New Lagrangian and Hamiltonian Methods in Field Theory

NEW Li%tIR4NGIAN

4I) d4iii1ILFONI41 IN

fHEUriY

G. Giachetta L. Mangiarotti G. SardanashviI3

World Scientific

NEW LAGRANGIAN

AND HAMILTONIAN METHODS IN FIELD THEORY

NEW LAGRANGIAN

AND HAMILTONIAN METHODS IN FIELD THEORY

G. Giachetta Univ. Camertno

L. Mangiarotti Univ Camerino

G. Sardanashvily Moscow State Univ.

World Scientific Singapore

*Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd.

P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661

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NEW LAGRANGIAN AND HAMILTONIAN METHODS IN FIELD THEORY Copyright ® 1997 by World Scientific Publishing Co. Pte. Ltd.

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ISBN 981-02-1587-8

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Preface This book addresses the application of jet manifold formalism to contemporary classical field theory. This is the natural development of the well-known geometric formulation of field theory, where classical fields are represented by sections of fibred manifolds. In mathematics, the theory of differential operators and the calculus of variations are expressed in terms of jet manifold, which, in field theory, provide the adequate mathematical language for Lagrangian and Hamiltonian formalisms. In general, the book considers only first order Lagrangian and Hamiltonian systems because most contemporary field models are of this type. Two main peculiarities of the jet formulation of field theory should be emphasized. Firstly, jets of fibred manifolds (when sections are identified by a finite number of terms of their Taylor series) form smooth finite-dimensional manifolds. Therefore, the dynamics of field systems is defined on finite-dimensional configuration and phase spaces. Secondly, jet manifolds provide the language of modern differential geometry to deal with general connections which are represented by sections of jet bundles. As a consequence, the dynamics of field systems include connections in a natural way. When analytical mechanics is seen as a field theory over a 1-dimensional base, we find a clear illustration of the role of connections. Dynamic, Hamiltonian, and frame connections are the main ingredients in this formulation. There are two main geometric models of classical field theory which exhaust all observable fundamental fields. These are the gauge theory on principal bundles, including Higgs fields, and the gravitation theory on natural and spinor bundles, including Dirac's fermion matter. In this book, we do not pretend to give a comprehensive description of these models, but to exhibit general methods of investigating classical field systems within the framework of the jet formalism. In the Lagrangian formalism, we use the first variational formula of the calculus of variations as the main tool for discovering the differential conservation laws, including the energymomentum ones, in gauge and gravitation theories. In this connection, spinor fields under deformations of a gravitational field are considered. v

vi

PREFACE

The Hamiltonian counterpart of the Lagrangian formulation of field theory is the covariant polysymplectic Hamiltonian formalism, where canonical momenta correspond to the derivatives of field functions with respect to all world coordinates, not only the temporal one. We investigate the relations between Lagrangian and Hamiltonian formalisms for a wide class of degenerate systems which include almost all contemporary field models. The reduction of the polysymplectic Hamiltonian formalism over a 1-dimensional base provides the adequate mathematical formulation of time-dependent Hamiltonian mechanics in a frame-covariant form. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. In the first two Chapters of the book, we summarize the relevant material on jet manifolds and connections, which is enough for physical applications. At the end of the book, a few topics on jet formalism are reviewed in a more general setting, in order to stimulate further investigations.

Contents Preface

V

Introduction

1

1

7

Fibred Manifolds 1.1

1.2 1.3 1.4 1.5 1.6

Immersion and submersion ........................ Fibred manifolds

. .... ............. ...........

8 12

Vector and affine bundles ......................... 21

Tangent bundles of fibred manifolds ........ ... ........ 25

Vector and multivector fields ....................... 29

Differential forms on fibred manifolds ..... ....... .. .. .. 33

2 Jet Manifolds and Connections

. ........... .............

43

43 First order jet manifolds 2.2 58 2.3 Connections Differentials and codifferentials. Identities ............... 68 2.4 75 2.5 . 2.6 82 2.7 Jets of principal bundles 2.8 Canonical principal connection ..... .... ....... ...... 89 2.1

Second order jet manifolds ........................ 52

.... .............. ............ ..

Composite connections .... ..... .. ...............

Second order connections ...... ...... ........ .. .. 80

. ....... .... ........... ..

97

3 Lagrangian Formalism 3.1

3.2 3.3 3.4 3.5 3.6 3.7

Technical preliminaries. Higher order jets ................ 98 The first variational formula .... .. .. .. .. .. . ..... . .. 101

Euler-Lagrange operators ....... .... .. .. .. ....... 108 Lagrangian polysymplectic structures

. .... .............

121

Lagrangian conservation laws ..... .... ....... ...... 128 Conservation laws in gauge theory ... .... ..... ........ 140 Conservation laws in gravitation theory .... .. .... . vii

.. .. .. 155

CONTENTS

viii

3.8 3.9

.. ... ........... .. ... . .. 172 ....................... 204

Gauge gravitation theory . Appendix. Gauge mechanics

4 Hamiltonian Formalism 4.1

4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

231

Symplectic structure ........................... 232 Polysymplectic structure

.

. ... .. .. .. ...... . .... .. 241 .. .. ....... .... .. 247 .. ..... .. .. .. ..... .. .. .. 252 .. ... .. .. .. ..... .. .. .. .. 261 .

Hamiltonian forms .... .. ..... Hamilton equations .... Degenerate systems ....

.............. ........ 281 ................ ........ 296

Quadratic degenerate systems Affine degenerate systems Hamiltonian conservation laws Vertical extension of polysymplectic formalism 4.10 Appendix. Hamiltonian time-dependent mechanics

... .... .. .. ... .... .... 301 ............. 305 .......... 308

5 Special Topics

333 ............................. 333 Jets of modules ...............

5.1

Higher order jets

5.2 5.3 5.4 5.5 5.6 5.7

Jets of submanifolds Infinite order jets Variational bicomplex Geometry of differential equations Formal integrability

....... ........ 347

................ ... ........ 362

............................. 373 ........................... 381 .................... 386

.. .... ....... .. .. ... .... ..

.

. 405

Bibliography

427

Glossary of Symbols

443

Index

449

Introduction In this book, we follow the geometric formulation of classical field theory, where fields are represented by sections of fibred manifolds. For instance in gauge theory,

these are principal and associated bundles. Jet manifold formalism enables us to extend this formulation to the dynamics of classical fields. As is well known, the theory of differential operators [26, 109, 1521 and the calculus of variations (see 113, 41, 77, 110, 174) and references therein) are expressed

in terms of jet manifolds. In brief, one can say that k-order jets are equivalence classes of sections of a fibred manifold Y -+ X which are identified by the values of the first k + 1 terms of their Taylor series at points of X. The key point is that the resulting space PY is equipped with a structure of a finite-dimensional smooth manifold, called the k-order jet manifold of Y -i X. Furthermore, jet manifolds provide the language for modern differential geometry to deal with general

connections [105, 127, 1671. Every connection I' on a fibred manifold Y -+ X is represented by a section of the affine jet bundle J'Y Y. The first two Chapters of the book summarize the relevant material on fibred and jet manifolds which is enough for the applications. In these Chapters, we consider first and second order jet manifolds which provide the appropriate formulation of first order Lagrangian and Hamiltonian formalisms. For the sake of convenience of physical applications, local coordinate expressions are widely used. Of course, they obey the appropriate transformation rules. Special attention is given to affine bundles, composite fibred manifolds Y -+ Z -' X, and connections. It should be emphasized that principal connections on a principal bundle P with a structure group G are also described by sections of the jet bundle PP --+ P which are equivariant under the canonical action of G on P. Then they are represented by sections of the fibre bundle C = J' P/C --+ X. Using this bundle, one can develop Lagrangian and Hamiltonian formalisms of gauge fields in the same way as for other fields.

We consider first order Lagrangian and Hamiltonian systems because most contemporary field models are of this type. This is not the case of General Relativity. 1

INTRODUCTION

2

However, Hilbert-Einstein Lagrangian density leads to second order Euler-Lagrange equations (see Remark 3.2.12). In first order Lagrangian formalism, a Lagrangian density is defined on the first order jet manifold J'Y, which plays the role of a finite-dimensional configuration

space of sections of Y -, X. We base our analysis of Lagrangian systems on the first variational formula (3.2.8) which provides the canonical decomposition of the L along a projectable vector field u Lie derivative

on Y - X. The first term of this decomposition contains the Euler-Lagrange operator EL associated with L. The second one is the divergence d,,T", which plays a prominent role in the study of differential conservation laws. We also give the intrinsic definition of Euler-Lagrange operators as differential operators of the variational type in terms of the variational bicomplex. It must be emphasized that Euler-Lagrange equations are not a unique type of equations met within the framework of Lagrangian formalism. Also considered are Cartan equations and Hamilton-De Donder equations, which arise in the framework of multisymplectic formalism (see Section 3.4 and Remark 3.4.4). Moreover, the Euler-Lagrange-Cartan operator (3.4.10) associated with the Cartan equations is the Lagrangian counterpart of the Hamilton operator within the framework of the Hamiltonian formulation of field theory (see Propositions 4.5.6, 4.5.10 and 4.5.11). In the case of almost regular Lagrangian densities (see Definition 4.5.13), the Cartan equations are equivalent to the constrained Hamilton equations (4.5.29) defined on the Lagrangian constraint submanifold of the phase space. The relations between

Cartan, Hamilton-De Donder, and constrained Hamilton equations are given by Propositions 3.4.4, 4.5.16, and 4.5.17. The first variational formula enables us to describe Lagrangian conservation laws in a unified way. On-shell, it leads to the weak identity dATA.

If the Lagrangian density L is invariant under the local l-parameter group of gauge vantransformations whose generator is the vector field u, the Lie derivative ishes. Then we obtain the weak conservation law dATA -_ 0 of the corresponding symmetry current T along the vector field u. This symmetry current is not defined uniquely, but depends on the choice of a Lepagean equivalent of the Lagrangian density L. Symmetry currents associated with different Lepagean equivalents differ from each other in a superpotential term.

INTRODUCTION

3

One says that a symmetry current is reduced to a superpotential if, on-shell, it takes the form

T" = W" + d"U''", where the term W" is expressed in terms of the variational derivatives of the Lagran-

gian density L and, therefore, vanishes on-shell, while U"'' = -U"". The term U'''' is called a superpotential (see Remark 3.5.5). Nother currents in gauge theory and the energy-momentum currents in gravitation theory reduce to a superpotential because the corresponding vector fields u depend on derivatives of parameters of gauge transformations [52, 67, 1651. Furthermore, in both cases, a superpotential U"'' depends on gauge parameters that guarantee the form-invariance of conservation laws under gauge transformations. Since conservation laws are linear in a vector field u, one can consider superposition of different conservation laws along different vector fields on Y. In particular, every symmetry current is a superposition of a Nother current along a vertical vector field and a stress-energy-momentum (SEM) current along a vector field r on X which gives rise to a vector field on Y. Accordingly, different lifts of r onto Y (e.g., by means of different connections on Y -+ X) lead to different SEM currents which differ from each other in Nother currents. In gravitation theory on natural bundles Y (e.g., tensor bundles), we have the

canonical lift z on Y of vector fields r on X. These lifts are generators of 1parameter groups of general covariant transformations of Y. We observe that, in General Relativity [148], in Palatini formalism [21], and in metric-affine gravitation theory [67, 68], the SEM currents reduce to the well-known Komar superpotential and its generalization (3.7.54). The difficulties arise in gauge gravitation theory in the presence of Dirac fermion

fields. The corresponding spin structure is associated with a certain gravitational field, and it is not preserved under general covariant transformations. To overcome this difficulty, one considers the universal covering bundle LX of the linear frame bundle LX, whose structure group is the universal covering group UL-(4,R) of the general linear group GL(4,R) [56, 151, 1721. One can think of LX -, X as being a universal spin structure because any Riemannian and pseudo-Riemannian spin structures are subbundles of this fibre bundle, which inherits the general covariant transformations of the frame bundle LX. As a consequence, we obtain the SEM conservation law in gauge gravitation theory, where the corresponding SEM current is also reduced to the generalized Komar superpotential [69, 1661.

INTRODUCTION

4

As an important application of jet formalism, we treat analytical mechanics as a particular case of field theory over a 1-dimensional base R, and observe that connections play a prominent role in this formulation. These are dynamic connections on the jet bundle J'Y -+ Y which correspond to dynamic equations, while connections on Y -' R define reference frames. The counterpart of Lagrangian formulation of field theory is (covariant) polysymplectic Hamiltonian formalism, where canonical momenta correspond to the derivatives of field functions with respect to all world coordinates, not only the temporal one [31, 86, 101, 163, 164]. As is well known, applied to field theory, the familiar symplectic technique takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space (see Remark 4.2.4). Polysymplectic Hamiltonian formalism is defined on the finite-dimensional Legendre manifold

11= V'Y ®(n'T'X), where V'Y denotes the vertical cotangent bundle of Y, and T'X is the cotangent bundle of X. This manifold is provided with the canonical polysymplectic form (4.2.5) which leads to the notion of Hamiltonian connections and Hamiltonian forms.

Every Lagrangian density L on the jet manifold J'Y determines the Legendre

map L of J'Y to 11. If a Lagrangian density is hyperregular (i.e, L is a diffeomorphism), Lagrangian and polysymplectic Hamiltonian formalisms are naturally equivalent. This is not the case of degenerate Lagrangian densities, while in general, we have a set of Hamiltonian forms associated with the same degenerate Lagrangian density. We then study the relations between Lagrangian and Hamiltonian formalisms for a wide class of semiregular degenerate Lagrangian densities (see Definition 4.5.8). If a Hamiltonian form is associated with L, the solutions of the corresponding Hamilton equations which live in the Lagrangian constraint space Q = L(J'Y) yield the solutions of the Cartan and Euler-Lagrange equations for L. Conversely, we need a family of associated Hamiltonian forms in order to exhaust all solutions of the Euler-Lagrange equations. Such a complete family certainly exists in the case of affine and almost regular quadratic Lagrangian densities. These classes of degenerate field systems are analyzed in detail because most contemporary field models belong to them. The 1-dimensional reduction of the polysymplectic Hamiltonian formalism provides the adequate Hamiltonian formulation of time-dependent mechanics. Its usual formulation requires a given splitting Y = R x M which, however, is broken by any

INTRODUCTION

5

time-dependent canonical transformation and reference frame transformation, including transformations between inertial frames. The field-like approach provides the frame-covariant formulation of time-dependent mechanics over the phase space II = V'Y. The main ingredient in this formulation is the canonical 3-form (4.10.13) which provides the phase space V'Y with a canonical (degenerate) Poisson structure. R define reference frames just as in the Lagrangian Complete connections on Y formulation of mechanics. Though we consider first order Lagrangian and Hamiltonian systems, higher and infinite order jets are also used. The last Chapter of the book addresses a few topics regarding these jets. In particular, we briefly review different notions of jets: the above-mentioned jets of sections of fibred manifolds, jets of modules, jets of submanifolds and jets of local diffeomorphisms of manifolds. Each of them may find its corresponding physical application. In particular, first order jets of submanifolds are suitable for formulating relativistic mechanics (see Example 5.3.5), while jets of local diffeomorphisms of manifolds provide the standard language for C-structures [104].

For the convenience of the reader, several mathematical facts are included as Remarks, thus making our exposition self-contained. The book is provided with a detailed Index and Glossary of symbols.

Chapter 1 Fibred Manifolds This and the next Chapters summarize the main notions on fibred manifolds, jet manifolds and connections which find application in classical field theory. The relevant material is presented in a fairly informal way. It is tacitly assumed that the reader has some familiarity with the basics of differential geometry [20, 83, 103, 170, 171, 186].

Throughout the book, all maps are smooth, i.e., of lass C°°, while manifolds are real, finite-dimensional, Hausdorff, second-countable and, hence, paracompact. Unless otherwise stated, we assume that manifolds are connected. We use the standard symbols ®, V, and A for the tensor, symmetric, and exterior products, respectively. The interior product (contraction) is denoted by J. By og are meant the partial derivatives with respect to coordinates with indices 8A. If M is a manifold, we denote by

7rM:TM -+M

and

it :T'M - M

its tangent and cotangent bundles, respectively. Given coordinates (z°) on M, they are equipped with the induced coordinates (?,k) and (z", ix) with respect to the holonomic bases {OA} and {dzA} for the tangent and cotangent spaces to M, respectively.

Given a manifold map f : M - M', by

Tf :TM -- TO is meant its tangent map. This has the coordinate expression

(z'", P) o T f = (P, OAP?),

fP = z"A - f,

7

CHAPTER 1.

8

FIBRED MANIFOLDS

relative to the induced coordinates (z", ia) and (zA, i'a) on M and M', respectively. The symbol C°°(M) denotes the space of smooth real functions on a manifold M.

Immersions and submersions

1.1

In this Section, we consider manifold maps of a particular type, namely, immersions and submersions, and treat them together in order to emphasize their dual nature. Let M and N be manifolds of dimensions m and n, respectively. Recall that by

the rank of a map f : M - N at a point p E M is meant the rank of the linear map Tpf : TTM -+Tf(p)N.

Suppose that f is of maximal rank at p E M. It follows that

m=n

m
.

Tpf is an isomorphism; Tpf is injective;

m > n = Tpf is surjective. Then f is said to be a local diffeomorphism, an immersion, a submersion at the point p E M, respectively.

Remark 1.1.1. Since the function p rankp f is a lower semicontinuous function, then Tpf is of maximal rank also on an open neighbourhood of p. The following results follow from the inverse function Theorem ((45), p.273).

THEOREM 1.1.1. Let f (i) If f is a local diffeomorphism at p, then there exists an open neighbourhood U of p such that f : U -. f (U) is a diffeomorphism onto the open set f (U) C N. (ii) The map f is an immersion at p if and only if there exist a (cubic) coordinate chart (U, (p) of M centred at p and a (cubic) coordinate chart (V, i,) of N centred at f (p) such that the following diagram U '0

v 1

I (_a,a)m " (_a,a).,

m

1.1. IMMERSIONS AND SUBMERSIONS

9

Of f o(p-1 : (x',...,z") H (x',...,xm,0,...,0) is commutative. If f is an immersion at p, then it is locally injective around p. (iii) The map f is a submersion at p if and only if there exist a (cubic) chart

(U, tp) centred at p and a (cubic) chart (f (U), ib) centred at f (p) such that the following diagram U

!

`-°

(-a, a)" x (-a,

I

I

a),"-n

pr,

(1.1.2)

f (U) - ' (-a, a)"

00f

oie-1

:

(a1 ,...,xm)

-+ (x ,...,2n) 1

is commutative. If f is a submersion at p, it is locally surjective around p. 0 Proof. For the proof, the reader is referred to [186], pp.24-29.

QED

DEFINITION 1.1.2. Let f : M -, N be a map. (i) The map f is said to be a local diffeomorphism [immersion,submersion] if it is a local diffeomorphism [immersion, submersion) at all points p E M. A local diffeomorphism and submersion are necessarily open maps, that is, they send open subsets of M onto open subsets of N. (ii) The pair (M, f) is said to be a submanifold of N if / is an injective immersion. The terminology imbedded submanifold is used if f is an open map. Equivalently, it is a homeomorphism onto f (M) equipped with the relative topology induced from

N. If (M, f) is an imbedded submanifold, the map / is said to be an imbedding. For the sake of simplicity, we will often identify (M, f) with f (M). If M C N, its natural injection will be denoted by iM : M - N. We will write M N for imbeddings. (iii) The triple (M, f, N) is called a fibred manifold if f is a surjective submersion.

M is the total space, N is the base space, f is the projection (or fibration) and Mq = f -I (q) is the fibre over q E N. If there is no danger of confusion, a fibred manifold (M, f, N) will be denoted f : M -, N or simply M -' N. 0 Example 1.1.2. If f is an immersion, then it is locally injective, but not necessarily

injective. For example, let M = R, N = R2 and let f : M N be the following immersion

CHAPTER 1.

10

R

FIBRED MANIFOLDS

R2

c

F:

Obviously, (R, f) is not a submanifold of R2.

Example 1.1.3. A submanifold which is not an imbedded submanifold is exemplified by the figure

R 0 f:

R'

c

f(0)

Clearly, f is not a homeomorphism onto its image equipped with the relative topology because f (U) is not an open subset of f (R) C R2 for a suitable open neighbourhood U of the point 0 E R. For a submanifold to be an imbedding, the following criteria are required. PROPOSITION 1.1.3. Let (M, f) be a submanifold of N.

(i) The map f is an imbedding if and only if, for each point p E M, there is a (cubic) coordinate chart (V, ti) of N centred at f (p) so that f (M) f1 V consists of all points of V with coordinates 1

m

A glance at the diagram (1.1.1) shows that an immersion f : M -- N is a local imbedding, that is, every point p E M has an open neighbourhood U C M such that the restriction f JU : U -+ N is an imbedding.

1.1. IMMERSIONS AND SUBMERSIONS

11

(ii) Suppose that f : M - N is a proper map, that is, the pre-images of compact sets are compact. Then (M, f) is a closed imbedded submanifold of N. In particular, this occurs if M is a compact manifold. (iii) Suppose that dim M = dim N. Then (M, f) is an open imbedded submanifold of N. O A standard way of constructing submanifolds is given, under suitable conditions, by taking the pre-images of submanifolds.

THEOREM 1.1.4. Let f : M -. N be a smooth map and (Q, g) a submanifold of N such that P = f -' (g(Q)) is non-empty. Suppose that Ti(p)N = T f(TpM) +Tg(T9-I(f ,))Q)

for any p E P. Then P can be provided with a manifold structure so that (P, ip) is a submanifold of M of dimension

dimP=dimM-dimN+dimQ. If (g, Q) is an imbedding, then there is a unique manifold structure on P such that (P, ip) is an imbedded submanifold of M. 0 Proof. For the proof, the reader is referred to [186], p.31.

QED

Since a point can be thought of as a 0-dimensional manifold, Theorem 1.1.4 has the following corollary. COROLLARY 1.1.5.

Let q E f(M) and P = f-'(q). If the tangent map Tf is

surjective at any p E P, then (P, ip) is a closed imbedded submanifold of M and

dim P = dim M - dim N. 0 Example 1.1.4. Let (M, f, N) be a fibred manifold. Then each fibre f-I (q) is a closed imbedded submanifold of M of dimension dim M - dim N.

Remark 1.1.5. Imbedded submanifolds appear also in connection with the following typical situations.

CHAPTER 1.

12

FIBRED MANIFOLDS

(i) Let f : M - N be a (smooth) map factorizing through a submanifold (P, h) of N, that is, f (M) C h(P). Then there is a unique map g : M P such that the diagram

P is commutative. If h is an imbedding, then g is smooth. (ii) The other case is concerned with extension of smooth functions. Let f

M ' N be an imbedding and let f (M) be closed in N. Let g E COD(M). Then there exists g` E COD(N) such that the diagram

Mf N \ R is commutative.

1.2

Fibred manifolds

Hereafter, 7r : Y

X denotes a fibred manifold with dim X = n and dim Y dimX = I (see Definition 1.1.2 (iii)). Example 1.1.4 implies that the fibres Y. _ 7r-'(X), x E X, are imbedded submanifolds of Y of dimension 1. 1 Unless otherwise stated, we assume I > 0, i.e., fibred manifolds with discrete fibres are not considered. I As we know from the diagram (1.1.2), the total space Y admits an atlas of charts, called fibred charts, with the following property. For any fibred chart (U, w), there is a chart (ir(U), t/i) of X such that the diagram

U - (-a, a)" x (-a, a)' C R'+" I

I

pri

a(U) -L (-a, a)" C R" is commutative. The set of functions

(xA=rAop, U`=r$ocQ) A=1,...,n, i=1,...,1,

1.2. FIBBED MANIFOLDS

13

where (r'', r) are the Cartesian coordinates on R"*", are said to be a fibred coordinate system (or simply fibred coordinates) in U. Note that (xa) is a coordinate system in 7r(U) C X. constitute a fibred coordinate atlas of Y whose The coordinate charts (U; transition functions

x" = fA(x'),

y" = f`(x'`,yj)

are compatible with the fibration Y - X. I For the sake of simplicity, a domain U of a fibred coordinate chart (U; xa, y') will not be specified if there is no danger of confusion. 1

Let V be a manifold. Then a local trivialization of a fibred manifold 7r : Y -. X with respect to the manifold V is an open covering {Ua} of X together with a family {0a} of diffeomorphisms

0a:7r-1(Ua)-4UaxV, called the trivialization maps, over Ua, such that the diagram

7r-1(U.)U. X V ,r l pry U.

is commutative for each U.-

DEFINITION 1.2.1. A fibred manifold 7r : Y -+ X, together with a manifold V, is called a fibre bundle if it admits a local trivialization with respect to V, called the X. The atlas %P _ {(U.,tP0)} is called a bundle typical fibre of the bundle Y atlas.

Given such a fibre bundle Y - X and its bundle atlas 'P = ((U., V).)), we have the collection of diffeomorphisms 'P

UanUpXV

,

UanUpxV

pr, f pr, UanUp

(1.2.1)

CHAPTER 1.

14

FIBRED MANIFOLDS

whenever U. n Up # 0. Let tfi0(x) denote the restriction of the trivialization map ipa to the fibre Y= and pop the map of U. n Up into the group of diffeomorphisms of V which is defined by

pap(x) = t/ia(x) o 'Pj' (x),

x E u. n us.

(1.2.2)

The maps pp are called the transition functions of the bundle atlas T, and satisfy the cocycle condition P.# (X) o pp,r(x) = p< (x),

x E U. n Up n U.r.

Let (Ua, 7Pa) be a local trivialization of a fibre bundle it : Y fibre V. There exists an associated fibred coordinate system

(xA=xaoir,y'=v`opr2oQ

X with a typical

(1.2.3)

in a neighbourhood of each point y E a-'(U0) which is determined by means of the trivialization map tP., a coordinate system (xA) of X around x = ir(y), and a coordinate system (v') of V around pr2 o rlia(y). The coordinates (1.2.3) are called bundle coordinates.

Remark 1.2.1. Let X and V be manifolds and let Y be a set. Assume that there is a surjection a : Y -+ X with the following properties.

There exist an open covering {Uo} of X and a collection {z/ia} of bijections such that the diagram is commutative Uo X V

W-1 (U.)

ir lfpr, U.

for each Ua.

The maps 0a o 0j' as in (1.2.1) are diffeomorphisms.

Then there is a unique manifold structure on Y for which it : Y -+ X is a fibre bundle with the typical fibre V and the bundle atlas ((U., 0.)) (183), p.39). Note that two bundle structures on a manifold Y are said to be equivalent if the corresponding bundle atlases are equivalent, that is, a union of these atlases is also a bundle atlas.

1.2. FIBRED MANIFOLDS

15

Example 1.2.2. Given a fibre bundle x : Y - X, there is a standard way to obtain a fibred manifold which is no longer a fibre bundle. One simply takes

a: Y'=Y\{y}- X where y is a point of Y. Obviously, the fibres of this fibred manifold are not diffeomorphic to each other. At the same time, there are examples of fibred manifolds whose fibres are diffeomorphic to each other, but they are not fibre bundles. Given R3 with coordinates (x, y, z), let us consider its open submanifold

Y = R3 \ ({xz = 1, y = 0} U {(0,0,0)}), and the projection a of Y onto the x-axis. Of course, this is a fibred manifold with fibres diffeomorphic to R2 \ {(0,0)}. However, 7r: Y -, X is not locally trivial over a neighbourhood of 0 E R. Indeed, for any open interval I = (-c, c) of the x-axis, the first homotopy group of 7r-I (I) is different from that of I x (R2 \ {(O, 0))). We have the following useful criterion for a fibred manifold to be a fibre bundle. THEOREM 1.2.2. Let it : Y - X be a fibred manifold. If x is a proper map, then a : Y X is a bundle. In particular, a fibred manifold with a compact total space is a bundle ([105], p.75). 0 A more complete relation between fibred manifolds and fibre bundles is given by Proposition 2.5.1. This involves the notion of an Ehresmann connection. The Cartesian product manifold

pri:XxV is a fibre bundle called the trivial bundle. THEOREM 1.2.3. Any fibre bundle over a contractible base is trivial ([1701, p.53).

0 By a local section of a fibred manifold (or a surjection) 7r : Y - X is meant

a map s : U -, Y of an open subset U of X into Y such that it o s = Id U. In particular, when U = X we refer to s as a global section or simply a section.

PROPOSITION 1.2.4. A surjection x : Y - X is a fibred manifold if and only if there exists a local section s of 7r : Y -a X passing through each y E Y. 0

CHAPTER 1.

16

FIBRED MANIFOLDS

Proof. If a local section through each y E Y exists, then the tangent map T,r is a surjection at y and, consequently, n is a submersion. The converse assertion follows immediately from Theorem 1.1.1 (ii).

QED

By virtue of Proposition 1.1.3 (i), the image s(U) of a local section s : U Y of a fibred manifold Y -' X is an imbedded submanifold of Y. If s is a global section of Y - X, then s(X) is a dosed imbedded submanifold of Y.

Remark 1.2.3. Let A be a (closed) subset of X. A smooth local section s of a : Y - X over A is defined to be the restriction to A of a (smooth) local section on an open set containing A. It may happen that a fibred manifold has no global section. We have the following well-known theorem.

THEOREM 1.2.5. Let Y -, X be a fibre bundle whose typical fibre is diffeomorphic to a Euclidean space R. Then every (smooth) local section s defined on a closed subset A of X can be extended to a global section of Y X. In particular, Y -+ X has always a global section (if we take A = 0) ([170), p. 55).

X. A fibred morphism between two fibred manifolds 7r : Y -i X and ir' : Y' -' X' is a pair of maps 9; : Y - Y' and f : X -' X' such that the diagram Henceforth, S(Y) denotes the set of global sections of a fibred manifold Y

Y'

Y__

*1

! *'

X I-X'

(1.2.4)

is commutative, i.e., 0 sends fibres onto fibres. In brief, we will say that (1.2.4) is a fibred morphism

4': Y - Y'

f

over f and, if f = Id X, then

9i:Y-Y x is a fibred morphism over X.

1.2. FIBRED MANIFOLDS

17

An isomorphism of fibred manifolds is a fibred morphism (1.2.4) such that 0 is a diffeomorphism. A fibred morphism [isomorphism] of Y - X to itself is called an endomorphism [automorphism]. An automorphism over Id X, is said to be a vertical automorphism.

A pair (Y, 0) of a fibred manifold Y -. X and a fibred morphism 4' : Y -e Y' over X is said to be a fibred submanifold of Y' - X if (Y, 0) is a submanifold of Y'. The following fact is a straightforward consequence of the diagram (1.1.1). For each y E Y, there exists a fibred chart (U, W) of Y about y with coordinate functions

(z ,...,zn ,y ,...,y) 1

1

!

and a fibred chart (V, 0) of Y' about 0(y) such that OIP-1

(2l,..., 2n,y1 ,...,yl)'-' (zL,...,zn,y ,...,yl,U,...,U). 1

A fibred imbedding 0 is sometimes termed a (bred monomorphism. Note that if (Y, 0) is a fibred submanifold of Y', then the restriction (4, 1y Y=) is a submanifold of the fibre Y. The following theorem provides useful criteria for an image and pre-image of a fibred morphism to be fibred submanifolds ([152], p.19). Let 4) : Y -' Y' be a fibred morphism over X. Given a global section s' of the fibred manifold Y' X such that s(X) C Im 4', by the kernel of the fibred morphism 0 with respect to the section s' is meant the pre-image

Ker,.4' = 4-1(s'(X))

(1.2.5)

of s(X) by -. THEOREM 1.2.6. If 4' : Y -+ Y' is a fibred morphism of constant rank, then

Im 4s and Ker,.4' of 0 with respect to the above-mentioned section s' are fibred submanifolds of Y' and Y, respectively. 0

Given a fibred manifold Y - X and a map f : X' X, the pull-back fibred manifold (or simply the pull-back) f'Y is a fibred manifold over X' with the total space

f'Y = { (z', y) E X' x Y;

ir (y) = f (z') )

and the projection pr1 : f'Y 9 (x', y) '-' z' E V.

CHAPTER 1.

18

FIBRED MANIFOLDS

Roughly speaking, the fibre of f'Y over a point x' E X' is that of Y over the point f (x) E X. If a : Y -i X is a fibre bundle, so is f* Y. Note that the projection pre : f'Y Y is a fibred morphism

f'Y

Y

1

I

pr

*

(1.2.6)

X' --aX I over f . Given a coordinate chart (U'; x") on X' and a fibred coordinate chart (U; xA, y')

of Y such that f (U') C a(U), then (U'; x'", y`) is a fibred coordinate chart on f'Y.

Lets : U -+ Y be a local section of the fibred manifold n : Y - X.

If

U' = f-'(U) is non-empty, we can define the pull-back section f's : U' -» f'Y by the relation f s(x') = (x', s o f (x')).

X yields the corresponding global In particular, every global section s of Y section f's of the pull-back f'Y -. X'. The composition of fibred manifolds Y -' Z and Z - X is obviously a fibred manifold w : Y-WYZ Z-4 X

(1.2.7)

such that the diagram

z

Y

x is commutative. It is called a composite fibred manifold.

If Y -, Z and Z - X are fibre bundles, so is the composition Y - X (see Remark 2.5.3). Dealing with the composite manifold (1.2.7), we use the fibred coordinates

Y (x" z°, y') .YZ

I

Z

(xa,zp)

*zx 1

X

(x")

(1.2.8)

1.2. FIBRED MANIFOLDS

19

where (x", z') are fibred coordinates on Z -' X. This means that the transition functions z' --" (x', za) do not depend on the coordinates y'.

Example 1.2.4. Let a : Y - X and a' : Y' - X be fibred manifolds over the same base X. Their fibred product

YxY' x over X is the composite fibred manifold

7r''Y-eY'-'X.

or

Example 1.2.5. Let 7r : Y -. X be a fibred manifold. Using the tangent map Ta : TY -+ TX, we obtain the following commutative diagram

TY

Tir,

TX *x

WY

(1.2.9)

1

y *+ X

A glance at this diagram shows that TY - X has two composite fibrations

TY-e TX -.X (xA, y', ±A, U`)'-' (x

)'-' (xA)

and

TY-' Y

-`+X

Let 1 be a fibred morphism between the fibred manifolds x : Y X and n' : Y' -, X'. Then the tangent map T4 : TY -' TY' is a fibred morphism with respect to both the composite fibrations of TY given above:

Tin

TY

TY'

I

I

TX WX

1

Tf

TWI

TX' l

X-L. X,

-XI

CHAPTER 1.

20

FIBRED MANIFOLDS

The following two assertions on composite fibred manifolds are useful in application to field theory.

PROPOSITION 1.2.7. Let h : X -* Z and g : Z -' Y be sections of the fibrations lrzx and xyZ, respectively. Then their composition

s=goh

(1.2.10)

is a section of the composite fibred manifold it : Y -. X (1.2.7). Conversely, if Tryz : Y -i Z is a fibre bundle whose typical fibre is diffeomorphic to a Euclidean space, then every global section s : X - Y is represented by a composition as in (1.2.10), where h = 7ryz o s and g : Z Y is an extension of the local section g : h(X) - Y which is defined by the diagram Y

9r t.

h(X) - - X

g(h(x)) = s(x),

PE X.

This is an immediate consequence of Theorem 1.2.5, since h(X) is a closed imbedded submanifold of Z.

PROPOSITION 1.2.8. Given the composite fibred manifold (1.2.7), let h : X - Z be a global section. Then the pull-back MY -+ X of irYz : Y - Z by h is a fibred submanifold of the fibred manifold it : Y X, as follows from the diagram h-Y `per Y

prl 1f ,r X 0

(1.2.1 1)

1.3. VECTOR AND AFFINE BUNDLES

1.3

21

Vector and affine bundles

In this Section we recall some basic properties of vector and affine bundles which we will need for what follows. DEFINITION 1.3.1. A vector bundle is a fibre bundle 7r : Y -+ X such that: the typical fibre V and all fibres Y= = 7r-1(x), x E X, are real finite-dimensional vector spaces;

there is a bundle atlas {(U.,><'')} whose trivialization morphisms V)a restrict to linear isomorphisms 0.(x) : Y. -i V for each x E UQ.

Remark 1.3.1. A vector bundle is a fibre bundle whose structure group is the general linear group GL(V). e By virtue of Theorem 1.2.5, every vector bundle 7r : Y - X has a global section. In particular, it admits the zero section 0 : X Y defined by 0(x) = 0= E Y. for each x E X. The set S(Y) of global sections of Y -, X is both a real vector space and a module over the ring C°°(X) of smooth functions on X. When dealing with a vector bundle Y, we use linear bundle coordinates (xA, y`) (1.2.3) associated with a bundle atlas' = {(U.,0.)} of Y. We have (Pr2 00-)(Y) = Ve,,

y = y'e;(x),

e;(x)

Here {e;} is a fixed basis for the typical fibre V of Y and (e'- (x)) is the fibre basis (the frame) for the fibre Y= of Y which is associated with the bundle atlas T. A morphism of vector bundles 4' : Y -- Y' is defined as a fibred morphism over f whose restriction 4s : Y. -e Y' to each fibre of Y is a linear map. It is called a linear bundle morphism over f. DEFINITION 1.3.2. Let 4' : Y Y' be an injective vector bundle morphism over X. We say that (Y, 4') is a vector subbundle of Y' X. Using Proposition 1.1.3 (i), one can show that every vector subbundle of a vector bundle is a dosed imbedded submanifold.

CHAPTER 1.

22

FIBRED MANIFOLDS

The following assertion is a corollary of Theorem 1.2.6.

PROPOSITION 1.3.3. If Y -. X and Y' -' X are vector bundles and 4' : Y - Y' is a linear bundle morphism over X of constant rank, then the image of 4' and the kernel Ker of 4' with respect to the zero section 6 of Y' -e X are vector subbundles of Y' X and Y -i X, respectively. 0 There are the following standard ways to construct vector bundles from given ones.

Let Y -. X be a vector bundle with a typical fibre V. By Y X is meant the dual vector. bundle with the typical fibre V' dual of V. Their interior product is defined as the fibred morphism over X

YxY' x

x R.

Let Y -' X and Y' - X be vector bundles with typical fibres V and V', respectively. Their Whitney sum Y®Y is a vector bundle over X with the typical fibre V ®V. Let Y X and Y' X be vector bundles with typical fibres V and V', respectively. Their tensor product Y ®Y' is a vector bundle over X with the typical fibre V ® V'.

Example 1.3.2. The tangent bundle 'sx : TX . X of a manifold X is a vector bundle over X such that, given an atlas {(U., W.)} of X, TX is provided with the holonomic atlas 'P = {(U.,0G.

(1.3.1)

The associated linear bundle coordinates are the induced coordinates (a-%) with respect to the holonomic frames {8a} in tangent spaces T=X. Their transition functions read a

i'a = 0Xµ i". The tangent bundle TX is a fibre bundle with the structure group GL(dim X, R).

1.3. VECTOR AND AFFINE BUNDLES

23

The cotangent bundle of X is the dual T'X of TX. It is equipped with the induced coordinates (ia) with respect to holonomic coframes {dz} dual of {aa}. Their transition functions read

z I' The tensor products

(®TX) (& (®T'X) of tangent and cotangent bundles are called tensor bundles. Given a holonomic atlas (1.3.1) of TX, they are provided with holonomic coordinates i4;..sk with the transition functions az"k , , az'°' ... 8z'°- ax" az az''1 Tip, aeOk'...vk

Given a map f : X - X', the tangent map T f is a linear bundle morphism of TX to TX'. If f is a diffeomorphism, we have also the linear bundle morphism T* f : T`X - T'X' over f, called the cotangent map. Its coordinate expression is 1 a)'At').

0T'f = (fe(e), a(a Let us turn now to affine bundles.

DEFINITION 1.3.4. Let i : Y - X be a vector bundle with a typical fibre V. An affine bundle modelled over -f : V -+ X is defined as a fibre bundle a : Y -, X whose typical fibre V is an affine space modelled over V, and the following conditions hold

All the fibres Y of Y are affine spaces modelled over the corresponding fibres

F. of Y. There is a bundle atlas { (U°1 t/i°) } whose local trivializations

W-' (U.) 4U°xV x \ 1f pr, U.

restrict to affine isomorphisms 0°(z) : Y, -a V for each x E U°.

(1.3.2)

CHAPTER 1.

24

FIBRED MANIFOLDS

A Remark 1.3.3. An affine bundle is a fibre bundle with the structure group GA(V) of affine automorphisms of V.

Let Y x Y' be the fibred product of two affine bundles Y -. X and Y' -. X x which are modelled over the vector bundles Y - X and Y - X, respectively. This product, called the Whitney sum, is also an affine bundle modelled over V ® 7. X

When dealing with an affine bundle, we use affine bundle coordinates (z', y`) induced in it-'(U0) by the local trivialization (1.3.2) and by the choice of an origin in V and a basis {ei} for the vector space V. Then (xa,V) are the vector bundle coordinates in Y'(U0) induced by the local trivialization (1.3.2) and by the choice of the same basis {e;} for V.

Example 1.3.4. Every vector bundle has a natural structure of an affine bundle. In particular, the tangent bundle TX of a manifold X has the natural structure of an affine bundle which is called the affine tangent bundle of X. Theorem 1.2.5 guarantees the existence of global sections of an affine bundle Y -. X. Every such global section s yields the fibred morphism (1.3.3)

X

4',(y) = y - s(x),

y E Y.

An affine bundle morphism between two affine bundles a : Y X' is a fibred morphism

X and 7T': Y' -+

Y-Y'

'lX f X,

l,r

(1.3.4)

whose restriction 0.: Y. -+ Yfi:> to each fibre Y. of Y - X is an affine mapping. The affine bundle morphism (1.3.4) uniquely determines the linear bundle morphism

YAP' 71

1 r

X - X,

(1.3.5)

1.4. TANGENT BUNDLES OF FIBBED MANIFOLDS

25

which is called the linear derivative of 4'.

DEFINITION 1.3.5. Let Y -' Y' be an injective affine bundle morphism over X. We say that (Y, 0) is an affine subbundle of Y' --+ X. 0 It is readily seen that, if (Y, 4') is an affine subbundle of Y' -, X, then (Y,4) is a vector subbundle of r : Y -+ X. It follows immediately that (Y,4') is a closed imbedded submanifold of Y'. The following assertion is a corollary of Theorem 1.2.6 and Proposition 1.3.3. PROPOSITION 1.3.6. If Y X and Y' -' X are affine bundles and 4' : Y - Y' is an affine bundle morphism over X of constant rank, then the image of 4' is an affine subbundle of Y' X modelled over the vector bundle Imp X. Let s: X - Y'

be a global section such that s'(X) C Im4'. Then the kernel Ker,4 of 4' with respect to s' is an affine subbundle of Y - X modelled over the vector bundle

Ker4 - X. 0 Remark 1.3.5. The situation considered in this remark often occurs. Let Y'- X be an affine bundle modelled over a vector bundle V - X. Let Y C Y' be an affine subbundle modelled over a vector bundle V - X. Assume that 7 is the Whitney sum of V and a complementary vector bundle Z X. Then one can easily verify X decomposes in the Whitney sum that the affine bundle Y' Y'=YED Z. x

1.4

Tangent bundles of fibred manifolds

Let Y - X be a fibred manifold coordinatized by (r", y`). Its tangent bundle TY is equipped with the corresponding induced coordinates a

_ 8xµ' (X-1, 0, x

,

V),

CHAPTER 1.

26

FIBRED MANIFOLDS

A glance at the transformation law (1.4.1) shows that the tangent bundle TY -. Y has the vector subbundle

VY = KerTi given by the coordinate relation 0 (see the diagram (1.2.9)). This subbundle, called the vertical tangent bundle, consists of vectors tangent to fibres of Y. It is provided with the induced coordinates (a, y', y') with respect to the holonomic fibre bases {8;}. Let T4i : TY -' TY' be the tangent map to a fibred morphism 4i : Y Y'. Its restriction V4i = T'F o ivy : VY -+ VY',

y"oVY= v

(1.4.2)

,

is a linear bundle morphism of vertical tangent bundles over 'F. The morphism (1.4.2) is called the vertical tangent map Wt. The vertical cotangent bundle V'Y Y of Y is defined as the vector bundle dual of the vertical tangent bundle VY -' Y. We will denote by {ay'} the fibre bases for V'Y which are dual of the fibre bases {8;} for VY. The comparison of the transformation law ay"

= m ayt

of {ay'} with the transformation law

dy" _ M dyi + f dx" of the holonomic coframes {d-c, dy'} for T'Y shows that V'Y fails to be a subbundle of the cotangent bundle T'Y of Y. At the same time, there is the canonical surjection

T'Y

y

V'Y,

xadxa + ycdy' '-' yisy'. With VY and V'Y, we have the following two exact sequences of vector bundles

over a fibred manifold Y - X: (I.4.3a)

0--Y

(1.4.3b)

1.4. TANGENT BUNDLES OF FIBRED MANIFOLDS

27

where all the morphisms are over Y.

Remark 1.4.1. It is easily observed that the surjection

lrT:TY Y YxTX,

(1.4.4)

is an affine bundle modelled over the pull-back vector bundle

pr2:VYx(YxTX)-YxTX. Y X X In general, there is no canonical splitting of the exact sequences (1.4.3a) and (1.4.3b). Every splitting

I:YxTX X

TY,

(1.4.5)

8aH8A+1"x8 of (1.4.3a) and the dual splitting

r : V'Y Y T'Y,

(1.4.6)

of (1.4.3b) correspond to the choice of a connection r on the fibred manifold Y - X. Here the local functions 1',, on Y are the connection parameters of I' (see Section 2.3).

Remark 1.4.2. For the sake of simplicity, we will denote the pull-backs

YxTX

and

x simply by TX and T'X.

YxT'X x

Vertical tangent bundles of many fibred manifolds are pull-back bundles.

DEFINITION 1.4.1. One says that a fibred manifold Y -, X admits a vertical splitting if there exists a linear bundle isomorphism

a: VY-ya'V=Y xV Y X

(1.4.7)

CHAPTER 1.

28

FIBRED MANIFOLDS

where V -' X is a vector bundle. O Fibred coordinates (z, y') of the fibred manifold Y -' X are called adapted to the vertical splitting (1.4.7) if the induced coordinates of the vertical tangent bundle VY take the form

y`=Vea where (zµ, V) are bundle coordinates of V. In this case, coordinate transformations of the coordinates y' are independent of the coordinates y'.

Example 1.4.3. A vector bundle Y

IVY

X has the canonical vertical splitting

Tr'Y.

The linear bundle coordinates of Y are adapted to this vertical splitting so that = y`.

Also an affine bundle Y modelled over a vector bundle V has the canonical vertical splitting

The affine bundle coordinates of Y are adapted to this splitting so that y' = y'. Note that the tangent functor T preserves algebraic structures of fibred mani-

folds. In particular, if Y - X is a vector bundle, so is TY - X. If Y - X is an affine bundle modelled over a vector bundle V -y X, then TY X. bundle modelled over the vector bundle TY

X is an affine

Let us now consider the composite fibred manifold

YMZ. ZM. X Z. (1.2.7). For the sake of simplicity, we denote by YZ the fibred manifold Y Accordingly, VYZ is the vertical tangent bundle of Y -. Z. It is a subbundle of the vertical tangent bundle VY -. Y. In the fibred coordinates (1.2.8), these two subbundles of TY are characterized as follows:

VY = {za,z°,y',±A = O,.i°'}, VZY = {z", zP, y`, i'' = O, i' = O, y' }.

1.5. VECTOR AND MULTIVECTOR FIELDS

29

Considering also the vertical tangent bundle VZ of Z -. X, we obtain the following commutative diagram

VYVUVZ !

I

YnLZ Z .Z.,

X and the following exact sequence of vector bundles over Y:

0-'VYz'VY-iYxVZ-O. z 1.5

(1.4.8)

Vector and multivector fields

Let M be a manifold. A vector field v on M is defined to be a global section of the

tangent bundle TM - M. The set T(M) of vector fields on M is both a module over the ring C°°(M) and a real Lie algebra with respect to the Lie bracket of vector fields.

Remark 1.5.1. Given a function f E C°°(M), the Lie derivative of f along a vector field v on M is the function

=vjdf on M. Then the Lie bracket of vector fields is the bilinear map (., .]

: T(M) X T(M) - T(M)

defined by the relation

Vf E C°°(M).

LH,,,.lf =

The well-known properties of the Lie bracket are immediately deduced from this definition. In particular, if v = va8a and u = uµ8µ, then [V, U]" = v '

0

-8uµ 8z"

uX 'd A

8z"

CHAPTER 1.

30

FIBRED MANIFOLDS

Let g : M -. N be a diffeomorphism. Then, for every vector field v on M, one can define a vector field

g.v=Tgovog' on N, called the push-forward of v by g. The push-forward is obviously a Lie algebra

isomorphism of T(M) onto T(N). If g is a diffeomorphism of M and g.v = v, one says that v is invariant under g.

Remark 1.5.2. Let us briefly recall the relation between vector fields and 1parameter groups of (local) diffeomorphisms. Let v be a vector field on a manifold M. A curve c in M is said to be an integral curve of the vector field v if

c=voc in the domain of c. For every point z E M, there is a unique integral curve c : (-e, e) -. M of v for some c > 0 such that c(0) = z. This statement follows from the well-known theorem on solutions of a system of ordinary differential equations ([45], p.283).

Let U C M be an open subset and e > 0 . By a local 1-parameter group of local diffeomorphisms of M defined on (-e, e) x U is meant a mapping

G:(-e,e)xU-M, G : (t, z) - Ge(z), which satisfies the following properties:

(i) for each t E (-e, e), Gt is a diffeomorphism of U onto the open set Gt(U) of M;

(ii) if t, s, t + s E (-e, e) and z, G,(z) E U, then Gt+,(z) = Gt O G,(z).

If the mapping G is defined on R x M and satisfies the conditions: (i') for each t E R, Gt is a diffeomorphism of M,

(ii') for all t, s E R and p E M, Gt+.(z) = Gt o G.(x).

1.5. VECTOR AND MULTIVECTOR FIELDS

31

then it is called a 1-parameter group of diffeomorphisms of M. Every local 1-parameter group of local diffeomorphisms Ge defines a vector field v on U by setting v(z) to be the tangent vector to the curve c(t) = Gt(z) at t = 0. Conversely, we have the following theorem.

THEOREM 1.5.1. Let v be a vector field on a manifold M. For each z E M, there exist e > 0, a neighbourhood U of z and a unique local 1-parameter group of local diffeomorphisms defined on (-c, e) x U which determines v ([1031, p.13). In brief, v is the generator of a local 1-parameter group of local diffeomorphisms G of M. If a vector field v is induced by a 1-parameter group of diffeomorphisms of M, then v is said to be complete. Note that if C is defined on (-e, e) x M for some c > 0, then v is complete. In particular, every vector field on a compact manifold is complete.

Let x : Y

X be a fibred manifold. The projection a allows us to define

different types of vector fields on Y.

A vector field u on a fibred manifold Y - X is said to be projectable if it projects over a vector field ux on X, i.e., the following diagram

Y- - TY I

I

X

TV

ux

is commutative. A projectable vector field has the coordinate expression u = U,8.% + u'8;,

ux = u''8a,

where u'' are local functions on X, and u' are local functions on Y. Projectable vector fields form the Lie subalgebra P(Y) C T(Y). A projectable vector field u on a fibred manifold Y is said to be vertical if it projects over the zero vector field ux = 0 on X. The set of vertical vector fields on Y is the Lie subalgebra V(Y) C P(Y).

Example 1.5.3. The vertical vector field v = y'8; on a vector bundle Y - X is called Liouville's field.

CHAPTER 1.

32

FIBRED MANIFOLDS

Remark 1.5.4. In accordance with Remark 1.5.2, every projectable vector field u on a fibred manifold Y defines a local 1-parameter group G of local diffeomorphisms

of Y. It is readily observed that elements Gt of this group are local fibred automorphisms of Y over local diffeomorphisms of X whose generator is the projection

ux of u on X. If u is a vertical vector field on Y, this is the generator of a local 1-parameter group of vertical automorphisms of Y. A multivector field t9 of degree r (or simply a r-vector field) on a manifold M is

a section of the skew-symmetric tensor product ATM M of the tangent bundle of M. It is given by the coordinate expression t9 = 1'tyal....4aa1 A ... A aa,.

By I V I is meant the degree of t9.

Let us denote by T,(M) the vector space of r-vector fields on M. In particular, T (M) = T(M). All multivector fields on M form the Z-graded algebra m = dim M.

T . ( M ) _ ®T (M),

This algebra is provided with the Schouten-Nijenhuis bracket

]sN :1(M) X T,(M) - T+.-1(M),

(1.5.1)

which generalizes the Lie bracket of vector fields as follows [17, 1821. Let 01,.. ., 0, and u be vector fields on M. The Lie derivative L of t91 A along the vector field u is given by the formula

L.(ty1A...At9,)=Et91A...A(U,t9j]A...At9,.

A10,

(1.5.2)

j.1

The Schouten-Nijenhuis bracket (1.5.1) is a unique local type extension of the Lie derivative (1.5.2) which satisfies the condition [u1 A ... A u,,w]sN = - (-1)j+1u1 A ... A u1 A . J.1

for anyul,...,u,. E T(M) and WET(M).

A U, A

(1.5.3)

33

1.6. DIFFERENTIAL FORMS ON FIBRED MANIFOLDS This bracket has the coordinate expression t9 =

V = va,...e.8o, A ... A 8a.,

t9a,...a.8a, A ... A 8a,,

[t9, V]SN = t9 * V + (-1)1#11vly * t9,

t9 * V =

I

I

19

I

V

I! (t9W 1....% -10JA01...a.8.%1 A ... A 8k_t A 8a1 A ... A 80.)

There are the relations 0, V]SN = (-1)1611vl IV, i9]SN, ]t/, t9 A VISN = IV, d]sN A V + (- 1)1"11101+Ield A (v, V]SN,

(_l)Wllai+I"IIV,t9 A V)SN + (-1)lall"I+1' 1[t9, V A V]SN +

(-1)IvIIdI+IvI]v, V A 191SN = 0.

Example 1.5.5. Let w = [w,w]SN

=

A8" be a bivector field. We have A8)2

(1.5.4)

Every bivector field w on a manifold M yields the associated bundle morphism u# : T' M -+ TM defined by z E M,

a,,8 E T= M,

(1.5.5)

w'(a) = w1(z)aµ8". A bivector field w whose bracket (1.5.4) vanishes is called a Poisson bivector field.

1.6

Differential forms on fibred manifolds

Let M be an m-dimensional manifold. Recall that an exterior r-form

4A ... A dza' on M is a section of the skew-symmetric tensor product Ar T'M - M. The 1-forms are usually called Pfafan forms.

CHAPTER 1.

34

FIBRED MANIFOLDS

Let us denote by IY(M) the vector space of exterior r-forms on a manifold M. This is also a module over the ring O°(M) = C°°(M). All exterior forms on M constitute the exterior Z-graded algebra

O'(M)

(-D 0,.(M)

r

with respect to the exterior product. The exterior differential d is the first order differential operator on 0'(M):

d : 0'(M) - D"'(M), A dza' A ... de".

dO = r It obeys the relations

dod=0, d(4 A a) = d(*b) A a + (-1)Iml0 A d(a).

Given a map f : M N, by f q is meant the pull- back on M of a form 4 on N by f which, for 1-forms 0, is defined by the condition

vJf'm(z) =Tf(v)J4(f(z)),

Vv ETsM.

We have the relations

f'(0 A a) = f'4 A f'a, df'.O = f'(dq5)

Contraction of a vector field u = 08, and an r-form 0 on M is given in coordinates by

'

uJ0 _ F, k:1 1

(r -

(-1)k-1

r!

u ak0a,...>,,...a.dZ"A...ndx

-Ak

A...Adz''=

1)!u"4,w,...°._,dz°' A ... A dz°'-'.

It satisfies the relations

0(u1,...,ur) = u*J ...u1Jo, uJ(4'Aa)=uJ4Aa+(-1)I#I4AuJa, (u,ulj4 = ujd(u'J4') - u'Jd(uj0) - 0(u, t1),

0 E D'(M).

1.6. DIFFERENTIAL FORMS ON FIBRED MANIFOLDS

35

Example 1.6.1. Let 11 be a 2-form on M. It defines the linear bundle morphism S1b : TM - T' M, V E TZM.

I Slb(v) d=d -vJil(z),

(1.6.2)

In coordinates, if S2 = zi1,,,,dz' A dz" and v = v"8", then

ft(v) = -f1,,,,v"dz". One says that Il is of constant rank k if the corresponding morphism (1.6.2) is of constant rank k. The 2-form is non-degenerate if its rank equals dim M (which is necessarily even). A closed non-degenerate 2-form is called a symplectic form. The Lie derivative L, the expression

of an exterior form 0 along a vector field u is given by

LuO = uJdi+d(uj0). It satisfies the relation

Lu(0Aa) =Lu46 na+0ALo. Let M be an oriented manifold provided with a non-degenerate pseudo-Riemannian metric with a signature i9: z

g E V O'(M),

g = ga"dz'' ®dz",

or equivalently s

g E V 7(M),

g = ga"8®®8".

The associated volume form is r1=

Ig(dz'A...Adz,

g=det(ga")

1 We denote the corresponding induced fibre metrics in tensor bundles by the same symbol g. t The Hodge star operator is the isomorphism

V(M)

Y(M)

CHAPTER 1.

36

FIBRED MANIFOLDS

defined by the condition

0A*o=aA*0= Its coordinate expression is 1/2

*dz"' n ... A dzar =

(In9 I

r)t Ea,...a.

,...r.,,,dz"'+, A ... A da'""

where a is the skew-symmetric Levi-Civita tensor with the component el... In particular, we have * * 40 = (-1)r(m-r)4.

= 1.

-0 E Dr(M),

,

*1i=(-1)''.

*1=17,

The codifferential 6 acting on exterior forms is defined as 6: SY(M) -+ OT_I (M),

bm = (-1)'++l+i, * d

46 E or(M),

(1.6.3)

for each r > 0. In particular, bf = 0 for f E O°(M). 2 In fact, the formula (1.6.3) does not require the orientability of M because the star operator occurs iterated. 1 The property d o d = 0 implies

1bob=0. Moreover, the following relation holds:

dmA*o - 0A*6a = d(OA*a), where 0,o E 0'(M). Recalling (1.6.3), one can say that 6 is the adjoint of d, and vice versa From (1.6.3) we easily see that the coordinate expression of b is dz1" A ... A

1

,(pi...Mr = o"" ... /

dz'r-1

Wa,....1r1 I9Iea(

I9I

a''...Mr_,).

(1.6.4)

1.6. DIFFERENTIAL FORMS ON FIBRED MANIFOLDS

37

Let K be the Levi-Civita connection on M associated with the metric g. Then using the well-known identities K;,°µ = - 2g"°(8a9°µ + 8µgva - 8°9av),

(1.6.5)

8a 900 = gaV K,A + g'Ka KA00

19=

8

191,

we obtain the covariant expression of (1.6.4) which is simply

where V is the covariant derivative relative to K, that is,

=

+ ... +

+

Let Y -. X now be a fibred manifold with fibred coordinates (x, y'). The pull-back of exterior forms on X by ir provides the inclusion ,r' : 0'(X) £V(Y). Elements of a'(D'(X)) C iD'(Y) are called basic forms on Y. They are given by the coordinate expression A ... A dXA-,

where 0a,...a, are local functions on X. Exterior forms

:Y-+ATX,

1

a

=TA...Adza.,

are local functions on Y, are said to be the semibasic or horizontal forms. A horizontal n-form is called the horizontal density. where

Remark 1.6.2. In the sequel, we will use the notation 1w=dz'A...Adz",

W.%=9.Jw,

Let us turn now to tangent-valued forms.

B.Jwa=w,.a

(1.6.6)

CHAPTER 1.

38

FIBRED MANIFOLDS

Elements of the tensor product £) (M) ® T(M) are called the tangent-valued r-forms

0: =Ti

,...%'d?n...ndz'-®8µ.

A tangent-valued 0-form is a vector field.

Example 1.6.3. There is one-to-one correspondence between the tangent-valued 1-forms 0 on a manifold M and the linear bundle morphisms over M

TM-+TM, : TTM 3 v'--. vJ4(z) E TM,

T'M

(1.6.7)

T'M,

T, *M 3 v' - m(z)Jv' E T, M.

(1.6.8)

In particular, the canonical tangent-valued 1-form

BM=dzA®8

(1.6.9)

on M corresponds to the identity morphisms (1.6.7) and (1.6.8). The space V (M)®T(M) of tangent-valued forms is provided with the F}olicherNUenhuis bracket (F-N bracket) which generalizes the Lie bracket of vector fields (105, 128). The F-N bracket reads

I, JFN : D'(M) 0T(M) x D'(M) 0 T(M) -e Dr+,(M) ®T(M), (1.6.10)

(Lnanf)®u+(-1)'(daAujf)®v+(-1)r(vJaAdf)®u, a E Dr(M),

6 E iY(M),

u,v E T(M).

Its coordinate expression is [W, a]FN =

r.s. 1

r'Yat...ar-,vBarQA.+,..J1r+. + SQ ar+q...A,+.BX1+14%,...a,)dzA' A ... Adz ,+. ® e,.,

46 E Or(M) 0T(M),

a E 17'(M) 0T(M).

1.6. DIFFERENTIAL FORMS ON FIBRED MANIFOLDS

39

1 For the sake of simplicity, the F-N bracket will be denoted simply by [., .1. 1

The F-N bracket makes O'(M) ® T(M) into the graded Lie algebra: [0411 =

(-1)I0II*I+1[1p,01,

(1.6.11)

[m, [+L, 811= [[0, VG1, e1 + (-1)I1Ir0I [+,, [m, ell,

(1.6.12)

0,,,b, B E i7'(M) ® T(M).

Given a tangent-valued form 0, the Nijenhuis differential on O'(M) ®T(M) is defined as the morphism 1d9 : a -- dda = [9, a],

Vol E D*(M) ® T(M).

By virtue of (1.6.12), it has the property dm[1G, 9] = [dm'', e] + (-1)I#II

I ['+G, d#9]

Example 1.6.4. If 0 = u is a vector field, the Nijenhuis differential reduces to the Lie derivative of tangent-valued forms [u, a] = (u°Asa,...". sall"VA2

+

A.8",u" )dz"' A... Adx"' ®8,,,

where a E O'(M) ® T(M). Let Y -, X be a fibred manifold. We consider the following subspaces of the space O'(Y) ®T(Y) of tangent-valued forms on Y: tangent-valued horizontal forms

0 E O'(X) ®T(Y), ds"t A ... A dx"'

®+ J11...J1r8i1

r

where,...", and 0,..." are local functions on Y; projectable tangent-valued horizontal forms

0 E i7'(X) ® P(Y) c 17'(X) ®T(Y),

dx'\- A ...A(x"-®+ ",...J1.Vf], OAP,

are local functions on X, while 0are local functions on Y;

CHAPTER 1.

40

FIBRED MANIFOLDS

vertical-valued horizontal forms E 0'(X) ® V(Y) C fl'(X) ®P(Y)44:Y-AT'X®VY,

A...Adxa'®8;. Example 1.6.5. Vertical-valued horizontal 1-forms Y

o=aadza®2, are termed soldering forms. For instance, let us take Y = TX. Due to the vertical splitting

VTX = TX x TX, every tangent-valued 1-form

0: X

T'X ®TX X

on X determines a soldering form

o : TX (1d- TX x(T'X ®TX) 25 T'X ® VTX X

X

TX

onTX. The spaces 0'(X) ®P(Y) and 0'(X) ®V (Y) are closed under the F-N bracket. Remark 1.6.6. We mention also the TX-valued forms

0:Y-nT'X®TX, Y

(1.6.13)

and V'Y-valued forms

0:Y-AT'X0 V'Y, 0=

A ... A d? ®avi.

(1.6.14)

1.6. DIFFERENTIAL FORMS ON FIBRED MANIFOLDS

41

It should be emphasized that (1.6.13) are not tangent-valued forms and (1.6.14) are not exterior forms. They exemplify vector-valued forms. Given a manifold M and a vector bundle E -' M, a vector-valued r-form on M is defined to be a morphism M

For instance, let f : M - N be a map and

a tangent-valued r-form on N.

Similarly to (1.6.1), one can define the pull-back

j'¢ : M XT'M®TN on M of the tangent-valued form 0 by the map f . This is a TN-valued r-form on M.

In particular, let Y - X be a fibred manifold. The pull-backs a,

over Y of tangent-valued forms 0 on X exemplify the TX-valued forms (1.6.13). a

Remark 1.6.7. Similarly to Example 1.6.3, there is one-to-one correspondence between the global sections of the fibre bundle

V'Y ®VY - Y

(1.6.15)

Y

and the linear bundle morphisms VY Y VY and V'Y Y V'Y over Y. For instance, IdVY corresponds to the canonical section (1.6.16)

of the fibre bundle (1.6.15).

Chapter 2 Jet Manifolds and Connections In this Chapter we will address those aspects of first and second order jet formalism which are important in physical applications. For a more comprehensive treatment

the reader is referred to [105, 127, 167]. Higher order and infinite order jets are considered in last Chapter. Here we are concerned with jets of sections of fibred manifolds, whereas other notions of jets are dealt with in Chapter 5.

First order jet manifolds

2.1

Given a fibred manifold a : Y X, let us consider the equivalence classes jss, x E X, of (local) sections s of Y -, X so that sections s and s' belong to the same class j=s if and only if Ts IT.x= Ts' IT.x Roughly speaking, these sections are identified s' E j.'s

b

s'(x) = s(x),

8as"(X) = 8as(x),

by their values and the values of their first order partial derivatives at the point x of X with respect to any fibred coordinates (a`,y') of Y around s(x). The equivalence class j.'s is called the first order jet of sections s at the point

XEX. Let us consider the set

J'Y = U j=s. zEX

43

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

44

We have the following canonical surjections

WI: J1Y 3 jms"x E X, 7ro: J'Y 3 j=s'- s(x) E Y

(2.1.2) (2.1.3)

which form the commutative diagram J1Y W". Y

x There are several equivalent ways to provide the set J'Y (2.1.1) with a manifold structure.

PROPOSITION 2.1.1. Let a fibred manifold Y -i X have an atlas of fibred coordinates (xa, y'). Then the set J'Y can be endowed with an atlas of the adapted coordinates (X.%, y', yi,),

(2.1.4)

(xA, y',Y.10U.s) = (xi', s'(x),

where the coordinates yA, have the transition functions

8x" (2.1.5) = 8x'a (Dµ + y'µ8t)y". The coordinates (2.1.4) make J'Y into an (n + I + nl)-dimensional manifold called the first order jet manifold of the fibred manifold Y -, X. 0 yea

Remark 2.1.1. In the physics literature, the coordinates ya are often called velocity coordinates or derivative coordinates.

The surjection J'Y -. X (2.1.2) is a fibred manifold, whereas the surjection J'Y - Y (2.1.3) is a fibre bundle. Thus, we have the composite fibred manifold

J'Y- Y - X. IfY -, X is a bundle, so is J'Y - X. Furthermore, a glance at the transformation law (2.1.5) shows that J'Y - Y is an affine bundle, called the jet bundle. It is modelled over the vector bundle

J'Y=T'X®vY-.Y,

(2.1.6)

2.1. FIRST ORDER JET MANIFOLDS

45

where T'X stands for the pull-back x'T*X.

Example 2.1.2. Let a: Y X and zr: Y' -+ X be two fibred manifolds over the same base X. Then the fibred morphisms

prl: YxY'--4Y and X

pre:

YxY'-Y'

x

induce the canonical isomorphism

J'YxJ'Y'. x

x

Example 2.1.3. If a fibred manifold Y -i X is trivial, the fibred jet manifold J'Y -i X is not necessarily so. Let Y = X x F be a trivial bundle. Then we have

J'Y = T'X OTF. Y

If Y -+ R" is a trivial bundle, so is J'Y -, R".

Let s : X -+ Y be a (local) section. Then the map x " j=s defines a (local) section

3 = J18: X -. J'Y,

(2.1.7)

(y', ya) o J's = (siW.0.%st(x)),

of the fibred jet manifold J'Y -i X. This is called the first order order jet prolongation (or simply the jet prolongation) of the section s. A section 3 of the fibred jet manifold J'Y - X is called integrable if it is the jet prolongation (2.1.7) of a section of Y - X. Let 44p: Y Y' be a fibred morphism between fibred manifolds Y X and Y' -. X' over a diffeomorphism f : X -. X'. For any (local) section s of Y - X, we have the induced (local) section

+.s=0oso f': X'-.Y'. Then there exists a unique fibred morphism

J10: J'Y -' J'Y'

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

46

over j characterized by the condition

J'4;oJ's=J'(4'.s)of for each (local) section s of Y -+ X. This fibred morphism is called the first order jet prolongation (or simply the jet prolongation) of 4'. J'4' is both an affine bundle morphism over the fibred morphism 4, and a fibred morphism over the diffeomorphism f as is seen from the diagram

Jiy J'* JIY/ Y

Y,

'1X

X'

(2.1.8)

IT'

The coordinate expression of J'4' is y' o J'4' = (0;'V + y'µ8j4'') 88

(2.1.9)

With the obvious meaning of the symbols used, the jet prolongation (2.1.8) has the properties

o 4') = J'q, o J'4',

J'(ldY) = IdJ'Y,

(2.1.10)

which say that J' is a covariant functor. If Y -- X is a bundle endowed with an algebraic structure, this algebraic structure may be inherited by the fibred jet manifold J'Y -. X due to the jet prolongations of the corresponding morphisms.

Example 2.1.4. For instance, if Y -. X be a vector bundle, so is J'Y

X.

Moreover, we have the canonical identification

J'Y',

(J'Y)'

(2.1.11)

where Y' -e X and WY)'

X are the bundles dual of Y

X and J'Y -' X,

respectively.

Example 2.1.5. Furthermore, let Y be a vector bundle and () the interior product

x r 0 () = 1%t1/i,

xR,

2.1. FIRST ORDER JET MANIFOLDS

47

where y' and yi are dual bundle coordinates on Y and Y', respectively, and r is the canonical coordinate of R. The jet prolongation of () is the linear bundle morphism

J'():J'YxJ'Y'-'T'X x R, x xµ O J' () = yµyi + y'y,,i.

Let Y -. X and Y' -. X be vector bundles and ® the bilinear bundle morphism

®:YXY'-- YY', yik

O ® =

yi_!k

The jet prolongation of 0 is the bilinear bundle morphism

Jig: J'Y X J'Y' - J' (Y ®Y'), ytµ O J'®= yµyk + y`y

.

Example 2.1.6. Let Y be an affine bundle modelled over a vector bundle Y, then J1Y X is an affine bundle modelled over the vector bundle J'Y -- X.

Example 2.1.7. Let Y Z X be the composite fibred manifold (1.2.8) with coordinates (x'',z ,y'). Let us consider the jet manifolds J'Z of Z - X, J'YY of Y -' Z and J'Y of Y - X. These are coordinatized respectively by (xA, zP, za),

(xA, zP, y', ya, yp),

(xA, zP, y',

y1)

There exists the canonical morphism

P:J'ZZJ'Yz-+J'Y,

(2.1.12)

P('h, jn(:)g) _ .7i (g o h)

yaoP=VA+ypza, where g and h are sections of the fibred manifolds Y

Z and Z

X, respectively

([167], p.113).

Assume that irYz : Y

J'arYz : J'Y - J'Z

Z is a vector bundle. It is easily seen that (2.1.13)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

48

is also a vector bundle. Accordingly, if Y - Z is an affine bundle modelled over Z, then (2.1.13) is an affine bundle modelled over the vector a vector bundle V

bundle J'V -. J' Z. Every vector field u on a fibred manifold Y - X can be lifted to a vector field on the jet manifold J'Y. This lifting is based on the existence of the canonical morphism r : J'TY TJ'Y. Let us note the following two facts: (i) The jet manifold J'TY is an affine bundle

J'TY 1JIT'l J'Y xx J'TX, y` Y

(A)

W)')

(2 y' yap

*I I

(

llM

which is modelled over the pull-back of J'VY over the base J'Y xx J'TX. This is an immediate consequence of Remark 1.4.1 and Example 2.1.7. (ii) Using coordinate transformation laws, one can easily show that the tangent bundle TJ'Y of the jet manifold J'Y is an affine bundle

TJ'Y c

(*'*') J'Y t

(X,y ,ya) X

X

TX,

J1 Ii

r

i

a

(XA'y ,ya,x

IY

which is modelled over the pull-back of VJ'Y over the base J'Y x x TX. It can be proved [127] that there is a unique affine morphism

J'TY

__r_+

I

TJ'Y I

(2.1.14)

J'YxJ'TXJ'YxTX x x such that:

r o J'Ts = TJ's for each (local) section s: X - Y; its linear derivative r (1.3.5) restricted to the fibres is the canonical isomorphism

VJ'Y = J'VY (2.1.15)

V,.o

VY

49

2.1. FIRST ORDER JET MANIFOLDS (X." y', y%, #i,?li,)

q (xA, y'' Y.1%, V, (0-0.

The coordinate expression of r is a,

(Z yi, yia, ±-x, yi,

yia)

o r = (X., y`, y`a, za , y (1/')a - yµ(s'`)a)

Let u be a vector field on a fibred manifold Y - X. Its first order jet prolongation is defined as the vector field

J'u=roJ'u:J'Y-+J'TY-TJ'Y, I J'u = u-'8a + uiai + (daui - yµdauµ)8; ,

(2.1.16)

on J'Y where d = 8,, + yi,a denote the operators of total derivatives. The vector field (2.1.16) projects onto the vector field u on Y. For instance, if u is a vertical field on Y X, we have

J'u: J'Y -. VJ'Y C TJ'Y, J'u=u'8i+daui8;. Remark 2.1.8. Since J'Y Y is an affine bundle modelled over the vector bundle J Y (2.1.6), the vertical tangent bundle VyJ'Y of J'Y -. Y admits the canonical splitting VyJ'Y = J'Y x JAY = J'Y x(T'X ® VY) C VJ'Y. Y

Y

JtY

(2.1.17)

As in (1.4.8), we have the exact sequence of vector bundles over J'Y:

iJ'YxVY-'0.

(2.1.18)

There are the following two canonical morphisms of the jet manifold J'Y into tensor bundles which enable us to handle jets as tangent-valued forms. (i) Given the jet manifold J'Y, there is a unique bundle monomorphism

A:J'YtiT'X®TY, J = dxA ®d,, = dx" ®(8a + ya8;),

(2.1.19)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

50

such that

AoJ's=Ts X. for any section s of the fibred manifold Y Note that the image A(J'Y) -, Y is an affine subbundle of the vector bundle Y

which is characterized by the coordinate conditions iµ' = bµ. This affine subbundle is modelled over the vector subbundle

T'X®VYcrX®TY. Y Y In particular, the affine bundle structure of J'Y - Y can be deduced from the canonical monomorphism A (2.1.19). (ii) The complementary bundle monomorphism to A,

0: J'Y '-' T'Y ®V Y,

(2.1.20)

8=9`®8,=(dy'-y;,dz'')®8i, is called the contact I -jet form. The image 9(J'Y) bundle of the vector bundle

Y of J'Y is an affine sub-

T'Y®VYV Y which is characterized by the coordinate conditions y'; = b.

Remark 2.1.9. For the sake of simplicity, we will often identify the jet manifold J'Y with its images under the morphisms (2.1.19) and (2.1.20). The canonical morphisms (2.1.19) and (2.1.20) can be viewed as the morphisms

A: J'YxTX 3 (.,8a)-da=88JAETY

(2.1.21)

0 : J'Y x V'Y 9 (.,ay') -. 0` = BJdy' E T'Y.

(2.1.22)

x

and Y

2.1. FIRST ORDER JET MANIFOLDS

51

The morphism (2.1.21) determines the canonical horizontal splitting of the pullback

J'Y x TY = A(TX) J® VY,

(2.1.23)

± 8a+y'0 =x''(ea+y'8i)+(yt-xlya)ai, and the corresponding splitting of the exact sequence (1.4.3a) lifted over J'Y. Similarly, the morphism (2.1.22) yields the dual canonical horizontal splitting of the pull-back

J'Y x T'Y = T'X ® O(V'Y), Y

(2.1.24)

J'Y

xadx'' + y,dy' _ (xA + y;ya)dx" + 1,;(dy'

- yad?),

and the corresponding splitting of the exact sequence (1.4.3b) lifted over J'Y.

Example 2.1.10. Let u be a vector field on a fibred manifold Y -' X. Its pull-back over J'Y is defined as the morphism 7rr*u:

The pull-back iro'u has the same coordinate expression u = uAOA + u'8; as u, but this is not a vector field on J'Y. Using the canonical splitting (2.1.23), we obtain the splitting Tf0'u = UH + UV,

I u''8a+u'8;=ua(8a+y;,88)+(u'-u"ya)8,.

(2.1.25)

In brief, we say that (2.1.25) is the canonical horizontal splitting of the vector field U.

Let 0 be an exterior 1-form on Y and iro'q5 its pull-back over J'Y. The canonical splitting (2.1.24) leads to the canonical horizontal splitting of the form it O: 7ro''o _ OH + 4v,

Fadx'' + q'dy' = (&a + ya0+)dx'' + q;(dy' - yadx' )-

Example 2.1.11. Let r be a global section of the jet bundle J'Y -, Y. Substituting the tangent-valued form

Aor=dxa®(O +r'a8;)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

52

in the canonical splitting (2.1.23), we obtain the familiar splitting (1.4.5) of the exact sequence (1.4.3a) by means of a connection I' on Y - X. Accordingly, substitution of the tangent-valued form

0oF=(dy'-I"adx%)®8; in the canonical splitting (2.1.24) leads to the dual splitting (1.4.6) of the exact sequence (1.4.3b) by means of a connection r on Y -' X.

2.2

Second order jet manifolds

X be a fibred manifold. Considering the first order jet manifold of the Let Y fibred manifold J'Y -e X, we define the repeated jet manifold J'J'Y. Given the coordinates (2.1.4) on J'Y, the repeated jet manifold J'J'Y is provided with the adapted coordinates (X ,?/',YA,?%µ,Y;")-

Denoting by iri the canonical projection J'J'Y -' J'Y, we have the following commutative diagram

J'J'Y J-i J'Y Nil

I

1

*o

o

where

(x',y',ya) 0in1 = (xA,y`,ya),

(2.2.1)

W, y', ya) o J' 7r = (x-1, y',YA)

(2.2.2)

The morphisms (2.2.1) and (2.2.2) provide the repeated jet manifold J'J'Y with two different affine bundle structures with respect to the base J'Y.

The projection lrlI : J'J'Y - J'Y is an affine bundle modelled over the vector bundle

T'X ® VJ'Y -e J'Y. Jly

(2.2.3)

2.2. SECOND ORDER JET MANIFOLDS

53

On the other hand, J'ao : J'J'Y - J'Y is an affine bundle (see Example (2.1.6) whose underlying vector bundle

J'(T'X ®VY) - J'Y

(2.2.4)

differs from (2.2.3).

Note that there is no canonical identification of these affine structures, but this is induced by the choice of a symmetric linear connection on X (see Proposition 2.6.1).

Taking the affine difference of the images of J'J'Y by J'iro and all over Y, we obtain the following fibred morphism over Y:

S1:J'J'Y-T'X®VY, Y (x", y', ll'a) o Sj = (x", y', lla - Y;),

(2.2.5)

whose coordinate expression follows at once from (2.2.1) and (2.2.2). The kernel of S1 defines the canonical affine subbundle J2Y

.

J'J'Y

J'Y of J'J'Y which is characterized by the coordinate conditions

This subbundle is called the sesquiholonomic jet manifold. The underlying vector

bundle of J2Y -a J'Y is

T'X ® VyJ'Y -' J'Y,

ily

where VyJ'Y C V J'Y (see (2.1.17)). The induced adapted coordinates on J2Y are denoted by (xA, y, ya, yW,).

The canonical splitting

T'X ®VyJ'Y ®T'X ® VY = VT'X ® VY ® XT'X ® VY J1Y JlY ily ily JIY

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

54

yields the following splitting of the sesquiholonomic jet manifold:

J'Y=J2Y ED XT'X®VY, JIY

YAµ = 2(YAµ +YW + 2(yµ - yµA),

where the subbundle J'Y C fi ' is characterized by the coordinate conditions I

yiy.

This subbundle is called the second order jet manifold of Y - X. The induced adapted coordinates of JPY are denoted by (2.2.6)

(x', y`, sex, y;,A),

where the symmetry condition the coordinates yW read

yaµ is understood. The transition functions of

Of Y:X

&XIJA

where

are the total derivatives and ya are given by the expression (2.1.5). Thereby, we have the following affine bundle monomorphisms over J'Y: JAY

J2Y

J'J'Y.

(2.2.7)

The second order jet manifold J2Y can also be seen as the union

Jay = U j=3, SEX

where j=s are the equivalence lasses of (local) sections s of Y

X which are

defined by the conditions

s' E j=s and 8a,,s^(x) = 8s(x),

s' E j=s

with respect to any fibred coordinates (xA, y') of Y coordinates (2.2.6), we have (xA, y', ya, y71µ) o

j.28

= (XA, s`(x), 8As`(x), OAµs'(x)).

X around s(x). In the

2.2. SECOND ORDER JET MANIFOLDS

55

There is the composite fibration

where the canonical projection 7r ? : J2Y -' J'Y is an affine bundle modelled over the vector bundle

JFY = VT'X ® VY -+ J'Y. JIY

The following diagram of canonical projections

J2YJ'Y / 1!

W2 / o 1

X4- Y 11

commutes.

If s : X -. Y is a (local) section, then the map J2s:x_-. j=s is a (local) section of the fibred manifold J2Y -- X. It is called the second order jet prolongation of s. The diagrams

J..

J2,'

J'J'Y

J2Y

J2Y `-`' J'Y and

X

.n,J1Jl, x

are commutative.

Remark 2.2.1. Let 3: X

J'Y be a (local) section defining the (local) section s = ao o3 of Y -- X. The following three facts are equivalent: 3 = J's;

J'I:X _J2YCJ'J'Y; J11: X -' J2Y C J'J'Y. 0

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

56

X and Y' - X' be fibred manifolds and 0 : Y -' Y' a fibred morphism Let Y over a diffeomorphism 1 : X - X'. One can consider the first order jet prolongation of the fibred morphism J'+ (2.1.8). By definition, this is the repeated jet prolongation

J'J'o : J'J'Y - J' J'Y' of the fibred morphism 4'. The morphism J'J'4b is compatible with the inclusions (2.2.7), that is, J2Y

J2.

J2Y ti J'J'Y 1

1

rjm

Pilo

1

J2YJ2Y ti J'J'Y, The induced morphisms J2+ and Js4' are called sesquiholonomic and second order jet prolongations of 0, respectively. We have the commutative diagram

J2Y.J'Y, JIY

+J'Y'

YY' I

1

X -L X' where J4 is an affine bundle morphism over J'4'. Of course, J2 like J1 (see (2.1.10)) is a covariant functor. Some properties of J2 should be recalled.

Let Y - X and Y' -+ X be two fibred manifold over the same base X. Then the fibred morphisms

pr1:YxY'-.Y

and

pr2:YxY'- Y'

induce the canonical fibred isomorphism

J2(YxY')Q! J2YxJ2Y'. x

x

2.2. SECOND ORDER JET MANIFOLDS

57

X is a vector bundle, so is 7r2 : JZY - X.

If Y

Let Y - X be an affine bundle modelled over a vector bundle V -. X. Then J2Y -+ X is an affine bundle modelled over the vector bundle J2Y X.

Remark 2.2.2. Let Y -. X be a fibred manifold. Then JZY has three vertical subspaces

VJ1yJ2Y = J2Y X J1Y = J2Y X VT'X ® VY C Vyf2Y C VJ2Y. J1Y

J1Y

J1Y

In particular, there is the following exact sequence (1.4.8) of vector bundles over JZY which are associated with the composite fibration J2Y JlY X:

0,VJ,yJ2Y_.VJ2Y-+J2Y x VJ1Y (see (2.1.15)), we have the canonical isomorphism

\f

V J2Y = J2V Y V".2

VY

W,Y4,Ya,U ,'YI,, I' 4A) «(xA,y4,1/`,I& y0,(1l')a,(0W), where J2VY is the second order jet manifold of the vertical tangent bundle VY -, X. Given the fibred manifold J1Y of vector bundles over J'Y:

X, let us consider the exact sequence (1.4.3a)

O - VJ'Y -TJ'Y -. J'Y x TX - 0. x

Its pull-back over J2Y splits canonically in the following way

fly xx TX -TJ'Y 1

J2Y

-, J'Y 1

(s." y" 14, i sr y'+ yai) 0 '\ = (xa> yi> Ya., i-\r ya' xar 4Ai").

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

58

Equivalently, we have the affine injection over J'Y

PY ,a. T'X 0 TJ'Y J1Y

J'Y

A=dz-\ ® da=da''®(8a+ya8.+ya"8;). The complementary morphism of A is

J2Y x TJ'Y e. VJ'Y JAY

(2.2.8)

J'Y

which, equivalently, can be seen as a vector-valued form

j2y - T'J'Y ® V J'Y, 0=0'08j+0'06{'', 0'=dy'-yadx", O =dya-y")dx". This is called the contact 2 -jet form. As for first order jet manifolds (see (2.1.14)), we have the affine morphism r J2TY - TJ2Y. As a consequence, every vector field u on a fibred manifold Y -+ X gives rise to the vector field

J2u = ua8a + u'8; + (dau' - y"dau")8, + (dau' - y'

(2.2.9)

on the second order jet manifold J2Y. This is called the second order jet prolongation of u. Several constructions on first and second order jets given above will be generalized to higher order jets in Section 5.1.

2.3

Connections

There are several equivalent definitions of a connection on a fibred manifold.

(A) A connection on a fibred manifold Y - X is the choice of a splitting I' of the exact sequence (1.4.3a), i.e., r

O---- VY--+TY=.YxTX -.0

(2.3.1)

2.3. CONNECTIONS

59

or the dual splitting of the exact sequence (1.4.3b). It follows that a connection r is a section

r:YxTX -TY,

(2.3.2)

(xa y' xa y') o r = (xa y` xa ra` xa) of the affine bundle (1.4.4) which is a linear morphism over both Y and TX. The local functions rA on Y are said to be the components of the connection r or the connections parameters with respect to the fibred coordinates (xA, y'). The image of Y x TX by the connection r defines the horizontal subbundle HY C TY which splits TY as follows:

TY=HY®VY,

(2.3.3)

x''aa + "O; = ±A(& + r;,&) + (v' - rn xa)a;.

(B) Given the horizontal splitting (2.3.3), by the same symbol r we will also denote the projection

r=pr2:TY Y VY,

or This projection obeys the condition

r Ivy= IdVY.

(2.3.4)

Conversely, every morphism r : TY - VY which has the property (2.3.4) defines a

connection on Y - X. (C) Every linear morphism r over Y (2.3.2) uniquely defines the tangent-valued semibasic form

Y-LT'X®TY Y (2.3.5) I

I

X -T'X ®TX x r=dx"®(Oa+IxtO,), I which projects onto the canonical tangent-valued form Ox (1.6.9) on X. Therefore, one can think of such a form r as being another definition of a connection on Y - X.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

60

(C') Given the splitting (2.3.1), the dual splitting of the exact sequence (1.4.3b) reads r

0-'Y xT'X'T'Y-.V'Y---0. Y X

Then, every connection r on Y -' X is represented by the vertical-valued form

F: Yev V'Y®VYr

T'Y®VY,

(2.3.6)

where 0y = dy' ® 8; is the canonical section (1.6.16). This form is given by the coordinate expression

r = (dy' - I"adx'') ®8;. (D) Within the framework of jet formalism, a connection r on a fibred manifold Y -+ X is defined as a section of the affine jet bundle J'Y -. Y, that is,

I:Y-J1Y, (2.3.7)

(yl, y', ya) r = (xA, y',

The equivalence of this definition with those given above has been observed in Example 2.1.11 (see also [167], p.146). It is an immediate consequence of the definition (D) that connections on a fibred X exist and form an affine space modelled over the vector space of manifold Y soldering forms on Y -+ X.

I Following Remark 2.1.9, we will often identify sections (2.3.7) of a jet bundle with tangent-valued forms (2.3.5) or (2.3.6). The concept of connection leads directly to the following two constructions. (i) Since a connection r on a fibred manifold Y -. X is a section of the affinc bundle J1 Y -. Y, it defines the fibred morphism

Dr:J1Y -T'X®VY, Dr:z z-Foiro(z),

(2.3.8)

zEJ'Y,

Dr=(ya-f"A)®8;, (see (1.3.3)). One can think of this morphism as being the first order differential operator on Y. It is called the covariant differential relative to the connection F.

2.3. CONNECTIONS

Ifs : X

61

Y is a (local) section, from (2.3.8) we obtain its covariant derivative

Ors= Dr0J's: X _T'X ®VY, vrs = (8xs` - I';a o s)dax ®8;. It is easily seen that the following conditions are equivalent:

vrs=0

.

J's=ros.

(2.3.9)

A (local) section s is said to be an integral section of the connection f if s obeys the conditions (2.3.9). PROPOSITION 2.3.1. Let s : X -+ Y be a global section. By virtue of Theorem 1.2.5, there exists a connection r such that s is an integral section of r. Given a vector field u on X, the contraction

u j vrs = vrs = ux((9xs' - r'a o s)8; is said to be the covariant derivative of s along the vector field u.

(ii) Let r be a connection on a fibred manifold Y -, X, and let u be a vector field on its base X. Then from the morphism r (2.3.5) we obtain the following vector field on Y:

ru=ujr:Y - HY C TY,

(2.3.10)

r=ux(8x+r"aa), which is called the horizontal lift of u by the connection r. Note that ru projects over u.

Let u, v : X -, TX be two vector fields. Let us compute the vector field

R(u, v) = - r[u, v] + [ru, rv]

(2.3.11)

on Y. It is readily observed that this is a vertical vector field given by the coordinate expression

R(u,v) = uxv'`R'x.

R'4, =8xrµ-80 r'a+r'x8,rµ-ria, q,.

(2.3.12)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

62

It follows that one can define a VY-valued semibasic 2-form on Y

R:Y-+AT'X®VY, Y

(2.3.13)

R=2R`a,,d?Adx"®8 called the curvature of the connection I' (see (2.4.8) for another definition of curvature).

Remark 2.3.1. The concept of curvature leads to important integrability conditions. The following conditions are equivalent.

The lifting

UET(X) .--I'UEP(Y) is a Lie algebra morphism.

The curvature R of the connection r vanishes identically, i.e., R = 0. The horizontal subbundle HY C TY is an involutive distribution (i.e., the Lie bracket of any two horizontal vector fields is again the horizontal one) and, hence, completely integrable (see Remarks 4.1.2 and 3.9.3). There exists a (local) integral section s of the connection r through any point

yEY.

Let us consider connections in relations to three important constructions, namely, those of Cartesian product, pull-back, and reduction. (i) Let Y -, X and Y' - X be fibred manifolds over the same base X. Let 1' be X and I" a connection on Y' - X. The product connection a connection on Y r x I" is the unique connection on the fibred manifold Y Y' -+ X such that the X diagram

J'YXJ'Y' (r,r)

YxY' x

J'(YXY') rxr

2.3. CONNECTIONS

63

is commutative. The product connection has the coordinate expression

r x r' = dx

A

®( a x+ r;,JW+ ra

),

(2 . 3. 14)

where (xA,y') and (x', y'") are fibred coordinates on Y and Y', respectively. (ii) Given a fibred manifold Y X, let f : X' X be a map. Let Y' = f'Y X' be the pull-back of Y - X. Every connection r on Y - X induces a connection r' on Y' X' called the pull-back of the connection of r with respect to f . Indeed, using the projection (1.2.6), we obtain the linear morphism

r':TY'(xY`') Y,xTYiIariY'xVY=VY' Y Y Y restricts to the identity on each fibre, it follows that over Y'. Since pre : Y' r'I vy, = Id VY'. Hence, r' is a connection on the fibred manifold Y' -. X'. The connection parameters 17. of 1" are given by r"a = (ra o prz)aa f a.

(2.3.15)

Let IT be the curvature of the pull-back connection V. Then we have

R':Y'-.AT'X'®VY', Y' R' =

n dx'e ®a;,

2

o Pr2)8afA8Qfµ,

(2.3.16)

where RA,, are components of the curvature R of the connection F. The relation (2.3.16) shows that R = przR. (iii) Let iy : Y -+ Y' be a fibred submanifold of a fibred manifold Y' X. Let r' be a connection on Y' -+ X. If there exists a connection r on Y --+ X such that the diagram

Y -r, 'Y

1

!

Toy

Y'--T'X ®TY'

is commutative, we say that r' is reducible (or that it restricts) to the connection

r. Let r' be a connection on a fibred manifold Y' -, X and r a connection on its fibred submanifold Y -+ X. Then the following conditions are equivalent:

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

64

C is reducible to I ;

Tiy(HY) = HY'(;,,(y), where HY C TY and HY' C TY' are the horizontal subbundles determined by r and V, respectively; for each vector field u E T(X), the vector fields r u and Vu are iy-related, that is,

Tiyofu= I"uoiy.

(2.3.17)

Let if be the curvature of a connection I" reducible to a connection r and let R be that of r. Then the following diagram Y

ATX®VY Y Id®Viy 1

Y'a+AT'X ®VY' Y0 is commutative. Indeed, if u, v E T(X), then we have

Tiy o [ru,l'v) = [I"u,I''v] o iy since the property of being iy-related is preserved by the Lie bracket. Hence the result follows from (2.3.11).

Example 2.3.2. Linear connections. Let r be a connection on a vector bundle Y - X. We say that I' is a linear connection if f : Y -+ JIY (2.3.7) is a linear bundle morphism over X. In linear bundle coordinates (x, y') of Y, the connection parameters of r read (2.3.18)

where r.%'j are local functions on X. Note that linear connections are principal connections. Linear connections always exist (see Remark 2.7.5). They form an affine space modelled over the linear space of linear soldering forms

Y- T'X®VY, o = oa`;yidx 0 8i,

2.3. CONNECTIONS

65

where oa'j are local functions on X. The curvature of a linear connection 1' (2.3.18) can be seen as a Y' ® Y-valued x 2-form on X, that is,

R = R,,,,';dz" n dx" ®ays ®8;,

aarv i - 8µI'a'i + I'ahirvch - rvhiraih Some standard operations with linear connections should be recalled. Let Y X be a vector bundle and I' a linear connection (2.3.18) on Y. Then, there is a unique linear connection I" on the dual vector bundle Y' -- X such that the following diagram is commutative:

J'YxJ'Y'J10 T'X xR x rx r 1

I

YxY' -.0 X xR x

(,Id)

where 6 is the global zero section of the cotangent bundle T'X. The connection I" is called the dual connection of I'. It has the coordinate expression I

rk. =

-raj .y>>

(2.3.19)

where (xa, yj) are the fibred coordinates on Y' dual of those on Y. For instance, a linear connection on the tangent bundle TX reads

K = dx'' ®(B% + K.%"X" ).

(2.3.20)

Accordingly, the dual connection K' on the cotangent bundle T'X is

K' = dxA ® (8a -

(2.3.21)

For the sake of simplicity, we denote these connections by the same symbol K and call them a linear connection on a manifold X.

I It should be emphasized that the expressions (2.3.20) and (2.3.21) differ in a minus sign from those used in most of the physics literature. l

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

66

Let Y - X and Y' -' X be vector bundles with linear connections r and r, respectively. Then the product connection (2.3.14) is the direct sum connection

r®ron YED Y'. Let Y - X and Y' - X be vector bundles with linear connections r and r, respectively. There is a unique linear connection r ® r on the tensor product Y ®Y' -+ X such that the diagram x

J'YXJlY'! JI(Y®Y') rxr

i

i

r®r

Yx x Y' ®- y 0y, X commutes. It is called the tensor product connection and has the coordinate expression

(r ® r)a = FAijyfk + rak* ',

(2.3.22)

where (xa, y'k) are bundle coordinates on Y ®Y' - X. X

Example 2.3.3. Affine connections. Let Y -p X be an affine bundle modelled over a vector bundle V - X. A connection r on Y -, X is said to be an affine connection if r : Y J'Y (2.3.7) is an affine bundle morphism over X. Affine connections are principal connections, and they always exist (see Remark 2.7.5).

Note that, for any affine connection r : Y -p J'Y, the corresponding linear derivative r : Y -, J'Y (1.3.5) uniquely defines the associated linear connection on

the vector bundle Y - X. Using affine bundle coordinates (xA, y') on Y, the condition that r is affine reads

r`a = rai;y' + ai,

(2.3.23)

where ra'1 and oa are local functions on X. The coordinate expression of the associated linear connection is Va

= ra

where (xA, y') are the associated linear bundle coordinates on Y.

2.3. CONNECTIONS

67

Note that the functions oa are not necessarily the components of a global section

of the vector bundle T'X ® Y - X. However, this is the case of a vector bundle Y X. Indeed, both the affine connection r (2.3.23) and the associated linear connection r are connections on the same vector bundle Y -- X, and their difference is a basic soldering form on Y. Thus, every affine connection on a vector bundle

Y -' X is the sum of a linear connection and a basic soldering form on Y -' X. In particular, let Y be the tangent bundle TX. Then we have the canonical basic soldering form o = 0x (1.6.9). The corresponding affine connections

r=K+6x,

(2.3.24) 6111

on TX, where K is an arbitrary linear connection (2.3.20) on TX, are called the Cartan connections. Given an affine connection I' on a vector bundle Y - X, let us denote by R and R the curvature of t and r, respectively. They are sections of the vector bundle

Yx(AT'X(& Y)-4Y. It is readily observed that R

+ T,

where the Y-valued 2-form

rr:X-AT'X®Y, x T = 27-AµdxA Adx" ®Oi,,s T'Aµ

Aµ

l+ A

t A µhµ7hsAh,

is the torsion of the connection r with respect to the basic soldering form o (see (2.4.17)).

In particular, for Cartan connections (2.3.24), we have

T : X - AT'X ®TX, T = 2,r,',xd? n dx" ®a,,, T,V A = K, A - K,, µ,

which is a familiar torsion of the linear connection K on TX.

(2.3.25)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

68

2.4 Differentials and codifferentials. Identities Since connections on fibred manifolds are represented by tangent-valued forms, one can apply the Fr licher-Nijenhuis bracket (1.6.10) in order to discover several important identities involving connections.

By virtue of (2.3.5), a connection r on a fibred manifold Y - X is an object I' E D'(X) ® P(Y) projected onto the canonical form

Ox E D'(X) ®T(X). The F N covariant differential associated with r is defined to be the Nijenhuis differential

dr :17.(X) ®P(Y) - D." (X) ®V(Y), dr-O,= [r,m],

O E Or(X) ®P(Y).

It has the property (2.4.1)

dr [46, TGJ = [dr-0, tbJ + (-1)1 1 [0, drtG] .

Let

0=a®uE1Y(X)0P(Y)

(2.4.2)

be a projectable tangent-valued r-form. Then from (1.6.10) we obtain the important formula

dr4=da0ur+(-1)raAdru,

(2.4.3)

where ur is the vertical part of u determined by the connection r, i.e.,

u = I'uX + ur,

uX = Tir o u.

The coordinate expression of (2.4.3) is dry

_ r! +I7I8i0A,...a. -

,., +

+ axe

(2. 4 . 4)

8,I"aA dx" A ... A dx - ®8i.

In the same manner, given a soldering form o E D'(X) 0 V(Y) on Y - X, the soldered differential associated with o is defined as

d, : D'(X) ®P(Y) _ [a, 01, 0 E X(X) 0 P(Y). D'+'(X) ®V(Y)4d,4,

2.4. DIFFERENTIALS AND CODIFFERENTIALS. IDENTITIES

69

Of course, the property (2.4.1) holds. If 0 is as in (2.4.2), we have

d,0=da®(uXJo)+(-1)'aAd,u, 40 _ *i

(2.4.5)

+ 8ioa0a,...a.)dz-' A tea' A ... A dTx ®8i.

Let us further suppose that the base X is oriented and that g is a pseudo-

Riemannian metric on X. The covariant codifferential associated with r is

br : O'(X) ® P(Y) - il'-'(X) ® V(Y), br= Cdr*.

(2.4.6)

Recalling the coordinate expression (2.4.4), we obtain aµ

bra _ - (rg

+

(2.4.7)

A... A dx'%--1 ®8i,

where

+

+ vama,...a, = 8a-a,...a. + KAY),,

,..a. +

and K is the Levi-Civita connection (1.6.5) of the metric g. Similarly, given a soldering form a, the soldered codifferential associated with a and g is

-(-1)r(

0'-' (X) ® V(Y),

6, : '0" (X) ® P(Y) I b, =

9AP

bed = - (r - 1)I

i

a

i a - BQoao,+ _, )dx-'

A dxal A ... A

We will apply covariant and soldered differentials and oodifferentials to obtain various objects [127, 1401.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

70

We first have the following equivalent definition of the curvature R of a connection 1', originally defined in (2.3.13), that is,

R=zdrr=1[I,I']:Y-.XT'X®VY.

(2.4.8)

The connection r and its curvature R satisfy the identities brI' = 0,

[R, R] = 0,

(2.4.9)

drR = jr, R] = 0,

(2.4.10)

brR = 0.

(2.4.11)

The first identity in (2.4.9) follows at once from (2.4.7), and the second from (1.6.11). The identity (2.4.10) is the (generalized) second Bianchi identity (or the homogeneous Yang-Mills equation). It is an immediate consequence of the graded Jacobi identity (1.6.12). We see from (2.4.4) that its coordinate expression is

F

I'',8,R

.-

(2.4.12)

0,

(Av)

the sum being cyclic over the indices A, p and P. In order to prove (2.4.11), take

0 E O'(X) ® V(Y),

E O'(X) ® V(Y).

Then from (1.6.10) and (1.6.3) we obtain the identity

[0,.01= -[0,.01.

(2.4.13)

Now (2.4.6), (2.4.8) and (1.6.12) yield brR = [R, R]. Then the result follows from (2.4.13). In the of gauge theories, (2.4.11) is the (generalized) charge conservation identity.

The Yang-Mills operator associated with r is defined as

brR:Y-T'X®VY. Its local expression, as follows from (2.4.6), (2.4.7) and (2.3.12), is

drR = (brR)adxA ®8 (brR)a = -gt V0Rpa, OQRpa = 8aR{pa + KQ"pR,a + KQ"aR4.y +

0

(2.4.14)

2.4. DIFFERENTIALS AND CODIFFERENTIALS. IDENTITIES

71

In the same manner, given a soldering form or, we define the soldered curvature

P=1d

p=

(2.4.15)

dxAAde®8.,

p0118joµ - oµ8jo.,.

As before, we have the identities b,o = 0,

[P, PI = 0,

dvP = [o, P) = 0,

b;p=0. The soldered Yang-Mills operator associated with o is

b,p:Y - T'X0VY,

(2.4.16)

boP = (bgp)%dXA ®8., os

Given a connection I' and a soldering form or, the torsion of 1' with respect to a is defined as 2

Its coordinate expression, as follows from (2.4.4) or (2.4.5), is r = (8ao,, + N,,Bjo',, -

.,oµ)dxa A dx" ®8;.

(2.4.17)

There is the (generalized) first Bianchi identity

drr = 4a = [R, of = -d, R.

(2.4.18)

No w let I, = I' + o. Then we have the important relations 2p, (2.4.19)

as follows from (2.4.8), (2.4.15) and (2.4.16).

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

72

Example 2.4.1. In particular, let Y - X be a vector bundle and r an affine connection on it. Then, recalling (2.3.23), we have

r=r+a, p=0, r= r, R=T+r. Note that a is a basic soldering form. Hence, p = 0 as (2.4.15) shows. Fom (2.4.12) we see that the second Bianchi identity d4R = 0 takes the form

E (B&RP'j + ra"jR,,,,'ii + ra`hRA,,"i) = 0.

(2.4.20)

(w) On the other hand, the first Bianchi identity (2.4.18) reads

d, +drr=0. Given a connection r and a soldering form a, the Ricci tensor of r with respect to a is defined as

r=b,R:Y-eT'X®VY,

(2.4.21)

r = rad? 0 0j, op

;

As before, let iv = r + a. Then from (2.4.19) and (2.4.21) we obtain

r'=r+b,p+b,r, brR' = brR+r' +brp+brr, which give some basic relations involving the Ricci tensor, the Yang-Mills operator, the soldered curvature and the torsion.

Suppose that the soldered curvature p and the torsion r vanish. Then we have R' = R and r' = r. Moreover, suppose that brR = 0. Then we obtain brR = 0.

Example 2.4.2. Let Y

X be a vector bundle and r an affine connection on Y. Then, recalling (2.3.23), we have

r=i'+a, 6,r

0,

brR=brR+r+brr.

2.4. DIFFERENTIALS AND CODIFFERENTIALS. IDENTITIES

73

The Ricci tensor, as (2.4.21) shows, is now a basic vector valued 1-form

r:X-T'X®Y, r = r'dx" ® et,

(2.4.22a)

ra _

(2.4.22b)

Note that R satisfies the free Yang-Mills equation brR = 0 if and only if does and r + Err = 0. When the torsion r vanishes, then the equation brR = 0 is equivalent to the equations &jW = 0 and r = 0.

Example 2.4.3. In particular, let Y = TX be the tangent bundle. The torsion r vanishes in the case of a Cartan connection 1'. It follows that a Cartan connection r satisfies the equation brR = 0 if and only if R = 0 and r = 0. Note that the expressions (2.4.22a) and (2.4.22b) become

r = r"d? ®O ,

(2.4.23a)

ra = -g" RpA"b,

(2.4.23b)

where

fi + KA"pK,,°,, - K,,"pKA°ry.

&p *p =

(2.4.24)

For instance, let us consider the case when r is a Cartan connection and K the Levi-Civita connection of a metric g on X. In this case, the curvature R is given are the Christoffel symbols (1.6.5). by the expression (2.4.24) where The curvature (2.4.24) satisfies the well-known identities VvRAj°p + VM&A°p + DAR,,,,°p = 0,

(2.4.25)

RA,,,p + R8AQ,, + RppaA = 0,

(2.4.26)

-R,,AQp = -RAµpo,

(2.4.27)

RA,.ep =

(2.4.28)

where

R.%,.$ = g.,RA"p,

(2.4.29)

and the symbol VA denotes the covariant derivative with respect to the Levi-Civita connection, i.e., VAR,w°p = BAR,,v°p + KA"pRµv°,y +

KA"µ-v°$ - K.%°-,R,,,,''# + K,ryvR,"*p.

(2.4.30)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

74

We see that (2.4.30) coincides with (2.4.14). The identity (2.4.25) is the second Bianchi identity which follows from (2.4.20);

while (2.4.26) is the first Bianchi identity; identities (2.4.27) imply that R is a 2form on X which takes its values into AT'X. The last identity (2.4.28) results from (2.4.26) and (2.4.27). This shows that R is a self-adjoint operator on AT'X. From (2.4.14), (2.4.29) and (2.4.30), we obtain = -9p°OpRepva,

(2.4.31)

where (2.4.28) is used. On the other hand, from (2.4.23b) we obtain 907ra = R',.%,

(2.4.32)

where (2.4.27) and (2.4.28) have been used. It follows that, in the particular case of a Cartan connection on TX, our definition of the Ricci tensor reduces to the standard one. As is well known, the Ricci tensor is symmetric, i.e., Rpapa = Rpapa.

This follows at once from (2.4.28). Using the second Bianchi identity (2.4.25), (2.4.31) and (2.4.32), we obtain the

important identity g 'V Rap&A = V.Rp9pa - V RpapA.

(2.4.33)

As a consequence of (2.4.33), we come to the equivalence of the free Yang-Mills equations for the Cartan connection I' and the free Einstein equations, i.e.,

6rR=0 a r=0. Recall that the Einstein tensor is defined as

Ca = r", Z

R,

(2.4.34)

where R = ra is the scalar curvature. Using (2.4.32) and the second Bianchi identity (2.4.25), we see that OAR = 20.,e%.

Hence, as is well known, the Einstein tensor (2.4.34) is divergence-free, i.e., OC", _ 0.

75

2.5. COMPOSITE CONNECTIONS

2.5

Composite connections

Throughout this Section, we will refer to a composite fibred manifold

Y!'-z z (2.5.1)

w \f wzX

X with adapted coordinates (x", za, y`) as in (1.2.8). The jet manifolds J'Z of Z -- X, J'Yz of Y --+ Z and J'Y of Y - X are equipped with the coordinates

W, -1,

W, zP, 4),

, FA, yp),

W, x°, V, 4, A),

respectively (see Example 2.1.7). We will now consider relations between connections on the fibred manifolds Z

X,Y-+Zand Remark 2.5.1. Let 'y=dx"® (8A+ yP\Op

be a connection on the composite fibred manifold Y - X and

r = dx" 0 (8A + raiP)

(2.5.2)

a connection on the fibred manifold Z -+ X. We say that the connection ry is projectable over the connection r if the diagram

®TY I

wrz !

Twyz

Z -LT'X ®TZ or, equivalently, the diagram

Y.".J'Y WYZ

I

j

lwvz

z r J'Z

are commutative. It is readily observed that the commutativity of these diagrams is equivalent to the condition ' = I a.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

76

Let

A=dx"®(8,,+A;,8;)+dzp®(8p+A,0;)

(2.5.3)

be a connection on the fibred manifold Y -. Z. Given a connection r (2.5.2) on Z -. X, the canonical morphism p (2.1.12) enables us to obtain a connection ry on Y -' X in accordance with the diagram

J'ZxJ'Y2_f.J'Y z (r,A) I

ZxY x

I" Y

This connection, called the composite connection, reads

1 y=dxa®(Ba+IaOp+(As+API',p,)8,).

(2.5.4)

Obviously, 7 is projectable over T.

Remark 2.5.2. An equivalent definition of a composite connection is the following. Let A and I' be connections as before. Then their composition r)

Y x TZ - TY Y x TX x z is the composite connection ry (2.5.4) on the composite fibred manifold Y - X. In brief, we will write

7=Aor.

(2.5.5)

In particular, let us consider a vector field r on the base X, its horizontal lift rr over Z by means of the connection r and, in turn, the horizontal lift A(I'r) of Fr over Y by means of the connection A. Then A(rr) coincides with the horizontal lift -yr of r over Y by means of the composite connection ry (2.5.5).

We will use this result to show that, if Y - Z and Z -. Y are fibre bundles, so

Remark 2.5.3. Ehresmann connections. Let zr : Y

X be a fibred manifold and I' a connection on it. Let t I- x(t) and t h- y(t) be smooth maps in X and Y, respectively. Then t '-. y(t) is called a horizontal lift of x(t) if

ir(y(t)) = x(t),

jc(t) E Hy(,)Y,

t E R,

77

2.5. COMPOSITE CONNECTIONS

where HY C TY is the horizontal subbundle associated with the connection I'. If, for each path x(t) (to < t < t1) and for any yo E a= '(x(to)), there exists a horizontal lift y(t) (to < t < t1) such that y(to) = yo, then 1' is called an Ehresmann connection ([83], p.314).

PROPOSITION 2.5.1. A fibred manifold is a fibre bundle if and only if it admits an Ehresmann connection ([83), p.314). 0

Let now Y -e Z - X be a composite fibred manifold where Y - Z and X are fibre bundles. These bundles admit Ehresmann connections A and t, Z respectively, whose composition is easily proved to be an Ehresmann connection. Hence, Y -, X is a fibre bundle. Every connection A (2.5.3) on the fibred manifold Y -e Z determines a splitting of the exact sequence (1.4.8) by restricting A to Y x VZ, that is, z

VY=(YXVZ)®VYzi

(2.5.6)

,Y8o + 0'84 = ±'(8y + A'n8;) + (y' - Af )8;.

Note that only the connection parameters A, (and not AA') are involved in the splitting (2.5.6). Using this splitting and the canonical morphism p (2.1.12), we obtain the morphisms

A:J'Y--eJ'ZXYit (x.,z ,y`,zXP,yi,) o A = (x,#,y',za, A' +Ayza), and

DA=J'YtiJ'YxJ'Y-'T'X®VYzCT'X®VY, DA = (yip - Aa - A,za)dxa ®8{.

(2.5.7)

One can think of the morphism (2.5.7) as being the first order differential operator on the composite fibred manifold Y -+ X. It is called the vertical covariant differential relative to the connection A.

Remark 2.5.4. The vertical covariant differential (2.5.7) can be defined also as

DA =przoD,:J'Y-eT'X®VY-T'X®VY2,

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

78

where D., is the covariant differential (2.3.8) relative to some composite connection (2.5.5), but it does not depend on specification of 1' and y.

Now, let h be a global section of the fibred manifold Z - X. The restriction h+Yz = Yzlh(x)

is an imbedded fibred submanifold

ih:Yh=h'Yz-- Y

(2.5.8)

of Y -' X as the diagram (1.2.11) shows. Note that VVYh=VYZIv"-

Given a connection A (2.5.3) on Y -+ Z, its pull-back (2.3.15) on Yh -' X reads At, = dxh ® (8 + [Ai o h + (A, o h)8,,hn]8;).

(2.5.9)

It is readily observed that the covariant differential

DA,, : J'Yh -' T'X ® VY", DA. = (ya - Aa o h - (A, o h)8hh')dxa ®a;,

(2.5.10)

relative to At, coincides with the restriction of the vertical covariant differential DA

(2.5.7) to J'ih(J'Yh) C J'Y. Now, let I' be a connection on Z X and let -y = A o r be the composition (2.5.5). Then it follows from (2.3.17) that the connection -y is reducible to the connection Ah if and only if the section h is an integral section of t, i.e.,

By virtue of Proposition 2.3.1, such a connection r always exists.

Let Y -+ Z - X be a composite fibred manifold where Y

Z is a vector

bundle. Let a connection

7=dxA0(OA+Iaap+AAtjy38,)

(2.5.11)

on Y -+ X be a linear morphism over the connection t on Z -+ X. The following constructions generalize the notions of a dual connection and a tensor product connection on a vector bundle.

2.5. COMPOSITE CONNECTIONS

79

(i) Let Y' - Z X be a composite fibred manifold where Y' -' Z is the vector bundle dual of Y -' Z. Given the projectable connection (2.5.11) on Y X over r, there exists a unique connection 7' on Y' -+ X, projectable over I', such that the following diagram commutes:

J'Y J'z X J'Y'

J'Z x(T'X x R)

I

I

z

YxY' z

0

rxaxid

ZxR

where 6 is the zero section of T'X. We term -y' the dual connection ofy over I'.

(ii) Let Y -' Z -+ X and Y' - Z -+ X be composite fibred manifolds where

Y -, Z and Y'

Z are vector bundles. Let -y and ' be connections (2.5.11) on X and Y' -i X, respectively, which are projectable over the same connection r on Z -+ X. There is a unique connection Y

170 ,'=dzA0

(2.5.12)

on the tensor product Y ®z Y' - X, which is projectable over I', such that the diagram

J'YBiz X J'Y' J' (Y ®Y') J1Z 'Yxy

I

YXY'

®

I

,ey

Y®Y'

is commutative. This is called the tensor product connection over r.

Example 2.5.5. Let r : Y - J'Y be a connection on a fibred manifold Y - X. Then, by virtue of the canonical isomorphism VJ'Y = J'VY, the vertical tangent map vr: VY -. VJ'Y defines the connection

vr:VY-+J'VY, v r = dxa ® (8a + r°a

+ 8j I"ay'

),

(2.5.13)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

80

on the composite fibred manifold VY -+ Y - X. This is called the vertical connection to r. Of course, the connection yr projects onto r, i.e., the diagram VY vr.JIVY 1

!

Y r J'Y is commutative. Moreover, yr is linear over r. Then there is the dual connection of yr over r on the composite fibred manifold V'Y Y -. X:

V'r : V'Y - J'V'Y,

v'r=d?o(BA+r'a

(2.5.14)

which is called the covertical connection to r. Note that, if Y X is a vector bundle [an affine bundle] and r is a linear [shine]

connection on Y - X, the connection (2.5.13) is the product connection adapted to the canonical vertical splitting of VY.

2.6

Second order connections

Let Y - X be a fibred manifold. A second order connection I' on Y . X is defined as a first order connection on the fibred manifold ir' : J'Y -' X, that is, this is a section of the affine bundle 7rij : J'J'Y J'Y:

f: J'' Y - J' J' Y, W, y , y11, A, y' ) o r = (X-" yi, /L fA, flo, where a and k are local functions on J'Y.

A second order connection r on the fibred manifold Y - X is said to be sesquiholonomic [holonomic] if it takes its values into the subbundle J2Y [J2Y] of J'J'V. We have the coordinate equality r`A=1/A

which characterizes a sesquiholonomic connection and the additional condition

2.6. SECOND ORDER CONNECTIONS

81

which characterizes a holonomic connection. Equivalently, a second order connection I' can be given as a TJ'Y-valued semiba-

sic 1-form on J'Y

ily

F

T'X ® TJ'Y, J1 Y

dxA®(aa+fta;+fsxµa;),

(2.6.1)

which projects onto 9x.

Remark 2.6.1. Recalling (2.3.13) and (2.1.6), we see that the curvature R of a first order connection r on Y -, X is a soldering form on J'Y:

R= a : J'Y --+ AT'X ® VY 'T'X ® VyJ'Y,

ily

ily

a= 2k,,,dx''06 , where VyJ'Y is the vertical tangent bundle of the affine jet bundle J'Y - Y which admits the vertical splitting (2.1.17). Every first order connection on a fibred manifold Y -' X gives rise to the second order one by choosing a symmetric linear connection on X.

The first order jet prolongation J'l' of a connection r on Y - X is a section of the repeated jet bundle J'iro (2.2.2), but not of r1l. Given a symmetric linear connection K (2.3.21) on X, one can overcome this difficulty by constructing the affine involution sK of J'J'Y over J'Y such that

SKOSK=IdJ'J'Y,

J'J'Y

J'J'Y (2.6.2)

W1I \f Jlxo

J'Y Let Y - X be a fibred manifold and K a symmetric linear connection on X. Using the canonical isomorphism (2.1.15), we obtain

T'X ® VJ'Y - J'T'X ® J'VY, J'Y ily

(2.6.3)

which is a linear morphism over J'Y. Moreover, the composition of (2.6.3) with (2.1.11) leads to the linear isomorphism over J'Y

T'X ® VJ'Y 3K J'(T'X ®VY), JiY

Y

W,1/1,1/A, SIA, V;,X) 0 3K = (x-', V, VX

1

x I V1.% - KA°µ Jr,),

(2.6.4)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

82

between vector bundles (2.2.3) and (2.2.4) associated with the affine bundle structures Ku and J'ao of J1J1Y, respectively. The following fact now can be easily proved.

PROPOSITION 2.6.1. Let K be a symmetric linear connection on X. Then there is a unique affine involution (2.6.2) of J'J1Y whose linear derivative is 3K (2.6.4). Its coordinate expression is

(y r1/,1lar$K=( ryryaryarbµA-Ka 0 Let I' : Y - J'Y be a connection on Y -' X and K be a symmetric linear connection on TX. Then I' gives rise to the second order connection I'K=8KOJ'I':J'Y--.J'J'Y,

rK = dxa ® (8,, + I°8 A +

y.,a,r,, + Ka°µ(U'v -

l'(2.6.5)

which is an affine morphism J'Y rK J'J'Y ,0

*rr 1

Y

r

1

J'Y

over the first order connection 1'.

2.7 Jets of principal bundles The main object under consideration here is the fibre bundle C = J'P/G whose sections are principal connections on a principal bundle P X with a structure Lie group C. 1 For the general theory of principal bundles we refer the reader to (103]. 1

Let ap : P -. X be a principal bundle with its structure group a real Lie group G.

2.7. JETS OF PRINCIPAL BUNDLES

83

One can say that P - X is a general affine bundle modelled over the trivial group bundle X x G so that

Rc: PxG - P,

R9:p.pg,

(2.7.1)

pEP, 9EC,

is the free transitive action of G on P on the right. A principal bundle P is equipped with a bundle atlas Wp = {(U0,0Q)} whose trivialization morphisms t4p: Trp' (Ua) -+ Ua x C

obey the condition pre 0 V). o R. = g o prz o V). ,

Vg E G.

Due to this property, every trivialization morphism trr uniquely determines a local section za : U. -+ P such that prz o t/ra o Z. = 1a,

where 1a is the unit element of G. The transformation rules for za read zp(x) = za(x)Pap(x),

x E v. n Up,

(2.7.2)

where pap are transition functions of the atlas Tp. Conversely, the family {(U.,z0)} of local sections of P which obey (2.7.2) uniquely determines a bundle atlas dip of

P. Note that the tangent functor T preserves a principal bundle structure. Given a principal bundle P - X, the fibre bundle TP - TX is a principal bundle

TRR:TP x T(X xC) -+TP TX

with the structure group TG = C x g,, where gi is the left Lie algebra of left-invariant vector fields on the group C.

A principal bundle P -' X admits the canonical trivial vertical splitting

a: VP=Pxgi such that a-' (em) are fundamental vector fields on P corresponding to the basis elements em of the Lie algebra gi.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

84

Taking the quotient of the tangent bundle TP -. P and the vertical tangent bundle VP of P by TRc (or simply by C), we obtain the vector bundles

TOP = TP/C,

and VOP = V P/C

(2.7.3)

over X. Sections of TOP - X are C-invariant vector fields on P, while sections of VOP -. X are C-invariant vertical vector fields on P. Hence, the typical fibre of VOP -+ X is the right Lie algebra g of the right-invariant vector fields on the group 0. The group G acts on this typical fibre by the adjoint representation. The Lie bracket of vector fields on P goes to the quotient by G and defines the Lie bracket of sections of the vector bundles TOP -. X and VcP X. It follows that VOP - X is a Lie algebra bundle, the gauge algebra bundle in the terminology of gauge theories, whose fibres are Lie algebras isomorphic to the right Lie algebra

gofG. Example 2.7.1. When P = X x G is trivial (e.g., C is an abelian group), we have

VcP=X xTG/C=X xg.

Example 2.7.2. Given a local bundle splitting of P, there are the corresponding for the Lie algebra g, local bundle splitting of TOP and VOP. Given the basis X such we obtain the local fibre bases {8", ep} for TOP -+ X and {en} for VOP that [ev,ea] = cgver,

where c;, are the right structure constants of C. If

C rl : X

TcP,

t=E"a"+eer,

17 =7f,9,.+7f ev,

are sections, the coordinate expression of their bracket is

It, n] = (e"on" - if8 ")a" + (S"a"nr - ,17"aaf +

et°n°)e,.

(2.7.4)

In contrast with the tangent functor T, the jet functor J' fails to preserve a principal bundle structure.

2.7. JETS OF PRINCIPAL BUNDLES

Let J' P be the first order jet manifold of a principal bundle P structure Lie group G. The jet prolongation

85

X with a

J'Ro: J'P x J'(X xG)-+J'P of the canonical action (2.7.1) brings the fibre bundle J' P - X into a general affine bundle modelled over the group bundle

J'(XxG)=Gx(T'X®gi) over X which, however, is not necessarily trivial. Hence, J'P -+ X fails to be a principal bundle in general.

Remark 2.7.3. A principal bundle structure is inherited by jet prolongation of principal bundles in terms of jets of manifolds (see Proposition 5.3.4).

Bearing in mind that the jet bundle PP -' P is an affine bundle modelled over the vector bundle

T'X®VP-P, P let us consider the quotient of the jet bundle PP - P by J' R0. We obtain the affine bundle

C=J'P/G-X

(2.7.5)

modelled over the vector bundle

Z`=T'X ®VcP-+X. Hence, there is the canonical vertical splitting

VC =Cx?7. x Remark 2.7.4. It is easily seen that the fibre bundle J' P - C is a principal bundle with the structure group G. It is canonically isomorphic to the pull-back

J'P=Pc=CxP-i C.

(2.7.6)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

86

In the case of a principal bundle P -, X, the exact sequence (1.4.3a) can be reduced to the exact sequence

O-,VcP-TcP-TX -0.

(2.7.7)

by taking the quotient with respect to the action of the group C. A principal connection A on a principal bundle P X is defined as a section A : P -+ J'P which is equivariant under the action (2.7.1) of the group C on P, that is,

J'R9oA=AoR9,

VgEG.

(2.7.8)

Turning now to the quotients (2.7.3), such a connection defines the splitting of the exact sequence (2.7.7). It is represented by the tangent-valued form

T'X ®TcP

A/

1

X " T'X ®TX A = dx' ®(8a + Aaeq),

(2.7.9)

where Aa are local functions on X.

On the other hand, due to the property (2.7.8), there is obvious one-to-one correspondence between the principal connection on a principal bundle P -, X and the global sections of the fibre bundle C - X (2.7.5), which is therefore called the bundle of principal connections.

Remark 2.7.5. An immediate consequence of this definition is that, by virtue of Theorem 1.2.5, principal connections on a principal bundle exist.

Remark 2.7.6. Let a principal connection on the principal bundle P --+ X be represented by the vertical-valued form A (2.3.6). Then the form -0-*+ T'P®gi 7: P-A +T*POVPld P

is the familiar 91-valued connection form on the principal bundle P. Given a local bundle splitting (Ut, z() of P, this form reads

"A =Op -Adxa®eq,

2.7. JETS OF PRINCIPAL BUNDLES

87

where Op is the canonical gi-valued 1-form on P, {e,} is the basis of 91, and Aa are local functions on P such that A (pg)Eq =

"Aq

(p)adg' 1(EV)

The pull-back zf4 of 7 over Ut is the well-known local connection 1 -form

At = -Aadxa ®eq,

(2.7.10)

o zf are local functions on X. where Al = It is readily observed that the coefficients Al of this form are precisely the coefficients of the form (2.7.9). Moreover, given a bundle atlas of P, the bundle of principal connections C is equipped with the associated bundle coordinates (x'', a°,) such that, for any section A of C --, X, the local functions

Al=aaoA are again the coefficients of the local connection 1-form (2.7.10). In gauge theory, these coefficients are treated as gauge potentials. We will use this term to refer to sections A of the fibre bundle C -, X.

The curvature FA of the principal connection A (or the strength of A) is the V0P-valued 2-form on X

FA: X-.A2T'X®VcP, FA = 2 FAr d? A dx" ®e 'ku

Faµ=[8a+Aaeq,8µ+Aµey]'=BaA;-OAA+

A%Aµ,

(2.7.11)

whose coordinate expression follows from (2.7.4). Let now

Y = (PxV)/G

(2.7.12)

be a fibre bundle associated with the principal bundle P X whose structure group G acts on the typical fibre V of Y on the left. Let us recall that the quotient in (2.7.12) is defined by identification of the elements (p, v) and (pg, g-'v) for all g E C. For short, we will say that (2.7.12) is a P-associated fibre bundle.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

88

As is well known, the principal connection A (2.7.9) induces the corresponding connection on the P-associated fibre bundle (2.7.12). If Y is a vector bundle, this connection takes the form

A=ds"®(8a+AP lpB;),

(2.7.13)

where Ip are generators of the representation of the Lie algebra g on V. This is called the associated principal connection or simply a principal connection on Y X.

In particular, a principal connection A yields a linear connection on the gauge algebra bundle VcP -+ X. The corresponding covariant derivative VA of a section

e=?epofVCP-,X reads VAC: X-.T'X®VcP, I VAt = (&c + cAaE°)dx" ®er If u is a vector field on X, the covariant derivative VAC of by

along u is simply given

V.C = uJV''f = [uJA,C], where A is the tangent-valued form (2.7.9). In particular, we have Vaeq = fppQAae,.

(2.7.14)

The covariant derivative V" is compatible with the Lie bracket of sections of

VcP - X, that is, VUA[C, n]

= fog E, n] + [C, Vy n]

for any vector field u : X -. TX and sections , n : X - VcP.

Remark 2.7.7. Let P

X be a principal fibre bundle with a structure Lie group C. Then the F-N bracket on O'(P) 0 P(P) is compatible with the canonical action Rc, and we obtain the induced F-N bracket on 9'(X) ®S(TOP), where S(TOP) is the vector space of sections of the vector bundle TcP - X. Recall that S(TcP) projects onto T(X). If A E 01(X) 0 S(TcP) is a principal connection as in (2.7.9), the associated F-N covariant differential is

dA : O'(X) 0 S(TTP) -# O'' (X) 0 S(VcP), dAO = [A, 01, 0 E Dr(X) 0 S(TcP).

(2.7.15)

2.8. CANONICAL PRINCIPAL CONNECTION

89

Note that, on or(X) 0 S(VVP), the differential dA coincides with the covariant differential relative to the linear connection VA on the vector bundle VIP - X whose connection parameters are given by (2.7.14). If 0 = a ® i; where a E Dr(X) and C E S(VVP), we have the formula

dA4 = da0C+ (-1)"a AVA which follows from (2.4.3).

Using the covariant differential (2.7.15), we can easily see that the curvature FA E D2(X) 0 S(VVP) of the connection A, as given in (2.7.11), reads

FA= 2dAA= I[A,A].

2.8

Canonical principal connection

This Section is devoted to the jet manifold J'C of the bundle of principal connections

C - X and the canonical connection on this fibre bundle. In gauge theory, the jet manifold J'C plays the role of a configuration space of gauge potentials.

Remark 2.8.1. Given the coordinates (x,aµ) of C, the jet manifold J'C of C is equipped with the adapted coordinates (x", aµ, at ).

Given a fibred manifold Y -+ X, let us recall the complementary morphism (2.1.20) written in the form

0:J'YYx TY -.VY,

(2.8.1)

Just as (2.1.23), it provides the canonical horizontal splitting of TY over J'Y. This splitting is nicely interpreted in the case of principal bundles [57, 58).

Let P -, X be a principal bundle with a structure Lie group C. Taking the quotient of (2.8.1) with respect to C, we obtain

CxT0P

V0P (2.8.2)

x 9(8a) _ -a"ep,

9(ep) = ep.

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

90

It follows that the exact sequence (2.7.7) admits the canonical splitting over C. Let us now consider the pull-back principal bundle Pc (2.7.6) whose structure group is G. Since

V0(CxP)=CxVGP, x

X

T0(CxP)=TCxT0P, x

x

(2.8.3)

we can interpret (2.8.2) as a principal connection

TC

TC x T0P x

(2.8.4)

C + aaep,

A(8p) = 8p,

A=dx"®(8a+aaep)+daa®8,a., on the principal bundle X

It follows that the principal bundle Pc carries the canonical principal connection given by (2.8.4). Following (2.7.11), we see that the curvature FA of A is given by 2

FA: C - AT'C®VGP, FA = (daa A dxa + 1 e a°,aµdx" A dam`) ®e,.

(2.8.5)

This is called it the canonical curvature, and its meaning is the following. Let A : X -- C be a principal connection on the principal bundle P -' X. Then the pull-back A' FA = FA,

(2.8.6)

is the curvature of the connection A. Remark 2.8.2. A consequence of the existence of the canonical principal connection

on Pc is that the vector bundle

CxVCP x

2.8. CANONICAL PRINCIPAL CONNECTION

91

is provided with the canonical linear connection such that the corresponding covariant derivative V is 0.%j Ve, = c'ygaper,

&,'j Veo = 0

(2.8.7)

(see (2.7.14)).

Let dA be the F-N covariant differential associated with A which acts on 0'(C)® S(To(C xX P)) (recall (2.7.15)). Of course, on 0* (C) ® S(VcP), dA coincides with

the F-N covariant differential associated with the linear connection V given by (2.8.7) (see Remark 2.7.7). We have

A E 0'(C) ® S(Tc(C x P)), A = dx'` ® (8,, + apep) + dap ®8 ,

(2.8.8)

FA = 2dAA = I[A A) E 02(C) ®S(1/VP),

dFA=dAFA=0, where the last equation is the (second) Bianchi identity relative to FA (recall (2.4.10)). Using (2.8.7), one can verify this identity directly. Note that, from (2.8.6) and (2.8.8) we obtain

dAFA=A'dAFA=0, i.e., the (second) Bianchi identity relative to FA (the differential dA have been defined in (2.7.15)).

Example 2.8.3. In particular, let us consider the trivial principal bundle P = X x R -, X. Then C = T'X -- X is the affine cotangent bundle, and principal connections on P are precisely 1-forms on X. The canonical connection A, the F-N covariant differential dA and the curvature reduce to

0 = iadsa, dA = d,

FA=S2=dOE n(T'X), 11=d? A d?,

(2.8.9)

that is, they are the Liouville form, the familiar exterior differential and the canonical symplectic form on the cotangent bundle T'X, respectively. 9

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

92

This Example shows that the bundle C X is in a sense a generalization of the cotangent bundle T'X -' X. Indeed, just as T'X carries the canonical symplectic form (2.8.9), in the same manner C does the canonical VaP-valued 2-form (2.8.5). Recall that, given a vector field u : X -' TX, its canonical lift it : T'X -. TTX (3.7.9) is uniquely determined by the equation 1z10 = -d(uJO),

(2.8.10)

and has the coordinate expression

U = u"a" - Xµa"u"a".

This is also called a Hamiltonian lift. The meaning of (2.8.10) is that we define actually a R-Lie algebra representation of T(X) into ? (T'X), the Lie algebra of projectable Hamiltonian vector fields over T'X. The generalization of this representation by means of the canonical curvature FA is of basic importance in gauge theories.

Let C : X - TAP be a section projected onto a vector field u on X. Using (2.8.2), we obtain the morphism over X

i.e., as (2.8.3) shows, a section of V0(C xX P) -. C. Then the equation u(JFA = -d(1;J9)

(2.8.11)

uniquely determines a vector field ut : C tations lead to

e = u"a" + e'eo,

TC projectable over u. Simple compu-

u = u"a",

uu = u"a" + uaa;,

(2.8.12)

One can think of (2.8.11) as being the definition of a projectable Hamiltonian vector field on C, and can write of E 74(C). We have u1(,n1 = [ut, t4,J,

Ve, ii E S(TcP).

(2.8.13)

Thereby, ?(C) is a Lie algebra. In particular, there is the subalgebra 74(C) C 74(C) of the vertical Hamiltonian vector fields on C: 1; E S(VcP), FU(

=

°,

ug E 74(C), u., = a"{' + cr aTV7.

(2.8.14)

2.8. CANONICAL PRINCIPAL CONNECTION Since VC = C xX

93

VcP C TC, we can write

ug=Ve:C--+VC, CES(VcP),

(2.8.15)

where V is the covariant derivative (2.8.7).

Using the jet lift (2.1.16), we obtain the jet prolongation of the vector field (2.8.12) over J'C (we continue to use the same symbol u() uE : J'C

TJ'C,

uE = uAaA + ua8; + uaµa;µ,

ua =8aµ

cc,aµ&

aj,,,aµu" + dp,a°,µ

-

-

(2.8.16)

where uj, is given as in (2.8.12). In particular, when £ E S(V0P), (2.8.16) reduces to

ut : J' C -+ V J'C, ut = U'49.'% + uaµa;'",

uAµ = dAu =

µS

c;gn.%.

+ gnyaAS ,

where the expression of ua is given in (2.8.14).

Remark 2.8.4. Let us now fix a principal connection A : X

C and study the

lifts defined by it. If u : X -+ TX is a vector field, then (2.7.9) determines the section

C=uJA: X -'TAP, C = uAaA + eep,

P = AauA.

Its Hamiltonian lift, as in (2.8.12), is U{ = UAvA + uAaA. ,

ua = 8 A,',uµ + cp,a°AMu' - (a,', - Aµ)8Au9.

Let v : X -. TX be another vector field and

n=vJA: X --'TSP. Moreover, put

w = [u,v]:X-+TX

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

94

and

Then we see from (2.3.12) and (2.7.11) that (2.8.17)

[f, n] = C + FA (u, v),

where FA(u, v) is the section P,

FA

FA(u, v) = uav"(OAA14 - B,,A;, + c;,AaA,°,)e

FA being the curvature of A. Recalling (2.8.13) and (2.8.15), from (2.8.17) we obtain [uu, u,,] = uC + V [FA (U, v)].

Let us try to interpret the lifting of ut as its horizontal lift uJ r for a connection r on the fibre bundle C X. For this purpose, let us consider a symmetric linear connection K on X and the linear connection VA induced by A on VCP - X given by (2.7.14). Let r be the X induced by K and A. Given the tensor product connection on T'X 0 VcP coordinates (xa, a-, 8'a,,) of J' (T'X (9 VcP), we have

r:T'X ®VCP -. J1(T'X ®VcP), .,4,or=-K°,,x +cygavA%. Using the fibred morphism

4;A :C-'T'X®VcP introduced in (1.3.3), from the following commutative diagram, we obtain the section

r:C-.J1C PC J1 J1(T'X ® VcP)

r1 C

I

".

T'X ® VCP

Iaa,, o r = ra,, = OAAµ + cy,nPA°,, -

-Ate) + cygAxA9.

(2.8.18)

2.8. CANONICAL PRINCIPAL CONNECTION

95

Of course, I' is an afiine morphism over X, i.e., an affine connection on the affine X, while the associated linear connection is r. Moreover, it is easily bundle C seen that A is an integral section of r, i.e.,

J'A=roA. The canonical curvature FA (2.8.5) can be seen in a slightly different way. Namely, there is a horizontal VCP-valued 2-form on J'C:

J'C-AT'X ®VcP

(2.8.19)

which satisfies the condition

.FoJ'A=FA for each principal connection A : X -. C. We have

.F=

Z,,dx"ndx"®e,, (2.8.20)

aµ;, + c'ygaaaF°,.

It is readily observed that (2.8.19) is an affine surjection over C and, hence, its kernel C+ = Ker8.1 with respect to the zero section

0:C-CxAT'X®VcP x is an affine subbundle of J'C -e C. Thus, we have the canonical splitting over C:

J'C=C+®C_=C+®(Cx T'X®VcP), aµ = 2 (aaµ + aµa - cpgaAQµ) + 2 (a4,

(2.8.21)

- Q'A +a°,Q°

The corresponding canonical projections are

S=pr,:J'C-'C+, S,' = 2 (aaµ + aµa - cygaaaN),

and pr2 =.F/2 (2.8.19).

(2.8.22)

CHAPTER 2. JET MANIFOLDS AND CONNECTIONS

96

Remark 2.8.5. The of iine subbundle C+ -' C is modelled over the vector bundle x which is a vector subbundle of

JC=T'X®VC-+C. C Recall that

VC =CxT'X0VCP. x Of course, also the vector part of (2.8.21) is a vector subbundle

CxAT'X®VCP --x

C

of J'C.

Remark 2.8.6. Let r : C -e J'C be a connection on the bundle of principal connections C - X. Then s o r is a C+-valued connection on C -i X which satisfies the condition (S o r)aµ - (s o r); ,a + cygnaa,°, = 0.

In particular, let us consider the affine connection r (2.8.18). Then we obtain the connection

s=sor:C-.ECJ'C, S,y, =

(BaAI, + 8NAa -

2 -2Ka"n(a - A') -

which has the property

SoA=SoJ'A.

ANua)

aaaM

(2.8.23)

Chapter 3 Lagrangian formalism We will limit our study to first order Lagrangian formalism, since most contemporary field models are described by first order Lagrangian densities. This, however, is not the case of the Hilbert-Einstein Lagrangian density in General Relativity which belongs to the particular class of second order Lagrangian densities leading to second order Euler-Lagrange equations (see Remark 3.2.12).

From the mathematical point of view, first order Lagrangian formalism is free from the ambiguities which are present in the higher order one [77]. For it the finite-dimensional configuration space of fields represented by sections s of a fibred manifold Y X is the first order jet manifold J'Y of Y. A first order Lagrangian density on J'Y is defined as a horizontal density

L:J'Y - A"7-X,

n=dim X.

Given fibred coordinates (xa, y`) of Y X and the corresponding adapted coordinates (x", y', ya) of J'Y, a Lagrangian density reads L = Gw,

w = dx'...dx,

where G is a local function on J'Y (see (1.6.6) for the notation). We will denote the Lagrangian momenta by n; = 8,' G.

We will use the first variational formula (3.2.8) which provides the canonical decomposition of the Lie derivative of a Lagrangian density along a projectable vector field on Y in accordance with the variational task, and thus leads to the EulerLagrange operator and differential conservation laws. Formulas (3.5.15), (3.5.17)

97

CHAPTER 3. LAGRANGIAN FORMALISM

98

and (3.5.25) are basic elements of our analysis of conservation laws, including those of energy-momentum, in field models.

3.1

Technical preliminaries. Higher order jets

Though our consideration is restricted to first order Lagrangian theory, we will occasionally appeal to higher order jets for the sake of completeness. Referring the reader to Chapter 5 for a detailed exposition, here we summarize only the relevant material on higher and infinite order jets. In brief, one can say that the r-order jet manifold J'Y of a fibred manifold

Y -+ X comprises the equivalence classes f.s, x E X, of sections s of Y - X identified by values of the first r + I terms of their Taylor series at the point x. There is the inverse system

of higher order jet manifolds J'Y. of Y -. X, the r-order jet manifold J'Y is Given fibred coordinates endowed with the adapted coordinates (xa,yX), 0
\Ia,...al)

modulo permutations. This differs from the union of collections

AE = (ak ...

ai)

where indices A and a are not permuted. We will use the compact notation BA = BAk o ... O OA1,

A = (Ak ...,\I).

Every section s of a fibred manifold Y the section (J's)(x) = j=s

of the fibred jet manifold J'Y - X.

(3.1.1)

X has the r-order jet prolongation to

3.1. TECHNICAL PRELIMINARIES. HIGHER ORDER JETS

99

X admits the r-orderjet lift Fu (4.6.34) Every projectable vector field u on Y over the r-order jet manifold FY of Y -. X. This is a projectable vector field on XY (see Definition 5.1.4). We have the inverse system Tr;-, Pr T,''+t T,ro Ttr of the Lie algebras P, of projectable vector fields on the higher order jet manifolds

J'Y. Remark 3.1.1. If a vector field u on a fibred manifold Y - X is the generator X (see of a (local) 1-parameter group [4 t] of local fibred automorphisms of Y Remark 1.5.2), then its r-order jet lift is the generator of the (local) 1-parameter group of the local fibred automorphisms of the J'Y - X, which are the r-order jet prolongations J'(bi (5.1.6) of 4N. There is the direct system -01

-12

x;+r.

of vector spaces 0 of exterior forms on XY. The limit D , of this direct system exists, and b y r ° are meant the corresponding monomorphisms D -. Oo,. In brief, one can say that the limit Oo, consists of all exterior forms defined on the finite order jet manifolds J*Y which are identified with respect to the pull-back prolongation.

1 For the sake of simplicity, we will denote the pull-backs 7r; o and 7r° exterior forms 0 E 0* by the same symbols 0. 1

of

The limit £) inherits the operations which are preserved by the pull-back procedure. Thus, Oa, is an exterior Z-graded algebra with respect to operations of the exterior product and the exterior differentiation. The important advantage of considering exterior forms on finite order jet manifolds as elements of the limit Oa, is that OL is generated locally by the basic forms dxa and the contact forms IPA = dJn - ya+nd?

0
In particular, the vector subspace Dc, C O:. of exterior s-forms admits the canonical decomposition 000 =

O;'®O;0-' ®...®i7.

CHAPTER 3. LAGRANGIAN FORMALISM

100

Elements of il00'-k are called k-contact forms. Elements of D'A, n = dim X, are 00 said to be k-contact densities. We denote by hk the k-contact projection

hk : JD' . ok;'-k,

k < s.

For instance, the horizontal projection ho : 0'00 - O;;' is given by

dxa'-' ds",

d y A F-' ya+AdxA,

0:51

Accordingly, the exterior differential on il:,, is decomposed into the sum

d=dH+dv of the horizontal differential dH and the vertical differential dv such that

d : ilk,.-k 00

d ilk'-k 00 dv : ilkm-k 00

ilk+l .-k 00

®ilk'-k+l

00

W ilk.&-k+1

ok+l,s-k ca

They have the homology properties dHdH = 0,

dvdv = 0,

dvdf + dHdv = 0.

We also have the relation

hood=dH Oho. In order to obtain coordinate expressions of the above operators, let us introduce the total derivative (3.1.2)

I

where summation is over all multi-indices A, 0 <1 A 1. It acts on exterior forms oA, o E O , by the rules da(mAo) =d,,(b) d.%(dc) =

Similarly to (3.1.1), we use the compact notation

dA=dak o...oda

A = (Ak...A,).

3.2. THE FIRST VARIATIONAL FORMULA

101

Then the horizontal differential dH is given by the local expression dHO = dxA A da(0),

(3.1.3)

4' E! ea,.

In particular, we have

dHf = dafdx'',

f E Do.,

da(dx") = 0,

d1(dxµ) = 0,

da(dyn) = dy`''+A, da(01A) = 84A,

dHdyA = dxa A dy;,+A, dHO = dam'' A B'a+A,

0 <1 A I, 0
Accordingly, the vertical differential has the properties

dvf =e"fen,

f E0D°,

0:5I A I,

dv (dx') = 0,

dvdy' = -d? A dya+A,

3.2

40A = 0.

The first variational formula

Here we will not study the calculus of variations in depth, but will recall only some basic notions. Our goal is the first variational formula (3.2.13). We will follow the standard formulation of the variational problem where "deformations" of sections of a fibred manifold Y -+ X are induced by local 1-parameter groups of local fibred automorphisms of Y over Id X [13, 110, 1741. Let N be an n-dimensional compact submanifold of X with the boundary ON. A vertical vector field u on Y X is called an admissible vector field if it vanishes on a neighbourhood of 7r-1 (ON). A section s of the fibred manifold Y -+ X is said to be a critical section of the variational problem for an exterior n-form p on J'Y if and only if

f s'Lp,,p = 0

(3.2.1)

N

for any admissible vector field u on Y - X. Here J'u is the jet prolongation (2.1.16) of the vector field u. I For the sake of simplicity, ifs is a section of Y

(fs)'0 of forms 0 on J'Y by s',. 1

X, we denote the pull-backs

CHAPTER 3. LACRANGIAN FORMALISM

102

Using the relation

Lri.p = d(J'ujp) + (J'ujdp), one can bring the functional (3.2.1) into the form

I s'LjiLp = N

I ON

I s'(J'ujdp),

(3.2.2)

N

where the first term in the right-hand side of (3.2.2) equals zero if u is an admissible vector field. However, it may happen that the relation (3.2.2) is not appropriate for the variational task because the second term in the right-hand side of it can depend on the derivatives of components of the vector field u, and can therefore contribute to the integral over the boundary ON.

Example 3.2.1. For instance, let L = Lw be a first order Lagrangian density on the jet manifold J'Y. We have s'(J'ujdL) = s'(u'OjL + dau`8;'G)w.

(3.2.3)

The second term in the expression (3.2.3) can be rewritten as

dau'8,"Gw = da(u'8, L)w - u'da(8, G)w. This contains an exact form which contributes to the boundary integral. It follows that the condition

s'(J'ujdp) = 0 is fitted on critical sections if, for any section s, the form s'(J'ujdp) depends on the components of the vector field u, but not on their derivatives. Exterior n-forms p which possess this property are the Lepagian forms. There are several definitions of Lepagian forms [110]. We will mention the following equivalent conditions for an exterior n-form p on the r-order jet manifold

J'Y to be a Lepagian form. For each projectable vector field ur on J'Y, the horizontal projection ho(u,.J dp) depends on the iro-projection of %. onto Y only.

For every vertical vector field ur on the jet bundle J'Y - Y, p obeys the relation ho(u.jdp) = 0.

3.2. THE FIRST VARIATIONAL FORMULA

103

Example 3.2.2. Every closed form on J'Y is obviously a Lepagian form. Example 3.2.1 shows that Lagrangian densities fail to be Lepagian forms in general. However, one can replace the variational problem for a first order Lagrangian density L with the variational problem for a Lepagian equivalent pL of L. This is defined as a Lepagian form on the (r + 1)-order jet manifold P+iY which satisfies the condition ho(PL) _=_L.

Example 3.2.3. Every Lepagian form p on PY is a Lepagian equivalent of the (k < r)-order Lagrangian density L = ha(p). If PL is a Lepagian equivalent of L, we have the equality

I s'L = f S* PL N

N

for any section s of Y -' X. It follows that the variational problem for a Lagrangian density L is equivalent to the variational problem for its Lepagian equivalent pL over

sections s of the fibred manifold Y -' X. Obviously, a section s of Y - X is a critical section of the variational problem for a Lagrangian density L if the relation

s'JouJdpL = 0

(3.2.4)

holds for any vertical vector field u on Y -+ X.

I Hereafter, by a critical section we mean just that one satisfying the relation (3.2.4).

Remark 3.2.4. If s is a critical section, the relation (3.2.4) holds also for an arbitrary projectable vector field u on Y

X.

PROPOSITION 3.2.1. Every r-order Lagrangian density has Lepagian equivalents

on J''-'Y (see [77, 112]). 0 In particular, a first order Lagrangian density has Lepagian equivalents on J'Y. Hereafter, we will limit our attention to this kind of Lepagian equivalents of first order Lagrangian densities.

CHAPTER 3. LAGRANGIAN FORMALISM

104

Example 3.2.5. Lepagian equivalents on J'Y of the zero Lagrangian density L = 0 are contact forms given by the local expressions 8 + c,;"'8) (3.2.5) po = A wv + X,

where c;"' = -c? are skew-symmetric local functions on Y and X is a (k > 1)contact form on J'Y (see [77]). Lepagian equivalents of a first order Lagrangian density L form an affine space modelled over the linear space of the Lepagian equivalents (3.2.5) of the zero Lagrangian density. One usually chooses the Poincar6-Cartan form

HL : J'Y - T'Y A (A1T'X),

(3.2.6)

HL=Cw+a;O'AW.%,

as the origin of this affine space. Then the general coordinate expression for a Lepagian equivalent of a first order Lagrangian density L reads

PL=HL

6'+cAw,+X.

(3.2.7)

Example 3.2.6. In particular, if a fibred manifold Y - X is equipped with a connection I' and a fibre metric

a:Y - VV'Y, one can set

s

/jay

- al)ypQJv8 'aa7

where R is the curvature of r and g is a metric on X.

Remark 3.2.7. In comparison with other Lepagian equivalents, the PoincarECartan form is a semibasic form on the jet bundle J'Y -. Y, and it possesses some essential peculiarities. Moreover, it is the Lagrangian counterpart of Hamiltonian forms in polysymplectic Hamiltonian formalism on fibred manifolds (see Chapter 4).

Let PL be a Lepagian equivalent of a first order Lagrangian density L. Given a projectable vector field u on Y, the first variational formula provides the canonical decomposition LriuL = ho(J'ujdp,) + hod(J'u1 pL)

(3.2.8)

105

3.2. THE FIRST VARIATIONAL FORMULA

of the Lie derivative of L along the first order jet lift of u (or simply along u). For any Lepagian equivalent pL of L, we have ho(J'ujdPt) = UVJEL,

(3.2.9)

where

uv = (uJO`)a = (u` - uAva)a: is the vertical part of the canonical horizontal splitting (2.1.25) of u over J'Y, and

EL : J2Y - T'Y A (AT'X), EL = (a; - daa; )LO' A w = 6;,C6' A w,

(3.2.10)

is the second order Euler-Lagrange operator associated with the Lagrangian density L. Its coefficients b;G are called the variational derivatives. Hence, the first variational formula (3.2.8) takes the form

LriuL = uvJEL + deho(J'uJPL),

(3.2.11)

where do is the horizontal differential (3.1.3). Furthermore, it is readily observed

that dyho(JiuJPo) = 0,

(3.2.12)

if po is a Lepagian equivalent (3.2.5) of the zero Lagrangian density. Therefore, we

can restrict our attention to the PoincarE-Cartan form pL = HL. Since HL is a semibasic form on the jet bundle J'Y -+ Y, we have

LjiuL = uvJEL +dyho(uJHt)

(3.2.13)

In view of the relation (3.2.9), critical sections of the variational problem of a Lagrangian density L satisfy the system of (variational) second-order Euler-Lagrange equations 01C - (8a + 0.%3% + aaaµs;8 )a; c = 0.

(3.2.14)

Remark 3.2.8. It may happen that the equations (3.2.14) fail to be differential equations in a strict mathematical sense (see Remark 3.3.11).

It should be noted that different Lagrangian densities L and L' lead to the same Euler-Lagrange operator EL = EL' if they differ from each other in a Lagrangian

CHAPTER 3. LAGRANGIAN FORMALISM

106

density Lo whose Euler-Lagrange operator EL. is equal to zero. Such a Lagrangian density is called variationally trivial.

Example 3.2.9. Lagrangian densities of topological field models are examples of variationally trivial Lagrangian densities (3.2.15) (see Example 3.6.10). PROPOSITION 3.2.2. A first order Lagrangian density Lo is variationally trivial if and only if Lo = ho(e)

(3.2.15)

where a is a closed n-form on the fibred manifold Y -+ X [110]. 0

Remark 3.2.10. Since J'Y - Y is an affine bundle, the De Rham cohomologies of J'Y coincide with those of Y [13]. Thus, every closed exterior form on J'Y is the sum of the pull-back of a closed form on Y and an exact form on J'Y. By virtue of Proposition 3.3.8, every exact form on J'Y is a Lepagian equivalent of a variationally trivial first order Lagrangian density. It follows that every closed form a on the jet manifold J'Y is a Lepagian equivalent of a variationally trivial first order Lagrangian density. In particular, if L is a Lagrangian density and pL, is its Lepagian equivalent, then the Lepagian form pl, + e is the Lepagian equivalent of the Lagrangian density

L' = L + ho(e)

which leads to the same Euler-Lagrange operator as L. For instance, L' = L if c is a contact form. Remark 3.2.11. One can consider the first order differential equations on the first

order jet manifold J'Y - X instead of the second order ones on Y

X. For

instance, let us consider the variational problem for the Poincar6-Cartan form HL (3.2.6) over sections 3 of the fibred jet manifold J'Y - X. By definition, a critical section -9 of this variational problem satisfies the relation

3'(uJdHt) = 0 for all vertical vector fields

u = u`8; + uµ8;

(3.2.16)

3.2. THE FIRST VARIATIONAL FORMULA

107

on J'Y -, X. It follows that it obeys the system of first order differential equations 8j a; (8a3' - 3O'a) = 0,

&G-(8a+*a8,+8,,3v"g )8;G+

8r(8' -Va)=0. a

(3.2.17a)

(3.2.17b)

These are called the (variational) Cartan equations.

Remark 3.2.12. Reduced second order Lagrangian densities. Let us consider second order Lagrangian densities on the second order jet manifold J2Y of Y which, however, lead to second order Euler-Lagrange equations. In Remark 4.4.8, there is an example of a first order Lagrangian density which yields first order Euler-Lagrange equations. A second order Lagrangian density L on J2Y has Lepagian equivalents on the third order jet manifold J3Y. In particular, the associated Poincare-Cartan form HL is uniquely defined and given by the coordinate expression

L&,]Awl. Let u be a projectable vector field on the fibred manifold Y order case, the first variational formula (3.2.13) is written as

uv

j HL)

(3.2.18)

X. In the second (3.2.19)

where

EL = (8; - d,,8" + ddA8;`'')G6' A w

(3.2.20)

is the 4th-order Euler-Lagrange operator. Let us consider a second order Lagrangian density L whose Euler-Lagrange operator EL (3.2.20) reduces to the second order one [111]. This happens if the associated Poincard-Cartan form HL (3.2.18) is defined on the first order jet manifold J'Y. This is the case of a Lagrangian density L which meets the conditions 8i°p8""G = 0,

(3.2.21)

(&8" -8;8j%)L=0.

(3.2.22)

The relation (3.2.21) means that such a Lagrangian density is linear in the coordinates y',,,,, i.e., it is given by the local expressions L = (G' + ir;`ay,',,,)w,

(3.2.23)

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108

where C and a;`" are local functions on J'Y. By virtue of the relation (3.2.22) and Remark 4.4.2, there exists a local horizontal

form 4 = O''wa on J'Y - X such that

Then let us consider the local form

a=HL - dO. This is a Lepagian equivalent of the local first order Lagrangian density L, = ho(v) = L - hod4

which leads to the same second order Euler-Lagrange operator on a given coordinate chart as the Lagrangian density (3.2.23). It should be emphasized that such a first order Lagrangian density is not globally

defined in general. At the same time, if the functions 7j" are independent of the coordinates y,', we can take

'0 = 4 0(4 -

(3.2.24)

where r is a connection on Y -. X. The form (3.2.24) is globally defined, and we obtain the first order Lagrangian density Li = L - dA[7r; "(yµ

which leads to the same second order Euler-Lagrange operator as (3.2.23).

3.3

Euler-Lagrange operators

For the sake of completeness, we are concerned below with some basic notions of the geometric theory of differential operators and differential equations. We refer the reader to Section 5.6 for a more complete exposition of this theory. The goal is to describe the Euler-Lagrange operators as differential operators of the variational type in a straightforward manner.

DEFINITION 3.3.1. Let Y -+ X be a fibred manifold, and I = dim Y - dim X. A system of k-order partial differential equations on Y - X is defined to be a closed fibred submanifold it of the fibred manifold JkY -+ X. 0

3.3. EULER-LAGRANGE OPERATORS

109

I For the sake of brevity, we will often call ! a differential equation. 1

Let JkY be provided with coordinates (?,IA), 0 <1 A 1< k, as before. There exists a local coordinate system (e'c, - ) on JkY such that a is given locally (in the sense of Proposition 1.1.3 (i)) by the system of equations

A = 1, ... ,

eA(z , YO = 0,

(3.3.1)

A differential equation 0- is said to be determined if codim(! = 1; overdetermined if codimlE > 1;

underdetermined if codimt < 1.

By a classical solution of a differential equation it on Y -. X is meant a (local) section s of Y -. X such that its k-order jet prolongation J's takes its values into

One usually considers differential equations to be associated with differential operators. There are several equivalent definitions of (non-linear) differential operators. We can start from the following. DEFINITION 3.3.2. Let Y - X and E -+ X be fibred manifolds, which are assumed to have global sections. A k-order E-valued differential operator on a fibred manifold Y - X is defined to be a section C of the pull-back fibred manifold

pre:Ey=Ex JkY -

JkY.

(3.3.2)

A

Given fibred coordinates (x'', y') of Y and (x'', 17) of E, the pull-back (3.3.2) is provided with the coordinates (?,y'E,17°),

0
There is one-to-one correspondence between the sections Jl= : JkY

Eye

roe =

(3.3.3)

CHAPTER 3. LAGRANCIAN FORMALISM

110

of the fibred manifold (3.3.2) and the fibred morphisms 14i : JkY -- E

(3.3.4)

over X. We have

0= pr1 o E,

E = (4;, Id JkY).

This correspondence leads to another equivalent definition of differential operators on Y -+ X.

X and E X be fibred manifolds as before. A fibred morphism PY - E over X is called a k-order E-valued differential operator on the fibred manifold Y -, X. DEFINITION 3.3.3. Let Y

It is easy to see that a differential operator 4' (3.3.4) sends each section s of X. The mapping Y -. X onto the section 4i o Jks of E

A.: S(Y) - S(E), I A+:sI. 4ioJks, r(x) = E°(?, &s''(x)),

0:51 E 1< k,

is said to be the standard form of a differential operator.

Remark 3.3.1. Usually, differential operators are defined when fibred manifolds X are vector bundles. However, this is not the case of gauge Y -# X and E theory where the bundle of principal connections C (2.7.5) is not a vector bundle. At the same time, almost all differential operators met in field theory (e.g., the EulerLagrange operators) take their values into subbundles of tensor bundles 'A T'Y X. Such differential operators are represented by exterior forms E (3.3.3) on jet manifolds of Y - X.

Let us turn now to Euler-Lagrange operators on a fibred manifold Y These take their values into the tensor bundle

T'Y A ("T'X),

X.

(3.3.5)

and can be intrinsically defined as 1-contact densities of the variational type on jet manifolds. Cf. Section 5.5 where the corresponding variational bicomplex is constructed. Here we are concerned only with the basic ingredients in this construction.

3.3. EULER-LAGRANGE OPERATORS

111

Let us consider the subspace ill," C 1)o, of k-contact densities on jet manifolds and the quotient (3.3.6)

Ek = fl,k,n/dHfloo -'.

PROPOSITION 3.3.4. The quotient Ek (3.3.6) is the complement of the subspace dy(ila,"-1) C 17;," [1801, that is,

0"00 =Ek®dHilkn-'. 00 0 The corresponding projection map Ek = rk(ilaon)

has the properties

rkody=0.

TkOTk=Tk,

It is defined as 1

k,n

where r is the operator given by the coordinate expression Jr(.0)

= (-1)JA10' A [dA(8ij0)] ,

0
which acts on contact densities 0 E il;' [12, 1801.

Example 3.3.2. In the case of 1-contact densities ¢ E D',", 00 we have

Iri(0) = (-1)IAIdn(0, )B' Aw

(3.3.7)

(see [12, 611). It should be emphasized that the subspace E, C D ' consists of

1-contact densities 0 which take their values into the tensor bundle (3.3.5), that is, they are given by the coordinate expression 0 = 4i;6' A w. Building on Proposition 3.3.4, let us consider the short exact sequence -e"-+ Ek -i 0,

CHAPTER 3. LAGRANGIAN FORMALISM

112

where

Iek= rkohk. It is a simple exact sequence because

JD"* = Kerek ® E. 00 Since

d(Kerek) C Kerek+i, we come to the following commutative diagram 0

0

1

1

1

Dn-1 d , fln00 00

Ker el - Ker e2

a.

ko!

M!

1

1

Ker ho -e Ker ho 1

0

0

iln+i a, Do

I

e,

1

on+2 00

(3.3.8)

.!

!

fl 00 -i

fl 00

E1-' E2

I

I

1

1

0

0

0

0

Its first and second rows are the subcomplexes of the De Rham complex. The last row -JH 00-1 00

d

d

00-00

c

c

(3.3.9)

is also a enchain complex, that is, El o dH = 0,

Ek+1 0 ek = 0.

(3.3.10)

This complex is called the spectral sequence.

Remark 3.3.3. Let us recall briefly the basic notions of homology and cohomology of complexes [22, 124). A sequence

0 .L Bo 4' B,4 ....---Bp.°t' ...

(3.3.11)

3.3. EULER-LAGRANGE OPERATORS

113

of Abelian groups Bp and homomorphisms 8p is said to be a chain complex if Op o Op+t = 0,

Vp E N,

that is, Im Op}1 C Ker 8p. The quotient

Hp(B.) = KerOp/ImOp+,

is called the pth homology group of the chain complex B. (3.3.11). The chain complex (3.3.11) is called exact at an element Bp if H,(B.) = 0. B. is an exact sequence if it is exact at each element. A sequence

0--.B0 0 B' 6'....a

Bpi...

(3.3.12)

of Abelian groups Bp and homomorphisms lP is said to be a cochain complex if

l"o6"

=0,

VpEN.

The pth cohomology group of the cochain complex B' (3.3.12) is the quotient

H'(B') = Kerb'/Imdp-'. The De Rham complex of exterior forms on a manifold M

exemplifies a cochain complex, whose cohomology group H"(M) is called the Dc Rham pth cohomology group. It is the quotient of the space of closed p-forms by the subspace of exact p-forms. Since Ek_1 C Stk-',n, the cochain morphisms ek of the complex (3.3.9) take the form

I ek=Tkodv=jrod,

(3.3.13)

where

S=rod

(3.3.14)

is the variational map (12, 61] which satisfies the homology rules

6 ob=0,

body=0.

(3.3.15)

CHAPTER 3. LACRANGIAN FORMALISM

114

In particular, the cochain morphisms e, and e2 take the explicit form I ei(ZW) = (-1)IAtdA(8°C)8' Aw, 1 ez(dk°' A w) = 1 [8"0;9',, A6' + (-1)1A19' A dA(8; 461)] A w,

where summation is over all multi-indices A, 0 <[ A I. They are called the EulerLagrange map and the Helmholtz-Sonin map, respectively. DEFINITION 3.3.5. Given a horizontal density L = Gw on a finite order jet manifold

J'Y -' X (i.e., an r-order Lagrangian density), the exterior form EL, = el(L) = b(L), EL = (-1)'AidA(8; G)9' Aw,

(3.3.16)

0 <) A I< r,

is called the Euler-Lagrange form associated with the r-order Lagrangian density L. This form is the 2r-order differential operator (3.3.3), and the corresponding morphism

Et : J''Y -. T'Y A (" T' X)

(3.3.17)

is called the Euler-Lagrange operator associated with L.

I Now, we will identify the notions of an Euler-Lagrange form and an EulerLagrange operator. I

In particular, if L is a first order Lagrangian density on the jet manifold J'Y, the operator (3.3.17) is the second order Euler-Lagrange operator (3.2.10). Remark 3.3.4. Remarks 3.2.12 and 4.4.8 show that the Euler-Lagrange operator associated with an r-order Lagrangian density may be of the order < 2r. In this case, the Euler-Lagrange form (3.3.16) is a pull-back of an exterior form on a (< 2r)-order jet manifold. Using the spectral sequence (3.3.9), one comes to the following general solution of the well-known inverse problem of the calculus of variations [12, 42, 94, 174].

DEFINITION 3.3.6. Differential operators which take their values into the tensor bundle (3.3.5) are called Euler-Lagrange-type operators. These are elements of the subspace El C D 00 .

3.3. EULER-LAGRANGE OPERATORS

115

In particular, the projection map ri (3.3.7) sends 1-contact densities onto EulerLagrange-type operators. Every Euler-Lagrange operator is obviously an EulerLagrange-type operator.

DEFINITION 3.3.7. An Euler-Lagrange-type operator E is said to be a locally variational operator if e2(E) = 1b(E) = 0.

PROPOSITION 3.3.8. In accordance with the relations (3.3.10), any dH-ecact (and,

consequently, exact) Lagrangian density is variationally trivial and every EulerLagrange operator is locally variational. The obstruction for a locally variational operator to be the Euler-Lagrange one lies in the non-zero cohomology group

H"+' = KerE2/Imel of the complex (3.3.9) at the element El. Since the columns of the diagram (3.3.8) are simple exact sequences, one may deduce the cohomology groups of the complex (3.3.9) from the cohomology groups of the first two rows of the diagram (3.3.8) (see Remark 5.5.1). In particular, the following assertion can be easily proved. PROPOSITION 3.3.9. If Y = R'+" - R", the spectral sequence (3.3.9) is exact, that is,

Kerei = Imdy,

Imek = Kerek+1

(see (42, 1801).

I It follows that the above-mentioned obstruction for a locally variational operator to be the Euler-Lagrange one is topological, but not algebraic. I Since the columns of the diagram (3.3.8) are simple exact sequences, the spectral sequence (3.3.9) can be regarded as a subcomplex of the cochain complex

10-R

i7°00 °"

... i7°,"-' ao

eH

il°" 00

-. D''" 6

00

a

O2,n 00

(3.3.18)

CHAPTER 3. LAGRANCIAN FORMALISM

116

where 6 is the variational map (3.3.14). The complex (3.3.18) is called the variational sequence. Its homology groups are equal to those of the spectral sequence (3.3.9), and in particular, it is locally exact in the sense of Proposition 3.3.9.

Example 3.3.5. Since the variational sequence (3.3.18) is locally exact, a local r-order Lagrangian density Lo is variationally trivial if and only if Lo = df(hoo) = hods

o E Or'

By virtue of Proposition 3.2.2 and Remark 3.2.10, each dosed form on J'Y is a Lepagian equivalent of a variationally trivial first order Lagrangian density.

Example 3.3.6. Since the variational sequence (3.3.18) is locally exact, every locally variational operator 6 is locally an Euler-Lagrange operator, that is, E = bL,

(3.3.19)

where L is a local Lagrangian density. For instance, let

E: J2Y-+T'YA(n7"X), E= &0'Aw, be a second order Euler-Lagrange-type operator. It is locally variational if and only

if b(E) = [(28; - da88 + dda8;'')&Bt A 0' +

(3.3.20)

-dA8J`&BNn

+

(8w&-8;"E;)B'µaA0']Aw=0. Then there is a coordinate atlas of the fibred manifold Y X such that, on each coordinate chart, E is in the form of (3.3.19). The condition (3.3.20) is equivalent to the following system of equations

8;& - B;E; -

E; + d,,da8;'& = 0,

8;&+2d8""&=0,

(3.3.21)

Indeed, if these equations hold, then a direct check shows that the condition (3.3.20) does so too, that is, the operator E is locally variational. Conversely, if the operator .6

3.3. EULER-LAGRANGE OPERATORS

117

takes the local form (3.3.19) on each coordinate chart, then it satisfies the equations (3.3.21).

It is readily observed, that the equations (3.3.21) represent the coordinate expression of the condition

r(J2uJdE) = 0, where u is an arbitrary vertical vector field on the fibred manifold Y is its second order jet prolongation (2.2.9) [61).

(3.3.22)

X and J2u

Example 3.3.7. The operator (3.6.32) in the Chern-Simons gauge model, which is locally associated with the Chern-Simons Lagrangian density, exemplifies a locally variational, but not variational operator.

We will use the condition (3.3.22) to discover the important relation between gauge symmetries of Lagrangian densities and those of Euler-Lagrange operators (see Remark 3.5.7). Let E be a second order Euler-Lagrange-type operator and u a projectable vector field on Y -» X. It can be proved in a straightforward manner [61) that

LnuE = r[d(uvj9) + (J2u)vJdE),

(3.3.23)

where (J2u)v is the vertical part of the vector field J2u. This relation is called the master equation. If the operator e is locally variational, it follows from (3.3.22) and (3.3.23) that L,r,,,E = 6(uvJE)

(3.3.24)

for every projectable vector field u on Y X. Let EL be the Euler-Lagrange operator associated with a first order Lagrangian density L. By virtue of the first variational formula (3.2.13) and the relation (3.3.15), the equality (3.3.24) is brought into the form I L,r.,,Et =

(3.3.25)

The following two assertions are immediate corollaries of the formula (3.3.25).

PROPOSITION 3.3.10. Let a first order Lagrangian density L be invariant under

a 1-parameter group of fibred automorphisms of Y - X whose generator is a projectable vector field u on Y, i.e, 0.

CHAPTER 3. LAGRANGIAN FORMALISM

118

Then the same is true for the associated Euler-Lagrange operator Et, i.e., 0.

PROPOSITION 3.3.11. Let an Euler-Lagrange operator Et associated with a first order Lagrangian density L be invariant under a 1-parameter group of fibred automorphisms of Y X whose generator is a projectable vector field u on Y. Then LJi L is a variationally trivial Lagrangian density. By virtue of Proposition 3.2.2, we have ho(e),

(3.3.26)

where a is a dosed n-form on the fibred manifold Y. O Let us return to differential equations. Let 4 be a k-order differential operator. Given a global sections of the fibred

manifold E - X such that s C Im 4), by the kernel of the differential operator 0 with respect to the section !is meant the kernel Ker;4 (1.2.5) of the fibred morphism

(3.3.4) with respect to s. If this kernel is a fibred submanifold of JkY - X, it is a k-order differential equation

t# = Ker;4), called the differential equation associated with the differential operator 0. 2 Obviously, not every differential operator leads to a differential equation. I The following condition is sufficient for a kernel of a differential equation to be a differential equation. PROPOSITION 3.3.12. Let the morphism 4i (3.3.4) have constant rank. By virtue of Theorem 1.2.6 (see also [26], p.396), its kernel (1.2.5) is a fibred submanifold of the fibred manifold JkY -' X and, consequently, it is a k-order differential equation. 0

Example 3.3.8. Let us consider the fibre bundle Y = R2 -, R with coordinates (x, y) and the fibre bundle E = R2

go 4'= Vy.

R with coordinates (x, y). Let

3.3. EULER-LAGRANCE OPERATORS

119

be a first order E-valued differential operator on Y. Its kernel with respect to the zero section of E is the set of points of J'Y having the coordinates y = 0 or yz = 0. Obviously, this set is not a submanifold of J' Y.

This is a typical example of a differential operator which does not lead to a differential equation. The following example shows that a differential equation may be associated with different differential operators.

Example 3.3.9. Let us consider the fibre bundle Y = R2 -' R coordinatized by (x, y) and the fibre bundle E = R3 R coordinatized by (x, 5, z-). Let

yo4'=y=+x,

z'o4=y=+x

(3.3.27)

be a first order E-valued differential operator. It is easily checked that this differential operator has constant rank equal to 2. Its kernel with respect to the zero section of E is the imbedded submanifold of J'Y given by the equation 4E = {(x, y, y=) E R3 ; y= + x = 0}.

(3.3.28)

This is a first order differential equation on the fibre bundle Y which is associated with the differential operator (3.3.27). This differential equation is also associated with the differential operator

yo4/=y(y:+x),

zoo=y:+x.

(3.3.29)

The kernel of 4' with respect to the zero section of the fibre bundle E coincides with that of the operator (3.3.27). At the same time, the rank of the operator 4' (3.3.29) fails to be constant. This equals 2 at points of the submanifold (3.3.28) and equals 3 at others. From now on, we will consider only those differential operators taking their values

into a vector bundle E X. In this case, one usually considers the kernel Kerb of 4, denoted simply by Ker 4, with respect to the global zero section 0 of the vector

bundle E -, X. If Kert is a fibred submanifold of JkY, we have the differential equation REm = Ker 4

CHAPTER 3. LAGRANCIAN FORMALISM

120

given locally by coordinate relations

E°(?,yE')=0,

a=1,...,dimE-dim X.

(3.3.30)

Remark 3.3.10. One should distinguish the system of equations (3.3.1) from that of equations (3.3.30). The first is written with respect to local coordinates on the domain JkY of a differential operator, whereas the latter is seen with respect to bundle coordinates on its codomain E. Let us consider Euler-Lagrange equations associated with Euler-Lagrange operators. Given a k-order Euler-Lagrange operator (3.3.16), if its kernel is a fibred submanifold of the jet manifold J"Y, we have the system of k-order Euler-Lagrunge equations written in the coordinate form (3.3.30) as I

(-1)JAidA(8"G) = 0.

In particular, let Ey be the second order Euler-Lagrange operator (3.2.10) associated with a Lagrangian density L. Then the corresponding system of the second order differential Euler-Lagrange equations is a fibred submanifold of J2Y -, X given by the coordinate relations

(8;-da8;)G=0.

(3.3.31)

Classical solutions of these equations are critical sections of the variational problem for the Lagrangian density L, and they satisfy the Euler-Lagrange equations (3.2.14).

Remark 3.3.11. Strictly speaking, not every Euler-Lagrange operator leads to differential Euler-Lagrange equations. Rom now on, we will assume that EulerLagrange equations and other equations which we deal with within the framework of Lagrangian and Hamiltonian formalisms are differential equations in accordance with Definition 3.3.1. We will return to these questions in Section 5.7 where the formal integrability of Yang-Mills equations is examined. An important advantage of Euler-Lagrange equations should be emphasized. In accordance with Propositions 3.3.10 and 3.3.11, these equations inherit gauge symmetries of a Lagrangian density, while the first variational formula (3.2.11) enables us to examine differential conservation laws on solutions of these equations (see Section 3.5).

3.4. LAGRANCIAN POLYSYMPLECTIC STRUCTURES

121

Lagrangian polysymplectic structures

3.4

The Cartan equations (3.2.17a) - (3.2.17b) associated with the Euler-LagrangeCartan operator (3.4.10), but not the Euler-Lagrange ones are the Lagrangian counterpart of the Hamilton equations in Hamiltonian field theory. Let us consider a polysymplectic structure on the configuration space J'Y of first order Lagrangian formalism which leads to these differential equations and this differential operator. This structure fails to be canonical, but depends on the choice of a Lagrangian density. Let

L:J'Y-'A"T'X be a first order Lagrangian density on J'Y. Let us consider the vertical tangent map V L to L. Since J'Y - ' Y is an affine bundle, V L yields the linear morphism

x(nT'X)

J'Y x(T'X

over J'Y and the corresponding morphism L : J1Y

V'Y ®(AT*X) ®TX

(3.4.1)

over Y. DEFINITION 3.4.1. The fibre bundle

R=V'Y®(A"T'X)®TX V'YY(i'T'X)

(3.4.2)

over Y is called the Legendre bundle. 0 Given fibred coordinates (xa, y`) of Y, the Legendre bundle (3.4.2) is provided with the induced coordinates (x", y`, p;), where the coordinates p, have the transition functions

p' = det(8x')

8sµ P;

(3.4.3)

In these coordinates, the morphism (3.4.1) reads

(x-',y',Pi) 0 L = V,1/',v; )

(3.4.4)

CHAPTER 3. LAGRANGIAN FORMALISM

122

It is called the Legendre map associated with the Lagrangian density L.

Remark 3.4.1. A Lagrangian density L is said to be regular if the associated Legendre map L is a local difeomorphism, that is,

det('O'G) # 0. A Lagrangian density L is called hyperregular if L is a diffeomorphism. The Legendre bundle H plays the role of a finite-dimensional phase space of fields represented by sections of the fibred manifold Y X (see next Chapter). The basic property of II is that it is provided with the canonical tangent-valued Liouville form which is given by the canonical monomorphism

e: H-°X'T'Y®TX,

(3.4.5)

e=-p,dy'AW®8a. Remark 3.4.2. The subtle point is that the coordinate expression p; dy' AWa

could not define any global exterior form on n because TX is not a subbundle of TY. It should be recalled that the exterior differential d cannot be applied to tangentvalued forms like (3.4.5).

DEFINITION 3.4.2. There is a unique TX-valued (n + 2)-form f2 on Ii such that the relation f2J46

= -d(ej()

holds for any exterior 1-form o on X. This form is called the polysymplectic form. It is given by the coordinate expression

fI=dp; Ady'AW®8a.

(3.4.6)

0 The tangent-valued Liouville form (3.4.5) and the polysymplectic form (3.4.6) provide the Legendre bundle n with the canonical polysymplectic structure.

3.4. LAGRANGIAN POLYSYMPLECTIC STRUCTURES

123

The pull-backs of these forms by the Legendre map L equip the jet manifold J'Y with the polysymplectic structure associated with the Lagrangian density L as follows.

We have the tangent-valued Liouville form

BL = Z'9: J'Y -, ''T'Y®TX, Y 1BL = -7r;`dy' A w ®8,,,

and the polysymplectic form associated with the Lagrangian density L

fIL = L'S2 : J'Y - "A2T'J'Y ®TX, Y

I Sl, = d7r, A dy$ A w ®8,,.

(3.4.7)

Contracting BL with the canonical form ) (2.1.19), we obtain the exterior Legendre form on P Y 3L = AJOL : J'Y - i AT'Y,

which is associated with the Lagrangian density L. Then the Poincare-Cartan form (3.2.6) associated with L is recovered as

HL=9L+L=ir,dy'Awa-7{Lw,

xL=7riyi,-G.

(3.4.8)

Applying the exterior differential to the Poincare-Cartan form HL and bearing in mind that dHL = 0, we find that dHL = EL + dv#L,

EL=diA,+dvL, where EL is the second order Euler-Lagrange operator (3.2.10) associated with L.

Let J'J'Y be the repeated jet manifold of Y - X with coordinates (xa, y41 ya, gAI YAA), and

r=dx"®(ea+f 8;+k8 )

CHAPTER 3. LAGRANGIAN FORMALISM

124

a second order connection (2.6.1) on Y -. X. DEFINITION 3.4.3. A second order connection £ is said to be a Lagrangian connec-

tion for the Iagrangian density L if it obeys the condition f rj nL = dHL,

(3.4.9)

where HL is the PoincarF-Cartan form associated with L. 0 Using the coordinate expressions (3.4.7) and (3.4.8), we find that (3.4.9) is equiv-

alent to (Da - y.%)

7r, = 0,

AL - Ban; - i 81r; - Nk,e; a; +

y;)B;,r, = 0.

In order to clarify the meaning of (3.4.9), let us consider the following first order Lagrangian density on the repeated jet manifold J1J'Y:

=L-S,JOL:J'J'Y- A"T'X, C+(va-ylx)lri, where S1 is given by (2.2.5). Its associated Euler-Lagrange operator reads

SE : J'J'Y -+T'J'Y A(A" 7- X), EZ = [(81L -dart; + 8,1r1'(

-

B; ar (J#,,

- yj,)dya) n

(3.4.10)

+y;,8;. It is readily observed that the condition (3.4.9) is equivalent to the one

Imf C KerS1 or CEo£=0. Ez, called the Euler-Lagrange-Cartan operator associated with the Lagrangian density L, leads to the first order differential equations

0- =KerSZCJ'J'Y on the fibred jet manifold J1Y -+ X. These are the Cartan equations at, rip (

- ytµ) = 0,

AL - darr; + (j - ya)8{7r." = 0.

(3.4.11a)

(3.4.11b)

3.4. LACRANCIAN POLYSYMPLECTIC STRUCTURES

125

Their classical solutions -9 : X - J'Y are solutions of the variational Cartan equations (3.2.16). Remark 3.4.3. The Euler-Lagrange-Cartan operator £L is the Lagrangian counterpart of the Hamilton operator within the framework of polysymplectic Hamiltonian formalism on fibred manifolds (see the relation (4.5.16)). In particular, we will show that a Lagrangian connection always exists for a hyperregular Lagrangian density as well as for a semiregular Lagrangian density which admits a weakly associated Hamiltonian form (see Remark 4.5.8).

Let us write the Euler-Lagrange-Cartan operator £E as an exterior form on the repeated jet manifold J'J'Y. We have

£i; = d(ai,HL) - AJIiISIL where A is now the affine monomorphism over J'Y:

A : J'J'Y

T'X ® TJ'Y, JLY

A=dx"®(8a+rAOi+yaµar) The restriction of £L to the sesquiholonomic jet manifold Y2Y defines the first order Euler-Lagrange operator EL : JAY JYT'Y

A (AnT'Y),

lEL=(8;-dAO;)G6'nW,

(3.4.12)

dA = 8a + y68. + y;,

This, in turn, is restricted to the second order Euler-Lagrange operator £c associated with a Lagrangian density L, as is illustrated by the diagram

J'J'Y L T'J'Y A (AT'X) J2Y

T'Y A (A'T'X) 11

1

J'Y -` T'YA(A'T'X)

CHAPTER 3. LAGRANCIAN FORMALISM

126

Strictly speaking, the first order Euler-Lagrange operator is not a differential operator. At the same time, its kernel

KerEtCJ1YCJ1J1Y defines a system of first order differential equations given by the coordinate expressions

ya - ya = 01

(3.4.13a) (3.4.13b)

which are called the first order Euler-Lagrange equations. As an immediate consequence of Remark 2.2.1, the first order Euler-Lagrange equations (3.4.13a) - (3.4.13b) are equivalent to the second order Euler-Lagrange equations (3.3.31), and represent the familiar first order reduction of the second order Euler-Lagrange equations. It is easily seen that the first order Euler-Lagrange equations (3.4.13a) - (3.4.13b)

(and consequently the second order ones (3.3.31)) are equivalent to the Cartan equations (3.4.11a) - (3.4.11b) on the integrable sections of J'Y -+ X. They are completely equivalent to the Cartan equations in the case of a regular Lagrangian density.

Remark 3.4.4. Hamilton-De Donder formalism. By definition, the PoincareCartan form HL (3.4.8) yields the fibred morphism HL over Y of the jet bundle J1Y -. Y to the fibre bundle

Zy =T'Y A(fl'T'X).

(3.4.14)

This is termed the Legendre morphism associated with HL. A glance at (3.4.2) and (3.4.14) shows that there is the exact sequence of fibre bundles

0-eHxnT'XtiZy -II-O, x where

azn:Zy - II is a 1-dimensional affine bundle modelled on the pull-back

HxXT'XIl. x

(3.4.15)

3.4. LAGRANCIAN POLYSYMPLECTIC STRUCTURES

127

The fibre bundle (3.4.14) is endowed with the induced coordinates (x' , y', pp , p), where the coordinate p has the transformation law

8x'

pr = det(ax,,.)(P -

GV W avi &N1j).

(3.4.16)

Relative to these coordinates, the morphisms NL and azn read

(p: , P) o HL = (W ;', G - r;' yµ),

(3.4.17)

Pi'oazn=pi. Then we have the useful relation

I L=irZ4oHL

(3.4.18)

between the Legendre map L and the Legendre morphism HL. The fibre bundle Zy (3.4.14) is equipped with the canonical exterior n-form given by the coordinate expression

IS=pw+p;dy'Awa

(3.4.19)

[31, 77] (see Example 4.2.2). It is readily observed that the Poincare-Cartan form HL is the pull-back of the canonical form S (3.4.19) by the associated Legendre morphism (3.4.17). The canonical form (3.4.19) provides the fibre bundle (3.4.14) with the multisymplectic structure characterized by the multisymplectic form

Oz = dE = dp; A dy' n w + dp n w.

(3.4.20)

In addition to the Legendre bundle II, the fibre bundle Zy is a possible candidate for a finite-dimensional phase space of fields represented by sections of the fibred manifold Y - X. We call it the homogeneous Legendre bundle. Hamiltonian formalism founded on Zy is Hamilton-De Donder formalism [4, 41, 54, 73, 77, 112]. Given a Lagrangian density L, let the image ZL of the configuration space J'Y by the Legendre morphism (3.4.17) be an imbedded subbundle iL : ZL '-+ Zy

of the fibre bundle Zy -e Y. It is provided with the pull-back De Donder form

oL = iL.

CHAPTER 3. LACRANGIAN FORMALISM

128

By analogy with the Cartan equations (3.2.16), the corresponding Hamilton-De Donder equations for sections r of the fibred manifold ZL - X are written as

r (uJdrt) = 0,

(3.4.21)

where u is an arbitrary vertical vector field on ZL -i X. To obtain an explicit form of the Hamilton-De Donder equations (3.4.21), one should substitute solutions ya(xi`, y`, pi l), C(x &, y`, p{ , p) of the equations

pi = a; (x', y', yµ), p = £(x'', y', v") - x. (x'', ,', i )ya

(3.4.22)

in the Cartan equations (3.2.16). If a Lagrangian density is regular, the equations (3.4.22) have a unique solution and the Hamilton-De Donder equations take the coordinate form

and are equivalent to the Cartan equations. If a Lagrangian density is degenerate, the equations (3.4.22) may admit different solutions or no solution at all. More can be said in the following case. PROPOSITION 3.4.4. Let the Legendre morphism HL : J'Y --4 ZL be a submersion.

Then a section 3 of J'Y X is a solution of the Cartan equations (3.2.16) if and only if HL o li is a solution of the Hamilton-De Donder equations (3.4.21) [77). 0

3.5

Lagrangian conservation laws

The first variational formula (3.2.13) provides us with the standard procedure for discovering differential conservation laws in field models. The main formulas are (3.5.15), (3.5.17) and (3.5.25). By a differential conservation law in field theory on a fibred manifold Y - X is meant a relation d(s'T) = 0

(3.5.1)

3.5. LACRANGIAN CONSERVATION LAWS

129

where T = TAwa is a horizontal (n - 1)-form (a current) on the configuration space

J'Y

andsisasectionofY -X.

The relation (3.5.1) is called a strong conservation law if it holds for all sections s of the fibred manifold Y -. X, whereas (3.5.1) is said to be a weak conservation law if it takes place only on solutions of field equations. I The symbol ";z:;" stands for weak identities. I

The first variational formula leads to a conservation law when a Lagrangian density is invariant under 1-parameter groups of gauge transformations.

Remark 3.5.1. Gauge transformations. In field theory on fibred manifolds, by an active gauge transformation is meant a fibred automorphism 4) (1.2.4) of a fibred manifold Y - X over a diffeomorphism / of its base X. Transformations X are called passive gauge transformations. of atlases of fibred coordinates of Y The relation between active and passive gauge transformation is the following. Let 4) be a fibred automorphism of Y -' X. Given an atlas _ {(U0,00)} of fibred coordinates on Y, there exists the atlas

1Y'={U;=-(U.), 0Q=too4 1}

(3.5.2)

of fibred coordinates on Y such that the fibred coordinates of points 4)(y) with respect to the atlas lP' (3.5.2) are those of points y with respect to the atlas 'P. It follows that the invariance under passive gauge transformations implies that under active gauge transformations.

I Unless otherwise stated, by gauge transformations are meant active gauge transformations. Z We will use the following notation:

AUT (Y) for the group of all fibred automorphisms of a fibred manifold Y X,

Aut (Y) for the group of fibred automorphisms of Y -' X over X, Diff (X) for the group of diffeomorphisms of X.

CHAPTER 3. LACRANCIAN FORMALISM

130

Automorphisms of Y - X over X are said to be vertical gauge transformations, whereas elements of AUT (Y) are called general gauge transformations. Obviously,

the group Aut (Y) of vertical gauge transformations is the kernel of the natural homomorphism AUT (Y) -' Diff (X). At the same time, a diffeomorphism f E Diff (X) of X does not necessarily give

rise to a fibred morphism of Y -' X. When that happens, we call such a lift of f E Diff (X) onto Y the horizontal lift. Example 3.5.2. Every diffeomorphism f E Diff (X) gives rise to the tangent automorphism T f E AUT (TX) of the tangent bundle TX of X and to the corresponding automorphisms of tensor bundles.

Example 3.5.3. Let us consider the fibred product Y X Y'

(3.5.3)

of fibred manifolds Y and Y' over the same base X. We have the inclusion

Aut (Y) x Aut (Y') - Aut (Y x Y) X

of groups of vertical gauge transformations. At the same time, a pair (4', 4'), 4> E AUT (Y), 4i' E AUT (Y'), is a general gauge transformation of the product (3.5.3) if 0 and V cover the same diffeomorphism of X. To discover differential conservation laws, let us consider only 1-parameter groups of gauge transformations.

1 That means we consider gauge transformations homotopic to the identity mor-

phism. 1 Recall that every 1-parameter group of diffeomorphisms of a manifold M defines

a complete vector u field on M, and vice versa. Moreover, one can think of any vector field on M as being the generator of a local 1-parameter group of local diffeomorphisms of the manifold M (see Remark 1.5.2). Accordingly, every 1-parameter gauge group [4's] C AUT (Y) of automorphisms of Y -+ X yields a projectable vector field u = UAOA + u`8;

(3.5.4)

3.5. LAGRANGIAN CONSERVATION LAWS

131

on Y -+ X, which is the generator of [40_]. It projects onto the vector field r =

08" on X which is the generator of the 1-parameter group [jfJ C Diff (X) of diffeomorphisms of X. Conversely, every projectable vector field (3.5.4) on a fibred

manifold Y -+ X is the generator of a local 1-parameter gauge group of local automorphisms of Y .

In particular, a vertical vector field u on Y -. X is the generator of a local 1-parameter group of local vertical automorphisms of Y - X.

Remark 3.5.4. Given a connection

on a fibred manifold Y -' X, every vector field r = TAB,, on X gives rise to the horizontal vector field

Tr=TJr=r"(ea+r'O1)

(3.5.5)

on Y (see (2.3.10)). Accordingly, the local 1-parameter group of local diffeomorphisms of X generated by the vector field r gives rise to the local 1-parameter group of local automorphisms of Y - X generated by the vector field (3.5.5).

Let u be a projectable vector field on a fibred manifold Y -. X and J'u = u" 8A + u'8; + (daub -

its jet prolongation (2.1.16) on J'Y. This is the generator of the first order jet prolongation J14 of the local 1-parameter gauge group [4itJ generated by u. Then the Lie derivative Lj,,,L of a Lagrangian density L on J'Y along Pu (or simply along u) is equal to zero if and only if L is invariant under the above-mentioned 1-parameter group of gauge transformations, that is,

J'4 L = L,

Vt E (-E, E).

Such gauge transformations are called invariant transformations. We have [BauaG + (u-%O.% + u'8; + (daub

(3.5.6)

The first variational formula (3.2.11) provides the canonical decomposition of the Lie derivative (3.5.6) in accordance with the variational task. In coordinates,

CHAPTER 3. LACRANGIAN FORMALISM

132

this decomposition reads

8auAG+[uAOA+u`8;+(d.,u`-y;,8Au")8;]G= (u' - y,,u")(8; - dab; )G dA[ir (u"y'" - u') - uaG - d"(ci (y,,u° - tO))],

(3.5.7)

where

b;G = (8; -

)G

(3.5.8)

are the variational derivatives (3.2.10), while the skew-symmetric functions c,"" correspond to different Lepagian equivalents (3.2.7) of the Lagrangian density L. On-shell, i.e., on the kernel (3.3.31) of the Euler-Lagrange operator CL, biG = (8, - dab; )G = 0,

(3.5.9)

the first variational formula (3.5.7) leads to the weak identity Lj1,.L

(3.5.10)

OAuAG + [u-,9.% + u'8; + (d,,u` - y' 8Au")8; ]G

-da[rr{ (u"yµ

- u') - uaG - d"(d '(y..u° - u'))],

where To = TpWX = -ho(J'uJPL),

is said to be the symmetry current along the vector field u. This current corresponds to the Lepagian equivalent PL of L. We say that (3.5.10) is a weak identity associated with the vector field u. If the Lie derivative Lj,,,L (3.5.6) vanishes, i.e., the strong equality 0

takes place, the weak identity (3.5.10) becomes the weak conservation law 0

dyho(J'u1 PL),

0

-da[1r (u"y" - u{) - u''G - d"(c'`"(y,,u° - u'))],

of the symmetry current 't (3.5.11) along the vector field u.

(3.5.12)

3.5. LAGRANCIAN CONSERVATION LAWS

133

The weak identity (3.5.12) leads to the differential conservation law

0 = -d(s'To) on solutions s of the Euler-Lagrange equations (3.2.14). This differential conservation law implies the integral conservation law

f s'Tpwa = 0,

(3.5.13)

8N

where N is a compact n-dimensional submanifold of X with the boundary ON. In view of the strong identity (3.2.12), we further limit our consideration to the Poincar6-Cartan form p,, = HL (3.2.6). In this case, the first variational formula (3.5.7) reads

8auAf- + [ua8a + u8; + (dau' - V 8au")8; ]L =

(3.5.14)

(u'-yu")(8;-dab;)G-da(a;(u"yµ-u')-u''G]. This identity leads on-shell to the weak identity

Lji L

dyho(uJ HL),

(3.5.15)

0.%u%,C +[u''8a+u'8;+(dau'-

-da[ii(u"y"-u')-u"L], where

T = Taws = -ho(uJ HL),

(3.5.16)

TA =ff:(u"y' -u')-u.'G, M is the symmetry current along the vector field u, which corresponds to the Poincar6Cartan form. If the Lie derivative Ljt4L vanishes, we have the weak conservation law 0 -_ dyho(uJ HL),

(3.5.17)

10 -_ -dA(,r (u"y,', - u') - UAL]

of the symmetry current T (3.5.16) along the vector field u.

Remark 3.5.5. Superpotential. Note that the symmetry current Tp in the weak identity (3.5.10) is not uniquely defined, but with the accuracy of an arbitrary dijclosed form. At the same time, its dependence on the vector fields u corresponding

CHAPTER 3. LAGRANCIAN FORMALISM

134

to different gauge transformations should be the same. The currents Tp (3.5.11) associated with different Lepagian equivalents possess this property. Indeed, they differ from each other in the dy-closed form

a = -e(y.,u"

-ho(J'ujpo) = dHa,

- u')wA

where po is a Lepagian equivalent (3.2.5) of the zero Lagrangian density. This difference may be significant if one examines integral conservation laws, where different Lepagian equivalents lead to different fluxes through a 2-codimensional surface.

It may happen that a symmetry current T- can be put into the form

= W + d1U = (WA + dU"

(3.5.18)

where the term W contains only variational derivatives (3.5.8), i.e., W .:s 0 and

U = U"`jA: J'Y

"A2T'X

is a horizontal (n - 2)-form on J'Y -+ X. Then one says that T reduces to the superpotential U [52, 67, 165).

It should be emphasized that, in this case, a symmetry current assumes the dH-exact form (3.5.18) on-shell only. Moreover, the equality

T-dHU=W(b;G)=0

(3.5.19)

is a combination of the Euler-Lagrange equations b;G = 0. If a symmetry current T reduces to a superpotential, the integral conservation law (3.5.13) becomes tautological. At the same time, the superpotential form (3.5.18) of T implies the following integral relation

f s'TAWA = N^-'

f s'U'

1,,a,

(3.5.20)

ON-1

where Nn"' is a compact oriented (n - 1)-dimensional submanifold of X with the

boundary 8N°-'. One can think of this relation as being a part of the EulerLagrange equations written in an integral form. Superpotentials are met with in gauge theory and in gravitation theory, where generators of gauge transformations depend on derivatives of gauge parameters.

Example 3.5.6. Let us consider conservation laws for Lagrangian densities which have the same Euler-Lagrange operator. It should be recalled that they differ from

3.5. LAGRANGIAN CONSERVATION LAWS

135

each other in variationally trivial Lagrangian densities. Let L and L' be different Lagrangian densities which lead to the same Euler-Lagrange operator EL = EL-. By Proposition 3.2.2, the first variational formula (3.5.14) and the weak identity (3.5.15) for L' differ from those for L in the strong identity dyho(uje)

(3.5.21)

where c is a closed n-form on Y. If the Lagrangian densities L and L' have the same symmetries, i.e., Lp ho(e) = 0,

the contribution -u j e of the strong identity (3.5.21) to the symmetry current (3.5.17) is not tautological (see Example 3.6.10 on topological field theories).

Remark 3.5.7. One distinguishes between invariant transformations and generalized invariant transformations [13, 110, 1741. An automorphism $ of a fibred manifold Y X is called a generalized invariant transformation if 4) preserves the EulerLagrange operator EL, but not necessarily the Lagrangian density L. For instance, gauge transformations in the Yang-Mills gauge theory are invariant transformations, whereas they are generalized invariant transformations in the Chern-Simons gauge model (see Example 3.6.9). Let us consider conservation laws in the case of generalized invariant transformations. Let L be a Lagrangian density and EL the associated Euler-Lagrange

operator. Let u be a projectable vector field on Y -e X, which is the generator of a local 1-parameter group of generalized invariant transformations, that is, L.nuEL = 0.

Then, in accordance with (3.3.26), we have ho(e),

where a is a closed n-form on the fibred manifold Y. In this case, the weak transformation law (3.5.15) reads ho(c) _- duho(uj HL),

CHAPTER 3. LAGRANCIAN FORMALISM

136

In particular, if e = da is an exact form, we obtain the weak conservation law 0

dgho(uJ HL - a).

(3.5.22)

Remark 3.5.8. Background fields. Background fields, which do not live in the dynamic shell (3.5.9), break conservation laws as follows. Let us consider the product

Yto,,=YxY'

(3.5.23)

of a fibred manifold Y with coordinates whose sections are dynamic fields, and a fibred manifold Y' with coordinates (x, yA), whose sections are background fields which take the background values YB

=.0B(x),

Y AB = Ba0B(x)

A Lagrangian density L is defined on the total configuration space J'Y,,,,,. Let u be a projectable vector field on Y,a which projects also onto Y' because gauge transformations of background fields do not depend on the dynamic ones. This vector field takes the coordinate form u = u"(x)8a + uA(x", yB)8A + u'(x", yB, yi)8,.

(3.5.24)

Substitution of (3.5.24) in (3.5.14) leads to the first variational formula in the presence of background fields A+ 8au"L+[uAOA+UABA+u'8 (dau' - y'"8au")8; ]L = (u" - ua)8AL + 7r-dA(uA - y' u") + (u` - y'auA)h,L - dAIir (u"yµ - u`) - uAL].

Then the following weak identity BAUAL + [uA8. + uAOA + u8 + (dauA -

(dau' - y"8au")8i ]C - (uA - ya ua)8AL + 1r-da(uA - yN u") -

dA[ir (u"y,, - u') - u"L) holds on the shell (3.5.9).

3.5. LACRANGIAN CONSERVATION LAWS

137

By construction, total Lagrangian densities L are usually invariant under gauge transformations of the product (3.5.23). In this case, we obtain the weak identity in the presence of background fields (uA - yµuµ)8AG + 7r da(uA - yµ uµ)

dA[7ri (uµyµ

(3.5.25)

Remark 3.5.9. The weak identity (3.5.25) can also be applied when the dynamic equations are not the Lagrangian ones, but are given by the coordinate expressions

(8,-da6;)G+F,=0, where F; are local functions on J'Y. It reads F,

-4 [1r (uµyµ - u') - uAG].

1 The weak identity (3.5.15), the weak conservation law (3.5.17) and the weak identity in the presence of background fields (3.5.25) are the basic ingredients in our analysis of differential conservation laws in classical field theory. I

It is easy to see that the weak identity (3.5.15) is linear in the vector field u. Therefore, one can consider superposition of weak identities (3.5.15) associated with different vector fields.

For instance, if u and u' are projectable vector fields on Y projecting onto the same vector field on X, the difference of the corresponding weak identities (3.5.15) results in the weak identity (3.5.15) associated with the vertical vector field u - u'. X, which projects Every projectable vector field u on a fibred manifold Y onto a vector field r on X, can be written as the sum

of a liftr of 7- over Y and a vertical vector field 0 on Y. It follows that the weak identity (3.5.15) associated with a projectable vector field u can be represented as the superposition of those associated with f and t9.

CHAPTER 3. LAGRANGIAN FORMALISM

138

In the case of a vertical vector field t9 = 6'8 on Y -, X, the weak identity (3.5.15) takes the form [08, + dafl'8; ]G su da(tr{,9 ).

(3.5.26)

In field theory, vertical gauge transformations describe internal symmetries. If a Lagrangian density is invariant under internal symmetries, we have the Not her conservation law 10

da(7ri t9')

of the Nother current T = -t91 HL,

(3.5.27)

The well-known example of a Nother conservation law is that in gauge theory of principal connections (see next Section). 1 Note that the Nother current (3.5.27) differs in the minus sign from the familiar

one in the physics literature. I A vector field r on X can be lifted onto the total space Y of a fibred manifold Y --+ X by means of a connection I' on Y. This lift is the horizontal vector field rr (3.5.5).

The weak identity (3.5.15) associated with the horizontal vector field rr takes the form I'',,) + (da (rµ1",,) 8Mr'`G + [r"8µ + r"18;

-dal7r4 r"(yµ -

It

(3.5.28)

- 6Nr"GJ.

The corresponding symmetry current (3.5.16) along zr reads

Tr = -ho(rr1 HL), I`fir = r"(tr'(yµ - I"µ) - b0, G).

This current is called the stress-energy-momentum (SEM) current relative to the connection 1' [55, 79, 67, 165).

3.5. LAGRANCIAN CONSERVATION LAWS

139

To discover SEM conservation laws, one may choose different connections on Y - X (e.g., different connections for different vector fields r on X or different connections for different solutions of the Euler-Lagrange equations). It is readily seen that the SEM currents relative to different connections r and C differ from each other in the Nother current (3.5.27) along the vertical vector field 0 = l" (110 - 1 µ)8j.

Example 3.5.10. Let all vector fields r on X be lifted onto Y by means of the same connection r on Y - X. The weak identity (3.5.28) can be rewritten as follows

rO{[8µ+r'µ8j+(8a1"µ+

BSI"µ)a'']G-da[7r;(I''µ-F1µ)+6 L]}

0.

Since this relation takes place for the arbitrary vector field r on X, it is equivalent to the system of weak equalities

dATr'v = M, - y)bjG

(8µ + r,,8j +

0,

where 1r'µ are components of the tensor

Tr = -ho(r1 HL) ='r'µdx" ®WA,

I TrAµ-rj(8µ-rµ)-bµG. This is called the SEM tensor relative to the connection r.

Example 3.5.11. If we choose the local trivial connection (I o) = 0, the SEM weak identity (3.5.28) takes the form 8µG TOa

-djo'µ, aj

(3.5.29) G,

where To-;, is the canonical energy-momentum tensor. Though it is not a true tensor, the weak identity (3.5.30) on solutions s of the Euler-Lagrange equations is well defined: ,C 0

µ8) + (3.5.30) 8(8 8t' (To-%j, O s) 0. This results from the SEM weak identity (3.5.28) when, for every solution s, we choose the connection r which has s as its integral section, i.e.,

r'NOs=8µs'.

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One can say that the weak identity (3.5.30) does not contain any Nother current. Note that, in gravitation theory on natural bundles T X (e.g., tensor bundles), we have the canonical horizontal lift of vector fields on X onto T (see Section 3.7).

This lift, under certain conditions, can be represented locally as a horizontal lift (3.5.5). The corresponding symmetry current is also called a SEM current.

3.6 Conservation laws in gauge theory An extensive literature is devoted to gauge theory (see [131] and references therein).

In this book (see Sections 2.7 and 2.8), we give a brief outline of the geometric basics of gauge theory of principal connections. Section 5.7 is devoted to formal integrability of the Yang-Mills equations, while in this Section, we will study Nother conservation laws and SEM conservation laws in gauge theory.

Let P -. X be a principal bundle with a structure Lie group G (or simply a G-principal bundle. In a gauge model with a symmetry group G, gauge potentials are identified with principal connections on the principal bundle P, while matter fields are represented by sections of fibre bundles associated with P. Here we will consider only gauge theory with unbroken symmetries. The reader is referred to Remarks 3.8.4 and 3.8.13 for the mathematical picture of spontaneous symmetry breaking. A principal connection on a principal bundle P -i X is defined as an equivariant connection A on P (see relation (2.7.9)). Hence, there is one-to-one correspondence between the principal connections on P -' X and the global sections of the bundle of principal connections C - X (2.7.5). Given a bundle atlas 'I" of P, this fibre bundle is provided with the bundle coordinates (xa,a' such that, for every section A of C,

I aµoA=Aµ are the coefficients of the local connection 1-form A on X (2.7.10) with respect to

the atlas V. The first order jet manifold J'C of the fibre bundle C is equipped with coordinates (z A, aa, aµa). Let

Y = (P x V)/G

(3.6.1)

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141

be a P-associated fibre bundle (2.7.12) with a standard fibre V on which the structure group G of P acts on the left. By (p] we denote restriction of the canonical morphism

P x V - (P x V)/G to {p} x V. Then, by definition of Y, we have

[PI(v) = (P9](9-,v) For the sake of convenience, we will write

vEV. Remark 3.6.1. In fact, Y (3.6.1) is the fibre bundle canonically associated with the principal bundle P. A fibre bundle Y -. X, given by the triple (X, V, 'I') of a base X, a typical fibre V and a bundle atlas 'P, is called a fibre bundle with a structure group C if C acts effectively on V on the left and the transition functions p>,p (1.2.2) of the atlas 'V take their values into the group C. Fibre bundles (X, V, C,'1') and (X, V', C, 4") with the same structure group C, which may have different typical fibres, are called

associated if the transition functions {pp} and {p'',,,} of the atlases 'Y and 'V', respectively, belong to the same element of the oohomology group H' (X; C) (see Remark 3.8.1). Any two associated fibre bundles with the same typical fibre are isomorphic to each other ([93], p.41), but their isomorphism is not canonical in general. A fibre bundle Y - X with a structure group C is associated with a C-principal bundle P -+ X. If Y is canonically associated with P as in (3.6.1), then

every atlas T p = {(Q., za) } of P canonically determines the associated atlas of Y

'I' = {(U0,0o(x) = [za(x)]-'); every automorphism of a principal bundle P yields the corresponding auto. morphism (3.6.3) of the P-associated fibre bundle (3.6.1). Unless otherwise stated, by a P-associated fibre bundle we mean the quotient (3.6.1).

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Every principal connection A on P - X yields the associated principal connection (2.7.13) on the associated vector bundle Y - X. This connection reads

A=dx''®(8a+Aa p8,). In physical applications, a P-associated fibre bundle is called a matter bundle. The total configuration space of a gauge model with unbroken symmetries is the product

JIytct=J'YXJ'C.

(3.6.2)

In gauge theory, several classes of gauge transformations are considered [131, 151,

168). By a gauge transformation of a principal bundle P is meant its automorphism d'p which is equivariant under the canonical action (2.7.1), that is, the diagram

P- R9 P +P I

I

P

P-+P R,

Ryo4'p_'PpoR., commutes for each 9 E C. This is called a general principal automorphism of P. Every general principal automorphism of P yields the corresponding automorphisms

pEP,

vEV,

(3.6.3)

of the P-associated bundle Y (3.6.1). For the sake of brevity, we will write

Oy : (P x V)/C -+ (4'p(P) x V)/C. General principal automorphisms 0 of the principal bundle P determine also the corresponding automorphisms

J'P/G

J'4p(J'P)/G

(3.6.4)

of the bundle of principal connections C 199].

'Ib obtain the Nother conservation laws, we will consider only vertical automorphisms of the principal bundle P, which are called principal automorphisms, or simply gauge transformations if there is no danger of confusion.

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143

Remark 3.6.2. Every principal automorphism of a principal bundle P is represented as +P(P) = PI (P),

PEI',

(3.6.5)

where f is a G-valued equivariant function on P, i.e.,

f (P9) = 9-'I(P)9,

Vg E C.

(3.6.6)

There is one-to-one correspondence between the functions f (3.6.6) and the global sections s of the group bundle

Pc = (P X G)/G,

(3.6.7)

whose typical fibre is the group C which acts on itself by the adjoint representation ([85], p.277). There is the canonical fibre-to-fibre action of the group bundle PG on any P-associated bundle Y:

PCxY -Y, x

((p, 9) - G, (P, v) . C) _+ (P, 9v) C,

dg E G, WE V.

Then, the above-mentioned correspondence is defined by the relation (s(7rpx (P), P) '-' PI (P)

It follows that principal automorphisms of a principal bundle P -. X with a structure group G form the group Gau(P) C Aut (P), called the gauge group, which is isomorphic to the group of global sections of the group bundle (3.6.7). A suitable Sobolev completion (see [3]) makes the gauge group into a Banach Lie group

\aai(P) ([131], p.151). Let £ be a C-invariant vertical vector field on a principal bundle P corresponding to a local 1-parameter group [+P] of principal automorphisms of P. We will call e a principal vector field. Recall one-to-one correspondence between the principal vector fields on P and the sections of the gauge algebra bundle VcP -, X (2.7.3). In the notation of Example 2.7.2, a principal vector field reads

{ = e'e,, where {ep} is the basis of the Lie algebra g.

(3.6.8)

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144

Remark 3.6.3. Global sections of the fibre bundle V0P - X form the infinitedimensional Lie algebra S(VcP), which is called the gauge Lie algebra A suitable Sobolev completion turns S(VcP) into the Banach Lie algebra of the gauge Banach

Lie group ti(P) of Remark 3.6.2. Therefore, one can think of the components P(x) of a principal vector field (3.6.8) as being gauge parameters. The principal vector fields (3.6.8) are transformed under the generators of gauge transformations by the adjoint representation given by the Lie bracket

e : t - [t', f] = 4,C iV ep,

C, e E S(V0P).

Accordingly, gauge parameters are changed by the coadjoint representation (3.6.9)

Given a principal vector field C (3.6.8) on P, the corresponding principal vector X, which corresponds to the (local) field on the P-associated fibre bundle Y 1-parameter group [-iPy] of principal automorphisms (3.6.3) of Y, reads

ev = t' y8i. Accordingly, the principal vector field on the bundle of principal connections C, which corresponds to the local 1-parameter group [fic] of principal automorphisms (3.6.4) of C, takes the form

cc = (8vf' + 46°,')8;

(3.6.10)

(see (2.8.14)).

Remark 3.6.4. Let us consider a local I-parameter group of general principal automorphisms [+p] of the principal bundle P - X whose generator is a projectable G-invariant vector field

t=rAOA+eep: X - T0P be the corresponding 1-parameter group of automor(see Example 2.7.2). Let X. The generator of [4'c] is the vector field phisms (3.6.4) of the fibre bundle C lc (2.8.12) on C which takes the coordinate form ICc = ra8., + (8,

+ c; aµCP - a,70;r'`)8; ,

(3.6.11)

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145

where fP are gauge parameters.

A principal vector field on the product C Y reads X

kvc = (8µC' + c9-PaµL )& + ?J,8 = (t48S + 74S )8A,

(3.6.12)

where the collective index A is used uy""8A = 6P8T ,

un 8A = c ,,aµ8; + p8{.

A Lagrangian density L on the configuration space (3.6.2) is said to be gaugeinvariant if the strong equality

Lj'frcL = 0 holds for every principal vector field C (3.6.8) [177]. In this case, the first variational formula (3.5.14) leads to the strong equality 0 = (t4CP + 7b"08,,e°)6AG + da1(up"CP + 71p"P8µ4P)aAA],

(3.6.13)

where SAG are the variational derivatives of L and dA = 8 + aPjµ8p + ya8i.

Due to the arbitrariness of gauge parameters C"(x), this equality is equivalent to the following system of strong equalities:

) = 0, u""bAL + dA(vp"aA) + t i = 0,

(3.6.14b)

4 A7rA + 4"7rA = 0.

(3.6.14c)

1"bAG + d,,(up"

(3.6.14a)

Remark 3.6.5. Substituting (3.6.14b) and (3.6.14c) in (3.6.14a), we obtain the well-known constraint conditions of the variational derivatives of a gauge-invariant Lagrangian density: 4bAL - d, (up""bAL) = 0.

On-shell, the first variational formula (3.6.13) leads to the weak conservation law

10 ft dA [(t4e + t4"8"P)ir ]

(3.6.15)

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146

of the Nother current (3.6.16)

Accordingly, the equalities (3.6.14a) - (3.6.14c) on-shell lead to the familiar Wither identities for a gauge invariant Lagrangian density L:

d,644A) -- 0,

(3.6.17a)

da(t4'`7rA) + U47rA -- 0,

(3.6.17b)

t4A7

+ t4'`7rA = 0.

(3.6.17c)

They are equivalent to the weak equality (3.6.15) due to the arbitrariness of the gauge parameters tP(x). A glance at the expressions (3.6.15) and (3.6.16) shows that the Nother conservation law and the Nother current both depend on gauge parameters. The weak identities (3.6.17a) - (3.6.17c) play the role of the necessary and sufficient conditions in order that the weak conservation law (3.6.15) be gauge-covariant, i.e., form-invariant under changing gauge parameters. This means that, if the equality (3.6.15) takes place for gauge parameters 1;, it does so for arbitrary deviations l; +b£

of C. Then the conservation law (3.6.15) is also covariant under gauge transformations, when gauge parameters are transformed by the coadjoint representation (3.6.9).

1 Thus, dependence of the Nother current on gauge parameters guarantees that the Nother conservation law is maintained under gauge transformations. 1

It is easily seen that the equalities (3.6.17a) - (3.6.17c) are not independent. In fact, (3.6.17a) is a consequence of (3.6.17b) and (3.6.17c). This reflects the fact that, in accordance with the strong equalities (3.6.14b) and (3.6.14c), the Nother current (3.6.16) is brought into the superpotential form (3.5.18): TA = et4 6AG - dµ(f°r+:'7rA) where the superpotential is Since a matter field Lagrangian does not depend on the derivative coordinates ate, the Not her superpotential (3.6.18)

3.6. CONSERVATION LAWS IN GAUGE THEORY

147

depends on the gauge potentials only. We have the corresponding integral relation (3.5.20), which reads

f s 9-'ma =

N-,

J s'(i)ww,,

(3.6.19)

aN-1

where N' ' is a compact oriented (n - 1)-dimensional submanifold of X with the boundary 8N"''. One can think of (3.6.19) as being the integral relation between the symmetry current (3.6.16) and the gauge field generated by this current. In the electromagnetic theory, the similar relation between an electric current and the electromagnetic field generated by this current is well known (see Example 3.6.7). In comparison with (3.6.19), this relation, however, is free from gauge parameters due to the peculiarity of Abelian gauge models. Remark 3.6.6. It should be emphasized that the superpotential form of the Wither current (3.6.16) is caused by the fact that principal vector fields (3.6.12) depend on derivatives of gauge parameters.

Example 3.6.7. Electromagnetic field. In gauge theory with an Abelian symmetry group G, one can take the Nother current and the Nother conservation law not depending on gauge parameters. Let us consider the electromagnetic theory, where

f'(y) = iy'.

a = U(1),

In this case, a gauge parameter t is not changed under gauge transformations as follows from the coadjoint representation law (3.6.9). Therefore, one can put, e.g., = 1. Then the Nother current (3.6.16) takes the form

T' = -u'7rA Since the group G is Abelian, this current (3.6.20) does not depend on gauge potentials and it is invariant under gauge transformations. We have TA = -iy1irf .

(3.6.20)

It is easy to see that T, under the sign change, is the familiar electric current of matter fields, while the Nother conservation law (3.6.15) is precisely the equation of continuity. The corresponding integral equation of continuity (3.5.13) reads

f ON

)wA = 0,

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148

where N is a compact n-dimensional submanifold of X with the boundary ON. Though the Nother current T (3.6.20) is expressed in the superpotential form

V = -b-,C + dUA, the equation of continuity is not tautological. This equation is independent of an electromagnetic field generated by the electric current (3.6.20) and it is therefore treated as the strong conservation law. When = 1, the electromagnetic superpotential (3.6.18) takes the form U'`"=W.A=---F"A,

where F is the electromagnetic strength. The corresponding equality (3.5.19) is precisely the system of Maxwell equations

d".F"A = iytir . Accordingly, the integral relation (3.6.19) is the integral form of the Maxwell equations. In particular, the well-known relation between the flux of an electric field through a closed surface and the total electric charge inside this surface is recovered.

Remark 3.6.8. Utiyama theorem. One can regard the strong equalities (3.6.14a) - (3.6.14c) as conditions of a Lagrangian density L to be gauge-invariant. Let us study these equations in the case of a Lagrangian density L : J1C -+ XT'X

(3.6.21)

for free gauge fields. Then the equations (3.6.14a) - (3.6.14c) read

Qa°,8, L + a',"8;"L) = 0,

(3.6.22a)

8q L + cmaa8;'L = 0,

(3.6.22b)

8P'L+8p L=0.

(3.6.22c)

Let us utilize the coordinates (aµ, Sx, Y ;x) (2.8.20), (2.8.22), which correspond to the canonical splitting (2.8.21) of the jet manifold J'C. With respect to these coordinates, the equation (3.6.22c) reads O= 0. "A

(3.6.23)

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149

Then the equation (3.6.22b) takes the form

aG

_ 0.

(3.6.24)

aaµ

A glance at the equations (3.6.23) and (3.6.24) shows that the gauge-invariant Lagrangian density (3.6.21) factorizes through the strength F of gauge potentials, i.e.,

J'C f-'

c\

C_

/r

n T+X

[25, 58]. Then the equation (3.6.22a) is written as

a

a" = 0,

which is the equivalent of gauge-invariance of the Lagrangian density Z. As a result, the conventional Yang-Mills Lagrangian density LYM of gauge potentials on the configuration space J'C in the presence of a background world metric

g on the base X reads LYM =

1 9 I w,

g=

(3.6.25)

where aC is a non-degenerate G-invariant metric in the Lie algebra of g and a is a coupling constant.

Let us turn now to SEM conservation laws in gauge theory. For the sake of simplicity, we will consider only gauge theory without matter fields. The corresponding Lagrangian density is the Yang-Mills Lagrangian density (3.6.25) on the jet manifold J'C. Given a vector field r on X, let B be a principal connection on the principal bundle P -+ X and TB = r"(0,% + Baep)

the horizontal lift of r onto P by means of the connection B. This vector field, in turn, gives rise to the vector field TB (3.6.11) on the bundle of principal connections C, which reads

I r8 = r-'Ox + (rA(aBa + C;aµB) - a"T1'(a0

(3.6.26)

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150

Let us discover the SEM current along the vector field 'ra (3.6.26) [61, 1651. Since the Yang-Mills Lagrangian density (3.6.25) depends also on a background pseudo-Riemannian metric g, we will consider the total Lagrangian density

L=

a = det(o,,,,),

J 0 1 w,

(3.6.27)

on the total configuration space J' (C Epa), where EPR is the bundle of pseudoX Riemannian fibre metrics (3.7.18) with coordinates (xa, a""). Given a vector field T on X, there exists its canonical lift (3.7.7) T = r 8a +

+ &TPc"°)aas

onto EPR C VT'X, which is the generator of a local 1-parameter group of general covariant transformations of EPR (see Remark 3.7.8). Thus, we have the canonical lift TB =

7-A8.%

+

cy

-8,.7-8(a0 - B;)]&,. +

+ 0OTP0'"°)8'0

of a vector field r on X onto the product C

EPR. For the sake of simplicity, we X will denote it by the same symbol TB. The total Lagrangian density (3.6.27), by construction, is invariant under gauge

transformations and general covariant transformations. Hence, its Lie derivative along the vector field J'TB equals zero. Then we can use the formula (3.5.25). On the Yang-Mills shell and the background field a'' = g""(x), this reads 0 (8vT°gl + 0,,r g"° - 8ag0T'')8.0G - dATB, where

TB =

+ cy a.B,a, - a;,,) + Ovrµ(aµ - BB)] -

(3.6.28)

is the SEM current along the vector field (3.6.26). This weak identity can be written in the form 0_- 8aT"tN j g 1- T"(,,0a)tp j g I- dATB,

are the Christoffel symbols (1.6.5) of g and

where tp

(g 1 = 2g"°8°sCYM

(3.6.29)

3.6. CONSERVATION LAWS IN CAUCE THEORY

151

is the metric energy-momentum tensor of gauge potentials (see Remark 3.7.9). We then have the relation tN

bµ.Cym.

19 I =

Let A be a solution of the Yang-Mills equations. Let us consider the lift (3.6.26) of the vector field T on X onto C by means of the principal connection B = A. In this case, the SEM current (3.6.28) reads

T'A oA=r"(tµoA) 19 The SEM weak identity (3.6.29) on the solution A takes the form

0-{,,Pa}(t.' o

I - da[(ty o A) I g I].

Thus, it leads to the familiar covariant conservation law

V,((t o A)

I -g J) = 0,

(3.6.30)

where V is the covariant derivative with respect to the Levi-Civita connection of the background metric g. Note that, in the case of an arbitrary principal connection B, the corresponding SEM weak identity (3.6.29) differs from (3.6.30) in the Nother conservation law 0 -where

`

Sc = Svar = (&,

y

+ ccgna,S )a:

r

S = r'`(Bµ - Aµ),

is the principal vector field (3.6.10) on C. It should be emphasized that, in order to obtain the covariant conservation law (3.6.30), the gauge invariance of the Lagrangian density (3.6.25) has been used. 1

Example 3.6.9. Chern-Simons gauge model. The Chern-Simons gauge model on a 3-dimensional base manifold X3 [19, 131) is an example where a Lagrangian density is not gauge-invariant. In this model, gauge transformations are generalized invariant transformations, but not invariant transformations (see Remark 3.5.7). As a consequence, the Nother current (3.6.16) is not conserved, and we have the conservation law (3.5.22). Moreover, neither is the SEM current conserved, though the

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152

Chern-Simons Lagrangian density (as well as Lagrangian densities of other topological field models) is independent of a background metric. Let p __+ X3 be a principal bundle with structure group a semisimple Lie group C, and let C X' be the bundle of principal connections with coordinates (x", aa). The Chern-Simons Lagrangian density fails to be globally defined on the configuration space J'C, but it is given on each bundle chart by the coordinate expression

(fL - 3aaa,°.)w,

Los =

(3.6.31)

where e°''" is the skew-symmetric Levi-Civita tensor and k is the coupling constant. It is readily observed that the Lagrangian density (3.6.31) is not gauge-invariant. At the same time, Los leads to the gauge-invariant and globally defined differential operator eLcs =

kamfle

F 9Q` A w.

(3.6.32)

We will call e,,, the Euler-Lagrange operator, though it is a locally variational, but not variational operator. Of course, gauge transformations in the Chern-Simons model keep the Euler-Lagrange operator (3.6.32), but not the Lagrangian density (3.6.31), invariant. In the Chern-Simons model, we have the above-mentioned weak conservation law (3.5.22), where the total conserved current is the sum of the standard Nother current plus the additional term as follows. Let Cc be the principal vector field (3.6.10) on the bundle of principal connections C X. We obtain

Ljif,Lcs =

kOfE°n",%C'aa"w.

(3.6.33)

In accordance with Proposition 3.3.11, the Lie derivative (3.6.33) is expressed in the form

ho(ds), where aµwJ1

is a2-formon C-.X.

3.6. CONSERVATION LAWS IN GAUGE THEORY

153

Hence, the weak identity (3.5.26) in the case of the principal vector field Cc (3.6.10) leads to the conservation law (3.5.22): 0

-da(TA + oa),

(3.6.34)

where 4.

1

k

amneAµcc

"au

is the Nother current. Furthermore, the conservation law (3.6.34) on the ChernSimons shell

5GCs = k a c mnea.rx- = 0 takes the superpotential form 0 -_

dUla),

eaµAtnam UAX = 24C mn a

Let us now turn to the SEM conservation law in the Chern-Simons model. Let r be a vector field on the base X and r8 its canonical lift (3.6.26) onto the fibre bundle C by means of a principal connection B. Remind that this is a vector field associated with a local 1-parameter group of general principal automorphisms

of C- X. Bearing in mind the corresponding transformation law of the term e°`w, we obtain I+,p;HLcs = aLeaaµ8o(T°B. )a'Hw.

The corresponding conservation law takes the form 0 -- -dA[T.B +

)a°p),

(3.6.35)

where TB is the SEM current (3.6.28) along the vector field ra. It follows that the SEM current of the Chern-Simons model is not conserved because the Lagrangian density (3.6.31) is not gauge-invariant. At the same time, we have the conservation law (3.6.35) of another quantity.

Example 3.6.10. Zbpological gauge models. Let us consider Lagrangian densities of topological gauge models which are invariant under the general principal

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154

automorphisms of the fibre bundle C. Though their Lagrangian densities are variationally trivial, the corresponding strong identities yield superpotential terms (see Example 3.5.6) when topological Lagrangian densities are added to the Yang-Mills one.

Let P --+ X be a principal bundle with a structure group G. Recall that the fibre bundle J1P -, C is also a G-principal bundle (see Example 2.7.4) provided with the canonical principal connection A (2.8.4) with the curvature FA (2.8.5). Let I(g) be the algebra of real G-invariant polynomials on the Lie algebra g of the group G. Then there is the well-known Weil homomorphism of 1(g) into the De Rham cohomology algebra H+(C,R) (see 1131]). By virtue of this homomorphism, every k-linear element r E 1(g) is represented by the cohomology class of the closed characteristic 2k-form r(FA) on C. If A is a section of C -+ X, we have

A'r(FA) = r(FA), where FA is the strength of A and r(FA) is the corresponding characteristic form on X.

Let n = dim X be even and let a characteristic n-form r(FA) on C exist. This is a Lepagian form which defines a gauge-invariant Lagrangian density Lr = ho(r(FA))

on the jet manifold J1 C. The Euler-Lagrange operator associated with Lr vanishes identically. Then, for every projectable vector field u on C, we have the strong equality

L.n.ho(r(FA)) = dnho(J1uJr(FA) If u = Cc is a principal vector field (3.6.10) on C, this equality takes the form

0 = dyho(J1uJr(FA)) For instance, let dim X = 4 and the group C be semisimple. Then the characteristic Chern-Pontryagin 4-form is

r(FA)=amn AFP'. This is the Lepagian equivalent of the Chern-Pontryagin Lagrangian density

L=

;w

of the topological Yang-Mills theory.

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

3.7

155

Conservation laws in gravitation theory

There are several approaches to discover energy-momentum conservation laws in gravitation theory. In contrast with infinite-dimensional instantaneous Hamiltonian formalism [23, 89, 117], we are here concerned with Lagrangian field theory, where the energy-momentum of gravity is treated as a peculiar Nother current [9, 21, 69, 91, 92, 166] or a Belinfante-Rosenfeld-like tensor [87, 79, 173]. Gravitation theory is formulated on fibre bundles T X, called natural bundles, which admit the canonical horizontal lift T of any vector field r on X. One can think of this canonical lift as being the generator of a local 1-parameter group of general covariant transformations of the fibre bundle 7'. We will examine the SEM conservation laws in gravitation theory along such canonical lifts of vector fields on X. Remark 3.7.1. The reader is referred to [105] for a detailed outline of the category

of natural bundles. By a natural bundle we will here call a fibre bundle T -- X which admits a canonical horizontal lift of any local diffeomorphism of its base X. Tensor bundles exemplify natural bundles. 1 Throughout the rest of this Chapter, by a manifold X is meant a 4-dimensional

manifold. This is assumed to be non-compact, orientable and, unless otherwise stated, parallelizable in order that a pseudo-Riemannian metric, a spin structure and a causal space-time structure can exist on it (60, 189J. Moreover, an orientation

on X is chosen. I Remark 3.7.2. In classical field theory, if cosmological models are not discussed, some conditions of causality should be satisfied (see [88]). A compact space-time does not possess this property because it has closed time-like curves. Every noncompact manifold admits a non-zero vector field and, as a consequence, a pseudoRiemannian metric ([46], p.167). A non-compact 4-dimensional manifold X has a spin structure if and only if it is parallelizable (i.e., the tangent bundle TX - X is trivial) [60]. Moreover, this spin structure is unique [7, 60). The orientability of the manifold is not needed for a pseudo-Riemannian structure and a spin structure to exist on it. This requirement, and the additional condition of time-orientability, seem natural if we are not concerned with cosmological models [46]. It should be noted that also paracompactness of manifolds has a physical reason. A manifold is

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156

paracompact if and only if it admits a Riemannian structure ([130]; [103], p.271).

A manifold X obeying the above-mentioned conditions is called a world manifold.

Accordingly, a linear connection and a fibre metric on the tangent and cotangent bundles of X are said to be a world connection and a world metric. The coordinate expressions of world connections on TX and T'X are respectively:

K = dx " ®( 8a +

(3. 7. 1 )

K'

(3. 7. 2 )

d? 0 (8,,

Remark 3.7.3. Unless otherwise stated, a coordinate atlas Tx = { (Uc, Ot) } of X and, hence, the corresponding holonomic bundle atlas

'T = {(U(,T¢C))

(3.7.3)

of the tangent bundle TX is assumed to be fixed. Let

xtx:LX - X be the principal bundle of oriented linear frames in the tangent spaces to a world manifold X (or simply the frame bundle). Its structure group is GL4 = GL+(4,R).

A world manifold X, by definition, is parallelizable if and only if the frame X is trivial and, consequently, it admits a global section, called a bundle LX frame field.

Unless otherwise stated, T denotes a fibre bundle associated with LX, e.g., a tensor bundle

T=(®TX)®((&' T'X).

(3.7.4)

Principal connections on the frame bundle LX and the associated connections on the LX-associated bundles T are also called world connections.

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

157

Given holonomic frames {8"} in the tangent bundle TX associated with the holonomic atlas (3.7.3), every element {Ha} of the frame bundle LX takes the form H. = H%0,.,

where H'`a is a matrix element of the group CL4. These matrix elements constitute the bundle coordinates J1

(x

,

H".),

iµ =

H a =

a

ata

H a,

of LX, in which, the canonical action (2.7.1) of CL4 on LX reads R9 : H"a I

11"69ba,

9 E CLL.

The frame bundle LX is equipped with the canonical R4-valued 1-form BLx which has the coordinate expression OLx = Ha"dx" & ta,

(3.7.5)

where {ta} is a fixed basis for R4 and Ha,, is the inverse matrix of H".. X belongs to the category of natural bundles. Every The frame bundle LX diffeomorphism f of X gives rise canonically to the automorphism

H'a) i-' (f" (x), Of " HP.)

(3.7.6)

of LX and, consequently, to the corresponding automorphisms (3.6.3) of the associated bundles T. These automorphisms are called the general covariant transformations or the holonomic automorphisms.

Example 3.7.4. If 7' = TX is the tangent bundle, f = T f is the familiar tangent map to the diffeomorphism f. If T = T'X is the cotangent bundle, 1= T' f is the cotangent map (see Example 1.3.2). We will denote the group of holonomic autornorphisrns by HOL(X) C AUT (LX). This is isomorphic to Diff(X). The lift (3.7.6) leads to the canonical horizontal lift T of every vector field T on

X onto the principal bundle LX and the associated fibre bundles. The canonical lift of r onto LX is defined by the relation

LOLx=0.

CHAPTER 3. LACRANCIAN FORMALISM

158

We have the corresponding canonical lift T = T"8" + [8 ,,T

a,

...a-

xp1...pk

+

V

-a-

L9

Bp,T x pz...pk - ...J

(3.7.7)

9i...A

of r onto the tensor bundle (3.7.4) and, in particular, the lifts

'r = T"8, +

(3.7.8)

onto the tangent bundle TX and

T` = T"8, - 8pr"i

(3.7.9) a71-0-

onto the cotangent bundle T'X. The bundle of principal connections on LX admits the canonical lift (3.7.40) of vector fields on X.

Remark 3.7.5. If a vector field r on X is complete, so is its canonical lift F.

Remark 3.7.6. If a vector field r is non-vanishing at a point x E X, then there exists a local symmetric linear connection K around x for which r is an integral section

Then the canonical lift z (3.7.7) can be found locally as the horizontal lift of r by means of this symmetric connection. One can consider the horizontal lift TK = T"(8a + Kap°i°app)

(3.7.10)

of a vector field r on X onto TX and the associated bundles by means of any world connection K. This is the generator of a local 1-parameter group of nonholonomic automorphisms of these bundles. Non-holonomic automorphisms of the frame bundle are met with in the gauge theory of the general linear group CL4 191]. Note that the lifts (3.7.8) and (3.7.10) were treated as generators of the gauge group of translations in the pioneer gauge gravitation models (see [90, 96] and references therein).

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

159

A glance at the expression (3.7.7) shows that the generators ,7 of general covariant

transformations (as well as the generators (3.6.12) of principal automorphisms in gauge theory) depend on the components r'` of the vector fields r (which play the role of gauge parameters) and their partial derivatives. The main peculiarity of the SEM conservation laws along these generators is that the corresponding SEM currents reduce to the sum of a superpotential term and a term which displays itself in the presence of a background world metric on X. Such a phenomenon takes place in General Relativity [1481, the Palatini model [21, 149], metric-affine gravitation theory and gauge gravitation theory [68, 69, 1661. We will start from tensor field theories which clearly illustrate this phenomenon. Let T be a tensor bundle (3.7.4) with coordinates (x',yA), where the collective index A is used: YA =_ 91 ...ak (3.7.11)

In this notation, the canonical lift z (3.7.7) onto T of a vector field r on X reads T = r''8a + UAaBfir°8A.

(3.7.12)

Remark 3.7.7. The expression (3.7.12) is the general form of the canonical lift of a vector field r on X onto a natural bundle T, when this lift depends only on the first partial derivatives of the components of r. Therefore, the results obtained below for tensor fields are also true for every such natural bundle T. Let L be a Lagrangian density on J'T which is invariant under general covariant transformations, i.e., L satisfies the strong equality

Ljt7L = 0. In the coordinate form (3.5.6), this equality reads r°)zrA - yQ 8sr°7rA = 0.

8°(7°G) + uA Opr°8AG +

(3.7.13)

The corresponding weak identity (3.5.17) takes the form 0

dA[7rA(yy r° - uAaO,r°) -r-,C].

(3.7.14)

Due to the arbitrariness of the gauge parameters r°, the equality (3.7.13) is equivalent to the system of strong equalities (3.7.15a)

SaG + uA06AG + d,,(uAair ) uAa7rA + uAQ7[A = 0,

=

Y.ArO ,

(3.7.15b) (3.7.15c)

CHAPTER 3. LAGRANGIAN FORMALISM

160

where bAG = (OA - d,,bw)G

are the variational derivatives (3.2.10). Substituting the relations (3.7.15b) and (3.7.15c) in the weak identity (3.7.14), we obtain the SEM conservation law 0

da[uAQ6AGr° + d,,(uAQ7rAr°)].

(3.7.16)

A glance at the expression (3.7.16) shows that, on-shell, the corresponding SEM current leads to the superpotential form (3.5.18), that is, `ET = uA;bAGT° +

where UTPA = uAQ7rAr°

(3.7.17)

is the SEM superpotential of tensor fields.

I It is readily seen that the SEM superpotential (3.7.17) emerges from the dependence of the canonical lift T (3.7.12) on the derivatives of the components of the vector field r. This dependence guarantees that the SEM conservation law (3.7.16) is maintained under general covariant transformations. 1

Let us now consider tensor fields, treated as matter fields, in the presence of a background pseudo-Riemannian metric on a world manifold X.

Remark 3.7.8. A pseudo-Riemannian metric g on a world manifold X is represented by a section of the fibre bundle

EPR = GLX/0(1,3),

(3.7.18)

where by CLX is meant the bundle of all linear frames in TX, and 0(1,3) is the complete Lorentz group. We will call EPR the metric bundle. Since X is oriented, the metric bundle is associated with the principal bundle LX of oriented frames in TX. The typical fibre of EPR is the quotient

CL(4,R)/0(1,3).

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

161

This quotient space is homotopic to the Crassmann manifold 0(4,3;R) and it is homeomorphic to the topological space RP3 x R', where R.P3 is the 3-dimensional real projective space ([151], p.164). For the sake of simplicity, we will often identify the metric bundle with an open subbundle of the tensor bundle

EPR C 'TX. It is equipped with coordinates (xa, a""). By a,,, are meant the components of the inverse matrix and o = det(a,,,,). We follow Remark 3.5.8 to obtain the SEM conservation law of tensor fields in the presence a background metric. The total configuration space is the jet manifold J I Y of the fibred product Y = T x EPiz.

x

This product is endowed with coordinates (xA, yA, a"), where we continue to use the compact notation (3.7.11). Liven a vector field T on X, its canonical lift onto Y reads f = 7-.18A + UAQB$T°8A + (O"T°a"A + O"TAa,)&Q .

(3.7.19)

Let a total Lagrangian density L on J'V be invariant under general covariant transformations, that is, Lj,TL = 0

(3.7.20)

for any vector field f (3.7.19).

1 Unless otherwise stated, we assume that Lagrangian densities do not depend on the derivative coordinates aa"" of a world metric. 1

Due to the arbitrariness of the gauge parameters T°, the equality (3.7.20) is equivalent to the system of strong equalities 8AG = 0, 6.0,C

uAQ

+ + uAo

(3.7.21 a)

uAobAG + d"(uAa A) = y 1rq, = 0.

(3.7.21b)

(3.7.2lc)

CHAPTER 3. LACRANCIAN FORMALISM

162

On the shell

a,=g'"'(x),

6AL=0,

the corresponding weak identity (3.5.25) takes the form

-

0

8.7-8g- - 8ag0 TA)BQpG -

da[XA(y rQ - uA$o

7.o)

(3.7.22)

- T"G].

Substituting (3.7.21b) and (3.7.21c) in (3.7.22), we obtain the SEM weak identity

0 ^ 8aT"tµ 19 I - T"{,,6a)t0 -'I g I IS

(3.7.23)

9 I + U" %bAITQ + d, (uAnrATQ)],

where tN

I g I = 2ga"8,,,,G

(3.7.24)

is the metric energy-momentum tensor of tensor fields.

Remark 3.7.9. Given a product YXEPR X

and a Lagrangian density L on the jet manifold of this product, the metric crnergymomentum tensor is defined to be tµ

Io

2o,

6""'C'

where b,,,,G are the variational derivatives with respect to the metric coordinates p'µ.

A glance at the expression (3.7.23) shows that, on-shell, the SEM current of tensor fields in the presence of a background pseudo-Riemannian metric is the sum TA

= T" O

VI

g l + d,,(u"Q,TQ)

(3.7.25)

of the metric energy-momentum tensor (3.7.24) of these fields and the superpotential (3.7.17). The latter does not contribute to the differential conservation law which takes the familiar form VAtµ-_ 0.

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

163

At the same time, if a metric field is dynamic, the metric energy-momentum tensor (3.7.24) vanishes on-shell. This is the phenomenon of "hidden" energy, which displays itself only in the presence of a background metric.

Example 3.7.10. Proca field. We will take a Proca field as an example of a tensor matter field. Proca fields are described by sections of the cotangent bundle

T = M. Their configuration space is the jet manifold J1T'X, coordinatized by (x , kµ, k,a), where k,, = i,, are the induced coordinates on T'X. The jet bundle J1T'X -. T'X is modelled over the pull-back bundle

T'X

X(®T'X) -.T'X.

Given a world connection K, the configuration space J'T'X admits the following splitting

J'T'X = S+ ®(T"X x AT* X), X

(3.7.26)

where S+ is an affine bundle modelled over the vector bundle

T'X X((T'X) - T"X (see Example 4.6.5). In coordinates, this splitting reads k,,,, = 2 (Sav + .FFa,,) (3.7.27)

S,, = k,,,, + k -

(3.7.28)

where

S°- K° -K° is the torsion (2.3.25) of the linear connection K. If a connection K is symmetric, we have

F,,,, = k,,,, Let us consider the relation (3.7.15c). In coordinates the splitting (3.7.26), it takes the form IMP OSw

0.

(3.7.29)

Sp) associated with

CHAPTER 3. LACRANCIAN FORMALISM

164

It follows that, in order to be invariant under general covariant transformations, the Lagrangian density of Proca fields must factorize through the morphism

.F : J'T'X-.T'X x n T'X. x

Indeed, the standard Lagrangian density of Proca fields in the presence of a background world metric g and a background world connection K takes the form LP =

2m2f P- kµka[

7

19 Iw.

(3.7.30)

Let us consider the SEM conservation law of Proca fields in the presence of a background world metric g when K is the Levi-Civita connection of g. Let r be a vector field on the base X and

r = r"8µ -

'5Z

its canonical lift (3.7.9) onto 7' X. The weak identity (3.7.23) for this vector field reads 0

19 I -

19 I -

da['rµtµ 19 I - k°r' 6''G - dµ(k°&0µ7°)),

where

µ

19 I =

7r"a r,,,,

- m2g"''kµk 19

6,,C p.

Hence, on-shell, the SEM current (3.7.25) of Proca fields in the presence of a background metric is the sum

P = r"tµ 19 I - dµ(k°ir' r°)

(3.7.31)

of their metric energy-momentum tensor and the superpotential

UPS" _ -k°7r' r°.

(3.7.32)

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

165

Let us now examine SEM conservation laws in dynamic gravitational models. In this Section we are concerned with General Relativity and metric-affine gravitation theory. The next Section will be devoted to gauge gravitation theory. In General Relativity, gravity is described by a pseudo-Riemannian metric and its Lagrangian density is the Hilbert-Einstein Lagrangian density. This is a second order Lagrangian density whose Euler-Lagrange operator reduces to a second order differential operator (see Remark 3.2.12). Thus, the configuration space of General Relativity is the second order jet manifold J2EPR of the metric bundle EPR. In this model, it is convenient to consider the metric bundle as an open subbundle of the tensor bundle

EPR C vT'X equipped with the coordinates (xA, a,,). The jet manifold .PEPR is coordinatized by (xA, g p, gw, 9µaaA)' The second order Hilbert-Einstein Lagrangian density on the configuration space J2EPR reads LHE = 2-0°µ00°Rve i 9 Ire

(3.7.33)

where Rµ

(2 (oµP011 + ovQsµ

- 0'1100v -

6000M) +

Qcr({µ`a}(v"o} 1 f l -2QQS(Qval+ + Opav - Q{9µv) ivQµl =

This Lagrangian density is linear in the second order derivative coordinates gaa and leads to the second order Euler-Lagrange operator and to the Poincarc-Cartan form (3.2.18), which lives on the first order jet manifold J1EPR.

Remark 3.7.11. Let us recall the useful relations 1 80, 8 _ a0 8 ao0

o_

o 80.0,

i9aw,

Let us consider Proca fields from Example 3.7.10 as the matter source of a metric gravitational field.

CHAPTER 3. LAGRANCIAN FORMALISM

166

The total configuration space of metric gravitational fields and Proca fields is the second order jet manifold J2Y of the fibred product

Y=T'XxEpa.

(3.7.34)

On this configuration space, the Lagrangian density of Proca fields is given by the expression (3.7.30) where the background world metric g"v is replaced with the metric coordinates o', and F is given by the expression (3.7.29). We have

Lp =

47O.F°pfrw -

2rri2g'`''k'ka)

o

1W.

(3.7.35)

The total Lagrangian density L is the sum of the Hilbert-Einstein Lagrangian density (3.7.33) and the Lagrangian density of Proca fields (3.7.35). The associated Poincare-Cartan form on the first order jet manifold J'Y reads

HL=HHE+ Hp, where Hp = (Cp - Tr1Mk>v)w + aaµdk,, A wA, 'e V, a 11

ly

and HHE is given by the following expression [111]: HHE

,l-K

(oM°oa9

I -al (o°MO."oCO({y.M}{O°6} -

_

o°O{9,°}daw) Awa]

Let r be a vector field on X and

f = T'Ba -

80r'k 8°

(3.7.36)

its canonical lift onto the fibred product (3.7.34). Since the Lagrangian densities LHE (3.7.33) and Lp (3.7.35) are invariant under general covariant transformations, the Lie derivative of their sum L by the second order jet lift J2T off (3.7.36) vanishes. Then the first variational formula (3.2.19) leads to the following SEM conservation law [148]: 0

da{2o,1°r"6°aG - r°kb" G + I(ra"VvrM - o""OVTA) - k, ir"ar°J}. dM(2K Io

(3.7.37)

3.7. CONSERVATION LAWS IN GRAVI"IA77ON Th EORY

167

A glance at the conservation law (3.7.37) shows that the corresponding SEM current reduces to the superpotential which is the sum of the Komar superpotential

UKµ"

-

zK

O I /OAvOvrv - Oµ"OvT-A)

(3.7.38)

of metric gravitational fields (106] and the superpotential (3.7.32) of Proca fields.

Z Note that, in comparison with the expression (3.7.31), the energy-momentum current of Proca fields in the presence of a dynamic metric field reduces to a superpotential only. Z

Let us turn now to SEM conservation laws in metric-affine gravitation theory [68, 166].

Remark 3.7.12. In rnetric-affine gravitation theory, gravity is described by a pseudo-Riemannian metric g and a world connection K on X. The reader is referred to [137, 91, 139] for a general formulation of this gravitational model and to [43, 91] for a study of its solutions.

Since world connections are associated with principal connections on the frame bundle LX, there is one-to-one correspondence between the world connections and the sections of the quotient fibre bundle

CK = J'LX/GL4.

(3.7.39)

We call this the bundle of world connections. With respect to the holonomic frames in TX, the bundle CK is provided with

the coordinates (x', kA".) so that, for any section K of CK - X, ka"a o K = Ka"a

are the coefficients of the world connection K (3.7.1). The bundle of world connections CK (3.7.39) admits the canonical horizontal lift

/c : J'LX/CL, -. J'J(J'LX)/CL,, of any diffeomorphism / of X and, consequently, the canonical lift T = Tµ8µ +

8s'r"k,,°v - i9,,T"kv°Q + Opsel8k

Qp

(3.7.40)

CHAPTER 3. LACRANCIAN FORMALISM

168

of any vector field r on X. For the sake of simplicity, let us utilize the compact notation T = T-ABA + (U 0007° + UAQPO$pT°)OA

for the lift (3.7.40), where

Y=kp A, u µ°$tv = btbob° µf7 s f Upo A7-kp 8

o

° t -kp °1rs5-k7 9bp

The configuration space of the metric-affine gravity is the first order jet manifold of the fibred product

EPRXCK.

This configuration space is equipped with the adapted coordinates

(xA,o°t9 ,kp°a,aAoB ,kAp° a) Remark 3.7.13. The jet manifold J'CK admits the canonical splitting (2.8.21). We will denote the corresponding projection F by R. This projection has the coordinate expression RAp°p = kAp°B - k,,a o + kp°tkAs8 - kA tkpt$.

(3.7.41)

It is readily observed that, if K is a section of CK -» X, then R o .1'K is the curvature (2.4.24) of the world connection K. We will again consider Proca fields as a matter source of the metric-affine gravity.

Then the total configuration space is the first order jet manifold J'Y of the fibred product

Y = T'X

X

EPR X CK.

(3.7.42)

The total Lagrangian density 1, on this configuration space is the sum L = LMA + Lp

(3.7.43)

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

169

of a metric-affine Lagrangian density LMA and the Lagrangian density (3.7.35) of Proca fields, where 1:;,,, is given by the expression (3.7.27) and (3.7.44)

We will assume that LMA factorizes through the curvature (3.7.41) and that it does not depend on the derivative coordinates aa°p of a world metric. Then the following relations take place: AY B B,CMA

V_1

(3.7.45)

= a ak 9

(3.7.46)

e

We also have the equalities a ARY

EAU o = 7rAt4 A9 ° ==

ay B

°

B

c ' .y. GMA - ncB c k°

Given a vector field r on a world manifold X, its canonical lift onto the product (3.7.42) reads

r" 8 - 8°r°k 8° + (a"

a°"

(uAQ0Or° + uAoµ8s1,r°)8A.

Let the total Lagrangian density L (3.7.43) be invariant under general covariant transformations, i.e., Lj,T(LMA + Lp) = 0.

(3.7.47)

Then, on-shell, the first variational formula (3.5.14) leads to the weak conservation law 0

da[an(y r°

- uAQa5r° - UAQ B"BT°) +

(3.7.48)

k°8,,r°) - raG], where

TM A' = it (y° ro - uAoO5r° - uAe 8ipr°) is the SEM current of the metric-affine gravity.

(3.7.49)

CHAPTER 3. LACRANCIAN FORMALISM

170

Due to the arbitrariness of the gauge parameters r", the equality (3.7.47) is equivalent to the system of strong equalities

8AG=0, 6P + 2aP"6°"G + u"a6AG - k.60L + d"(w"Au"a - k°ir"a) -

(3.7.50)

(ya 7r,O + k,7r'") = 0, [(uA, 8A + UAry8Aa),CMA + (O'.y° + 2k.y7r`°)Gp]8°6T" = 0,

(3.7.51)

7r(A y) = 0,

(3.7.52)

where 6°,,G, bAG and Of, are the corresponding variational derivatives.

Remark 3.7.14. It is readily observed that the equality (3.7.51) holds owing to the relation (3.7.46) and to the fact that the Lagrangian density Lp factorizes through .F. The equality (3.7.52) holds due to the relation (3.7.45). Substituting the term y.7roA + k°"7rfl" from the expression (3.7.50) in the SEM conservation law (3.7.48), we bring this conservation law into the form 0 -_ -da[2a-'"r°6°"G + uAar°bAG - k°T°b"G 7rAUAoB07° + d"(rra"°O)8sT° 1+

d"(7rquA,1)T°

(3.7.53)

-

d"(k°7r"AT°)].

Note that the last term in this expression is precisely the divergence of the superpotential (3.7.32) of Proca fields. After separating the variational derivatives, the SEM conservation law (3.7.53) of the metric-affine gravity and the matter Proca fields leads to the superpotential form

0 a -da[2a4`T°6°"G + (k"aryb a"G - k,,°°b"°-%G - !c°° yb v'G)T° +

6o"G8"T° - d"(&°"G)T° +

r° - °vT))]r

where the SEM current on-shell reduces to the generalized Komar superpotential UMA"a I

k , T°).

=

Remark 3.7.15. We can rewrite the superpotential (3.7.54) as UMAW = 2 BGMA (DVTa

8R"°

+ S°°T°),

(3.7.54)

3.7. CONSERVATION LAWS IN GRAVITATION THEORY

where D is the covariant derivative relative to the connection torsion (3.7.44) of this connection.

171

and

is the

Remark 3.7.16. Let us emphasize that matter Proca fields do not contribute to the total superpotential (3.7.54). The corresponding term -d,,(k°7r'`AT°)

in the expression (3.7.53) disappears because of dependence of the Lagrangian den-

sity Lp on the torsion (3.7.44). As will be seen later, also in gauge gravitation theory fermion fields do not contribute to the total SEM current because of their interaction with a torsion.

Example 3.7.17. Let us consider the Hilbert-Einstein Lagrangian density LHE = Z-R I o Iw, R

in the metric-affine gravitation model. Then the generalized Komar superpotential (3.7.54) comes to the Kornar superpotential (3.7.38) if we substitute the Levi-Civita connection Let us generalize this example by considering the Lagrangian density LPL = f (R) Ira Iw,

where f(R) is a real polynomial function of the scalar curvature R. In the case of a symmetric connection, we reobtain the superpotential Of I a I (7 DpT, UPL'`A = OR

of the Palatini model [21].

Example 3.7.18. It is readily observed that, in the local gauge where the vector field r is constant, the SEM current of metric-affine gravity (3.7.49) leads to the canonical energy-momentum tensor `AMA = (xA"9°kw

. - 6.LMA)T°

This tensor was suggested in order to describe the energy-momentum complex in the Palatini model (44, 144, 149].

CHAPTER 3. LACRANCIAN FORMALISM

172

3.8

Gauge gravitation theory

There is an extensive literature on gauge gravitation theory (see [11, 91, 96, 139,160] and references therein). We here consider the SEM conservation law in gauge grav-

itation theory. Difficulties arise because of the presence of Dirac fermion fields. The key point is that the Dirac spinor bundles over a world manifold are not preserved under general covariant transformations. To overcome this difficulty, we will consider the universal spin structure on a world manifold, based on the two-fold universal covering group CL. [56, 118, 151, 172]. It admits the canonical horizontal lift (3.8.60) of vector fields on a world manifold X. The goal is the SEM conservation law (3.8.67) of the metric-affine gravity and Dirac fields.

Remark 3.8.1. Homotopy and homology. Let us first recall some basic notions of homotopy theory [82, 1881. Application of this theory to fibre bundles is based on the following two facts. Let Y

X be a principal bundle. Let f, and f2 be two mappings of a manifold

Z to X. If these mappings are homotopic, the pull-back bundles flY -. Z and f2Y - Z are isomorphic ((170], p.53). The De Rham cohomology groups H'(M) of a paracompact manifold M are isomorphic to the (tech oohomology groups H'(M,R) with coefficients in R ([93], p.37). This isomorphism enables us to represent characteristic classes of principal bundles as the De Rham cohomology classes of characteristic exterior forms expressed in terms of principal connections (51, 1311 (see Remark 3.6.10).

Continuous maps f and f of a topological space Z to a topological space Z' are said to be homotopic if there is a continuous map

g: (0, 1) xZ- Z' whose restriction to {0} x Z [{1} x Z] coincides with f [f']. The set of equivalence classes of homotopic maps Z -- Z' is denoted by a(Z, Z'). The topological spaces Z and Z' are called homotopically equivalent or simply homotopic if there exist

mappings f : Z -+ Z' and g : Z' -. Z such that g o f is homotopic to the identity map Id Z, and fog is homotopic to Id V. In particular, a topological space is called contractible if it is homotopic to one of its points. For instance, Euclidean spaces Rk are contractible.

3.8. GAUGE GRAVITATION THEORY

173

Let (Sk, a) be a k-dimensional sphere and let a E Sk be a point. Let us consider the set of equivalence classes of homotopic maps of Sk to a topological space Z which sends a onto a fixed point b of Z'. If Z is pathwise connected this set does not depend on the choice of a and b, and one can talk about the set 7rk(Z) of equivalence classes

of homotopic maps Sk - Z. This set can be provided with a group structure, and is called the kth homotopy group of the topological space Z. The homotopy groups 7rk> (Z) are always Abelian, while the first homotopy group 7rj(Z) is also known as

the fundamental group of Z. By 7ro(Z) is denoted the set of pathwise connected components of Z. Let us bear in mind that a manifold is pathwise connected if and only if it is connected. A topological space Z is said to be p-connected if it is pathwise connected and its homotopy groups 7rk
for more complicated constructions of topological spaces we refer the reader to the van Kampen theorem ([381, p.63). Note that, given a group G1 and Abelian groups

G2i G3, ..., there exists a connected (cell) topological space Z with wk(Z) = Gk (k = 1, 2, ...). Homotopy groups of topological spaces are homotopic invariants in the sense that they are the same for homotopic topological spaces. However, it may happen that non-homotopic topological spaces have the same homotopy groups. Other homotopic invariants of topological spaces are homology and cohomology groups. While there are different homology theories, one usually refers to singular homology and cohomology [37, 48, 82] and to (tech cohomology [47, 931. Here we are briefly concerned with (tech cohomology. Let it = {U;)1E1 be an open covering of a paracompact topological space Z. La us consider a function ' which associates to each (p+ I)-tuple (io,... , ip) of indices in I such that U,, fl ... fl U;, 0 0 an element of an Abelian group K. One can think of 0 as being a K-valued function on the set U,, fl ... fl Uy. These functions form an Abelian group Bp(11, K). Let us consider the cochain morphism

lP:Bp(U,K)

B;+'(1.1, K),

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174

[p+1

_

(6')(i0,... Iip{.1) = L(- 1)k4.(io,...,lk,...tp}1), k=O

where tk means that the index ik is omitted. One can check that

o'01P =0. Thence, we have the eocllain complex

...-.BP(U,K)--'P-+ B'(U,K),.-. (see Remark 3.3.3), and its oohomology groups

HP(il K) = Ker 6P/Im 6P-'

can be defined. Of course, they depend on an open covering U of the topological space Z. Let if be a refinement of the covering U. Then there exists a homomorphism P(1.1; K) -. HP(U'; K).

One can take the direct limit of the groups HP(U, K) with respect to these homomorphisms, where U runs through all open coverings of Z ([931, p.27). This limit is the pth Oech cohomology group HP(Z; K) of Z with coefficients in K. For paracompact and second countable manifolds which we deal with, the tech cohomology groups coincide with the singular cohomology groups ([471 pp.248,285)

and, as was mentioned above, the tech cohomology groups with coefficients in R coincide with the De R.ham cohomology groups. Note that, in the same manner, the oohomology group H'(Z; C) of Z with coefficients in a non-Abelian group C can be defined [93].

In conclusion, let us recall the following isomorphism

H'"(Z x Z'; K) _ E Hk(Z; K) 0 H'(Z'; K)

(3.8.1)

k+!-m

for the eohomology groups of the product Z x Z' if K is a field ([481, p.84). We describe Dirac spinors as follows [36, 150, 156] (see [27, 1181 and references therein for a general Clifford algebra technique). Let M be the Minkowski space equipped with the Minkowski metric which reads

q=diag(1,-1,-1,-1)

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with respect to a basis {e°} of M. Let C1,3 be the complex Clifford algebra generated by elements of M. This is the complexified quotient of the tensor algebra ®M of M by the two-sided ideal generated by elements

e®e'+e ®e-2ri(e,e') E ®M,

e,e' E M.

Remark 3.8.2. The complex Clifford algebra C13 3 is isomorphic to the real Clifford algebra R2,3, whose generating space is RS equipped with the metric

diag(1,-1,-1,-1, 1). Its subalgebra generated by elements of M C R5 is the real Clifford algebra R,,3. A spinor space V is defined as a minimal left ideal of C1,3 on which this algebra acts on the left. We have the representation 7 : M 0 V --+ V,

(3.8.2)

'Y(e°) =,Y,,,

of elements of the Minkowski space M C C13 3 by the Dirac 7-matrices on V. The explicit form of this representation depends on the choice of the minimal left ideal V of C1,3. Different ideals lead to equivalent representations (3.8.2). The spinor space V is provided with the spinor metric

a(v, v) = (Tiv + V'v) = 2 (v+7°v' + v'+ry°v),

(3.8.3)

2

since the element e° E M satisfies the conditions (e°)+ = CO,

(e°e)+ = e0e,

Ve E M.

By definition, the Clifford group G1,3 consists of the invertible elements 1, of the real Clifford algebra R1,3 such that the inner automorphisms defined by these elements preserve the Minkowski space M C R1,3, that is, 1,e11 =1(e),

e E M,

(3.8.4)

where I is a Lorentz transformation of M. Thus, we have an epimorphism of the Clifford group G1,3 onto the Lorentz group O(1, 3). Since the action (3.8.4) of the Clifford group on the Minkowski space M

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is not effective, one usually consider its pin and spin subgroups. The subgroup Pin(1, 3) of C13 3 is generated by elements e E M such that 1)(e, e) = ±1. The even part of Pin(1,3) is the spin group Spin(1,3). Its component of the unity L. = Spin°(1, 3) = SL(2, C) is the well-known two-fold universal covering group zL : L.

L = L./Z2,

Z2 = {1, -1},

(3.8.5)

of the proper Lorentz group L = S0°(1, 3).

Recall that L is homeomorphic to RP3 x R3 ([821, p.27). The Lorentz group 1, acts on the Minkowski space M by the generators (3.8.6)

Lab`d = ri.abb - 77Li6;.

Remark 3.8.3. The generating elements e E M, ri(e, e) = ±1, of the group Pin(1, 3) act on the Minkowski space by the adjoint representation which is the composition

e : v - eve -I = -v + 2

e, e)

e,

e, V E R',

of the total reflection of M and the reflection across the hyperplane el = {w E M; 17(e, W) = 0}

which is perpendicular to e with respect to the metric n in M. By the well-known Cartan-Dieudonnb theorem, every element of the pseudo-orthogonal group 0(p, q) can be written as a product of r < p + q reflections across hyperplanes in the vector space RP" ([118], p.17). In particular, the group Spin(1, 3) consists of the elements of Pin(1,3) which are an even number of reflections of M. The epimorphism of Spin(1, 3) onto the Lorentz group SO (1, 3) and the epimorphism (3.8.5) are defined by the fact that elements e and -e of M determine the same reflection of M across

the hyperplane el = (-e). The Clifford group C,,3 acts on the spinor space V by left multiplications

GO -3 1,:v- 1,v,

VEV.

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177

This action preserves the representation (3.8.2), i.e.,

7(lM ® l,V) = 47(M 0 V). The spin group L. acts on the spinor space V by means of the generators Lab =

4

[Y., 7b]

(3.8.7)

Since

this action preserves the spinor metric (3.8.3).

Let us now consider a bundle of Minkowski spaces MX - X over a world manifold X. By definition, this is a fibre bundle with the structure group SO(1, 3). This bundle is extended to the bundle of Clifford algebras CX with fibres C2X generated by the fibres M=X of the fibre bundle MX [14, 157]. CX is a fibre bundle with the structure group Aut(Ct,s) of inner automorphisms of the Clifford algebra C1,3. This structure group is reducible to the Lorentz group SO(1, 3) and, of course, the bundle of Clifford algebras CX contains the subbundle MX of the generating Minkowski spaces. However, CX does not necessarily contain a spinor subbundle because a spinor subspace V of C1,3 is not stable under inner automorphisms of C1,3. As was shown [15, 157], the above-mentioned spinor subbundle SM exists if the transition functions of CX can be lifted from AutC1,3 to CL1,3. This agrees with the usual conditions of existence of a spin structure.

The bundle MX of Minkowski spaces must be isomorphic to the cotangent bundle T'X for sections of the spinor bundle SM to describe Dirac fermion fields on a world manifold X. In other words, we should consider a spin structure on the cotangent bundle T'X of X [118]. There are several almost equivalent definitions of a spin structure on a world manifold X [7, 14, 92, 118]. A Dirac (or pseudo-Riemannian) spin structure on a world manifold X is said to be a pair (P., z.) of an L.-principal bundle P. - X and a principal bundle morphism z.: P. - LX

(3.8.8)

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178

of P. to the frame bundle LX -+ X. More generally, one can define a spin structure on any vector bundle E -. X ([118], p.80). Then the definition above applies to and the fibre metric the particular case in which E is the cotangent bundle in T'X is a pseudo-Riemannian metric. Example 3.8.11 will exhibit a Riemannian spin structure on X. Since the hmomorphism L. -. CL4 factorizes through the epimorphism (3.8.5), every bundle morphism (3.8.8) factorizes through a morphism of P. to some principal subbundle of the frame bundle LX whose structure group is the proper Lorentz group L. It follows that the necessary condition for existence of a Dirac spin structure on X is that the structure group CL4 of LX is reducible to the Lorentz group L.

I From the physics viewpoint, it means that the existence of Dirac's fermion matter implies the existence of a gravitational field. 1

Remark 3.8.4. G1H-structure. Let us recall some basic notions. Let rrpx

:

P

X be a principal bundle with a structure group C, and let H be a closed Lie subgroup of C. We have the composite fibre bundle

P -i P/H -+ X,

(3.8.9)

E=P/HEX

(3.8.10)

where

is a P-associated fibre bundle with the typical fibre C/H on which the structure group C acts naturally on the left, and

PE=P-°4P/H

(3.8.11)

is a principal bundle with the structure group H (1103], p.57). One says that the structure group C of a principal bundle P is reducible to a Lie subgroup H if there exists a principal subbundle P" of P with the structure group H. This subbundle is called a reduced C1H-structure (74, 104, 191].

Note that in [74, 104], the authors are concerned with reduced structures on the frame bundle LX. This notion is generalized to an arbitrary principal bundle in [191]. In [74], a reduced structure is restricted to a monomorphism of a given principal bundle P - X with a structure group H to the principal frame bundle LX. Thereby, only isomorphic CL(4,R)1H-structures are considered. The set of these

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179

structures is in bijective correspondence with the group Gau(LX) (see Propositions 3.8.4 and 3.8.5). Let us recall the following two theorems.

THEOREM 3.8.1. A structure group G of a principal bundle P is reducible to its closed subgroup H if and only if P has an atlas V p with H-valued transition functions ([1031, p.53).

Given a reduced subbundle Ph of P, such an atlas eY is defined by a family of local sections {z0} which take their values into Ph. THEOREM 3.8.2. There is one-to-one correspondence

Ph = 7r-:(h(X)) between the reduced 11-principal subbundles Ph of P and the global sections h of the quotient fibre bundle P/I1 -» X ([103], p.57). Given such a section h, let us consider the restriction h* FE (1.2.11) of the Hprincipal bundle PE (3.8.11) to h(X) C E. This is an H-principal bundle over X ([103], p.60), which is isomorphic to the reduced subbundle Ph of P.

In general, there are topological obstructions to the reduction of a structure group of a principal bundle to its subgroup. In accordance with Theorem 1.2.5, the structure group C of a principal bundle P is always reducible to its dosed subgroup H, if the quotient C/H is homeomorphic to a Euclidean space Rk. PROPOSITION 3.8.3. In this case, all H-principal subbundles of P are isomorphic to each other as H-principal bundles ([170], p.56). In particular, a structure group C of a principal bundle is always reducible to its maximal compact subgroup H since the quotient space C/H is homeomorphic to a Euclidean space ([170], p.59). It follows that there is one-to-one correspondence between the equivalence classes of C-principal bundles and those of If-principal bundles if H is a maximal compact subgroup of C [93, 1701. In particular, this is the case of GL(n,R)- and 0(n)-principal bundles as well as of GI.+(n,R)- and 80(n)-principal bundles.

PROPOSITION 3.8.4. Every vertical principal automorphism + E Gau(P) of the principal bundle P X sends an 11-principal subbundle Ph onto an isomorphic H-principal subbundle Phi.

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180

Proof. Let 10 = {(U0,

),p".p},

za(x) = zp(x)p".hp(x),

X E UQ fl Up,

be an atlas of the reduced subbundle P", where za are local sections of P" -. X and p"p are the transition functions. Given a vertical automorphism + of P, let us provide the reduced subbundle P"' = 4,(P") with the atlas 'ph' = {(U., zo'), pop}

determined by the local sections

Z!, =A oza of P"' - X. Then it is readily observed that p' p(X) = p p(x),

x E U. (I Up.

QED

Conversely, let two reduced subbundles P" and PW of a principal fibre bundle P be isomorphic to each other as H-principal bundles, and $ : P" P"' be an isomorphism. Then, 40 is extended to a principal automorphism of P. 13 PROPOSITION 3.8.5.

Proof. The isomorphism d> determines a C-valued function f on P" given by the relation Pf (P) = +(P),

PEP".

Obviously, this function is H-equivariant. Its prolongation to a G-equivariant function on P is defined to be f (pg) = g 'f (P)g,

p E P",

g E C.

In accordance with the relation (3.6.5), this function defines a principal automorQED phism of P whose restriction to P" coincides with 0. Given a reduced subbundle P" of a principal bundle P, let

Y" = (P" X V)/H

(3.8.12)

3.8. GAUGE GRAVITATION THEORY

181

be the associated fibre bundle with a typical fibre V. Let P" be another reduced subbundle of P which is isomorphic to P", and

Y"' = (P"' x V)/H The fibre bundles Y" and Y"' are isomorphic, but not canonically isomorphic in general.

PROPOSITION 3.8.6. Let P" be an H-principal subbundle of a C-principal bundle P. Let Y be the P"-associated bundle (3.8.12) with a typical fibre V. If V carriers a representation of the whole group C, the fibre bundle Y" is canonically isomorphic to the P-associated fibre bundle

Y=(PxV)/C. 0 Proof. Every element of Y can be represented as (p, v) G, p E P". Then the desired isomorphism is

e=*

QED It follows that, given an H-principal subbundle P" of P, any P-associated fibre bundle Y with the structure group G is canonically equipped with a structure of the P"-associated fibre bundle Y" with the structure group H. Briefly, we can write

Y=(PxV)/C^_-(PhxV)/H=Y". However, if P" 34 P"', the P"- and P-associated bundle structures on Y are not equivalent. Indeed, given bundle atlases Th of P" and I "' of P", the union of the associated ati'ases of Y has necessarily G-valued transition functions between the charts from W" and %P"'.

Since a world manifold X is assumed to be parallelizable, the structure group CL4 of the frame bundle LX is obviously reducible to the Lorentz group L. The corresponding L-principal subbundle (or simply the Lorentz subbundle) is denoted by L"X, and is said to be a Lorentz structure.

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182

By virtue of Theorem 3.8.2, there is one-to-one correspondence between the L,

principal subbundles L"X of LX and the global sections h of the quotient fibre bundle

ET = LX/L,

(3.8.13)

called the tetrad bundle. This is an LX-associated fibre bundle with the typical fibre GL4/L. Since the group CL4 is homotopic to its maximal compact subgroup SO(4) and the proper Lorentz group is homotopic to its maximal compact subgroup SO(3), the quotient CL,,/L is homotopic to the Stiefel manifold 411(4,1; R) = SO(4)/SO(3) = S3

([170], p.33) and it is homeomorphic to the topological space S3 x R'. The fibre bundle (3.8.13) is the two-fold covering of the metric bundle EPR (3.7.18). Its global sections are called the tetrad fields.

Remark 3.8.5. In gravitation theory, a pseudo-Riemannian metric g and a tetrad field h are usually identified with a physical gravitational field. At the same time, there are gravitational models where a physical gravitational field is described by an "effective" metric which differs from the geometric one [123].

are isoSince X is parallelizable, any two Lorentz subbundles LhX and morphic to each other. It follows that, by virtue of Proposition 3.8.5, there exists a vertical bundle automorphism E Gau(LX) which sends L"X onto L"X. The the fibre bundle ET -' X transforms the associated vertical automorphism 4 Pr tetrad field h into the tetrad field h'. Every tetrad field h defines an associated Lorentz atlas 41" _ ((UC, z,')) of LX such that the corresponding local sections z' of the frame bundle LX take their values into the Lorentz subbundle f)'X. Given a Lorentz atlas W", the pull-back

h" 0 t, = zch'Ol x = /iadza ®t,

(3.8.14)

of the canonical form 0j.x (3.7.5) by a local section z' is said to be a (local) tetrad form. The tetrad form (3.8.14) determines the tetrad coframes h" = h',(x)dxµ,

x E Uc,

(3.8.15)

3.8. CAUCE GRAVITATION THEORY

183

in the cotangent bundle T'X. These coframes are associated with the Lorentz atlas

V

The coefficients hµ of the tetrad form and the inverse matrix elements (3.8.16)

hQ = SQ o zh

are called the tetrad functions. Given a Lorentz atlas W", the tetrad field h can be represented by the family of tetrad functions {hQ}. In particular, we have the well-known relation g = ha 0 h°nab,

9;_ --haµhb gl

ab

,

between the tetrad functions and the metric functions of the corresponding pseudoRiemannian metric g : X -e Epa.

Remark 3.8.6. Since a world manifold X is assumed to be parallelizable, it admits global tetrad forms (3.8.14). In the general case of a manifold X provided with a Lorentz structure, there also exists a Lorentz atlas such that the temporal tetrad form h° is globally defined. This is a consequence of the fact that the Lorentz group L is reducible to its maximal compact subgroup SO(3) and, therefore, there exists an SO(3)-principal subbundle LQX C LhX C LX, called a space-time structure. The corresponding global section

of the quotient fibre bundle L"X/SO(3) - X with the typical fibre R3 is a 3dimensional spatial distribution FX C TX on X. Its generating 1-form written relative to a Lorentz atlas is precisely the global tetrad form h° (160). We then have the corresponding space-time decomposition

TX=FX0NF, where NF is the 1-dimensional fibre bundle defined by the tetrad frame h° = ho8M. In particular, if the generating form h° is exact, the space-time decomposition obeys Hawking's condition of stable causality [88].

Given a tetrad field It, let LhX be the corresponding reduced Lorentz subbundle. Since X is non-compact and parallelizable, the principal bundle L"X can be extended uniquely (up to autornorphisms) to a L,-principal bundle P" X [60). We have the principal bundle morphism

z":Ph-LhXCLX

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184

over X such that

zh o Rs = R.,(s),

V9 E L.

This is an h-associated pseudo-Riemannian spin structure on a world manifold. We will call P' the h-associated principal spinor bundle. Note also that every Lorentz atlas {z,) of LX gives rise to an atlas of the principal spinor bundle P". Let us consider the L"X-associated fibre bundle of Minkowski spaces

MhX = (LhX X M)/L = (P" x M)/L,

(3.8.17)

and the P^-associated spinor bundle

S"=(P"xV)/L

(3.8.18)

called hereafter the h-associated spinor bundle. By virtue of Proposition 3.8.6, the fibre bundle MX (3.8.17) is isomorphic to the cotangent bundle

T'X

(L"X x M)/L = MhX.

(3.8.19)

Then there exists the representation

I 7h:T'X ®S"=(Ph X (M® V))/L,-(P"x7(M®V

(3.8.20)

of covectors on X by the Dirac 7-matrices on elements of the spinor bundle S". Relative to an atlas {zc} of P" and the associated Lorentz atlas {z = zh o zo} of LX, the representation (3.8.20) reads

Y'(7h(h° ®v)) =7°AByB(v),

v E Sz,

where yA are the corresponding bundle coordinates of S", and h° are the tetrad coframes (3.8.15). For brevity, we can write h° = 7h (h°) = 7

dx'' = 7h (dx") = h;7°.

Sections s" of the h-associated spinor bundle S" (3.8.18) describe Dirac fermion fields in the presence of the tetrad field h. Indeed, let At, be a principal connection on S", and let

DAs : J'S" - TeX 0 s S", DA,, _ (VA A

-

0 8A,

3.8. GAUGE GRAVITATION THEORY

185

be the corresponding covariant differential (2.3.8), where

VSh=ShxSh. X

The first order differential Dirac operator is defined on S" by the composition Dh = 7h o DA,, : J'S" -y T'T07 Sh 7_S h]

(3.8.21)

yA C A, = h.(-Y')AD(ye - I abALabAny°) Remark 3.8.7. The spinor bundle Sh is a complex fibre bundle with a real structure group over a real manifold. Of course, one can regard such a fibre bundle as the real one. In particular, the jet manifold j I Sh with coordinates (x", yA, ya) is defined as usual.

The h-associated spinor bundle Sh is equipped with the fibre spinor metric

ah:S"xSh-.R, x ah(v, v) = 2 (v+ry°ti + v'+ry°v),

V, t/ E Sh.

Using this metric and the Dirac operator (3.8.21), one can define the Dirac Lagrangian density on J'S" in the presenceof a background tetrad field h and a background connection Ah on S" as

La:JIS"-,AT'X, Lh = [ah(iDh(w), w) - rnah(w, w)1 h° A ... A h3,

w E J'Sh.

Its coordinate expression is ,Ch =

ha[y,+e(7°'Y°)AU(ya -

(b A - 2AA

2Aa"L,.b"cyc)

-

(3.8.22)

myA(7°)Aeye} det(hµ).

Remark 3.8.8. Spin connections. Note that there is one-to-one correspondence between the principal connections, called spin connections, on the h-associated prin-

cipal spinor bundle P" and the principal connections, called Lorentz connections, on the L-principal bundle L"X as follows. Let us first recall the following theorem ([1031, p.79).

CHAPTER 3. LACRANCIAN FORMALISM

186

THEOREM 3.8.7.

Let P - X and P - X be principle bundles with structure

groups C' and C, respectively. Let 4) : P -+ P be a principal bundle morphism over X with the corresponding homomorphism C -- C. For every principal connection A' on P', there exists a unique principal connection A on P such that T4) sends the horizontal subspaces of A' onto the horizontal subspaces of A. 0 It follows that every principal connection Ah = dx" ®(8a + 2 Aaabeb)

(3.8.23)

on P^ defines a principal connection on L"X which is given by the same expression (3.8.23). Conversely, the pull-back zpwA on P" of the connection form WA of a Lorentz connection Ah on L"X is equivariant under the action of group L. on Ph and, consequently, it is a connection form of a spin connection on P". In particular, the Levi-Civita connection of a pseudo-Riemannian metric g gives rise to the spin connection with the components A,,ab

= r)"ha(Baltk - h'(,\14,,))

(3.8.24)

on the g-associated spinor bundle S9. In gauge gravitation theory, Lorentz connections are treated as gauge potentials associated with the Lorentz group. At the same time, every world connection K on a world manifold X defines a spin connection on any h-associated principal spinor

bundle P". This enables us to reduce gauge gravitation theory to metric-affine gravitation theory in the presence of Dirac fermion fields 169, 166).

Note that, in accordance with Theorem 3.8.7, every Lorentz connection Ah (3.8.23) on a reduced Lorentz subbundle L"X of LX induces a world connection K (3.7.1) on LX whose coefficients are KA" = ItkOah"k + r1k ha"hk,Aaa.

At the same time, every principal connection K on the frame bundle LX defines a Lorentz connection K" on an L-principal subbundle L h X as follows. It is readily observed that the Lie algebra of the general linear group CL4 is the direct sum e(CL4) = 9(L) ® m

3.8. GAUGE GRAVITATION THEORY

187

of the Lie algebra g(L) of the Lorentz group and a subspace m such that ad(l)(m) C m,

`dl E L.

Let WK be a connection form of a world connection K on LX. Then, by the wellknown theorem ((103], p.83), the pull-back on L"X of the g(L)-valued component wL of WK is a connection form of a principal connection Kh on the reduced Lorentz subbundle L"X. To obtain the connection parameters of Kh, let us consider local connection 1-forms of the connection K with respect to a Lorentz atlas 4+" of LX given by the tetrad forms h°. This reads 2 *WK = KabkdxA (9 ebk,

KAbk = -h.BAhk +

where {ebk) is the basis of the Lie algebra of the group CL4. Then, the Lorentz part of this form is precisely the local connection 1-form of the connection Kh on L"X. We have 1

(3.8.25)

AA °b = (7h µ - rlk°h°,)(OAhk - k

If K is a Lorentz connection Ah, then obviously Kh = Ah. The connection Kh on L"X given by the local connection 1-form (3.8.25) defines the corresponding spin connection on S" 1 Kh = dxA ® (8A +

(7!

hp - ill-hb)(8Ahk - hkKa"v)LbAB198A),

(3.8.26)

4

where Lb are the generators (3.8.7) [69, 166]. Such a connection has been considered in [5, 154, 179]. Substituting the spin connection (3.8.26) in the Dirac operator (3.8.21) and the Dirac Lagrangian density (3.8.22), we obtain a description of Dirac fermion fields in the presence of arbitrary linear connection on a world manifold, not only of the

Lorentz type. . One can use the connection (3.8.26) in order to obtain a horizontal lift onto S" of a vector field r on X. This lift reads r7"h,°,)(BAItk

rK,, = rABA + 9Ta(Yjkbhµ

-

- h1kKA"v)L°bAByB8A.

(3.8.27)

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188

Moreover, we have the canonical horizontal lift

r49.% + 1(ihµ -,i hµ)(ra8Ahk - hk8rµ)Lb"eal 8A

(3.8.28)

of vector fields r on X onto the h-associated spinor bundle S". Remark 3.8.9. To construct the canonical lift (3.8.28), one can write the canonical

lift of r on the frame bundle LX with respect to a Lorentz atlas *" and take its Lorentz part. Another way is the following. Let us consider a local non-vanishing vector field r and a local world symmetric connection K for which r is an integral section (see Remark 3.7.6). The horizontal lift (3.8.27) of r by means of this connection is given by the expression (3.8.28). In a straightforward manner, one can check that (3.8.28) is a well-behaved lift of any vector field r on X. The canonical lift (3.8.28) is brought into the form

T = r{} - (i7kbhN 4

°hbµ)I? V "e L ab"BYBVA,

where r{} is the horizontal lift (3.8.27) of r by means of the spin Levi-Civita connection (3.8.24) of the tetrad field h, and are the covariant derivatives of r relative to the same Levi-Civita connection. This is precisely the Lie derivative of spinor fields described in [53, 107]. The canonical lift (3.8.28) fails to be a generator of general covariant transformations because it does not involve transformations of tetrad fields. To define general covariant transformations of spinor bundles, we should consider spinor structures

associated with different tetrad fields. The difficulties arise because, though the principal bundles L"X and L"X are isomorphic, the associated structures of bundles of Minkowski spaces MIX and M"'X (3.8.19) on the cotangent bundle T'X are not equivalent (see Remark 3.8.4). As a consequence, the representations -11, and ,y". (3.8.20) for different tetrad fields h and h' are not equivalent [160, 164]. Indeed, let V = tµdxP = taha = t' hja,

be an element of T'X. Its representations ry" and -y" (3.8.20) read 'yh(t*) = to'y° = tµha-'°,

'Yh'(t*) = t'ta = tµh'a-fa

3.8. GAUGE GRAVITATION THEORY

They are not equivalent because no isomorphism condition

7"'(t*) =

189

of S" onto S" can obey the

dt' E T'X.

It follows that every Dirac fermion field must be described in a pair with a certain tetrad (gravitational) field. We thus observe the phenomenon of symmetry breaking in gauge gravitation theory which exhibits the physical nature of gravity as a Higgs field [160]. The goal is to describe the totality of fermion-gravitation pairs.

Remark 3.8.10. All spin structures on a manifold X which are related to the two-fold universal covering groups possess the following two properties [84].

Let P -+ X be a principal bundle whose structure group C has the fundamental group a, (C) = Z2. Let d be the universal covering group of C. 1. The topological obstruction to the existence of a G-principal bundle P - X covering the bundle P - X is given by the tech cohomology group H2(X; Z2) of X with coefficients in Z2. Roughly speaking, the principal bundle P defines an element of H2(X; Z.2) which must be zero so that P -+ X can give rise to P -. X. 2. Non-equivalent lifts of P -+ X to G-principal bundles are classified by elements of the tech cohomology group HI (X; Z2). In particular, the well-known topological obstruction to the existence of a Riemannian spin structure (see Example 3.8.11) and a pseudo-Riemannian spin structure is the second Stiefel-Whitney class w2(X) E H2(X; Z2) of X ([118], p.82). In the case of 4-dimensional non-compact manifolds, all Riemannian and pseudoRiemannian spin structures are equivalent [7, 60).

Example 3.8.11. Riemannian spin structure. Let us consider spin structures on Riemannian manifolds. Let X be an arbitrary 4-dimensional oriented manifold.

The structure group CL4 of the frame bundle LX is reducible to the maximal compact subgroup SO(4) because the quotient CL4/SO(4) is homeomorphic to the Euclidean space R10. It follows that a Riemannian metric gR, represented by a section of the quotient fibre bundle ER = LX/SO(4) - X, always exists on a manifold X. The corresponding SO(4)-principal subbundle L9X is called a Riemannian structure on X.

CHAPTER 3. LAGRANGIAN FORMALISM

190

Given two different Riemannian metrics gR and g'R on X, the corresponding of LX are isomorphic as SO(4)-principal SO(4)-principal subbundles L9X and bundles (see Proposition 3.8.3). To introduce a Riemannian spin structure, one can consider the complex Clifford algebra C4,0 which is generated by elements of the vector space R4 equipped with the Euclidean metric [27, 118]. The corresponding spinor space VR is a minimal left ideal of C40 0 provided with a Hermitian bilinear form. The spin group is Spin(4) which is the two-fold universal covering group of the group SO(4), and is isomorphic to SU(2)®SU(2) ([34], p.430).

Let us assume that the second Stiefel-Whitney class w2(X) of X vanishes. A Riemannian spin structure on a manifold X is defined as a pair of a Spin(4)-principal

X and a principal bundle morphism z. of P. to LX. Since such a bundle P. morphism factorizes through a bundle morphism

zy:P.-+L9X for a Riemannian metric 9R, this spin structure is a gR-associated spin structure. We will denote the corresponding gR-associated principal spinor bundle by P-9. All these bundles on a 4-dimensional manifold X are isomorphic [7].

Example 3.8.12. Universal spin structure. The group GL4 is not simplyconnected. Its first homotopy group is ai(GL4) = xi(SO(4)) = 72 ([82], p.27). Therefore, GL4 admits the universal two-fold covering group GL4 such

that the diagram

UL-4 -. GL4 (3.8.29)

Spin(4) -i SO(4) is commutative [91, 118, 151, 172].

A universal spin structure on X is defined as a pair of a CL4-principal bundle LX X and a principal bundle morphism

LX

X

3.8. GAUGE GRAVITATION THEORY

191

[39, 151, 1721. There is the commutative diagram

LX -. LX (3.8.30)

1 1

Pg - LgX for any Riemannian metric gR [151, 1721. Since the group CL4 is homotopic to the group Spin(4), there is one-to-one correspondence between the non-equivalent universal spin structures and non-equivalent

Riemannian spin structures [172]. In our case, all universal spin structures as well as the Riemannian ones are equivalent. The group CL4 has finite-dimensional representations, but its spinor representation is infinite-dimensional (91, 1451. Elements of this representation are called world spinors, and their field model has been developed (see [911 and references therein). At the same time, the following procedure enables us not to exceed the scope of standard fermion models. Let us consider the commutative diagram

LX s-+ LX (3.8.31)

ER

and the composite fibre bundle

LX - ER -: X, where LX -, ER is a Spin(4)-principal bundle. For each pseudo-Riemannian metric On, the restriction of the Spin(4)-principal bundle LX - ER to gR(X) C ER is isomorphic to a gR-associated principal spinor bundle P9 (see Remark 3.8.4). Therefore, the diagram (3.8.31) is said to be the universal Riemannian spin structure. Let us consider the composite spinor bundle

S. ER -+ X,

(3.8.32)

ER is the spinor bundle associated with the Spin(4)-principal bundle where S LX -, ER. Then, whenever On is a Riemannian metric on X, the sections of the spinor bundle SO associated with the principal spinor bundle Pg from the commutative diagram (3.8.30) are in bijective correspondence with the sections s of the

CHAPTER 3. LAGRANGIAN FORMALISM

192

composite spinor bundle (3.8.32) which project onto gR, that is, 7rsr o s = gR (see Remark 3.8.13). In order to relate this model to the above-mentioned model of world spinors, let us note that the total space S of the spinor bundle (3.8.32) has the structure of a fibre bundle which is associated with the GL4-principal bundle LX - X and whose typical fibre is the quotient (UL-4 X VE)/Spin(4)

(3.8.33)

(see Remark 3.8.14). Then, a morphism of the quotient (3.8.33) to the spin representation space of the group UL-4 yields the corresponding morphism of the composite spinor bundle (3.8.32) to the GL4-associated bundle of world spinors.

The construction above on composite fibre bundles illustrates the standard description of spontaneous symmetry breaking in gauge theories where matter fields admit only exact symmetry transformations [159, 1621.

Remark 3.8.13. Spontaneous symmetry breaking. Spontaneous symmetry breaking is a quantum phenomenon. In classical field theory, spontaneous symmetry breaking is modelled by classical Higgs fields. In gauge theory on a principal bundle P - X, the necessary condition for spontaneous symmetry breaking to take place is the reduction of the structure group G of this principal bundle to a dosed subgroup

H of exact symmetries [96, 99, 146, 1781. The topological obstructions to this reduction has been discussed in Remark 3.8.4. Higgs fields are described by global

sections h of the quotient fibre bundle E = P/H - X (3.8.10). In accordance with Theorem 3.8.2, the set of Higgs fields h is in bijective correspondence with the set of reduced H-principal subbundles Ph of P. Given such a subbundle Ph, let

Yh = (Ph X V)/H

(3.8.34)

be the associated fibre bundle with a typical fibre V which admit a representation of the group H of exact symmetries, but not the whole symmetry group G. Its sections describe matter fields in the presence of the Higgs fields h. In general, the fibre bundle Y' (3.8.34) is not associated (see Proposition 3.8.3) or canonically associated (see Remark 3.6.1) with other H-principal subbundles Ph' of P. It follows that, in this case, V-valued matter fields can be represented only by pairs with Higgs fields. The goal is to describe the totality of these pairs (sh, h) for all Higgs fields.

3.8. GAUGE GRAVITATION THEORY

193

Let us consider the composite fibre bundle (3.8.9) and the composite fibre bundle

YEM E

X

(3.8.35)

where Y --+ E is the fibre bundle

Y = (PxV)/H associated with the H-principal bundle PE (3.8.11). Given a global section h of the fibre bundle E --+ X (3.8.11), let

Yh = (P" x V)/H

(3.8.36)

be a fibre bundle associated with the H-principal subbundle P" of P. There is the canonical injection

ih:Y"=(P"xV)/H--+Y over X whose image is the restriction

h'Y = (h'P x V)/H of the fibre bundle Y -' E to h(X) C E, i.e., ih(Y") = 7rrE(h(X))

(3.8.37)

Then every global section Sh of the fibre bundle Y" corresponds to the global section ihosh of the composite fibre bundle (3.8.35). Conversely, every global section s of the composite fibre bundle (3.8.35) which projects onto a section h = iryE os of the fibre bundle E -+ X takes its values into the subbundle ih(Y") C Y in accordance

with the relation (3.8.37). Hence, there is one-to-one correspondence between the sections of the fibre bundle Y" (3.8.36) and the sections of the composite fibre bundle (3.8.35) which cover h. Thus, it is precisely the composite fibre bundle (3.8.35) whose sections describe the above-mentioned totality of pairs (s", h) of matter fields and Higgs fields in gauge theory with broken symmetries [159, 164]. The feature of the dynamics of field systems on composite fibre bundles consists in the following. Let the composite fibre bundle Y (3.8.35) be provided with coordinates (x, an, y'), where (x, a') are bundle coordinates of the fibre bundle E -, X. Let

AE=dxA®(O +A;,O,)+da'"®(8,"+Am8i)

(3.8.38)

CHAPTER 3. LAGRANGIAN FORMALISM

194

E. This connection defines the be a principal connection on the fibre bundle Y splitting (2.5.6) of the vertical tangent bundle VY and leads to the vertical covariant differential (2.5.7) which reads

DA =dx"®(ya-Aa-Ama )8;.

(3.8.39)

The operator (3.8.39) possesses the following important property. Given a global section h of E - X, its restriction

Dh=DAEOJ'ih:J'Y"-+T'X®VYh, Dh=dz"®(ya-Aa-Am8"h'")8;,

(3.8.40)

to Yh is precisely the familiar covariant differential relative to the principal connection At, = dz" ® [8" + (Am 8"h"' + A;,)8;]

on the fibre bundle Y" - X which is induced by the principal connection (3.8.38) on the fibre bundle Y - E ([1031, p.81).

Thus, we may construct a Lagrangian density on the jet manifold J'Y of a composite fibre bundle which factorizes through DA, that is,

L: J'Y T)+T'X ®VYE -. T'X.

(3.8.41)

Remark 3.8.14. The total space of the composite fibre bundle Y - X (3.8.35) has the structure of the P-associated bundle

Y = (P x (C x V)/H)/G, where the elements (p, g, v) and (pab, b''g, a-'v) for all a E H and b E G are identified. Its typical fibre is the quotient (C x V)/H of the product G x V by identification of the elements (g, v) and (ag, a 'v) for all a E H. The group G act on this typical fibre by the rule

X admits the In particular, if the typical fibre V of the composite fibre bundle Y action of the group C, these two bundle structures on Y are equivalent. 9

3.8. GAUGE GRAVITATION THEORY

195

Let us turn now to fermion fields in gauge gravitation theory, basing our consideration on the following two facts. PROPOSITION 3.8.8. The L-principal bundle

Pl, = GL4 - CL4/L

(3.8.42)

is trivial.

Proof. In accordance with the classification theorem ([1701, p.99), a C-principal bundle over an n-dimensional sphere S" is trivial if the homotopy group 7rn_1(G) is trivial. The base space Z = GL4/L of the principal bundle (3.8.42) is homeomorphic to S3 x R'. Let us consider the morphism f, of S3 into Z, f,(p) = (p,0), and the S. Since L is homeomorphic to RP3 x R3 pull-back L-principal bundle fl PL and or2(L) = 0, this bundle is trivial. Let fz be the projection of Z onto S3. Then, the pull-back L-principal bundle fs (f 1 PL) - Z is also trivial. Since the composition fi o f2 of Z into Z is homotopic to the identity morphism of Z, the Z is equivalent to the bundle Pi, ([170], p.53). It follows that bundle fs (fi P1,) QED the bundle (3.8.42) is also trivial. PROPOSITION 3.8.9. As in (3.8.29), we have the commutative diagram

CL4 -, GL4

0 GL4 to the Lorentz Proof. The restriction of the universal covering group GL4 group L C CL4 is obviously a covering space of L. Let us show that this is the universal covering space. Indeed, any non-contractible cycle in GL4 belongs to some subgroup SO(3) C GL4 and the restriction of the covering bundle UL-4 -, CL4 to SO(3) is the universal covering of SO(3). Since the proper Lorentz group is homotopic to its maximal compact subgroup SO(3), its universal covering space QED belongs to GL4. Following Example 3.8.12, let us consider the universal spin structure LX -' X. It is unique since X is parallelizable. In virtue of Proposition 3.8.9, we have the

CHAPTER 3. LACRANCIAN FORMALISM

196

commutative diagram

LX - LX Ih

L^X

for each tetrad field h (see also [56]). It follows that the quotient EX IL. is precisely the quotient Dr (3.8.13) so that there is the commutative diagram

LX -i LX (3.8.43) F_r

By analogy with the diagram (3.8.31), the diagram (3.8.43) is said to be the universal Dirac (pseudo-Riemannian) spin structure. We have the composite fibre bundle LX

- Fr

X,

(3.8.44)

where LX -+ F.r is an L.-principal bundle. The universal Dirac spin structure (3.8.43) can be regarded as the L.-spin structure on the bundle of Minlaowski spaces

EM=(LX x M)/L- Dr associated with the L-principal bundle LX -. ET. Since the frame bundle LX and the fibre bundle P1, (3.8.42) are trivial, the fibre bundle EM -' Dr is also trivial. Hence, it is isomorphic to the pull-back F.r

x

T'X.

(3.8.45)

Since the fibre bundle F.r - X is trivial, the fibre bundle EM is isomorphic to the trivial bundle of Minkowski spaces over the product S3 x R' x X. It follows that the set of non-equivalent spin structures on the bundle EM is in bijective correspondence with the cohomology group H' (S3 x R' x X; Z2) ((118], p.82). Since the cohomology group Hl (S3; Z2) is trivial and a spin structure on S3 is unique (40], one can show that non-equivalent spin structures on EM are classified by elements of the cohomology group HI (X; Z') and, consequently, by non-equivalent spin structures on X. It follows that the spin structure (3.8.43) on the fibre bundle EM is unique.

3.8. CA UGE GRAVITATION THEORY

197

Following the general procedure of describing spontaneous symmetry breaking in Remark 3.8.13, let us consider the composite spinor bundle

S

Fr M. X,

(3.8.46)

where

S = (LX x V)/L. is the spinor bundle S -+ E.r associated with the L.-principal bundle LX -+ E.r. Given a tetrad field h, there is the canonical isomorphism

ih:S''=(PhxV)/L.- (h'LX xV)/L. of the h-associated spinor bundle Sh (3.8.18) onto the restriction h'S of the spinor bundle S -. Esr to h(X) C Dr. Then, every global section sh of the spinor bundle Sh corresponds to the global section ih o sh of the composite spinor bundle (3.8.46). Conversely, every global section s of the composite spinor bundle (3.8.46) which projects onto the tetrad field h takes its values into the subbundle ih(Sh) C S.

Let the frame bundle LX - X be provided with a holonomic atlas (3.7.3) and the principal bundles LX -+ Fir and LX -. Dr have the associated atlases (z,", U.) and {zz = z" o z,, UL}. With these atlases, the composite spinor bundle S is equipped with the bundle coordinates (x'', a;, y'l), where (', a;) are coordinates of ET such that aQ are the matrix components of the group element (TIC o zj(a), we have a E U., aEX (a) E Uc. For each section h of

(aa o h)(x) = ha(x), where h; (x) are the tetrad functions (3.8.16). The composite spinor bundle S is equipped with the fibre spinor metric as(v, v') = 2(v+ry°v' + z/+7°v),

7rst(v) = 7rsE(v )

Since the fibre bundle of Minkowski spaces EM -+ Er is isomorphic to the pullback bundle (3.8.45), there exists the representation

ryE : T'X ® S = (LX x (M ® V))/L. - (LX x ry(M ® V))/L. = S given by the coordinate expression

jx'' = 7E(dx'') = so-y".

(3.8.47)

CHAPTER 3. LACRANGIAN FORMALISM

198

Restricted to h(X) C F.r, this representation recovers the morphism ryh (3.8.20).

Using this representation, one can construct the total Dirac operator on the composite spinor bundle S as follows. Since the fibre bundles which make up the composite fibre bundle (3.8.44) are trivial, let us consider a principal connection AE (3.8.38) on the L.-principal bundle LX -, ET given by the local connection 1-form AE =

(Aa°bdx-' + Akabdok)

® Lab,

(3.8.48)

where Arab

(nkaQa _

2

Aµab

`°aµ)a"Ka".,

= 2 (71 C, - r/k6BN

(3.8.49)

and K is a world connection on X. We choose this connection because of the following properties. The principal connection (3.8.48) defines the associated spin connection

As = dx' 0 (8 +

A%GbLabAByBOA) + dok ® (8N + Z ANabLbAByBBA)

(3.8.50)

on the spinor bundle S - E'r. Let h be a global section of ET X and S' the restriction of the bundle S -. Err to h(X). It is easily seen that the restriction of the spin connection (3.8.50) to Sh is precisely the spin connection (3.8.27). The connection (3.8.50) yields the first order differential operator DAs (2.5.7) on the composite spinor bundle S X, and reads

DAs : J'S - T'X O S, DAs = dxk 0 [ya - i (Aaab + ANabo' )LabAByBJOA = dx" ®(ya -

1

(3.8.51)

@ (rltbaµ _ 17b°ou)( 01 k - okKa"v)LabAByBJOA

The corresponding restriction Dr, (3.8.40) of the operator DAs (3.8.51) to J1Sh C J'S recovers the familiar covariant differential on the h-associated spinor bundle X relative to the spin connection (3.8.27). Sh

3.8. GAUGE GRAVITATION THEORY

199

Combining (3.8.47) with (3.8.51), we obtain the first order differential operator

D=7EoD: J'S-+T'X ®S -i S,

(3.8.52)

FIT Qn7o8A

yB o D =

WA - 4

Qµ - nk°ap)(° k - QkKaµv)L6AcUC],

X. One can think of 1) as being the total Dirac operator on S because, for every tetrad field h, the restriction of D to J'S" C J'S is exactly the Dirac operator Dh (3.8.21) on the h-associated spinor bundle S" in the presence of the background tetrad field h and the spin connection (3.8.27). on the composite spinor bundle S

1 It follows that gauge gravitation theory reduces to a model of the metric-affine

gravity and Dirac fermion fields. 1

The total configuration space of this model is the jet manifold J'Y of the fibred product

Y=CKXS

(3.8.53)

where CK is the bundle of world connections (3.7.39). It is provided with coordinates (x", o;, kµ°6, yA),

Let JEY denote the first order jet manifold of the fibre bundle Y -' Dr. This fibre bundle can be provided with the spin connection

Ay : Y-JEY-JES, Ay = dxa ® (8,, + AaobLA ByBOA) + dog ® (0, + Aµ°bL ByBoA),

(3.8.54)

where Aµ°b is given by the expression (3.8.49) and Ahab

=-

(nkboµ _ 77

0,,b

2

Using the connection (3.8.54), we obtain the first order differential operator

Dy:J'YT'X®S, DY = &x''

]ea

4

(,7kbQµ - n1-Qµ)( a.P %k - Q,kAµv)LbA8yB]8A'

(3.8.55)

CHAPTER 3. LAGRANCIAN FORMALISM

200

and the total Dirac operator

Dy=7EODy:J'Y- T'X®S

(3.8.56)

S,

ET

YB

o DY =

0,n1'aBA [YA

- 4 (ea,. - nk'ob)(ak - akka",.)L bACyc],

on the fibre bundle Y X. CK, the restrictions of the spin connection Ay (3.8.54), Given a section K : X the operator Dy (3.8.55) and the Dirac operator Dy (3.8.56) to the pull-back K'Y are exactly the spin connection (3.8.50) and the operators (3.8.51) and (3.8.52), respectively.

The total Lagrangian density on the configuration space J'Y of the metric-affine gravity and fermion fields is the sum (3.8.57)

L = LMA + LD

of a metric-affine Lagrangian density LMA (R A $, a'),

ovY = oav

vvynoa

,

and the Dirac Lagrangian density LD = [ay(iD(w), w) - mas(w, w)]a° A ... A o3,

w E J'S,

and ay is the pull-back of the fibre spinor metric as onto the fibre bundle Y -' (CK x E.r). Its coordinate expression is

where o' = ,CE)

= { 2av (yA1(7°7°)AB(y1 - 4 (nkao (y aX+A -

nk°aµ)(aak 4

myA(7°)AB?}

_a j, I

akkaµv)L6Bcf) -

- akka"v)yCL.+bCA(7°7°)ABYB] -

(3.8.58)

a = det(ow)

It is readily observed that

8k

+ 8kG a

- 0,

that is, the Dirac Lagrangian density (3.8.58) depends only on the torsion (3.7.44) of a world connection. Let us turn now to general covariant transformations.

3.8. GAUGE GRAVITATION THEORY

201

Since a world manifold X is parallelizable and the universal spin structure is unique, the GL4-principal bundle LX -. X as well as the frame bundle LX admits the canonical lift of any diffeomorphism f of the base X. This lift is defined by the commutative diagram LX

LX

LX - - LX

LI where t is the holonomic bundle automorphism of LX (3.7.6) induced by f [39). The associated morphism of the spinor bundle S (3.8.46) is given by the relation

Dr is equivariant, this is a fibre-to-fibre automorphism of the bundle S over the canonical automorphism 4i£ of the LX-associated tetrad bundle Dr - X (3.8.13) induced by the diffeomorphism f of X. Thus, we have the commutative diagram of general covariant transformations of the spinor bundle S: S -. S

£4.

I

I

X . X Accordingly, there exists a canonical lift Ts onto S of every vector field r on X. The goal is to find its coordinate expression. Difficulties arise because the tetrad coordinates a; of Dr depend on the choice of an atlas of the bundle LX -, Er. Therefore, non-canonical vertical components appear in the coordinate expression of T.

A comparison with the canonical lift (3.7.19) of a vector field r onto the metric bundle EPR shows that a similar canonical lift of r onto the tetrad bundle ET takes the form fE = raOA +

8M ear p

+ Q 84-

(3.8.59)

CHAPTER 3. LAGRANCIAN FORMALISM

202

where the terms Q obey the condition (Q;ae + Q;ob Mob = 0.

The term Q;aµ is the above-mentioned non-canonical part of the lift (3.8.59). Let us consider a horizontal lift of the vector field iE onto the spinor bundle

S - E by means of the spin connection (3.8.50). It reads

ASS = raaa + a"ea: a

+

µ)a,(49"Tµ

- KaµvrV)(LbAByBBA+ L.+AByA8B) +

8

`` aBc + 4Qk(rlkeaµ - nk°a°)(L"ABYBOA + Moreover, following Remark 3.8.9, we obtain the desired canonical lift of r onto S: Ts = r" 8,, +

(3.8.60)

1 8 (kb 7 A +aB + 4Qk n - +I lab)µ (LbAayBBA + L+ as syA ), Q` aa" which can be written in the form

rs = raaa +

8L9 + acp Qi"t(+Ikba, _ qC6sµ

-Lab-aa 8o + L,bAByBaA +

where L,bd,, are the generators (3.8.6). The corresponding total vector field on the fibred product Y (3.8.53) reads

fy=T+t9, f = r.ax + S'rµa"" ', +

(3.8.61)

8Ar" k,,° . - 8µr%°A + 8µpr°)

4Qk(n"sµ - fJ"P°aµ)[-Ledad8a.11

8k w°A

+ L bAByB8A + L.+AByAOB].

Its canonical part 7 (3.8.61) is the generator of a local 1-parameter group of general covariant transformations of the fibre bundle Y, whereas the vertical vector field i9

203

3.8. GAUGE GRAVITATION THEORY

is the generator of a local 1-parameter group of principal (Lorentz) automorphisms of the fibre bundle S E. By construction, the Dirac Lagrangian density (3.8.57) obeys the relations

L.n,LD = 0. LJI LMA = 0,

(3.8.62)

Lj,TLD = 0.

(3.8.63)

The relation (3.8.62) leads to the Nother conservation law. Let us analyse the equalities (3.8.63) in order to obtain the SEM conservation law of the metric-affine gravity and fermion fields. Using the compact notation, let us rewrite the vector field f (3.8.61) in the form =7-08µ + 0,,7-"ff.'

+ (uAQ807° + 8µ 800

8pT°)8A.

Due to the arbitrariness of the functions r°, the equalities (3.8.63) lead to the strong equalities O°GMA + 20Pµ6OLMA + 1LAQbAGMA + d (i1LAQ) = U°V GMA

(3.8.64)

and bQLD + y v tQ + 80° Oaa` + BALpu°0

(3.8.65)

k

ID aµ + 8LD_A Z/° +

04

8LD + + VOA I

where

ota=o;8e. 0

We also have the relations (3.7.45), (3.7.46) and

_

8LD '8k-x-µ.

8LD

(3.8.66)

87a:#

The corresponding SEM conservation law reads 0 S: -da[8A1CMA(yo7-° 80°D

-

(89T°aa - 0°KTµ) +

uA0a007.°

- uAa , y°7_°

OyAA

+

.o) 8JCD

-

Ay AT° - TAC).

OJ

(3.8.67)

CHAPTER 3. LAGRANCIAN FORMALISM

204

Substituting the term V 88GMA in (3.8.64), and the term

OLD 0"811D 80O ac + W,9 °

8GD Y.+ + "MAA

in (3.8.65), in (3.8.67), we bring this conservation law into the superpotential form

0 rz -da[2a4T°bw,G + (k"",,b a'G b"°"G8"T°

- dd(&°"G)T° + d"(xva°a(8vT° - kv°vr°))] -

+ (3.8.68)

(80

da[ 8GDa.1'+ 8GD a" 8 T° . a Sex. a) " ]

In virtue of the relations (3.8.66), the last term in the expression (3.8.68) vanishes, that is, neither fermion fields nor Proca fields contribute to the superpotential. It follows that the SEM conservation law (3.8.67) leads to the form (3.5.18), where U is the generalized Komar superpotential (3.7.54). Thus, one could say that the generalized Komar superpotential (3.7.54) appears to be a universal superpotential for gravitation models.

3.9

Appendix. Gauge mechanics

The usual formulation of time-dependent mechanics implies a splitting Y = R x M of the event manifold Y and the corresponding splittings R x T'M and R x TM of phase and configuration spaces (see [18, 33, 49, 119, 135, 143] and references therein). Here we describe Lagrangian mechanics in a frame-free form as the particular case of Lagrangian field theory when the event space Y is a fibred manifold over a 1dimensional base R. The main ingredients in this formulation are a connection on the event fibred manifold Y - R which is a reference frame and a dynamic connection on the jet bundle J'Y Y which is associated with a dynamic equation.

A. Fibred manifolds over R As is well known, classical Newtonian systems are described by an event space Y which is stratified over R, that is, Y is equipped with the global Newtonian time

t:Y-R

(3.9.1)

with dt # 0 everywhere on Y. Thus Y is a fibred manifold over R (but not necessarily a fibre bundle).

3.9. APPENDIX. GAUGE MECHANICS

205

Remark 3.9.1. Hereafter, the base R is parameterized by the coordinates t with the transition functions e = t+oonst. Relative to these coordinates, R is equipped with the standard vector field 8t and the standard 1-form dt, which is also the volume element on R.

When dealing with the event space Y, we will always use fibred charts (U; t, compatible with the fibration (3.9.1).

it

Its Let J1 Y be the first order jet manifold of the fibred manifold Y coordinates will be denoted by (t, y', y;). The canonical morphism (2.1.19) takes the form

A : J'Y - TY,

J = at + yi8{.

It is easy to see that the affine jet bundle P Y - Y is modelled on the vertical tangent bundle JAY = VY.

Remark 3.9.2. For the sake of simplicity, we will often identify J' Y with the corresponding subbundle of TY.

The corresponding splitting of the vertical tangent bundle VyJ'Y of J'Y - Y reads

a:VyJ1Y-J'YxVY,

a(8{) = A.

(3.9.2)

In this case, the exact sequence (2.1.18) takes the form a-1

0-+VyJ1Y 4VJ1Y `4J1YYX VY-.0. Hence, we obtain the following endomorphism of VJ'Y:

v=i0

1o7rv,

v(80 = 81"

(3.9.3)

v(8i) = 0,

which obeys the condition v o v = 0. Using the contact I -jet form 0 (2.1.20) and the corresponding morphism

J1Y x TY - - VyJ1Y, Y

at

6

a

{8t

4

&j ,

CHAPTER 3. LAGRANGIAN FORMALISM

206

we can extend the endomorphism (3.9.3) to the tangent bundle TJ'Y in accordance with the diagram

TJ'Y -° . VyJ'Y -TJ'Y (3.9.4)

VJ'Y 0 VJ'Y y(O) = 8;,

I v(8e) = -yi&,,

v(t9=0.

This is called the vertical endomorphism, which inherits the property v o v = 0. The transpose of the endomorphism v is

T'J'Y .T'J'Y, v;-(d-t) -=O,

v'(dy{) = 0,

v'(dy') _ t9= dal' - yidt,

v'ov'=0. Using the endomorphism v', one can introduce the vertical exterior differential dd = v'd

(3.9.5)

acting on the exterior algebra of forms on J'Y. For example, let ! be a function on J'Y. Then we have

dd! = Of ;. 80 A connection on the event space Y - R of a mechanical system is given by a section r of the jet bundle J'Y Y. In accordance with Remark 3.9.2, it is represented by the vector field (3.9.6)

on Y which is the horizontal lift of the standard vector field 8; on R by means of the connection r. Obviously, connections on the fibred manifold Y -' R are curvature-free connections.

Remark 3.9.3. Curvature-free connections. Recall that every connection r on a fibred manifold Y - X, by definition, yields the horizontal distribution r(TX) C TY (2.3.3) on Y. It is generated by horizontal lifts

,r=T"(aa+I°a8;)

3.9. APPENDIX. GAUGE MECHANICS

207

onto Y of vector fields r = r10% on X. The associated Pfaffian system is locally generated by the forms (dy' - I''adxa). Note that the horizontal distribution I'(TX) is involutive if and only if r is a curvature-free connection (see Remark 2.3.1). By virtue of Theorem 4.1.5, the horizontal distribution defined by a curvaturefree connection is completely integrable. The corresponding foliation on Y is transver-

sal to the foliation defined by the fibration it : Y -, X. It is called the horizontal foliation, and its leaf through a point y E Y is defined locally by an integral section sy of the connection r through y. Conversely, let Y admit a horizontal foliation such that, for each point y E Y, the leaf of this foliation through y is locally defined

by a section sy of Y - X through y. Then, the map

r(y) = j;sy,

ir(y) = x.

is well defined. This is a curvature-free connection on Y. COROLLARY 3.9.1. There is one-to-one correspondence between the curvature-free

connections and the horizontal foliations on a fibred manifold Y - X. O Given a horizontal foliation on Y --* X, there exists the associated atlas of fibred coordinates (xa, y') of Y such that every leaf of this foliation is locally generated by the equations y' =const. and the transition functions y' - y"(y') are independent of the base coordinates xa [29]. This is called the atlas of constant local trivializations. Two such atlases are said to be equivalent if their union is also an atlas of constant local trivializations. They are associated with the same horizontal foliation. COROLLARY 3.9.2. There is one-to-one correspondence between the curvature-free

connections r on a fibred manifold Y -, X and the equivalence lasses of atlases of constant local trivializations of Y such that i°a = 0 relative to the coordinates of the corresponding atlas. 0

B. Dames of reference In accordance with Remark 2.3.1, every connection on a fibred manifold Y -, R defines a horizontal foliation on Y - R. Its leaves are integral curves of the vector

CHAPTER 3. LAGRANGIAN FORMALISM

208

field (3.9.6). The corresponding Pfaffian system is locally generated by the forms (dy' - r'dt). There exists an atlas of constant local trivializations such that r= at relative to the associated coordinates. These coordinates are called adapted to r.

I A connection r: Y -' J'Y on Y -. R is called a reference frame (or simply a frame) on Y. I Given a frame I', we obtain the associated splitting

TY=R®VY, pr : TY

VY,

Pr(8i) = -08,,

Pr(8i) = 8;,

and the dual injection

V'Y Y T'Y, pr(dl}) =

r'(i) = >9i = dy' - r'dt.

The restriction of pr to J'Y C TY leads to the covariant differential ,i e°--r. ye-r, J'YIVY, pros=c-roc:I- VY.

Here c : I - Y is a (local) section of Y

(3.9.7)

R and c : I - J'Y is its first order jet

prolongation.

One can think of Or o c as being the relative velocity of the motion c with respect to the frame r. Note that Vr o c vanishes identically if and only if c is an integral section of r. The coordinate expression of Vr o c is

pros=oc)8{,

c'=y`oc,

or, simply,

Vroc=L"O,

c' =a'0c,

(3.9.8)

if the coordinates are adapted to r. A connection on a fibred manifold r on Y -' R is said to be complete if the horizontal vector field (3.9.6) is complete.

PROPOSITION 3.9.3. Every trivialization of Y - R defines a complete connection. R defines a trivialization Y = R x M. Conversely, every complete connection on Y

3.9. APPENDIX. GAUGE MECHANICS

209

The vector field (3.9.6) reduces to the vector field 01 relative to the coordinates associated with this trivialization.

Proof. Every trivialization of Y - R defines a one-parameter group of vertical isomorphisms of Y -. R, and hence a complete connection. Conversely, let r be a complete connection on Y - R. The vector field r (3.9.6) is the generator of a 1-parameter group Gr which acts freely on Y. The orbits of this action are of course the integral sections of r. Hence we obtain a projection

ar : Y -+ Y/Gr = M,

(3.9.9)

where M is the configuration space with respect to the frame r. This projection, together with the projection Y -' R, defines a trivialization

Y=RxM.

(3.9.10)

QED

Remark 3.9.4. It follows that the fibred manifold Y --+ R is a fibre bundle if and only if there exists a complete connection r on Y. In this case, Y automatically trivial since the base R is contractible.

R is

Different complete frames lead to different trivializations (3.9.10) which differ from each other in projections (3.9.9).

Let r be a complete connection. Restricting the tangent map Tar : TY -. TM to the vertical tangent bundle VY, we obtain the following isomorphism over 7rr:

VY ATM I

I

Y -' M *r We define the observed motion with respect to the frame r as

Then its velocity er : I - TM can be canonically identified with the relative velocity (3.9.7) which coincides with (3.9.8) relative to the coordinates adapted to the frame

r.

CHAPTER 3. LAGRANCIAN FORMALISM

210

Extending this construction to the jet manifold J1Y, we obtain the following diagram

J'Y (`'yamr) R x TM ! Y

1

\(t,wr)

RxM

t if pr' R which, in the adapted coordinates, simply reads (t,y',ye)'-'

yi).

C. Dynamic equations Let J2Y be the second order jet manifold of Y -' R provided with fibred coordinates (t, yt, yi, y`a). Let c : I - Y be a motion. Its second order jet prolongation is denoted by c : I - J2Y. We then have the diagram

I'-- R where J2Y - P Y is an affine bundle modelled on the vector bundle

VYJ'Y = J'Y Y VY

J'Y.

(3.9.11)

Note that VJ.YJ2Y = j2y X VY C TJ2Y. Y

There is the canonical splitting (2.2.8) of the tangent space TJ'Y over JPY:

J2Y x TJ'Y = R ® VJ'Y,

Y fly 0: J2Y x TJ'Y - V J'Y,

J'Y t

{

(3.9.12)

3.9. APPENDIX. GAUGE MECHANICS

211

We can obtain from (3.9.12) the following affine injection

J2Y JYTJ'Y, (t)y', yt, yu) '-' (t, y', yt,1, 2/' = ya yt = YD.

(3.9.13)

A dynamic equation f is defined to be a section of the affine bundle J'Y -+ J'Y. Using the canonical injection (3.9.13), f can be seen as the vector field (3.9.14)

I e =8

on J'Y. It is characterized by the conditions v(C) = 0,

dt jC = 1,

where v is the endomorphism (3.9.4). A dynamic equation a can be regarded as a holonomic connection on the fibred

manifold J'Y -' R. It induces the splitting

TJ'Y = R ® VJ'Y,

(3.9.15)

Jay

where the horizontal line bundle over J'Y is precisely the trivial bundle generated by the nowhere vanishing vector field (3.9.14). Recalling (3.9.11), we can define the covariant differential VC associated with as the affine morphism JAY °-° VYJ'Y,

ya'-' yit - Ci.

(3.9.16)

If c : I - Y is a motion, we obtain the following splitting

c=Coe+V oC,

(3.9.17)

where a,, = 0E o c is called the (absolute) acceleration of the motion c with respect to C. Its coordinate expression is

VY

/1 I.11 Y

d'=x'oc, (3.9.18) a,

= (i' - ` o c)8{.

Recall that solutions of the dynamic equation t are the motions c such tha c =1: o c. Then these solutions can be equivalently characterized as the motions c with the zero acceleration a, = 0, i.e., geodesics.

CHAPTER 3. LAGRANGIAN FORMALISM

212

There is another consequence of J2Y - J'Y being an affine bundle. Given a dynamic equation £, any other dynamic equation S can be written in a unique way as

c=e+f,

f :J'Y---VyJ'YCTJ'Y,

where the vertical vector field f acquires the meaning of a force. One may think of f as being an external force acting on the system £. The resulting motions c will no longer be geodesics of C because they satisfy the equation

ac= foe, but they are geodesics of n.

D. Dynamic connections Let us consider the first jet manifold J'' J'Y of the affine bundle J'Y Y. The adapted coordinates on JyJ'Y are (NA,yi,It ,), where we use the compact notation

(y° = t). Since J'Y - Y is affine, so is Jy' J'Y - Y, modelled over the vector bundle JyVY - Y. A connection

-Y: J'Y -. J .J'Y, (?, V" Aid ° 7 = (y" yi, ?'i,),

(3.9.19)

on the jet bundle J'Y - Y is called a dynamic connection, and is represented by the tangent-valued form (2.3.5) which reads

I 7=dy''®hA=dy''®(8a+

(3.9.20)

In particular, let us consider an affine dynamic connection y, that is, (3.9.19) is an affine morphism over Y: J'Y --- '_+ J'yJ'Y

Y

yu0 7=='Ykao+2 yll where the connection coefficients, are local functions on Y.

(3.9.21)

3.9. APPENDIX. GAUGE MECHANICS

213

It is easily seen that there is one-to-one correspondence between the shine dynamic connections ry and the linear connections K on the tangent bundle TY such that Ka°,, = 0. In particular, we may consider symmetric affine connections ry such that 'ykav = -Y'µ,,.

Contracting J'Y C TY with T*Y, we obtain from (3.9.20) the following affine morphism ri over J'Y

j2y

J'yJ'Y

ry`A J'Y

A-77 =of + AM, A This enables us to obtain a dynamic equation f, from the arbitrary dynamic connection ry. In coordinates, we have

J'Y ~ J'J'Y,

'Y

J1Y - J2Y,

I yet°57=4=-T°+ &,

(3.9.23)

In particular, if y is affine as in (3.9.21), we obtain the equation £r = 'Y yiyj + (-A +'Y." )yi + Yoo,

which reduces to = 7 jyiyi + 2'Ytkoy4 + 700

if y is symmetric. Using (3.9.20), we can associate with the equation C,, (3.9.23) the vector field

,y =8t+ylg8.+O8k=ho +yih It lives in the horizontal subbundle HJ'Y C TJ'Y of ry spanned by the horizontal fields h.,:

TJ'Y = HJ'Y ® VYJ'Y, JlY

8a=ha-'8k,

8;=8;.

(3.9.24)

In fact, f,, is the unique dynamic equation belonging to the horizontal subbundle

HJ'Y -. J'Y.

CHAPTER 3. LAGRANGIAN FORMALISM

214

Let c : I - Y be a motion and c : I - J2Y C TJ'Y. Then, according to (3.9.24), c admits the canonical splitting into horizontal and vertical parts, which coincides with (3.9.17). Thus, we have

ac:I - VY, c=t,roc+a,,, oC)ak=(ck-ryooe-('y'oc)e)ak,

4=( -

where a, is the acceleration of c with respect to ry. The geodesics of -y are precisely the solutions of the associated dynamic equation f,, i.e., c = lry o c.

Remark 3.9.5. Let ry be a connection on J'Y -e Y and r : Y -' J'Y a reference frame. The covariant differential of r with respect to ry is

vr=J'I'-yor:Y--.T'Y®VY, yr = varkdya ® ak,

(3.9.25)

vark=aark - ryaor.

Now suppose that a motion c is an integral curve of r, i.e., e = r o c. Recalling (3.9.22), we see that

c=tloJyroc. Thus, if yr = 0 (i.e., r is autoparallel with respect to ry), we obtain

c=f7oJyroc=poryoroc=cot, and, hence, c is necessarily a geodesic of -jA reference frame r all of whose integral curves are geodesics of -y is called a geodesic reference frame (with respect to -y). We have seen that, if r is autoparallel, then it is geodesic. The condition for r to be geodesic is

Vrr=rjyr=o, ra(OAF-ryaor)=o,

(3.9.26)

r°= 1.

Using the canonical projection T'Y -, V'Y, we can cut dt away from (3.9.25) and obtain the spatial covariant differential Vr of r (with respect to ry), that is,

Vr:Y-V'Y®VY, Vr = v{I'kdyi ®ak. It is easily seen that the following two conditions are equivalent:

3.9. APPENDIX. GAUGE MECHANICS

215

vr=o; r is geodesic, i.e., vrr = 0 and or = o. Let us consider the jet prolongation J'r (2.1.16) of the vector field r. It reads

Pr = r"a" + drka°E. Since the condition (3.9.26) takes the form

r"a"rk=for, we obtain the formula

Thus, r is geodesic if and only if

J'rIr=ciIr or, equivalently, the restriction jr 'Ir takes its values into the horizontal subbundle HJ1Y of the connection y. Let 'y be a dynamic connection. Then, using the horizontal vector fields ha given by (3.9.20), we define RAM = ha ryN - hµ rya =

- 87M + rya

yµ &7e

(3.9.27)

which are local functions on J'Y. Then the curvature R of ry is defined as (3.9.28)

R= 2Rxkdy" Ady" ®8k = (2R,',dy' Adyi + Ro)dt Ady') 08k. Contracting J'Y C TY with T'Y in (3.9.28) and then cutting dt away, we obtain the tensor

p:J'Y-iV'Y®VY, p = pi dy' 0 ak, to be used in the sequel.

pk = (R,Jyi + III),

(3.9.29)

CHAPTER 3. LACRANGIAN FORMALISM

216

Remark 3.9.6. Let 7 be a dynamic equation with connection coefficients '. Then

t

rr

=8;+ i81, '

(3.9.30)

at,

is a local basis for the vector fields on J'Y. It is easily seen that the dual basis is dt,

t=

dy` - yedt,

'y = dye - 7odt -

(3.9.31)

As we will see, given a dynamic connection 'y, it is sometimes convenient to deal with the non-holonomic bases (3.9.30) and (3.9.31). For example, using the forms 7i, we can express the vertical projection associated with 7 and denoted by the same symbol -y as

-y: J'Y --- TJ'Y 0 VyJ'Y,

7 = 71®8;.

(3.9.32)

Of course, (3.9.32) provides another equivalent definition of the connection 7.

We have seen that a dynamic connection 7 on the affine bundle J'Y -' Y gives

rise to a dynamic equation f, : J'Y -. J2Y. In fact, this is the unique dynamic equation belonging to the horizontal subbundle HJ'Y c TJ'Y determined by the connection 7. Conversely, given any dynamic equation : J'Y - J2Y, we will show that, necessarily, f = C., for some dynamic connection 7.

LEMMA 3.9.4. Let C : J'Y J2Y be a dynamic equation. Then a induces the following involution If of the vertical tangent bundle VJ'Y - J'Y:

VJ'Y VJ'Y, k

it(8f) _ -8; - te8k

it(8;) = 8;.

0 Proof. Let

u=a'8;+6{8; be a vertical vector field on J'Y -, R. Set

It(u)=(£,vuj-v[C,u):J'Y-VJ'Y,

3.9. APPENDIX. GAUGE MECHANICS

217

where v is the endomorphism (3.9.4). The result immediately follows.

QED

The involution If splits VJ'Y in the following way

VJ'Y = Ht ® VyJ'Y, JY

hr=8,+2

8kr

where hi is a local basis for the subbundle HH C VJ'Y. Note that HH canonically isomorphic to Vy J' Y - J' Y:

J1Y is

Hf -4 VyJ'Y hi

I

Of.

J'Y From the splitting (3.9.15), we obtain the following:

TJ'Y = R ® VJ'Y = HJ'Y ® VyJ'Y, JlY

J1 Y

HJ1Y = R ® H. J1Y

Setting

h = 8a + y8k,

70 = e - 7jkYt',

18k

7i = 2

we see that there are two equivalent local bases for HP Y, namely

(3.9.33)

and {h.,}

defined in (3.9.33). Of course, HJ'Y C TJ'Y is the horizontal subbundle of a dynamic connection 7 whose connection coefficients are -. In fact, (3.9.33) shows

that t _

,.

Moreover, we deduce from (3.9.33) that the map f *-+ 7 f is injective, i.e., if 7f = 7C, then the dynamic equations f and ( coincide. However, not every dynamic connection 7 is of the type 7 = 7f for a dynamic equation £. PROPOSITION 3.9.5. Let y be a dynamic connection as in (3.9.19). Then we have 7 = 7f for a dynamic equation C : J'Y -+ JPY according to (3.9.33) if and only if the connection coefficients 7a satisfy the following condition 070 ole

+

(3.9.34)

CHAPTER 3. LAGRANCIAN FORMALISM

218

Proof. The condition (3.9.34) follows immediately from (3.9.33) by taking the partial derivative along the coordinate yy:

o 2ry; - 2 1, -rya. Then, taking the partial derivative of

Conversely, let y satisfy (3.9.34) and (3.9.23), we obtain 8pf,k

ayt

= 0)0 +

t by"

-yk = 2?;`

where we used (3.9.34). The result immediately follows.

QED

A dynamic connection -y which satisfies the condition (3.9.34) is said to be symmetric. Note that a symmetric connection ry satisfies the following symmetry property (3.9.35)

Remark 3.9.7. Let ry be an arbitrary dynamic connection (3.9.19). Let t; = Ch be the associated dynamic equation (3.9.23), i.e., `k = 'Yu +-tikyi. According to (3.9.33), we can associate with the following symmetric connection

l

atk 1

k

0 10

)

k - `k

_*Y,

at

ayt

(3.9.36)

We can define the torsion T of the connection ry in the following way Y

T= Tkdy ®ak,

Tt _ yjk

- & t - vayt yj = 2(,yt - -, ) aLOA:

,/ l k

We see that 'y=.= if and only if T = 0. In particular, let -y be an affine connection (3.9.21), i.e., 7a = 7Jk10 +jEl'

(3.9.37)

219

3.9. APPENDIX. GAUGE MECHANICS

Then the condition (3.9.34) is equivalent to yal. = yy. The symmetric connection corresponding to (3.9.37) is

__

= 2(.y + yam). 1

Every dynamic connection y on the affine bundle J'Y connection Vy (2.5.13) on the composite fibre bundle

VyJ'Y.J'Y-.Y.

Y induces the vertical

(3.9.38)

This connection is projectable over -y and is given by

VYJ'Y-J''VYJ'Y

(U11, 1/i0,

ya, ) I

I

J'Y

JYJ'Y

i

i1, i

Uu) 0 Vy = (ya, Ui,

c

Note that J'' VyJ'Y = VyJyl J'Y -+ Jy' J'Y is a vector bundle and that Vy is a linear morphism over y.

Actually, since J'Y -y Y is an affine bundle, the vector bundle VyJ'Y - J1Y is equipped with a linear connection y whose coefficients are

Jle

0.

(3.9.39)

Here 'V stands for the covariant derivative associated with IF. The connection Vy on the composite fibre bundle (3.9.38) is the composition Vy ='y o y (2.5.4) of the connections y and 'y on J1Y - Y and VyJ'Y - J'Y, respectively. In particular, suppose that -y is affine as in (3.9.37). Then y has the associated linear connection defined on VY -, Y, that is, 1 = 731J.

(3.9.40)

Recalling (3.9.2), it is clear that the induced linear connection (3.9.39) on VyJ'Y J'Y is precisely the pull-back of the linear connection (3.9.40).

CHAPTER 3. LAGRANCIAN FORMALISM

220

E. Lagrangians and connections Let us consider a first order Lagrangian density of a mechanical system

L=Gdt, function a Lagrangian. L Its fibre derivative defines the Legendre map (3.4.1):

p,oL=a,=OC,

(3.9.42)

where (t, y{, p,) are the holonomic coordinates on V* Y. Using again the fibre deriva-

tive of L, we obtain the symmetric tensor

C:J'Y-+V'Y®V'Y, G,j = 8{aj.

(3.9.43)

The Poincare-Cartan form HL (3.2.6) associated with L is the 1-form on J'Y defined as

Hj, = d. L + Gdt : J'Y -+ T'Y, HL = a,dy` - (irill', - G)dt,

(3.9.44)

where we used the vertical exterior differential do (3.9.5). The Euler-Lagrange operator EL (3.2.10) associated with the Lagrangian G reads

,01,=4,Ldy:

6L=dd8C-8,G=Gjiyia+O,,ryi+8,i,-OC. Note that this expression differs in minus sign from (3.2.10). As we know, 6L is an affine morphism over J'Y. Since

VvJ'Y_J'YYVY, EL, can be also seen as the affine morphism over J'Y

J2Y . V'J'Y (3.9.45)

JIY

3.9. APPENDIX. GAUGE MECHANICS

221

In particular, a Lagrangian G is regular if and only if the Euler-Lagrange operator is an affine isomorphism (3.9.45). Equivalently G, which is a symmetric tensor on J'Y, must be non-degenerate, i.e., it must induce an the vertical bundle VyJ'Y isomorphism between VyJ'Y and VyJ'Y:

wily

+ wily J'Y

b:

a regular Lagrangian. Then, as is clear from (3.9.45), the kernel of Cl, uniquely determines a dynamic equation _ CL given by C:J'Y J2Y,

C;kCk = -8,a; - 8,ay.1 + 8;G.

(3.9.46)

Using the covariant derivative Vt (3.9.16) associated with the dynamic equation C, we obtain the relation

VyJ'Y

VyJ'Y

viL 'ale,, J'Y EL=boVt. Then, given a motion c : I - Y , we have

bone=CL-C, where a,, is the acceleration of c (3.9.17). In other words, EL o c is the acceleration of the motion c expressed in covariant terms. Thus, the motion c satisfies the Lagrange equations, i.e., £L o c = 0 if and only if c is a geodesic of C = Cl,.

Remark 3.9.8. Note that G is a mass tensor and not merely a geometric dimensionless tensor. Indeed, while a,, is a true acceleration, b o a< has the dimension of a force.

Taking the partial derivative of (3.9.43) with respect to yi, we obtain

ac;k k

a£k

a27r;

air;

azn;

h

ar,

CHAPTER 3. LAGRANGIAN FORMALISM

222

It follows that

` S ' Gif + Gik73k + Gjk

k

i = 0,

l atk

S , 7i = 28pi

(3.9.47)

and (3.9.48)

where CC

G,

!G-,., +

at

G hi

yi + a

j

a7Cj 7r

i

£k,

+

G,k"Ii .

Note the presence in the identity (3.9.47) of coefficients ryk of the symmetric dynamic connection y determined by the dynamic equation £ = eL according to (3.9.36). The linear connection 'y (3.9.39) induced by -y on the vector bundle V,J1Y -a J'Y can be introduced, and the covariant derivative VVG of G can be J1Y computed. Indeed, C is a tensor defined on the vector bundle VyJ'Y equipped precisely with the covariant derivative V, and C is a vector field on the base J'Y. From (3.9.39) we see that OEdy, =

(c + ftl

yi)dy`' = 7kdy,.

Hence, we obtain

VVCij = . Gij + Gik'yj + Gkj'yi .

Thus, the identity (3.9.47) means that the compatibility condition VG = 0 is satisfied. We have proved the following.

PROPOSITION 3.9.6. Let C be a regular Lagrangian and G the corresponding (nondegenerate) mass tensor. Moreover, let -y be the symmetric dynamic connection on

J'Y -. Y determined by the dynamic equation

C=Ct:J1Y-J2Y.

3.9. APPENDIX. GAUGE MECHANICS

223

Then, if 0 denotes the induced covariant derivative on the vertical tangent bundle VyJ'Y - J'Y, the following compatibility condition holds [711: C. Gif

G,,t_yik +Gk17'ik

= 0.

(3.9.49)

F. Newtonian and Lagrangian systems The concept of Newtonian system is basic in the scheme of classical mechanics. It is characterized by the following three ingredients: (i) a fibred manifold t : Y R , where t is the absolute time; (ii) a Riemannian mass metric C on the vertical bundle VyJ'Y J'Y, i.e.,

G: J'Y -' V'Y®V'Y,

G = G,,dx` ®dxt,

Y

where Gi; = G;i are local functions on J'Y which satisfy the symmetry property, that is, OGi; &yk

= 8Gk f},,; i

(3.9.50)

note that (3.9.50) here is assumed as an independent hypothesis; (iii) a symmetric connection ry on the affine bundle J'Y Y. As we know from (3.9.34) and (3.9.35), the connection coefficients the identities

7k = i0

419

+pi,k ayt

ay',

=

of ry satisfy

(3.9.51) .

Denoting by C the dynamic equation associated with ry, we have k

ki

I LqSk

741

=2

ft['

Using the linear connection I? induced by ry on the vector bundle VyJ'Y J'Y, we require that also the compatibility condition (3.9.49) holds. Hereafter, a Newtonian system is denoted by the triplet (Y,G,'y) . Proposition 3.9.6 may be reformulated by saying that, given a regular Lagrangian we can associate with it a Newtonian system (Y, C, -y) where G is given by (3.9.43)

CHAPTER 3. LAGRANCIAN FORMALISM

224

and 7 is the symmetric connection determined by the dynamic equation £ (3.9.46).

Let -y be a (symmetric) dynamic connection and G a mass metric tensor. Then we can define the following 2-form over J1 Y:

u') = G(7u, vi') - G(-fu, vu'),

I

where u, 1/ are vector fields on J1Y, v is the vertical endomorphism (3.9.4) and y is the vertical projection (3.9.32). It is easily seen that the local expression of w.r is

c = G.37' A s',

(3.9.52)

19' =dyi- yjdt.

7'=dye-7odt-7ndy",

Note that wy has maximal rank and

C7jw1-0,

C =er

A direct computation from (3.9.52) shows that G, + Gk-t, + Gkj7i )dt A 7' A $' + aGij 7' A 19' A 7k +

dw,

(3.9.53)

aylk

)7'A19' ABk+GkpjdtAt9'At9'+ e

Ghj(h. - 7k)O A 19i A 19k,

where pf has been defined in (3.9.29) and we use the local basis (3.9.31). Suppose that the first two terms in (3.9.53) vanish identically (i.e., (Y, C, 7) is

a Newtonian system). Then also the third term in (3.9.53) vanishes identically. Indeed, by taking the partial derivative of (3.9.47) with respect to yk, we obtain 8G;j + 8Gt j 7k + G &y Bpi 8yk

Wit,

V

h _ BCij7kh 7. 8 e`

_ _ 02C. _ i8

O2 GiJ

V,

_ 14

_

02G.2

(RWth -

h_

- 8C;k 7jh - Cih v8bi/j

The assertion follows from the symmetry properties (3.9.50) and (3.9.51). Recalling (3.9.27), we see that the last term in (3.9.53) can be rewritten as follows Ghj(h. 7k )f A 19' A

8k

=

(Cu,R,"kj + GjhlRk + GkkR% )19' A 19' A 19k.

6

3.9. APPENDIX. GAUGE MECHANICS

225

Thus, we have proved the following. PROPOSITION 3.9.7. The triplet (Y, G, -y) is a Newtonian system if and only if dw., = G;kpjdt A 1 A 0 + G (Gu,Rk + GfhR4 + GkhR )t9A tV A 0.

1 It should be emphasized that the property of (Y, G, y) to be a Newtonian system is expressed in terms of the curvature of the symmetric connection ry. 2 LEMMA 3.9.8. Let y be a symmetric dynamic connection. Then the following

identity holds

8._3pk 591- V

8pkk

0 Proof. It easily follows from (3.9.27), (3.9.29) and the properties (3.9.51).

QED

From this lemma we obtain the following corollary. COROLLARY 3.9.9. Let ry be a symmetric dynamic connection and G a mass metric satisfying the symmetry property (3.9.50). Suppose that (3.9.54)

G{kpj = Gfkp{ .

Then the following cyclic identity holds: G,AR" + Gj.%Vt + GkARjt = 0. O

Proof. The relation (3.9.54) leads to BG,h pk

-B

P,%

i

= G,h Bpi

- Gik04

ft1k

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226

and, hence, OG h

Pjh -

BCjh

OTC

Pi +

BGkh h

P; -

OCth h

0614

Pk + V;

.,e

e

Ph-

BGhk h

8A Pj = e

-3(C;hRkj + GjhR4 + GkhRj;) The result follows immediately.

QED

PROPOSITION 3.9.10. Let (Y, G, ry) be a Newtonian system. Then the following two conditions are equivalent (711: (i) w, is a closed form, i.e.,

dw, = 0;

(3.9.55)

(ii) the symmetry property (3.9.54) holds, i.e., (3.9.56)

C,hpj = GjhP°.

A Newtonian system (Y, C, ry) for which the condition (3.9.55) or, equivalently, (3.9.56) holds is said to be a Lagrangian system. Using -y and G, we can define the following Euler-Lagrange type operator E(f)

= e.):

V'J'Y

VyJ'Y V(

J2Y

E(t)=0oVE,

& (f)=G;k(yu-tk)

Now let G : J'Y -. R be a Lagrangian. In fact, G denotes a sheaf of local Lagrangians on J'Y (see Remark 3.9.9). Recalling the definition of the PoincardCartan form HL given in (3.9.44), we can easily see that its differential dHL satisfies the following identity dHL = [6;G -

(v'u - e')[dt A 0 +

ry' A t9j + (h; irj)19' A t9j.

Recalling (3.9.48), we deduce from (3.9.52) that the following conditions are equivalent:

3.9. APPENDIX. GAUGE MECHANICS

227

the sheaf C defines a global Euler-Lagrange operator EL and

-'L = CM;

(3.9.57)

the sheaf C defines a global form dHL and dHL = w.,.

(3.9.58)

Note that the sheaf C also defines a global mass tensor

8a;

02C

In particular, all the local Lagrangians of the sheaf G are regular. Under the equivalent conditions (3.9.57) or (3.9.58), since dw, = 0, the triplet (Y, G, 'y) is a Lagrangian system. We have already met an example of a Lagrangian system determined by a unique global Lagrangian G according to Proposition 3.9.6. Note that the condition dw, = 0 implies the condition (3.9.57) or, equivalently, (3.9.58). Thus, a Lagrangian system is characterized by the equivalent conditions (3.9.57) or (3.9.58). PROPOSITION 3.9.11. Let (Y, G, ry) be a Lagrangian system, i.e., such that dw,. = 0. Then the condition (3.9.58) holds.

Proof. There exists a local 1-form 0 = adt + ;O + rycry{,

around each point of J'Y such that

d4 = w,,,

(3.9.59)

where a, A and ryi are local functions on J'Y. By computing dO from (3.9.59) and using (3.9.52), we see that

87:_0 8"

8y

It follows that there exists a local function A on J1Y such that

dJ _ (f \)dt + (h{ - \)O + 7t8i +

f = fr

Note that we have used the duality between the bases (3.9.30) and (3.9.31). Setting dA, we obtain

Vi

= ,r, 6' + Gdt,

it _

h, A,

G = a - f A.

(3.9.60)

CHAPTER 3. LAGRANGIAN FORMALISM

228

Actually, t/i is precisely the Poincar6-Cartan form associated with L. Indeed, we have

dt/'=dq= [biC7iAt9t+(h;

t

a t

G

as defined in (3.9.60) provides the sheaf of local Lagrangians associated

QED

with (Y, G, -y).

Remark 3.9.9. Of course, the sheaf L is not unique. Let C be another sheaf of local Lagrangians associated with (Y, C, y). Setting X = C - L, it is easily seen that

e;xi - axt = 0,

X = xiar, + Xo,

eixi - aixo = 0,

where xA are local functions on Y. Equivalently, x is a sheaf of closed local 1-forms

on Y, that is, X=XAdp",

dx=0.

In particular, we have Hu = Ht + X.

G. Conservation laws Let L be a Lagrangian (3.9.41) on J'Y. 1b obtain differential conservation laws, we use the first variational formula (3.2.13) of the calculus of variations (see Section 3.5). Let us consider conservation laws along a vector field u = u`8e + ui8i,

ut = 0, 1,

on Y -, R. Then we have LJy,L = (J'uJdC)dt = (u°8i + ui8i + dtu`8;)Ldt,

(3.9.61)

3.9. APPENDIX. GAUGE MECHANICS

229

and the first variational formula (3.2.13) takes the coordinate form

J'uJdL = (ui - u`ye)(8; - dt8;)L - dt`,

(3.9.62)

where

Z = ,q(utyy - ui) - utL

(3.9.63)

is the current along the vector field u. On-shell, the first variational formula (3.9.62) leads to the weak identity

J'uJdL -_ -dtT.

(3.9.64)

If the Lie derivative Lj' L vanishes, we have the conservation law

0 ~. -dt[7r{(uty - u') - utL]. This is brought into the differential conservation law 0

-

(iri o c(u'8tci - u' o c) - ut G o c)

jj

on solutions c of the Lagrange equations. A glance at this expression shows that, in mechanics, the conserved current (3.9.63) plays the role of a first integral of motion.

Every symmetry current (3.9.63) along a vector field u (3.9.61) on Y can be represented as a superposition of the Nother current along a vertical vector field u, where uu = 0, and of the energy current along the horizontal vector field r (3.9.6) [50].

If u is a vertical vector field, the weak identity (3.9.64) reads

(u'8i + dtu'8; )L -- dt(ir u'). If the Lie derivative of L along u equals zero, we have the integral of motion T = wiui.

In the case of the horizontal vector field r (3.9.6), the weak identity (3.9.64) takes the form (8t + I" 8i + d*I"811)L = -dt(ii(ye - I'i) - L),

(3.9.65)

where

Tr=iri(2 -I'')-L

(3.9.66)

CHAPTER 3. LAGRANGIAN FORMALISM

230

is the energy function with respect to the frame I'.

Remark 3.9.10. With respect to the coordinates adapted to r, the energy conservation law (3.9.65) takes the familiar form 8,G = -dc(ir y

- C).

(3.9.67)

Example 3.9.11. Let C be a regular Lagrangian and l; = CL the dynamic equation determined by L as in (3.9.46). Then, since

, o0,

dTr dt

oC=L(Tr,

we obtain

LtTr = -Lj,rL. Thus, Tr is a first integral of the dynamic system if and only if

LjtrL = 0.

Chapter 4 Hamiltonian Formalism Here we will follow the notation of the previous Chapter. Unless otherwise stated,

by Y - X is meant a fibred manifold over an n-dimensional base with fibred coordinates (xa, y'). This Chapter deals with the polysymplectic Hamiltonian formalism defined on the Legendre bundle

lI=AT'X®V'Y®TX Y Y

(4.0.1)

over a fibred manifold Y, which is coordinatized by (xA, y`, p, ). Recall that every Lagrangian density L on the configuration space J'Y of the first order Lagrangian field theory induces the Legendre map L (3.4.1) of J'Y to II. This morphism takes the form

pp oL=B;'G which shows that the Legendre bundle (4.0.1) is a natural n-dimensional generalization of a phase space of symplectic formalism [31, 64, 86, 101, 161, 162). Moreover, polysymplectic Hamiltonian formalism applied to fibre bundles Y R over a 1-

dimensional base R leads to the adequate geometric formulation of Hamiltonian time-dependent mechanics (see Section 4.10). If a Lagrangian density is hyperregular, polysymplectic Hamiltonian formalism is equivalent to the Lagrangian one. We will concentrate on the relations between Lagrangian and polysymplectic Hamiltonian formalisms in the case of non-regular systems [65, 163, 164, 190).

231

CHAPTER 4. HAMILTONIAN FORMALISM

232

4.1

Symplectic structure

In this Section we summarize the basic notions of symplectic geometry which will be needed in the sequel [2, 6, 120, 182]. Let Z be a manifold. A Jacobi bracket (or a Jacobi structure) on Z is defined as a bilinear map

C°°(Z) X C°°(Z) 9 (f,9)

{f,9} E C°°(Z),

where C°° is the space of real smooth functions on Z. This map, by definition, satisfies the following conditions:

(Al) {g, f } = -(f, g} (skew-symmetry), (A2) (f, {g, h)) + {g, {h, f } } + {h, {f, gj) = 0 (Jacobi identity), (A3) the support of (f, g) is contained in the intersection of the supports of f and g. PROPOSITION 4.1.1. Every Jacobi bracket on a manifold Z is uniquely defined in accordance with the relation

{f, 9} = w(df,d9) +uJ(fd9 - 9df)

(4.1.1)

by a pair of a vector field u and a bivector field w on Z such that Law = 0,

[w, w] = 2u A w

(4.1.2)

[102, 122, 132]. o

Example 4.1.1. Taking w = 0, every vector field u on a manifold Z defines the Jacobi bracket (4.1.1). The relations (4.1.2) are obviously satisfied.

The Jacobi bracket (4.1.1) with u = 0 is said to be a Poisson bracket. According to (4.1.2), a bivector field won a manifold Z provides a Poisson structure if it meets the condition [w, w] = 0,

that is, it is a Poisson bivector field (or simply a Poisson bivector) (see Example 1.5.5). A manifold Z equipped with a Poisson bivector w is called a Poisson manifold (Z, w).

4.1. SYMPLECTIC STRUCTURE

233

Besides the conditions (Al - A3), the Poisson bracket

{f,9} = w(df,d9)

(4.1.3)

satisfies also the Leibniz rule

{h, fg} = {h,f}g+ f{h,g}.

(4.1.4)

A Poisson structure defined by a Poisson bivector w is said to be regular if the

associated morphism 0 : T'Z -. TZ (1.5.5) has constant rank. Hereafter, by a Poisson structure we mean a regular Poisson structure. Note that there are no pull-back or push-forward operations of Poisson structures by manifold maps in general. The following assertion deals with Poisson projections, whereas Theorem 4.1.7 is concerned with Poisson injections. PROPOSITION 4.1.2. Let (Z, w) be a Poisson manifold and 7r : Z The following properties are equivalent:

Y a projection.

for every pair (f, g) of functions on Y and for each point y E Y, the restriction of the function { f o a, g o a} to the fibre it-1(y) is constant;

there exists a Poisson structure on Y for which it is a Poisson morphism. If such a Poisson structure exists, it is unique ([1201, p.116). 0 DEFINITION 4.1.3. Given a function f on a Poisson manifold (Z, w), the image

1Of=w*df,

Of=w&-a'f

of its differential of by the morphism wp is called the Hamiltonian vector field of f .

0 The Hamiltonian vector field t9 f, by definition, obeys the relation I

t 9 fidg = { f, g}

(4.1.5)

for any function g on Z. It is easy to see that [1f,19g[ = I9{f.g}

(4.1.6)

234

CHAPTER 4. HAMILTONIAN FORMALISM

This relation provides the set of Hamiltonian vector fields with a Lie algebra structure. Using (4.1.4) and (4.1.6), one can show that

(Le.w)(df,dg) =,9hJd{f,g} - {t9h14f,g} - {f,t9hjdg} = 0. It follows that every Hamiltonian vector field is the generator of a local 1-parameter group of automorphisms of the Poisson manifold (Z, w).

The values of all Hamiltonian vector fields at all points of Z constitute the characteristic distribution of the Poisson manifold (Z, w). In virtue of the relation (4.1.6), this distribution is involutive. Remark 4.1.2. Distributions. Let Z be an m-dimensional manifold. Recall that a k-codimensional smooth distribution T on Z is defined as a subbundle of rank m - k of the tangent bundle TZ. A smooth distribution T is said to be involutive if [u, u'J is a section of T, whenever u and u' are sections of T.

Let T be a k-codimensional distribution on Z. Its annihilator T' is a kdimensional subbundle of T'Z called the Pfafan system. It means that, on a neighbourhood U of every point z E Z, there exist k linearly independent sections 01,. . . , Ok of T' such that

T iu= nKer0,.

i

Let AT denote the ideal of the exterior algebra O'(Z) which is generated by sections

of T. PROPOSITION 4.1.4. A smooth distribution T is involutive if and only if the ideal AT is a differential ideal, i.e., d(AT) C AT ([186J, p.74). o Note that, given an involutive k-codimensional distribution T on Z, the quotient

TZ/T is a k-dimensional vector bundle, called the normal bundle of T. There is the exact sequence 0

T TZ -- TZ/T - 0.

(4.1.7)

Given a fibred manifold Y - X, its vertical tangent bundle VY exemplifies an involutive distribution on Y. In this case, the exact sequence (4.1.7) is precisely the exact sequence (1.4.3a). A connected submanifold N of the manifold Z is called an integral manifold

of a distribution T on Z if the tangent spaces to N belong to the fibres of this

4.1. SYMPLECTIC STRUCTURE

235

distribution at each point of N. Unless otherwise stated, by an integral manifold we mean an integral manifold of maximal dimension, equal to dimension of T. An integral manifold N is called maximal if there is no other integral manifold which contains N. THEOREM 4.1.5. Let T be a smooth involutive distribution on Z. For any point z E Z, there exists a unique maximal integral manifold of T passing through z ([186], p.75).

In view of this fact, involutive distributions are also called completely integrable distributions.

COROLLARY 4.1.6. Every point z E Z has an open neighbourhood U which is a domain of a coordinate chart (z',... , z"') such that the restrictions of T and T' to U are generated by the m - k vector fields

a (7Z1 I

8 ... 18zm-k

and the k Pfaffian forms

dzm-k+',... , dzm respectively.

It follows that integral manifolds of an involutive distribution form a foliation. Let us recall that a k-codimensional foliation on a m-dimensional manifold Z is a partition of Z into connected leaves F1 with the following property. Every point of Z has an open neighbourhood U which is a domain of a coordinate chart (z°) such that, for every leaf F1, the connected components of F, fl U are described by the equations zm-k+l

= const.,

z"' = const.

[97, 155]. Note that leaves of a foliation fail to be imbedded submanifolds in general. For instance, every submersion 7r : Z - X defines a foliation on Z whose leaves

are the fibres x-'(x), x E X. Every nowhere vanishing vector field u on a manifold Z defines a l-dimensional involutive distribution on Z. Its integral manifolds are the integral curves of u. In virtue of Corollary 4.1.6, around each point z E Z, there exist local coordinates (z', ... , z") such that u is given by U

8

8z,.

CHAPTER 4. HAMILTONIAN FORMALISM

236

Let us turn to the characteristic distribution generated by Hamiltonian vector fields. We have the following theorem.

THEOREM 4.1.7. The characteristic distribution of a Poisson manifold (Z, w) is completely integrable. The Poisson structure induces the symplectic structures on leaves of the corresponding foliation of Z ([1821, p.26), which is therefore called a symplectic foliation. 0 Recalling that a 2-form 11 on a manifold Z is called presymplectic if it is dosed, a presymplectic form fl is said to be symplectic if it is non-degenerate (see Example 1.6.1). A manifold Z equipped with a symplectic [presymplectic] form is said to be a symplectic [presymplectic] manifold. The symplectic foliation admits the adapted coordinates described in Corollary 4.1.6. Moreover, one can choose these coordinates in such a way to bring the Poisson bracket in the following canonical form ([182] p.29;[187]). PROPOSITION 4.1.8. For any point z of a Poisson manifold, there exists a coordinate system i

k

in a neighbourhood of z such that

{y',yi} = {py,pj} = {y',z°} = {pi,z°} = {za,z"} =0,

{p;,yi} =6;. (4.1.8)

These coordinates are called canonical coordinates. 0 In canonical coordinates (4.1.8), the Poisson bracket (4.1.3) takes the form

0109

01,19

A Poisson structure is called non-degenerate if wr is an isomorphism. In this case, the Poisson bivector w is non-degenerate and defines the corresponding symplectic form Cl on Z given by the relation (4.1.9). A non-degenerate Poisson structure can exist only on an even-dimensional manifold.

4.1. SYMPLECTIC STRUCTURE

237

PROPOSITION 4.1.9. On every even-dimensional manifold Z, there is the one-toone correspondence between the symplectic forms 11 and the Poisson bivectors w in accordance with the equalities I w(O, a) = Sl(wrO, waa),

0,a E O'(Z),

11(t9, V) = W(flb6, Slbv),

(4.1.9)

t9,v E V'(Z),

(see relations (1.5.5) and (1.6.2)) [121). C3

The equalities (4.1.9) take the coordinate form 0-ow"', = 6.1. In canonical coordinates, we have Sl = dpi A dy',

w=

8, n

Example 4.1.3. Let M be a manifold with coordinates (y') and PM its cotangent bundle provided with the holonomic coordinates (y', p;). The cotangent bundle 7M is equipped with the canonical symplectic form In = dpi A dy'

(4.1.10)

and the canonical Liouville form 1B=prdy'.

Furthermore, for every closed 2-form 0 on M, the form Sl + 0 is also a symplectic form on T' M. The canonical symplectic form (4.1.10) plays a fundamental role in view of Darboux's theorem ([1201, p.135). This theorem is an immediate consequence of Proposition 4.1.8 and Proposition 4.1.9.

THEOREM 4.1.10. Let (Z,11) be a symplectic manifold. Each point of Z has an open neighbourhood U which is the domain of a canonical coordinate chart

(y'r... IY"Ipl,...,N) such that the symplectic form fl has the coordinate expression (4.1.10) on U. 13

CHAPTER 4. HAMILTONIAN FORMALISM

238

The notion of a Hamiltonian vector field on a symplectic manifold may be reformulated as follows.

DEFINITION 4.1.11. A vector field 0 on a symplectic (or presymplectic) manifold (Z, fl) is said to be locally Hamiltonian [Hamiltonian] if the form t9JS1 is closed [exact]. 0 As an immediate consequence of this definition, we find that:

a vector field t9 is locally Hamiltonian if and only if it is an infinitesimal symplectomorphism, that is, L6Sl = d(t9JQ) = 0;

a vector field 0 is Hamiltonian if and only if it is a Hamiltonian vector field in accordance with Definition 4.1.3, i.e. t9 = t9j, where

=-t9,J$1,

t9,=OfO;-O,ft .

Remark 4.1.4. Generalized Poisson structure. Different generalizations of the Poisson structure have been suggested. In particular, it seems natural to consider a multivector field w on M in order to introduce the multibracket f1,...,fk

atjl'

akfk

of k functions on M [8, 95, 176]. In particular, the multibracket of n functions on an n-dimensional manifold X can be given by the relation A)w_X411A...Adfn.

A different way is to extend the Poisson bracket given by a bivector field w to multivectors and differential forms. Let T.(M) be a Z-graded algebra of multivector fields on a manifold M. Let us introduce the operation tDt9

= = [W, VISN,

4.1. SYMPLECTIC STRUCTURE

239

where 1., .]SN is the Schouten-Nijenhuis bracket (1.5.3). This operation has the homology property ws = 0 [182]. Let 19 and v be multivectors. There is defined the bracket [19, v]w = -[rW, v]SN,

which has the property [19, v1w = _(_l) roII

I [v,191w

- w([O, v])

This bracket is graded skew-commutative on the quotient T.(M)/t (T.(M)). If the bivector w on a manifold M is non-degenerate, i.e., M is a symplectic manifold, it defines an isomorphism TM «- T'M. Using the Schouten-Nijenhuis bracket of multivectors, one can then construct a bracket of differential forms [138]. In the general case, we have the homomorphism

00(al,

. , at) = (- l)k0(wtlal, .,Oak), w(00) _ -wd(dd).

4 E ilk(M),

Let us consider the operator

bw=wJ od - dowJ

(4.1.11)

on the exterior algebra D'(M) [108, 182], and recall that the contraction of multivectors and exterior forms is

(,tl A... At9k)J0=19kJ...t9 Jm The operator bw (4.1.11) is related to the operator iu by the formula (CV)J0 = i9j(6w0) + (-1)kbw(t9J-0),

VV E Ok(M),

Vi9 E Tk_,(M).

The corresponding bracket of exterior forms is defined to be

ilk(M) x Y(M) _ ilk+,-l(M), v}w = (6..O) A o + (-1)"01.0 A (bwo) - bw(.0 A o).

This bracket has the properties (.0, a). = (-1)"01101(a,0)w, (_1)101Q01-0{0, {ti,©}w}w

+

(-1)10"(10"-1-){O, {0,Q)w1w y

({0,Q)w) = [00, W U1SN

(-1)"0100"-1){a,

= 0,

{9,O)w)w +

(4.1.12)

CHAPTER 4. HAMILTONIAN FORMALISM

240

In particular, the bracket (4.1.12) of 1-forms reads {0,

Loea - L0,.0 - d(w(4, a))

(4.1.13)

or

{0, a}. = wrOJda - OaJdO + d(w(*b, a)).

It provides O'(M) with a Lie algebra structure such that

wr:iD'(M)

T(M)

is a lie algebra homomorphism ,ur({m, a}) = [ 0 0 ,0 a ]-

The relation of the bracket (4.1.13) to the Poisson bracket (4.1.3) is the following {d/, dg}w = d{ f, g}.

The operator &, (4.1.11) obeys the equality

do& +6 od=0 that leads to the formula

d{m,a}. = -{dO,a}. - (-1)NI{0,da},,. Then the bracket

{0,a}d = -(dgf,a)w can be introduced (28, 138]. This bracket is graded skew-commutative {0, a}d = (-1)101b1{a, O}d

on the quotient iO'(M)/dil'(M).

(4.1.14)

4.2. POLYSYMPLECTIC STRUCTURE

4.2

241

Polysymplectic structure

Let Y - X be a fibred manifold and n _. Y the Legendre bundle (4.0.1). By the Legendre fibred manifold we will mean the fibration n - X. Thus, we have the composite fibration 'lrnx = n o any : 11

Y - X.

(4.2.1)

Given fibred coordinates (?, y') on the fibred manifold Y - X and the holonomic coordinates (zA,±A) and (xa,ia) of the bundles TX and T'X, respectively, the Legendre bundle (4.0.1) is equipped with an atlas of holonomic coordinates with the transition functions

p'; = det(

Ox,

OXIA

Txwpj . (4.2.2) 8x') 8y' 8xµ These coordinates are compatible with the composite fibration (4.2.1) and are linear bundle coordinates on the vector bundle 11 - Y. We will call them the canonical coordinates. 1 One can think of p; as being the momentum coordinates of momenta of field functions with respect to all world coordinates. 1 There are the following canonical morphisms:

the bundle isomorphism over Y

inn

V'Y ("A-IT'X),

in : (xA, y{, pi) ''-' p; ay` A wa,

(4.2.3)

where {dy'} are the fibre bases for the vertical cotangent bundle V'Y of Y; the bundle monomorphism over Y 9 : IT

Y 'TT'Y ®TX,

(4.2.4)

9 : (x\, V, p,) '--. -p, dy' A w,& Ok, which defines the tangent-valued Liouville form (3.4.5) on the Legendre bundle

n.

CHAPTER 4. HAMILTONIAN FORMALISM

242

Let us recall that the polysymplectic form

St=dp, Ady'Aw®8,,

(4.2.5)

on II is defined as a unique TX-valued (n + 2)-form on II such that the relation

ft1 0 = -d(ej 0) holds for any exterior 1-form 0 on X (see Definition 3.4.2). I The tangent-valued Liouville form (4.2.4) and the polysymplectic form (4.2.5)

provide the Legendre bundle lI (4.0.1) with the polysymplectic structure. I

Example 4.2.1. Let Y -. R be a fibre bundle over X = R with coordinates (t, y'). It is readily observed that the corresponding Legendre bundle is the phase space II = V'Y, coordinatized by (t, y', p,), of time-dependent mechanics (see Section 4.10). The polysymplectic form (4.2.5) on this phase space reads

St=dp;ndy'Adt®88. This form cannot be contracted to the familiar exterior symplectic form if transformations of pi and y' depend on the temporal coordinate t. This Example shows that the polysymplectic form (4.2.5) fails to be the straightforward n-dimensional generalization of the canonical symplectic 2-form. Such a generalization is provided by a multisymplectic form 1100, 134].

Example 4.2.2. Multisymplectic structure. Let M be an m-dimensional manifold with coordinates (za). Consider the fibre bundle

AT'M- M of exterior k-forms on M. It is coordinatized by (z",pA) where A = (A, < ... < A,) are multi-indices of the length I A 1= k. The manifold AT'M is equipped with the canonical exterior k-form a defined by the relation

u j...u,Je(p) = Its coordinate expression is

9=Epa,...4dz'' A

pE

kT'M,

u, E T,(AT'M).

4.2. POLYSYMPLECTIC STRUCTURE

243

where the sum is over all multi-indices A. The exterior differential d6 of this form is the (k + 1)-symplectic form

am = de = E dP,,,...,,, A dz''' A ... Adz" 4 A

which belongs to the class of multisymplectic forms [134]. If k = 1, d9 is the familiar symplectic form on the cotangent bundle T' M.

A diffeomorphism 4) of the manifold ATOM is said to be a multisymplectic d i f f e o m o r p h i s m if it preserves the canonical multisymplectic form, i.e., 4 dO = de.

It appears that multisymplectic diffeomorphisms have much simpler structure than do symplectic ones.

Recall that, given the tangent space Tp(AT'M) to the manifold ATOM at a point p, its subspace Wp is a Lagmngian subspace if dO lw,= 0 and if Wp is maximal

in the lattice of subspaces with this property. Lagrangian subspaces constitute the Lagrangian distribution We C T(AkTM) on the manifold ATOM. It easily observed that, if k > 1, this is precisely the vertical tangent bundle V(AT'M) of the fibre bundle ATOM -. M. This distribution is obviously integrable, and its leaves are the fibres of the fibre bundle ATOM -- M. Every multisymplectic diffeomorphism must preserve the Lagrangian distribution associated with the multisymplectic form dO. It follows that k-multisymplectic diffeomorphisms are fibred morphisms of the bundle of k-forms ATOM M. It can be proved that, if 4) is a multisymplectic diffeomorphism, then 0 decomposes as a semidirect product of a holonomic automorphism of the fibre bundle ATOM - M and an afline morphism

P-- P + r(((P)),

p E ATOM,

(4.2.7)

where r is a closed k-form on M [134]. Let Y X now be a fibred manifold over a (1 < n)-dimensional base X. Let us consider the canonical form 6 (4.2.6) on the bundle AT'Y of n-forms on Y. Let

Zy =TOY A("A'T'X)

(4.2.8)

be the homogeneous Legendre bundle introduced in Remark 3.4.4. It is equipped with coordinates (x', y', pi', p), where the coordinate p has the transformation law

P' = det(ax,v)(P- Bv:Ox"

CHAPTER 4. HAMILTONIAN FORMALISM

244

The canonical bundle monomorphism over Y

iz : T'Y A (AIT'X)

AT'Y

yields the pull-back form iz6 on Zy which is precisely the canonical form

=pv+p,dy'Awa

(4.2.9)

(3.4.19) on Zy. This is the reason why the exterior differential f 1z = dp A w + dp, A dy' A 1.s,

(4.2.10)

of this form is also called a multisymplectic form [771. R is the fibre bundle from Example 4.2.1, the multisymplecIn particular, if Y tic form (4.2.10) leads to the canonical symplectic form

f2z=dpAdt+dppAdy' on the cotangent bundle T'Y of Y. The multisymplectic diffeomorphisms 4, of the bundle ATY -+ Y, which keep of the bundle Zy -' Y, the image iz(Zy) C ART-Y, induce automorphisms which preserve the multisymplectic form (4.2.10) on Zy. Every 4'z decomposes as a semidirect product of a holonomic automorphism of the fibre bundle Zy - Y over a bundle morphism of Y X and an affine morphism (4.2.7) where p E Zy and r : Y Zy is a closed form. In conclusion, let us mention also the vector-valued symplectic form defined on the frame bundle LX over a manifold X. It is the exterior differential of the canonical form OLx (3.7.5) [56, 1471.

Similarly to Example 4.2.2, an automorphism + of the Legendre bundle lI is said to be a polysymplectic (or canonical) automorphism if $ preserves the polysymplectic form Cl (4.2.5). A polysymplectic automorphism obeys the following coordinate relations

t

84>i tom'

ice; ice'

W, 0XV 4 ox -

814 W3

of ft'

8yJ 8yk

p ,, j

9yk

=

0,

' = 6k6."

v r, 80' 81/11 &Y,

=

0.

(4 . 2 . 11 a) (4 .2 . 11b)

(4.2.1 lc)

4.2. POLYSYMPLECTIC STRUCTURE

245

It is easily justified that the set of polysymplectic automorphisms includes: (i) holonomic automorphisms of the Legendre bundle II _Y induced by bundle morphisms of Y - X (their coordinate expression is (4.2.2)) and (ii) the affine morphisms

4 En,

9 --. 4 + r(1rny(9)),

where

r=r,(y)dy'Aw®8a is a section of II - Y such that, for each exterior 1-form 0 on X, the form rJ xnx( = r; (y)q A(x)dy' A w

is closed. Actually, one can show that, if n > 1, every polysymplectic automorphism is a semidirect product of such kind of morphisms. The example dim Y = dim X + I illustrates this fact as follows.

Example 4.2.3. If dim Y = dim X + 1, the relation (4.2.1 la) reads 8j? 8 y '

8p" 8PP

8p 8y' 8p" 8pµ

=o

If we assume that 1/ = yoO depends on the momenta pa, then det(TO) must be equal to zero and, consequently, 4) is not an automorphism. It follows that polysymplectic

automorphisms are compatible with the fibration II - Y. Then, it is not difficult to show that they take the above mentioned form. . l From now on we will consider only holonomic coordinates (4.2.2) on fl. I

Remark 4.2.4. Instantaneous Hamiltonian formalism. Using the multisymplectic structure on the homogeneous Legendre bundle Zy Y; one can obtain the instantaneous Iiamiltonian formulation of field theory in terms of infinitedimensional symplectic spaces [78). Let Y - X be a fibred manifold and N an (n-1)-dimensional imbedded compact submanifold of X which may be treated as a Cauchy hypersurface. There always exist local coordinates (x°, x°) of X so that the hypersurface N is locally defined by

CHAPTER 4. HAMILTONIAN FORMALISM

246

the coordinate relation x° = 0. By YN is meant the restriction of the fibred manifold Y

to N C X.

Recall that S(Y) denotes the space of sections s of Y. Accordingly, the symbol S(YN) stands for the set of restrictions of sections of Y - X to N. Note that, being completed in appropriate Sobolev topology, the space S(YN) becomes a smooth infinite-dimensional manifold [3, 131]. It may be coordinatized by local functions

S' = y' o s,

S E S(YN).

The tangent space T,S(YN) to S(YN) at a section s of YN -+ N is defined as the set of sections u of the vertical tangent bundle VYN -+ N which cover s. Similarly, the cotangent space T;S(YN) to S(YN) consists of sections of the bundle

nA' T* N ® V'YN -. N YN

which cover s. In adapted coordinates, an element e E T,S(YN) reads

e=e,Jy'®"o,

"o=Boj(J.

The natural contraction of u E T,S(YN) with e E T, S(YN) is given by the integral

uJe =

IN u(x)Je(x).

Let us consider the fibred manifold Zy - X (4.2.8) and let ZN denote its restriction to N. The space S(ZN) of sections of ZN - N possesses the induced fibration S(ZN) -, S(YN). It is the space S(ZN) which could play the role of an infinite-dimensional phase space in instantaneous Hamiltonian formalism. Let E be the canonical form (4.2.9) and nz the multisymplectic form (4.2.10)

on Zy. Let us introduce the corresponding forms . and fl on S(ZN). For every r E S(ZN) and u, v E T,.S(ZN), they are defined by the relations

uJ=(r) = I r'(uJE), N

vJuJfl(r) = Ir'(vJuJf z) N

Relative to the coordinates

S'=yo r,

a= pa or,

R,

4.3. HAMILTONIAN FORMS

247

these forms are written as ma(r) = f II,°dR' ®wb, N

f2(r) = f dli,° AdR' ®oro. N

The form f2, however, fails to be symplectic because of a non-trivial kernel. For example, 1) vanishes on elements

TS(ZN)9u+up. This difficulty is overcome by symplectic reduction as follows.

PaoPosrrloN 4.2.1. The quotient S(ZN)/Kerfl is canonically isomorphic to the cotangent bundle T'S(YN) endowed with the canonical symplectic structure (78). 13

Roughly speaking, the mapping

: S(ZN) -+ S(ZN)/Ker 5 removes the momentum p and the spatial momenta p;'2'3. Relative to the canonical coordinates (S', P,) of T'S(YN), this morphism reads

P;oC=lt,°.

Hamiltonian forms

4.3

Given the Legendre bundle 11 over a fibred manifold Y - X, let us recall the exact sequence

zy-.n-.o,

(4.3.1)

where

nzn : Zy -+ n

(4.3.2)

CHAPTER 4. HAMILTONIAN FORMALISM

248

is an affine bundle over II with 1-dimensional fibres. DEFINITION 4.3.1. Let h be a section of the fibre bundle (4.3.2). Then the pull-back

H=h'E:H-AT'Y,

(4.3.3)

H = p; dy` A wa - ?{w,

of the canonical form -E on Zy by h is called the polysymplectic Hamiltonian form (or simply the Hamiltonian form). The exterior differential of the Hamiltonian form (4.3.3) is the pull-back

dH = h'SlZ over II of the multisymplectic form (4.2.10).

Example 4.3.1. Let

r=dXA®(aa+r1a,) be a connection on the fibred manifold Y

X. Hence, we have the splitting

r:V'Y-T'Y, of the exact sequence (1.4.3a). Then r also yields splitting

hr: 11 -'ZY, hr :p;ay'®wasp;dy'AWA-p;r`aw, of the exact sequence (4.3.1). It follows that every connection r on the fibred manifold Y -. X defines the Hamiltonian form HP =h*r=,

Hr =p;dy'Awa-p;r'aw,

(4.3.4)

on the Legendre bundle H.

Example 4.3.2. Let us consider the fibre bundle Y -. R as in Example 4.2.1. In this case, a Hamiltonian form reads

H = pidy' - Ndt,

4.3. HAMILTONIAN FORMS

249

that coincides with the well-known Poincare-Cartan integral invariant of timedependent mechanics [6].

PROPOSITION 4.3.2. Hamiltonian forms on II constitute an affine space modelled over the linear space of horizontal densities

H=7{w:II-XT'X on fI

(4.3.5)

X. They are called Hamiltonian densities. 0

Proof. The proof is based on the fact that the affine bundle (4.3.2) is modelled over the pull-back vector bundle

fI x AT'X -- TI. x

QED It means that, if H is a Hamiltonian form and If is a horizontal density (4.3.5), then H - H is also a Hamiltonian form. Conversely, if H and H' are Hamiltonian forms, their difference H - H' is a Hamiltonian density (4.3.5). Proposition 4.3.2 and Example 4.3.1 lead to the following. COROLLARY 4.3.3. Every Hamiltonian form on the Legendre bundle R admits the decomposition

H=Hr -Hr=p;dy'Awa-p r w-??rw,

(4.3.6)

where r is a connection on Y -. X. Given another connection r' = r + o, where o is a soldering form on Y, we have

Hr = Hr - P; caw.

Moreover, every Hamiltonian form admits a canonical decomposition as follows. We mean by a Hamiltonian map any bundle morphism y

J'Y,

bao4'=4'%(9),

9E11,

(4.3.7)

CHAPTER 4. HAMILTONIAN FORMALISM

250

over Y. Its composition with the canonical morphism (2.1.19) yields the bundle morphism Y

r

represented by the TY-valued I-form 4, = dx" ®(Oa + 4ia(4)8:)

(4.3.8)

on the Legendre bundle n -i Y.

Example 4.3.3. Let r be a connection on Y - X. Then, the composition

roirny:n - Y- JlY, dxa®(8a+PAO,),

(4.3.9)

is a Hamiltonian map. Conversely, every Hamiltonian map 4) : II -' J'Y yields the associated connection

ro=$o0 on Y - X, where 0 is the global zero section of the Iegendre bundle n - Y. In particular, we have

r = r.

PROPOSITION 4.3.4. Every Hamiltonian form H on the Legendre bundle II -+ Y defines the associated Hamiltonian map

JIY, yaoH=B'87i. H: 11

(4.3.10)

0 Proof. The vertical tangent map VH of the morphism H (4.3.3) defines the linear morphism

VH:VU-+ T'Y

4.3. HAMILTONIAN FORMS

251

over Y. Therefore, it can be represented by the section

VH=ap; ®dy'Aw,,-dp; ®w OP, of the fibre bundle

V'll®XT'Y-+f1. n

After natural contractions, this section becomes the section

VH=(dy'-

dx-%)®8,

of the pull-back

11x(T'Y®VY)

11.

This represents the Hamiltonian map (4.3.10) of IT to the jet manifold J'Y considQED ered as an affine subbundle of T'Y ® VY.

COROLLARY 4.3.5. Every Hamiltonian form H on the Legendre bundle fI -+ Y determines the associated connection

f,, =HOO

onY -+X. In particular, we have 1,/Nr=1',

where H. is the 1-lamiltonian form (4.3.4) associated with the connection r on Y -+X. COROLLARY 4.3.6. Every Hamiltonian form (4.3.6) admits the canonical splitting

H = Hr,, - H. A

(4.3.11)

CHAPTER 4. HAMILTONIAN FORMALISM

252

The following assertion generalizes Example 4.3.1. PROPOSITION 4.3.7. Every Hamiltonian map (4.3.7) represented by the form (4.3.8) on n defines the associated Hamiltonian form

Ho = 0J 9 = pi dy' A wa - p;

(4.3.12)

where 9 is the tangent-valued Liouville form (4.2.4).

In particular, if

Hg = H, then H = Hr for a connection I' on Y.

4.4

Hamilton equations

Hamilton equations in symplectic mechanics are the equations of integral curves of Hamiltonian vector fields. Hamilton equations in polysymplectic Hamiltonian formalism are the equations of integral sections of Hamiltonian connections as follows. Let J'n be the first order jet manifold of the Legendre fibred manifold n -- X. It is equipped with the adapted fibred coordinates

(i-1,y',pi,y,,vil)We have the commutative diagram

J'n".!! J'Y n

4 lily

I

Y

oJ'arny=Y,'. DEFINITION 4.4.1. A connection

7 = dxa ®(OA + 7a8; + 7 8,)

on the Legendre fibred manifold n - X is said to be a Hamiltonian connection if the exterior form 7J fI = dp; A dy' A wa - (7adp; -

nw

(4.4.1)

4.4. HAMILTON EQUATIONS

253

is closed.

Example 4.4.1. Every connection r on a fibred manifold Y - X gives rise to the connection

r=dx''®[8 +ri(y)a;+

(4.4.2)

on the Legendre fibred manifold II - X, where K is a symmetric linear connection (3.7.1) on X. Due to the isomorphism (4.2.3), the connection (4.4.2) is constructed as follows. It is the tensor product

r=(rxK)®V'r

(4.4.3)

over r of the product connection r x K on the pull-back

Y xX A'T'X - X and the covertical connection V'r to r (2.5.14) on the vertical cotangent bundle X. Since the connections r x K and V'r are linear connections over r, V'Y X, by their tensor product (4.4.3) is well defined. The connection (4.4.2) on II construction, projects onto the connection I' on Y - X. The connection I'' (4.4.2) obeys the relation

rJn = d(I'J19). It follows that [' is a Hamiltonian connection. 1 Thus, Hamiltonian connections always exist on the Legendre fibred manifold II -- X, and every connection r on Y -' X gives rise to a Hamiltonian connection

onH - X.l It is easily observed that a connection ry on the fibred manifold n - X is a Hamiltonian connection if and only if y satisfies the conditions - a, rya = 0,

(4.4.4)

; - 0. '-em, = 0,

(4.4.5)

19371 +aa';=0.

(4.4.6)

a;,

t

CHAPTER 4. HAMILTONIAN FORMALISM

254

Using the relation (4.4.6), we find that the second term in the right-hand side of the expression (4.4.1) is a closed form. Then, in accordance with the relative Poincar6 lemma ([1], p.69), this expression is brought locally into the form 7J S2 = d(p, dy' Awa - 9-4w) = dH.,,

(4.4.7)

where 4% is a local function on f1 such that

7.t = -ax-

7a =

Remark 4.4.2. Relative Poincard lemma. Let us consider the vector space R'" x R" with the Cartesian coordinates (q', za). Let ilk denote the vector space of exterior k-forms on R' x R". Recall that there exists the homotopy operator

H:JDkdoH+Hod= ldiOk, [2, 186]. Let

0=wnw, be an exact (r + n)-form on R' x R". Then, 0 is brought into the form

0=donw, where o is an (r - 1)-form on R'" x R". By analogy with the explicit form of the homotopy operator ([2], p.118), this form is defined by the relation i

v,_,J...viJu,,J...u1J(a(z)Aw)= f tr_1[u_j1...v,1u"1...uiJzJ*(tz)]dt, 0

v;ERn, u,ER",

tER, ZER-xR".

Indeed, it is easy to check that

d(o(z) A w) = f j (t'm(tz))dt =,O(z). 0

Given a connection r on the fibred manifold Y expression (4.4.7) can be written as

H.,=Hr-7'{rw,

X, the local form H., in the

4.4. HAMILTON EQUATIONS

255

where 7Irw is a local horizontal density on II - X. In accordance with Proposition 4.3.2, it follows that H, defines the local section

h,: (xa, y%, pi) '-' (2, y', Pi , p

of the fibre bundle Zy -, II, i.e., Hr is a local Hamiltonian form. Thus, we have proved the following.

PROPOSITION 4.4.2. For every Hamiltonian connection ry on the Legendre fibred

manifold II -. X, there exists a local Hamiltonian form H in a neighbourhood of each point q E II such that

Let us formulate the converse assertion.

DEFINITION 4.4.3. The Hamilton operator EH for a Hamiltonian form H on the Legendre fibred manifold II -' X is defined to be the first order differential operator Efr : J' II -y

"A1

T' II,

EH=dH - St= ((y;,-8a71)dpi -(Pai+8,7l)dy'JAw,

(4.4.8)

where SZ = dp, A dy' A wa + pa,dy' A w - y, dpi A w

is the pull-back of the polysymplectic form 12 (4.2.5) onto J'II. Here we have used the canonical morphism (2.1.21)

J'IIxTX - J'IlxTll, II yn 8a "- 8a + Yi 0i + P.P%A'

and the natural contractions.

A glance at the expression (4.4.8) shows that the Hamilton operator EH is an affine morphism over ll of constant rank. Thereby, its kernel is an affine subbundle eH = KerEH

II

(4.4.9)

CHAPTER 4. HAMILTONIAN FORMALISM

256

of the jet bundle J'fI - R which is given by the coordinate relations ya

(4.4.10a)

pxA,

(4.4.10b)

This affine bundle is modelled over the vector subbundle of the vector bundle

T'X ® VII - lI,

(4.4.11)

which is defined by the coordinate relations

0,; = 0

(4.4.12)

with respect to the fibre coordinates ;) on (4.4.11). Since 0-H (4.4.9) is an affine subbundle, it is a closed imbedded submanifold of the fibred manifold J'fI - X and, therefore, is a system of first order differential equations on fl X in accordance with Definition 3.3.1. DEFINITION 4.4.4. The first order differential equations (!H (4.4.9) are called the Hamilton equations for the Hamiltonian form H on the Legendre bundle II. 0

Remark 4.4.3. A glance at the Hamilton operator (4.4.8) shows that polysymplectic Hamiltonian forms may be considered modulo closed forms since closed forms do not make any contribution to the Hamilton operator. Since the subbundle eH (4.4.9) is affine, it always admits a global section ry. Any such section is a connection on 11 --+ X which meets the condition

CH Oy=0.

This condition takes the form

7J1E = W.

(4.4.13)

It follows that every connection on II - X which takes its values into the Hamilton equations (1H is a Hamiltonian connection. DEFINITION 4.4.5. A Hamiltonian connection 7 on the Legendre fibred manifold II - X is said to be associated with a Hamiltonian form H if -y obeys the relation (4.4.13). O

4.4. HAMILTON EQUATIONS

257

Thus, we have proved the following.

PROPOSITION 4.4.6. Every Hamiltonian form on the Legendre bundle f1 has an associated Hamiltonian connection. 0

We have the equations of a Hamiltonian connection associated with a given Hamiltonian form: (4.4.14) (4.4.15)

By the equation (4.4.14), every Hamiltonian connection 'y for a Hamiltonian form H satisfies the relation J'anyo'y=H,

(4.4.16)

7

where H is the Hamiltonian map (4.3.10). It projects onto the connection rH on Y - X associated with the Hamiltonian form H. We have the commutative diagram

Y

A glance at the equations (4.4.15) shows that there is a set of Hamiltonian connections associated with the same Hamiltonian form H. They differ from each other in soldering forms o on fI -+ X which obey the equations ojf2 = 0,

oa=0,

'a M=0,

(4.4.17)

and take their values into the subbundle (4.4.12) of the vector bundle (4.4.11). Consequently, it is sufficient to have a general solution of the equations (4.4.17) and

to find a particular solution of the equations (4.4.14) - (4.4.15) in order to obtain all Hamiltonian connections associated with a given Hamiltonian form H.

CHAPTER 4. HAMILTONIAN FORMALISM

258

Example 4.4.4. If n = 1, the equations (4.4.17) have evidently only the zero solution. Consequently, there always exists a unique Hamiltonian connection associated with a given Hamiltonian form.

Example 4.4.5. Let us find the expression of a Hamiltonian connection for a Hamiltonian form H when a fibred manifold Y - X admits the vertical splitting

VYQ5 YxY

(4.4.18)

x

and, in the coordinates adapted to this splitting, the Hamiltonian density H in the canonical splitting (4.3.11) of H depends on the momentum coordinates only. Let y' be coordinates on Y adapted to the splitting (4.4.18). It means that transformation laws of the holonomic coordinates y' of VY and Pi of V'Y do not depend on y'. Consequently, the transformation law of the corresponding momentum coordinates p; of the Legendre bundle H X also are independent of y'. If there is the splitting (4.4.18), every Hamiltonian map 4s (4.3.7) defines the connection 'y, on the Legendre fibred manifold II -' X which is given by the expression

7m = d? ® (8a + 4ia8; + (-8j45;,p; +

Ka°°p)Y1

with respect to the coordinates adapted to the splitting. Here K is a symmetric linear connection on X. In particular, we have -rr = r, where I' is the connection (4.4.2).

Let H be a Hamiltonian form which satisfies the above-mentioned requirement and H the associated Hamiltonian map. Then, the connection -yp = dx'' ® f 8a + 8.-%1i8; + (-8j8 ?ip;` + K.%" p7 -

defined by N is a Hamiltonian connection associated with the Hamiltonian form H. A classical solution of the Hamilton equations (4.4.9), by definition, is a section

r of the Legendre fibred manifold H -, X such that its jet prolongation J'r takes its values into the kernel of the Hamilton operator Ey (4.4.8). Then, r satisfies the differential equations Bar'

(4.4.19a)

Bart

(4.4.19b)

4.4. HAMILTON EQUATIONS

259

Every integral section J'r = ry o r of a Hamiltonian connection ry associated with a Hamiltonian form H is a classical solution of the corresponding Hamilton equations. Conversely, if r is a global solution of the Hamilton equations (4.4.19a) - (4.4.19b) for a Hamiltonian form H, there exists an extension of this solution J'r : r(X) - J' 1I to a Hamiltonian connection which has r as an integral section. Substituting J'r in (4.4.16), we obtain the identity

I J'(7rn'or)=9 or

(4.4.20)

for every classical solution r of the Hamilton equations.

Remark 4.4.6. It may happen that the Hamilton equations (4.4.9) for a Hamiltonian form H have no classical solution through a given point q E H.

Remark 4.4.7. Note that the Hamilton equations (4.4.19a) - (4.4.19b) can be introduced without appealing to the Hamilton operator. They are equivalent to the relation

r'(uJdH) = 0

(4.4.21)

which is assumed to hold for any vertical vector field u on II X. The Hamilton equations (4.4.21) are similar to the Cartan equations (3.2.16). It is so because the Poincarc-Cartan form is the Lagrangian counterpart of a Hamiltonian form.

Remark 4.4.8. Liven a Hamiltonian form H (4.3.3) on the Legendre fibred maniX, let us consider the Lagrangian density

fold II

a

LH=(pcya-x)w

(4.4.22)

on the jet manifold J'll. It does not depend on the velocity coordinates gla. It is readily observed that the Poincar6-Cartan form HL (3.2.6) of the Lagrangian density (4.4.22) coincides with the Hamiltonian form H, and the Euler-Lagrange operator (3.2.10) for the Lagrangian density LH is precisely the Hamilton operator EH (4.4.8) for the Hamiltonian form H. As a consequence, the first order EulerLagrange equations for LH are equivalent to the Hamilton equations for H. Note that (4.4.22) exemplifies a first order Lagrangian density which leads to a first order Euler-Lagrange operator.

CHAPTER 4. HAMILTONIAN FORMALISM

260

Remark 4.4.9. The Cauchy problem. The system of Hamilton equations (4.4.19a) - (4.4.19b) has the standard form Sob(x, 4))ea41 = f. (X, -0)

for the Cauchy problem or, to be more precise, for the general Cauchy problem since the coefficients S.1b depend on the variable functions 0 in general [116). Here 0 is a compact notation for the field functions r' and r; . However, the characteristic form ca E R,

det(S,' bcx,),

of this system fails to be different from zero for any ca. One can overcome this difficulty as follows.

Let us single a local coordinate x' out and replace the equations (4.4.19a) with the equations air' = a17{, dia'a7{ = daa17[,

A # 1,

(4.4.23)

The systems (4.4.23) and (4.4.19b) have the standard form for the Cauchy problem with the initial conditions

r'(x) _

0'(x7,

r" (x) = Wi (XI),

aar'=8j,7{,

'\ 01,

(4.4.24)

on a local hypersurface S of X transversal to coordinate lines x'. are solutions of the Cauchy problem (of class Cl) for the equations (4.4.23) and (4.4.19b) with the initial conditions (4.4.24), they satisfy the equations (4.4.19a). 0 PROPOSITION 4.4.7. If r' and

Proof. The proof is standard ([116], p.28). If x is a point of an open neighbourhood of the point x' E S, we have the relations

r'(x) = f a,7Ws + ¢{(x'), 0

4.5. DEGENERATE SYSTEMS

261

x1

8ar'(x) = J 8,8ifds+O.,.P(x') _ 0

f 0,81'Wds +

8a9i(x),

a

1.

0

The statement can be extended to analytic functions.

QED

Thus, in order to formulate the Cauchy problem for the Hamilton equations in polysymplectic Hamiltonian formalism, one should single a one of the coordinates out and consider the system of equations (4.4.23) and (4.4.19b).

Degenerate systems

4.5

This Section is devoted to the relations between polysymplectic Hamiltonian formalism and Lagrangian formalism when a Lagrangian density is degenerate. The main

peculiarity of these relations consists in the fact that there is a set of Hamiltonian forms associated with the same degenerate Lagrangian density. Given a fibred manifold Y X, let H be a Hamiltonian form on the Legendre bundle II over Y and H the corresponding Hamiltonian map (4.3.10). Let L be a Lagrangian density on the jet manifold J1Y of Y and L the corresponding Legendre map

L:J'YYH, peoL=7l. The associated Lagrangian and Hamiltonian systems are characterized by the diagram

n L

i

L

Jl y L 11 which fails to commute in general, that is,

Loi#IdH and HoLOIdJ'Y.

(4.5.1)

CHAPTER 4. HAMILTONIAN FORMALISM

262

Only if the Legendre map L is a diffeomorphism, does a Lagrangian system meet necessarily a unique equivalent Hamiltonian system such that the associated Hamiltonian map is the inverse diffeomorphism j? = L-1 (see Example 4.5.5). Remark 4.5.1. It follows that, when the Legendre map is regular at a point, a local Lagrangian system on an open neighbourhood of this point has the equivalent local Hamiltonian system. In order to keep this local equivalence in case of degenerate Lagrangian densities, one may require that the image L(J1Y) of the configuration space contains all points where the Hamiltonian map H is regular. Let

Q =

denote the image of the configuration space J1Y by the Legendre map. Following the terminology of constraint theory, we call Q a Lagrangian constraint space or simply a constraint space. We will see that all Hamiltonian counterparts of solutions of the Euler-Lagrange equations live in the Lagrangian constraint space. Unless otherwise stated, we regard Q as a subset of II without a manifold structure.

DEFINITION 4.5.1. Given a Lagrangian density L on J'Y, a Hamiltonian map 4b: II -. J1 Y is called associated with L if L o 4) IQ= Id Q,

IP; (q)

= 8;

(4.5.2)

qEQ.

0

Remark 4.5.2. It follows that, given a Hamiltonian map 4) associated with the Lagrangian density L, a point q E II belongs to the Lagrangian constraint space if and only if pi (q) = 8; C(x", N', V,x(q))

(4.5.3)

DEFINITION 4.5.2. We say that a Hamiltonian form H is weakly associated with a Lagrangian density L if it is of the following type:

H=H,+4)'L

(4.5.4)

263

4.5. DEGENERATE SYSTEMS

where' is a Hamiltonian map associated with L and Hm is the Hamiltonian form (4.3.12) associated with +. 0 PROPOSITION 4.5.3. If a Hamiltonian form H is weakly associated with a Lagrangian density L, the following relations hold:

H IQ=,b Q, H IQ= H'HL,

(4.5.5) (4.5.6)

IQ,

H(q) = p"c3,h(q) - C(x", y`, &A' ((9)),

q E Q,

(4.5.7)

where 1-!I, is the Poincare-Cartan form (3.2.6). 0

Proof. The proof of (4.5.5) and (4.5.6) is straightforward; (4.5.7) is deduced from QED (4.5.2) and (4.5.6). Note that, unless otherwise stated, all objects are defined on the whole Legendre bundle II and their restriction to Q means only that their values at the points Q C 11 are considered. To overcome this difficulty, we can narrow the class of Hamiltonian forms related to a given Lagrangian density.

DEFINITION 4.5.4. Let H be a Hamiltonian form (4.5.4) weakly associated with a Lagrangian density L. We say that H is associated with L if

H=

(4.5.8)

O

An equivalent definition is the following.

DEFINITION 4.5.5. A Hamiltonian form H is said to be associated with a Lagrangian density L if H satisfies the relations

A

L o l IQ= Id Q,

(4.5.9a)

H = Hjl +

(4.5.9b)

CHAPTER 4. HAMILTONIAN FORMALISM

264

The relations (4.5.9a) and (4.5.9b) have the coordinate expressions I

(q) = 8; C(e,Y',8ax(q)),

q E Q,

(4.5.10)

and

?l =8µ?{ - G(xA, y{, 8a7i).

(4.5.11)

The latter is the equality (4.5.7) which now holds on the whole Legendre bundle 11.

At the same time, the relation (4.5.6) remains true at the points of the constraint space Q only. Acting on both sides of the equality (4.5.11) by the exterior differential, we obtain the relations

8,.H(q) = -(8vG)(xµ,p,8i

8f(q) = -(8;G)(x"'1li,

-H(q})

q E Q,

q E Q,

(PC - (arc) (e, V, - 8a?i))8`µ8Q?{ = 0.

(4.5.12)

(4.5.13) (4.5.14)

A glance at the relation (4.5.14) shows that:

the condition (4.5.9a) is a corollary of (4.5.9b) if the Hamiltonian map H is regular det(88,8µ7{) IQ96 0

at all points of the constraint space Q, the Hamiltonian map H is non-regular outside the Lagrangian constraint space Q in accordance with Remark 4.5.1.

Example 4.5.3. Let L = 0 be the zero Lagrangian density. In this case, the Lagrangian constraint space is Q = 0(Y), where 6 is the canonical zero section of the Legendre bundle II Y. The condition (4.5.2) is trivially satisfied by every Hamiltonian map. Hence, any Hamiltonian form (4.3.12) is weakly associated with the zero Lagrangian density. At the same time,

4.5. DEGENERATE SYSTEMS

265

the Hamiltonian forms associated with L = 0 must obey additionally the condition (4.5.11) which takes the form

7= pµ&µf{ These are the Hamiltonian forms Hr (4.3.4). There exist Lagrangian densities which do not possess associated Hamiltonian forms defined everywhere on the Legendre bundle H.

Example 4.5.4. Let Y be the bundle R2 R with coordinates (x, y). The jet manifold J'Y = R3 and the Legendre bundle H = R3 with coordinates (x,y,y2) and (x, y, p), respectively. Put G = exp y=.

The corresponding Legendre map reads

poL=expysIt follows that the Lagrangian constraint space Q is given by the coordinate relation

p > 0. This is an open subbundle of the Legendre bundle, and L is a diffeomorphism of J'Y onto Q. Hence, there is a unique Hamiltonian form H on Q which is associated and weakly associated with L. It reads

l = p(ln p - 1). This Hamiltonian form, however, fails to be smoothly extended to II.

Now, let us investigate the relations between the equations in the Lagrangian formalism and its polysymplectic Hamiltonian counterpart in accordance with the jet extension J'n

J1z

I

JIP J'J'Y

Iz

J'J'Y'a J'n

of the diagram (4.5.1). Let us recall the coordinate expressions

(y;,,c J'H = da=8+y;,8J+p;,j8y,

da8,,l),

CHAPTER 4. HAMILTONIAN FORMALISM

266

and a

i

a

(PiryrPµi)0J

1

a

as=as+i a;+Y,pa; Note that, if -y is a Hamiltonian connection for the Hamiltonian form H, the composition J' JI o -y takes its values (U,1/µ) o J' H o 'r = (&µW, M ,,*H)

into the sesquiholonomic subbundle JAY of the repeated jet manifold J'J'Y. Example 4.5.5. Let us start from a hyperregular Lagrangian density, i.e., when the Legendre map L is a diffeomorphism. In this case, the Lagrangian formalism and the polysymplectic Hamiltonian formalism are equivalent. If a Lagrangian density L is hyperregular, there always exists a unique associated and weakly associated Hamiltonian form

H = Hi_, + L-''L. The corresponding Hamilton map (4.3.10) is the diffeomorphism H = L-' as well as its first order jet prolongation J'H:

J'LoJ'1 =IdJ'fI. PROPOSITION 4.5.6. Let L be a hyperregular Lagrangian density and H the associated Hamiltonian form. The following relations hold: HL, = L-H,

(4.5.15)

EZ = (A)-EH, Eu = (J'JJ)'ee,

(4.5.16) (4.5.17)

where EN is the Hamilton operator (4.4.8) for H and Ey is the Euler-LagrangeCartan operator (3.4.10). O The proof is straightforward.

A glance at (4.5.6) and (4.5.15) shows that the PoincarA-Cartan form is the Lagrangian counterpart of the Hamiltonian forms (4.3.6), whereas the Lagrangian counterpart of the Hamilton operator is the Euler-Lagrange-Cartan operator 81 (3.4.10).

4.5. DEGENERATE SYSTEMS

267

In particular, if 7 is a Hamiltonian connection for the associated Hamiltonian form H, then the composition J' H o ry takes its values into the kernel of the EulerLagrange-Cartan operator Cl; (more exactly, in the kernel of the first order EulerLagrange operator E, (3.4.12)), that is, J'Ho-yoL is a Lagrangian connection for L. Conversely, if I' is a Lagrangian connection for L, then J' L o -y o ft is a Hamiltonian one. This proves the following assertion. PROPOSITION 4.5.7. Let L be a hyperregular Lagrangian density and H the associated Hamiltonian form. (i) Let r : X 17 be a solution of the Hamilton equations (4.4.19a) - (4.4.19b) for the Hamiltonian form H. Then, the section

s=7rnyor of the fibred manifold Y -, X is a solution of the second order Euler-Lagrange equations (3.2.14), while its first order jet prolongation

s=/-/or=J's satisfies the Cartan equations (3.2.17a) - (3.2.17b). (ii) Conversely, if a section -s of the fibred jet manifold J'Y -' X is a solution of the Cartan equations (3.2.176) - (3.2.17b), the section

r=Los of the Legendre fibred manifold II -, X satisfies the Hamilton equations (4.4.19a) - (4.4.19b).

It follows that, given a hyperregular Lagrangian density, there is one-to-one correspondence between the solutions of the second order Euler-Lagrange equations

(and, consequently, of the Cartan equations) and the solutions of the Hamilton equations of the associated Hamiltonian form. In the case of a regular Lagrangian density L, the Lagrangian constraint space Q is an open subbundle of the Legendre bundle II Y. If Q 34 II, an associated Hamiltonian form fails to be defined everywhere on n in general (see Example 4.5.4). At the same time, an open constraint subbundle Q can be provided with the pullback polysymplectic structure with respect to the imbedding Q F-+ H, so that we may restrict our consideration to Hamiltonian forms on Q. If a regular Lagrangian density is additionally semiregular (see Definition 4.5.8), the associated Legendre

268

CHAPTER 4. HAMILTONIAN FORMALISM

morphism is a diffeomorphism of J'Y onto Q and, on Q, we can recover all results true for hyperregular Lagrangian densities. Contemporary field models are almost never regular. Hereafter, we restrict our consideration to semiregular Lagrangian densities. DEFINITION 4.5.8. A Lagrangian density L is called semiregular if the pre-image L'' (q) of any point q of the Lagrangian constraint space Q is a connected subman-

ifold of J'Y. 0 This notion of degeneracy is most appropriate in order to study the relations between solutions of Euler-Lagrange and Hamilton equations [161, 190].

PROPOSITION 4.5.9. All Hamiltonian forms weakly associated with a semiregular

Lagrangian density L (if they exist) coincide with each other at the points of the Lagrangian constraint space Q:

HIQ=H'IQ Moreover, the Poinear6-Cartan form HL (3.2.6) for L is the pull-back

HL = L'H,

(ri a - G)rv = ?i(x", pf

)w,

(4.5.18)

of any weakly associated Hamiltonian form H by the Legendre morphism L. O

Proof. Let u be a vertical vector field on the jet bundle J'Y - Y. If u takes its values into the kernel Ker TL of the tangent morphism to L, it is easy to see that

LVHL=0, where L. is the Lie derivative with respect to u. Hence, the Poincarb-Cartan form HL for a semiregular Lagranglan density L is constant on the connected pre-image L''(q) of each point q E Q. Then results follow from (4.5.6). QED

Remark 4.5.6. Note that the Hamilton operators of Hamiltonian forms in Proposition 4.5.9 do not necessarily coincide at points of Q because of the derivatives of these forms which are present in the expression (4.4.8).

4.5. DEGENERATE SYSTEMS

269

Remark 4.5.7. Example 4.5.3 shows that the Hamiltonian forms associated with a Lagrangian density take quite a specific form outside the Lagrangian constraint space Q. Indeed, the condition (4.5.9b) rigidly restricts the arbitrariness of these forms on a neighbourhood of Q. For instance, let H be a Hamiltonian form associated with a semiregular Lagrangian density L. Substituting this condition in (4.5.18), we obtain

(ir o H - P, )aax = x(x", yi, 7r; o H)

- x(x", Eli, Pi)

(4.5.19)

at every point of R\Q. Let us assume that the constraint space Q is given locally by the equations p; = 0 where (3;') is a subset of the coordinates (pi) and p; are the remaining ones. Then the relation (4.5.19) takes the form

'Pp) = hw'z*'O'&P) Using this relation, one can show that, if the Hamiltonian density 71 is an analytic function in the momentum coordinates at a point q E Q, then it is an affine function in the coordinates pi around q. For example, let Y be the fibre bundle R3 RZ with coordinates (x', x2, y). The jet manifold P Y Y and the Legendre bundle 17 over Y are equipped with coordinates (xl, x2, yr yl, y2) and (x', x2, y, pl, p2), respectively. Let G=

1

(yl )2w.

(4.5.20)

This Lagrangian density is semiregular. The associated Legendre map reads

PIoL=Yi, p2oL=0. The corresponding constraint space Q consists of the points with the coordinate p2 = 0. Hamiltonian forms associated with the Lagrangian density (4.5.20) are given by the expression H = PAdy A W.% - [2 (p1)s + c(xI, x', y)p2]

,

where c(xl , x2, y) is an arbitrary function. These Hamiltonian forms are affine in the momentum coordinate p2. 0

CHAPTER 4. HAMILTONIAN FORMALISM

270

Let H be a Hamiltonian form weakly associated with a semiregular Lagrangian density L. Acting by the exterior differential on the relation (4.5.18), we obtain the equality (ya - Oax o L)dar; Aw - (8,L + A. (x o L))dy Aw = 0

(4.5.21)

or

?10 L) = 0,

8;r,( -B,xoL)-(OC+(O;x)oL)=.0. Using the equality (4.5.21), one can extend the relation (4.5.16) (but not necessarily the relation (4.5.17)) I ei = (J' L)'CH

(4.5.22)

to semiregular Lagrangian densities. This relation enables us to extend Proposition 4.5.7 (i) also to Iamiltonian forms associated with semiregular Lagrangian densities. PROPOSITION 4.5.10. Let a section r of the Legendre fibred manifold 11 X be a classical solution of the Hamilton equations (4.4.9) for a Hamiltonian form H weakly associated with a semiregular Lagrangian density L. If r lives in the constraint space Q, the section

s=irnyor of the fibred manifold Y --+ X satisfies the second order Euler-Lagrange equations, while its first order jet prolongation

's =llor=J's obeys the Cartan equations (3.2.17a) - (3.2.17b). 0

Proof. Put s = H o r. Since r(X) C Q, then

r=Lo

J'r=J'LoJ'3.

If r is a solution of the Hamilton equations, the exterior form EH vanishes at points

of Jlr(X). Hence, the pull-back form EL = (J'L)'EH vanishes at points J's(X). It follows that the section y of the fibred jet manifold J'Y -, X obeys the Cartan

4.5. DEGENERATE SYSTEMS

271

equations (3.2.17a) - (3.2.17b). By virtue of the relation (4.4.20), we have 's = J's. Hence, s is a classical solution of the second order Euler-Lagrange equations. QED

Remark 4.5.8. In accordance with the relation (4.5.22), if ry is a Hamiltonian connection for the associated Hamiltonian form H, then the composition J'H o ry takes its values into the kernel of the Euler-Lagrange-Cartan operator El, that is, J' H o 7 o L is a Lagrangian connection for L. It follows, that a semiregular Lagrangian density which has a weakly associated Hamiltonian form always admits a Lagrangian connection. s Proposition 4.5.7 (ii), however, must be modified as follows.

PROPOSITION 4.5.11. Given a semiregular Lagrangian density L, let a section 3 of

the fibred jet manifold J'Y -. X be a solution of the Cartan equations (3.2.17a) - (3.2.17b). Let H be a Hamiltonian form weakly associated with L so that the associated Hamiltonian map satisfies the condition

IHoLos= J'(apos).

(4.5.23)

Then, the section

r=Los, ri = 7ri (x'1,V , TJa),

r`=s`,

of the Legendre fibred manifold fI -+ X is a solution of the Hamilton equations (4.4.19a) - (4.4.19b) for H. Proof. The Hamilton equations (4.4.19a) hold in virtue of the condition (4.5.23). Using the relations (4.5.21) and (4.5.23), the Hamilton equations (4.4.19b) are brought into the Cartan equations (3.2.17b):

daa; os=-(8;l)oLo38µfloLo3)8;io3+8Cos= (8,s' - sµ)8{aj" o 3 + 8;G o 3.

QED

Remark 4.5.9.

Let 33

: X -. J'Y be a solution of the Cartan equations for a

Lagrangian density L. In accordance with the relation (4.4.20), the condition

HoLos=J'(aoo3)

(4.5.24)

CHAPTER 4. HAMILTONIAN FORMALISM

272

is necessary in order that L o -9 be a solution of the Hamilton equations. Propositions 4.5.10 shows that, if H is a Hamiltonian form weakly associated with a semiregular Lagrangian density L, every solution of the corresponding Hamilton

equations which lives in the constraint space Q yields a solution of the Cartan equations and the second order Euler-Lagrange equations for L. At the same time, the condition (4.5.23) is the obstruction to a solution 9 of the Cartan equations from being a solution of the Hamilton equations. COROLLARY 4.5.12. Let a solution 3 of the Cartan equations obey the condition (4.5.23) for a Hamiltonian form weakly associated with a semiregular Lagrangian density L. In accordance with Propositions 4.5.10 and 4.5.11, its projection iro o3 onto Y is a solution of the second order Euler-Lagrange equations. This Corollary provides a solution of the so-called "second order equation" problem in the case of semiregular Lagrangian densities [18, 76]. Given a degenerate Lagrangian density, there are obviously solutions of the Cartan equations which are not solutions of the Hamilton equations. At the same time, one may consider a set of Hamiltonian forms associated with a degenerate Lagran-

gian density in order to exhaust all solutions of the second order Euler-Lagrange equations.

Example 4.5.10. Let L = 0. This Lagrangian density is semiregular. Its EulerLagrange equations come to the identity 0 = 0. Every section s of the fibred X is a solution of these equations. Given a section s, let I' be manifold Y a connection on Y such that s is its integral section. The Hamiltonian form 11r (4.3.4) is associated with L, and the Hamiltonian map Hr satisfies the relation (4.5.23). The corresponding Hamilton equations have the solution

r=L0Jls, r{ =s{,

rt =0.

We will say that a family of Hamiltonian forms H associated with a Lagrangian density L is complete if, for each solution s of the second order Euler-Lagrange equations, there exists a solution r of the Hamilton equations for a Hamiltonian form 11 from this family so that s = xny o r. Let L be a semiregular Lagrangian

4.5. DEGENERATE SYSTEMS

273

density. Then, in virtue of Proposition 4.5.11, such a complete family of associated Hamiltonian forms exists if and only if, for every solution s of the Euler-Lagrange

equations for L, there is a Hamiltonian form H from this family such that the relation (4.5.23) holds.

Remark 4.5.11. A complete family of Hamiltonian forms associated with a given Lagrangian density fails to be defined uniquely. For instance, Example 4.5.10 shows that the Hamiltonian forms (4.3.4) constitute a complete family associated with the zero Lagrangian density, but this family is not minimal. Lagrangian densities of field theories are almost regular as a rule. Therefore, let us consider Lagrangian densities of this type. DEFINITION 4.5.13. A Lagrangian density L is said to be almost regular if-

the Lagrangian constraint space Q -- Y is a dosed imbedded subbundle iq : Y; U of the Legendre bundle H Q

the Legendre map L : J'Y

Q is a submersion with connected fibres.

An almost regular Lagrangian density, by definition, is semiregular. PROPOSITION 4.5.14. Let L be an almost regular Lagrangian density. On an open

neighbourhood of U of each point q E Q, there exists a complete family of local Hamiltonian forms associated with L 1190). 0

Proof. Given a point q E Q, let (2a', y, p{) be local coordinates in some open neighbourhood of q. Since the rank of the Legendre morphism Z is constant, one can select a maximal subset

of the coordinates ya so that the equations

a= 8L can be solved for a = ea(xm,y',pip, where

v

are the remaining coordinates. Substituting (4.5.25) in the equation

_' 8G

CHAPTER 4. HAMILTONIAN FORMALISM

274

we obtain ' = &"(XU, Y" pip, 4),

where lj play the role of local coordinates of the constraint space Q. For every section s of Y - X, the Hamiltonian map

X = 0(X",,

(4.5.26)

satisfies (4.5.2). The corresponding local Hamiltonian form (4.5.4) satisfies (4.5.8).

Hence, it is associated with L. Given a section s of Y - X, the Hamiltonian map (4.5.26) obeys the relation (4.5.23). It follows that the local Hamiltonian forms (4.5.26) constitute a complete family.

QED

The example below shows that a complete family of associated Hamiltonian forms may exist when a Lagrangian density is not necessarily semiregular. Example 4.5.12. Let Y be the bundle R2 --+ R1 in Example 4.5.4, with coordinates

(z, y). Put G = 3(y:)3.

The associated Legendre map reads p o L = y=.

(4.5.27)

The corresponding constraint space Q is given by the coordinate relation p > 0. It fails to be a submanifold of fl. There exist two associated Hamiltonian forms H+ = pdy - 3p3V2dx,

H- = pdy +

2 p3/2

3 to the two different solutions on Q which correspond

y=f and y=-f of the equation (4.5.27). They form a complete family. Thus, we have shown the following.

All solutions of the Hamilton equations which correspond to solutions of the second order Euler-Lagrange equations live in the Lagrangian constraint space.

4.5. DEGENERATE SYSTEMS

275

Solutions of the same Euler-Lagrange equations correspond to solutions of different Ilarnilton equations in general. We may conclude that, roughly speaking, the Hamilton equations involve some additional conditions in comparison with the second order Euler-Lagrange equations.

Therefore, let us separate a part of the Hamilton equations which are defined on the Lagrangian constraint space Q when L is an almost regular Lagrangian density. Given an almost regular Lagrangian density, let us assume that the fibred manifold

L:J'Y-+Q has a global section. In accordance with Theorem 1.2.5, this section can be extended

to a Hamiltonian map : II - J'Y which is associated with L. This guarantees the existence of global Hamiltonian forms weakly associated with L. Let

HQ = i4 H

(4.5.28)

be the restriction of a Hamiltonian form H weakly associated with L to the constraint space Q. In virtue of Proposition 4.5.9, this restriction is uniquely defined, and HL = L'IIQ. We call (4.5.28) the constrained Hamiltonian form. For sections r of the fibred manifold Q -' X, we can write the equation

r'(ugJdHq) = 0,

(4.5.29)

where ug is an arbitrary vertical vector field on Q X (65, 1641. It is called the constrained Hamilton equation. These equations fail to be equivalent to the Hamilton equations restricted to the constraint space Q. PROPOSITION 4.5.15. For any Hamiltonian form H weakly associated with the almost regular Lagrangian density L, every solution r of the Hamilton equations which lives in the constraint space, i.e., r : X - Q is a solution of the constrained Hamilton equations (4.5.29). 0

Proof. For any vertical vector field uq on Q -. X, the vector field Tiq(uq) is obviously a vertical vector field on fl -+ X. Then we have

r'(uQJdllq) = r'(ugJiQdH) = r'(Tiq(uq)JdH) = 0

CHAPTER 4. HAMILTONIAN FORMALISM

276

if r is a solution of the Hamilton equations (4.4.21) for the Hamiltonian form H. QED

In brief, we can identify a vertical vector field uQ on Q -. Y with its image TiQ(uQ) and can bring the constrained Hamilton equations (4.5.29) into the form

r'(uQJdH) = 0,

(4.5.30)

where r is a section of Q X and uQ is an arbitrary vertical vector field on Q -. X. Using the equation of the Lagrangian constraint space (4.5.3), it is easy to show that a vertical vector field

u=u`01+u;BA on f1 -. X is tangent to Q if and only if u satisfies the equations -(O;O," G + OP

.C8Aµ7{)u' + (6i6 - OkB; G8µ8,1,?{)u = 0.

PROPOSITION 4.5.16. A section -3 of J1Y - X is a solution of the Cartan equations

(3.2.16) if and only if L o 3 is a solution of the constrained Hamilton equations (4.5.29).

Proof. Let uQ be a vertical vector field on Q -+ X. Since L is a submersion, there is a vertical vector field v on J1 Y - X such that

TLov=uQ. For instance, v is the horizontal lift of u by means of a connection on the fibred Q. Let a section S : X - J' Y be a classical solution of the Cartan manifold J1 Y equations (3.2.16). Then we have (L o 3)' (uQ j dHQ) = 3'(v jdHL) = 0.

(4.5.31)

It follows that the section L o 3 : X - Q is a solution of the equations (4.5.29). The converse is obtained by running (4.5.31) in reverse, bearing in mind that the QED restriction of any vector field v on J1Y to 3(X) is projectable by L. PROPOSITION 4.5.17. The constrained Hamilton equations (4.5.29) are equivalent to the Hamilton-De Donder equations (3.4.21).

4.5. DEGENERATE SYSTEMS

277

Proof. Let H be a Hamiltonian form weakly associated with L and h the corresponding section of the fibre bundle Zy - II. This section yields the morphism

The morphism hq does not depend on the choice of H. This is a section of the fibre

bundle Zy - II over Q C Il, i.e., (4.5.32)

azn o hQ = Id Q. Moreover, we have

HQ = iQH = iQ(h'?) = hQ=-, whenever 1/ is a Hamiltonian form weakly associated with the Lagrangian density L. In accordance with the relation (4.5.18), we have HL = hq o L,

(4.5.33)

where HL is the Legendre morphism (3.4.17) associated with the Poincare-Cartan form H. Substituting (3.4.18) in (4.5.33), we obtain HL = hQ o zzn o HL.

it follows that hq o 7rzn I

zi=

Id iL(ZL)

(4.5.34)

where iL(ZL) = HL(J'Y) is the image of the Legendre morphism HL. A glance at the relations (4.5.32) and (4.5.34) shows that there is the fibred isomorphism *znaL

ZL-'Q Ij'ohQ over Y. Since HQ = hh_ and ?L = iLE, we have

t/q = (ii' a hq)'-L,

L = (wzn o iL)'HQ.

Hence, the Hamilton-De Donder equations (3.4.21) are equivalent to the constrained Hamilton equations (4.5.29). QED

CHAPTER 4. HAMILTONIAN FORMALISM

278

The Propositions 3.4.4, 4.5.16 and 4.5.17 give the relations between Cartan, Hamilton-De Donder and constrained Hamilton equations when a Lagrangian density is almost regular in accordance with Definition 4.5.13.

Example 4.5.13. Hamiltonian systems on composite manifolds. An interesting example of a degenerate field system is furnished by a Hamiltonian system on a composite fibred manifold (4.5.35)

Y

This is the case of gauge models with spontaneously broken symmetries (see Remark 3.8.13) and gauge gravitation theory.

Let the composite fibred manifold (4.5.35). have the coordinates (x", a"', yi), where (x", a') are fibred coordinates on the fibred manifold E -, X. Let

AE = d? 0 (8 + A8i) + dam ®(8,n + A;n8i)

(4.5.36)

be a connection on the fibred manifold Y -' E. Recall that such a connection defines the splitting VY = VYE ®(Y x VE), Y

E

(4.5.37)

+ O"Om = (Eli - A;na'")8; + om(8m +

Using this splitting, one can construct the first order differential operator

DA = d? ® (ya - A'A' - A;naa )8i,

(4.5.38)

on the composite fibred manifold Y. Let a Lagrangian density L on the jet manifold J'Y factorize through DA as in (3.8.41). It means that L depends on the velocity coordinates oµ only through the differential operator (4.5.38). Such a Lagrangian density is degenerate, and we have the constraint 7rm + An7r = 0.

The Dirac Lagrangian density (3.8.58) of gauge gravitation theory is of this type. Let us consider an associated Hamiltonian system. The Legendre bundle fI over the composite fibred manifold Y (4.5.35) is coordinatized by a

i

a

(x , a-, y , Pmo i).

4.5. DEGENERATE SYSTEMS

279

The horizontal splitting (4.5.37) yields the corresponding splitting

fl = "T'X ®TX ®[V'YE ®(Y x V'E)] Y

Y

(4.5.39)

E

Y

Y. Given the splitting (4.5.39), the Legendre bundle II can be provided of II with the coordinates

(4.5.40)

compatible with this splitting. These coordinates, however, are not canonical. Note that, given a global section h of the fibred manifold E X, the submanifold

0m = hm(x),

(4.5.41)

Pm = 0

of the Legendre bundle II is isomorphic to the Legendre bundle n' over the restriction Y' = h'Y of the fibred manifold Y E to h(X) C E. Let the composite fibred manifold Y be provided with the composite connection

A=dxA®[8. +f,"",am+(A;nf'a +Aa)8;] (2.5.4) defined by the connection AE (4.5.36) on Y - E and by the connection r = dxa.® (8a + r;,'8m)

on E

X. Relative to the coordinates (4.5.40), every Hamiltonian form on the

Legendre bundle II (4.5.39) is given by the expression A wa -

H = (P; dy` +

(4.5.42)

[Pi Aa + Pm r-% + i(e' a-, V, M, lei AW, where

aAi+ arm Pm A = Pi A'+ aPi Pm a A

,r,-.

The corresponding Hamilton equations read

8,,p; = -P; + 8 Am(1'a` + 8. 7i)] - a?i, 8AYI = A;, + AWa + 8AIN + ffln,

(4.5.43b)

aaPm = -p; (amA;, + Bmlln(I'a + 8avm = ra + 8a ?{.

(4.5.43d)

)] - Pn8mr;,

(4.5.43a)

(4.5.43c)

CHAPTER 4. HAMILTONIAN FORMALISM

280

Let the Hamiltonian form (4.5.42) be associated with a Lagrangian density L on J'Y which factorizes through the differential operator (4.5.38). Then, the Hamiltonian density jiw is independent of the momenta p and the Lagrangian constraint space reads

In this case, the equation (4.5.43d) reduces to the gauge-type condition (4.5.44)

independent of the momentum coordinates.

In particular, let us consider such a Hamiltonian system in the presence of a background field h(x) (e.g., h is a Higgs field or a gravitational field) which is a section of the fibred manifold E --: X. Substituting the gauge-type condition (4.5.44)

into the equations (4.5.43a) - (4.5.43b) and restricting them to the submanifold (4.5.41), we obtain the equations

Sap; = -p; 8;[Aa + A,Bahm[ - Al?,

(4.5.45)

Bay' = A;, +

for sections of the Legendre fibred manifold fI" - X of the fibred manifold Y" endowed with the connection

Ah=dx''®[8a+(A'oh8ahm+Aaoh)8;] (2.5.9). Equations (4.5.45) are the Hamilton equations corresponding to the Hamiltonian form

Hh = P, dy' A wa - [P{ Aha +( e, h"'(x), Y', #j r Irm = 0)]w on II" which is the pull-back of the Hamiltonian form (4.5.42) on II. In particular, the above construction provides the Hamiltonian description of gauge models with spontaneous symmetry breaking in Remark 3.8.13. 9

4.6. QUADRATIC DEGENERATE SYSTEMS

4.6

281

Quadratic degenerate systems

The Lagrangian densities of field models are almost always quadratic or affine in the derivatives of field functions. Gauge theory exemplifies a model with a degenerate quadratic Lagrangian density, whereas fermion fields are described by the affine one. In this Section, we obtain the complete families of Hamiltonian forms associated with almost regular quadratic Lagrangian densities. These Hamiltonian forms are affine and quadratic in the momentum coordinates. The key point of our consideration is the splitting of the configuration space J1 Y in the dynamic sector and the gauge one, which coincides with the kernel of the Legendre map. As an immediate consequence of this splitting, a part of the Hamilton equations reduces to gauge-type conditions, independent of the momentum coordinates.

Given a fibred manifold Y - X, let us consider a quadratic Lagrangian density which has the coordinate expression

L=GW, G = I a,'` yayl + b ya + c,

(4.6.1)

where a, b and c are local functions on Y. The associated Legendre map reads

P: oL=afyN+b;.

(4.6.2)

LEMMA 4.6.1. The Lagrangian density (4.6.1) is semiregular. O

Proof. If q E Q, the system of linear algebraic equations (4.6.2) for yiµ, has solutions which form an affine space modelled over the linear space of solutions of the homogeneous linear algebraic equations

0=a,'jYµ, where V. are bundle coordinates of the vector bundle T'X 0 VY.

QED

Let us assume that the Lagrangian density L (4.6.1) is almost regular. The Legendre map (4.6.2) is an affine morphism over Y. It defines the corresponding linear morphism Y

Pi 0 T = j' Upi

Y

CHAPTER 4. HAMILTONIAN FORMALISM

282

whose image lQ is a linear subbundle of the Legendre bundle TI -. Y. Accordingly, the Lagrangian constraint space Q, given by the equations (4.6.2), is an affine subHence, Q - X has a global section. For the bundle of 1I -. Y modelled over sake of simplicity, let us assume that it is the canonical zero section 0(Y) of II -- Y. Then Q = Q. The kernel

Ker L = L-' (0(Y))

of the Legendre map with respect to 0(Y) is an affne subbundle of the jet bundle J'Y -. Y, which is modelled over the vector bundle

KerL = L '(0(Y)) C 7"X ®YY. Then there exists a connection I1:Y

Ker L,

,1 a.,mr, + b"

(4.6.3) (4.6.4)

X which takes its values into Ker L. With this connection, the Lagrangian on Y density (4.6.1) can be brought into the form G

Yµ - 1~µ

For instance, if the Lagrangian density (4.6.1) is regular, the connection (4.6.3) is a unique solution of the algebraic equations (4.6.4). PROPOSITION 4.6.2. There exists a linear map

v: i1-+T'X®VY,

over Y such that

Loaoiq=iq. 13

(4.6.5)

283

4.6. QUADIZZ.ATIC DEGENERATE SYSTEMS

Proof. The map (4.6.5) is a solution of the algebraic equations (4.6.6)

and B = i . After The matrix 4.j' is symmetric with respect to the indices A diagonalization, this matrix has non-vanishing Components aAA, A E I. Then a solution of the equations (4.6.6) takes the form UAA' = 0,

aAA = aAA i

A34 WE I,

while the remaining components are arbitrary. In particular, there is a solution 1

aAB = 0,

aAA = aAA'

B 76 A,

AEI.

(4.6.7)

This solution satisfies the relation

a = aoZoa.

(4.6.8)

Further on, by a is meant (4.6.7). If the Lagrangian density (4.6.1) is regular, the QED map (4.6.5) is uniquely determined by the equations (4.6.6).

The connection (4.6.3) and the map (4.6.5) play a prominent role in the construction below. PROPOSITION 4.6.3. We have the splitting

J'Y = Ker L ®Im(a o L),

(4.6.9)

Y

ya = [ya - a,(akJ y,, + br)] + [a"' (a 1rµ + k)),

of the configuration space J'Y. It follows that, since T and a are linear morphisms, their composition Z o a is a surjective submersion of II onto Q. PROPOSITION 4.6.4. There is also the splitting 11 = Ker a ®Q, P;=(P±-a10

0,µ o fl

(4.6.10)

+ [a j aµvl>k1

CHAPTER 4. HAMILTONIAN FORMALISM

284

of the Legendre bundle II -' Y. COROLLARY 4.6.5. Every vertical vector field u = u'8; + u, 8j on the Legendre fibred manifold 11 -+ X admits the decomposition u = (u - uQ) + uQ, ua

(4.6.11)

= [ui - a' Paf'ukl + [a

t ask ukl,

where uQ = u48, + a'.Oappaukua

is a vertical vector field on the constraint space Q - X. Given the linear map a (4.6.5) and the connection r (4.6.3), let us consider the affine Hamiltonian map

4) P + a : fI J1 Y, `pa = r, (y) + aV ,

(4.6.12)

where I' is the Hamiltonian map (4.3.9). By the very definition of o, this Hamiltonian map satisfies the condition (4.5.2), where L is the Legendre map (4.6.2). Thence, the corresponding Hamiltonian form H (4.5.4) is weakly associated with the Lagrangian density (4.6.1). It reads H = Ps dy' A wa - [r %(Pp - b') + 2a'jpi Pt - c]w.

(4.6.13)

This Hamiltonian form H satisfies the condition (4.5.8), and, thus, is associated with the quadratic Lagrangian density (4.6.1).

Remark 4.6.1. The Hamiltonian form (4.6.13) is quadratic in the momentum coordinates p; . At the same time, it becomes affine on Kera in accordance with Remark 4.5.7.

We aim to show that the Hamiltonian forms (4.6.13) parameterized by connections r (4.6.3) constitute a complete family. Given the Hamiltonian form (4.6.13), let us consider the Hamilton equations X. They read (4.4.19a) for sections r of the Legendre fibred manifold fI

Jts=(P+a)or,

s=any or,

(4.6.14)

4.6. QUADRATIC DEGENERATE SYSTEMS

285

or

Dart = aid where

Dar`=aar`-(Fos)a is the covariant derivative relative to the connection t. With the splitting (4.6.9), we have the surjections

S=pr,:J1Y-+KerL,

(4.6.15)

S:Y.,\~ya-o (akj`yN+br), and

F = prz : J1Y Im(r o L), a a(aki j" + bk). ya

(4.6.16)

With respect to these surjections, the Hamilton equations (4.6.14) break In the following two parts:

SoJ's=I'os,

(4.6.17)

VAr' = a i't (ak1 aµrl + bk),

and

FoJ's=aor, ik

vµ

(4.6.18)

a = Ork. ik a +bk)

The Hamilton equations (4.6.17) are independent of the canonical momenta rk and play the role of gauge-type conditions. Moreover, for every section s of the fibred manifold Y -. X (in particular, for every solution of the second order Euler-Lagrange equations), there exists a connection 1' (4.6.3) such that the equation (4.6.17) holds. Indeed, let 1" be a connection on Y X whose integral section is

s. Put

I' =So r, r = ";, - as (akf r" + bk

CHAPTER 4. HAMILTONIAN FORMALISM

286

In this case, the Hamiltonian map (4.6.12) satisfies the relation (4.5.23) for s, i.e.,

4)0LoJ's=J's. Thence, the Hamiltonian forms (4.6.13) constitute a complete family. The Hamiltonian forms from this family differ from each other only in the connections r (4.6.3) which lead to the different gauge-type conditions (4.6.17). It follows that the equations (4.6.17) are the additional conditions which make the Hamilton equations differ from the constrained Hamilton equations (4.5.29).

PROPOSITION 4.6.6. For every Hamiltonian form H (4.6.13), the Hamilton equations (4.4.19b) and (4.6.18) restricted to the constraint space Q are equivalent to the constrained Hamilton equations (4.5.29) (or (4.5.30)). 0 Proof. In accordance with the decomposition (4.6.11) of a vertical vector field u on the Legendre fibred manifold 11 -- X, the constrained Hamilton equations (4.5.30) take the form

r'(a;jAaj8afdH) = 0,

(4.6.19a)

r'(O;JdH) = 0.

(4.6.19b)

The equations (4.6.19b) are obviously the Hamilton equations (4.4.19b) for H. Bear-

ing in mind the relations (4.6.4) and (4.6.8), one can easily bring the equations (4.6.19a) into the form (4.6.18). COROLLARY 4.6.7.

QED

By virtue of Proposition 4.5.16, a section if of J1Y - X

is a solution of the Cartan equations (3.2.17a) - (3.2.17b) for the almost regular Lagrangian density (4.6.1) if and only if Lob is a solution of the constrained Hamilton

equations (4.4.19b) and (4.6.18). 0 It follows that the equations (4.6.17) are responsible for the obstruction condition (4.5.23) on solutions 3 of the Cartan equations for the Lagrangian density (4.6.1) to provide solutions of the Hamilton equations and the second order Euler-Lagrange equations. It is readily seen that the equations (4.6.18) do not contribute to this obstruction condition. If r = Loa, they hold for solutions 3 of the Cartan equations (3.2.17a).

LEMMA 4.6.8. Let 8 be a classical solution of the Cartan equations for the almost regular Lagrangian density L (4.6.1). Let go be a section of T'X ® VY - X which

4.6. QUADRATIC DEGENERATE SYSTEMS

287

takes its values into KerI and projects onto s = it o g. Then the sum 3 +lo over Y is also a solution of the Cartan equations.

Proof. The proof is an immediate consequence of the relation

r= Los= Lo(s+so). QED Remark 4.6.2. Thus, we may say the following about the gauge-type freedom of the Cartan equations for almost regular quadratic Lagrangian densities. By analogy with gauge theory, let us call gauge-type class the pre-image.F-' (z) of every point 3 E Im(u o L).

Then the rnorphisrn S o 1/ o L determines the representative (S o Il o L) (Z) E

of the gauge-type class

in accordance with the diagram

ll(Q) C J'Y

'\f Q

This representative does not coincide with z in general. Accordingly, we can speak about gauge-type classes of solutions of the Cartan equations which differ from each other in sections 36 from Lemma 4.6.8. The corresponding gauge-type condition for solutions of the Cartan equations is the modification

So%10Lo3=So3,

(4.6.20)

of the Ilarnilton equations (4.6.17), which have the form

Sot/=SoJ'(rtoos).

(4.6.21)

The condition (4.6.20) selects a particular solution of the Cartan equations from a gauge-type class. In contrast with (4.6.20), the relation (4.6.21) is a condition on sections of the fibred manifold Y - X. One can think of it as being the gauge-type condition on solutions of the Euler-Lagrange equations. 9

CHAPTER 4. HAMILTONIAN FORMALISM

288

Example 4.6.3. Gauge theory. We here follow the notation of Section 3.6. Let P - X be a principal bundle with a structure group C. Gauge theory of principal connections on P -, X is described by the degenerate quadratic Lagrangian density (3.6.25) on the first order jet manifold J'C of the bundle C -. X (2.7.5) of principal connections. Therefore, its polysymplectic Hamiltonian formulation may follow the general procedure for models with degenerate quadratic Lagrangian densities. The peculiarity of gauge theory consists in the fact that the splittings (4.6.9) and (4.6.10) of configuration and phase spaces are canonical.

Let C and J' C be provided with the coordinates (x`, a%) and (x , as , aµ,,), respectively. The configuration space J'C of gauge theory admits the canonical splitting (2.8.21), i.e.,

J'C=C+®C-=C+®(CxkT'X®VGP), a

_ (aµa

(4.6.22)

'gaµa°,) + 2 (aw - an, + 4ygaµa°,),

with the corresponding projections

S: J'C - C+, f : J'C - C_,

S;A = a;,A + aa - 4oaNa°,

(4.6.23)

Ja1 = a AA - a +cyoaµaa.

(4.6.24)

The Yang-Mills Lagrangian density (3.6.25) on this configuration space is LYM =

l 9 w1

g = det(9,w),

(4.6.25)

where ac is a non-degenerate C-invariant metric in the dual of the Lie algebra of g, g is a pscudo-Riemannian metric on X, and e is a coupling constant (see Remark 3.6.8). The finite-dimensional phase space of gauge theory is the Legendre bundle

lrnc:H-»C,

H=AT'X®TX®[Cx(4.6.26)

This is endowed with the canonical coordinates (x, aa, pm'). The Legendre bundle 11 (4.6.26) admits the canonical splitting

l1= II+ ®H_, = P;,,"'1 + Pm`

(4.6.27)

+ Pt) + 2 NO. - P."`).

4.6. QUADRATIC DEGENERATE SYSTEMS

289

The Legendre map associated with the Lagrangian density (4.6.25) takes the form

PM,`a)oLYM=0,

(4.6.28a)

&I o LyM = e 2am g1-gA0 ,01 91.

(4.6.28b)

A glance at this morphism shows that Ker LyM = C+,

and the Lagrangian constraint space is

Q = LYM(J'C) = IL. Obviously, Q is an imbedded submanifold of 11, and the Lagrangian density LYM is almost regular. Accordingly, the canonical splittings (4.6.22) and (4.6.27) are similar to the splittings (4.6.9) and (4.6.10), respectively, and the corresponding surjections (4.6.23) and (4.6.24) are exactly the surjections (4.6.15) and (4.6.16), respectively. Therefore, we can follow the general procedure described above in order to construct a complete family of Hamiltonian forms associated with the Yang Mills La grangian density (4.6.25). Let us consider connections Con the fibre bundle C - X which take their values into Ker L, i.e.,

r: C I

aµ

C+,

- l JA

(4.6.29) 644491

0.

Given a symmetric linear connection K on X, every principal connection B on the principal bundle P - X gives rise to the connection i'B : C - C+ (2.8.23) such

that

I'ao13=SoJ'B. It reads re%J, = 2 [O Ba + BaBN -

C,r(aaB.+aµBa)]-KA".(aa-Bp)

(4.6.30)

CHAPTER 4. HAMILTONIAN FORMALISM

290

Given the connection (4.6.30), the corresponding Hamiltonian form (4.6.13) is

HB=p*'"daµAwa-p;"Ie w-xrtiw,

e ac

xrM = 4

(4.6.31)

9

It is associated with the Lagrangian density LYM. Note that, in contrast with the Lagrangian density LYM, the Hamiltonian forms (4.6.31) fail to be gauge-invariant, whereas the constrained Hamiltonian form (4.5.28) 1

HQ = %HB = pr'"(daN A wa + 2c!a,,a9w) - 7.1rMw

(4.6.32)

is so. Here, by gauge transformations are meant automorphisms of the Legendre

bundle ll -. C over C which are induced by gauge automorphisms of the fibre bundle C -+ X. The corresponding principal vector fields on II X read

fu = (8"e' + ,

a9,CP)8; - cwCPpr'`8.'%"

(see the expression (4.8.2)). Given the Hamiltonian form HB, the corresponding Hamilton equations for sections r of the Legendre fibred manifold II -+ X consist of the equations (4.6.28b) and the equations

8.%rµ + 8ra = 2rBr ,), a B.Prga") + 8ar,v =

(4.6.33) (4.6.34)

The Hamilton equations (4.6.33) and (4.6.28b) are similar to the equations (4.6.17) and (4.6.18), respectively. The Hamilton equations (4.6.28b) and (4.6.34) restricted to the constraint space (4.6.28a) are precisely the constrained Hamilton equations (4.5.29) for the constrained Hamiltonian form Hq (4.6.32), and they are equivalent to the Yang-Mills equations for a gauge potential

A=iriicor. Different Iamiltonian forms HB lead to different equations (4.6.33). The equation (4.6.33) is independent of canonical momenta, and is precisely the gauge-type condition (4.6.17):

l'BoA=SoJ'A.

4.6. QUADRATIC DEGENERATE SYSTEMS

291

A glance at this condition shows that, given a solution A of the Yang-Mills equations, there always exists a Hamiltonian form HB (e.g., HB.A) which obeys the condition (4.5.23), i.e.,

HB0LY,yoJ'A=J'A. It follows that the Hamiltonian forms HB (4.6.31) parameterized by principal connections B constitute a complete family. It should be emphasized that the gauge-type condition (4.6.33) differs from the familiar gauge conditions in gauge theory which single out a representative of each gauge coset (with the accuracy to Gribov's ambiguity). Namely, if a gauge potential A is a solution of the Yang-Mills equations, there exists a gauge conjugate potential A' which is also a solution of the same Yang Mills equations and satisfies a given gauge condition. At the same time, not every solution of the Yang-Mills equations is a solution of the system of the Yang-Mills equations and a certain gauge condition. In other words, there are solutions of Yang Mills equations which are not singled out by the gauge conditions known in gauge theory. In this sense, this set of gauge conditions is not complete. In gauge theory, this lack is not essential since one can think of all gauge conjugate potentials as being physically equivalent, but not in the case of other constraint field theories, e.g., that of Proca fields (see Example 4.6.5). Within the framework of the polysymplectic Hamiltonian description of quadratic Lagrangian systems, there is a complete set of gauge-type conditions in the sense

that, for any solution of the Euler-Lagrange equations, there exists a system of Hamilton equations equivalent to these Euler-Lagrange equations and a supplementary gauge-type condition which this solution satisfies. In gauge theory where gauge conjugate solutions are treated physically equivalent, one may replace the equation (4.6.33) by a condition on the quantity

(S o J' A)aµ =

(8AAM + 8,,A;, -

2 Yang-Mills equations. In particular, which supplements the

9"µ(S o J`A)a,. = a'(x)

recovers the familiar generalized Lorentz gauge condition.

Example 4.6.3 shows that the main ingredients in gauge theory are not directly related with the gauge invariance property, but are common for field models with

CHAPTER 4. HAMILTONIAN FORMALISM

292

degenerate quadratic Lagrangian densities. In order to illustrate this fact clearly, let us compare the gauge-invariant model of electromagnetic fields with that of Proca fields.

Example 4.6.4. Electromagnetic fields. For the sake of simplicity, let X be the flat Minkowski space with the Minkowski metric

,7 = diag(1, -1, -1, -1). In gauge theory, electromagnetic potentials are identified with principal connections on a principal bundle P -, X with the structure group U(1). In this case, the gauge algebra bundle (2.7.3) is isomorphic to the trivial linear bundle

VIP=X4xR. The corresponding bundle of principal connections C (2.7.5) with coordinates (x', a,,) is an affine bundle modelled over the cotangent bundle The finite-dimensional configuration space of electromagnetic potentials is the jet bundle J'C C modelled over the pull-back tensor bundle

Z`=®TOXxC-.C. x The canonical splitting (4.6.22) of J1C is

PC = C+ ®(X T'X

X

C),

(4.6.35)

where C+ -, C is an affine bundle modelled over the pull-back symmetric tensor bundle

Z°+ = VT'X x C. X

Relative to the adapted coordinates (x), aµ, aa,,) on J'C, the splitting (4.6.35) reads 1

acv. =

2

(S4 +

a()µ) + a1)µ1

For any section A of C -* X, we find that

is the familiar strength of an electromagnetic field.

4.6. QUADRATIC DEGENERATE SYSTEMS

293

On the configuration space (4.6.35), the conventional Lagrangian density of electromagnetic fields is written as

16rrV

LE

(4.6.36)

The finite-dimensional phase space of electromagnetic theory is the Legendre bundle

n= (AT-X (&TX ®TX) x C, x equipped with the canonical coordinates (x-, aµ, a"). With respect to these coordinates, the Legendre morphism associated with the Lagrangian density (4.6.36) reads

pi`i o LE = 0,

(4.6.37a)

Pt

(4.6.37b)

l

0 LE =

In accordance with the Example, the Hamiltonian forms H9 = P"da" A wa - p',pra,,w - 7{Ew,

(4.6.38)

rBa" = 2(e"Ba +OOB"), -7rJ?,,T?J%P01a1 P11101,

parameterized by electromagnetic potentials B, are associated with the Lagrangian density (4.6.36), and constitute a complete family. Given the Hamiltonian form HB (4.6.38), the corresponding Hamilton equations consist of the equations (4.6.37b) and the equations

aar" + a"ra = O.A. + a"Ba, Sara" = 0.

(4.6.39) (4.6.40)

On the constraint space (4.6.37a), the equations (4.6.37b) and (4.6.40) reduce to the Maxwell equations in the absence of matter sources. At the same time, the equation (4.6.39), independent of canonical momenta, plays the role of a gauge-type condition discussed in Example 4.6.3. .

Example 4.6.5. Proca fields. The model of massive vector Proca fields (see Example 3.7.10) is a degenerate field theory which is similar to the electromagnetic one, but without the gauge invariance property.

CHAPTER 4. HAMILTONIAN FORMALISM

294

Recall that Proca fields are represented by sections of the cotangent bundle T'X. The finite-dimensional configuration space of Proca fields is the jet bundle J'T'X - T'X with coordinates (x", k,,, k.%), modelled over the pull-back tensor bundle

®T'X x TX - TX.

(4.6.41)

On the Minkowski space X, the Lagrangian density (3.7.30) of Proca fields looks like the electromagnetic one (4.6.36) minus the mass term, i.e.,

Lp = Lr - Im2rp k kaw.

(4.6.42)

It is almost regular. The finite-dimensional phase space of Proca fields is the Legendre bundle

f= AT*X®TX®TX x T'X, x

equipped with the holonomic coordinates (x", kp, p"P). With respect to these coordinates, the Legendre morphism associated with the Lagrangian density (4.6.42) takes the form p(A) o LP = 0,

0,01o LP = _

(4.6.43a) 1

V7

pfup,

(4.6.43b)

We have

KerLP=VT'XxT'X x

and

Q = AT'X ®(A TX) x T'X, P(4) = 0.

Following the general procedure describing quadratic degenerate systems, let consider the map a (4.6.5): k,,,, o a = -2rrqu,q,,ppl" 1 ,

where k.,,, are the fibred coordinates on the fibre bundle (4.6.41). Since

Ima = AT'X x T'X x

us

295

4.6. QUADRATIC DEGENERATE SYSTEMS and

Kera = AT'X ® (V TX) xT'X,

x one can perform the corresponding splitting (4.6.9) of the configuration space

J'T'X = VT'X ® XT'X, ka" = 2 (Sa" +

k(.%,.) + klapl,

and the splitting (4.6.10) of the phase space

n = [AT'X ®(V TX)] ®Q, T*X

p1" = p(Av) + P1>01.

Let us consider connections on the cotangent bundle T'X taking their values into Ker Lp. Bearing in mind that K = 0 on the Minkowski space X, we can write every such connection as

r = dxa ® (3 + where 0 _ ta,.d? ®8p is a symmetric soldering form on T'X. By analogy with the case of electromagnetic fields, it suffices to take the connections rB = dxa ®(aa + (epBa + OxBp)&], 2 - . X. Then it is readily observed that the Hamiltonian where B is a section of T'X

forms

IIB = pa"dk,, AWa - pA"rBw

- ?{pw,

iip = i{E + 1 Met p`k"kv, are associated with the Lagrangian density Lp (4.6.36) and constitute a complete family.

Given the Hamiltonian form HB, the corresponding Hamilton equations for sections r of the fibre bundle II -+ X consist of the equations (4.6.43b) and the equations aarp + L9, r.\ =

Sara" = - I m2W-r,,.

a"Ba,

(4.6.44) (4.6.45)

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296

On the constraint space (4.6.43a), the equations (4.6.43b) and (4.6.45) are precisely

the constrained Hamilton equations (4.5.29), and they are supplemented by the gauge-type condition (4.6.44). At the same time, one may replace (4.6.44) with a certain condition on the quantity aar,, +

e.g., with the generalized Lorentz gauge condition

a(x). In contrast with the case of electromagnetic fields, no such condition, however, is compatible with all physically non-equivalent solutions of the Euler-Lagrange equations for Proca fields. For instance, the Lorentz gauge condition 7"Aa,,ra = 0

is compatible with the wave solutions.

4.7 Affine degenerate systems Let us turn now to an affine Lagrangian density which has the coordinate expression

L=Lw,

L=b;ya+c,

(4.7.1)

where b and c are local functions on Y. The associated Legendre map takes the form

p, oL=b;.

(4.7.2)

We have the commutative diagram

J1Y rQCII Y

b=b,wA®dy$,

where Q = b(Y) is the image of the section b of the Legendre bundle 17 -, Y. Clearly, the Lagrangian density (4.7.1) is almost regular.

4.7. AFFINE DEGENERATE SYSTEMS

297

Let I' be an arbitrary connection on the fibred manifold Y -' X and f the associated Hamiltonian map (4.3.9). This Hamiltonian map satisfies the condition (4.5.2), where L is the Legendre morphism (4.7.2). Let us consider the Hamiltonian form (4.5.4) corresponding to f. It reads

H = Hr +LoI'=p;dy'Awa-(p; -b;)I"aw+ao,

(4.7.3)

and is associated with the affine Lagrangian density (4.7.1). This Hamiltonian form is affine in the canonical momenta. The corresponding Hamiltonian map is

yao%=I"A.

(4.7.4)

It follows that the Hamilton equations (4.4.19a) for the Hamiltonian form H reduce to the gauge-type condition Bar' = I'a,

whose solutions are integral sections of the connection r.

Conversely, for each section s of the fibred manifold Y - X, there exists a connection r on Y whose integral section is s. Then, the corresponding Hamiltonian map (4.7.4) obeys the condition (4.5.23). It follows that the Hamiltonian forms (4.7.3) parameterized by connections r on the fibred manifold Y -, X constitute a complete family.

Example 4.7.1. Metric-affine gravity. The metric-affine gravitation theory with the Hilbert-Einstein Lagrangian density exemplifies an affine degenerate model. We will follow the notation of Section 3.7. The total configuration space of metric-affine gravity is

'EPR X 1CK,

(4.7.5)

where Emit is the bundle of pseudo-Riemannian metrics (3.7.18) and CK is the bundle of world connections (3.7.39). This configuration space is equipped with coordinates QQ Q QQ Q (x:a kvQ+Da kaµQ)

Let us recall that a°Q o g = g°Q are the metric functions of g for any section g of EPR - X, while k,,°Q o K = KN°# are the components of a world connection K for any section K of CK --4 X.

CHAPTER 4. HAMILTONIAN FORMALISM

298

Accordingly, the total phase space of metric-affine gravity is the product

11=IIEx11c x of the Legendre bundles IIE - EPR and l1c - CK. It is equipped with the adapted coordinates (aa , o ap , ka

d,, p) a

p+ app

On the configuration space (4.7.5), the Hilbert-Einstein Lagrangian density of General Relativity reads LHe =

2K01*Raa°11

(4.7.6)

v 1w,

R,,,,°,@ = ka°p - kA,,°p

+

It is affine in the coordinates ka,°p, and is independent of the coordinates o.%0 The corresponding Legendre morphism is given by the expressions

o LHE

2K

(? - boo)

I0

.

(4.7.7)

,

which define the Lagrangian constraint space. X, one Using the whole set of connections on the fibre bundle EpR X CK can construct the complete family of Hamiltonian forms (4.7.3) associated with the Lagrangian density (4.7.6). However, it suffices to consider its following subset. Let K be a world connection and 2lka°CkVefp - k°eka`p +

k,,`SKi e +

e-

8,X% 'p +

K(a`v)(ke°p - Ke°p)

the corresponding connection (4.6.30) on the bundle CK. Let K' be a symmetric world connection which induces the corresponding connection on the bundle EPR of pseudo-Riemannian metrics. On the fibre bundle EPR X CK -, X, we then have the connection

l9 a =

.\ Kaa eep

x'p +Kaeaac,

I',°p = 1'Kw°0 + RK1rap,

4.7. AP PINE DEGENERATE SYSTEMS

299

where RK is the curvature of the world connection K. The corresponding Hamiltonian form (4.7.3) is

p''.PdkY p) A WA - HHEW, KAa`o°`) + p1oprKAv°p + 7111E = ppa(Ka so`p + HIIE =

(4.7.8)

It is associated with the Lagrangian density LHE. The corresponding Hamilton equations for a world metric g and a world connection k read 03Ag°p = Ka s9`p + Kasag°`,

(4.7.9a)

UAk,, p = I'KAy°p + RKAv°p,

(4.7.9b)

aAPap = -P 8Ko`° - p'Ko`e + K (RK°s - 29pRK)

(4.7.9c)

19 I,

(4.7.9d)

5AP'°p = ph°I°`kp. - pl'"1 spkr`° + p('IV )°l Kp`

- P("Y).OK,`°

The I Iarnilton equations (4.7.9a) and (4.7.9b) are independent of momenta and, consequently, reduce to the gauge-type condition (4.7.4). In accordance with the canonical splitting (4.6.22) of J1CK, the gauge-type condition (4.7.9b) breaks in two

parts (4.7.10)

RKAv°p,

8,.(Ka°p - kA°$) + av(KA°p

- kA°p) -

KK°p)

kA`$K + k, °$KA°e -

-

(4.7.11)

0.

It is readily observed that, for a given world metric g and a world connection k, there always exist world connections K' and K such that the gauge-type conditions (4.7.9a), (4.7.10) and (4.7.11) hold (e.g., K' is the Levi-Civita connection of g and K = k). It follows that the Hamiltonian forms (4.7.8) constitute a complete family. Being restricted to the constraint space (4.7.7), the Hamilton equations (4.7.9c) and (4.7.9d) take the form (4.7.12)

(11K°0

- 29agRK) 19 I = 0, 1)°(v'--g9°p) - 66D.,(v/91) 9Ap(k°vA - kA"°)

-

-

9°p(ka"°

- / A) -

6°9Ap(kAu, - kyPA)l - 0,

(4.7.13)

CHAPTER 4. HAMILTONIAN FORMALISM

300

where Dag°g = 8agwe

-

kag,ag°'.

Substituting the equation (4.7.10) in the equation (4.7.12), we obtain the Einstein equations

R,, - g0pR = 0. The equation (4.7.13) is the equation for torsion and non-metricity terms of the connection

In the absence of matter sources, it admits the well-known solution

D.e = V°gg', where V. is an arbitrary covector field corresponding to the projective freedom [70, 158).

Example 4.7.2. Dirac fermion fields. The Lagrangian density (3.8.22) of Dirac fermion fields in the presence of a background tetrad field h and a background spin connection A is affine in the velocity coordinates y.. Let us find a complete family of Hamiltonian forms associated with this Lagrangian density. Let S" be an h-associated spinor bundle (3.8.18). The Legendre bundle over S" is the pull-back

n,=AT'X®TX®S"', sib

where by S" -+ X is meant the dual of S" with the canonical coordinates (e, yA, pA).

X. This Legendre bundle is provided

For the sake of simplicity, let us consider Dirac fermion fields on a flat Minkowski space when hN = dµ. The Legendre map associated with the Dirac Lagrangian density (3.8.22) is

P=A=

(4.7.14)

YµA

+A =, A These relations define the Lagrangian constraint subspace of the Legendre bundle r1,.

4.8. IIAMILTONIAN CONSERVATION LAWS

301

Given a background spin connection A, any connection on the bundle S" is represented by the sum A + S, where

S=SAdx"®8A is a soldering form on S" X (which is not necessarily linear). Then the Hamiltonian forms (4.7.3) associated with the Lagrangian density (3.8.22) read HS

(1µ/

dyA

+

Ns =

A

7{sw,

(4.7.15)

y8Av BAF} + myA

where A, 8 = I A°r"LOhAB.

The corresponding Hamilton equations consist of the equations 0011A+ = 1,+Aµ BA + S+,,A,

(4.7.16a)

(9,1 PA = -PBA,,BA - (Pa - 7rg)8ASN - my8(7e)BA i a 2S C(7 I )CA,

(4.7.16b)

and the equations for the components yA and p+A. The equation (4.7.16a) and the conjugate equation for yA imply that a solution r is an integral section for the connection A + S on the spinor bundle S". It follows that the Hamiltonian forms (4.7.15) constitute a complete family. On the constraint space (4.7.14), the equation (4.7.16b) reads 81,

= -1r A,%BA - mye(7°)BA - ZS,c(7s?'")CA

(4.7.17)

Substituting (4.7.16a) in (4.7.17), we obtain the familiar Dirac equation.

4.8

Hamiltonian conservation laws

To obtain the conservation laws within the framework of Hamiltonian formalism, it is convenient to go back to Remark 4.4.8.

CHAPTER 4. HAMILTONIAN FORMALISM

302

Given a Hamiltonian form H (4.3.3) on the fibred Legendre manifold 11 -, X, let us consider the Lagrangian density

L,, = (

(4.8.1)

VA, - 7i)w

on the jet manifold J'11. We have mentioned that the Poincar6-Cartan form HL of the Lagrangian density (4.8.1) coincides with the Hamiltonian form H, while the Euler-Lagrange operator for Lv is precisely the Hamilton operator ER for H. Then we will follow the standard procedure describing differential conservation laws

in Lagrangian formalism (see Section 3.5), and apply the first variations) formula (3.2.11) to the Lagranglan density (4.8.1) [165). In accordance with the canonical lift (3.7.7), every projectable vector field

u = u"8 + u18t on the fibred manifold Y - X gives rise to the vector field I u = u"8,. + uu8c + (-8iuf 4 - 8µuµpi + 8ruAp(`)8ia on the Legendre bundle 11

(4.8.2)

Y. Then we have

I L 11 = Li, LN = (-uc8tf - 8,,(uµii) - u,a8a7{ + pi 8au`)w.

(4.8.3)

Z It follows that the Hamiltonian form H and the Lagrangian density LX have the same gauge symmetries. 1

Remark 4.8.1. Given the splitting

W=pill-iir (4.3.6) of a Hamiltonian form H, the Lie derivative (4.8.3) takes the form p; ([8a + r Oi, u)j - [8a + I'ca8i, uJ°T;,)w (8,,um?ir + uJd7?r)w)

where [., .) is the Lie bracket of vector fields.

-

(4.8.4)

(4.8.5)

4.8. HAMILTONIAN CONSERVATION LAWS

303

In the case of the vector field is (4.8.2) and the Lagrangian density LH (4.8.1), the first variational formula (3.2.11) takes the form

8,,(uµ?{) - u; 8afi + p; 8au' = -(u` - y,,uµ)(pa; + 8,7i) + (-8cu'4 - 8,uµp; + 8,,uAp - k;u')(ya - 8'a7i) da[p; (8,,?lu" - u') - uA(p; 8µ7i - N)]. On the shell (4.4.10a) - (4.4.10b), this identity reads

-U1(9,W - 8µ(uµ7{) - 4a.*%' i + p, 8au'

-

(4.8.6)

da[p; (8µ7{uµ - u`) - uA(p, 8µ7i - 7i)].

If L,,,;,/.ll = 0, we obtain the weak conservation law O;zz -da[p, (uµ8'µ7{ - u') - u"(p;`8,,7i - 7i)]w

(4.8.7)

of the current 1. = pi (uµ8µ7{ - u') - u''(p; 8µ7{

(4.8.8)

On solutions r of the Hamilton equations (4.4.19a) - (4.4.19b), the weak equality (4.8.7) leads to the weak differential conservation law 0

x

There is the following relation between differential conservation laws in Lagrangian and Hamiltonian formalisms.

PROPOSITION 4.8.1. Let a Hamiltonian form H be associated with a semiregular Lagrangian density L. Let r be a solution of the Hamilton equations (4.4.19a) (4.4.19b) for ii which lives in the Lagrangian constraint space Q. Let s = any or be the corresponding solution of the second order Euler-Lagrange equations for L so that the relation (4.5.23) holds, that is,

%//oL0J's=J's (see Propositions 4.5.10 and 4.5.11). Then, for any projectable vector field u on the fibre bundle Y -+ X, we have

t(r) = T (Y or),

` (L o Js) = T(s),

(4.8.9)

CHAPTER 4. HAMILTONIAN FORMALISM

304

where T is the current (3.5.16) on J'Y and T is the current (4.8.8) on It 0 Proof. The proof follows from the relations (4.5.10), (4.5.11) and (4.5.18). QED

In particular, let u = u'8; be a vertical vector field on Y - X. Then the Lie derivative L ,-,H (4.8.4) takes the form

L;,H = (p [8 + I"a8;,u]' - uJd?ir)w. The corresponding current (4.8.8) reads

to = -u'p;Let r = r-A% be a vector field on X and TA

R% + 17A)

its horizontal lift onto Y by means of a connection r on Y -' X. In this case, the weak identity (4.8.6) takes the form - (8,. + rµ8, - pi 8t I", 0 )fir + pA R'%µ 2e -da7lrA,,,

?(4.8.10)

where the current (4.8.8) reads 71ra = .r'1'=ra" = 7-P (p;8µ?ir

- bµ(p;

Zr)).

The relations (4.8.9) show that, on the Lagrangian constraint space Q, the current (4.8.10) can be treated as the Hamiltonian SEM current relative to the connection

r. In particular, let us consider the weak identity (4.8.6) when the vector field u on lI is the horizontal lift of a vector field r on X by means of a Hamiltonian connection

on II - X which is associated with the Hamiltonian form H. We have u = r'(e,, + 8;,xa; + ;8a). In this case, the corresponding SEM current reads ' _ -ra(p; 8',,?{ - 71),

(4.8.11)

and the weak identity (4.8.6) takes the form -8,,N + d,% (p-,% 8'µ?{)

(4.8.12)

80 W, 8'a?{ - 7{).

A glance at the expression (4.8.12) shows that the SEM current (4.8.11) is not conserved, but we can write the weak identity -8,,,% + d

%

8,aN -

- N)]

0.

This is exactly the Hamiltonian form of the canonical energy-momentum conservation law (3.5.29) in Lagrangian formalism.

4.9. VERTICAL EXTENSION OF POLYSYMPLECTIC FORMALISM

305

4.9 Vertical extension of polysymplectic formalism By analogy with the BRS generalization of mechanics [80, 81], the vertical extension of polysymplectic Hamiltonian formalism, developed in this Section, is a preliminary

step toward its BRS quantization.

Given a bundle Y -. X, let us consider its vertical tangent bundle VY with coordinates for sections of Y

We will show that polysymplectic Hamiltonian formalism X is naturally extended to Hamiltonian formalism for sections

ofVY - X. The Legendre bundle (4.0.1) over VY -e X is

Ilvy = V'VY ®(nYT'X) with coordinates (za, y, y', q; , v;).

Remark 4.9.1. Let us consider the fibre bundles TT'X and T'TX. Given bolonomic coordinates (z`,pa) on T'X and (?,v) on TX, these bundles are provided with the coordinates (za, p,,, z'', Pa) and (zx, va, xa, A%), respectively. By inspection of the coordinate transformation laws, one can show that there is the isomorphism 71 "X 25 7'*f'X,

pa

+6.%,

Pa

of these bundles over TX (see also [35] and [101], p.63). Given a fibred manifold Y -. X, a similar isomorphism between the fibre bundles VV'Y and V'VY over VY takes place. In the holonomic coordinates (z,`,y',pp) on V'Y and (zA, y', v{) on VY, this isomorphism reads

VV'Y

V'VY,

pi .vi, A -0j.

(4.9.1)

PROPOSITION 4.9.1. In virtue of the bundle isomorphism (4.9.1), there exists the bundle isomorphism over VY !!vy ^- VH, where (x `, y`, p; , y', P;) are the coordinates on V H. 0

CHAPTER 4. HAMILTONIAN FORMALISM

306

We will use the compact notation

ai=8yi,

a'a=

,

8v=Y

One can develop Hamiltonian formalism on the Legendre bundle [Ivy by analogy

with that on II. The vertical bundle VII is endowed with the canonical polysymplectic form I Av = [dP; Ady'+dp{ Ady'] Aw®aa, whose coordinate expression is maintained under holonomic transformations of the

composite bundle VII - 11 -. Y. PROPOSITION 4.9.2. Let 7 be a Hamiltonian connection on R associated with a Hamiltonian form H (4.3.6). Then, the vertical connection V7 (2.5.13) is a Hamiltonian connection associated with the Hamiltonian form !I v = (Pi dal' - y'dp,) A wa - hvw,

(4.9.2)

IN-V = av11 _ (U'ai + P: aa)n,

on Vfl. O Proof. Liven the Hamiltonian connection

7=dx"®(aµ+7µ8i+7µia'a), 7µ = aµfc,

Tai

it is easily seen that the vertical connection V ,y = dx" ® [eµ + 7µa; + 7µ;8'a + 8v7µ6; + 8v70'a`a]

obeys the Hamilton equations for the Hamiltonian form (4.9.2), i.e.,

7µ=atµxv=vr Tai = -arty =

7µ=aµfv=OV0, Tai =

-aixv = -avairc. QED

4.9.

VERTICAL EXTENSION OF POLYSYMPLECTIC FORMALISM

307

In particular, given the splitting

,H =Pir'a+jIr relative to a connection r on Y -. X, then we have the splitting

NV=P;r'a-1!'(-P;8,F)+8vjf with respect to the lift f (4.4.2) of I' onto II -+ X. Note that the Hamiltonian form Hv (4.9.2) can be also obtained in the following 11 (3.4.14), let us consider the vertical tangent way. Given the fibre bundle Zy bundle VZ of Z --+ X with coordinates

(xA,y',Pi,Ry',P;)P) It is provided with the canonical form 1=v=Pw+P,dy'nwa-y'dp; nwa,

whose expression is maintained under holonomic coordinate transformations. Let H = WE, where h is a section of the fibre bundle Zy -, H. Then we have

Hv = (Vh)'Ev, where V h : VII - V Z is the vertical tangent map of h. Remark 4.9.2. One can use also the form Ev+d(y'p{) Awa since the form d(y'p) A wa is well behaved.

We now turn to the vertical extension of Lagrangian formalism on J'Y to the configuration space VJ'Y = J'VY provided with coordinates

(x.,y',ya,y',lla). Given a Lagrangian density L on J1Y, let us consider the Lagrangian density

1.v=pr20VL:VJ'Y-+hT'X, Lv = 8vL = (y'8s + y{a); )G,

on VJ1Y. Then the variational derivatives

b,Lv=b;L=O

(4.9.3)

CHAPTER 4. HAMILTONIAN FORMALISM

308

recover the second order Euler-Lagrange equations (3.3.31). The Lagrangian density (4.9.3) yields the Legendre map Lv

VJ'Y vYVf,

P, =O{Zv=7f{, P: =8y7r;a.

Conversely, the Hamiltonian form Hv (4.9.2) on VII determines the Hamiltonian map

Hv=VH:VII-YVJ'Y, pa = 8.%?lv = 8;,71,

ira = 8y8'a?l.

PROPOSITION 4.9.3. Given a Lagrangian density L, let the Lagrangian constraint

space Q be a fibred submanifold Q -e Y of the Legendre bundle II

Y. If a

Hamiltonian form H is associated with the Lagrangian density L, the Hamiltonian form Hv is associated with the Lagrangian density Lv. 0

Proof. If Q is a fibred submanifold of II, the relation (4.5.9a) takes the form

Lo///oiQ=iQ. Then the corresponding vertical tangent morphism satisfies the relation

VLoVHoViQ=ViQ. The relation (4.5.9b) for Hy reduces to the relation (4.5.13) for H.

4.10

QED

Appendix. Hamiltonian time-dependent mechanics

There is an extensive literature on autonomous Hamiltonian mechanics phrased in terms of symplectic geometry (2, 6, 85, 1201. Its standard example is a mechanical of a manifold M. system whose phase space is the cotangent bundle The usual formulation of time-dependent Hamiltonian mechanics just as the Lagrangian one requires a given splitting Y = R x M of the event manifold Y and the corresponding splitting R x T'M of the phase space V *Y. These splittings, however,

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

309

are broken by any time-dependent canonical transformation and any reference frame transformation, including transformations of inertial frames.

Here we continue to describe time-dependent mechanics as a particular field theory, when the event space Y is a fibred manifold over R [18, 32, 164]. Then the 1-dimensional reduction of polysymplectic Hamiltonian formalism provides the adequate mathematical formulation of time-dependent Hamiltonian mechanics on the Legendre bundle 11 = V'Y -4 Y.

(4.10.1)

The main ingredients in this formulation are: (i) the canonical 3-form (4.10.13) which provides the phase space V'Y with the canonical Poisson structure, (ii) connections on Y -' R which define reference frames (see Section 3.9), and (iii) Hamiltonian connections whose integral sections are solutions of the Hamilton equations. Let us emphasize the following essential peculiarities of time-dependent Hamiltonian mechanics in comparison with the symplectic one. The canonical Poisson structure on a phase space of time-dependent mechanics is degenerate.

A 1-larniltonian is not a function on a phase space. As a consequence, the evolution equation is not reduced to a Poisson bracket, and integrals of motion cannot be defined as functions in involution with a Hamiltonian.

l-lamiltonian and Lagrangian formulations of time-dependent mechanics are equivalent only in the case of hyperregular Lagrangians. A degenerate Lagrangian admits a set of associated Hamiltonians none of which describes the whole mechanical system given by this Lagrangian.

We will follow the notation of Section 3.9. Here, we assume that Y - R is a fibre bundle. Given a trivialization

Y = R x M,

(4.10.2)

we have the corresponding splittings of the configuration and phase spaces

J'Y = R x TM,

(4.10.3)

11=RxT'M.

(4.10.4)

CHAPTER 4. HAMILTONIAN FORMALISM

310

Recall that 8j and dt are the standard vector field and the standard 1-form on R, respectively.

Remark 4.10.1. Throughout this Section, the fibration Y -, R is once for all. This, however, is not the case of relativistic mechanics whose description requires formalism of jets of submanifolds (am Example 5.3.5).

A. Canonical Poisson structure A Poisson structure is an important ingredient in many constructions of classical and quantum mechanics. The Legendre bundle V'Y of time-dependent mechanics is provided with the canonical Poisson structure as follows. Let (t, y') be coordinates on Y -+ R. Then the Legendre bundle V'Y is equipped with the holonomic coordinates (t,y',A), while the first order jet manifold J'V'Y of V'Y - R is coordinatized by (t, y', pi, yie, pa). Let us consider the homogeneous fibre bundle Zy = T'Y (4.2.8) with coordinates (t, y', pi, p). It possesses the canonical form (4.2.9), which is the Liouville form

_=pdt +pdy+,

(4.10.5)

and the canonical form (4.2.10), which is the symplectic form

i2z=dpAdt+dpmAdy'.

(4.10.6)

The corresponding Poisson bracket on the space COD(T'Y) of functions on T'Y reads

{f, g) = & f0c9 - e°9&f + 8' f 8,g - 8'gOi f .

(4.10.7)

Let us consider the subspace of C°°(T'Y) which comprises the pull-backs of functions on V'Y by the projection T'Y - V'Y. It is easily seen that this subspace is closed under the Poisson bracket (4.10.7). By virtue of Proposition 4.1.2, there exists the canonical Poisson structure

I {f,9}v = 8'f8i9 - 8'98if

(4.10.8)

on V'Y induced by (4.10.7). The corresponding Poisson bivector on V'Y is vertical with respect to the fibration V'Y -+ R, and reads

w"=0,

wij=0,

w'1=1.

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311

Since the rank of w is constant, the Poisson structure (4.10.8) is regular. It is obviously degenerate. Given the Poisson bracket (4.10.8), the Hamiltonian vector field t9 j of a function

f on V'Y is defined by the relation (4.1.5), i.e.,

{f,9}v = t9jjdg,

9 E C°°(V'Y).

It is the vertical vector field

1+9,=8'f0i-eif8`

(4.10.9)

on V'Y - R. Thus, the characteristic distribution of the Poisson structure (4.10.8) is precisely the vertical tangent bundle VV'Y of V'Y - R. In accordance with Theorem 4.1.7, this Poisson structure defines the symplectic

foliation on V'Y which coincides with the fibration V'Y - R. Furthermore, a on glance at the bracket (4.10.8) shows that the holonomic coordinates V'Y are exactly the canonical coordinates (4.1.8) for the Poisson structure (4.10.8). The symplectic forms on the fibres of V'Y -, R are the pull-backs

0c=dpiAdy' of the canonical symplectic form on the typical fibre TM of V'Y - R with respect to trivialization morphisms [30).

B. Canonical polysymplectic structure The I'oisson structure (4.10.8) can be introduced in a different way. The Legendre bundle V'Y (4.10.1) admits the canonical polysymplectic form (4.2.5) which reads

SZ=dpiAdy'Adt08,.

(4.10.10)

Following general polysymplectic formalism, we say that a connection

-y=dt®(A+ry'i%+7;8') on the Lcgendre bundle V'Y R is Hamiltonian if the exterior form ryJl is closed. A connection y is Hamiltonian if and only if ry obeys the conditions (4.4.4) - (4.4.6) which now take the form

8'- - Y,1. = 0, A7j - 8j'Yi = 0, 83ry`+0'1j=0.

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As in Example 4.4.1, we observe that every connection 1' = dt 0 (8t + I18i) on the fibre bundle Y -. R gives rise to the Hamiltonian connection which coincides with the covertical connection

t''=V'1'=dt®(8t+1"8i-8fl''ppOi)

(4.10.11)

(2.5.1 4) on V'Y. We then have

V'1'Jf2 = dHr, Hr = p;dy' - pit" dt.

(4.10.12)

Th e polysymplectic form (4.10.10) defines the canonical closed 3-form

A=Ajdt, A=dp, Ady'Adt,

(4.10.13)

on the Legendre bundle V'Y. The canonical forms 11 (4.10.10) and A (4.10.13) on V'Y can be seen on the same footing as follows.

PROPOSITION 4.10.1. Let u be a vector field on V'Y - R projected onto the standard vector field 8t on R. This vector field obeys the relation

d(uJA) = 0

(4.10.14)

if and only if u is the horizontal lift

Try=Bt+ry'8;+ryi8'

(4.10.15)

of the standard vector field 8t on R by means of a Hamiltonian connection ry on

V'Y -yR. o Proof. It is readily observed that I 7J0 = r,JA.

QED Every connection 'y on the fibre bundle V'Y - R is a curvature-free connection (see Remark 3.9.3). By virtue of Proposition 2.3.1, such a connection defines a

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

313

horizontal foliation on V'Y -. R. Its leaves are the integral curves of the horizontal lift (4.10.15) of 88 by y. DEFINITION 4.10.2. The vector field r., (4.10.15) which obeys the condition (4.10.14)

is said to be a locally Hamiltonian horizontal vector field

Given the canonical form A (4.10.13), every function f on V'Y defines the corresponding Hamiltonian vector field t9f (4.10.9) by the relation

It9fJA=dfndt. Then the Poisson bracket (4.10.8) is recovered by the condition

{f, g}vdt =

9Jt9i1A.

DEFINITION 4.10.3. Vertical vector fields t9 on V'Y -+ R which satisfy the condition (4.10.17) are called locally Hamiltonian vector fields.

Locally Hamiltonian horizontal vector fields r., constitute an of lne space modelled over the linear space of locally Hamiltonian vector fields t9. Using the decomposition

ry=yr+t9®dt,

(4.10.16)

where I' is a connection on Y - R, one can show that every closed form r.,JA on V'Y -. R is exact. Indeed, let us consider a vertical vector field 19 on V'Y - R such that the form t9JA is closed, i.e., d(t91 A) = 0.

(4.10.17)

It is easily seen that t9J A takes the form or A dt, where or is a 1-form. Every closed 2-form a n dt on V'Y is exact. It is an immediate consequence of the isomorphism (3.8.1) of the De Rharn cohomology groups. In accordance with the relative Poinear6 lemma (see Remark 4.4.2), the condition (4.10.17) implies that locally

t9JA=dfAdt.

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314

C. Hainiltonian forms DEFINITION 4.10.4. A 1-form H on the Legendre bundle V'Y is called a locally Hamiltonian form if

r,JA = dH for a connection -y on V'Y - R. In particular, Hr (4.10.12) is a Hamiltonian form. There is obviously one-to-one correspondence between the Hamiltonian connections and the locally Hamiltonian forms considered throughout modulo dosed forms. PROPOSITION 4.10.5. difference

Given the locally Hamiltonian forms H., and H.,., their

a=H,,-H,., (r,-r,.)JA=da, is a 1-form on V'Y such that the 2-form a A dt is closed since d(a A dt) = da A dt = 0

and, consequently, exact. In accordance with the relative Poincare lemma, this condition implies that

a = fdt + dg,

(4.10.18)

where f and g are local functions on V'Y. DEFINITION 4.10.6. Following Definition 4.3.1, by a Hamiltonian form H on the Legendre bundle V'Y is called the pull-back !I = h* S" = p;dy` -1idt,

(4.10.19)

of the Liouville form ° (4.10.5) on T'Y by a section h of the fibre bundle T'Y V'Y. Remark 4.10.2. Given trivializations (4.10.2) of Y R and (4.10.4) of V'Y - R, the Hamiltonian form (4.10.19) is the well-known Poincar6-Cartan integral invariant

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

315

of time-dependent mechanics [6]. However, if a trivialization of Y R is not fixed, the Hamiltonian g{ in the expression (4.10.19) is not a function (see (4.10.20)).

As in the polysymplectic case, any connection r on Y R defines the Hamiltonian form Hr (4.10.12) on V'Y, and every Hamiltonian form on V'Y admits the splitting

H = pdy' - Ndt = p;dy' - (pj * +?Lr)dt,

(4.10.20)

where I' is a connection on Y R and fr is a function on V'Y. Hamiltonian forms on V'Y constitute an affine space modelled over the linear space of functions on V'Y (see Proposition 4.3.2). Then it follows from the splitting (4.10.16) and Proposition 4.10.5 that every locally Hamiltonian form H., is a Hamiltonian form locally in the sense that, in a neighbourhood of every point q E V'Y, the form H.r coincides with the pull-back of the Liouville form E on V'Y by the local section

(t, y', R) '-' (t, y`, pi, P = -Pi" + I) of T'Y

V'Y, where f is a local function on V'Y (see (4.10.18) and Proposition 4.4.2). In particular, every locally Hamiltonian form admits the splitting (4.10.20) locally.

The converse assertion is the following.

PROPOSITION 4.10.7. For any Hamiltonian form H on the Legendre bundle V'Y, there exists a unique Hamiltonian connection -fm on V'Y - R such that Try,, JA=dH.

0 Proof. The Hamilton operator ,6.q (4.4.8) on the Legendre bundle V'Y - R reads

Cy : i'v'y - XT'V'Y, E.t = dH - !Z = [(yt'

- 8'7- )dp: - (pa + 8,f)dy'] A dt,

where 52 = dpi A dy' + pttdy' A dt - y'dpti A dt

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316

is the pull-back of the canonical form 0 (4.10.10) onto J'V'Y (see Definition 4.3.4). The kernel of this Hamilton operator is the system of first order differential Hamilton equations

y; = 8'?{,

-8;l

(4.10.21a) (4.10.21b)

on V'Y. It is the image of the section -y,, = dt ® (88 + 8`NO; - 0i7{8')

(4.10.22)

of the jet bundle J'V'Y -+ V'Y which is a Hamiltonian connection on V'Y - R for the Hamiltonian form H.

QED

The classical solutions of the Hamilton equations (4.10.21a) - (4.10.21b) are the integral sections of the associated Hamiltonian connection (4.10.22) or, equivalently, the integral curves of the vector field

Iru = 8t + 8`718 - 8,7{8'

(4.10.23)

which is the horizontal lift of the standard vector field 8, on R by means of the connection (4.10.22).

DEFINrrION 4.10.8. A locally Hamiltonian vector field r1H = rH associated with a Hamiltonian form H is called the Hamiltonian horizontal vector field. 0 Horizontal Hamiltonian vector fields TH (4.10.23) form an affine space modelled over the linear space of Hamiltonian vector fields (4.10.9). The horizontal Hamiltonian vector field (4.10.23) satisfies the relations

r11JH=pi8'fl-H=(H],

(4.10.24)

TuJdH = 0. With a Hamiltonian form H (4.10.20) and the corresponding horizontal Hamiltonian vector field rH (4.10.23), we have the Hamilton evolution equation dil,f = 1-HJ4f = (8, + 8'H81-

(4.10.25)

on functions on the Legendre bundle V'Y. Substituting a classical solution of the Hamilton equations (4.10.21a) - (4.10.21b) in (4.10.25), we obtain the time evolution of the function f.

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

317

Given the splitting (4.10.20) of a Hamiltonian form H, the Hamilton evolution equation (4.10.25) is brought into the form duef = 8ef + (1"8, - 8,r'p,O') f + {Zr, 7f }v.

(4.10.26)

A glance at this expression shows that the Hamilton evolution equation in time-

dependent mechanics does not reduce to the Poisson bracket. This fact may be relevant to the quantization problem. The second term in the right-hand side of the equation (4.10.26) remains classical.

Remark 4.10.3. Given the canonical Poisson structure (4.10.8) on the Legendre bundle V'Y, one can consider the generalized Poisson bracket {.,.},, (4.1.12) on the exterior algebra t)'(V'Y) or the bracket {.,.}d (4.1.14) on the quotient 1'(V'Y)/dT'(V'Y). In particular, the generalized Poisson bracket (4.1.12) of two Hamiltonian forms H and H' reads

(11, ll'}w = py(8'7{' - 8'f)dt.

D. Prosymploctic structure Besides the canonical Poisson structure, the phase space V'Y of time-dependent mechanics may be provided with presymplectic and contact structures which, however, are specified by the choice of a Hamiltonian form H. By definition, a Hamiltonian form H is the pull-back H = h'E" of the Liouville

form (4.10.5) by a section h of the fibre bundle T'Y - V'Y. Accordingly, its differential

d/1= (dp, + 8{fdt) A (dy' - 8'fdt) is the pull-back h'Sl2 of the symplectic form (4.10.6). It is a presymplectic form of constant rank 2m since the form (dll )m = (dpi A dy')m - m(dp, A dy')m-' A df A dt

(4.10.27)

is obviously nowhere vanishing. However, every locally Hamiltonian vector fields uH

on V'Y with respect to the presymplectic form dH (i.e., uHJdH must be closed) is proportional to the horizontal Hamiltonian vector field rH (4.10.23).

CHAPTER 4. HAMILTONIAN FORMALISM

318

Remark 4.10.4. Let ry be a connection on V'Y - R and r, the corresponding horizontal vector field (4.10.15). The 2-form r.,JA is presymplectic if and only if ry is a Harniltonian connection.

E. Contact structure Let us recall some basic notions.

DEFINITION 4.10.9. Given a (2m + 1)-dimensional manifold Z, a contact form on Z is defined as a 1-form 0 such that 0 A (d0)m 96 0

everywhere on Z. The pair (Z, 0) is called a contact manifold. 0

A manifold Z equipped with a contact form 0 is orientable, and 0 A (0)", is a volume element. The exterior differential dO of a contact form 0 is a presymplectic form.

The following assertion is a variant of the well-known Darboux theorem ([120), p.288).

THEOREM 4.10.10. Let (Z, 0) be a (2m + 1)-dimensional contact manifold. Every

point z of Z has an open neighbourhood U which is the domain of a coordinate chart (z°, ... , z2"`) such that the contact form 0 has the local expression m

0 = dz° -

zm+'dz`

on U. 't'hese coordinates are called Darboux's coordinates. 0 A contact form on an odd-dimensional manifold generates the Jacobi bracket as follows.

PROPOSrI'ION 4.10.11. Let 0 be a contact form on Z. There exists a unique nowhere vanishing vector field E on Z such that

EJ0 = 1,

EJdO = 0.

This is called the Reeb vector field of 0 ((120), p.291). 0

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

319

Relative to Darboux's coordinates, the Reeb vector field reads E = 80. PROPOSITION 4.10.12. Every contact form 0 on an odd-dimensional manifold Z yields the associated Jacobi structure on Z. It is defined by the Reeb vector field E of 0 and by the bivector field w such that

wjO = 0,

w1bjd0 = -(i0 - (EJO)0)

(4.10.28)

for every 0 E O1(Z) [132].

Relative to Darboux's coordinates, the Jacobi structure (4.10.28) reads

{f,9} _ (am+i9aif -++f8i9) + (9alf - fao9), i=1

where m

f = E zm+i am+if + f, i=1

in

9 = E .tm++8m+i9 + 9. i=1

Let us turn now to Hamiltonian forms on the Legendre bundle V'Y (4.10.1).

PROPOSITION 4.10.13. The Hamiltonian form (4.10.11) is a contact form if the function [7{] (4.10.24) nowhere vanishes [120).

Proof. Since the horizontal Hamiltonian vector field TH (4.10.23) is nowhere vanishing, the condition H A (dH)m 34 0 is equivalent to the condition T,,J(1l A (dH)m) =

(THJH)(dH)m = [M(dH)m & 0.

The result follows because the form (dH)m (4.10.27) is nowhere vanishing.

QED

Remark 4.10.5. To make [%] everywhere different from zero, one may add an exact form (e.g., the form cdt, c =const.) to H. For instance, the Hamiltonian form Hr (4.10.12) is not a contact form since [71] = 0, but the equivalent form Hr - dt, where [7{J = 1, is so. 0

Given a Hamiltonian form H, let the function [N] be nowhere vanishing so that H is a contact form. The corresponding Reeb vector field reads

EH = [7{]-irH.

(4.10.29)

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320

By virtue of Proposition 4.10.12, we have the associated Jacobi bracket defined by the Reeb vector field (4.10.29) and by the bivector field wy on V'Y derived from the relations

wil(0,.)] H = 0,

wH(4', )}dH = -(0 - (EH}O)H)

for any 1-form 4 on V'Y [1321. We find w,,(4,, a) _ ¢'a; - a{4,, + P:a'EHJO -

where 0, a are arbitrary 1-forms on V'Y. The corresponding Jacobi bracket on functions on V'Y reads {f,g}u = wH(q5,u) + EHj (fd9 - 9af) = {f,9}v + J?{]''(g`dHgf - fdHe9), where {f, g) v is the canonical Poisson bracket (4.10.8) and

f =740if -f,

9=P:a'9-g.

Givers a contact Hamiltonian form H, one can consider also the Jacobi bracket

{f,9}E = [NJ-'(fdH,9 - 9dH,f) defined by the Reeb vector field EH alone.

F. Canonical transformations In contrast with the (n > 1) polysymplectic case, canonical transformations in time-dependent mechanics are not compatible with the fibration V'Y - Y. DEFINITION 4.10.14. By a canonical automorphism is meant an automorphism p R which preserves the canonical Poisson structure over R of the fibre bundle V'Y

(4.10.8) on V'Y, that is,

{f op,gop}v=({f,g}v)op. O

It is easily seen that an automorphism p of V'Y R is canonical if and only if p preserves the canonical form A (4.10.13) on V'Y, that is,

A=p'A.

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

321

The bundle coordinates on V'Y - R are called canonical if they are canonical for the Poisson structure (4.10.8). Canonical coordinate transformations satisfy the relations Oil i ay, aPi OPk

_

0P', BY, aPk OPi

-- 0, 0

a a i 19y" OPi a

ask

OM W i/ aV OPk

By definition, the holonomic coordinates on V'Y are canonical coordinates. Accordingly, holonomic automorphisms Pi H P; IV Pi

(4.10.30)

of the Legendre bundle V'Y -' Y induced by the vertical automorphisms of Y -' R are also canonical. PROPOSITION 4.10.15. Canonical automorphisms send Hamiltonian connections onto Hamiltonian connections (and consequently locally Hamiltonian forms onto locally l larniltonian forms).

Proof. The proof is based on the relation TP(-rr) = r,(,7),

where y is a connection on V'Y -e R and ry is the horizontal vector field (4.10.15). If y is a Hamiltonian connection such that

rjA=dH, we have

rvc_,)1A = (P-')'(r,JA) = d((P-1)'H)

QED A glance at the relation (4.10.14) shows that each locally Hamiltonian horizontal vector field r,, is the generator of a local 1-parameter group G., of canonical automorphisms of V'Y R. This leads to the following assertion.

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322

PROPOSITION 4.10.16. Let 'y be a complete Hamiltonian connection on V'Y -+ R. There exist canonical coordinate transformations which bring all components of ry to zero, i.e., ry = dy ®81. O

Proof. Let Vo Y be the fibre of V'Y - R at the point 0 E R. Then canonical coordinates of VO Y dragged along integral curves of the complete vector field rr satisfy the statement of the proposition. FYom the physical viewpoint, these coordinates are the initial values of the canonical variables. QED In particular, let H be a Hamiltonian form (4.10.20) such that the corresponding horizontal Hamiltonian vector field Ty (4.10.23) is complete. By virtue of Proposition 4.10.16, there exist canonical coordinate transformations which bring the Hamiltonian 71 into zero. Then the corresponding Hamilton equations reduce to the equilibrium equations

yi=0,

Pa=0.

Example 4.10.6. Let us consider 1-dimensional motion with constant acceleration a with respect to the coordinates (t, y). The corresponding Hamiltonian form and the Hamiltonian connection read

- ay, ryy=p, 2

ryp=a.

(4.10.31)

This Hamiltonian connection is complete. The canonical coordinate transformation

y,=y-Pt+

ate2

,

Tf =P - at

brings Lite components of the connection (4.10.31) to zero.

Example 4.10.7. Let us consider the 1-dimensional oscillator with respect to the same coordinates. The Hamiltonian form and the Hamiltonian connection of this system read = 2 (P' + y2), 'Yy = P,

.yr = -y.

(4.10.32)

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

323

This Hamiltonian connection is complete. The canonical coordinate transformation

y'= y cost - paint,

p' = p cos t + Y sin t

brings the components of the connection (4.10.32) to zero.

Note that any Hamiltonian form H can be locally brought into the form where ?{ = 0 by local canonical coordinate transformations. It should be emphasized that canonical automorphisms do not send Hamiltonian forms onto Hamiltonian forms in general. Let !I be a Hamiltonian form (4.10.19) on V* Y. Given a canonical automorphism p, we have

d(p'11 - H) = 0. It follows that

p'!!-H=dS, where S is a local function on V'Y. We can write locally

p'H = pidpi - W o pdt. Then the corresponding coordinate relations read

8'S = PAP. - pi, 8'S = Pi8'P',

%'-x=p8p'-BSS. Taken on the graph Do = {(q, p(q)) E V'Y X V'Y)

of the canonical automorphism, the function S plays the role of a local generating function. ior instance, if the graph A. is coordinatized by (t, yi, y'i), we obtain the familiar expression

1-i' - 71 = 88S(t,yi,j) Example 4.10.8. The holonomic morphisms (4.10.30) admit locally the generating function

5(t,Y'J'pi) = 1t(t,IMP,

-

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324

Let us turn now to the Hamilton evolution equation and its splitting (4.10.26). Let the connection r in the expression (4.10.26) be a complete connection on the fibre bundle Y -. R. Then there exist holonomic coordinate transformations which bring the Hamiltonian evolution equation (4.10.26) into the familiar Poisson bracket form dfut = 811 + {11, f}v.

G. Lagrangian Poisson structure In contrast with the Legendre bundle V`Y, the configuration space J'Y of timedependent mechanics does not possess any canonical Poisson structure in general. A Poisson structure on J1 Y depends on the choice of a Lagrangian G. Let .11 Y be provided with coordinates (t, y, yi) as in Section 3.9, and let r be a Lagrangian (3.9.41). The notation 7r1 = 8,G,

GO = 8;88G

should be recalled. Let A be the canonical 3-form (4.10.13) on the Legendre bundle V-Y. Its pullback by the Legendre map L (3.9.42) reads

AL =LOA =da,Ady'Adt. By means of AL, every vertical vector field 19=1918;+918,

on J'Y . R yields the 2-form VJAL = {[e'Cj ++91(Ojir1- &irj)jdy' ->9'GfidV } A dt.

If the Lagrangian G is regular, this is one-to-one correspondence. Indeed, given any 2-form 0 = (.O1dy' +

dyy) A dt

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

325

on J'Y, the algebraic equations i3'Gji +79i(8j7ri - 8i7rj)

-t9icji = j have a unique solution 79i =

+

77' =

0i7rk)J.

In particular, every function f on J'Y determines a vertical vector field

t9 f = -(G-')li8t fai +

on J'Y

(G-'yi(ei f + (G'')k,.&n

f(8kifi

-

(4.10.33)

R in accordance with the relation

t9fj1lL=dfAdt. Then the Poisson bracket If,g)Ldt = 79g)'t9fJCIL,

(4.10.34)

f,9 e C°°(J'Y),

can be defined on functions on J'Y, and reads

If, g)L = [(G-)'j + (8n7lk (8n7rk

-

,,pp

L)TM')(

f8,9 - 8g8jf) +

f Ojg.

The vertical vector field 191 (4.10.33) is the Hamiltonian vector field of the function f with respect to the Poisson structure (4.10.34). In particular, if the Lagrangian C is hyperregular, that is, the Legendre map is a diffeomorphism, the Poisson structure (4.10.34) is obviously isomorphic to the Poisson structure (4.10.8) on the phase space V'Y. The Poisson structure (4.10.34) defines the corresponding symplectic foliation

on J'Y which coincides with the fibration J1Y - It. The symplectic form on the leaf J' Y of this foliation is f1i = dire A dyi (183). The configuration space J1Y of time-dependent mechanics can be also provided

with an L-dependent presymplectic structure. This is the exterior differential diIL = dirt A dyi - d(p;y` - C) A dt

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326

of the Poincar6-Cartan form HL (3.9.44), which is the pull-back of the canonical symplectic form flz (4.10.6) on the fibre bundle T'Y by the Legendre morphism HL (3.4.17).

H. Degenerate Lagrangian systems Following Section 4.5, we can establish the relations between Lagrangian and Hamiltonian formulations of time-dependent mechanics. From the mathematical point of view, these formulations are not equivalent when Lagrangians are degenerate. born the physical viewpoint, velocities are physical observables in classical mechanics, whereas momenta are physical observables in quantum mechanics. I The key point is that a mechanical system described by a degenerate Lagrangian appears to be a multi-Hamiltonian constrained system within the framework of Iiarniltonian formalism. I

Let Y -+ R be an event bundle, V'Y the phase space and J'Y the configuration space of time-dependent mechanics. Let us recall that a Hamiltonian form H on V'Y is said to be associated with a Lagrangian C on JIY if H obeys the conditions

L o Hfq = Idq, Q = L(J'Y), //i,- H = L o H.

(4.10.35a) (4.10.35b)

It is called weakly associated if (4.10.35b) takes place only at points of the Lagrangian constraint space Q. If a Lagrangian C is hyperregular, there exists a unique Hamiltonian form associated with C. If a Lagrangian C is semiregular (see Definition 4.5.8), all Hamiltonian forms associated with C (if they exist) coincide on the Lagrangian constraint space Q, and the Poincare-Cartan form HL is the pull-back (4.5.18) of any such a Hamiltonian form If by the Legendre map L. In accordance with Propositions 4.5.10 and 4.5.11, if H is a l larniltonian form associated with a semiregular Lagrangian C, every solution of the corresponding Hamilton equations which lives in the Lagrangian constraint space Q yields a solution of the Lagrange equations for C. At the same time, to exhaust all solutions of the Lagrange equations, one must consider a complete family (if it exists) of Hamiltonians associated with C.

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

327

Let G be an almost regular Lagrangian (see Definition 4.5.13) and HQ the constrained Hamiltonian form (4.5.28) on the Lagrangian constraint space Q, which is

an imbedded subbundle of the Legendre bundle V'Y -. Y. Then the constrained Hamilton equations (4.5.29) for HQ admit all solutions of the Cartan equations for G (see Proposition 4.5.16). In the spirit of well-known Cotay's algorithm for analyzing constrained systems in symplectic mechanics [16, 75], the Lagrangian constraint space Q plays the role of the primary constraint space. However one has to apply this algorithm to each Hamiltonian form H weakly associated with a Lagrangian C. If C is semiregular, all these Hamiltonian forms coincide with each other on Q, but not the corresponding horizontal Hamiltonian vector fields (4.10.23). The necessary condition for a local solution of the Hamilton equations for a Hamiltonian form H to live in the Lagrangian constraint space Q is that the horizontal Hanmiltoriian vector field ry (4.10.23) is tangent to Q at some point of Q. Given a Hamiltonian form H weakly associated with G, we can express this condition in the explicit form

pt = a;G(t, yi, Mn), (a, + 01%a; - a,?ia') jd(p; - O C(i, y', acre)) = 0.

(4.10.36a) (4.10.36b)

The equation (4.10.36a) is the coordinate expression of the relation (4.10.35a), and can be taken as the equation of the Lagrangian constraint space Q. The equation (4.10.36b) requires that the vector field ry is tangent to Q at a point with coordinates

(t,y',pi) In particular, one can apply the description of the quadratic degenerate systems in Section 4.6 to those in time-dependent mechanics. Note that, since Hamiltonians in time-dependent mechanics are not functions on a phase space, we cannot apply to them the well-known analysis of the normal forms [24] (e.g., quadratic Hamiltonians in symplectic mechanics [6]).

I. Conservation laws and integrals of motion In autonomous mechanics, an integral of motion, by definition, is a function on the phase space whose Poisson bracket with a Hamiltonian is equal to zero. This notion cannot be extended to time-dependent mechanics because the Hamiltonian evolution equation (4.10.26) is not reduced to the Poisson bracket. In Section 3.8 we have studied conservation laws in Lagrangian mechanics. To discover conservation laws within the framework of Hamiltonian formalism, let us

CHAPTER 4. HAMILTONIAN FORMALISM

328

consider the Lagrangian (4.4.22) on JI V'Y (165], and apply the first variational formula (3.2.13) to it. Given a vector field (3.9.61) on the event bundle Y, its lift u (4.8.2) onto the phase space V'Y reads

u`8'+u'8;-Auip;8`,

u

u`=0,1.

As a particular case of the weak identity (4.8.6), we have

-wait - u'8tf + pcdtu'

dt(-p;u' + ut7{)

(4.10.37)

for the current I It

= -ptu' + ut7{.

(4.10.38)

In the case of a vertical vector field u, where u' = 0, this transformation -law leads to the weak equality 8,%

dtpt.

In the case of the horizontal lift fi (4.10.15), the weak identity (4.10.37) takes the form

-8tf - r'8tn + pdtl'' st, -dtxr, where ? r = it - pt1'' is the Hamiltonian function in the splitting (4.10.20). The following assertion is a particular case of Proposition 4.8.1. PROPOSITION 4.10.17. Given an event bundle Y -+ R, let a Hamiltonian form H on the Legendre bundle V'Y be associated with a semiregular Lagrangian G on J'Y. Let r be a solution of the Hamilton equations (4.10.21a) - (4.10.21b) for H which lives in the Lagrangian constraint space Q and c the associated solution of the Lagrange equations for G so that the conditions (4.5.23) are satisfied. Let u be the vector field (4.5.23) on Y - R. Then, we have

`t(r) = `I(H or),

''t(Z o J'c) ='1(c),

where T is the current (3.9.63) on J1Y and 'I is the current (4.10.38) on V'Y. COROLLARY 4.10.18. The Hamiltonian counterpart of the Lagrangian energy function Tr (3.9.66) in the sense of Proposition 4.10.17 is the Hamiltonian function 7{r in the splitting (4.10.20).

4.10. APPENDIX. HAMILTONIAN TIME-DEPENDENT MECHANICS

329

Therefore, we can treat ?{r as the energy function with respect to the frame I'. In particular, if I" = 0, we obtain the well-known energy conservation law 8t?{

dth,

which is the Hamiltonian variant of the Lagrangian one (3.9.67).

J. Unified Lagrangian and Hamiltonian formalism The relations between Lagrangian and Hamiltonian formalisms described above are broken under canonical transformations if the transition functions y' - y" depend on moments. The following construction enables us to overcome this difficulty. Given an event bundle Y R, let V'J'Y be the vertical cotangent bundle of J'Y R with coordinates

(" yi' yi, yi' ii')

and J' V'Y the jet manifold of V'Y

R with coordinates

(t, yi, pi, Vii, pa).

PROPOSITION 4.10.19. There is the isomorphism

11 = V'J1 Y = J1V'Y,

yi

pu,

pi,

(4.10.39)

over J' Y.

Proof. The isomorphism (4.10.39) can be proved by comparing the transition functions of the coordinates (yi,y;) and (pi,pu).

QED

Due to the isomorphism (4.10.39), one can think of II as being both the Legendre bundle over the configuration space J'Y and the configuration space over the phase

space V'Y. Hence, the spaceII can be utilized as the unified configuration and phase space of joint Lagrangian and Hamiltonian formalism.

Remark 4.10.9. In connection with this, note that, according to [10, 133, 1811, the dynamics of an autonomous mechanical system described by a degenerate Lagrangian G on TM is governed by a differential equation on T'M generated by due to the canonical diffeomorphism between MM and T'TM (see Remark 4.9.1).

CHAPTER 4. HAMILTONIAN FORMALISM

330

The manifold n is equipped with coordinates (t, y', yl, pa, pi), where (y', pti) and (y' , pi) are canonically conjugate pairs. The space II is endowed with the canonical form (4.10.13) which reads A = (dpti A dy' + dpi A dy,) A dt = dt(dpi A dy' A dt),

(4.10.40)

where

dt=Ot+yl,8i+p 0' is the total derivative. The corresponding Poisson bracket (4.10.8) takes the form { f' 9 }v =

Of Og _ Og Of _ Og Of Of Og apt, V + OR BV apt. 8y' OA 8yi

(4.10.41)

It is readily verified that the canonical form (4.10.40) and, consequently, the Poisson bracket (4.10.41) are invariant under the transformations of II which are jet prolongations of the canonical automorphism of V'Y. I Let

H = ptidy` + p+dyi - x(t, Y`, Yt, pti, pi)dt

be a Hamiltonian form (4.10.20) on n. The corresponding Hamilton equations (4.10.21a) - (4.10.21b) read dty'

=

(4 . 10 . 42 a)

dtyt' =Bpi,

(4.10.42b)

dtpi =

(4.10.42c)

dips

(4.10.42d)

Substitution of (4.10.42a) in (4.10.42b) and of (4.10.42c) in (4.10.42d) leads to the equations &H

dt 0N 8341

&H

8p; , Off[

ay,

(4. 10. 43a)

(4.10.43b)

4.10. APPENDIX. HAMILTONIAN TIMEDEPENDENT MECHANICS

331

which look like the Lagrange equations for the "Lagrangian" ?{. Though it is not a true Lagrangian, one can put

?f = -Ly + dt(p I"), whenever Ly is a Lagrangian on J' Y. Then the equations (4.10.43a) - (4.10.43b) are equivalent to the Lagrange equations for the Lagrangian Ly on J'Y. However, their solutions fail to be solutions of the corresponding Hamilton equations (4.10.42a)

- (4.10.42d) in general. This illustrates the fact that solutions of the Hamilton equations (4.10.42a) - (4.10.42d) are necessarily solutions of the Lagrange equations (4.10.43a) - (4.10.43b), but the converse is not true. To give a unified Hamiltonian-Lagrangian picture, let us consider the Hamiltonian form

H = pudyt + pcdyi - (dt?fn + (piye - ?in) - Ly)dt,

(4.10.44)

where Ly is a semiregular Lagrangian on the configuration space J'Y and Hn is a Hamiltonian form on V'Y associated with Ly. The corresponding Hamilton equations (4.10.42a) - (4.10.42d) read

day' _ f,

(4.10.45a)

dal< ,'

(4 . 10 .45b )

= d, 9Hn + ya - 6Hn 8p,

-n

dsp+ dep

= -d

a

- Pi + 8

Oh n + ay,

81l'

(4 . 10. 45 c)

,

+ 04 .

5'

(4 . 10 .45d )

Using the relations (4.5.9a) and (4.5.12), one can show that solutions of the Hamilton equations (4.10.21a) - (4.10.21b) for the Hamiltonian form Hn which live in

the Lagrangian constraint space Ly(J'Y) C V'Y are solutions of the equations (4.10.45a) - (4.10.45d).

Now let us consider the Lagrange equations (4.10.43a) - (4.10.43b) for the Hamiltonian (4.10.44). They read day '

d<>h

- Mnn = 0 , 8Ly

( 4 . 10 . 46a)

8?{n

8Ly

- dtNii --0V - Chi

(4 . 10 . 46b )

332

CHAPTER 4. HAMILTONIAN FORMALISM

In accordance with Proposition 4.5.11, every solution of the Lagrange equations for the Lagrangian Gy such that the relation (4.5.23) holds are solutions of the equations (4.10.46a) - (4.10.46b). In particular, if the Lagrangian Gy is hyperregular, the equations (4.10.46a) (4.10.46b) and the equations (4.10.45a) - (4.10.45d) are equivalent to the Lagrange equations for Ly and the Hamilton equations for an associated Hamiltonian form.

Chapter 5 Special topics This Chapter is devoted to a few topics on higher order and infinite order jet formalisms which are involved in different constructions of the calculus of variations and field theory. Note that there are two main approaches to jet formalism. The geometric one define jets as equivalence classes of sections of a fibred manifold and, more generally, as equivalence classes of submanifolds which have a contact of some order. Within the framework of the algebraic approach, the notion of jets of modules is basic. The overlap of these two jet machineries lies in jets of sections of vector bundles. We have observed that it is convenient to call into play the infinite order jet machinery in order to describe finite order dynamic systems. We follow the pragmatic approach to the calculus in infinite order jets when the algebraic limits of objects on finite order jet manifolds are considered [42, 109, 180, 184]. This prevents us from the specification of a manifold structure on the infinite order jet space which is not a well behaved smooth manifold [13, 174, 1751-

5.1

Higher order jets

In this Section, the basics of the calculus in the higher order jets are recalled. Let us begin with the familiar geometric notion of jets of sections of fibred manifolds [105, 127, 167).

Remark 5.1.1. We will follow the notation of Section 3.1. Recall that, given fibred coordinates (ac., y') of a fibred manifold Y -+ X, we use the multi-index A, I A [= r, for collections of numbers (J1,...J11) modulo permutations. By A + E is meant the

333

CHAPTER 5. SPECIAL TOPICS

334

collection

A+ E= (A,...Alak...a,) modulo permutations, while AE denotes the union of collections (A,... Aiak...a,),

AE =

where the indices A, and of are not permuted. Recall the symbol of the total derivatives

d.% :D*-'O7 1, da = 8a + E YA+a

,

,

where the sum is taken over all collections A, 0 <1 A 1. Here by 0 is meant the algebra of exterior forms on J'Y. The following relations hold [da, dQ] = 0, da(q5 A a)

d,(4,) A o+ 46 A da(a),

=

d.%(d4) = d(dab).

In contrast with the partial derivatives 8,,, the total derivatives have the coordinate transformation law

da =

9XA

dµ.

We will use the notation 8A = 8a. o ... o 8,,,

dA = da, o ... o dx,,

A = 0,...X0.

DEFINITION 5.1.1. The r-order jet space J'Y of sections of a fibred manifold

Y -+ X (or simply the r-order jet space of Y

X) is defined to be the union

U is VEX

of the equivalence classes 9=s of sections s of Y so that sections s and s' belong to the same equivalence class 2s if and only if

s"(x),

9As`(x) = 8Ad (x),

0 <) A I< r.

5.1. HIGHER ORDER JETS

335

0 In brief, one can say that sections of Y - X are identified by the r + 1 terms of their Teylor series at points of X. Of course, the particular choice of a coordinate atlas does not matter for this definition. There are several equivalent ways in order to provide the jet space JkY with a manifold structure. They lead to the following result. PROPOSITION 5.1.2. Given an atlas of fibred coordinates (XA, y') of a fibred manifold Y -y X, the r-order jet space FY of Y is endowed with an atlas of the adapted coordinates

(x,yA),

0
(5.1.2)

W, yn) o S = (XA, 00i W),

possessing the following transition functions

ax

YA+A = a,a dvyA

(5.1.3)

0 The coordinates (5.1.2) bring the set J'Y into a smooth manifold of finite dimension

Ia- i!(n - 1)!

It is called the r-order jet manifold of the fibred manifold Y X. The coordinates (5.1.2) are compatible with the natural surjections

7ri : J'Y - JlY which form the composite fibration 7rr:

with the properties IrAk 0

7rrk= 7rAr,

It 1

0 7rr A-= 7rr.

CHAPTER 5. SPECIAL TOPICS

336

A glance at the transition function (5.1.3) when I A I= r shows that the fibration

is an affine bundle modelled over the vector bundle

VT'X ® VY -y J'-'Y.

(5.1.4)

Remark 5.1.2. To introduce higher order jet manifolds, one can use the construction of the repeated jet manifolds. Let us consider the first-order jet manifold J'J'Y of the fibred manifold XY X, provided with the adapted coordinates (x",ye,yaA),

I A I< r.

There are the following bundle morphisms

Ja;_1 : J'J'Y -+ J'J'-'Y, J'-'Y, P; : J' J'Y

y

o Jir;_1 = y1,

yAA o P: = y%+A,

I A I< r - 1,

IAI < r - 1.

Their affinc difference over J'''Y is the r-order Spencer morphism

VJ''1Y,

IAI< aA0Sr=4A-ya+A, where VJ'''Y is the vertical tangent bundle of the fibred manifold J'-'Y - X. i

The kernel of the r-order Spencer morphism Sr is the r-order sesquiholonomic jet manifold J'+'Y coordinatized by

(x", ye, y'),

IAI< r,

I E I= r.

We have the chain of fibred monomorphisms

Jr+'Y --4.7"+'Y -

J'J'Y.

Hereafter, we will identify the jet manifolds J'+'Y and J'+'Y with their images in

J'J'Y.

Remark 5.1.3. Following Remark 5.1.2, one can consider the r-order jet manifold

J'J'Y of the fibred manifold J'Y - X with coordinates

(x", y6),

IAI
IEI
5.1. HIGHER ORDER JETS

337

There is the canonical monomorphism

vk,.: J'+kY ti Jr Jky

(5.1.5)

given by the coordinate relations YEA o ark = 2/E+A.

In the calculus in r-order jets, we have the r-order jet prolongation functor Jk such that, given fibred manifolds Y and Y' over X, every fibred morphism $ : Y Y' over a diffeomorphism f of X admits the r-order jet prolongation to the morphism

Jr,b : J'Y 9 f.8 s-+ j*.(.) (() o s o f -') E J'Y'

(5.1.6)

of the r-order jet manifolds.

Example 5.1.4. Let Y -, X and Y' - X be two fibred manifolds over the same base X. Then the fibred morphisms

pr,:YxY'-Y X and

pr2:YxV'-.Y' X induce the canonical fibred isomorphism

J'(YxY')-J'YxJ'Y'. x x The jet prolongation functor is exact. If 4b is injection ]surjection], so is J'+. The jet prolongation functor also preserves an algebraic structure. In particular, if Y -+ X is a vector bundle, so is J'Y -. X. If Y - X is an affine bundle modelled over the vector bundle Y - X, then J'Y X is an affine bundle modelled over

the vector bundle JrY - X. Every section s of a fibred manifold Y -- X admits the r-order jet prolongation to the section (J's)(x) = Zs

CHAPTER 5. SPECIAL TOPICS

338

of the fibred jet manifold J'Y -+ X. Such a section of JrY - X is called an integrable section. The following integrability conditions hold. LEMMA 5.1.3. Let 3 be a section of the fibred manifold JTY - X. Then, the following conditions are equivalent:

(i) a = J's, where s is a section of Y (ii) J1-3: X J'+'Y,

X,

(iii) J13: X - J"+'Y. 0 Proof. The condition (ii) takes the coordinate form "A =

OJ-iA,

0
It follows that 3 = J'(iro o s).

QED

Given the k-order jet manifold JkY of Y -- X, there exists the canonical fibred morphism r(k) : JkTY -- TJkY. over

JkY X JkTX - JkY x TX x x whose coordinate expression is

(z",

0:51 A J< k,

yn, ?, A) o r(k) = (x", yn, ?, (y;)A

where the sum is taken over all partitions E + we have the canonical isomorphism over JkY

= A and 0 <J 41 J. In particular,

r(k) : JIVY -. VJkY, (?")A =1. o r(k).

As a consequence, every projectable vector field u = uµO,, + Uc

(5.1.7)

5.1. HIGHER ORDER JETS

339

on a fibred manifold Y -+ X has the following k-order jet lift to the vector field on

Jky: Jku = r(k) o Jku :

jky - TJkY,

Jku=uAOA+ui8i+u;,e;,

ua+A = dAun -1lµ+Aaauµ,

(5.1.8)

0
In particular, the k-order jet lift (5.1.8) of a vertical vector field on Y vertical vector field on JkY - X due to the isomorphism (5.1.7).

X is a

DEFINITION 5.1.4. A vector field u,. on an r-order jet manifold PY is called projectable if, for any k < r, there exists a vector field Uk on JkY -' X such that Uk o Irk = T7rk O u..

Example 5.1.5. A vector field on J'Y which is vertical with respect to some fibration J'Y J'-'Y is obviously a projectable vector field on J'Y. Let us denote by Pk the vector space of projectable vector fields on the jet manifold JkY. It is easily seen that P, are Lie algebras and that the morphisms T7rk, k < r, constitute the inverse system PO

PI 4- ...

Pr_ I T+W- Pr

(5.1.9)

of these Lie algebras.

PROPOSITION 5.1.5. The k-order jet lift (5.1.8) is the Lie algebra monomorphism of the Lie algebra Po of vector fields on Y X to the Lie algebra Pt of projectable vector fields on JkY such that [13, 174]

T7rk(J'u) = Jku o Irk.

(5.1.10)

The jet lift Jku (5.1.8) is said to be an integrable vector field on JkY. Every projectable vector field on J'Y is decomposed into the sum u, = JJT7rp(ur) +vr

CHAPTER 5. SPECIAL TOPICS

340

of the integrable vector field J4T7ra(ur) and the vector field v, which is vertical with

respect to a fibration "Y - Jr-kY. Every exterior form 0 on the jet manifold JAY has the lift to the pull-back form akt+" 0 on the jet manifold Jk+Y. Let ilk be the vector space of exterior forms on the jet manifold JAY. We have the direct system

LAX-+Oo

(5.1.11)

where OX and 0 are the vector spaces of exterior forms on X and Y respectively. The subsystem of (5.1.11) is the direct system

flo(X)

-O0

1I Oo ' .... "_', or

...

of the rings of real smooth functions ilk = C°°(JkY) on the jet manifolds JkY. Thence, one can think of (5.1.11) as being the direct system of left Ok-modules. The inverse system (5.1.9) of the Lie algebras of projectable vector fields and the direct system (5.1.11) of the modules of exterior forms are defined for any finite order r. These sequences admits the limits for r -. oo in the category of Lie algebras and in that of modules respectively, thus leading to the concept of infinite order jets (see Section 5.4). We have the exact sequences 0

VJkY ti TJkY - TX x JkY - 0,

0

JkY x T'X x

(5.1.12)

x

TJkY -. V *JkY

0,

(5.1.13)

of vector bundles over JAY. In general, they have not a canonical splitting, but their pull-backs over Jk+' split canonically due to the following canonical bundle monomorphisms over JAY: A(A)

: JA+'Y c

T'X 0 TJkY, Jlly

dx'' ®(8A +

0:51 A I< k,

ya+A8'),

(5.1.14)

and

0iki : Jk+'Y

JkY

T' JkY ® VJkY, JI,Y

O(k) = E(dyn - UA+AdxA) ®

,

(5.1.15)

5.1. HIGHER ORDER JETS

341

where the sums are taken over all collections of multi-indices A, I A I< k. The morphism (5.1.15) is called the contact k -jet form. The components 0`A = dye

-

(5.1.16)

tl'A+ad.T'',

of the contact k-jet form (5.1.15) are called the contact forms. The monomorphisms (5.1.14) and (5.1.15) yield the fibred monomorphisms over Jk+'Y A(k)

: TX x Jk+'Y c-+TJtcY x Jk+'Y, X

(5.1.17)

JkY

and

6(k): V'JkY x _T*JkY x Jk+'Y jky JkY

(5.1.18)

These morphisms split the exact sequences (5.1.12) and (5.1.13) over Jk+'Y and define the canonical horizontal splittings of the pull-backs lrk+"7. JkY = !(,t) (TX

x J+' Y) (D V JkY, J

+ E YYOi = a"(8,, +

(5.1.19)

r

ye+AB+ } + EM

-

y'a+A)8",

and 7rk+i.7.. Jky =

G 9(k)(V*JkY x Jk+1y),

Jk+1Y

xadxa +

JkY

21i dyn = (xa + E wya+A)&A +

(5.1.20)

3l: OA

where summation are over all multi-indices A, 0:51 A I<- k. In accordance with the canonical splitting (5.1.19), the pull-back

Uk:Jk+'Y-.JkY- TJkY over Jk+'Y of any vector field uk on JY admits the canonical splitting Ti = u11 + UV = u"(8A + E YA+A8) + E(ul - tAya+A)8".

In virtue of the canonical horizontal splitting (5.1.20), every exterior 1-form .0 on JkY admits the canonical splitting of its pull-back

CHAPTER 5. SPECIAL TOPICS

342

where h040 is a semibasic 1-form on Jk+'Y, while h1o takes its values into Im B(k). DEFINITION 5.1.6. An exterior 1-form 46 on the jet manifold JkY is called a contact 1-form if its pull-back sk+''O over Jk+'Y takes its values into the image Im9(k+1) of the monomorphism (5.1.18).

DEFINITION 5.1.7. Let Ck be the ideal of the algebra ilk of exterior forms on JkY which is generated by the contact 1-forms on JkY. This ideal comprises the exterior products of contact 1-forms and arbitrary exterior forms on JkY. Its elements are called contact forms on the jet manifold JkY, and Ck is said to be the ideal of contact forms.

The ideal Ck of contact forms is a module over the ring of smooth functions on JkY. Its Pfalfian system is locally generated by the contact forms OX, I A (< k, (5.1.16).

It is easily verified that the pull-back irkk+` 0 of every contact form on the jet manifold JkY onto Jk+'Y is also a contact form. Hence, we have the direct system

of left 0 modules *. ,4. w C1 4 C2 2,... + C. The ideal Ck defines the smooth Cartan distribution CJkY C TJkY of the dimension

m

(n+k- 1)! --I)!

k! (n

on the jet manifold JkY. This comprises the elements ofTJkY which are annihilated by contact 1-forms. Its fibres CgJkY C TQJkY, q E JkY, are called the Cartan subspaces. The ideal Ck of contact forms fails to be a differential ideal since d8X, I A (= k-1, are not contact forms. It follows that the Cartan distribution CJCY is not completely integrable.

Remark 5.1.6. Since the obstruction for the ideal CJ." Z to be differential lies only in the forms O)), (A (= k - 1, the limit of the direct system (5.1.11) has not this defect.

5.1. HIGHER ORDER JETS

343

Though CJkY has no integral manifolds of maximal dimension, it is easily seen that, given a section s of the fibred manifold Y -+ X, the image of its k-order jet prolongation Jks in JkY is an integral manifold of CJ*Y. Moreover, in virtue of the Lemma 5.1.3, any n-dimensional integral manifold B of the Cartan distribution on JkY such that 7r*(B) is diffeomorphic to X is the image of an integrable section 3 = Jks of the fibration JkY -+ X. This proves the following assertion. PROPOSITION 5.1.8. An exterior form .0 on the jet manifold JkY is a contact form if and only if its pull-back ro onto the base X by means of any integrable section r of JkY - X is equal to zero.

Thus, one can say that images of integrable sections of the fibration JkY -. X constitute a complete family of integral manifolds of the Cartan distribution on JkY. They are called k-order Cartan manifolds. It follows that the Cartan subspace CgJkY of the tangent space TgJkY at a point q E J4Y is the linear envelope of the tangent spaces to all k-order Cartan manifolds passing through q. Remark 5.1.7. Cartan manifolds play the role of classical solutions of systems of partial differential equations in the calculus in jets. Obviously, they do not exhaust all types of integral manifolds of Cartan distributions (see 11091). Several constructions of the calculus in higher order jets seem more natural if one follows the algebraic definition of jets. Given a module P over a commutative ring A,

the modules 3k(P) of k-order jets of the module P are defined in an algebraic way (see next Section). The relation of this algebraic notion to the above-mentioned geometric definition is based on the well-known duality between the category of vector bundles over a manifold X and that of projective modules over the ring C°°(X). It means that, if P is the module of sections of a vector bundle Y - X, then the modules 3k(P) of k-order jets of the module P are represented by sections of the fibre bundles which are exactly the flbred jet manifolds JIY X introduced above in a geometric way. We will meet below several examples of sequences of fibre bundles and modules. These are the Spencer sequences.

Remark 5.1.8. Spencer sequences. Let P be a module over a commutative ring A. Let n P be the skew symmetric k-tensor product of P and v P the symmetric k-tensor product of P.

CHAPTER 5. SPECIAL TOPICS

344

Let s denote the operation of symmetrization

s:p,®...®pk- Pp'=

1

k.

p°t®...®p°.

where the sum is taken over all permutations (al, - - , 7k) of the collection of numbers (1,

-, k). By a is denoted the operation of alternation. These operations are A-

module morphisms. The Spencer operator is defined as

I b:Vl'(&nP-. V' P® A P,

(5.1.21) b:(plV...VprV...Vpk)®(q'A...Agm),__.

V... Vp' V...pk)0(prnq' A...n F(p' r where by p? is meant that the element p' is omitted. It is a homology operator, i.e.,

bob=0. Thence, we have the complex

0_

P_kv'P®P 6-4 .. kV

(5.1.22)

called the k-order standard Spencer complex. It is easy to check that the standard Spencer complex is exact. The generalization of the Spencer complex (5.1.22) is the complex

where E is a A-module.

In particular, let P be the module O'(X) of sections of the cotangent bundle T'X of a manifold X over the ring D°(X) of real smooth functions on X. Then P is the module Dt(X) of exterior m.-forms on X and A:

6kX = VD'(X) is the module of sections of the symmetric tensor bundle

SkX = V"T'X.

(5.1.23)

5.1. HIGHER ORDER JETS

345

In this case, the (k < n)-order Spencer complex (5.1.22) takes the form

U - 6kX 6 ... 6k--x 017-(X)

o,

... flk(X)

where the Spencer operator (5.1.21) is given by the coordinate expression (5.1.24)

(-1) SE6k-'"X®17'"(X),

IA I=k-m-1,

with respect to the induced coordinates on TX. We also have the corresponding sequence of the tensor bundles

0 - SkX 6y ... Sk--X 0'A 7-X 6 ... AT*X

()

(5.1.25)

where the Spencer operator (5.1.21) is given by the expression similar to (5.1.24): Ck,...um+

b=

I A I= k - m - 1.

The following construction will play a prominent role in the sequel. X, Let Y -+ X be a vector bundle. Given a section -9 of the vector bundle JkY let us consider the section irk_i o 's of the vector bundle J'1-1Y -+ X and then the section

s = J'(ik-1 o s). I A I < k,

si,A = 8.%3"A,

of the vector bundle J' J" Y - X. The sections 3 and s take their values into the same fibres of the affine bundle J'Jk-'Y - J'-'Y. Therefore, their difference can be identified with a section of the vector bundle

VJk-'Y ® Tex, Jk-'Y

which is isomorphic to the pull-back

(J"-'Y ®T'X) x Jk-'Y, x

x

since Y -. X is a vector bundle. Furthermore, let 6 be the global zero section of the vector bundle Jk-'Y -+ X. Then, we have the bundle monomorphism of the fibre bundle

Jk-'Y ®T*X -.X

(5.1.26)

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346

onto the subbundle

(Jk-'Y (&T'X) x 0(X) C (J"-'Y ®T'X) x J"-Y. x

x

x

x

Since the projection 7rkt_i o (s --9) takes all its values into 0(X), we can identify the section 9 - 3 with a section of the bundle (5.1.26). Thus, we have the morphism 1Qk : 3k(y) - a7k-' (Y) ®V (X),

(5.1.27)

Bk :3'- J' (7rkt_ l 03) -3, of the space 3k(Y) of sections of the vector bundle JkY

X into the space

3k-'(Y) ® E)'(X) of sections of the fibre bundle (5.1.26). Since J'-'Y -, X is a vector bundle, one can choose the bases e" for the fibres of this vector bundle and can identify the induced coordinates VA with the fibre Then, the vector bundle (5.1.26) can be endowed with coordinates coordinates (xa, yn a),

0
(5.1.28)

with respect to the holonomic bases {dxa®O; },

0<1 AI
of its fibres. Relative to the coordinates (5.1.28), the morphism (5.1.27) reads I

ek(8) = (OANA - 3`a+A)0 ®dxA ,

0<1 AI
(5.1.29)

A glance at the coordinate expression (5.1.29) shows that, in virtue of the Lemma

5.1.3, the kernel of the morphism (5.1.27) consists of the integrable sections of JkY X. Hence, there is the short exact sequence 0

3°(y) J" 3k(y) -ik-3k-,(Y) 0,01m,

(5.1.30)

where by Jk is meant the morphism induced by the k-order jet prolongations of sections of Y -. X. This is the vector space monomorphism. Using the operation of alternation a, one can easily prolong the sequence (5.1.30) by means of the morphisms pk-,,, ® a as follows:

3k-I (y) ®D' (X) --. 3k-2(y) ® O2(X) - .. .

5.2. JETS OF MODULES

347

As a result, we obtain the sequence 3° ( y )

p

sk(y)

... 3°(y) (& Dk(X)

... k-m(y) ®i7""(X) S

(5 . 1 . 31)

(5.1.32)

p,

where the morphisms Sk+I are given by the expressions

k(

Sm+1

i

(5.1.33)

`S) A,u,...Qr-&m+t

-9 E 3k-m(Y) ®Dm(X ),

I A I= k - m -1,

with respect to the corresponding coordinates

of the fibre bundle

Jk-my®('T`X) - X. In particular, Sl = pk. It is easy to see that the morphisms (5.1.24), and obey the homology rule $kt+1 0

k

S,kn+1

are similar to the Spencer operators

p.

Hence, the sequence (5.1.31) is a complex, called the well-known k-order Spencer complex of the module 3°(Y) of sections of a vector bundle Y -' X. The morphisms (5.1.33) are called the Spencer morphisms. The Spencer complex (5.1.31) exemplifies a sequence that has the same structure as the standard Spencer complex (5.1.22).

5.2

Jets of modules

Within the framework of the algebraic approach, we will show that the representative

object of k-order linear differential operators acting on a module P is the module 3k(P) of k-order jets of P. In other words, every k-order linear differential operator on P with values into a module Q is given by composition:

of the k-order jet prolongation functor of P to Zlk(P) which is the k-order linear differential operator, and

CHAPTER 5. SPECIAL TOPICS

348

a module homomorphism of 3k(P) into Q.

This construction can be extended to nonlinear differential operators acting on sections of fibred manifolds. These operators are defined directly in terms of jets. We start from some basic elements of the differential calculus in modules [109). Suppose that A is a commutative R-ring with a unit element 1 E A. Later, we will choose that A to be a ring of real smooth functions on a manifold.

Let P and Q be left (or right) A-modules. The set Hom (P,Q) of R-module homomorphisms of P into Q is endowed with the A-bimodule structure by the rule (a4S)(p) = a 41(p),

(b*a)(p) = i(a -p),

a E A,

p E P,

(5.2.1)

denotes the multiplication operations in the A-modules P and Q. Let us consider the difference 5 of left and right multiplications in the A-bimodule where

Hom (P, Q):

60(4)=aO-q5*a,

0EHom(P,Q),

VaEA.

Let us set bo,...o.=6fto...o63.

DEFINITION 5.2.1. An element A E Hom (P, Q) is called an s-order linear differen-

tial operator from the A-module P into the A-module Q if Soo...a.(A) = 0

for arbitrary collections of s + I elements of A. 0 Let us denote by Diff ,(P, Q) the set of s-order linear differential operators from P to Q. It is endowed with the A-bimodule structure (5.2.1). By definition, there is the bimodule monomorphism

Diff,(P,Q) C Diffk(P,Q),

k>s.

Remark 5.2.1. At the same time, one must distinguish the A-bimodule Diff .(PQ) from the same set provided separately with the left A-module structure and the right Q), A-module structure. We denote these modules by Diff , (P, Q) and Diff 8 -(PQ), respectively.

5.2. JETS OF MODULES

349

Let us denote the module Diff ,(A, Q) simply by Diff ,(Q), and consider the morphism

Z.: Diff , (Q) - Q,

x,(0)0(1),

1 E A.

This morphism is an s-order differential operator on the right module Diff ; (Q), and a 0-order operator on the left module Diff -(Q). PROPOSITION 5.2.2. For any differential operator A E Diff-(P, Q), there exists the unique homomorphism

fa : P -+ Diff, (Q), [fo(P)](a) = A(a - P),

Va E A,

such that the following diagram commutes fo

P - Diff , (Q) Q

The correspondence A -4 f a defines the isomorphism

Horn; (P, Diff ; (Q)) = Diff

(5.2.2)

In other words, every differential operator from a A-module P to a A-module Q is represented by a morphism of P into the module of differential operators from A to Q. It follows that the A-module Diff; (Q) is the representative object of the functor

Diff; (.,Q) : P - Diff; (P, Q). Accordingly, the representative homomorphism of the composition of r-order and s-order differential operators is the composition homomorphism

c,,,.: Diff a (Diff , (Q)) - Diff ,+,(Q).

CHAPTER 5. SPECIAL TOPICS

350

This is defined by the requirement that the following diagram Commutes

D, Diif, (Dill, (Q)) -- Diff, (Q) Cs,r

I

I

1r

Zs+r Diff a+r(Q)

-''

Q

Let us dwell on the composition homomorphisms cj,r and on the first order linear differential operators. We will denote (Diff -,(Q))" = Diff , (... Diff - , ( Q ) A : ...).

For any a E A, we have the map

as : (Diff -,(Q))" -+ (Diff -,(Q))"-', a * = (Di * a)(A) = ZI(A * a) = (A * a)(1) = 0(a), A E (Diff -,(Q))", A(a) E (Diff i (Q))k-`. DEFINITION 5.2.3. The first order differential operators 8 from A to Q which obey the rule

8(aa') = a8(a) + a'O(a),

da, a' E A,

are called the derivations of the algebra A with values into the A-module Q. 0 The derivations 8 constitute the submodule D(Q) of the left A-module Diff i (Q). There exists also the R-module injection i : D(Q)

(5.2.3)

Diff -1(Q),

but Im i fails to be a right A-submodule of Diff i (Q). It is easily seen that a first order differential operator A belongs to Im i if and only if

D(0)=0(1)=0. Thence, we have the short exact sequence of R-modules

1 0 - 0 (Q)

Diff

(Q)

- 0,

5.2. JETS OF MODULES

351

called the first order Spencer sequence of differential operators.

Remark 5.2.2. Let i : P

Q be a A-submodule of the A-module Q. Any P_ valued derivative 8 of A yields the Q-valued derivation i o 8 of A, and we obtain the homomorphism of the left A-modules 81 i : 0(P) -' a(Q).

(5.2.4)

Difficulties arise if P fails to be an A-submodule of Q just as in the case of the injection (5.2.3).

Let us apply the derivation functor (5.2.4) to the injection (5.2.3). The composition a(DiIf , (Q)) comprises the derivations of A with values into the right A-module Diff -(Q). They, by definition, satisfy the condition

(aa')*8= (a*8)*a'+(a'*8)*a. Let us consider the composition 0(0(Q)), where 0(Q) is is thought of as a left Amodule. Its elements satisfy the condition

(aa') * 8= a'(a*8)+a(a'*8). Then, it is easily verified that the intersection

02(Q) = o(o(Q)) n a(Diff -(Q)) consists of those elements of a(Diff -,(Q)) which obey the relation

a*(a *(9)=a'*(a* 8). This is a left A-module. As a consequence, we obtain the following chain of Rmodule morphisms

02(Q) -e' a(Diff i (Q)) - (Diff (Q))2

' Diff z

(Q)

(5.2.5)

Set up inductively

o (a. (P) n a(Diff; (P))".

Applying the functor (5.2.4) to the chain (5.2.5) k-2 times, one obtains the k-orde Spencer sequence of differential operators

0 -' ak(Q) -~Dk-1(Diff i (Q))s-.... k -I

s

ak-m(Diiff m// \Q)) ' ... Diff k (Q)

`' Q

0,

(5.2.6)

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352

where

Sk m ok-m(Dlffm(Q))

ak-.n-i(Diff'n(Q))

-m-1(Diff,n+1(Q)) SAO:

=Dk.

THEOREM 5.2.4. The Spencer sequence (5.2.6) is a complex, that is, Sk o Sk-+1 = 0.

It is exact at the term 0k-1(Diff -,(Q)) ((1091, p.10,11). 0

Let us turn now to the modules of jets. Given an A-module P, let us consider the tensor product A ® P of R-modules provided with the A-module structure b(a ® p) = (ba) ® p.

(5.2.7)

For any b E A, we set bb(a ®p) = a ®(b' p) - (ba) ®p

Denote by pk+i the submodule of A ® P generated by all elements of the type 6110o ... o bb' (a ®p).

There are the natural injections

µ'Cµ',

r>s.

(5.2.8)

DEFINITION 5.2.5. The module of k-order jets of the A-module P (or simply a jet

module) is defined to be the quotient e(P) of A ® P by µk+t. 0 Besides the module structure induced by (5.2.7), the k-order jet module 3k(P) admits also the A-module structure given by the multiplication b * (a ® p mod µk+') = a ® (b . p) mod i'+'.

We will denote the k-order jet module with respect to this multiplication by 3.'(P).

5.2. JETS OF MODULES

353

There is the following R-module homomorphism

Jk : P -, 3k(P),

Jk(p) = 10 pmodµk+i,

(5.2.9)

and ZJk(P) is generated by the elements Jk(p), p E P. Note that the k-order jet functor (5.2.9) is the k-order differential operator.

Example 5.2.3. The first order jet module 3' (P) consists of elements a®p modulo the relations

The jet modules possess the following natural properties. As a corollary of (5.2.8), there exist the epimorphisms

: 3°+'(P) -' 3'(P)

4++

(5.2.10)

such that the diagrams

P

3'(P)

\-.

.

3'+`(P) are commutative ([109], p.15). The epimorphisms (5.2.10) constitute the inverse system

3'-I(P)

3'(P)

... "°+P.

Given the repeated jet module 38(3k(P)), there exists the homomorphism Qsk :

3s+k(P) y 3'(3k(P)),

which is defined by the requirement that the following diagram commutes

k 3k(P) J.+w

IP 1

3s+k(P)

J. . +

3'(3k(P))

CHAPTER 5. SPECIAL TOPICS

354

For any differential operator A E Diff, (P,Q) there is a unique homomorphism f° : 3'(P) - Q such that the diagram PROPOSITION 5.2.6.

Q

is commutative. 0 Proof. The proof is based on the following fact ([1091, p.13). Let h E HomA(A P, Q) and

a:!'3p--'a®pEA®P. Then, we obtain bb(h o a) (p) = h(6b(a ®p))

QED The correspondence A '' f° defines the isomorphism Hom A (3'(P), Q) = Diff ; (PI Q).

(5.2.11)

This isomorphism means that the jet module 3'(P) is the representative object of the functor

Diff; (P,.) : Q -i Diff; (P,Q). Let us now find the representative object of the derivation functor ak : Q

ak(Q)-

(5.2.12)

Let us consider the modules of jets 3'(A) of the ring A. We will denote them simply by 3'. In the case of 3', the isomorphism (5.2.11) leads to the one Hom A (3', Q) = Diff ; (Q).

(5.2.13)

Remark 5.2.4. The module 3' can be provided with the structure of a commutative algebra with respect to the multiplication aJ'(b) x a'J'(b') = aa'J'(bb').

5.2. JETS OF MODULES

355

There is the monomorphism

I ik : A - 3k(P),

ik(a) = a ®1 mod µk+,.

(5.2.14)

In particular, the algebra a' consists of the elements a ® b modulo the relations

a®b+b®a=ab®1+1®ab.

(5.2.15)

Let us consider the injection

i,:A-3' given by the expression (5.2.14) and the quotient

Oi = 3'/Im ii. PROPOSITION 5.2.7. The morphism

d':AL is a derivation of A. 0 Proof. There is the injection

d1(a) --. J'(a) - il(a) E 31. Then, using the relations (5.2.15), one find in the explicit form that

d'(ba)= 1®ba-ba®1=b®a+a®b-ba®1-ab®1 = bd'a + ad'b.

QED It is easy to see that the A-module O1 is generated by the elements d(a), while the morphism (5.2.14) is a monomorphism of Ol into 3' which splits the exact sequence

CHAPTER 5. SPECIAL TOPICS

356

Propositions 5.2.6 and 5.2.7 lead to the isomorphism

Hom-(01,Q) = 0(Q).

(5.2.16)

In other words, any Q-valued derivation of A is represented by the composition h o d1, h E Hom-(O1,Q) due to the property d1(1) = 0. Example 5.2.5. If Q = A, the isomorphism (5.2.16) leads to the duality condition

Hom-(01iA) = o(A).

(5.2.17)

The isomorphism (5.2.16) can be extended to higher order derivations. PROPOSITION 5.2.8. Let us define the modules Ok as the skew tensor products of the R-modules 01. There is the isomorphism Horn

(Ok, Q) = ak(Q)

(5.2.18)

0 The isomorphism (5.2.18) shows that the module OA: is the representative object of the derivation functor (5.2.12). One can use the isomorphisms (5.2.2), (5.2.11), (5.2.13) and (5.2.18) in order to obtain several relations involving jet modules. The relations

Hom-(3; 0 P, Q) = Hom-(P, Hom -(3', Q)) _ Hom-(P, Diff: (Q)) = tiff-(P, Q) = Hom-(3'(P),Q) imply the isomorphism

3'(P)=3:0 P.

(5.2.19)

Recall that the tensor product of modules is endowed with the module structure (5.2.7).

The similar relations lead to the isomorphism

Hom-(3'(Ok),Q) = ak(Diff '(Q)),

5.2. JETS OF MODULES

357

where ak(Diff -(Q)) is a left A-module. It means that 31(Ok) is the representative object of the composite functor

Q - Ok(Diff -(Q)).

D,t(Diff

Hence, the morphism

ok(Q) - Dk-L(Dlff -(Q)) implies the homomorphism

h' : 31(Ok-I) - Ok. As a consequence, the operators of exterior differentiation

dk=hkoJ1:Ok-1-Ok

(5.2.20)

can be defined. They form the De Rham sequence dk+l

...

(5.2.21)

which is a complex.

Building on the De Rham complex (5.2.21), one can introduce the homomorphisms

Sk-m : 3k(Ok-m-1) - 3k-L(OA:,)

which are defined by the requirement that the diagram

-,

S4_*y

y/ 3k(Ok-m-1) _

3k-1(Ok-m) -...

sl[

Ok (5.2.22)

Jkt I

Ok-m-1

d

Ok-m

- ... _d Ok

is commutative. The upper row of the diagram (5.2.22) is called the Spencer 3-sequence. Since 3k(.) are generated by Im Jk, this is a complex. It is related to the Spencer sequence of differential operators (5.2.6) by the isomorphisms Hom A (31(Ok), Q) = ak(Diff i (Q))

Using the isomorphisms (5.2.19):

3'(Ok)®P=38. 0Ok®P=3'(P)®Ok,

CHAPTER 5. SPECIAL TOPICS

358

one can bring the Spencer complex (5.2.22) into the complex

0 -+ P 'k 3k(P) 8' ... 3k-m(P) ®Om * ... P004 - 0,

(5.2.23)

where the morphisms Sk_i are given by the following expressions S,". (Jk-m+

(p) ®0) = k-m(p) ® d"4,

S,kn(Jk-.n+l (a)p ®0)

= Jk-m(a)Sm(P ®-0),

p E P, pE

-0 E 0m-1, 3k-,n+1

(5.2.24)

aEA.

The complex (5.2.23) is called the Spencer complex of the module P. Let us turn now to the case when A is the ring >;)°(X) of smooth functions on a manifold X.

To obtain a geometric realization of the modules over the ring O°(X) by the modules of sections of vector bundles, we should restrict our consideration to the subcategory of the geometric modules. DEFINITION 5.2.9. The O°(X)-module P is called a geometric module if

nu-P

zEX

where by p is meant the maximal ideal of functions vanishing at the point x E X. O

For the sake of brevity, one can say that the geometric modules over .O°(X) are those whose elements are defined only by their values at points of the manifold X. There is the functor

P - P/zEX n up from the category of all i7°(X)-modules into the category of the geometric modules. Moreover, if P and Q are geometric modules, then so are Diff k(P, Q), 0k(P) and

3k(P). It follows that the functors Diff k(P, .), 8* and Jk are representable in the category of geometric modules. They are exact in this category. Hereafter, we will consider only geometric O°(X)-modules. Then, the relation between algebraic and geometric approaches to jets is based on the following identifications [109].

5.2. JC'1 S OP MODULES

359

Every left £)°(X)-module P is identified with the module of sections of a vector

bundle Y - X. The module 0(17°(X)) is identified with the i°(X)-module T(X) of vector fields on the manifold X;

The module Ol coincide with the module 17' (X) of 1-forms on X in virtue of the duality condition (5.2.17).

The modules of jets 3k(P) are identified with the modules 3"(Y) of sections of the corresponding jet bundles JkY X of the vector bundle Y -. X. Furthermore, the operators dk (5.2.20) go over to the familiar exterior differential of exterior forms on X. The derivation functor 0' (5.2.12) sends every element q of a module Q onto the operator of differentiation along q. The jet functor Jk (5.2.9) is exactly the familiar functor of k-order jet prolongation. Namely, if q is a section of the vector bundle Y X corresponding to the module Q, then Jk(q) is the k-order jet prolongation of q. Obviously a number of equivalent statements are found within the frameworks of both geometric and algebraic approaches to the calculus in jets. Each of these approaches has its own advantages.

Example 5.2.6. The operation which has no counterpart is in the geometric picture of jets is the possibility to change a rings. Namely, in contrast with the functor of jet prolongation (5.1.6), fibred morphisms 4) and their jet prolongations over non-

diffeomorphic transformations f of a base X are considered. Let f : X' - X be a manifold morphism which is not necessarily a diffeomorphism. It induces the R-algebra homomorphism

f' : O°(X) -. D0(X1) whose image is the pull-back of !7°(X) onto X'. This moomorphism yields the following two functors of changing rings.

Every 330(X')-module P' of sections of a vector bundle Y' -. X' can be considered as a O°(X)-module by the rule a P''Ld f'(a)pr,

a E 0°(X),

p' E P.

CHAPTER 5. SPECIAL TOPICS

360

It is a geometric module. In particular, the ring O°(X') can be provided with the structure of a D°(X)-module.

Every geometric O°(X)-module P can be brought into the geometric 0'(X')module

0°(X') ® P.

(5.2.25)

O0(X)

X. The manifold morphism / : X' - X yields its mapping into the O°(X')-module P represented by sections of the pull-back bundle f'Y -- X', but the image of this For instance, let P be a 00(X)-module of sections of a vector bundle Y

mapping fails to be a D°(X')-submodule of P in general. At the same time, this is a D°(X)-submodule of the D°(X')-module (5.2.25). Changes of rings lead to the corresponding changes of modules of jets (see Definition 5.2.5).

Example 5.2.7. In the geometric description of jets, one can obtain coordinate expressions of different algebraic constructions, e.g., the coordinate expressions of the morphisms Sk_{ (5.2.24) which constitute the Spencer complex (5.2.23). Let Y -. X be a vector bundle provided with fibred coordinates (x', Y% and let (xa, yX) be the adapted coordinates (5.1.2) of the k-order jet manifold JkY of Y. We consider the Spencer complex (5.2.23) of the module P = 3°(Y) of sections of

the bundle Y - X. Let 3k(Y) be the O°(X)-module of sections of the vector bundle JkY - X. Each element 3 of 3k(Y) takes the local form

F ex'sa, a

where d are elements of O°(X) and sa are sections of the vector bundle Y - X. Let us calculate S, . By virtue of the relation (5.2.24), we have S; (-g) _

Jk''so ® dc° _ E 8

In the corresponding coordinates this expression reads

Jk-Iso 0 dx'`.

A I< k - 1, (5.1.28) of J"-'Y 0 M,

(A,,. o Ski) (a) = AAA - yN+A = 8,,c'8Asi,

A 15 k - 1.

(5.2.26)

5.2. JETS OF MODULES

361

By a similar way, one can obtain the coordinate expression of the morphism St:

A I< k - 2, (5.2.27)

Mjw o Sz)

where s is a section of the fibre bundle J'-'Y ®T'X. One can check directly that

Szk o I=0. A glance at the expressions (5.2.26) and (5.2.27) shows that the Spencer complex (5.2.23) of the module d°(Y) of sections of the vector bundle Y -+ X is exactly the Spencer complex (5.1.31).

THEOREM 5.2.10. In the category of the geometric modules, the Spencer complex (5.2.23) (and, in particular, the Spencer complex (5.1.31)) is exact, that is,

ImS, =KerSk., for all 0 S 1:5 k. O Proof. We refer to ([109], p.30) for a detailed proof which is based on the following QED assertion.

LEMMA 5.2.11. If Y --# X is a vector bundle, so is the fibration JkY - X. In this case, there exists the short exact sequence over X

SkX ®Y `+ JkY . Jk-'Y -+ 0, where the morphism a is given by the coordinate expression

yn=0, if 0
if

I A1=k,

with respect to the coordinates (x,VA) of SkX 0 Y. 0

CHAPTER 5. SPECIAL TOPICS

362

5.3

Jets of submanifolds

The notion of jets of sections of a fibred manifold is generalized to jets of submanifolds of a manifold, when this manifold has no fibration or its fibration is not fixed (109, 141].

DEFINITION 5.3.1. Let Z be a manifold of dimension m + n. The k-order jet of m-codimensional submanifolds of Z at a point z E Z is defined as the equivalence class [S]s of m-codimensional imbedded submanifolds of Z which pass through z and which are tangent to each other at z with order k > 0. 0 In other words, two submanifolds

is:ScZ, and is.:S through a point z E Z belong to the same equivalence class [S]z if and only if the k-tangent rnorphisms

Tkis : TkS ti TkZ and Tkis, : TkS' c+ TkZ have the same image at the point z E Z, that is, 'I'kis

Ir;s' .

Remark 5.3.1. In fact, the definition of the k-order jet JS]* of submanifolds involves only local properties of submanifolds around the point z E Z.

The union U (S];,

k > 0,

(5.3.1)

sEZ

J,nZ = Z. of jets [S]; is said to be the k-order jet manifold of the m-codimensional submanifolds of Z . It can be provided with a manifold structure as follows.

Let Y - X be an (m + n)-dimensional fibred manifold over an n-dimensional base X and 4i be an imbedding of Y into Z. Then there is the natural injection Jk4i : JkY j=s'-' [S]k(8(Z)),

S = Im (0 o s),

(5.3.2)

5.3. JET'S OF SUBMANIFOLDS

363

where s are sections of Y - X. PROPOSITION 5.3.2. The injection (5.3.2) defines a chart on J,' Z. Such charts cover the set JmZ, and transition functions between these charts are differentiable. They provide the set JmZ with the structure of a finite-dimensional manifold. 0

Proof. The proof is based on the fact that, given a submanifold S C Z which belongs to the jet [S]k, there exist a neighbourhood U. of the point z and the tubular neighbourhood Us of S n Us so that the fibration Us - S n Us takes place. It means that every jet [S] lives in a chart of the above-mentioned type. Later, we will describe these charts and the corresponding transition functions in explicit form.

QED

Lowering the order of tangency, one obtains the natural surjections (5.3.3)

J. "Z,

Pk-c : JmZ

Po:J,Z-Z, which are fibre bundles. However, the fibre bundle

4-1: JmZ-J,,, 11+ is not afline. There is no simple relation, as in the case of jets of fibred manifolds, between the higher order jets Jam,+1Z and the repeated jets Hereafter, we will use the following coordinate atlases on the jet manifolds JmZ of submanifolds of Z. Let Z be endowed with a manifold atlas with coordinate charts (z^),

A= 1,...,n + m.

(5.3.4)

Though by definition, is diffeomorphic to Z, let us provide J,Z with the Was obtained by replacing every chart (z") on a domain U C Z with the

n+m _ (n+m)! m

n!m!

charts on the same domain U which correspond to the different partitions of the collection (z A) in collections of n and m coordinates. We denote these coordinates by (xA,y`),

A = 1,...,n,

i = 1,...,m.

(5.3.5)

CHAPTER 5. SPECIAL TOPICS

364

The transition functions between the coordinate charts (5.3.5) of JO Z associated with the coordinate chart (5.3.4) of Z are reduced simply to exchange between coordinates x and y, and read

xal=y", ..., V1 =x-1 .

95min(n,m),

...,

T -+1 = xAq+1' V q+1 = 0 +1 .

54 =X4, ... , ix. = r .... V- = x

(5.3.6)

M, .

Ttansition functions between arbitrary coordinate charts of the manifold JI.Z take the form (5.3.7)

XA=?() ,

yf =f,(zll,U)

Remark 5.3.2. If S C Z is an imbedded submanifold which belongs to the jet class (S];, there exist an open neighbourhood U. of Z E Z and a coordinate chart (xa, y') (5.3.5) which cover U. so that s n u, is given by the coordinate relations

yi = and

(x", yA)([S]) = (XA(Z), BAS (x"(Z)))

Given the coordinate atlas (5.3.5) of the manifold J.* Z, the k-order jet manifold JmZ of Z is endowed with the atlas of coordinates (xA, yn ),

(5.3.8)

( A 1:5 k,

modulo permutations. However, the transition functions between these coordinate charts differ from those in where by A is meant a collection of indices (5.1.3).

Using the formal total derivatives (5.1.1), one can write the transformation rules for the coordinates (5.3.8) in the explicit form. Given the coordinate transformations (5.3.7), it is easy to find that

4=

d

(5.3.9)

5.3. JETS OF SUBMANIFOLDS

365

For instance, we have

ya = [(.. +

J

)J

(ate

(5.3.10)

The transformation rules for the higher order jet coordinates VA can be obtained inductively by the rule

ya+A = (4-9N 1IA'

(5.3.11)

The transformation law (5.3.11) is compatible with the surjections (5.3.3). It follows that these surjections are fibrations, and JmZ -' Jm 'Z are fibred manifolds, including JmZ - J°mZ. It is easy to see that the transformation law (5.1.3) is a particular case of the coordinate transformations (5.3.11) when the transition functions g" (5.3.7) are indepcrident of the coordinates V. At the same time, a glance at the expression (5.3.10) shows that, in contrast with (5.1.3), transformations of the coordinates (5.3.8) fail to be affine. Hence, the fibred manifolds JmZ -' Jm 'Z are not affine bundles in general.

Example 5.3.3. In order to illustrate this fact, let us calculate the coordinate transformations (5.3.10) if the transition functions (5.3.7) take the form (5.3.6) when only two coordinates y' and x' are exchanged, that is,

y' = x'.

x' = y1,

(5.3.12)

In this case, the operator (5.3.9) reads d d

d

d,r- y1ai1, d

1d

-dx°+yoai1,

ar#1.

Applying this operator toy and x`, we obtain

a#1,

i0=1P+J/oy, ro_-1

0=yv+g.,Y11

I=yiyi Hence, the coordinate transformations of 1-order jets read

z' Ji

1

= 1l17,

__

1 yr,

ye = - ya 1

Y!

-o

o

yo = Y.

-

Y. 1l11

(5.3.13)

CHAPTER 5. SPECIAL TOPICS

366

They are defined on the overlap of charts of the coordinates (ia`, y , ya) and (x", y1, yµ) where yi 96 0,

yi 34 0.

Arbitrary coordinate exchange (5.3.6) can be obtained as composition of the coordinate transformations (5.3.12) - (5.3.13). Note that the transition functions (5.3.13) do not depend on the coordinates (x, y) of the manifold Z. It follows that the fibrations (5.3.3) are fibre bundles. Moreover, it is readily observed that an arbitrary coordinate transformation (5.3.7) can be represented by composition of the coordinate exchanges (5.3.12) and fibred coordinate transformations

sa = xa,

y" = f"(x",y').

(5.3.14)

Therefore, it suffices to check that a coordinate expression in the calculus in jets of submanifolds is maintained under the fibred coordinate transformations (5.3.14) and the coordinate exchanges (5.3.12) in order to check that it is preserved under general coordinate transformations (5.3.7).

Example 5.3.4. Given a manifold Z, let us consider the 1-order jet bundle JJZ of m-codimensional submanifolds of Z. As an immediate consequence of Definition 5.3.1, there is one-to-one correspondence between the jets [S1= at a point z E Z and the n-dimensional vector subspaces of the tangent space TZ: IS]= - ?R% + ya(IS]=)a1)

The fibre bundle

Z

(5.3.15)

possesses the structure group C L(n, m; R) C GL(n + m; R),

of linear transformations of the vector space R'"+" which transform its subspace R" into itself. Its typical fibre is the Grassmann manifold Q3(n, m; R) = CL(n + m; R)/CL(n, m; R)

of n-dirnensional vector subspaces of the vector space R"'+".

5.3. JETS OF SUBMANIFOLDS

367

In particular, if n = 1, the fibre coordinates ya (5.3.8) of the fibre bundle (5.3.15)

with the transition functions (5.3.13) are exactly the standard coordinates of the projective space RP'".

Example 5.3.5. When n = 1, the formalism of jets of submanifolds provides the adequate mathematical description of relativistic mechanics as follows. Let Z be a (3 + 1)-dimensional manifold equipped with an atlas of coordinates (z°, z'), i = It 2, 3, (5.3.5) with the transition functions (5.3.7) which take the form z°

z' - z (z°, z').

' z (z°, z2),

(5.3.16)

The coordinates z° in different charts of Z play the role of temporal Coordinates, whereas z' are the spatial ones. We consider arbitrary coordinate transformations (5.3.16), not only the Lorentz ones. Let J3 Z be the jet manifold of 1-dimensional submanifolds of Z. This is provided

with the adapted coordinates (z°, z', 4) (5.3.8). Then one can think of zo as being the coordinates of 3-velocities. Their transition functions are obtained as follows. Given the coordinate transformations (5.3.16), the total derivative (5.3.9) reads d-,o = dso(z°)dso =

(+)dao.

In accordance with the relation (5.3.10), we have z° = d-.b(z°)d,o(z) =

OZO

e

azo (OP 19V +4 5z=k) az° + Z45z

The solution of this equation is

8z z0

+ z0J

8x°

or / 8z + z 8zJ

8z°

8z

° 8zk

This is the transformation law of 3-velocities, which illustrates that the jet bundle J'Z -i Z is not affine, but projective. To obtain the relation between 3-velocities and 4-velocities, let us consider the tangent bundle TZ equipped with the induced coordinates (z°, z', z°, z'). One can think of the coordinates (z°, i') as being 4-velocities. Then we have the morphism over Z (5.3.17)

zaop=i'/i°.

CHAPTER 5. SPECIAL TOPICS

368

It is readily observed that the coordinate transformation laws of zo and zf/V are the same. Note that the similar morphism R4- RP4 provides the projective space RP4 with the standard coordinate charts. The morphism (5.3.17) is a surjection. Let us assume that the tangent bundle is provided with a pseudo-Riemannian metric g and Q, C TZ is the hyperboloid given by the relation

=1,

µ,L=0,1,2,3.

The union of these hyperboloids over Z Q:=Q+UQ-

Q=aU is Lite union of two connected imbedded subbundles of TZ. Then the restriction of the morphism (5.3.17) to each of this subbundle is an injection of Q into JJZ. Let us consider the image of this injection in the fibre of J3Z over a point z E Z. There are coordinates (z°, z;) in a neighbourhood around z such that the pseudoRiemarnnian metric g(z) at z takes the form

g(z) = diag(1, -1, -1, -1). In this coordinates the hyperboloid Q, C TZ is given by the relation E(it)2

(Z°)2 -

= 1.

i

This is Lite union of the subsets Qt where z° > 0 and Q; where z° < 0. The image p(Q,) is given by the coordinate relation

E(zo) < 1. From Lite physical viewpoint, this relation means that 3-velocities are bounded in accordance with Special Relativity.

Remark 5.3.6. Preconnections. Let Y

X be a (m + n)-dimensional fibred manifold and J1Y the 1-order jet manifold of the fibred manifold Y - X. Let J,;,Y be the 1-order jet manifold of m-codimensional subspaces of the manifold Y. The image of J'Y into J,; Y by the injection (5.3.2) is the affine subbundle of the bundle (5.3.15). Its fibre at a point y E Y consists of the n-dimensional vector subspaces

5.3. JETS OF SUBMANIFOLDS

369

of the tangent space TY whose intersections with the vertical tangent space V5Y is the zero vector. Note that, generalizing the notion of connections on a fibred manifold, one may treat global sections of the bundle (5.3.15) as the so called preoonnections on the manifold Z [141]. By virtue of the well-known theorem ([931, p.57), if such a pre-

connection r exists, its image r(Z) in the tangent bundle TZ -i Z is a vector subbundle of TZ with the structure group GL(n; R). The quotient TZ/I'(Z) is also a vector bundle with the structure group GL(m; R). We have the decomposition

TZ = r(z) ®TZ/r(Z), which can be treated as the horizontal splitting with respect to the preconnection r by analogy with the horizontal splitting with respect to a connection on a fibred manifold.

However, it should be emphasized that, since JI.Z fails to be an affine bundle, preconnections do not constitute an affine space. Preconnections on an arbitrary manifold Z fail to exist (1170], p.144). Definition 5.3.1 of jets of submanifolds does not provide for jets of n-dimensional submanifolds of an n-dimensional manifold Z. Jets of this type are widely known due to their application to study of G-structures [104, 155].

Given an n-dimensional manifold Z, the k-order jet of the manifold Z at a point z E Z is the equivalence class [gZ]; of imbeddings gz of the pair (R", 0) into the pair (Z, z), where two imbeddings i and t" are said to be equivalent if the composition gil o g'Z and the identity map IdR" have the same derivatives up to order k at 0. 0 DEFINI'T'ION 5.3.3.

The union

J°Z,U [gz]; of the jets [9z]s over Z is said to be the k-order jet prolongation of the manifold Z.

Example 5.3.7. Let Z = R" and z = 0. Then the jets [gJ E JJR" constitute the group with multiplication defined by the composition of jets (Y10'.100, = [go #']o

CHAPTER 5. SPECIAL TOPICS

370

This is a Lie group. For instance,

JoR" = CL(n;R).

The group JoR" acts naturally on J0 1Z on the right by the law [9Z]s - [9J0 = [9z o 91'.,

[9Z)'.' E Jo Z,

[9[0 k E

jokRr.

PROPOSITION 5.3.4. The natural surjection J0 1Z -' Z is a principal bundle with the structure group JOkRn ([1551, p.56). 0

For instance, the jet bundle Jo Z - Z is exactly the principal linear frame bundle LZ with the structure group CL(n;R). Its elements are linear frames on the manifold Z. Similarly, one can think of elements of the bundle JO *Z Z as being k-order frames on the manifold Z.

Remark 5.3.8. Cartan distribution. Let us consider briefly the Cartan distributions C(Jnk,Z) on the jet manifolds of jets of m - codimensional submanifolds of a manifold Z . Their particular case are the Cartan distributions CJkY on the jet manifolds JkY of sections of a fibred manifolds Y - X that we have considered above. Integral submanifolds of C(J,kZ) play the role of classical solutions of differential equations defined on a manifold Z.

The contact 1-forms on the jet manifold JcY of sections of a fibred manifold X have been introduced by means of the canonical splitting (5.1.20) of the pull-back irk+"'T*JkY (see Definition 5.1.6). In the case of jets of submanifolds, there is no such a canonical splitting. Y

Civen a coordinate chart (5.3.8) of the k-order jet bundle submanifolds of a manifold Z, let us consider the which is generated by the local forms 0A = dyn - ya+Adxa,

of m-codimensional of local 1-forms

I A 1< k.

The elements of this module will be called the local contact forms. LEMMA 5.3.5. Transition functions between the coordinate charts (5.3.8) of the jet send local contact forms onto local contact forms. 0 manifold

5.3. JETS OF SUBMANIFOLDS

371

Proof. It suffices to check this fact in the case of the transition functions (5.3.14) and (5.3.12). The first case is obvious since (5.3.14) are fibred coordinate transformations. The second case is verified by the following direct calculations. We have - l'l'1+Adxl = dUA - IIA+edg* - 9A+1dU' _

6n = drA -1/a+Ad 1E

I

MEd

E+

- 1-b11d-dxi dyl =

Adx° + 2A gE

JEI_JAI

E

idxl -

(

- lli dam') MA)de

di g1.

bt dz

QED Building on this Lemma, we can give the following definition of contact forms on the jet bundle J,' ,Z. DEFINITION 5.3.6. A 1-form on the jet bundle JmZ is called a contact 1-form if it is a local contact 1-form with respect to any coordinate chart (5.3.8) on JmZ. Let of exterior forms on us denote by O(J,k,Z) the ideal of the Z-graded algebra JmZ which is generated by the contact 1-forms. This ideal comprises the exterior products of contact 1-forms and arbitrary exterior forms on JmZ. Its elements are called the contact forms on JmZ.

In particular, let us consider the k-order jet bundle manifolds of the manifold Y and the injection (5.3.2):

of n-dimensional sub-

JkldY:JkY -This is an imbedding onto a dense open set in JmY. Let O(JknY) be the ideal of contact forms on J,kAY. Then the pull-backs on JkY of contact forms on are obviously the contact forms on JkY which generate the ideal Ck of contact forms on JkY It is easily observed that the pull-back pkt+" 9 on the jet bundle Jm+'Z of every contact form on the jet bundle JknZ is also a contact form. Hence, we have the direct system

L o(JJZ) ...

O(JJZ)

CHAPTER 5. SPECIAL TOPICS

372

The ideal

defines the smooth distribution

of dimension

(n+k- 1)! k!(n-1)! in the tangent bundle TJmZ which is annihilated by O(JknZ). DEFINITION 5.3.7. The distribution C(J,kZ) is called the Cartan distribution. It is given in coordinate form by the equalities .JOA=O,

IAI
for all elements of this distribution. 0 The ideal O(Jk,Z) of contact forms fails to be a differential ideal, that is,

d(O(J'Z)) it O(J' Z) because dOA, I A 1= k - 1, are not contact forms. Hence, the Cartan distribution is not an involutive distribution, and therefore it does not satisfy the conditions of the Frobenius theorem. It follows that the Cartan distribution does not possess integral manifolds of maximal dimension in general. At the same time, there are n-dimensional integral manifolds of the Cartan distribution as follows. Let

is:S --+Z be an n-dimensional imbedded submanifold of Z. There exists the imbedding

Jkis : S -

(5.3.18)

given by the coordinate expression yA' o J*is(z) = va+E([SI.) = BAy ([S[:),

A I<_ k,

I E 1< k.

Its image JSk is said to be the k-order jet prolongation of a submanifold S. It is easy to see that the pull-back Jkis'O on JSk of any contact form 0 on J.' Z vanishes, and so JSk is an n-dimensional integral manifold of called the k-order Cartan manifold. It follows that the image of the tangent morphism

TJkis : TS

TJk,Z

5.4. INFINITE ORDER JETS

373

to the morphism (5.3.18) belongs to the Cartan distribution C(J4Z). This is called the Cartan subspace. Similar to the Cartan subspaces of the Cartan distribution CJkY on the jet manifolds JkY, the Cartan subspace of the tangent space TgJJZ to the jet manifold JmZ at a point q E Jk.Z is the linear envelope of the tangent spaces TQJSI to all k-order Cartan manifolds through q.

5.4

Infinite order jets

As was mentioned above, several constructions in the calculus of jets of sections of a fibred manifold Y - X get completeness if the limit of the inverse system (5.1.9) of algebras of projectable vector fields and the limit of the direct system (5.1.11) of modules of exterior forms on finite order jet manifolds J'Y are considered. Intuitively, one can think of elements of these limits as being the objects defined on the projective limit of the inverse system

...J'-'Y'x'-1

-01

1'Y

...

(5.4.1)

of finite order jet manifolds J'Y. By a projective limit of the inverse system (5.4.1) is meant a set J°°Y which obeys the conditions [136]:

there exist surjections

ir°° : J°°Y - X,

Ira : J°°Y - Y and 7r : J°°Y

for any k;

the diagrams J°°Y

Jky

xk r

J'Y

are commutative for any k and r < k.

JkY; (5.4.2)

CHAPTER 5. SPECIAL TOPICS

374

The projective limit exists. It is called the infinite order jet space. This space consists of those elements

(... , qi, ... , qj, ...),

qi E J'Y,

gj E J1Y,

of the Cartesian product JkY k

which satisfy the relations

qi=7r(gj),

for allj>i. Thus, elements of the infinite order jet space J°°Y really represent 00-jets j.18 of local sections of Y -i X. These sections belong to the same jet j. 's if and only if their Taylor series at a point x E X coincide with each other. Remark 5.4.1. It is dear that JODY is the projective limit of the inverse subsystem of (5.4.1) which starts from any finite order J'Y. For the sake of simplicity, we will denote sometimes r = 0 for Y and r = 0 for X.

The set J°°Y is provided with the weakest topology such that the surjections (5.4.2) are continuous. The base of open sets of this topology in J°°Y consists of pre-images of open subsets of JkY, k = 0,..., under the mappings (5.4.2). This topology is paracompact and admits smooth partitions of unity The space J°°Y can be also provided with some kind of a manifold structure, but it fails to be a well-behaved manifold (174, 1751. At the same time, a wide class of differentiable objects on J°°Y can be introduced [13, 1741. The procedure is the following.

At first, smooth functions on J°°Y are defined. A function

f:J°°Y-.R is said to be of class C°° if, for every q E JY, there exists a neighbourhood U of q and a smooth function

f(") : JkY - R for some k such that

f I U= P) o Irk Iu

5.4. INFINITE ORDER JETS

375

Vector fields on JY are introduced as derivations of smooth functions. Then, exterior forms on J°°Y are defined as the objects dual to vector fields. The above-mentioned limits of the inverse system (5.1.9) of Lie algebras of projectable vector fields and of the direct system (5.1.11) of modules of exterior forms on finite order jet manifolds J'Y belong to the classes of vector fields and exterior forms defined on the infinite order jet space. In this sense, elements of these limits are indeed the objects on J°°Y. The projective limit P. of the inverse system (5.1.9) exists. Its definition is a repetition of that of J°°Y. This is a Lie algebra so that the surjections

Tirk are Lie algebra morphisms which constitute the commutative diagrams

P. Txk f 1 Txr Pk

A. Tx,k

Let us consider the direct system (5.1.11) of R-modules Ok of exterior forms on finite order jet manifolds JkY, i.e., (5.4.3)

The limit $D Q of this direct system, by definition, obeys the following conditions [136):

for any r, there exists an injection

0; r

the diagrams

are commutative for any r and k < r.

CHAPTER 5. SPECIAL TOPICS

376

Such a direct limit exists. This is the R-module which is the quotient of the direct sum Dk k

with respect to identification of the pull-back forms ir,.

c#=ir *?,

0E0*,

oEOk

if 46 = Ak a. In other words, 0;, consists of all the exterior forms on finite order jet manifolds module pull-back identification.

Remark 5.4.2. Obviously, DL is the direct limit of the direct subsystem of (5.4.3) which starts from any finite order r.

Further on, we will denote the image of Dr in 0 by flr and the elements rr,'# of Oa, simply by 0. Remark 5.4.3. The differential calculus in Oa, is formulated in terms of filtered operators (109]. The R-module D:. possesses the structure of the filtered module as follows. Let

us consider the direct system of the commutative R-algebras of smooth functions on the jet manifolds FY:

170 ' 17° '-.... 00

D°(X)

... ,

Its direct limit D;, exists. This is the R-algebra filtered by the R-algebras irk 'D4:

D;,=UD4, k

Then, 0 D has the filtered D°-module structure given by the D4-submodules Ok of 0.

DEFINITION 5.4.1. An endomorphism A of Oa, is called a filtered morphism if there exists i e N such that 0 I Dk is the homomorphism of Dk into Dk+i over the injection 00 - D4+j for all k. In particular, every direct system of endomorphisms {ryk} of DA; such that

40 0 7'='Yfo4

5.4. INFINITE ORDER JETS

377

for all j > i has the direct limit y in filtered endomorphisms of Da,. If all yk are monomorphisms [epimorphisms], then y is also a monomorphism [epimorphism]. As a consequence, the following theorem holds ([1361, p.391). THEOREM 5.4.2. The operation of taking homology groups of a chain and cochain complexes commutes with the passage to the direct limit. 0

The operations of the exterior product A and the exterior differential d have the direct limits on fl;,. We will denote them by the same symbols A and d, respectively.

They provide D.. with the structure of a Z-graded algebra:

fla, _ ® D-O,,, "Wo

where D'O are the direct limits of the direct systems

flmx

170

"° . OM -... Dm'_+ fl

+1 - .. .

of R-modules flm of exterior m-forms on r-order jet manifolds FY. One calls elements of D' the exterior m-forms on the infinite order jet space. The familiar relations of a Z-graded algebra take place:

0' A D C O d:O00 -0'00', dod=0. Remark 5.4.4. As a consequence, we have the following cochain complex of exterior forms on the infinite order jet space

0-In

d+

,Kerd--'n4O0

(5.4.4)

.

Let us consider the cohomology group Hm(D:,) of this complex. By virtue of Theorem 5.4.2, this is isomorphic to the direct limit of the direct system of homomorphisms

H'(O) -. H'(D;+l) - .. . of the oohomology groups H'"(fl;) of the cochain complexes

0 O+KerdDor

O'

--

d+fl; --- 0,

l=dimJ'Y.

CHAPTER 5. SPECIAL TOPICS

378

The cohomology groups H'(£),,), m > 0, of such a cochain complex coincide with of the De Rham complex of the sheaves of germs the cohomology groups of exterior forms on the jet manifold JTY. The following assertion completes our consideration of oohomology of the complex (5.4.4).

PROPOSITION 5.4.3. The De Rham cohomology H'(J'Y) of jet manifolds JrY coincide with the De Rham cohomology HO(Y) of the fibred manifold Y -+ X [131.

0 Proof. It follows from the fact that the jet bundles XY J "Y are affine bundles which have the same De Rham cohomology than its base. QED It follows that the cohomology groups Hm(fla,), m > 0, of the cochain complex (5.4.4) coincide with the De Rham cohomology groups Hm(Y) of Y X. Let us recall the following natural splittings on infinite order jets. There is the canonical decomposition Ilea,=0000's ®0oc-'®...®0oa'oJ

(5.4.5)

where elements of are called k-contact and (s - k)-basic forms. We denote by ht the k-contact projection

hk : 0m - Ok,m-k,

k < m.

(5.4.6)

I It follows that, in the calculus in infinite order jets, any exterior form on a finite order jet manifold can be expanded in horizontal and contact forms only. I Accordingly, the exterior differential on ilo, is decomposed into the sum

d=dy+dv

(5.4.7)

of the horizontal differential dH and the vertical differential dv. These are defined as follows:

d : ilk,a 00 ka dH :17 k,a

dv : i7

I)k+',a ®ilk,a+1 00

00

k,a def

k,a+1

Ooo

k+1,'

oo

,

dH I il

k,a dd

dv I O00

k,a

prs o d 1000

k,a

pr, o d I O

5.4. INFINITE ORDER JETS

379

for all k, s. The operators dm and dv obey the standard relations dH(4iAa) = dm(o) A a + (-1)1010 AdH(a), dv(6 A a) = dv(40) A o + A dv(a),

46,a e boo,

and they are the homology operators:

dHodj=0,

dvodv=0,

dv o dH + dH o dv = 0.

(5.4.8)

We also have

hood=dHoho. Though we do not introduce a manifold structure on the infinite order jet space,

the elements of the direct limit 0 can be considered in the coordinate form as follows.

Let U be the domain of a fibred coordinate chart (U; x", y') of a fibred manifold

Y - X. Let Ur = (7ro')-'(U)

be the domain of the corresponding coordinate chart of the bundle JrY - Y. One can repeat the above procedure for the modules ilu, of the exterior forms defined on Ur and obtain their direct limit 0 (U). PROPOSITION 5.4.4. There exists the R-module homomorphism of 0a, to 0:.(U).

0 Proof. For every r, we have the R-module homomorphism =u.

: flr --+ 0u,

which sends every exterior form on JY onto its pull-back on U,.. Then there exists the above-mentioned homomorphism

iu' : 0* -0:(U) so that the diagram

Or ' t7 D'.

o0

iu. 1

0u.. - V". 000(U)

CHAPTER 5. SPECIAL TOPICS

380

commutes for any order r ([136], p.386).

QED

Elements of Oa,(U) can be written in the familiar coordinate form. The basic 1-forms &T-" and the contact 1-forms 9E = d y E - yE+,,dx-%,

0
.

constitute the set of generating elements of the filtered 00.(U)-module fl:.(U). In particular, we have the coordinate expression (3.1.3) for the horizontal differential dH = dxa A dA(0),

dHO = dx" A da(O),

0 E Ou,

where

d,,=d,-

0
(5.4.9)

are the formal total derivatives in infinite order jets.

Remark 5.4.5. Though the sum in the expression (5.4.9) is taken with respect to an infinite number of collections A, the operator (5.4.9) is well behaved since, given any form 0 E flu, the expression d,,(4) always involves a finite number of the terms O' only.

The reader is referred to Sections 3.1 and 5.1 for the explicit expressions for operators dy, dv and d,,. Given an atlas ((U; x", yi) } of fibred coordinates of Y - X, let us consider the module of exterior forms on infinite order jets Da,(U) for every coordinate chart (U; xa, y`) of this atlas. We have the following corollary of Proposition 5.4.4. COROLLARY 5.4.5. Every element 0 of the module Oo, is uniquely defined by the collection of elements {4u } of the modules Oa,(U), together with the corresponding

rules of coordinate transformations. 0 Further on, we will utilize the coordinate expressions for exterior forms on infinite order jets, without specifying the coordinate domain U. One can say that an object given by a coordinate expression as an element of each module Oa,(U) is also globally defined if its coordinate form is preserved under the corresponding coordinate transformations.

5.5. VARIA77ONAL BICOMPLEX

381

5.5 Variational bicomplex In Section 3.3, the application of the variational sequence to Lagrangian formalism has been discussed. Here, we obtain explicit expressions for the corresponding projection and variational operators. We restrict our consideration to the variational sequence in the calculus in infinite order jets [42, 180). In comparison with the finite order variational sequence [113, 114, 115, 1851, the essential simplification is that, if the order of jets is not bounded, there is decomposition (5.4.5) of exterior forms on jet manifolds into contact and basic forms. At the same time, we are allowed not to fix the order of the objects (e.g., Lagrangian densities and Euler-Lagrange operators) which we obtain from the analysis of the infinite order variational sequence. Building on the homology properties (5.4.8) of the horizontal and vertical differentials dy and dv, one can construct the following commutative diagram [42, 175, 180]: 0

0

1

1

R

R

1

0. ilc(X)

I'

1

JDO,O

17n-1X di

0 - 0_(X)

O'a 00

-

...

(-1)kdu!

dHI

(5.5.1)

du11)kdH

- ...

dv

dv

v

ilOn-1 dv

*°°.

0

1

eve... dvi

00

d1

di

0

dHI

Do,_

-1 0

...

ilk,n-1 dv _ _, .. . 00

(-1)kdn 4

k,n ± .. .

00

7k1

Eo

Ek

I

I

0

0

where, by definition, Ek = Oko0'n/dyi7;,'n-' 0

(5.5.2)

CHAPTER 5. SPECIAL TOPICS

382

Since all columns and rows of this diagram are complexes, it is called a bioomplex.

The quotient Ek, k > 0, (5.5.2) in the bottom row of

PROPOSITION 5.5.1.

the bicomplex (5.5.1) is isomorphic to the complement rk(D'-") of the subspace

dH(D',"-') C D". o Proof. Later, we will construct the projection operator rk such that 0, 0 E

0 = r,O + 40,

(5.5.3)

for any 0 E D;;", and we show that rk o dH = 0.

QED

Let us consider the operator

rk= krlD00

(5.5.4)

E(-1)t"IB' A [dA(e; J0)],

mED

,

(5.5.5)

where the sum is taken over all multi-indices A of length 0
due to the relation 0a,a, Aw = -dy(9`a,...a._,) Awa.,

(5.5.6)

for all basic elements O of the decomposition 0£1... k4COE,

A... A9" Aw

(5.5.7)

of forms 0 E O . It follows immediately that the relation (5.5.3) takes place. It is easily observed that the relation (5.5.6) is maintained under coordinate transformations. Hence, the operator (5.5.5) is globally defined on exterior forms from

00111".

Given an exterior form 0 (5.5.7), the operator rk (5.5.4) takes the explicit form k

.ml

Er IG

r;....-4i>

A ... A BE-'+E.-, A e.--.V'+E.f. A ... A B`"`+Ek A W,

5.5. VARIATIONAL BICOMPLEX

383

where the second sum is taken over all of partitions

Er=E

051E:I5iErl,

of the collection E. LEMMA 5.5.2. We have 't(1 E Qoon-

(1-1. o dH)(') = 0,

(5.5.8)

13

Proof. Given a form il

...Eke£11

W = W it ...ik

A ... A

AWo,

we find

dn,P=(-1)kId«+G«'..:ik0

A ... A gt

A...A9i +... k

A ... A gik + ...] A W

0

Let us calculate (rk o dH)(ik). The term To of dHVt is brought into 11

1

rk(To)=E( k Oil +E, A ... A9 -',

A0 +I

A...

0w+Ek A

For every term of this decomposition characterized by the partition

Er=

+...+-r

of the collection E there exists the corresponding term in rk(Tr) where A = a+ in the expression (5.5.4) and

Er+cc

+

+

such that their sum is equal to zero since

r

CHAPTER 5. SPECIAL TOPICS

384

For every term in Tk(',) with A = Ef,&, and

there exists the corresponding term in Tk(lkt) with A = a + Ej and

such that their sum is equal to zero. As a consequence, the relation (5.5.8) holds. QED

The relations (5.5.3) and (5.5.8) lead to the equality Th(0) = (7k o 7-k) (0) + (Tk o dH)(0) = (Tk o Tk)(0),

that is, the operator Tk (5.5.4) on £) ," is really a projection operator. Thus, Proposition 5.5.1 is proved. Let us consider the short exact sequence

C"`+k __+00O k -+ Ek ' 0,

(5.5.9)

where

ek=Tkohk is the projection operator and C"+k = Kerek. By virtue of Proposition 5.5.1, the exact sequence (5.5.9) is simple since we have the decomposition

Dk," =&® 0

dHO'"-1

00

LEMMA 5.5.3. We have d(C n+k) c C,"+k+1

0 Proof. If o = do E

S)k+1,",

0 E C"+k,

then

o = dH', 0 E

SZk+I,n-1'

5.5. VARIATIONAL BICOMPLEX

385

or

'PE D'kn-1-

o = (dv o dH)(?P) _ (-dl o dy)(0),

As a consequence, we can define the following sequence. 0

--

1

1

0

0

1

1

1

1

1

0

C°-1 d,

C- d + Cn+l -. Ca+2 - .. . 1

^-1 a 0-., Dn+1

0o

D-+2

1.1

dp

1

ell

DO,`- E 00

1

(5.5.10)

00

00

-'

C21

_'...

I

I

I

I

0

0

0

0

Its rows are complexes. Since its columns are simple sequences and ER_1 c 011-1,4 , we find

ek=rkody. It is easy to verify that (Fk+1 o ek)(0) = (rk+i o dv o rk o dv)(#) = (m+l o dv)(dv.0 - d,,4') = 0,

0 E Ek-1 C Dk-1,

0E

Dkn-1.

Remark 5.5.1. Since columns of the diagram (5.5.10) are exact sequences, we have the following exact sequence of the cohomology groups of its rows

-- H(D") _-- H(C") Q124], p.47).

H(E1) - H(O;;)

.. .

CHAPTER 5. SPECIAL TOPICS

386

5.6

Geometry of differential equations

This Section is devoted to the geometric analysis of differential equations and the notion of their formal integrability (26, 109, 152, 153]. In the next Section, the formal integrability criteria will be we be applied to the Yang-Mills and EinsteinYang-Mills equations of field theory. Let us recall some general notions on differential equations. DEFINITION 5.6.1. Let Z be an (m + n)-dimensional manifold. A system of k-order

partial differential equations in n variables on Z is defined to be a closed smooth submanifold f of the k-order jet bundle J,kaZ of n-dimensional submanifolds of Z. In brief, we will call a system of partial differential equations simply a differential equation. Unless otherwise stated, by a solution of a differential equation is meant a classical solution in accordance with the following definition.

DEFINITION 5.6.2. A classical solution of a differential equation is is said to be an

ndimensional k-order Cartan manifold B of the Cartan distribution on Jk Z, that is, the k-order jet prolongation B = JSk of an n-dimensional submanifold S of Z such that JSk C C Given a point q E !, there always exist integral manifolds JSk passing through q. If such an integral manifold belongs to !, it is obviously tangent to the equation 0- at q with any finite order. Therefore, the necessary (but not sufficient) condition for a differential equation iE to possess a classical solution through a point q E E is existence of an integral manifold JS' tangent to 19 at q with any finite order.

Remark 5.6.1. In this case, it may be hoped that, if all the objects under consideration are analytic, there exists an analytic solution of the equation it in a neighbourhood of the point q. These speculations lead to the notion of formal integrability of differential equations. To establish the criteria of formal integrability, the higher order jet prolongation of differential equations must be examined. Given a k-order differential equation E C J1Z, let us construct its s-order proSince there are not simple relations between longation into the jet bundle

387

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

the repeated jet bundles JJJ,knZ and the higher order jet bundles J, +'Z, one cannot perform this prolongation directly. However, if there exists a k-order Cartan manifold JSk tangent to is at a point q E e with an order s, the s-order prolongation of

JSk to the (k + s)-order Cartan manifold JSk+' sends the point q E e C J ,Z into DEFINITION 5.6.3. Let e be a k-order differential equation on a manifold Z. The s-order jet prolongation of the differential equation it, by definition, is the subset

01 C (Pk+')-'(e) C which comprises the points (S],kn+, such that JSk is tangent to

at pk

with

order > s. O If a differential equation 4E has a classical solution through a point q E 1E, this point gives rise to an element of every finite order jet prolongation I!(') of the differential equation . It follows that the necessary condition for a differential equation

i to admit a solution through everyone of its point is that the mappings Pk' I e(') : $(') --+ e.

(5.6.1)

are surjections.

The mappings (5.6.1), however, are neither surjective nor injective in general. Moreover, it may happen that the set (!(') is not a manifold. Remark 5.6.2. If PE(') is a smooth submanifold of the jet bundle Jk,,+'Z, then

and the mapping (5.6.1) is a manifold morphism. If this morphism is a surjection, then the submanifold is a (k + s)-order differential equation, by virtue of Definition 5.6.1. Obviously, this differential equation reduces to the lower order differential equation (I in the sense that there is bijective correspondence between classical solutions of the equation ! and its s-order jet prolongation (!('). If additionally every tangent vector to the differential equation a is tangent to some classical solution of !, then the mapping (5.6.1) is a submersion. The above-mentioned construction finds easier and more direct applications in the case of differential equations in n variables if these variables span a given manifold X.

CHAPTER 5. SPECIAL TOPICS

388

DEFINITION 5.6.4. Given a fibred manifold Y -+ X, a closed smooth fibred submanifold 0- C JkY over X is called a k-order differential equation over X. 0

Classical solutions of such a differential equation are integral sections of the X. Taking into consideration only these Cartan submanifolds fibration JkY of JkY, we can appeal to Definition 5.6.3 in order to construct the s-order jet prolongation of the differential equation a over X into the (k+s)-order jet manifold Jk+'Y of Y [109]. However, more explicit description of this prolongation can be obtained as follows [152].

Remark 5.6.3. Given a fibred manifold 7r : Y -. X, let try

:N

C-.+

X

be a submanifold [imbedded submanifold] of X. Then, (irk)-I (N) is a submanifold [imbedded submanifold) of JkY. Moreover, let us denote by

7rN:YN--'N the restriction jN'Y of Y over N. We have the natural surjection

IN : (irk)-`(N) -' kYN, for each section s of Y -, X.

INVxs) = jx(s o IN),

Given a k-order differential equation i< C JkY over X, let us consider the repeated jet manifold ok : J'JkY -+ JkY.

(5.6.2)

One can think of (oho)-I (e) C J'Jky

as being the s-order jet prolongation of the equation ! (in the spirit of Definition 5.6.3) by means of sections of the fibration (5.6.2). In accordance with the abovementioned Definition 5.6.3, the s-order jet prolongation of e, however, should be performed by means of sections of the fibration Jk+'Y - X, but not (5.6.2). DEFINITION 5.6.5. The s-order jet prolongation of the differential equation E over

X is the subset

I ei'i = (d)-i(e)n Jk+.Y.

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

389

If 0) is a smooth submanifold, then (e(+))(r) = LF(&+r)

0 A differential equation ! is called regular if all finite-order jet prolongations i(') of (£ are also differential equations. The above considerations motivate the following notion of formal integrability of differential equations over X.

DEFINITION 5.6.6. A regular k-order differential equation 0- is called formally integrable if the morphisms

4+.+l I (!(,+i) : e(,+I)

e(+),

s>0,

are fibred manifolds. 0 Example 5.6.4. Let a fibred manifold Y -a X be endowed with fibred coordinates (xA, y') and its k-order jet manifold JkY be provided correspondingly with the coordinates (x", yA) (5.1.2). Let C JkY be a regular k-order differential equation. If it is described locally by the system of equations E"(xa, ysA) = 0,

A = 1, ... , codim(!,

(5.6.3)

then its s-order jet prolongation 0) is given by the system of equations

E"=0,

d d,,E" = 0,

(5.6.4)

Let us bear in mind this Example in order to understand better some constructions of the theory of differentials equations given below.

In particular, it is easily seen that, ifs > 1, the equations (5.6.4) are linear with respect to the highest order jet variables y& I A I= k + s. Equations of this type

CHAPTER 5. SPECIAL TOPICS

390

are called quasilinear equations. They are almost completely characterized by their symbols.

DEFINITION 5.6.7. The symbol of a differential equation e C JAY at a point q E JkY is the vector space GQkI = Toe n Volrk_I,

(5.6.5)

where Vr k_1 is the vertical tangent bundle of the affine bundle JAY

Jk_'Y. 0

Remark 5.6.5. In particular, if (5.6.6)

Csg/k) = V91rk_1

C JAY, one can say that this equation is of order < k - 1. Indeed, the equality (5.6.6) takes place if and only if all functions (5.6.3) are independent of the highest order jet variables yA, I A 1= k. for every point q of the k-order differential equation

Recall that the fibration JkY

J"-'Y is an affine bundle modelled over the

vector bundle (5.1.4) and, therefore, its vertical tangent bundle admits the canonical vertical splitting

V7rk_I = JAY X (VY®SkX) JA-'Y

(5.6.7)

Y

where SkX denotes the symmetric tensor bundle (5.1.23).

DEFINITION 5.6.8. The differential equation I C JkY is called quasilinear if its symbols G(,k) are the same for all points of the same fibre of the fibre bundle JkY

Jk-1Y o A glance at Example 5.6.4 leads to the following assertion. PROPOSITION 5.6.9.

If 0- is a regular differential equation, its finite order jet

prolongations are quasilinear differential equations ([109], p.163). 0

Henceforth, we will consider differential equations associated with differential operators.

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

391

It is intuitively clear that any differential operator is constructed from linear differential operators. The isomorphism (5.2.11) shows that the representative object of k-order linear differential operators acting on a module P is the module 31(P) of k-order jets of P. Then the natural generalization of the notion of linear differential operators to "nonlinear" differential operators or simply to differential operators is the composition:

of the k-order jet prolongation functor of P to 3k(P) which is a k-order linear differential operator, and of some mapping of Zik(P) into Q which is not a module homomorphism.

Usually, one considers differential operators in the case of geometric modules when P and Q are the modules of sections of vector bundles Y -- X and E -+ X'. However, fibre bundles in physical applications, e.g., in the gauge theory are not necessarily vector bundles. Therefore, we consider the case of a fibred manifold Y - X possessing the vertical splitting

VY=YxY x

(5.6.8)

and of a vector bundle E X, though some basic notions can be formulated on a more general level, when Y and E are arbitrary fibred manifolds. Recall that, in accordance with Definition 3.3.2, a k-order E-valued differential operator on Y X is defined to be a section of the pull-back fibred manifold P(k) :

Ey = E

x

JkY

JkY.

(5.6.9)

Given fibred coordinates (x", y`) of Y and (x", y) of E, the pull-back (5.6.9) is provided with the coordinates

(xa,A, V), 0
(5.6.10)

There is one-to-one correspondence between the sections

E:JkY-+Ey,

b

(5.6.11)

of the fibred manifold (5.6.9) and the fibred morphisms over X

+ : JkY

E.

(5.6.12)

CHAPTER 5. SPECIAL TOPICS

392

As a consequence, differential operators on Y --+ X are equivalently defined as fibred

morphism JkY -+ E over X. Every such a morphism 4i (5.6.12) sends a section s of Y - X onto the section 4 o Jks of E X. The mapping

A#

I - 4 0 Y's,

(5.6.13)

E

U (x) =

E I< k,

is the standard form of differential operators.

Note that, as was manifested above, differential operators (5.6.13) factorize through the linear differential operator Jk : s " Jks and some mapping -9 -o 4 o 3 of the set of sections of the fibred manifold JkY - X to the set of sections of the fibred manifold E -+ X. DEFINITION 5.6.10. Let A be a E-valued s-order differential operator (5.6.13) on the fibred manifold Y -+ X. The mapping Oak)

: s -o Jk(A(s))

(5.6.14)

is called the k-order prolongation of the differential operator A. One can think of the k-order prolongation (5.6.14) as being a J'E-valued (s + k)order differential operator on Y. This is given by the k-order jet prolongation of the morphism $A: O(p = Jk4 A o 0k., A(k) : a'-- Jk4'p o J'+ks,

(5.6.15)

where ok, is the monomorphism (5.1.5). Accordingly, the prolonged operator Y') is represented by a section of the pull-back (JkE)Y'+k) -e J'+kY

of JkE over J'+kY. At the same time, the following construction enables us to describe the prolongation (5.6.14) as a section of the k-order jet manifold

Jk r+k

J'+kY

of the pull-back Wk of E over J'+kY

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

393

Given fibred manifolds Y - X and E - X, let E'.. be the pull-back (5.6.9) of E over JkY and J'EE the r-order jet manifolds of this pull-back. Given the fibred coordinates (5.6.10), this jet manifold is provided with the adapted coordinates 0
(x'',>

0 <_I E1 I<_ k.

(5.6.16)

Lets be a section of the fibration JkY - X. Then there exist a natural isomorphism of the pull-back s Ehk. - X to the fibred manifold E -. X and the corresponding isomorphisms of the r-order jet manifolds J'(s Ey) to the jet manifolds J'E. In accordance with Remark 5.6.3, we have the natural surjection J'(s Eky)

(P(k)')-'

and the corresponding surjection

rE,

ss: a) VI(z)oJ = (U A

(5.6.17)

of (8(Z)),

where 9 are sections of Ey - JkY. Example 5.6.6. We first consider the particular case E = Y. Let

Yy=YXJkY be the pull-back (5.6.9) of Y over JkY and P(k) : JkYy -y Jky

(5.6.18)

be the k-order jet manifold of this pull-back. There exists a section rk of the fibration (5.6.18) which takes the coordinate expression rq'A o r

kI) = {xu: VAW.'3),

if

q > 0,

if q=01

(5.6.19)

with respect to the adapted coordinates (5.6.16) of the jet manifold JkYy. In brief, one can say that the non-zero coordinates of rk read A G rk = UA.

Note that the section rk given by the coordinate expression (5.6.19) is globally defined since the transition functions of the coordinates y'i (5.6.16) are independent

CHAPTER 5. SPECIAL TOPICS

394

of any coordinates y ,. In should be emphasized that the section rk (5.6.19) is a unique one of the fibration (5.6.18) such that

4orko8=-g

(5.6.20)

for any section 3 of J*Y - X. Indeed, the equality (5.6.20) takes place by virtue of the expressions (5.6.19) and (5.6.17). Since JkY - J11-'Y is an affine bundle modelled over the vector bundle (5.1.4), one can generalize the proof of the uniqueness condition in the case of a vector bundle Y -. X (see [1091, p. 62) to an arbitrary fibred manifold Y in a straightforward manner. Let now Y - X be a vector bundle. In this case, the fibration (5.6.18) is also a vector bundle. We consider the Spencer morphism Si (5.1.27) of the R-module 3k(Yk) of sections of the bundle (5.6.18) into the R-module 3k-'(YY) ®O'(J'Y) of sections of the bundle

Jk''YY ®T'JkY

JkY.

(5.6.21)

JkY

Given the section rk of the bundle (5.6.18), its image

ek = Sl (rk)

(5.6.22)

by the morphism (5.1.27) is called the universal Cartan element. It takes the coordinate form

ek=[dyfn-ye+adx1®8:,

IAI
with respect to the coordinates (5.1.28) of the vector bundle (5.6.21).

Let us turn now to the general case of differential operators on sections of a fibred manifold Y -' X with values into the set of sections of a fibred manifold E X. Let 0 be a s-order differential operator in the standard form (5.6.13). THEOREM 5.6.11. There is a unique section rk(0) of the fibration

P(k+')k : Jk rk _, Jk++Y

(5.6.23)

which satisfies the relation iT o rk(1) o 3 = 0(k)(3)

for any section 3 of the fibration J'+kY

(5.6.24)

X. 0

395

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

Proof. With respect to the coordinates (5.6.16) (where I A I +q 5 k, I Er I5 s + k) on the jet manifold JkEy ', the section rk(O) is given by the the following non-zero coordinates A=(,\,... a,
I E I<- s,

U A o rk(O) = d,,1... da.Sn(x", Y ),

(5.6.25)

The uniqueness of the section (5.3.9) satisfying the condition (5.6.20) is proved just QED as the uniqueness of the section rk (5.6.19) in Example 5.6.6.

Let E -' X be a vector bundle. Just as for the definition of the universal Cartan element (5.6.22), let us consider the Spencer morphism S1 (5.1.27) of the Rmodule 3k(Ey+k) of sections of the bundle (5.6.23) into the R-module 3k-1(Ey k) )1(J*+kY) of sections of the fibre bundle

Jk-'EE Let A be a s-order differential operator (5.6.13) and rk(O) the associated section (5.6.24). Put

6k(O) = S; (rk(A)),

6k(p) _ d(dA, ... da.So) -dada, ... da,Eodxa ®8n,

I A I<- k -1.

Example 5.6.7. Comparing the expression (5.6.25) with the expression (5.6.19), we observe that rk = rk(Id Y).

Example 5.6.8. Let 0 be a (k - 1)-order differential operator on sections of a fibred manifold Y - X with values into the ring O°(X), that is,

E=RxX and

IEI
-1X+Edx-'), E

0
(5.6.26)

CHAPTER 5. SPECIAL TOPICS

396

is a section of the jet bundle

J'(R x JkY) X

JkY.

(5.6.27)

The bundle (5.6.27) is precisely the affine cotangent bundle of the jet manifold JkY,

and the Cartan element 91(0) (5.6.26) is a Cartan 1-form on JkY. Note that, building on Example 5.6.8, one can suggest a different definition of the contact forms which is equivalent to Definition 5.1.6. DEFINITION 5.6.12. Let us consider the module of exterior 1-forms on the jet manifold JkY over the ring of smooth functions on JkY which is generated by the forms 91(0) (5.6.26) for all (k -1)-order differential operator on sections of a fibred

manifold Y - X with values into the ring O°(X) of the smooth functions on X. Elements of this module are called contact 1-forms on JkY. O Accordingly, the Cartan distribution on the jet manifold JkY of a fibred manifold Y -+ X is defined as follows. DEFINITION 5.6.13. The Cartan subspace or the Cartan planeCgJkY of the tangent space TgJkY at a point q E JkY, by definition, comprises the vectors v E TgJkY such that

vJ91(A) = 0 for all D°(X)-valued (k - 1)-order differential operators on Y. The distribution q i-4 CgJkY, by definition, is the Cartan distribution on JAY. O Now, let us define the symbol of a differential operator (5.6.12). We assume that

E -. X is a vector bundle and that the fibred manifold Y - X admits the vertical splitting (5.6.8). Remark 5.6.9. The pull-back fibred manifold (5.6.9) over JkY is also the pull-back JkY Ey = Ey ' JkxlY -

of the fibred manifold Ey ' - Jk-'Y onto JkY. It follows that every section 6 of the fibred manifold EY -+ JkY is a fibred morphism

l; : JkY

Eyk

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

397

over J1-1Y. Accordingly, every differential operator + (5.6.12) can be represented by a fibred morphism

k!

JkY -°+ IE

*

'k- 1

Jk'IY

(5.6.28)

x

over Irk-1.

Let 4) be a s-order differential operator (5.6.12). Let us consider the vertical tangent morphism

VI : V7r;_1 - VE

(5.6.29)

to the fibred morphism (5.6.28). Here, by V, ;_1 is meant the vertical tangent bundle

to the affine bundle J'Y - J''IY. The morphism (5.6.29) is brought into the form

V4): J'YJ x-'Y(VY®S'X) -VE Y due to the vertical splitting (5.6.7) of Va;_1. Then, in view of the vertical splitting (5.6.8) of VY and the canonical vertical splitting of VE, it leads to (5.6.30)

DEFINITION 5.6.14. The symbol a(4) of the differential operator 0 (5.6.12) is the morphism

o(O)=przoVO:J'Yx(Y®S'X)-E.

(5.6.31)

The restriction oa($) of the morphism (5.6.31) to {q} x V. ® SsX, q E J'Y, x = 7r'(q), is called the symbol of the differential operator 'F at a point q E JkY :

04(x) : {q} x (Y® ® S'X) - E_.

(5.6.32)

0 Let (xA,yiE) be the familiar coordinates of JkY, (x,',yE) the coordinates of S'X OF, and (xa,V) the coordinates of E. Then the symbol o(E,) (5.6.31) reads

V o+=Sa(X."YE),

IEISs,

E V 5CIO(x",y ). J!Iak

(5.6.33)

CHAPTER 5. SPECIAL TOPICS

398

DEFINITION 5.6.15. A differential operator 4' (5.6.11) is called quasilinear if oq(4') = 0'q1 ($)

for all points q and q' such that 4_1(q) = irk-' (d)A

In particular, the symbol (5.6.31) of a k-order quasilinear differential operator reduces to the linear bundle morphism

0(41) : J"-'Y x V 0 SkX -i E X x

(5.6.34)

of the vector bundle (5.1.4). It follows that the k-order quasilinear differential operator 41 is an affine bundle morphism of the affme bundle JkY -+ J"-'Y such that

(q) - +(q') = u(4')(q - 9') if ir_1(q) = irk-' (q').

(5.6.35)

It is easily seen that the differential operator 4 is quasilinear if and only if the derivatives

in the expression (5.6.33) are independent of the highest order jet coordinates y'E, I E 1= k. This, in turn, takes place if the differential operator Em (5.6.11) is linear in these coordinates. In particular, all finite order jet prolongations of a differential operator are quasilinear operators. Symbols of the jet prolongations of differential operators can be introduced in a similar way,. Given a s-order differential operator (5.6.12) 4', let 4'(k) be its k-order jet prolongation (5.6.15). This is a fibred morphism

J'+kY ± *:+:-1

!

JkE !

J,+k-'Y + _ Jk' E

Pk-,

(5.6.36)

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

399

over the (k - 1)-order jet prolongation 4ii"i of 41. Let V4 (k) be the vertical tangent map V4i(k) : V41+1-1

V4_1

(5.6.37)

of the fibred morphism 0(k) (5.6.36). Owing to the corresponding vertical splittings of Vw.'+k_ VY and Vifk_ the morphism (5.6.37) is brought into the form

V4(k) : J'+ky X(y®S'+kX) - JkEX(E®SkX). The symbol a(4 )) of the k-order jet prolongation 10) of a s-order differential operator 4) is the morphism o V4Vk) : J'+kY x

(Y®St+kX) -s E ® SkX.

In fact, it can be written in the form

Io(Vk)) : J'Y x(Y®S'+kX) -e E®SkX

(5.6.38)

because all finite order jet prolongations of a differential operator are quasilinear differential operators. With respect to the coordinates (xA, y'E) of JkY, (XI, V'E) of SkX ®Y, and (x-a, VA) of E ® SkX, the symbol a(4)i0) (5.6.38) reads

(5.6.39)

Accordingly, the symbol (5.6.38) at a point q E J'Y is

o9(4VI) : {q} x V. ® S= kX - E. ® S' X,

x = 7rk(q).

(5.6.40)

Using the Spencer sequence of tensor bundles (5.1.25) and the symbols of the

CHAPTER 5. SPECIAL TOPICS

400

operators (5.6.32) and (5.6.40), one can construct the commutative diagram 0

0

1

oo(ilk))

g:®5=+kX

Vs®S= k-'X0nX

I E=®S=X

E. ®S=-'X ®T=X (5.6.41)

E. ® S.k-X ®nT=X

V ®S= k mX ®

F=®S=X ®nT=X

E_ ®nT=X

I

I

0

0

The both columns of this diagram are exact Spencer sequences. Let us turn now to differential equations associated with differential operators which take values into vector bundles E - X.

Recall that the differential equation (!i associated with a k-order differential operator 45, by definition, is the kernel

ei = Ker4

(5.6.42)

of 4s with respect to the global zero section 0 of the bundle E -, X, provided this kernel is a fibred submanifold of JkY -+ X. This differential equation can be expressed by the equality

4 1 of = 0, where Ei is the section (5.6.11) of the bundle Ey -' JkY. This equality takes the coordinate form

E'i(xa,y) = 0.

i = 1,...,dimE.

(5.6.43)

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

401

For instance, it is readily observed that the differential equation f-a associated with a differential operator $ is quasilinear if and only if t is a quasilinear differential operator. PROPOSITION 5.6.16. If a s-order differential equation ee is associated with a

s-order differential operator +, then the k-order jet prolongation e() of eo is a differential equation associated with the the k-order jet prolongation AV (5.6.14) of the differential operator Om ([109], p.164). The submanifold iEk) C J'+kY can be defined by the condition (5.6.44)

rk(,&o) I*k) = 0

where rk(0+) is the section (5.6.24) of the bundle JkE"' - Jk+,Y from Theorem 5.6.11.

be given by the coordinate equalities (5.6.43). Let the differential equation Then, with a glance at the coordinate expression (5.6.25) for rk(A,), the condition (5.6.44) takes the coordinate form I da, ... dA,E.'(x, E) = 0,

A = (.\1... Ar
(5.6.45)

similar to the equations (5.6.4) in Example 5.6.4. In particular, it is readily verified that the finite order jet prolongations of the differential equation e# are quasilinear differential equations in accordance with Proposition 5.6.9. Now, let us consider the criteria of formal integrability of differential equations associated with differential operators.

LEMMA 5.6.17. Let em be a differential equation associated with a s-order differential operator $ and id) be the k-order jet prolongation of Itv. If the symbol a($(k+1)) of the jet prolongation 4(k+0 (5.6.38) of the operator 4' is a surjection, then the morphism I e(k+l) : e(k+1) , e(k) (5.6.46) ir:++k+l

is a surjection.

Proof. The proof is based on the following consideration. Given a point q E 0+k), let I E J'+k+lY be a point over q, that is, q. We have Vk)(q) = 0,

CHAPTER 5. SPECIAL TOPICS

402

but

0

in general. Let

' =,q + e,

f E q X V. ®S=X,

7r'+k(q)

= x,

be another point over q. In virtue of the relation (5.6.35), we can write 40+1)(YY,d) - 410(k+1)(o = Q(4t(k+1))(e).

The point' belongs to e( +) if there exists an element a such that -4)(k+l)(q) = and this takes place since o(VA:+1)) is a surjection.

QED

Remark 5.6.10. Note that, if the symbol oq(4 )) is a surjection, then the k-jet prolongation a k) C J'+kY of the differential equation e c J'Y is transversal to J'+k-IY at q E e+k) a the fibre of the bundle J'+kY One can apply the following homological constructions in order to investigate the surjection condition. PROPOSITION 5.6.18. Let 4 + be a differential equation associated with a s-order differential operator 4) and gy(fk) the k-order jet prolongation of .. Then, the symbol G' (5.6.5) of eo coincide with Kergo(4S) and the symbols G'+k of e(k) coincide with Keroo(4s(k)). o

Proof. If the differential equation (! is given by the equalities (5.6.43), the condition that the vector

V91r%13v=V,

I=I=k,

is tangent to . is given by the equations vJd4 = 0, IF4-k

814

(5.6.47)

5.6. GEOMETRY OF DIFFERENTIAL EQUATIONS

403

Comparison of the expression (5.6.47) with the expression (5.6.33) shows that this vector v belongs to Kerw(+). The equalities

G.+k = Ker6a(4Vk`))

(5.6.48)

can be verified in a similar way.

QED

Building on Proposition 5.6.18, one can bring every line of the diagram (5.6.41) into the exact sequence 0 ..._. GQ+k-+R ®n T=X

E: ®SZ-'"X

(&A7z

V_ ®

X -+

Ss+k-mX

(5.6.49)

B=+'i`-'" ®n 7= X - 0,

where B`+k-m denotes the Kokero(4i(k-'")). Then the following columns can be added to the diagram (5.6.41) from the left and from the right: 0

0

0-+

I

1

G9'+k

B=

-+0

1

0-+ cQ k-1®7ix

Oq(*(k-'))0Id

B=-1®T=X ---0 1

(5.6.50)

0 --

Ga+k-", 0A 7s X ...... °q(4(

)Old

... - Be-" ®77= X . 0 1

0-+

GQ®/T=X 0

cq(O)®Id

B2®nT=X 0

The left column in the diagram (5.6.50) is called the Spencer b-sequence of the differential equation f-. Both columns in the diagram (5.6.50) are complexes and their oohomologies are related to each other as follows (1109], pp.167-169).

CHAPTER 5. SPECIAL TOPICS

404

Let us denote the column complexes in the diagram (5.6.50) by Ker a, Y, E and Koker a, respectively. Then, the diagram (5.6.50) takes the form of the sequence of complexes

0-+Kera-Y-'E-'Kokera-0. Since the complexes Y and E are exact, there exists the relation H(k) (Koker a) = H(, )2(Ker a)

(5.6.51)

between the cohomology groups H('k)(Kera) of the complex Kera and the cohomology groups H(*k)(Kokera) of the complex Kokera. As a consequence of the relation (5.6.51), we have the following.

(i) H(k) (Kera) = 0.

(ii) If H )(Keru) = 0 and the symbol aq(4

'>) is a surjection, then so is

aq($Ikl), i.e., B= = 0.

PROPOSITION 5.6.19. If the symbol aq(4i(1)) is a surjection and H(k)(Kera) = 0 for all k > 2, then all symbols aq(''I--0) are surjections.

Proof. The statement is proved by induction since the item (ii) above for the diagram (5.6.50) when k = r is the condition of the same item (ii) for the diagram QED (5.6.50) when k = r + 1. DEFINITION 5.6.20. A differential equation e c J'Y is called 2-acyclic if H(k) (Kera) = 0

forallk>2atallpoints gE 1E. In virtue of Lemma 5.6.17 and Proposition 5.6.19, if a differential equation 0 c J'Y is 2-acyclic and the symbol aq(fi(l)) is a surjection, then all morphisms (5.6.46) for k > 0, are surjections. Hence, to satisfy Definition 5.6.6 of formal integrability , one can require additionally that r.+' is a surjection and all morphisms (5.6.46) are submersions. We can refer to the Cartan - Kiihler theorem (see 11691; [109), p.170).

THEOREM 5.6.21. Let e c J'Y be a differential equation so that:

5.7. FORMAL INTEGRABILITY

405

the morphism 7r,+1 I $(1) : e(1) --

is a surjection;

the differential equation t C J'Y is 2-acyclic; the symbol oq(4i(1)) is a surjection.

Then, the differential equation (! is formally integrable.

Proof. The proof is based on the fact that the symbols G( s+k) are vector bundles over e(k).

QED

If t is a quasilinear differential equation, the following criterion of formal integrability is useful ([1521, p.65).

THEOREM 5.6.22. Let (E C J'Y be a quasilinear differential equations. If (i) the morphism Jr.+1Ie(1):(!(1)

is a surjection, (ii) the differential equation ( C J'Y is 2-acyclic, (iii) the symbol G('+1) is a vector bundle over il, then (I is formally integrable.

0

5.7 Formal integrability If X is a real-analytic manifold, Y and Y' are real-analytic fibred manifolds over

X, 4i : JkY -' Y' is a real-analytic morphism and s' : X -i Y' is a real-analytic section, then the differential equation defined in (5.6.42) is said to be analytic. Given an analytic differential equation 1E of order k, we are interested in finding its convergent power series solutions in a neighbourhood of any point x E X. We

call a point of (!('), s > 0, a formal solution of a of order k + s and a point of (c(O0) = proj lim 0-(') a formal solution. Of course, the construction of analytic solutions of 0 demands a preliminary step. This consists in seeking whether a

CHAPTER 5. SPECIAL TOPICS

406

formal solution of any order > k can be prolonged to a formal solution. A sufficient condition is obviously that the maps 7rk+;+1 : e(+}1)

- 4('), s >

0, are surjective.

(5.7.1)

Then the following important theorem [125, 1261 guarantees the existence of convergent power series solutions for analytic differentials equations satisfying (5.7.1). THEOREM 5.7.1. Let e be an analytic differential equation. Let x E X and is surjective for all n > s. Then, for every s > 0. If Ak+, : =k+n+>> point p E (!( k+'), there exists an analytic solution s : U C X -' Y of ( over a =k+n)

neighbourhood U of x such that j=+'s = p. In general, a direct check of (5.7.1) is not simple. Nevertheless, there are criteria which allow us to verify the surjectivity of all maps (5.7.1) in a finite number of steps. One of such a criterion is the formal integrability of the differential equation. Theorem 5.6.22, combined with Theorem 5.7.1, leads to the existence of analytic solutions of analytic quasi-linear differential equations. The second condition of Theorem 5.6.22 refers to the vanishing of some of Spencer's cohomology groups (see Definition 5.6.20). Here we replace this condition with a stronger one, namely, (ii') for all q E tE there exists a quasi-regular basis of for Glk) at q. This means the following. Let X E X and (X.), 1 < A < n, be a basis for TX, and (0A) is the dual basis for T'X. We denote by kI TTX the subspace of vk 7 X spanned by Bµ' V ... V 90k with j + 1 < p, < ... < pk < n. For every q E it, let us define

®(VY)y, where x = zrk(y) and y = iro(q). One says that (Xa) is a quasi-regular basis for C)

at gif n-l

dim(G(k+l))g = dim(G(k))g +

dim(G(k))g.i

(5.7.2)

The condition (ii') corresponds to the involutivity of the symbol G(k) of 0. A differential equation 11 is said to be involutive if it is formally integrable and its symbol 19 is involutive.

5.7. FORMAL INTEGRABILITY

407

We will study formal integrability of Yang-Mills equations in the presence of a scalar (matter or Higgs) field. Let P -' X be a principal fibre bundle (dim X > 1) with a compact structure Lie group C of dimension d, and let C X the bundle (2.7.5) of principal connections

on P. The standard coordinates on C, J'C and PC are (xa, aa), (xA, aa, a and (x'`, aa, a;, a;..%), respectively. Given a section A : X connection on P, we write

)

C, i.e., a principal

(xA, aa) o A = (x', A;,),

where A;, are local functions on X. Its curvature is FA (2.7.11). Let V be an m-dimensional vector space on which G acts as a transformation group and E -+ X the corresponding P-associated vector bundle. Sections 0 : X -

E of this bundle are scalar fields. Standard coordinates on E, J'E and J2E are denoted by (xa, w'), (xa, cp',,pµ) and (xA,,p', wµ,

respectively. Locally we write

(x., G`) o 0 = (x 0i), where 0 are local functions on X. Let p : g End(V) be the Lie algebra representation induced by the action of C on V. As we know, a principal connection A induces linear connections on the vector

bundles VGP -e X and E - X. We denote both of them by the same symbol VA. Their connection parameters are determined by the following equations (see (2.7.14)):

VAeq = cygAadxa ®e.,

(5.7.3)

VAe, _ -p,JAadxa ® e,

(5.7.4)

respectively. Here (e') is a basis for V and p,, =< e', p(e.)e f >. The scalar field couples minimally to the connection through the covariant derivative _YA,m = VAO : X -, T'X ®E,

7A,m = i ,dxa ®e;,

y a = eat' - ppiAa4.'.

(5.7.5)

Let g be a metric on X, and h and k the inner products on g and V, respectively, such that the adjoint representation and the representation of C on V are unitary. We will study the geometric structure of the following system of partial differential equations

VA*FA= JA".,

(5.7.6a)

VA

(5.7.6b)

* 'YA,# = 0,

CHAPTER 5. SPECIAL TOPICS

408

where * is the Hodge operator and JA,O : X A T'X ® VQP is the current. In coordinates, they read 8,,(

I9IF:")- I9Ie, AaF;"+ 191Py?i`=0,

8a( I9I7;) +

(5.7.7a)

(5.7.7b)

I9Ip 1Aa-y; = 0,

where

F,." = hr,9a°g"aF 'Yi =

(5.7.8)

,

(5.7.9)

k+i94".

The current JA,O is given by

JA,O=-P**1'A,O, (5.7.10)

JA,O=- I9IP:1I''Yiaa®e, where p' is the mapping

p' : 0*(X) ®S(E') -s 0 *(X) ®S(VVP), whose definition is exhibited by the local expression (5.7.10).

Let Y = C ® E be the Whitney sum of the fibre bundles C -' X and E -y X, which is affine bundle modelled over the vector bundle

Y=T'X®VcP®E

X.

Let us consider the differential operator 4 i : J2Y - "AIT*X ®VVP E D

®E',

4-(j=A,j=0) = [VA * FA(x) - JA,O(x),

VA

*'YA,O(X)]

(5.7.11)

for all x E X and sections (A,0) X - Y. From (5.7.8) - (5.7.10), (2.7.11) and (5.7.5) it is easily seen that 4i is quasi-linear. According to (5.6.42), we define

0- = Kerb __ Ker 4s c J2Y,

(5.7.12)

X. A pair (A, 0) where 0 is the zero section of "Al T'M ® VIP ®%nT'X ® E* formed by a principal connection A : X C and a scalar field 0 : X - E is a solution of the field equations (5.7.6a) - (5.7.6b) if and only if its second order prolongation (j2A, jsm) : X - J2Y takes values into .

5.7. FORMAL INTEGRABILITY

409

In what follows we will show that (5.7.12) is an involutive differential equation. Let us first verify that a is a fibred submanifold of J2Y X, that is, a differential equation of the second order. Let o(4;):2V

T'X®Y-+'A'T'X®VCP®AT'X®E'

(5.7.13)

be the symbol of 4). In coordinates (xA, uj,,,µ, v,,µ) on V T'X OF, we find from (5.7.8)

- (5.7.10), (2.7.11) and (5.7.5) that I9I

a(4i) . vvµ'-'

(5.7.14)

u0AA),

(9Ikj g v'aµ).

(5.7.15)

Note that the symbol a(4i) is constant along the fibers of J'Y

X, and is the

direct sum

a(4>)=a(V*F)ED a(V *'y)

(5.7.16)

of the two symbols

a(V*F):VT'X®T'X®VeP and

a(V *'Y) : VT*X ®E - AT'X ®E'. Here V * F and V *,y denote the Yang-Mills and scalar field operators, respectively. One can easily check that a(V * F) is determined by the following composition of morphisms 2

V T'X®T'X®V0P+T'X®VGP+ n'T'X®VVP,

(5.7.17)

where C acts as the identity on VcP, while

(a V,6 ®7) = 29(a, ,5)'y - 9(a, 7)f - 9(,6,7)a

(5.7.18)

for every a, ,B, y E 7" X, and q is the tensor product morphism of the Hodge operator on X and the metric isomorphism on VCP induced by h. The other symbol a(V *.y)

is the tensor product of the metric isomorphism between E and E' induced by k and the morphism I9IW,

(5.7.19)

CHAPTER 5. SPECIAL TOPICS

410

for every a, P E T'X. LEMMA 5.7.2. The symbol o(4') is a surjective morphism. 0

Proof. We show that both o(V * F) and o(V * y) are surjective morphisms. For every x E X, let (dx') be an orthonormal basis of T=X, i.e.,

if A jtµ,

g(dxa, dxµ) = 0,

g(dx', dx") = ±1,

if A = µ.

Let us consider the equations 29(a,Q)y - 9(a,y)# - 9(Q,y)a = dxA,

1<,\
Their solutions can be easily found, e.g., -y = g°°dx' and a = 0 = 1/f dxµ with ,u # A. Hence, C is surjective. Since n is an isomorphism, o(V * F) is surjective. The surjectivity of o(V * y) is evident.

QED

An immediate consequence of this lemma is that t C J2Y is a fibred submanifold

over X and ai(e) = J1Y. Now we consider the first order prolongation of +, i.e.

4si11:J3Y-J1 (n'T'X®VVP®nT'X®E'). We see from (5.6.36) and (5.7.11) that

=

U.1 (VA

* FA - JA,m), 7:(VA * ,yA,m))

(5.7.20)

for every x E X and every section (A, 0) : X -' Y. Let 3

nT'X®E')

V T'X0 17 -

be the first order prolongation of the symbol (5.7.13). Its coordinate expression can be obtained directly from (5.6.39), (5.7.14) and (5.7.15):

o(VO) uiw.µ''' yarµ'-'

j9jh,.9'9"1(u,"gvao l

g"'

A' ,

(5.7.21) (5.7.22)

411

5.7. FORMAL INTEGRABILITY

where (x", ua,,,A, v..,,,) is the standard chart of V T'X ® V. Moreover, we see that

a(00) = a((0 * F)i>) ® 0,((0 where

a((0 * F)(')) : V T'X ®T'X ®V0P -+ T'X ®%\' T'X ®VcP and

a((0*F)W'W):VT'X®E-+T'X®XT'X®E' are the first order prolongations of the symbols of the Yang-Mills and scalar field operators, respectively. In accordance with the previous notation, we denote the kernel of a(4)(1)) by C3.

Next we will show that the conditions (i) and (iii) of Theorem 5.6.22 hold. Let A : X -+ C be a principal connection and let

: J'( %P®

A'T'X ®VVP) -A' T'X ®VVP

be the morphism (over X) corresponding to the operator of covariant differentiation with respect to A, i.e., (5.7.23)

WA(3 O) = (VAO)(x)

f o r every x E X and e v e r y section 0 : X have

", ' T X ® VIP. Recalling (5.7.3), we

'PA : (xA, e; , 8ve.) '-+ (xA, &0 - c'yAaei ).

Note that '@A is a linear morphism, and its symbol a(%P):T'X®ni is precisely the wedge product. Moreover, a(WY) does not depend on the connection A. For the sake of brevity, let us denote-by the same symbol o(*) the morphism

0,(T):7'X®(n17'X®VZP9AT'X®E')-AT'X0

VZP

defined as the composition of a('P) with the natural projection onto the first factor, i.e.,

T'X ®(nn1 T'X ®VVP (D AT'X ®E') -

T'X nn-1 T'X ®Vc' P

rx ®Vc' P.

CHAPTER 5. SPECIAL TOPICS

412

The following lemma shows us that G3 is a vector bundle. LEMMA 5.7.3. The sequence

O°

C3. VT'X®V°i iT'X®(n'T'X®VZP(DnT'X®E') A"7'X®VcP-'0

is exact. 0 Proof. Note that this sequence decomposes into two the following two sequences:

0 ---(G, )3 -- V T' X ®T' X ®VcP °«v*F-+)'1'l T'X ®n' T'X ®VZP °

®VVP-0

and 0

(Ga)3 . VT'X ®E-4T'X ®X T'X (&E'--,-0.

Let us show that o((V * ry)(')) is surjective. Using (5.7.19) and neglecting E and E' in the tensor products, we find that a((V * 7)('))(a V $ V b) = 2fg(a, $)b + 9(a, b)fl + 9($, b)a) 0

I91 W

for all a,,6 and b E T'X. Let (dx") be an orthonormal basis for T= X, X E X. Then the equations

o((V*ry)('))(aV$Vb)=dxA®w, 1
uao,.µ I-'

u, ) _

Fjgjh,.(g'9µAu6µ - 9P`s9'upaµo) = 0 so that Ima(V*F)(i) C Kera(%P). Now we show that Imo((V*ry)(')) D Kera('I'). Actually, we must prove the inclusion Im(Coi) D Ker (a('P) on), which is equivalent to the above one since Ti : T'X ® VcP - n'T'X 0 VIP is an isomorphism. Recalling (5.7.18) and neglecting VIP and V'P in the tensor products, we find that

eoi: vT'X ®T'X

®T'X

413

5.7. FORMAL INTEGRABILITY is given by

oi(aV flV7(9 6) = 2g(a,0)^y(& 6 +29(a,,y)Q®6 +29(/3,7)a®6 g(a, 6) J 3 V y - 9(f, b)a V -y - g(-y, b)a V 0,

for all a, 6, ry, b E T* X. In particular, we have o i(a V a V a ® b) = 6a ® [g(a, a)b - g(a, 6)a}.

On the other hand, Ker(a(W) o p) is generated by tensors of the type µ ®µ, with g(µ,µ) = 0, and µ ®p, with 9(µ, p) = 0, 9(µ, µ) 910, 9(P, p) 96 0, µ, p E T'X. Then it suffices to solve the equations 6a ®[g(a, a)b - 9(a, 6)a] = µ ®µ, 6a ®[9(a, a)b - 9(a, 6)a) = µ ® p. These are solved by a = µ,

b such that - 6g(µ, b) = 1

and

a=µ, b= 69(µ,µ) P. respectively. Obviously a(*) is a surjective morphism. Therefore the lemma is QED proved.

Let us now verify the condition (i) of Theorem 5.6.22. LEMMA 5.7.4. The map ir 2

is surjective.

Proof. Let q E (E and (A, 4A) : X -, Y a section s u c h that q = (j=A, j ). Let us consider

e-1 o 4'(1)(7iA,ji0) E T=X ®(nn'T=X ®(V4'P)s ®' T=X ®E:).

Since 40) is quasi-linear, the fibre (61))q is not empty iff c' o 4(')(j=A, j=-0) E Im a(4O) or equivalently, by Lemma 5.7.3, if and only if a('P)oE 'o4;i'i(j=A, j=0) _ linear, we obtain 0. FYom (5.7.20), (5.7.23) and since a(Q1) o E- I, 4I0) (j.3 A,

= WA1 .(VA * FA - JA,*)) _ (VAVA * FA - DAJA,#)(X)

CHAPTER 5. SPECIAL TOPICS

414

Now using the differential identities VAVA * FA = 0,

_V AJA,* = P*VA * -fA,*,

we find that a,('P) o E

1 o -0) jjA, 7z0) = (P*V A * 7A,*)(x),

and the result follows by using the field equation (5.7.6b). The first of the above identities is a well-known identity involving the curvature of the connection, whereas

the other is a consequence of the unitarity of the representation of G on V QED

Remark 5.7.1. Let us consider the model where a self-interaction potential U : E -+ "T'X of scalar fields is present. Then the equation (5.7.6b) takes the form

VA1'A,*=f*, where f

DU : E -- "A T'X ® E' denotes the fiber derivative. In this case the

precedent discussion holds true apart from Lemma 5.7.4. Indeed, now we have

co') 0'e-1 0 0111 j:A, j=b) = (p*f*)(x)

and, hence, 0(1) -+ I is surjective if and only if p* f : E - nT'X ® VIP vanishes identically. Let us consider the Lie algebra representation

'=:X-VoP

-+

uw:E-+VE

induced by p : g - End(V). Locally it reads [611

E=

er

i--+

ua = e'is

a;

As one can easily verify,

C., U =< POLE' >' where f.,, denotes the Lie derivative and <, > is the natural contraction between VcP and VIP. It follows that Lemma 5.7.4 holds whenever the scalar field potential

U is gauge-invariant, i.e., L.. U = 0 for every =-: X - V0P. Finally, let us show that the above condition (ii') holds.

5.7. FORMAL INTEGRABILITY

415

LEMMA 5.7.5. For every q E if there exists a quasi-regular basis of

for G'

atq. 0 Proof. Let q E it and let (dxa) be an orthonormal basis of T=X where x = ir'(q). We have the following dimension counting: dim (G2)q,n_1 = d,

(5.7.24)

[(n-j)(2-j+1) _ 1J(nd+m), 1 < j
(5.7.25)

dim (G')q = dim (G3)q =

[n(n2

1)

- 1](n d + m),

n3 + 3n6' - 4n

(5.7.26)

(n d + m) + d.

(5.7.27)

The proof of (5.7.24) goes as follows. By definition (G2)q,n_i = Kerv(4')n_1i where Q(4i)n_1:2'V17iX®,7__

is the restriction of u(+) to

%n-1

n'TsX®(VVP)=®nT=X®E=

7=X ®Y=. Using (5.7.18) and (5.7.19), we find

S,, -i(dxn V dxn ®dx') = 2gnndx'- 2gnAdXn,

1< A < n,

Sn-1(dXnVdXn)=29nn®w,

thus showing that dim (Keren_1) = 1 and dim (Ker(n_1) = 0. Thence,

dim(G')q,n_1 =dim(KerSn_1)d+dim(Ker(n_1)m=d. To prove (5.7.25), let us note that

v(4i)j : 4TiX ®Y=

®(VcP)= ®nTZX ®E_,

1 < j < n - 2, is surjective and, hence, the sequence 0 -' (G2)gd

24 7=X ®Y=

f n/\1T=X ®(VVP)= ®'T=X ®Es -+0 is exact. The equality (5.7.26) is an immediate consequence of Lemma 5.7.2. With regard to (5.7.27), from Lemma 5.7.3 we find dim (G3)q =

(n 2) 3

(nd + m) - dim [Im u(4iill)J =

CHAPTER 5. SPECIAL TOPICS

416

(nd + m) - dim (Kera(fl] = (

n+2 -n](nd+m)+d= 3 /

n3+3n2-4n(nd+m)+d. 6

It follows that n-I

dim (C2)g + F dim (C2)o,j = j=1 (n(n21)

j)(n

- 1](nd+m)+

- 11(nd+m)+d,

j=I and, since

nZ (n-j)(n-j+1) J-1

n3-n-6

2

6

n-1

n3 + 3n2 - 4n (nd

we find

dim (G2)g + E dim (C2)ga = j=1

6

+ m) + d = dim ((3)9.

QED Summarizing the discussion above, we have shown that the differential equation (5.7.12) is involutive and, hence, formally integrable. Now let us assume that the principal bundle P -+ X and the metric g of X are analytic. Then it (5.7.12) is an analytic differential equation and Theorem 5.7.1 leads us directly to the following theorem.

X and the metric g be analytic. THEOREM 5.7.6. Let the principal bundle P For any x E X there is an analytic solution (actually different analytic solutions) (A, 0) of the differential equation (5.7.12) over a neighbourhood of x. 0 Let us now investigate formal integrability of Einstein-Yang-Mills equations.

Let E C V T'X

X be the bundle of metrics on X of fixed signature (62], E let V9 be the

provided with coordinates (xA, ga,,). For every metric g : X

5.7. FORMAL INTEGRABILITY

417

corresponding Levi-Civita connection. Its connection parameters are the Christoffel symbols, i.e., 1

ra°0 = 291(809aµ + 8a90Y - 8µ99a)

If U E T'(X), we will denote by div9u the 1-form (div9u)µ = Dauµ.

(5.7.28)

Given a principal connection A : X - C, we can define a covariant derivative O9Au of VcP-valued tensor fields u E 7,(X) ® S(V0P). If

u=uµ;.:.,. 8a, ®...®aa,®dx'' ®...®dx"' ®e9, then O9'Au E T,+, (X) ®S(V0P) is given locally by V9,A ua1..a.' = V9 ua,...a7 + µt...µ. v µ1...Y.

At ua1...ar4

v µt...µ.

The Ricci morphism reads

r: JOE - VT'X, r = 2rµ dxm V dx°,

rµv = aQrY°v - 8µr:,, + r0°srµsv - rjrQov.

(5.7.29)

Note that r is quasi-linear, and its symbol

Q(r) : VT'X ®VT'X - VT'X over E (note that

VE

=E

X

X

V T' X) is defined by

o(r)(9, a V'8 ®'y V 6) = 2(-2g(a, l)'7 V b - 2g(-y, 5)a V Q +

g(fl,b)a V -t + g(C".0 V 6 + g(8, ')a V 6 + g(a, 6)8 V ryJ

for every metric g : X - E and a, Q, ry, b E TX. Let

e:VT'X -VT'X

(5.7.30)

CHAPTER 5. SPECIAL TOPICS

418

be the linear morphism over E given by

e(g, a V p) = a V 0 - g(a,f)g,

for every g: X -' Eanda, ,6ET'X. LEMMA 5.7.7.

V T'X

Let n 0 2. For every metric g : X -' E the morphism e(g)

V T'X over X is an automorphism. 0

Proof. In the coordinates (x", u),") of V T'X, we easily find E(g) : u.y,'-. ua" -

1

(5.7.31)

Then 1

9"(uxw - 29a"9a$ua$) = (1 - n/2)ga"ua" and, hence,

ua"- I 9a"9°puop=0

g.""u".=0

u.'. =0. QED

The Einstein morphism is the morphism over X

G:JzE

VT'X

defined by

G=eor. Clearly C is quasi-linear, and its corresponding symbol is the morphism over E

a(G):VT'X®VT'X -.VT'X given by a(G) = e o a(r).

Since we are concerned with interaction between gravitational and gauge fields, the total fibre bundle is the fibred product

x

5.7. FORMAL INTEGRABILITY

419

Let h be a metric in the Lie algebra of G such that the adjoint representation is orthogonal. Then the Einstein-Yang-Mills (EYM) equations are G(g) + Ag = T(g, A), VA * F(A) = 0.

(5.7.32a) (5.7.32b)

Here A E R is a constant, * is the Hodge operator on X and T is the energymomentum tensor of the Yang-Mills field:

T: ExJ'C--'vT'X, M 2TT,, =

1

gaoF°11µ,

where F, 'O has been defined in (5.7.8). We easily see from (5.7.3) and (5.7.31) that locally the equations (5.7.32a), (5.7.32b) read raµ - 2ga,,g°0r0s + Ag,,. = Zo, oa ( I91 F'.") = aa( I91 Fr'``) -

(5.7.33a) I9Ic9, A9 F,- = 0.

(5.7.33b)

Let us define

A'T'X®V,P, 4'j:g,j:A) = (G(g) + Ag - T(g, A), VA * F'(A))(x),

for every a E X, (g, A) : X

Y. A section (g, A) : X

Y is a solution of the

EYM differential equations if and only if D(j2g, j2 A) = 0. According to (5.6.42), we define

= Ker -t C J2Y.

(5.7.34)

As (2.7.11), (5.7.29), (5.7.33a), and (5.7.33b) show, 1 is a quasi-linear morphism, and its symbol (5.7.35)

over

J'Y is given by a(4')=a(G)®a(O*F).

(5.7.36)

CHAPTER 5. SPECIAL TOPICS

420

Here

Y=vT'X®T'X®VcP is the vector bundle associated with Y - X. The linear morphism

0(0*F): 'T'X®T'X®VcP-- A'T'X®VVP is defined in (5.7.17)

LEMMA 5.7.8. If n > 3, the symbol 0(45) is surjective. 0

Proof. We show that both a(G) and 0(V * F) are surjective. Since 0(G) = e o 0(r) and owing to Lemma 5.7.7, we only need to show the surjectivity of 0(r). Let (dx") be an orthonormal basis of T=X, x E X, with respect E, i.e., to a metric g : X g(dx", dx") = 0, if A g(dxA, dx") = ±1, if A = ,u.

If 1 < A, µ, v < n are different from each other, then, using (5.7.30), we find o(r)(g, dx" V dx" ® dx" V dx") = -g""dx" V dx".

(5.7.37)

If A 0 µ, then 0(r)(g, dx" V dx" ® dx" V dx") = -g""dx" V dx" - g""dx" V dx".

Choosing 1 < a, 8, ry < n different from each other, we easily see that gO°dx° V dx° + 90°dx° V dx°,

with p 34 a,

g'ridxf V dxa + ggPdx'' V dx",

along with (5.7.37), form a basis of VT=X. Hence 0(r) is surjective. The surjectivity of 0(0 * F) has been shown in the proof of Lemma 5.7.2. QED It follows from the Lemma 5.7.8 that IE

and a differential equation. Moreover,

ir2(e) = J'Y.

X is a fibred submanifold of J2Y -a X

5.7. FORMAL INTEGRABILITY

421

Now we consider the first order prolongation of t, i.e.,

f(') : J3Y ' J'(VT'X ®&A'T'X ®VV'P), +(1)

(7=9,7=A) = (.'(G(9) + Ag - T(9, A)), 7=(VA * F(A))).

We deduce from (5.7.36) that its symbol is the morphism

VT'X ®Y -. T'X ®(V T'X ([)n1T'X ®VVP) over

J'Y given by a(4('))

= a(G(1)) ® a(O * F(')),

(5.7.38)

where

a(G(')):VT'X®VT'X -T'X ®VT'X and

a(O*F'(')):VT'X®T'X®V0P- T'X®nA'T'X®Vv'P are the symbols of the first order prolongation of the Einstein operator C and the Yang-Mills operator V * F, respectively. To verify the conditions (i) and (iii) of theorem 5.6.22, we will introduce the following morphisms over J'Y: 2

J'(VT'X) - T'X, ,,i(j=9,j.'u) = div9u

(5.7.39)

and

02 : J'(A'T'X ®VVP) - AT'X ®VVP, 02 (A (x), j:8) = (V AO) (x),

(5.7.40)

for every xEX,(g)A):X- Y, In (5.7.39), u E T1 (X) is the tensor equivalent to u obtained by using the metric g. Note that (5.7.39) and (5.7.40) are both linear morphisms. Their symbols are given by

a 0 0 V -0 = 9(a, fl)7 + 9(a, '1)f3

CHAPTER 5. SPECIAL TOPICS

422

and

a(02): TX0A'T'X0VVP-"T'X®VVP, a(+/)(a ® 0) = a n 0.

A corollary of the following lemma is that 63 = Kera(0)) is a vector bundle. LEMMA 5.7.9. If n > 3, then the sequence over J1Y

®(VT'X (Dnn17'X ®VaP)

°(0 OW42)T'X ®nT'X 0 V.P.0 is exact. 0

Proof. Bearing in mind (5.7.38), we see that the sequence decomposes in the following ones:

T'X®VT'X°

0

T'X -.o

and

0. (Ga)3 -

VT'X 0M

0 4A-'T'X ®VVP

°-''T'X ® ,,'P - 0. The exactness of the first sequence is proved in [59], while the exactness of the second one has been already shown in the proof of Lemma 5.7.3.

QED

LEMMA 5.7.10. If n > 3, then the map 7[23 : E(1) -+ it is surjective. o

Proof. Let q E iE and let (g, A) : X - Y be a section such that q = (jzg, j=A). Let us consider C-1

0O(1)(j.3g,j:A) E T=X ®(V 7=X ® A 1TX ® (VVP):).

Since fi(1) is quasi-linear, the fibre (f-M), is not empty if and only if e-1 1 0 (1)(?sg,jjA) E Ian (

or, owing to the previous Lemma, if and only if

a(iGi) (D 0'(02) o E 1 0 -0)(jg,j:A) = 0.

5.7. FORMAL INTECRABILITY

423

Since (5.7.39) and (5.7.40) are linear morphisms, this relation is equivalent to

V11 ®0i o t(')(7:g,j:A) = 0. We have

'i ED 0204'(')(j:g,j=A) =

0i ®t(j:(G(g) + Ag - T(g, A)), j=(V A * F(A))) _ VAVA (div9(G(g) + * F(A)(x)). T(g, A))(x), Using the Bianchi identity div9G(g) = 0, the charge conservation identity VAVA F(A) = 0 [129], and the identity div9g = 0, which holds since V9 is the Levi-Civita connection of g, the above relation yields (-div9T(g, A)(x), 0).

'01 ®02 c

After some straightforward calculations, we obtain

V:T: = Vµ( jgjFe)FpV jg1_'

(5.7.41)

Since the equation (5.7.33b) is identically satisfied on (!, the result follows from (5.7.28)

QED

The identity (5.7.41) is a consequence of the gauge-invariance and general covariance properties of the field system under consideration [61, 621. Finally, let us show that the condition (ii') holds. LEMMA 5.7.11. If n > 3, then for every q E 1E there is a quasi-regular basis for T.2(q)X for C2 at q. 0

Proof. Let q = (j=g, j=A) and let (dr) be a g-orthonormal basis of T=X. We have the following dimension counting whose proof will be given below: dim (C2),,.-, = n + d, di m (G2)g, =

1<j
di m (G2)q =

[

n(n 2

=

(

(n

di m (G3)q

(5.7.42)

[(n - j)(2 - j + 1)

2)

3

-

1

1)

][ n(n

2

1)

- n][ n(n2+ 1)

][n(n

2 1) + n d],

(5 . 7 . 43)

+ n d],

( 5 . 7 . 44 )

+ nd ] + n + d.

( 5 . 7 . 45 )

CHAPTER 5. SPECIAL TOPICS

424

Therefore, n-1

dim (G2)q + > dim (G2)q.i

i-I

j)(n j+ 1)

E

- 1J(n(n2 1) +redj+m+d.

i=0

(n- j)(n- j+1)

= n3+3n2+2n-6

2

j=0

6

we find that "-1 dim (G2)q + E dim (G2)ga =

n3+3n2-4n n(n+1) 6 2 (

+ ndJ + n + d,

i=1

and this is equal to (5.7.45).

QED

We have shown that the EYM differential equation (5.7.34) is involutive. It follows that, if the principal bundle P -' X is real analytic, then the EYM differential equation is analytic and Theorem 5.7.1 leads to the following result. THEOREM 5.7.12. Let the principal bundle P -+ X be real analytic. Then for every q E 41 there is an analytic section (g, A) : U C X - Y over a neighbourhood U of x = ir2(q) such that (g, A) is a solution of the EYM equations and (j=g, j=A) = q.

0 Now we prove (5.7.42)-(5.7.45). Let us first prove (5.7.42). By definition, (C2)q.n-1 = Ker(v(4i)q.n-t), where n-1

2,n-1 0,(4i)q,n-1

V T=X ®Ys -+ v'7-.X ®n T=X ®(VVP)Z

:

is the restriction of the symbol (5.7.35) to we find 2

2

a

o (r)q,n-1(V dx 0 V dx) _

2,V n-1

T=X ®Y=. From (5.7.30) and (5.7.18)

if.A=n

0,

-gnn dxa V

gA 2

dxn, if ,\

n

5.7. FORMAL INTEGRABILITY 2

1(V dxn ®dxa V dx'`) _ 0

2

425

0,

1-gdz'Vde,

ifA=nor p = n, A 14 I1 if A O n, µ# n, A Vtµ

ifA=n

Cc.- 1(V dxn ®dxa) = 2gnndxa, if A 54 n. These relations show that

dim ((72)9,n- l = dim [Ker v(r)q,n_ 1 ] + dim [Ker eq,n_ 1 ] = n + d.

To prove (5.7.43), let us go back to the proof of Lemma 5.7.8. It is easily seen that the morphism 0'(4))q j :

X ®Y= - VT=X ® n1 T=X

1 < j < n -2,

is surjective and, hence, (5.7.43) holds. The relations (5.7.44) and (5.7.45) are immediate consequences of Lemmas 5.7.8 and 5.7.9.

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Glossary of symbols AUT (Y), 129

dv, 100

Aut (Y), 129

dr, 68

C°°(M), 8

dA, 100

CK, 167

dA, 49

CX, 177

dr, 380

C1,3, 175

d 68

C4,01 190

{dy'}, 26

Ck, 342

8A, 98

CJkY, 342

a(Q), 350

C(JmZ), 372

D2(Q), 351

6,208

(P), 351

c',k, 353

Diff,(P,Q), 348

cam, 84

Diff (Q), 349

c,,., 350

Diff

(P, Q), 348

Dr, 60

Diff

(P, Q), 348

D, 199

(Diff i (Q))k, 350

Dh, 185

Ey, 109

dH, 100

CH, 255 443

GLOSSARY OF SYMBOLS

444

EL, 114

Hr, 249

Cr, 124

HP(B.), 113

EL, 125

HP(B'), 113

e('), 387

HP(M), 113

e4, 118

HP(Z; K), 174

FA, 87

HH(O 0), 377

FA, 90

hk, 100

f';, 34

HOL(X), 157

17

Jk, 353

{f,g}y, 310

J'u, 99

G,,3, 175

J's, 45

G1H, 178

J2s, 55

Gau(P), 143

J's, 338

GL4i 156

JIY, 43

9R, 189

J2Y, 54

9,84

J'Y, 334

91,83

JOOY, 373

[gz]s, 369

J,,Z, 362

H,250

J'J1Y, 52

H, 249

J2Y, 53

HL, 126

J'+lY, 336

HQ, 275

J'4,, 46

GLOSSARY OF SYMBOLS

445

J24), 56

0,340

jr, 4i337

D;, 99

ills, 43

D , 375

j:s, 54

D ;, 377

jrs, 374

0k#-k, 378

3', 354

O' (M), 34

3k(P), 352

O'(M), 34

3;(P), 352

99

3k(Y), 346

P(Y), 31

Ker$, 119

Pin(1, 3), 176

Ker,4i, 17

[p), 141

L, 176

Rc, 83

L, 122

Raµ, 62

L., 176

R1,3, 175

L., 32

R2,3, 175

LX, 156

rk, 393

LX, 190

rk(L), 394

LhX, 181

s _;, 347

MhX, 184

Sk -, 352

01,355

SkX, 345

Ok, 356

S(Y), 16

O(JmZ), 371

SO°(1, 3), 176

446

GLOSSARY OF SYMBOLS

Spin(1,3), 176

V7r k_1, 390

Spin°(1, 3), 176

V(Y), 31

Spin(4), 190

W2 W, 189

[SJ;, 364

Y", 78

T, 234

ZL, 127

`1, 133

Zy, 126

TP, 164

x4, 183

Tr, 138 'r, 139

r, 58

'1o, 132

r, 65

Tf, 7

ru,251

ToP, 84

I', 80

T(M), 29

ru, 61

T ,(M), 32

r ®1", 66

T.(M), 32

r ®r', 66

t", 162

r x r', 62

Up, 164

Om, 110

VY, 26

bo, 348

V-Y, 26

352

v r, 80

VcP,

84

VyJ1Y, 49

348

0,122 8k, 394

GLOSSARY OF SYMBOLS

441

OL, 123

PL, 103

9LX, 157

EPR, 160

9M, 38

ER, 189

9(k), 341

ET, 182

9h,341

a(4?), 397

oxiX., 99

a(4)), 399

123

aq(4i), 397

e(k+i), 341

-rH, 316

AL, 324

z, 157

)'(k), 341

[4 t], 130

A(k+I), 341

Wh, 182

µk+', 352

12°, 35

µm, 358

QL, 123

L, 127

oz, 127

II, 121

(0,122

ir', 44 vol, 44

Ors, 61

52

Ors, 61 V t, 211

*, 348

7r

(Z, Z'), 172

Index canonical coordinates, 226 241, 321 canonical energy-momentum tensor, 139 canonical horizontal splitting, 51, 341 of a vector field, 51 of an exterior form, 51 canonical tangent-valued 1-form, 38 Cartan connection, fi7 Cartan distribution, 342, 372, 396 Cartan equations, 107, 124 Cartan manifold, 372 higher order, 343 Cartan plane, 396 Cartan subspace, 342, 373 tech cohomology group, 174 chain complex, 113 charge conservation identity, 70 Chern-Simons gauge model, 151 classical solution, 191 386

absolute acceleration, 211 acceleration with respect to a dynamic connection, 214 affine bundle, 23 tangent, 24 affine bundle coordinates, 24 affine bundle morphism, 24 affine subbundle, 25 associated atlas, L41 associated fibre bundle, 87 canonically, 141

atlas of constant local trivializations, 207

automorphism, 17 canonical, 320 holonomic, 157 vertical, 17 base space, 9 bicomplex, 382 bundle atlas, 13 bundle coordinates, 14 bundle of Clifford algebras, 127 of Minkowski spaces, 177 of principal connections, 86 of world connections, 167

Clifford algebra, 175. Clifford group, 176 coadjoint representation, L44 cochain complex, 113 cocycle condition, 14 codifferential, 36 cohomology group, L13 complete family of Hamiltonian forms, 272

canonical coordinate transformations,

complete family of integral manifolds,

322 449

INDEX

450

343

components of a connection, 59 composite connection, 76 composite fibred manifold, 18 connection, 58 affine, 66 complete, 208 covertical, 80 dual, 65, 73 linear, 64 projectable, 75 reducible, 63 second order , 80 holonomic, 80 sesquiholonomic, 80 vertical, 80 connection form, 86 local, 87 connection parameters, 59 conservation law, 128, 133 covariant, 151 integral, L33 strong, 129 weak, 129, 132 contact form, 318, 341, 371 1-jet, 50 2-jet, 58 k-jet, 341 local, 370 contact manifold, 318 contraction, 34 cotangent map, 23 covariant codifferential, 69 covariant derivative, 61 along a vector field, 61

covariant differential, fi0 current, 123 curvature, 62 70

canonical, 912

Darboux's coordinates, 318 De Donder form, 127 De Rham cohomology group, 113 De Rham complex, 113 in infinite order jets, 377 De Rham sequence of an algebra, 357 derivation functor, 351

derivations with values in a module, 350

derivative coordinates, 44

differential equation, 19 386 2-acyclic, 403

associated with a differential operator, 118 determined, 109 formally integrable, 389 overdetermined, l09 quasilinear, 390 regular, 389 undetermined, 109 differential ideal, 224 372 differential operator, L09 linear, 348 quasilinear , 39$ standard form, 110 Dirac operator, 185 direct limit of endomorphisms, 377 direct sum connection, 66 direct system of endomorphisms, 376 distribution, 234 completely integrable, 235

451

INDEX involutive, 234 dynamic equation, 211

Ehresmann connection, 77 energy function, 230 energy-momentum tensor, 162 equation of continuity, 147 equivalent fibre bundles, 14 Euler-Lagrange equations, 105 first order, 126 higher order, 120 Euler-Lagrange form, 114 Euler-Lagrange map, 114 Euler-Lagrange operator, 105 first order, 125 higher order, 114 Euler-Lagrange-Cartan operator, 124 Euler-Lagrange-type operator, 114 event space, 204 exterior algebra, 34

fibration, 9 fibre, 9 fibre basis, 21 fibre bundle, 13 with a structure group, 141 fibre metric, L04 spinor, 185 fibred chart, 12 fibred coordinate system, 13 fibred endomorphism, 17 fibred manifold, 9 fibred monomorphism, 17 fibred morphism, 1fi over X, 16 fibred submanifold, 17

filtered algebra, 376 filtered module, 376 filtered morphism, 376 first Bianchi identity, 71 first order jet, 43 first variational formula, 104 in the presence of background fields, 136

F-N bracket, 38 F-N covariant differential, fib foliation, 235 symplectic, 23fi force, 211 form

basic, 37 contact , 99 k-contact, 100 exterior, 33 horizontal, 37 Pfafflan,33 semibasic, 37 soldering, 40 tangent-valued, 38 horizontal, 39 projectable, 39 vector-valued, 41 horizontal, 40 frame, 21 higher order, 320 frame bundle, 156 frame field, 156 Frolicher-Nijenhuis bracket, 38 functor of changes of rings, 359 fundamental group, 113

gauge algebra bundle, 84

INDEX

452

gauge group, 130, 1.43 local, 1.31

gauge Lie algebra, 144 gauge parameter, 144 gauge potentials, 87 gauge transformation, 1.25 active, 1.22 general, 1311

passive, 129 vertical, 134 gauge-type class; 287 gauge-type condition, 287 gauge-type freedom, 287 general affine bundle, 83 general covariant transformation, 157 geometric module, 358 group bundle, 143

Hamilton equations, 256 Hamilton evolution equation, 316 Hamilton operator, 255 Hamilton-De Donder equations, 128 Hamiltonian, 315 Hamiltonian connection, 252 Hamiltonian density, 249 Hamiltonian form, 248 associated, 263 constrained , 275 weakly associated, 263 Hamiltonian horizontal vector field, 316 Hamiltonian lift of a vector field, 92 Hamiltonian map, 249 associated with a Lagrangian density, 262 Hamiltonian vector field, 233 Helmholtz-Sonin map, 114

Higgs field, 192

Hodge star operator, 35 holonomic atlas, 22 holonomic ooframe, 23 holonomic coordinates, 23 holonomic frame, 22 homogeneous Legendre bundle, 127 homogeneous Yang-Mills equation, 74 homology group, 113 homotopic maps, 172 homotopic topological spaces, 172 homotopy group, L73 homotopy operator, 254 horizontal density, 37 horizontal differential, 144 horizontal distribution, 206 horizontal foliation, 207 horizontal lift, 134 canonical, 157 of a path, 7fi of a vector field, 61 horizontal subbundle, 59 ideal of contact forms, 342 imbedding, 9 immersion, 9 infinite order jet space, 374 instantaneous Hamiltonian formalism, 246

integral curve, 34 integral manifold, 234 maximal, 235 of maximal dimension, 235 integral of motion, 229 integral section of a connection, 61 interior product, 22

INDEX

invariant transformation, 131 generalized, 135 inverse problem, 114 isomorphism of fibred manifolds, 11 Jacobi bracket, 232 Jacobi structure, 232

jet bundle, 44 jet manifold, 44 higher order, 98, 334 repeated, 52 second order, 54 sesquiholonomic, 53 higher order, 336 of submanifolds of a manifold, 362

jet of a module, 352 jet prolongation, 337 of a differential operator, 392 of a differential equation, 387 of a manifold, 369 of a morphism, 46 second order, 56 of a section, 45 higher order, 338 second order, 55 of a submanifold, 372 of a vector field, 49 higher order, 339 second order, 58 kernel of a differential operator, 118 kernel of a fibred map, 11 Komar superpotential, 167 generalized, 110 Lagrangian, 220 Lagrangian connection, 124

453

Lagrangian constraint space, 262 Lagrangian density, 97 affine , 296 almost regular, 273 gauge-invariant, 145 hyperregular, 122 quadratic , 281 regular, 122 semiregular, 268 variationally trivial, 106 Lagrangian distribution, 243 Lagrangian function, 220 Lagrangian subspace, 243 Lagrangian system, 226 left Lie algebra, 83 Legendre bundle, 121 Legendre fibred manifold, 241 Legendre form, 123 Legendre map, 122 Legendre morphism, 126 Lepagian equivalent, 103 Lepagian form, 102 Lie bracket, 29 Lie derivative of a function, 29 of a multivector field, 32 of an exterior form, 35 of a tangent-valued form, 39 linear bundle coordinates, 21 linear bundle morphism, 21 linear derivative of an affine map, 25 Liouville form, 237 tangent-valued, 122 associated with a Lagrangian density, 123

INDEX

454

Liouville's field, 31 local diffeomorphism, 9 local one-parameter group of diffeomorphisms, 30

local trivialization, 18 locally Hamiltonian form, 314 locally Hamiltonian horizontal vector field, 313

locally Hamiltonian vector field, 313 locally variational operator, U.S Lorentz atlas, 182 Lorentz connection, 185 Lorentz structure, 181 Lorentz subbundle, 181

mass metric, 223 mass tensor, 222 master equation, 111 matter bundle, L42 metric bundle, L60 motion, 2f)8 multisymplectic diffeomorphism, 243 multisymplectic form, 127, 243 multisymplectic structure, 122 multivector field, 32

natural bundle, 155 Newtonian system, 223 Nijenhuis differential, 39 normal bundle, 234 Nother conservation law, 131 Nother current, 138 Nother superpotential, 146 observed motion, 212 open map, 9

Pfaffian system, 234 pin group, 176 Poincar6-Cartan form, 1114 Poisson bracket, 232 Poisson bivector, 33, 232 Poisson manifold, 232 Poisson structure, 232 canonical, 310 non-degenerate, 236 regular, 233 polysymplectic automorphism, 244 polysymplectic form, 122 associated with a Lagrangian den-

sity, 123 polysymplectic structure, 242 preconnection, 369 presymplectic form, 236 presymplectic manifold, 236 principal automorphism, L42 general, 142 principal bundle, 82 principal connection, 8fi associated, 88 canonical, 90 principal vector field, 143 Proca field, 163 product connection, 62 product of fibred manifolds, 12 projection, 9 proper map, 11 pseudo-Riemannian metric, 16f1 pull-back connection, 63 pull-back fibred manifold, 17 pull-back form, 34 pull-back of a vector field, 51

INDEX pull-back section, 18 push-forward of a vector field, 30

rank of a map, 8 reduced principal subbundle, 179 reduced structure, 178 reduction of a structure group, 178 reference frame, 2(18

relative Poincare lemma, 254 relative velocity, 208 relativistic mechanics, 367 representative object, 349 Ricci tensor, 72 Riemannian metric, 1.89 Riemannian structure, 189 right Lie algebra, 84 Schouten-Nijenhuis bracket, 32 second Bianchi identity, 74 generalized, 70 second Stiefel-Whitney class, 189 section, 18 critical, 111 global, 15 integrable, 45 SEM current, 138 SEM tensor, 189 soldered codifferential, 69 soldered curvature, 71 soldered differential, 68 soldered Yang-Mills operator, 71 space-time decomposition, 183 space-time structure, 183 spatial covariant differential, 214 spatial distribution, 183 spectral sequence, 112

455

Spencer complex, 344 of a module, 358 of sections, 34? Spencer morphisms, 347 r-order, 33fi Spencer operator, 344 Spencer 3-sequence, 357 Spencer 6-sequence of a differential equation, 403 Spencer sequence of differential operators, 351 higher order, 351 spin connection, 185 spin group, 1.76

spin structure Riemannian, 190 pseudo-Riemannian, 177 h-associated , 184 universal, 190 pseudo-Riemannian, 196 Riemannian, 191 spinor bundle, 184 principal, 184 spinor metric, L75 spontaneous symmetry breaking, 192 standard 1-form, 205 standard vector field, 205

strength, 87 subbundle, horizontal, 52 submanifold, 9 imbedded, 9 submersion, 9 superpotential, 134 symbol of a differential equation, 390 symbol of a differential operator, 397

INDEX

456

symmetric affme connection, 213 symmetric dynamic connection, 218 symmetry current, 132 symplectic form, 35 canonical, 237 symplectic manifold, 236 symplectomorphism, 238

tangent map, 7 tensor bundle, 23 tensor product connection, 66 79 tensor product of vector bundles, 22 tetrad bundle, 182 tetrad coframe, 183 tetrad field, 182 tetrad form, 182 tetrad frame, 182 tetrad function, 183 topological space k-connected, 123 contractible, 172 simply connected, 123 torsion, 71 of a dynamic connection, 218 total derivative, 49, 1W in infinite order jets, 384

total Dirac operator, 199 total space, 9 transition functions, 14 trivial bundle, 15 trivialization map, 13 universal Cartan element, 394 variational derivative, 145 variational map, 113 variational sequence, 116

vector bundle, 21 dual, 22 vector field admissible, 141 complete, 31 horizontal, 131 integrable, 339 projectable, 3L 339 vertical, 31 vector subbundle, 21 velocity coordinates, 44 vertical cotangent bundle, 26 vertical covariant differential, 77 vertical differential, 11X4

vertical splitting, 27 vertical tangent bundle, 26 vertical tangent map, 26 volume form, 35 Whitney sum of affine bundles, 24 Whitney sum of vector bundles, 22 world connection, 156 world manifold, 156 world metric, 156 world spinor, 191 Yang-Mills Lagrangian density, 149 Yang-Mills operator, 70