it the Schrodinger representation, which was described in Section 5.2. For G = SL2(IR), it is relatively easy to determine the unitary dual G (see, for example, LANG [L4], where however an error occurs: known equivalent representations are counted twice). The Weil representation, which appears in Section 5.3, plays here a special role in that it is a projective representation of SL2(R). Induced representations. With the use of induction, one may construct, in great generality, representations of the group G from representations of its subgroups. To this end. let H C G be a closed subgroup and K C G a subgroup such that
HxK-
is a topological isomorphism. Further, let
a:H - AutV
D.3. On the construction of representations
183
be a finite-dimensional continuous representation. Then the associated induced representation
r = Indgo of G is given, in that G operates by right translation on a space fl (a), which is spanned by the functions
0:G--+ V with 0 (hg) = A ( h )
(h) 0 (g) for all h E H, g E G
and 11 0 II2:=
I0(k)I2dk < oo, JK
where A = AH is the modular function of H, that. is, the function with
dr(xy) = IH(x) dry for all x, y E H, whenever dry is a right-invariant Haar measure on H. The factor A1/2 works so that r turns out to be unitary whenever a is. One of the most significant theorems from representation theory is the subrepresentation theorem, which is a statement that all interesting representations are captured as subrepresentations of known induced representations. As an example, the Schrodinger representation is relatively easy to realize as an induced representation (see, for example, BERNDT [Be]). The induced representations were particularly strongly propagated by the efforts Of MACKEY, [M].
The adjoint and coadjoint representations. To every Lie group G there belongs a Lie algebra g = Lie G, and also a natural representation of G on g, called the adjoint representation Ad. It arises from the fact that through conjugation K9(go) = e9-Ag(go) = ggog-1
a diffeomorphism of G is defined, and so, for every g E G, K9 induces maps
of the tangent spaces T..G. In particular, for go = e, TeG is mapped to itself. It is written as Adg := (Kg)se = (Log-"Y.e.
The action
GxTG
TeG,
(g, Y) Ad9Y, is defined via the identification of TeG and g, because K9,gz = Kg, K92 is a representation of G on g, in which the operators are also written as Ad(g).
One may also consider the contragredient representation, as in Section D.1, on the dual space g' of g to this representation. This is called
184
D. Representations of Groups
the coadjoint representation of G, and is of particular significance for symplectic geometry. Since for all go E G, the rcg(go) are differentiably dependent on g, the infinitesimal representation d(Ad) =: ad associated to Ad can be built as was described above under the infinitesmal method. It is not hard to see (see KIRILLOV ([Ki], p. 97)) that then
adX(Y) = [X,Y].
Bibliography
[AM]
Abraham, R., and Marsden, J.E., Foundations of Mechanics, Benjamin-Cummings, Reading, MA, 1978.
[Ae]
Aebischer, B., Borer, M., Kahn, M., Leuenberger, Ch., and Reimann, H.M., Symplectic Geometry, Prog. Math., vol. 124, Birkhauser, Basel, 1994.
[A)
Arnold, V.I., Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
[Ar]
Artin, E., Geometric Algebra, Interscience Publ., New York, 1957.
[Be]
Berndt, R., Darstellungen der Heisenberggruppe and Thetafunktionen, Hamburger Beitri ge zur Mathematik, Heft 3, 1988.
[BeS]
Berndt, R., and Schmidt, R., Elements of the Representation Theory of the Jacobi Group, Prog. Math., vol. 163, Birkhhuser, Basel, 1998.
[BS)
Berndt, R., and Slodowy, P., Seminar uber Darstellungen von SL2(F) and GL2(F), F em lokaler Korper, Hamburger Beitrage zur Mathematik, Heft 20, 1992.
[BI]
Blair, D., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math., vol. 509, Springer, Berlin, 1976.
[Bo)
Bourbaki, N., Elements de Mathdmatique, AlgCbre. Chapitre III: Algcbre multilineaire. Hermann, Paris, 1948.
[C)
Cartan, H., Differential forms, Houghton-MitHin, Boston, 1970.
[Cal
Cartier, P., Quantum Mechanical Commutation Relations and Theta Functions, in Algebraic Groups and Discontinuous Groups (A. Borel and G.D. Mostow, editors) Proc. Symp. Pure Math., vol. 9, Amer. Math. Soc., Providence, RI, 1966, pp.361-383.
[Ch]
Chern, S.S., Complex Manifolds without Potential Theory, Van Nostrand, Princeton, NJ, 1967.
[Cel
Chevalley, C., Thdorie des groupes de Lie, Hermann, Paris, 1951.
[CH]
Courant, R. and Hilbert, D., Methods of Mathematical Physics, Volume 1, Interscience, New York, 1953. 185
Bibliography
186
[D]
[dCj [E]
[EG]
Deligne, P., Equations Difflrentielles a Points Singuliers Raquliers, Lecture Notes in Math., vol. 163, Springer, Berlin, 1970. do Carmo, M.. Riemannian Geometry, Birkhauser, Boston, 1992. Eichler, M.. Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1966. El Gradechi, M.A., Theories classiques et quantiques sur l'espace-temps anti-de
Sitter et leurs limites a courbure null. These de Doctorat, University Paris 7, Paris, 1991. [FL]
Fischer, W., Lieb, I., Funktionentheorie, 7. Auflage. Vieweg, Braunschweig-
[F]
Forster. 0., Analysis 1, 3. Aufage. Vieweg, Braunschweig--Wiesbaden. 1984. Forster, 0., Lectures on Riemann surfaces, Springer, New York, 1981. Hermann, Paris, Godement, R., Topologie algebrique et theorie des
W iesbaden. 1994. [FRF] [Go]
1958.
[GW] [Gbj [Gr] [GS]
[HR] [HZ]
[Jai [K]
[K1] [K2] [Ki]
[Kn]
[Li] 1L21
[L3[ [L4] [L5]
[LL] [LV]
Goodman, R., and Wallach, N.R., Representations and invariants of the classical groups, Cambridge University Press, 1998. Greub, W., Multilinear Algebm, Springer, New York 1997. Gromov, M., Pseudoholomorphic Curves in Symplectic Manifolds, Invent. Math. 82 (1985), 307-347. Guillemin, V., and Sternberg, S., Symplectic Techniques in Physics, Cambridge University Press, 1984. Holmann, H., and Rummler, H., Altcrnierende Difjerentialformen, Bibbliogr. Inst., Mannheim, 1972. Hofer, H., and Zehnder, E., Symplectir. Invariants and Hamiltonian Dynamics, Birkhauser, Basel, 1994. Jacobson, N., Basic Algebra, Freeman, San Francisco, 1974. Kahler, E., Uber eine bemerkensurerte Ifermitesche Metrik, Abh. Math. Sem. Univ. Hamburg 9 (1933) 173-186. Kihler, E., Einfiihrung in die Theorie der Systeme van Differentialgleichungen, Teubner, Leipzig. 1934. Kiihler, E., Der inhere Difjerentialkalkul, Rendiconti die Matematica 21 (1962) 425 523, Kirillov, A.A., Elements of the Theory of Representations, Springer, Berlin, 1976. Knapp, A.W., Representation Theory of Semisimple Groups, Princeton University Press, 1986. Lang, S., Algebra, Addison-Wesley, Menlo Park, NJ, 1984. Lang, S. Fundamentals of Differential Geometry, Springer, New York, 1999, Lang, S., Linear Algebra, Springer, New York, 1985. Lang, S., SL2(R), Springer, New York, 1989. Lang, S., Undergraduate Analysis, Springer, New York, 1983. Landau, L.D., and Lifschitz, E.M., The Classical Theory of Fields, AddisonWesley, Reading, MA., 1961. Lion, G., and Vergne, M., The Weil Representation, Maslov Index and Theta Series, Prog. Math., vol.6, Birkhauser, Basel, 1980.
Bibliography
187
[M]
Mackey. G.W., Unitary Group Representations in Physics, Probability, and Number Theory, Benjamin/Cummings, Reading, MA, 1978.
[Mal
Marcus, M., Finite Dimensional Multilinear Algebra (in two parts), Marcel
[ML]
Dekker, New York, 1977. Mac Lane, S., Homology, Springer, Berlin, 1963.
[MS]
[Mu]
[Mull
McDuff, D., and Salamon, D.A., Introduction to Symplectic Topology, Oxford University Press, 1995. Mumford, D., Algebraic Geometry, I. Springer, New York 1976. Mumford, D., Tata Lectures on Theta II, Prog. Math., vol. 28, Birkhiiuser, Boston, 1984.
[Mu2)
Mumford, D., Tata Lectures on Theta III, Prog. Math., vol. 97, Birkhauser,
[Sal
Boston, 1991. Satake, I., Algebraic Structures of Symmetric Domains, Iwanami Shoten. Tokyo, and Princeton University Press, Princeton, 1980.
(Schl
Schottenloher, M., Geometrie and Symmetrie in der Physik, Vieweg, Braun-
[Sill (Si2l [SMI
schweig/Wiesbaden, 1995. Siegel, C.L., Symplectic Geometry, Academic Press, New York, 1964. Siegel, C.L., Topics in Complex Fhnction Theory, Vol. III. Wiley-Interscience, New York, 1973. Siegel, C.L., and Moser, J.K., Lectures on Celestial Mechanics, Springer, Berlin, 1971.
[We]
Souriau, J.-M., Structure of Dynamical Systems, A Symplectic View of Physics, Prog. Math., vol. 149, Birkhauser, Boston, 1997. Sternberg, S., Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ. 1964: 2nd ed., Chelsea. New York, 1983. Vaisman, I., Symplectic Geometry and Secondary Characteristic Classes, Prog. Math., vol. 72, Birkhauser, Boston, 1987. Wallach, N.R., Symplectic Geometry and Fourier Analysis, Math. Sci. Press, Brookline, MA, 1977. Weil, A., Varidtt;s Kohleriennes, Hermann Paris, 1957
[Well
Weil, A., Sur certain groupes d'op6rateurs unitaires, Acta Math. 111 (1964)
[So]
[St] [V]
[Wa]
143-211. [W]
Weyl, H. The Classical Groups. Princeton University Press, 1946.
[Wil
Wigner, E., Group Theory and its Application to the Quantum Mechanics of
[Wo]
Atomic Spectra, Academic Press, New York, 1959. Woodhouse, N., Geometric Quantization. Clarendon Press, Oxford, 1980.
Index angular momentum, 104 atlas of a vector bundle, 145 automorphic factor, 181
connection, 48, 158 integrable, 161 contact manifold, 86 weak, 85 contravariant or covariant vector, 149
basis complex unitary, 28 dual, 11
[-related unitary, 29 real unitary, 28 related symplectic, 22 symplectic, 12 Bianchi identity, 161 bundle, characteristic, 87 cotangent, 45, 147 differentiable vector, 146
coordinates, symplectic, 36 cotangent space, 142 covariant derivative, 158 curvature matrix, 161 derivation, 141, 164 difeomorphism, 139 symplectic, 83, 84 differentiable function , 139 differentiable section, 146 differential form closed, 155
tangent, 147
covariant derivative of a, 162
vector, 144
degree 1, 148 exact, 155
canonical form, 12 characteristic line fields, 91 Christoffel symbol, 159 coadjoint orbit, 110 coadjoint cocycle, 96 coadjoint orbit, 54 coboundary operator, 52, 172 cocycle identity, 96 cocycle relation, 145 cohomology groups De Rbarn, 175 of a group, 172 of a Lie algebra, 52, 174 of a manifold, 175 configuration space, 1, 45 conjugation, 53
qth degree, 153
reduced representation, 153 skew symmetric representation, 154 differential system, 56 involutive, 57 differentiation, exterior, 154 dual space, 11 energy function, 74
equations of motion for a charged particle, 75
Euler-Lagrange equation, 2 exponent map, 168 fiber derivative, 102
Row, 41, 156
189
Index
190
form
closed of type (1,1), 49 Kkhler, 50 left invariant q, 52 Liouville, 45 of type (1,1), 49 presymplectic, 85 symplectic, 9, 35 fundamental duality, 72 exact sequence, 78, 83, 111 geodesic, 159
group
general linear, 24 Heisenberg, 166 Jacobi, 120 orthogonal, 24 symplectic, 15, 24 unitary, 24, 28
Hamilton's equations, 3 Hamilton-Jacobi equation, 4 Hamiltonian equations, 74 Hamiltonian function, 3 time dependent, 89 Hamiltonian G-space, 94 Hamiltonian system, 74 completely integrable, 108 harmonic oscillator, 104 Heisenberg algebra, 117 Heisenberg group, 116, 117 horizontal section, 161 hyperbolic pair, 19 hyperbolic plane, 19 infinitesimal generator, 94, 97 inner product, 6, 13, 72 integral curve, 6 integral manifold, 57 intertwining operator, 177 isotropy group, 22 Jacobi identity, 163
Lie derivative of a differential form, 52, 72, 156 of a function, 71 of a vector field, 72, 165 Lie group, 165 manifold hermitian, 49 Kiihlcr, 45 of the solution of constant energy, 109 real, 1 Riemannian, 101 smooth, 35 symplectic, 35 map, differentiable, 138 moment, 94 symplectic, 36 map of
tangent and cotangent spaces, 143 vector fields and 1-forms, 149 matrix, symplectic, 16 metric Fubini- Study, 64 hermitlan, 26, 28 Kiihler, 47 PoincarE, 33 Riemannian, 48, 151
w-compatible pseudohermitian, 26 Minkowski space, 75 moment map Ad' -equivariant, 96, 105 momentum, 103 morphism, 55 of differentiable manifolds, 138 stable, 17
strongly stable, 17 symplectic, 14 multiplication exterior, 154
inner, 156 Mumford's criterion, 48
normal form of bilinear forms, 10 of contact forms, 86 observable, 7
Lagrange distribution, 132 Lagrange function, 2, 102 left translation, 166 Legendre transformation, 3 lemma Poincar6's, 155
Schur's, 180 Lie algebra, 163 Lie bracket, 7, 163
one parameter subgroup, 168 orientation, 12 orthogonal, 13 phase space, 3, 45, 88, 99 reduced, 105 Poincard's lemma, 155 Poisson bracket, 79, 80 position function, 101
Index
potential of a Kiihler form, 50, 66 prequantization, 130 primary quantities, 7, 130 principle of least action, 2 pseudoholomorphic curve, 68 quantization, 7, 112, 129 full, 130 radical, 18
rank of a bilinear form, 10 regular energy plane, 88 representation
adjoint, 54, 95, 183 coadjoint, 54, 95, 184 continuous, 179
contragredient, 178 induced, 183 infinitesimal, 112, 182 irreducible, 112, 178 left regular, 180 linear, 177
of a Lie algebra, 182 projective, 121, 179 Schrodinger, 118 Schrodinger Weil, 122 unitary, 179 Weil, 121
right translation, 166
scalar product Euclidean, 26 hermitian, 24 Siegel upper half plane, 30, 32 skew hermitian operator, 113 space affine, 23
complex projective, 63, 109, 137 hermitian, 28 homogeneous, 22 hyperbolic, 19 reduced, 105
structure canonical Euclidian, 24 canonical symplectic, 24 compatible complex, 25 complex, 2.5, 46
contact, 85 differentiable linear, 146 hermitian. 26 positive compatible complex, 30 structure constants, 164 submersion, 144
191
subspace coisotropic, 19, 21 isotropic, 19, 21 Lagrangian, 19, 21 real Lagrangian, 31 symplectic, 19 suspension of a vector field, 90 symplectic capacity, 69 invariant, 17, 21, 70 operation, 54, 93 radius, 69 reduction, 58, 104 space, standard, 14 transvection, 17
symplectomorphism, 15, 36
tangent field of ,b/' along Ft, 36 tangent space, 35, 139 tensor, 150
Riemannian fundamental, 152 theorem Darboux's, 42, 85 de Rham's, 175 Darboux's, 36 Flrobenius', 57 Groenewold- van Hove, 124 Gromov's, 68 Jacobi's, 83 Liouville's, 76
of Kostant and Souriau, 52, 60 of Stone and von Neumann, 118 Witt's, 21 unit ball, 49 vector analytic, 182 smooth, 113 vector field characteristic, 87 differentiable, 146 Hamiltonian, 6, 74 left invariant, 166 local Hamiltonian, 77 vector space Kiililer-, 26 positive Lagrangian, 30 symplectic, 10 Hamiltonian, 6 volume form, 12
Wirtinger calculus, 49
Symbols
Symplectic vector spaces (V, w)
symplectic vector space
14
W1 to WCV
the orthogonal space relative to w
22
radW=wnwl
radical of W
22
Wred = W/rad W
the symplectic space associated to W
23
L=L1CV
Lagrangian subspace
23
C(V)
collection of Lagrangian spaces L C V
26
T(L)
={L'EC(V),L®L'=V}
27
J=J(V,w)
space of w-compatible positive complex
structures J
s7n = Sp, (R)/U(n) Siegel upper half-space h(v, w) = g(v, w) + iw(v, w)
JEAutV
36
.7(R' , w0)
36
hermitian, Riemannian and outer form
32, 55
complex structure (J2 = -id)
29
Symplectic manifolds rn.EM
point of a differentiable real manifold of dimension 2n
cp:U-4R2o V(m) _ (q, p)
chart of a neighborhood U C M symplectic standard coordinates
144
40
193
Symbols
194
f E .F(M) X E V(M) a E 1l (M)
differentiable function on M differentiable vector field on M (= C(TM))
147
differentiable exterior q-form on Al
161
W E i22(M)
symplectic form on M, in particular
39
wo = E. dqi A dpi
the standard form
40
the Liouville form
49
W#
fundamental duality fl'(M) = V(M)
78
Wb
inverse mapping to w#
78
F:M-.M'
diffeomorphism in i F(m) = m' mapping of the tangent vectors F.,,,(X,,,) = X;,,, mapping of the 1-forms F;,,(a;,,,) = a,,, inner product of X E V(M) with a E 11 (M) Lie derivative off E.F(M) by X E V (M)
146
19
pi dgi
Fetrn
i(X)a Lxf Lxa
155
151
151
17, 78 77
Lie derivative of a E f24(M)
LxY = [X, Y]
by X E V(M) Lie derivative of Y E V(M) by X E V (M)
56, 164 78, 173
V
connection
166
Vxa, VxY
covariant derivative
166
Ft
flow of X E V(M) Poisson bracket of f, h E.F(M)
45, 165 85, 86
Hamiltonian vector field to f E ,F(M) infinitesimal generator of X E 8 = Lie C and the group operation ¢
80
/:Gx M-+M
with O(g, m) = gin = 0s(m) _ m(9)
99
NO = 990
right translation of g with go
174
Ago = 909
left translation of g with go
174
r.9.) = 9o990 1
conjugation with go
57
4,
moment map
100
Lie group
173
associated Lie algebra
174
continuous representation of a Lie group G
187
associated contragredient representation
186
(f, h} Xf E Ham (M) X el
99, 103
Representations
Symbols
195
d7r
associated infinitesimal representation
Ad
(of g = Lie G) representation of a Lie algebra g adjoint representation Ad (g) = (K9).
120, 190
fr
(thus =: Ad9) coadjoint representation Ad* (9) = (Ad(g-i))
57, 191
Ad'
(thus = Adg_1)
58, 191
190
7rS
Schrodinger representation of the Heisenberg 126
7rw
group H(R) Weil representation (projective representation) of SL2(R)
130
irsw
Schrodinger-Weil representation (projective
v
representation) of the Jacobi group GJ(R) part of the quantization mapping, A, assigning to each f E .7° a self-adjoint operator f
g
130 120
