Generalized Hamiltonian Formalism for Field Theory - CiteSeerX

Generalized Hamiltonian Formalism for Field Theory (World Scientific, Singapore, 1995 ) G. SARDANASHVILY

Preface Classical field theory utilizes traditionally the language of Lagrangian dynamics. The Hamiltonian approach to field theory was called into play mainly for canonical quantization of fields by analogy with quantum mechanics. The major goal of this approach has consisted in establishing simultaneous commutation relations of quantum fields in models with degenerate Lagrangian densities, e.g., gauge theories. In classical field theory, the conventional Hamiltonian formalism fails to be so successful. In the straightforward manner, it takes the form of the instantaneous Hamiltonian formalism when canonical variables are field functions at a given instant of time. The corresponding phase space is infinite-dimensional. Hamiltonian dynamics played out on this phase space is far from to be a partner of the usual Lagrangian dynamics of field systems. In particular, there are no Hamilton equations in the bracket form which would be adequate to Euler-Lagrange field equations. This book presents the covariant finite-dimensional Hamiltonian machinery for field theory which has been intensively developed from 70th as both the De Donder Hamiltonian partner of the higher order Lagrangian formalism in the framework of the calculus of variations and the multisymplectic (or polysimplectic) generalization of the conventional Hamiltonian formalism in analytical mechanics when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only time. Each approach goes hand-in-hand with the other. They exemplify the generalized Hamiltonian dynamics which is not merely a time evolution directed by the Poisson bracket, but it is governed by partial differential equations where temporal and spatial coordinates enter on equal footing. Maintaining covariance has the principal advantages of describing field theories, for any preliminary spacetime splitting shades the covariant picture of field constraints. Contemporary field models are almost always the constraint ones. In field theory, if a Lagrangian density is degenerate, the Euler-Lagrange equations are underdetermined and need supplementary conditions which however remain elusive in general. They appear automatically as a part of multimomentum Hamilton equations. Thus, the universal procedure is at hand to canonically analize constraint field systems on v

vi

the covariant finite-dimensional level. This procedure is applied to a number of contemporary field models including gauge theory, gravitation theory, spontaneous symmetry breaking and fermion fields. In the book, we follow the generally accepted geometric formulation of classical field theory which is phrased in terms of fibred manifolds and jet spaces.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v 1

1 Geometric Preliminary 7 1.1 Fibred manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Jet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 General connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Lagrangian Field Theory 2.1 Lagrangian formalism on fibred manifolds . . . . . . . . . . . . . . . 2.2 De Donder Hamiltonian formalism . . . . . . . . . . . . . . . . . . . 2.3 Instantaneous Hamiltonian formalism . . . . . . . . . . . . . . . . . .

39 40 47 50

3

Multimomentum Hamiltonian Formalism 3.1 Multisymplectic Legendre bundles . . . . . 3.2 Multimomentum Hamiltonian forms . . . . 3.3 Hamilton equations . . . . . . . . . . . . . 3.4 Analytical mechanics . . . . . . . . . . . . 3.5 Hamiltonian theory of constraint systems . 3.6 Cauchy problem . . . . . . . . . . . . . . . 3.7 Isomultisymplectic structute . . . . . . . .

4 Hamiltonian Field Theory 4.1 Constraint field systems . . . 4.2 Hamiltonian gauge theory . . 4.3 Electromagnetic fields . . . . 4.4 Proca fields . . . . . . . . . . 4.5 Matter fields . . . . . . . . . . 4.6 Hamilton equations of General 4.7 Conservation laws . . . . . . .

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97 97 103 113 115 118 121 125

viii

5 Field Systems on Composite Manifolds 5.1 Geometry of composite manifolds . . . . . . 5.2 Hamiltonian systems on composite manifolds 5.3 Classical Berry’s oscillator . . . . . . . . . . 5.4 Higgs fields . . . . . . . . . . . . . . . . . . 5.5 Gauge gravitation theory . . . . . . . . . . . 5.6 Fermion fields . . . . . . . . . . . . . . . . . 5.7 Fermion-gravitation complex . . . . . . . . . Bibliography

CONTENTS

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131 . 132 . 136 . 138 . 140 . 144 . 149 . 153 157

Introduction At present, there are three different Hamiltonian approaches to classical field theory: • the standard Hamiltonian machinery in fashion of the instantaneous Hamiltonian formalism, • the De Donder Hamiltonian formalism in the framework of the calculus of variations, • the multimomentum Hamiltonian formalism which is the polysymplectic generalization of the conventional Hamiltonian formalism in mechanics to fibred manifolds over an n-dimensional parameter space X, not only R. Constraint field systems are the main targets of the Hamiltonian approaches to field theory. All contemporary realistic field models meet constraints. These are gauge theory, Dirac fermion fields, Higgs fields and gravity. Their Lagrangian densities are never regular. One is hoped that, in field theory as like as in mechanics, the Hamilton equations will be most appropriate. In the straightforward manner, the standard Hamiltonian formalism applied to field theory comes to the instantaneous Hamiltonian machinery where canonical coordinates are field functions at a given instant of time.21 As a consequence, this formalism leads to dynamics which is played out on infinite-dimensional phase spaces.1,21,30 When applied to constraint field systems, it follows the Dirac-Bergman procedure generalized to infinite-dimensional symplectic manifolds.4 The major goal of the instantaneous Hamiltonian formulation of field theory consists in establishing simultaneous commutation relations for quantum fields. In contrast with the conventional Hamiltonian machinery, the De Donder Hamiltonian formalism and the multimomentum Hamiltonian one exemplify the generalized Hamiltonian dynamics where evolution is not merely a time evolution directed by the Poisson bracket, but it is governed by partial differential equations where temporal and spatial coordinates enter on equal footing. Both of these Hamiltonian approaches are covariant, for the first order canonical momenta correspond to partial 1

2

INTRODUCTION

derivatives of fields with respect to all the world coordinates, not only the temporal one. Their geometric frameworks lie in different multisymplectic or polysymplectic generalizations of symplectic geometry in analytical mechanics. We follow the generally accepted geometric description of classical fields by sections of fibred manifolds π:Y →X whose base X is regarded generally as an n-dimensional parameter space. Lagrangian and Hamiltonian formalisms on fibred manifolds utilize the language of jet spaces. Recall that, given a fibred manifold Y → X, the k-jet space J k Y of Y comprises the equivalence classes jxk s, x ∈ X, of sections s of Y identified by the first (k + 1) terms of their Taylor series at points x ∈ X. One exploits the well-known facts that: (i) the k-jet space of sections of a fibred manifold Y is a finitedimensional smooth manifold and (ii) a k-order differential operator on sections of a fibred manifold Y can be described as a morphism of J k Y to a vector bundle over X. As a consequence, the dynamics of field systems is played out on finite-dimensional configuration and phase spaces. Moreover, this dynamics is phrased in the geometric terms due to the 1:1 correspondence between sections of the jet bundle J 1 Y → Y and connections on the fibred manifold Y → X. The present book covers only first order Lagrangian and Hamiltonian systems, for the most of contemporary field models are described by first order Lagrangian densities. Only classical gravity with nonspin matter sources utilizes traditionally the second order Hilbert-Einstein Lagrangian density. One can reduce its order by either discarding a certain divergence term or using the Palatini variables when a world metric and a world connection are regarded on equal footing. Moreover, the contemporary concept of gravitational interaction is founded on the gauge gravitation theory with two types of gravitational variables.39,43,50 These are tetrad gravitational fields and gauge gravitational potentials identified with reduced Lorentz connections. They correspond to different matter sources: the energy-momentum tensor and the spin current. At present, all Lagrangian densities of classical and quantum gravity are of the first order in these fields. In the framework of the first order Lagrangian formalism, the finite-dimensional configuration space of sections of a fibred manifold Y → X is the jet manifold J 1 Y coordinatized by (xµ , y i , yµi ) where (xλ , y i ) are fibred coordinates of Y and yµi are so-called derivative coordinates or velocities. A first order Lagrangian density on the configuration space J 1 Y is

INTRODUCTION

3

represented by an exterior form L = L(xµ , y i , yµi )ω, ω = dx1 ∧ ... ∧ dxn . In the calculus of variations, also the Lepagean equivalents of a Lagrangian density play the prominent role.5,14,20,25,29,32,34 We follow the De Donder-Weyl approach to the calculus of variations when the first order Lepagean equivalent is choosen to be the Poincaré-Cartan form ΞL = πiλ dy i ∧ ωλ − πiλ yλi ω + Lω, πiλ = ∂iλ L,

ωλ = ∂λ cω.

It is the Poincaré-Cartan form which is the Lagrangian counterpart of Hamiltonian forms in both the De Donder Hamiltonian formalism and the multimomentum Hamiltonian one. In the framework of the first order Lagrangian formalism, there exist two different morphisms called the Legendre morphisms which lead to different candidates for a phase space of fields. b of (i) The Poincaré-Cartan form ΞL yields the associated bundle morphism Ξ L 1 the configuration space J Y to the bundle n−1

Z = ∧ T ∗X ∧ T ∗Y over Y . This bundle is endowed with the corresponding induced coordinates (xλ , y i , pλi , p) such that b = (π µ , L − π µ y i ). (pµi , p) ◦ Ξ L i i µ

This carries the canonical form Ξ = pω + pλi dy i ∧ ωλ and the corresponding multisymplectic form dΞ. The bundle Z is the phase space of the De Donder Hamiltonian formalism.2,7,9,10,13,20,29,31,34 In case of mechanics when X = R, the form Ξ reduces to the Liouville form Ξ = −Edt + pi dy i

4

INTRODUCTION

of the homogeneous formalism where E is the energy variable.18 The De Donder Hamiltonian machinery however fails to be formulated intrinsically and is derived from the Lagrangian formalism on fibred manifolds. As a consequence, it has no advantages of describing constraint field systems, otherwise the multimomentum Hamiltonian formalism which this book is devoted to. (ii) Given a Lagarngian density L, we have the associated Legendre morphism b L of the configuration space J 1 Y to the Legendre bundle n

Π = ∧ T ∗X ⊗ T X ⊗ V ∗Y Y

Y

over Y . This bundle is provided with the corresponding induced coordinates (xλ , y i , pλi ) such that b = πµ. pµi ◦ L i

This is the phase space on which the multimomentum Hamiltonian formalism is founded. The Legendre bundle Π carries the generalized Liouville form θ = −pλi dy i ∧ ω ⊗ ∂λ and the corresponding multisymplectic form Ω = dpλi ∧ dy i ∧ ω ⊗ ∂λ . In case of X = R, the forms θ and Ω recover respectively the standard Liouville form and the standard symplectic form in analytical mechanics. Note that, in the multimomentum Hamilton formalism, the multisymplectic geometry has been developed as generalization of the symplectic structure generated by the classical Poisson brackets in Hamiltonian dynamics,26,28 but finally, without appealing to spaces of states.7,22,51−52,53 In the literature, the forms like Ω are often termed the polysymplectic forms22 in contrast with the multisymplectic form Ξ. Building on the multisymplectic form Ω, one can develop the intrinsic formulation of the multimomentum Hamiltonian formalism in terms of Hamiltonian connections.22,52,53 The Hamiltonian connections play the role similar to the Hamiltonian vector fields in the symplectic geometry. We shall say that a connection γ on the fibred Legendre manifold Π → X is a Hamiltonian connection if the exterior form γcΩ is closed. Then, a multimomentum Hamiltonian form H on Π is defined to be an exterior form such that dH = γcΩ

(0.1)

INTRODUCTION

5

for some Hamiltonian connection γ. The crusial point consists in the fact that every multimomentum Hamiltonian form admits splitting f ω = pλ dy i ∧ ω − Hω H = pλi dy i ∧ ωλ − pλi Γiλ ω − H Γ λ i

(0.2)

where Γ is some connection on the fibred manifold Y . Moreover, every multimomentum Hamiltonian form itself sets up the associated connection on Y and, so meets the canonical splitting (0.2). One can think of the splitting (0.2) as being a workable definition of multimomentum Hamiltonian forms. Given a multimomentum Hamiltonian form H [Eq.(0.2)], the equality (0.1) leads to the first order partial differential equations ∂λ ri (x) = ∂λi H, ∂λ riλ (x) = −∂i H for sections r of the fibred Legendre manifold Π → X. We call them the Hamilton equations. Note that the phase space Z of the De Donder Hamiltonian formalism is the 1-dimensional affine bundle over the Legendre bundle Π,7 and there is the 1:1 correspondence between the global sections of this bundle and the multimomentum Hamiltonian forms H on the Legendre manifold Π. Every multimomentum Hamiltonian form H on Π is the pullback of the canonical form Ξ on Z by such a section. b of the If a Lagrangian density is hyperregular (i.e., the Legendre morphism L 1 configuration space J Y to the phase space Π is diffeomorphism), the Lagrangian formalism on fibred manifolds and the multimomentum Hamiltonian formalism are naturally equivalent to each other. In this case, there exists the unique multimomentum Hamiltonian form such that the first order Euler-Lagrange equations and the corresponding Hamilton equations are equivalent, otherwise in case of degenerate Lagrangian densities. In field theory, if a Lagrangian density is not regular, the system of the EulerLagrange equations becomes underdetermined and requires a supplementary gaugetype condition. In gauge theory, these supplementary conditions are the familiar gauge conditions. Such a gauge condition is introduced by hand and singles out a representative from each gauge coset. In general case, the gauge-type conditions however remain elusive. In the framework of the multimomentum Hamiltonian formalism, they appear automatically as a part of the Hamilton equations. The key point consists in the fact that, given a degenerate Lagrangian density, one must consider a family of different associated multimomentum Hamiltonian forms in order that solutions of the corresponding Hamilton equations exaust solutions of the

6

INTRODUCTION

Euler-Lagrange equations. For a wide class of degenerate Lagrangian densities, such complete families exist at least locally, and the adequate relations between Lagrangian and multimomentum Hamiltonian formalisms can be given. In particular, we spell out the models with degenerate quadratic and affine Lagrangian densities. The most of contemporary field models belong to these types. As a consequence, the tools are at hand to canonically analize constraint field systems on covariant and finite-dimensional levels. The present book is not concerned with multimomentum quantum field theory. This has been hampered by the lack of satisfactory commutation relations between multimomentum canonical variables.7,22 At the same time, the multimomentum Hamiltonian formalism may be extended to quantum field theory if one considers chronological forms, but not commutation relations.54

Chapter 1 Geometric Preliminary In the framework of the geometric approach to field theory, classical fields are identified with sections of fibred manifolds. In gauge theory, these are fibre bundles with a structure group. Therefore, the most of differential geometric methods utilized in contemporary field theory are based on principal bundles and principal connections. The literature on this subject is extensive. Our approach is phrased in terms of jet manifolds, without appealing to transformation groups. This Chapter aims to summarize the necessary prerequisites on jet manifolds and general connections.34,35,53,55 All morphisms throughout the book are differentiable mappings of class C ∞ . Manifolds are real, Hausdorff, finite-dimensional, second-countable (hence paracompact) and, unless otherwise stated, connected. Note that these are usual conditions that one requires of manifolds in the framework of the differential geometric approach to field theory. Since manifolds are paracompact, we can refer to the well-known theorems of extension of morphisms and existence of global sections of bundles. Connected manifolds simplify several mathematical constructions. In the book, many relations are specified in terms of local coordinates, but one can always justify that the particular choice of a coordinate system is not matter for them, unless otherwise stated, all entities are globally defined. The conventional symbols ⊗, ∨ and ∧ are utilized for the tensor, symmetric and exterior products respectively. By c is meant the interior product of a multivector on the right and a form on the left. The symbol ∂BA denotes the partial derivative with respect to coordinates with indices B A. Given a manifold M with an atlas of local coordinates (z λ ), the tangent bundle T M of M [resp. the cotangent bundle T ∗ M of M ] is provided with the atlas of 7

8

CHAPTER 1. GEOMETRIC PRELIMINARY

induced coordinates (z λ , z˙ λ ) [resp. (z λ , z˙λ )] relative to the holonomic fibre bases {∂λ } [resp. {dz λ }]. If f : M → M 0 is a manifold mapping, by T f : T M → T M 0, ∂f λ z˙ 0λ ◦ f = α z˙ α , ∂z is meant the tangent morphism to f . Throughout the book, we handle the following types of manifold mappings f when the tangent morphism T f to f meets the maximal rank. These are immersion, submersion and local diffeomorphism if f is both immersion and submersion. Recall that a mapping f : M → M 0 is called immersion [resp. submersion] at a point z ∈ M when the tangent morphism T f to f is injection [resp. surjection] of the tangent space Tz M to M at z to the tangent space Tf (z) M 0 . A manifold mapping f of M is termed immersion [resp. submersion] if it is immersion [resp. submersion] at all points of the manifold M . A triple f : M → M 0 is called the submanifold [resp. the fibred manifold or simply the fibration] if f is both immersion and injection [resp. both submersion and surjection] of M to M 0 . A submanifold which also is a topological subspace is termed an imbedded submanifold. In particular, every open subset U of a manifold M is endowed with the manifold structure such that the canonical injection iU : U ,→ M is imbedding. Given a manifold product, we denote by pr 1 and pr 2 the canonical surjections pr 1 : A × B → A,

1.1

pr 2 : A × B → B.

Fibred manifolds

Throughout the book, by π:Y →X

(1.1)

is meant a fibred manifold over an n-dimensional base X. We shall utilize symbols y and x for points of Y and X respectively. In field theory, X makes the sense of a parameter space, e.g., a world manifold. Coordinates of X will be denoted by xλ . We shall refer to the notations ω = dx1 ∧ ... ∧ dxn ,

ωλ = ∂λ cω.

1.1. FIBRED MANIFOLDS

9

A fibred manifold Y → X, by definition, is provided with an atlas of fibred coordinates (xλ , y i ), λ

xλ → x0 (xµ ), i

y i → y 0 (xµ , y j ),

(1.2)

compatible with the fibration (1.1). A fibred manifold Y → X is called locally trivial if there exist an open covering {Uξ } of the base X and a fibred coordinate atlas of Y over its open covering {π −1 (Uξ )}. In other words, all points of a fibre π −1 (x) of Y belong to the same fibred coordinate chart. By a differentiable fibre bundle (or simply a bundle), we mean a locally trivial fibred manifold (1.1) provided with a family of equivalent bundle atlases Ψ = {Uξ , ψξ }, ψξ : π −1 (Uξ ) → Uξ × V, where V is the standard fibre of Y . Recall that two bundle atlases are called equivalent if their union also is a bundle atlas. If Y → X is a bundle, the fibred coordinates (1.2) of Y are assumed to be bundle coordinates associated with a bundle atlas Ψ = {ψξ } of Y , that is, y i (y) = (v i ◦ pr 2 ◦ ψξ )(y),

π(y) ∈ Uξ ,

where v i are coordinates of the standard fibre V of Y . Given fibred manifolds Y → X and Y 0 → X 0 , by a fibred morphism is meant a fibre-to-fibre manifold mapping Φ : Y → Y 0 over a manifold mapping f : X → X 0 . If f = Id X is the identity morphism of X, the fibred morphism Φ is termed briefly the fibred morphism Y → Y 0 over X. X In particular, let XX denotes the fibred manifold X → X. Given a fibred manifold Y → X, a fibred morphism XX ,→ Y over X, by definition, is a global section of Y → X. It is a closed imbedded submanifold. More generally, let N be an imbedded submanifold of X. A fibred morphism NN → Y over N ,→ X is called a local section (or simply a section) of the fibred manifold Y → X over N . For each point x ∈ X, a fibred manifold, by definition, has a section over an open neighborhood of x. Remark. In accordance with the well-known theorem, if a fibred manifold Y → X has a global section, every section of Y over a closed imbedded submanifold

10


N of X is extended to a global section of Y due to the properties which are required of a manifold. 2 If fibred morphism Y → Y 0 over X is a submanifold, Y → X is called the fibred submanifold of Y 0 → X. Fibred imbeddings and fibred diffeomorphisms are usually termed monomorphisms and isomorphisms respectively. Given a fibred manifold Y → X, every manifold mapping f : X 0 → X yields the pullback fibred manifold f ∗ Y → X 0 (or simply the pullback) comprising all pairs (y, x0 ) ∈ Y × X 0 ,

π(y) = f (x0 ),

together with the surjection (y, x0 ) → x0 . In other words, the fibre of f ∗ Y at a point x0 ∈ X 0 consists with the fibre of Y at f (x) ∈ X. Every section s of the fibred manifold Y → X gives rise to the corresponding pullback section (f ∗ s)(x0 ) = ((s ◦ f )(x0 ), x0 ),

x0 ∈ X 0 ,

of the pullback f ∗ Y → X 0 . In particular, if the mapping f is a submanifold of X, the pullback f ∗ Y is called the restriction Y |f (X 0 ) of the fibred manifold Y to the submanifold f (X 0 ) ⊂ X or the portion of Y over f (X 0 ). The product of fibred manifolds π : Y → X and π 0 : Y 0 → X over X, by definition, is the total space of the pullbacks ∗

π∗Y 0 = π0 Y = Y × Y 0. X

A composite fibred manifold (or simply a composite manifold) is defined to be composition of surjective submersions πΣX ◦ πY Σ : Y → Σ → X.

(1.3)

It is the fibred manifold Y → X provided with the particular fibred coordinate atlases: (xλ , σ m , y i ) 0λ

λ

(1.4)

µ

x = f (x ), m σ 0 = f m (xµ , σ n ), i

y 0 = f i (xµ , σ n , y j ),

(1.5)

where (xµ , σ m ) are fibred coordinates of the fibred manifold Σ → X and transformation laws of coordinates σ m are independent of coordinates y i .


11

In particular, let T Y → Y be the tangent bundle of a fibred manifold Y → X. Its total space is the composite manifold T Y → T X → X. Given the fibred coordinates (1.2) of Y , the corresponding induced coordinates of T Y are (xλ , y i , x˙ λ , y˙ i ). The tangent bundle T Y → Y of a fibred manifold Y has the subbundle V Y = Ker T π which is called the vertical tangent bundle of Y . This subbundle consists of tangent vectors to fibres of Y → X. It is provided with the induced coordinates (xλ , y i , y˙ i ). The vertical cotangent bundle V ∗ Y → Y of Y , by definition, is the vector bundle dual to the vertical tangent bundle V Y → Y . It is not a subbundle of T ∗ Y . The fibre bases for V ∗ Y dual to the fibre bases {∂i } for V Y will be denoted {dy i }. Compare the transformation laws ∂y 0 i j dy , ∂y j ∂y 0 i j ∂y 0 i µ 0i dy = dy + µ dx . ∂y j ∂x i

dy 0 =

With V Y and V ∗ Y , we have the following exact sequences of bundles over a fibred manifold Y → X: 0 → V Y ,→ T Y → Y × T X → 0, Y

0 → Y × T ∗ X ,→ T ∗ Y → V ∗ Y → 0. Y

X

(1.6a)

X

(1.6b)

For the sake of simplicity, we shall further denote the products Y × T ∗X

Y × T X, X

X

and other pullbacks involving T X and T ∗ X by the symbols T X and T ∗ X respectively. Every splitting Y × T X ,→ T Y, X

Y

∂λ 7→ ∂λ + Γiλ (y)∂i ,

12


and V ∗ Y ,→ T ∗ Y dy i 7→ dy i − Γiλ (y)dxλ , of the exact sequences (1.6a) and (1.6b), by definition, corresponds to a certain connection Γ on the fibred manifold Y → X, and vice versa. Let Φ : Y → Y 0 be a fibred morphism over f . The tangent morphism T Φ to Φ reads λ i (x˙0 , y˙0 ) ◦ T Φ = (∂µ f λ x˙ µ , ∂µ Φi x˙ µ + ∂j Φi y˙ j ).

This is both a linear bundle morphism over Φ and a fibred morphism over the tangent morphism T f to f . Its restriction to the vertical tangent subbundle V Y of T Y yields the vertical tangent morphism V Φ : V Y → V Y 0, i y˙0 ◦ V Φ = ∂j Φi y˙ j .

(1.7)

Vertical tangent bundles of fibred manifolds utilized in field theory possess the following simple structure as a rule. One says that a fibred manifold Y → X admits vertical splitting if there exists the linear isomorphism α : V Y →Y ×Y Y

(1.8)

X

where Y → X is a vector bundle. The fibred coordinates (1.2) of Y are called adapted to the vertical splitting (1.8) if the induced coordinates of the vertical tangent bundle V Y take the form (xµ , y i , y˙ i = y i ◦ α) where (xµ , y i ) are bundle coordinates of Y . In this case, coordinate transformations y˙ i → y˙ 0i are independent of the coordinates y i . In particular, a vector bundle Y → X has the canonical vertical splitting V Y = Y × Y.

(1.9)

X

An affine bundle Y → X modelled on a vector bundle Y → X has the canonical vertical splitting V Y = Y ×Y . X

(1.10)


13

It is readily observed that linear bundle coordinates of a vector bundle and affine bundle coordinates of an affine bundle are always adapted to the canonical vertical splittings (1.9) and (1.10) respectively. We shall refer to the following fact. Lemma 1.1. Let Y and Y 0 be fibred manifolds over X and Φ : Y → Y 0 a fibred morphism over X. Let V Φ be the vertical tangent morphism to Φ. If Y 0 admits vertical splitting 0

VY0 =Y0×Y , then there exists the linear bundle morphism V Φ : V Y →Y × Y

0

(1.11)

Y

over Y given by the coordinate expression i

y 0 ◦ V Φ = ∂j Φi y˙ j . 2 By differential forms (or symply forms) on a fibred manifold, we shall mean exterior, tangent-valued and pullback-valued forms. Recall that a tangent-valued r-form on a manifold M is defined to be a section φ = φµλ1 ...λr dz λ1 ∧ . . . ∧ dz λr ⊗ ∂µ r

of the bundle ∧ T ∗ M ⊗ T M. In particular, tangent-valued 0-forms are vector fields on M . Remark. There is the 1:1 correspondence between the tangent-valued 1-forms on M and the linear bundle morphisms T M → T M or T ∗ M → T ∗ M over M : θ : M → T ∗ M ⊗ T M, θ : Tz M 3 t 7→ tcθ(z) ∈ Tz M, θ : Tz∗ M 3 t∗ 7→ θ(z)ct∗ ∈ Tz∗ M. For instance, Id T M corresponds to the canonical tangent-valued 1-form θM = dz λ ⊗ ∂λ , on the manifold M . 2

∂λ cθM = ∂λ ,

(1.12)

14

CHAPTER 1. GEOMETRIC PRELIMINARY r

Let Λ T ∗ (M ) be the sheaf of exterior r-forms on a manifold M and T (M ) the sheaf of vector fields on M . Tangent-valued r-forms on M constitute the sheaf r

∗ Λ T (M ) ⊗ T (M ).

This sheaf is brought into the sheaf of graded Lie algebras with respect to the Frölicher-Nijenhuis (F-N) bracket which generalizes the familiar commutation bracket of vector fields as follows. The F-N bracket is defined to be the sheaf morphism r

s

r+s

∗ ∗ ∗ Λ T (M ) ⊗ T (M ) × Λ T (M ) ⊗ T (M ) → Λ T (M ) ⊗ T (M ),

[φ, σ] = −(−1)rs [σ, φ] = [α ⊗ u, β ⊗ v] = α ∧ β ⊗ [u, v] + α ∧ Lu β ⊗ v − (−1)rs β ∧ Lv α ⊗ u +(−1)r (vcα) ∧ dβ ⊗ u − (−1)rs+s (ucβ) ∧ dα ⊗ v, r

α ∈ Λ T ∗ (M ),

s

β ∈ Λ T ∗ (M ),

u, v ∈ T (M ),

where Lu and Lv are the Lie derivatives of exterior forms. We have the coordinate expression [φ, σ] = (φνλ1 ...λr ∂ν σλµr+1 ...λr+s −(−1)rs σλν1 ...λs ∂ν φµλs+1 ...λr+s − rφµλ1 ...λr−1 ν ∂λr σλνr+1 ...λr+s +(−1)rs sσλµ1 ...λs−1 ν ∂λs φνλs+1 ...λr+s )dz λ1 ∧ . . . ∧ dz λr+s ⊗ ∂µ . Given a tangent-valued form φ, the Nijenhuis differential is defined to be the sheaf morphism dφ : σ 7→ dφ σ = [φ, σ].

(1.13)

In particular, if φ = u is a vector field, the Nijenhuis differential du recovers the Lie derivative of tangent-valued forms Lu σ = [u, σ] = (uν ∂ν σλµ1 ...λs − σλν1 ...λs ∂ν uµ +sσλµ1 ...λs−1 ν ∂λs uν )dxλ1 ∧ . . . ∧ dxλs ⊗ ∂µ . Remark. The Nijehuis differential (1.13) can be extended to exterior forms σ on a manifold M by the rule dφ σ = φcdσ + (−1)r d(φcσ) = (φνλ1 ...λr ∂ν σλr+1 ...λr+s +(−1)rs sσλ1 ...λs−1 ν ∂λs φνλs+1 ...λr+s )dz λ1 ∧ . . . ∧ dz λr+s .

(1.14)


15

In particular, if φ = θM , the familiar exterior differential dθM σ = dσ is reproduced. 2 On a fibred manifold Y → X, the following particular subsheafs of exterior and tangent-valued forms are usually considered: • exterior horizontal forms r

φ : Y → ∧ T ∗ X; • tangent-valued horizontal forms r

φ : Y → ∧ T ∗ X ⊗ T Y, Y

φ = dx

λ1

∧ . . . ∧ dxλr ⊗ (φµλ1 ...λr ∂µ + φiλ1 ...λr ∂i );

• tangent-valued projectable horizontal forms φ = dxλ1 ∧ . . . ∧ dxλr ⊗ (φµλ1 ...λr (x)∂µ + φiλ1 ...λr (y)∂i ) projected onto tangent-valued forms φX = dxλ1 ∧ . . . ∧ dxλr ⊗ φµλ1 ...λr (x)∂µ on X where φX ◦ π = T π ◦ φ; • vertical-valued horizontal forms r

φ : Y → ∧ T ∗ X ⊗ V Y, Y

φ = φiλ1 ...λr dxλ1 ∧ . . . ∧ dxλr ⊗ ∂i .

16


Vertical-valued horizontal 1-forms φ : Y → T ∗ X ⊗ V Y, Y

φ=

φiλ dxλ

⊗ ∂i ,

on Y → X are called soldering forms. By pullback-valued forms on a fibred manifold Y → X are meant the morphisms r

φ : Y → ∧ T ∗ Y ⊗ T X,

(1.15)

Y

φ = dxλ1 ∧ . . . ∧ dxλr ⊗ φµλ1 ...λr (y)∂µ , and r

φ : Y → ∧ T ∗ Y ⊗ V ∗ Y.

(1.16)

Y

It must be noted that the forms (1.15) are not tangent-valued forms on Y since the pullback Y × T X is not a subbundle of T Y . In particular, we shall refer to the pullback π ∗ θX of the canonical form θX on the base X by π onto Y . This is a pullback-valued horizontal 1-form on Y → X which we denote by the same symbol θX : Y → T ∗ X ⊗ T X, Y

λ

θX = dx ⊗ ∂λ .

(1.17)

As about the pullback forms (1.16), let us point at the canonical bundle monomorphism n

n+1

∧ T ∗ X ⊗ V ∗ Y ,→ ∧ T ∗ Y, Y i

Y

ω ⊗ dy 7→ ω ∧ dy i .

(1.18)

All horizontal n-forms on a fibred manifold Y → X are called horizontal densities.

1.2

Jet spaces

The key ingredients in the jet machinery which this Section briefs are the following. • Jet spaces of section of fibred manifolds are finite-dimensional manifolds. • One can express jets in terms of familiar tangent-valued forms.

1.2. JET SPACES

17

• There exists the canonical horizontal splitting of the exact sequences (1.6a) and (1.6b) over jet manifolds. We exploit the multi-index Λ, | Λ |= k, for symmetrized collections (λ1 ...λk ). By Λ + λ is meant the symmetrized collection (λ1 ...λk λ). Definition 1.2. The k-order jet space J k Y of sections of a fibred manifold Y → X is defined to comprise all equivalence classes jxk s, x ∈ X, of sections s of Y so that sections s and s0 belong to the same class jxk s iff i

∂Λ si (x) = ∂Λ s0 (x),

0 ≤| Λ |≤ k.

2 In other words, sections of Y → X are identified by the k + 1 terms of their Teylor series at points of X. It is readily observed that the particular choice of a coordinate atlas is not matter for the definition above. There are several equivalent ways to provide J 1 Y with a manifold structure. The result is the following. Given fibred coordinates (1.2) of the fibred manifold Y → X, the k-order jet space J k Y is endowed with the atlas of the adapted coordinates (xλ , yΛi ),

0 ≤| Λ |≤ k,

which bring J k Y into a finite-dimensional smooth manifold satisfying the conditions which we require of a manifold. The k-order jet manifold J k Y possesses the natural surjections πlk : J k Y → J l Y which form the composite fibration πk : J k Y → J k−1 Y → ... → Y → X. In particular, the first order jet manifold J 1 Y of sections of the fibration Y → X (or simply the jet manifold of Y ) consists of the equivalence classes jx1 s, x ∈ X, of sections s of Y so that sections s and s0 belong to the same class jx1 s iff T s |Tx X = T s0 |Tx X , that is, i

si (x) = s0 (x),

i

∂µ si (x) = ∂µ s0 (x).

18


There are the natural surjections π1 : J 1 Y 3 jx1 s 7→ x ∈ X, π01 : J 1 Y 3 jx1 s 7→ s(x) ∈ Y

(1.19) (1.20)

which form the composite manifold π1 = π ◦ π01 : J 1 Y → Y → X. The surjection (1.20) is a bundle. We call it the jet bundle. If Y → X is a bundle, so is the surjection (1.19). Given the fibred coordinates (1.2) of the fibration Y → X, the first order jet manifold J 1 Y of Y is provided with the adapted coordinate atlas (xλ , y i , yλi ), (xλ , y i , yλi )(jx1 s) = (xλ , si (x), ∂λ si (x)), ∂y 0 i j ∂y 0 i ∂xµ 0i y λ = ( j yµ + µ ) 0 λ . ∂y ∂x ∂x

(1.21)

(1.22)

A glance at the transformation law (1.22) shows that the jet bundle J 1 Y → Y is an affine bundle. It is modelled on the vector bundle T ∗ X ⊗ V Y → Y.

(1.23)

Y

Remark. It is readily observed that J 1 (Y × Y 0 ) = J 1 Y × J 1 Y 0 . X

X

2 The second order jet manifold J 2 Y of a fibred manifold Y → X is defined to be the union of all equivalence classes jx2 s of sections s of Y → X which are identified by their values and values of their first order and second order partial derivatives at points of X. The second order jet manifold J 2 Y is endowed with the adapted coordinates i i ), (xλ , y i , yλi , yλµ = yµλ

yλi (jx2 s) = ∂λ si (x), i yλµ (jx2 s) = ∂µ ∂λ si (x). In applications to field theory, we handle first order and second order jet manifolds as a rule.

1.2. JET SPACES

19

Let Y and Y 0 be fibred manifolds over X. Every fibred morphism Φ : Y → Y 0 over a diffeomorphism f of X gives rise to the morphism J k Φ : J k Y 3 jxk s 7→ jfk(x) (Φ ◦ s ◦ f −1 ) ∈ J k Y 0 of the k-order jet manifolds. The morphism J k Φ is called the k-order jet prolongation of Φ. In particular, the first order jet prolongation (or simply the jet prolongation) of Φ reads J 1Φ : J 1Y → J 1Y 0, i

y 0 λ ◦ J 1 Φ = ∂λ (Φi ◦ f −1 ) + ∂j (Φi yλj ◦ f −1 ).

(1.24)

It is both an affine bundle morphism over Φ and a fibred morphism over the diffeomorphism f . Remark. Jet prolongations of fibred morphisms satisfy the chain rule J 1 (Φ0 ◦ Φ) = J 1 Φ0 ◦ Φ. If Φ is injection (resp. surjection), so is J 1 Φ. 2 In particular, every section s of a fibred manifold Y → X admits the k-order jet prolongation to the section (J k s)(x) = jxk s of the fibred jet manifold J k Y → X. For instance, the first order jet prolongation of s to the section s = J 1 s of the fibred jet manifold J 1 Y → X reads (y i , yλi ) ◦ J 1 s = (si (x), ∂λ si (x)). Evidently, not every section s of J 1 Y → X is the jet prolongation of some section of Y → X. We call J 1 s the holonomic section of the fibred jet manifold J 1 Y → X. Every vector field u = uλ ∂λ + ui ∂i on a fibred manifold Y → X has the jet lift to the vector field u = r1 ◦ J 1 u : J 1 Y → J 1 T Y → T J 1 Y, u = uλ ∂λ + ui ∂i + (∂λ ui + yλj ∂j ui − yµi ∂λ uµ )∂iλ ,

(1.25)

20


on the fibred jet manifold J 1 Y → X where J 1 T Y is the jet manifold of the fibred manifold T Y → X and r1 is the canonical fibred morphism r1 : J 1 T Y → T J 1 Y, y˙ λi ◦ r1 = (y˙ i )λ − yµi x˙ µλ , over J 1 Y × T Y . In particular, there exists the canonical isomorphism Y

V J 1 Y = J 1 V Y, y˙ λi = (y˙ i )λ ,

(1.26)

where J 1 V Y is the jet manifold of the fibred manifold V Y → X and V J 1 Y is the vertical tangent bundle of the fibration J 1 Y → X. As a consequence, the jet lift (1.25) of a vertical vector field u on the fibred manifold Y → X consists with its first order jet prolongation u = J 1 u = ui ∂i + (∂λ ui + yλj ∂j ui )∂iλ onto the fibred jet manifold J 1 Y → X. These results are generalized to case of higher order jets. Given the k-order jet manifold J k Y of Y → X, there are the canonical fibred morphism rk : J k T Y → T J k Y and the canonical isomorphism V J kY = J kV Y where J k T Y and J k V Y are the k-order jet manifolds of the fibred manifolds T Y → X and V Y → X respectively, and V J k Y is the vertical tangent bundle of the fibration J k Y → X. As a consequence, every vector field u on a fibred manifold Y → X has the k-order jet lift to the vector field u = rk ◦ J k u : J k Y → T J k Y, u = uλ ∂λ + ui ∂i + uiΛ ∂iΛ , i i uiΛ+λ = (∂λ + yΣ+λ ∂iΣ )uiΛ − yΛ+µ uµ ,

0 ≤| Σ |≤ k,

on J k Y . If Y → X is a bundle endowed with some algebraic structure, this algebraic structure also has the jet prolongation onto the jet bundle J 1 Y → X.

1.2. JET SPACES

21

In particular, if Y → X is a vector bundle, J 1 Y → X does as well. Let Y be a vector bundle, Y ∗ → X the dual to Y and hi the linear fibred morphism hi : Y × Y ∗ → X × R, X

r ◦ hi = y i yi , where (y i ) and (yi ) are the dual bundle coordinates of Y and Y ∗ respectively. The jet prolongation of hi is the linear fibred morphism J 1 hi : J 1 Y × J 1 Y ∗ → T ∗ X × R, X i

X

1

x˙ µ ◦ J hi =

yµi yi

+ y yiµ .

Let Y → X and Y 0 → X be vector bundles and ⊗ the bilinear fibred morphism ⊗ : Y × Y 0 → Y ⊗ Y 0, X i k

X ik

X

y ◦⊗=y y . The jet prolongation of ⊗ is the bilinear fibred morphism J 1 ⊗ : J 1 Y × J 1 Y 0 → J 1 (Y ⊗ Y 0 ), X

X

yµik

1

◦J ⊗=

yµi y k

+

X

y i yµk .

If Y is an affine bundle modelled on a vector bundle Y , then J 1 Y → X is an affine bundle modelled on the vector bundle J 1 Y → X. Application of the jet formalism to differential geometry is founded on two canonical morphisms. Proposition 1.3. There exist the following bundle monomorphisms: (i) the contact map λ : J 1 Y ,→ T ∗ X ⊗ T Y, Y

Y

λ

λ = dx ⊗ ∂bλ = dxλ ⊗ (∂λ + yλi ∂i ),

(1.27)

(ii) the complementary morphism θ1 : J 1 Y ,→ T ∗ Y ⊗ V Y, Y

Y

b i ⊗ ∂ = (dy i − y i dxλ ) ⊗ ∂ . θ1 = dy i i λ

2

(1.28)

22


These canonical morphisms enable us to handle jets as familiar tangent-valued forms. Moreover, one can start elaboration of the jet manifold machinery from these monomorphisms. In particular, the relation λ ◦ J 1s = T s for every section s of a fibred manifold Y → X holds. The canonical morphisms (1.27) and (1.28) give rise to the bundle monomorphisms b : J 1 Y × T X 3 ∂ 7→ ∂b ∈ J 1 Y × T Y, λ λ λ

(1.29)

b i ∈ J Y × T ∗Y θb1 : J 1 Y × V ∗ Y 3 dy i 7→ dy

(1.30)

X

Y 1

Y

Y

b i [Eq.(1.28)] the vertical over J 1 Y . We call ∂bλ [Eq.(1.27)] the total derivative and dy differential. The morphism (1.29) determines the canonical horizontal splitting of the pullback b X) ⊕ V Y, J 1 Y × T Y = λ(T Y

(1.31)

J 1Y

x˙ λ ∂λ + y˙ i ∂i = x˙ λ (∂λ + yλi ∂i ) + (y˙ i − x˙ λ yλi )∂i . Similarly, the morphism (1.30) does the dual canonical horizontal splitting of the pullback J 1 Y × T ∗ Y = T ∗ X ⊕ θb1 (V ∗ Y ), Y

(1.32)

J 1Y

x˙ λ dxλ + y˙ i dy i = (x˙ λ + y˙ i yλi )dxλ + y˙ i (dy i − yλi dxλ ). In other words, over J 1 Y , we have the canonical horizontal splitting of tangent and cotangent bundles of Y and the corresponding canonical splitting of the exact sequences (1.6a) and (1.6b). As an immediate consequence of the splittings (1.31) and (1.32), we get the following canonical horizontal splittings of • a projectable vector field u = uλ ∂λ + ui ∂i = uH + uV = uλ (∂λ + yλi ∂i ) + (ui − uλ yλi )∂i on a fibred manifold Y → X;

(1.33)

1.2. JET SPACES

23

• an exterior 1-form σ = σλ dxλ + σ i dy i = (σλ + yλi σi )dxλ + σi (dy i − yλi dxλ ); • a tangent-valued projectable horizontal form φ = dxλ1 ∧ . . . ∧ dxλr ⊗ (φµλ1 ...λr ∂µ + φiλ1 ...λr ∂i ) = dxλ1 ∧ . . . ∧ dxλr ⊗ [φµλ1 ...λr (∂µ + yµi ∂i ) + (φiλ1 ...λr − φµλ1 ...λr yµi )∂i ] and, e.g., the canonical 1-form b i⊗∂ θY = dxλ ⊗ ∂λ + dy i ⊗ ∂i = λ + θ1 = dxλ ⊗ ∂bλ + dy i i i λ λ i = dx ⊗ (∂λ + yλ ∂i ) + (dy − yλ dx ) ⊗ ∂i .

(1.34)

The splitting (1.34) implies the canonical horizontal splitting of the exterior differential d = dθY = dH + dV = dλ + dθ1 .

(1.35)

Its components dH and dV act on the pullbacks φλ1 ...λr (y)dxλ1 ∧ . . . ∧ dxλr of exterior horizontal forms φ on Y X by π01 onto J 1 Y . In this case, dH makes the sense of the total exterior differential dH φλ1 ...λr (y)dxλ1 ∧ . . . ∧ dxλr = (∂µ + yµi ∂i )φλ1 ...λr (y)dxµ ∧ dxλ1 ∧ . . . ∧ dxλr , whereas dV is the vertical differential dV φλ1 ...λr (y)dxλ1 ∧ . . . ∧ dxλr = ∂i φλ1 ...λr (y)(dy i − yµi dxµ ) ∧ dxλ1 ∧ . . . ∧ dxλr . e is an exterior horizontal density on Y → X, we have If φ = φω e i ∧ ω. dφ = dV φ = ∂i φdy

24


There exist the following second order generalizations of the contact map (1.27) and the complementary morphism (1.28) to the second order jet manifold J 2 Y : (i)

λ : J 2 Y ,→ T ∗ X ⊗ T J 1 Y, 1 J Y

J 1Y

i λ = dx ⊗ ∂bλ = dxλ ⊗ (∂λ + yλi ∂i + yµλ ∂iµ ), λ

(ii)

(1.36)

θ1 : J 2 Y ,→ T ∗ J 1 Y ⊗ V J 1 Y, 1 J Y i

J 1Y

i dxλ ) ⊗ ∂iµ . θ1 = (dy − yλi dxλ ) ⊗ ∂i + (dyµi − yµλ

(1.37)

The contact map (1.36) determines the canonical horizontal splitting of the exact sequence 0 → V J 1 Y ,→ T J 1Y → J 1 Y × T X → 0. 1 1 J Y

J Y

X

In particular, we get the canonical horizontal splitting of a projectable vector field u on J 1 Y over J 2 Y : i u = uH + uV = uλ [∂λ + yλi + yµλ ] i +[(ui − yλi uλ )∂i + (uiµ − yµλ uλ )∂iµ ].

(1.38)

Building on the morphisms (1.36) and (1.37), one can obtain the horizontal splittings of the canonical tangent-valued 1-form θJ 1 Y on J 1 Y : θJ 1 Y = dxλ ⊗ ∂λ + dy i ⊗ ∂i + dyµi ⊗ ∂iµ = λ + θ1 and the exterior differential d = dθJ 1 Y = dλ + dθ1 = dH + dV .

(1.39)

They are similar to the splittings (1.34) and (1.35). The contact maps (1.27) and (1.36) are the particular cases of the monomorphism λ : J k+1 Y ,→ T ∗ X ⊗ T J k Y, λ

λ = dx ⊗ (∂λ +

JkY i yΛ+λ ∂iΛ ),

0 ≤| Λ |≤ k.

(1.40)

The k-order contact map (1.40) determines the canonical horizontal splitting of the exact sequence 0 → V J k Y ,→ T J k Y → J k Y × T X → 0. X

1.2. JET SPACES

25

In particular, we get the canonical horizontal splitting of a projectable vector field u on J k Y → X over J k+1 Y as follows: i i u = uH + uV = uλ (∂λ + yΛ+λ ∂iΛ ) + (uiΛ − yΛ+λ uλ )∂iΛ ,

0 ≤| Λ |≤ k.

This splitting is the k-order generalization of the splittings (1.33) and (1.38). To introduce higher order jet manifolds, one can use the chain of repeated jet manifolds. Definition 1.4. The repeated jet manifold J 1 J 1 Y is defined to be the first order jet manifold of the fibred jet manifold J 1 Y → X. 2 Given the coordinates (1.21) of J 1 Y , the repeated jet manifold J 1 J 1 Y is provided with the adapted coordinates i i (xλ , y i , yλi , y(µ) , yλµ ), i

y 0 λµ =

∂y 0 iλ j ( j yαν ∂yα

+

(1.41)

∂y 0 iλ j y ∂y j ν

+

∂y 0 iλ ∂xν ) . ∂xν ∂x0 µ

There are the following two repeated jet bundles: (i)

(ii)

π11 : J 1 J 1 Y → J 1 Y, yλi ◦ π11 = yλi ,

(1.42)

J 1 π01 : J 1 J 1 Y → J 1 Y, i yλi ◦ J 1 π01 = y(λ) .

(1.43)

Their affine difference over Y yields the so-called Spencer bundle morphism δ := J 1 π01 − π11 : J 1 J 1 Y → T ∗ X ⊗ V Y, Y

Y

i x˙ λ ⊗ y˙ i ◦ δ = y(λ) − yλi .

The kernel of this morphism is the affine subbundle Jb2 Y → J 1 Y, i y(λ) = yλi ,

(1.44)

of the repeated jet bundles (1.42) and (1.43) which is modelled on the vector bundle 2

⊗ T ∗ X ⊗ V Y. J 1Y

26


It is called the sesquiholonomic jet manifold. Given the coordinates (1.41) of J 1 J 1 Y , the sesquiholonomic jet manifold Jb2 Y coordinatized by i (xλ , y i , yλi , yλµ ).

The second order jet manifold J 2 Y , in turn, is the affine subbundle of the bundle (1.44) given by the coordinate condition i i yλµ = yµλ .

It is modelled on the vector bundle 2

∨ T ∗ X ⊗ V Y. J 1Y

Thus, we have the chain affine bundle monomorphisms J 2 Y ,→ Jb2 Y ,→ J 1 J 1 Y over J 1 Y together with the canonical splitting 2

Jb2 Y = J 2 Y ⊕ (∧ T ∗ X ⊗ V Y ), J 1Y

(1.45)

Y

1 i 1 i i i i + yµλ ) + (yλµ − yµλ ). yλµ = (yλµ 2 2 In case of higher order jets, let us consider the repeated jet manifold J 1 J k Y of the fibred k-order jet manifold J k Y → X. It is provided with the adapted coordinates i (xµ , yΛi , yΛλ ),

| Λ |≤ k.

Just as in case of k = 1, there exist two fibred morphisms of J 1 J k Y to J 1 J k−1 Y . Their affine difference over J k−1 Y is the k-order Spencer morphism J 1 J k Y → T ∗ X ⊗ V J k−1 Y J k−1 Y

where V J k−1 Y is the vertical tangent bundle of the fibration J k−1 Y → X. Its kernel is the k-order sesquiholonomic jet manifold Jbk+1 Y coordinatized by i ), (xµ , yΛi , yΣλ

| Λ |≤ k,

| Σ |= k.

Proposition 1.5. There exists the chain of fibred monomorphisms J k+1 Y ,→ Jbk+1 Y ,→ J 1 J k Y

(1.46)

1.2. JET SPACES

27

together with the canonical splitting k−1

Jbk+1 Y = J k+1 Y ⊕ (T ∗ X ∧ ∨ T ∗ X ⊗ V Y ).

(1.47)

Y

JkY

2 Similarly, the higher order jet prolongations of fibred morphisms can be constructed by means of repeated jet prolongations of the less order ones. Let Φ : Y → Y 0 be a fibred morphism over a diffeomorphism of X and J 1 Φ its first order jet prolongation (1.24). One can consider the first order jet prolongation J 1 J 1 Φ of the fibred morphism J 1 Φ. It is readily observed that the restriction of the morphism J 1 J 1 Φ to the second order jet manifold J 2 Y of Y consists with the second order jet prolongation J 2 Φ of the fibred morphism Φ. In particular, the repeated jet prolongation J 1 J 1 s of a section s of Y → X is a section of the fibred jet manifold J 1 J 1 Y → X. It takes its values into J 2 Y and consists with the second order jet prolongation J 2 s of s: (J 1 J 1 s)(x) = (J 2 s)(x) = jx2 s. In general, the following integrability conditions hold. Lemma 1.6. Let s be a section of the fibration J k Y → X. Then, the following conditions are equivalent: (i) s = J k s where s is a section of Y → X, (ii) J 1 s : X → Jbk+1 Y , (iii) J 1 s : X → J k+1 Y . 2 Outline of proof: The condition (ii) takes the coordinate form siλ1 ...λm µ = ∂µ siλ1 ...λm ,

0 ≤ m < k.

It follows that s = J k (π0k ◦ s). 2 Note that the jet manifold machinery is naturally extended to complex bundles over real manifolds. Now, we shall briefly concern differential operators expressed in jet terms. In order to describe the Euler-Lagrange operators and the Hamilton operators in field

28


theory, we must generalize the conventional notion of differential operators as follows. Let Y and Y 0 be fibred manifolds over X and J k Y the k-order jet manifold of Y. Definition 1.7. A fibred morphism E : J kY → Y 0

(1.48)

over X is called the k-order differential operator (of class C ∞ ) on Y . It sends every section s of the fibred manifold Y to the section E ◦ J k s of the fibred manifold Y 0 . 2 Recall that, in accordance with the standard definition of differential operators, Y and Y 0 are assumed to be vector bundles. Building on Proposition 1.5 and Lemma 1.6, we can describe reduction of higher order differential operators to the first order ones. Proposition 1.8. Given a fibred manifold Y → X, every first order differential operator E 00 : J 1 J k−1 Y → Y 0

(1.49)

X

on the fibred k-order jet manifold J k−1 Y → X implies the k-order differential operator E = E 00 |J k Y on the fibred manifold Y → X. Moreover, if any one first order differential operator on J k−1 Y → X exists, every k-order differential operator (1.48) on Y → X represents the restriction of a certain first order differential operator (1.49) on J k−1 Y to the k-order jet manifold J k Y . 2 Outline of proof: Because of the monomorphisms (1.46) every fibred morphism J k Y → Y 0 can be extended to a fibred morphism J 1 J k−1 Y → Y 0 . This extension however is not unique in general. 2 In particular, every k-order differential operator (1.48) yields the morphism E 0 := E ◦ pr 2 : Jbk Y → J k Y → Y 0 X

(1.50)

1.2. JET SPACES

29

where pr 2 is the surjection corresponding to the canonical splitting (1.47). We call E 0 [Eq.(1.50)] the sesquiholonomic differential operator associated with E. For every section s of the fibred manifold Y → X, the equality E 0 ◦ J 1 J k−1 s = E ◦ J k s holds. Moreover, let s be a section of the fibred (k − 1)-order jet manifold J k−1 Y → X such that its jet prolongation J 1 s takes its values into the sesquiholonomic jet manifold Jbk Y . In virtue of Lemma 1.6, there exists a section s of Y → X such that s = J k−1 s and E 0 ◦ J 1 s = E ◦ J k s.

(1.51)

Hereafter, we consider extensions of a k-order differential operator E [Eq.(1.48)] to first order differential operators (1.49) only via its extension to the associated sequiholonomic differential operator (1.50). Reduction of k-order differential operators to the first order ones implies reduction of the associated k-order differential equations to the first order differential equations as follows. Let a fibred manifold Y 0 → X from Definition 1.8 be a composite manifold b we shall denote the canonical Y 0 → Y → X where Y 0 → Y is a vector bundle. By 0, 0 zero section of Y → Y . Let a k-order differential operator E [Eq.(1.48)] on the fibred manifold Y be a fibred morphism over Y . Then, its kernal Ker E is defined b )) of 0(Y b ) ⊂ Y 0 by E. We shall say that a section s of to be the preimage E −1 (0(Y Y → X satisfies the associated system of k-order differential equations if J k s(X) ⊂ Ker E.

(1.52)

As a shorthand, we shall write E ◦ J k s = 0. Let a k-order differential operator E on Y → X be extended to a first order differential operator E 00 on the fibred k-order jet manifold J k−1 Y → X. Let s be a section of J k−1 Y → X. We shall say that s is a sesquiholonomic solution of the corresponding system of first order differential equations if J 1 s(X) ⊂ Jbk Y, J 1 s(X) ⊂ Ker E 00 .

(1.53)

30


Proposition 1.9. The system of the k-order differential equations (1.52) and the system of the first order differential equations (1.53) are equivalent to each other. 2 Outline of proof: In virtue of Eq.(1.51), every solution s of Eqs.(1.52) determines the solution s = J k−1 s

(1.54)

of Eqs.(1.53). Conversely, every solution s of Eqs.(1.53) takes the form (1.54) where s is a solution of Eqs.(1.52). 2 In field theory, we have the example of equivalent systems of first order and second order Euler-Lagrange equations.

1.3

General connections

One can introduce connections on fibred manifolds Y → X in several equivalent ways. For every point y ∈ Y , a connection on Y → X sends every tangent space vector τ λ ∂λ to the base X at x = π(y) to the tangent vector τ λ (∂λ + Γiλ (y)∂λ ) to Y at y. We here define connections on fibred manifolds in jet terms. Definition 1.10. A first order jet field (or simply a jet field) on a fibred manifold Y → X is defined to be a section Γ of the affine jet bundle J 1 Y → Y . A first order connection Γ on a fibred manifold Y is defined to be a global jet field Γ : Y → J 1 Y, yλi ◦ Γ = Γiλ (y),

(1.55)

on Y → X. 2 By means of the contact map λ [Eq.(1.27)], every connection Γ [Eq.(1.55)] on a fibred manifold Y → X can be represented by a projectable tangent-valued horizontal 1-form λ ◦ Γ on Y which we shall denote by the same symbol Γ = dxλ ⊗ (∂λ + Γiλ (y)∂i ), ∂y 0 i j ∂y 0 i ∂xµ 0i Γ λ = ( j Γµ + µ ) 0 λ . ∂y ∂x ∂x

(1.56)

1.3. GENERAL CONNECTIONS

31

Substituting the form Γ [Eq.(1.56)] into the canonical horizontal splittings (1.31) and (1.32), we obtain the familiar horizontal splitting x˙ λ ∂λ + y˙ i ∂i = x˙ λ (∂λ + Γiλ ∂i ) + (y˙ i − x˙ λ Γiλ )∂i , x˙ λ dxλ + y˙ i dy i = (x˙ λ + Γiλ y˙ i )dxλ + y˙ i (dy i − Γiλ dxλ )

(1.57)

of tangent and cotangent bundles of Y with respect to the connection Γ on Y . Conversely, every horizontal splitting (1.57) determines a certain tangent-valued form (1.56) and, consequently, a global jet field on the fibred manifold Y → X. Example. Let Y → X be a vector bundle. A linear connection on Y reads Γ = dxλ ⊗ [∂λ + Γi jλ (x)y j ∂i ].

(1.58)

2 Since the affine jet bundle J 1 Y → Y is modelled on the vector bundle (1.23), connections on a fibred manifold Y → X constitute the affine space modelled on the linear space of soldering forms on Y . In other words, if Γ is a connection and σ = σλi dxλ ⊗ ∂i is a soldering form on a fibred manifold Y , then Γ + σ = dxλ ⊗ [∂λ + (Γiλ + σλi )∂i ] is a connection on Y . Conversely, if Γ and Γ0 are connections on a fibred manifold Y , then i

Γ − Γ0 = (Γiλ − Γ0 λ )dxλ ⊗ ∂i is a soldering form on Y . One introduces the following basic forms involving a connection Γ and a soldering form σ: • the curvature of Γ: 1 1 i R = dΓ Γ = Rλµ dxλ ∧ dxµ ⊗ ∂i = 2 2 1 (∂λ Γiµ − ∂µ Γiλ + Γjλ ∂j Γiµ − Γjµ ∂j Γiλ )dxλ ∧ dxµ ⊗ ∂i ; 2

(1.59)

32


• the torsion of Γ with respect to σ: 1 Ω = dσ Γ = dΓ σ = Ωiλµ dxλ ∧ dxµ ⊗ ∂i = 2 j i i (∂λ σµ + Γλ ∂j σµ − ∂j Γiλ σµj )dxλ ∧ dxµ ⊗ ∂i ;

(1.60)

• the soldering curvature of σ: 1 1 ε = dσ σ = εiλµ dxλ ∧ dxµ ⊗ ∂i = 2 2 1 j i (σλ ∂j σµ − σµj ∂j σλi )dxλ ∧ dxµ ⊗ ∂i . 2

(1.61)

In particular, the curvature (1.59) of the linear connection (1.58) reads i Rλµ (y) = Ri jλµ (x)y j ,

Ri jλµ = ∂λ Γi jµ − ∂µ Γi jλ + Γk jλ Γi kµ − Γk jµ Γi kλ .

(1.62)

We have the relations Γ0 = Γ + σ, R0 = R + ε + Ω, Ω0 = Ω + 2ε.

(1.63)

A connection Γ on a fibred manifold Y → X yields the first order differential operator DΓ : J 1 Y 3 z 7→ z − Γ(π01 (z)) ∈ T ∗ X ⊗ V Y,

(1.64)

Y

DΓ = (yλi − Γiλ )dxλ ⊗ ∂i , on Y . It is called the covariant differential relative to the connection Γ. The corresponding covariant derivative of sections s of Y reads ∇Γ s = DΓ ◦ J 1 s = [∂λ si − (Γ ◦ s)iλ ]dxλ ⊗ ∂i .

(1.65)

A section s of a fibred manifold Y → X is called the integral section for a connection Γ on Y if Γ ◦ s = J 1 s,

(1.66)


33

that is, its covariant derivative relative to Γ vanishes: ∇Γ s = 0. Consider now several constructions involving linear connections on vector bundles. Let Y → X be a vector bundle and Γ a linear connection (1.58) on Y . On the dual vector bundle Y ∗ → X, there exists the linear connection Γ∗ = dxλ ⊗ [∂λ − Γj iλ (x)yj ∂ i ] called the dual connection to Γ. Example. A linear connection K on the tangent bundle T X of a manifold X and the dual connection K ∗ to K on the cotangent bundle T ∗ X read Kλα = −K α νλ (x)x˙ ν , ∗ Kαλ = K ν αλ (x)x˙ ν .

(1.67)

2 Let Y and Y 0 be vector bundles over X. Given linear connections Γ and Γ0 on Y and Y 0 respectively, they determine the tensor product connection i jk (Γ ⊗ Γ0 )ik + Γ0 λ = Γ jλ y

k

jλ y

ij

on the tensor product Y ⊗Y 0 → X X

of bundles Y and Y 0 . The construction of the dual connection and the tensor product connection can be extended to connections on composite manifolds (1.3) when Y → Σ is a vector bundle. Let Y → Σ → X be the composite manifold (1.3). Let J 1 Σ, J 1 YΣ and J 1 Y the first order jet manifolds of the fibred manifolds Σ → X, Y → Σ and Y → X respectively. Given fibred coordinates (xλ , σ m , y i ) (1.4) of Y , the corresponding adapted coordinates of the jet manifolds J 1 Σ, J 1 YΣ and J 1 Y are respectively (xλ , σ m , σλm ), i (xλ , σ m , y i , yeλi , ym ), λ m i m i (x , σ , y , σλ , yλ ).

(1.68)

34


A connection A on the composite manifold Y → Σ → X is called projectable onto a connection Γ = dxλ ⊗ [∂λ + Γm λ (σ)∂m ] on the fibred manifold Σ → X, if the following diagram commutes J 1π

YΣ J 1 Y −→ J 1Σ

A

6Γ

6

Y

−→ Σ πY Σ

Such a connection takes the coordinate form i A = dxλ ⊗ [∂λ + Γm λ (σ)∂m + Aλ (y)∂i ].

Let now Y → Σ be a vector bundle and Y ∗ → Σ → X the composite manifold where Y ∗ → Σ is the vector bundle dual to Y → Σ. Let a connection i j A = dxλ ⊗ [∂λ + Γm λ (σ)∂m + A jλ (σ)y ∂i ]

on Y → X projectable onto a connection Γ on Σ → X be a linear morphism over Γ. On Y ∗ → X, there exists the dual connection j i A∗ = dxλ ⊗ [∂λ + Γm λ (σ)∂m − A iλ (σ)yj ∂ ]

(1.69)

projectable onto Γ. Let Y → Σ → X,

Y0 →Σ→X

be composite manifolds where Y → Σ and Y 0 → Σ are vector bundles. Let A and A0 be connections on Y and Y 0 respectively which are projectable to the same connection Γ on the fibred manifold Σ → X and are linear morphisms over Γ. On the tensor product Y ⊗ Y 0 → Σ → X, Σ

there exists the tensor product connection i jk A ⊗ A0 = dxλ ⊗ [∂λ + Γm + A0 λ ∂m + (A jλ y

projectable onto Γ.

k

jλ y

ij

)∂ik ]

(1.70)


35

In particular, let Γ be a connection on a fibred manifold Y → X. Due to the canonical isomorphism (1.26), the vertical tangent morphism V Γ [Eq.(1.7)] to Γ sets up the connection V Γ : V Y → V J 1 Y = J 1 V Y, ∂ ∂ V Γ = dxλ ⊗ (∂λ + Γiλ i + ∂j Γiλ y˙ j i ), ∂y ∂ y˙

(1.71)

on the composite manifold V Y → Y → X which is projectable onto the connection Γ on Y . It is a linear bundle morphism over Γ: VΓ

V Y −→ J 1 V Y ?

Γ

?

Y −→ J 1 Y

The connection (1.71) yields the dual connection (1.69) V ∗ Γ = dxλ ⊗ (∂λ + Γiλ

∂ ∂ − ∂j Γiλ y˙ i ) i ∂y ∂ y˙ j

(1.72)

on the composite manifold V ∗Y → Y → X which also is projectable onto the connection Γ. Let us turn now to second order connections on fibred manifolds. Definition 1.11. A second order jet field [resp. a second order connection] Γ on a fibred manifold Y → X is defined to be a first order jet field [resp. a first order connection] on the fibred jet manifold J 1 Y → X. 2 Note that a second order connection is a section of the repeated jet bundle π11 [Eq.(1.42)], but not J 1 π01 [Eq.(1.43)]. Relative to coordinates (1.41) of the repeated jet manifold J 1 J 1 Y , a second order jet field Γ is given by the expression i

i

i i (y(µ) , yλµ ) ◦ Γ = (Γ(µ) , Γλµ ).

Using the contact map (1.36), one can represent it by the horizontal tangent-valued 1-form i

i

Γ = dxµ ⊗ (∂µ + Γ(µ) ∂i + Γλµ ∂iλ )

(1.73)

36


on the fibred jet manifold J 1 Y → X. A second order jet field Γ on Y is termed a sesquiholonomic [resp. holonomic] second order jet field if it takes its values into the subbundle Jb2 Y [resp. J 2 Y ] of J 1 J 1 Y . We have the coordinate equality i

Γ(µ) = yµi for a sesquiholonomic second order jet field and the additional equality i

i

Γλµ = Γµλ for a holonomic second order jet field. Given a first order connection Γ on a fibred manifold Y → X, one can construct a connection on the fibred jet manifold J 1 Y → X as follows. The first order jet prolongation J 1 Γ of the connection Γ on Y is a section of the repeated jet bundle (1.43), but not the bundle π11 [Eq.(1.42)]. Let K ∗ be a linear symmetric connection (1.67) on the cotangent bundle T ∗ X of X: ∗ Kλµ = K α λµ x˙ α , ∗ ∗ Kλµ = Kµλ .

There exists the affine fibred isomorphism rK of J 1 J 1 Y such that rK ◦ rK = Id J 1 J 1 Y , i i i i i (yλi , y(µ) , yλµ ) ◦ rK = (y(λ) , yµi , yµλ + K α λµ (yαi − y(α) )).

One can verify the following transformation relations of the coordinates (1.41): i

i

y 0 µ ◦ rK = y 0 (µ) , i

i

i

i

y 0 (µ) ◦ rK = y 0 µ , y 0 λµ ◦ rK = y 0 µλ + K 0

α

0i λµ (y α

i

− y 0 (α) ).

Hence, every connection Γ on a fibred manifold Y → X gives rise to the connection JΓ = rK ◦ J 1 Γ, JΓ = dxµ ⊗ [∂µ + Γiµ ∂i + (∂λ Γiµ + ∂j Γiµ yλj + K α λµ (yαi − Γiα ))∂iλ ], on the composite jet manifold J 1 Y → Y → X.

(1.74)


37

The connection JΓ is projectable to Γ in accordance with the commutative diagram JΓ

J 1 Y −→ J 1 J 1 Y π01

π11

?

?

1

Y −→ J Y Γ

It is an affine morphism over Γ The connection (1.74) is by no means the unique connection on J 1 Y → X which one can construct from a given connection Γ on Y → X. Note that the curvature R [Eq.(1.59)] of a first order connection Γ on a fibred manifold Y → X determines the soldering form i σ R = Rλµ dxµ ⊗ ∂iλ

(1.75)

on the fibred jet manifold J 1 Y → X. Also the torsion (1.60) of a first order connection Γ with respect to a soldering form σ on Y → X and the soldering curvature (1.61) of σ set up soldering forms on J 1 Y → X.

38


Chapter 2 Lagrangian Field Theory In the framework of the geometric approach to classical field theory, the dynamics of fields identified with sections of fibred manifolds Y → X is phrased in terms of jet manifolds. This Chapter is devoted to the Lagrangian formalism on fibred manifolds, its De Donder Hamiltonian derivation and the instantaneous Hamiltonian formalism stemmed from the previuos ones. The jet approach to Lagrangian systems has been mainly stimulated by the calculus of variations where the Lepagean equivalents of a Lagrangian density play the prominent role. The literature on this subject is extensive.5,14,20,25,29,32,34,54,56 Point out basic structural ambiguities in the higher order Langrangian formalism when the Cartan forms, Legendre morphisms etc. fail to be uniquely defined. Bearing in mind physical applications, we shall restrict our consideration to the first order Lagrangian systems whose configuration space is the jet manifold J 1 Y of Y . Here, we are not concerned deeply with the variational principle and the calculus of variations, but aim to detail different types of field equations which one handles in the first order Lagrangian formalism on a fibred manifold Y → X. They are the Cartan equations, the De Donder-Hamilton equations and three types of EulerLagrange equations: • the algebraic Euler-Lagrange equations for sections of the repeated jet bundle J 1J 1Y → J 1Y , • the first order differential Euler-Lagrange equations for sections of the fibred jet manifold J 1 Y → X, • the second order differential Euler-Lagrange equations for sections of the fibred manifold Y → X itself. 39

40

CHAPTER 2. LAGRANGIAN FIELD THEORY

To introduce the Euler-Lagrange equations, we start with the notion of the EulerLagrange operator, for it is the sesquiholonomic first order Euler-Lagrange operator which is the Lagrangian counterpart of the Hamilton operator in the multimomentum Hamiltonian formalism. Building on this fact, we shall get relations between solutions of the Hamilton equations and the first order Euler-Lagrange equations.

2.1

Lagrangian formalism on fibred manifolds

Let π:Y →X be a fibred manifold provided with fibred coordinates (xλ , y i ) [Eq.(1.2)]. Unless otherwise stated, the dimension of X is n > 1, for case of n = 1 possesses essential percularities and corresponds to the time-dependent mechanics.11,17 In jet terms, a first order Lagrangian density is defined to be a bundle morphism n

L : J 1Y → ∧ T ∗X over Y . It is viewed as an exterior horizontal density L = L(xµ , y i , yµi )ω

(2.1)

on the fibred jet manifold J 1 Y → X. The jet manifold J 1 Y thus plays the role of a finite-dimensional configuration space of fields represented by sections of Y → X. We further use the notation πiλ = ∂iλ L. Remark. In field theory, all Lagrangian densities are polynomial forms relative to velocities yλi . Note that a polynomial form of degree k on a vector space E is defined to be a linear form on the tensor space k

R ⊕ E ⊕ ... ⊕ (⊗ E). Given an affine space E modelled on a vector space E, polynomial forms on E are factorized by morphisms E → E. Since the jet bundle J 1 Y → Y is affine, every Lagrangian density of field theory factors as n

D

L : J 1Y → T ∗X ⊗ V Y → ∧ T ∗X Y

(2.2)

2.1. LAGRANGIAN FORMALISM ON FIBRED MANIFOLDS

41

where D is the covariant differential (1.64) relative to some connection on the fibred manifold Y → X. 2 With a Lagrangian density L, the jet manifold J 1 Y carries the the generalized Liouville form θL = −πiλ dy i ∧ ω ⊗ ∂λ

(2.3)

and the Lagrangian multisymplectic form ΩL = dπiλ dy i ∧ ω ⊗ ∂λ .

(2.4)

They are pullback T X-valued forms on J 1 Y → X. Thus, the Lagrangian formalism on fibred manifolds leads naturally to the notion of multisymplectic structure when momenta are canonically conjugates with all velocities yµi . The multisymplectic structure on the configuration space J 1 Y however is not canonical since the generalized Liouville form (2.3) and the multisymplectic form (2.4) depend upon option of a Lagrangian density. In the framework of the multimomentum Hamiltonian formalism, this structure will be recovered as the pullback of the canonical myltisymplectic structure on the corresponding Legendre manifold. Remark. Point out the existing ambiguities in the concept of multisymplectic (or polysymplectic) structure. The multisymplectic structure utilized for the multimomentum Hamiltonian formalism differs from the one of the De Donder Hamiltonian machinery, and both of them do not belong to the category of multisymplectic manifolds by G.Martin.38 2 Given a first order Lagrangian density L [Eq.(2.1)], its Lepagean equivalent ΞL is an exterior n-form on the jet manifold J 1 Y such that J 1 s∗ ΞL = L ◦ J 1 s

(2.5)

for all sections s of Y . We shall follow the so-called De Donder-Weyl approach to the calculus of variations which is based upon the Cartan forms as Lepagean equivalents. In first order theory, Cartan forms consist with the more particular Poincaré-Cartan forms. It is the Poincaré-Cartan form which is the Lagrangian counterpart of the multimomentum Hamiltonian forms in the multimomentum Hamiltonian formalism. At the same time, unless n = 1, there exist other first order Lepagean equivalents.20,25

42

CHAPTER 2. LAGRANGIAN FIELD THEORY

In the first order Lagrangian formalism, the Poincaré-Cartan form associated with a Lagrangian density L always exists and is uniquely defined. This is the exterior horizontal n-form on the jet bundle J 1 Y → Y which is given by the coordinate expression ΞL = πiλ dy i ∧ ωλ + πω, π = L − πiλ yλi .

(2.6)

With ΞL , we have the following first order differential Cartan equations for sections s of the fibred jet manifold J 1 Y → X: s∗ (ucdΞL ) = 0

(2.7)

where u is an arbitrary vertical vector field on J 1 Y → X. In the coordinate form, these equations read ∂jµ πiλ (∂λ si − siλ ) = 0, ∂i L − (∂λ +

sjλ ∂j

+

(2.8a)

∂λ sjµ ∂jµ )∂iλ L

+

∂i πjλ (∂λ sj

−

sjλ )

= 0.

(2.8b)

Solutions of the Cartan equations (2.7) extremize the action functional Z

s∗ ΞL .

X

If a section s of J 1 Y → X is the jet prolongation of a section s of Y → X, the form s∗ ΞL comes to the familiar Lagrangian form n

L(s) := L ◦ J 1 s : X → ∧ T ∗ X. The corresponding action functional reads Z

L(s).

X

This functional is proved to be stationary at a section s iff the jet prolongation J 1 s of s satisfies the Cartan equations (2.7). It means that, on holonomic sections s = J 1 s, the Cartan equations are equivalent to the differential Euler-Lagrange equations. In particular, if a Lagrangian density L is regular, the Cartan equations are equivalent to the Euler-Lagrange equations since all solutions of the Cartan equations are holonomic in virtue of Eq.(2.8a) and Lemma 1.6. We shall introduce the Euler-Lagrange equations as conditions for the kernal of the Euler-Lagrange operator.

2.1. LAGRANGIAN FORMALISM ON FIBRED MANIFOLDS

43

The Euler-Lagrange operator can be defined intrinsically as a second order differential operator n+1

EL : J 2 Y → ∧ T ∗ Y Y

of the variational type.3,16 Remark. Let J 2k Y be the 2k-order jet manifold of a fibred manifold Y → X. An Euler-Lagrange-type operator is defined to be a 2k-order differential operator n

E : J 2k Y → ∧ T ∗ X ⊗ V ∗ Y Y

on Y . This operator is termed locally variational if δE = 0 where δ is the variational operator constructed as follows. Let us consider a pullbackvalued horizontal density n

α : J k Y → ∧ T ∗ X ⊗ V ∗ J k Y, i

α = ω ⊗ (αi dy +

JkY αiΛ dyΛi ),

0 0. It is an open subbundle of the Legendre bundle. On Q, we

82

CHAPTER 3.

MULTIMOMENTUM HAMILTONIAN FORMALISM

have the associated multimomentum Hamiltonian form H = pλ dy ∧ ωλ − Hω, 1 H = p1 (ln p1 − 1) + (p2 )2 , 2 which however is not extended to Π. 2 Contemporary field theories are almost never regular, but their Lagrangian densities are nonetheless well-behaved. They are both semiregular and almost regular as a rule. Proposition 3.22. All multimomentum Hamiltonian forms associated with a semiregular Lagrangian density L coincide on the Lagrangian constraint space Q: H |Q = H 0 |Q . Moreover, the Poincaré-Cartan form ΞL [Eq.(2.6)] for L is the pullback b ΞL = H ◦ L,

πiλ yλi − L = H(xµ , y i , πiλ ),

(3.62)

of any associated multimomentum Hamiltonian form H by the Legendre morphism b 2 L. Outline of proof: Let u be a vertical vector field on the jet bundle J 1 Y → Y. If b of the tangent morphism to L, b it is easy to u takes its values in the kernal Ker T L see that Lu ΞL = 0 where Lu is the Lie derivative with respect to u. Hence, the Poincaré-Cartan form ΞL for a semiregular Lagrangian density L is constant on the connected preimage b −1 (q), of each point q ∈ Q. The statement then issues from Eq.(3.50). 2 L Coinciding on the Lagrangian constraint space Q, the multimomentum Hamiltonian forms associated with a semiregular Lagrangian density differ from each other outside Q. The condition (3.44b) restricts rigidly the arbitrariness of these forms at points of Π \ Q. Substituting this condition into Eq.(3.62), we obtain c − pλ )∂ i H = H(xµ , y i , π λ ◦ H) c − H(xµ , y i , pλ ) (πiλ ◦ H i λ i i

3.5. HAMILTONIAN THEORY OF CONSTRAINT SYSTEMS

83

at each point of Π\Q. It follows that, at a point which does not belong the constraint space, a multimomentum Hamiltonian form associated with a semiregular Lagrangian density is written H = pλi dy i ∧ ωλ − Hω, f i , pλ ), H = pλi ΓH iλ + pµj ΓH jµ + H(y i where pµj are coordinates which obey the relation c pµj 6= πjµ ◦ H

and pλi are the remaining ones. It is affine in momenta pµj . The Hamilton equations (3.35a) corresponding to the coordinates pµj are reduced to the gauge-type conditions ∂µ rj = ΓH jµ independent of momenta. Example. Let Y be the bundle R3 → R2 coordinatized by (x1 , x2 , y). The jet manifold J 1 Y and the Legendre bundle Π over Y are coordinatized by (x1 , x2 , y, y1 , y2 ) and (x1 , x2 , y, p1 , p2 ) respectively. Set the Lagrangian density 1 L = (y1 )2 ω. 2

(3.63)

It is semiregular. The associated Legendre morphism reads b =y , p1 ◦ L 1 2 b p ◦ L = 0.

The corresponding constraint space Q consists of points with the coordinate p2 = 0. Multimomentum Hamiltonian forms associated with the Lagrangian density (3.63) are given by the expression 1 H = pλ dy ∧ ωλ − [ (p1 )2 + c(x1 , x2 , y)p2 ]ω 2

84

CHAPTER 3.


where c is arbitrary function of coordinates x1 , x2 and y. They are affine in the momentum coordinate p2 . 2 The relation (3.62) implies the identity i λ b b (yλi − ∂λi H ◦ L)dπ i ∧ ω − (∂i L + ∂i H ◦ L)dy ∧ ω = 0.

(3.64)

In particular, we have the equality [yλi − ∂λi H(xµ , y j , πjµ )]∂iλ ∂kα L = 0

(3.65)

similar to Eq.(3.46). Building on Eq.(3.64), one can reproduce Eq.(3.56) b ΛL = EH ◦ J 1 L,

but not Eq.(3.57). It enables one to extend the part (i) of Proposition 3.21 to multimomentum Hamiltonian forms associated with semiregular Lagrangian densities. Proposition 3.23. Let a section r of the Legendre manifold Π → X be a solution of the Hamilton equations (3.35a) and (3.35b) for some multimomentum Hamiltonian form H associated with a semiregular Lagrangian density L. If r lives on the constraint space Q [Eq.(3.45)], the section c◦r s=H

of the fibred jet manifold J 1 Y → X satisfies the first order Euler-Lagrange equations (2.15a) and (2.15b). 2 Outline of proof: Since r ⊂ Q, we have b ◦s r=L

and can reproduce Eqs.(3.60). 2 The assertion (ii) from Proposition 3.21 however must be modified as follows. Proposition 3.24. Given a semiregular Lagrangian density L, let a section s of the fibred jet manifold J 1 Y → X be a solution of the first order Euler-Lagrange equations (2.15a) and (2.15b). Let H be a multimomentum Hamiltonian form associated with L so that its momentum morphism satisfies the condition c◦L b ◦ s = s. H

(3.66)


85

Then, the section b ◦s r=L

of the fibred Legendre manifold Π → X is a solution of the Hamilton equations (3.35a) and (3.35b) for H. It lives on the constraint space Q. 2 Outline of proof: The Hamilton equations (3.35a) hold in virtue of the condition (3.66). The equation (3.35b) is brought to the Euler-Lagrange equation (2.15b) b ◦s=∂L◦H c◦L b ◦s=∂L◦s ∂bλ πiλ ◦ J 1 s = −∂i H ◦ L i i

due to Eq.(3.51). 2 Lemma 3.25. For every pair of sections s of J 1 Y → X and r of Π → X which satisfy either Proposition 3.23 or Proposition 3.24, Eqs.(3.59) where s is a solution of the second order Euler-Lagrange equations (2.16) remain true. 2 Proposition 3.23 and Proposition 3.24 show that, if H is a multimomentum Hamiltonian form associated with a semiregular Lagrangian density L, every solution b 1Y ) of the corresponding Hamilton equations which lives on the constraint space L(J yields a solution of the Euler-Lagrange equations for L. At the same time, to exaust all solutions of the Euler-Lagrange equations, one must consider a family of different multimomentum Hamiltonian forms associated with L. Example. To illustrate these circumstances, let us consider the zero Lagrangian density L = 0. This Lagrangian density is semiregular. Its Euler-Lagrange equations come to the identity 0 = 0. Every section s of the fibred manifold Y → X is a solution of these equations. Given a section s, let Γ be a connection on Y such that s is its integrale section. The multimomentum Hamiltonian form HΓ [Eq.(3.21)] is associated with L. The corresponding momentum morphism satisfies Eq.(3.66). The Hamilton equations have the solution b ◦s r=L

given by the coordinate expression r i = si , riλ = 0.

(3.67)

86

CHAPTER 3.


2 We shall say that a family of multimomentum Hamiltonian forms H associated with a Lagrangian density L is complete if, for any solution s of the first order EulerLagrange equations (2.15a) and (2.15b), there exists a solution r of the Hamilton equations (3.35a) and (3.35b) for some multimomentum Hamiltonian form H from this family so that b ◦ s, r=L c ◦ r, s=H s = J 1 (πΠY ◦ r).

(3.68)

Let L be a semiregular Lagrangian density. Then, in virtue of Proposition 3.24, such a complete family of associated multumomentum Hamiltonian forms exists iff, for every solution s of the Euler-Lagrange equations for L, there is a multimomentum Hamiltonian form H from this family such that Eq.(3.66) holds. The complete family of multimomentum Hamiltonian forms associated with a given Lagrangian density fails to be defined uniquelly. For instance, the multimomentum Hamiltonian forms (3.21) constitute the complete family associated with the zero Lagrangian density, but this family is not minimal. At the same time, we consider only solutions of the Hamilton equations which live on the Lagrangian constraint space (3.45). Therefore, we can restrict ourselves to local multimomentum Hamiltonian forms and the Hamilton equations which are defined on some open e of the Lagrangian constraint space Q. Being an open imbedded neighborhood Q e inherits the multisymplectic structure of Π insubbundle of the Legendre bundle, Q cluding multisymplectic canonical transformations which keep the constraint space Q. We shall investigate existence of complete families of associated multimomentum Hamiltonian forms when a Lagrangian density is almost regular. In this case, the Lagrangian constraint space Q, by very definition, is an imbedded subbundle of the Legendre bundle and the Legendre morphism J 1Y → Q Y

(3.69)

is a fibred manifold. Proposition 3.26. Let L be an almost regular Lagrangian density. (i) On an open neighborhood of each point q ∈ Q, there exist local multimomentum Hamiltonian forms associated with L.


87

(ii) Let L be still a semiregular Lagrangian density. On an open neighborhood of each point q ∈ Q, there exists a complete family of local associated multimomentum Hamiltonian forms. 2 Outline of proof: Given a point q ∈ Q, let (xλ , y i , pλi ) be local coordinates of some open neighborhood of q. Owing to the constant degeneracy rank of the Legendre b one can select the maximal subset y i of the coordinates y i so that the morphism L, λ λ equations pλi =

∂L ∂y iλ

can be resolved for y iλ = φiλ (xµ , y j , pµj , y jµ )

(3.70)

where y jµ are the remaining coordinates. Substituitng Eq.(3.70) into the equation pλi =

∂L , ∂y iλ

we obtain pλi = pλi (xµ , y j , pµj ) where pµj play the role of local coordinates of the constraint space Q. For every section s of Y → X, the momentum morphism i

Φλ = φiλ (xµ , sj , pµj , y jµ = ∂µ sj ), Φiλ = ∂λ si

(3.71)

satisfies Eqs.(3.48) and (3.49). Thereby, the associated local multimomentum Hamiltonian form (3.47) is associated with L. Given a section s of Y → X, the momentum morphism (3.71) meets Eq.(3.66). It follows that, if a Lagrangian density L is semiregular, the above-mentioned multimomentum Hamiltonian forms constitute locally the complete family. 2 In field theories with quadratic and affine Lagrangian densities, the complete families of the associated globally defined multimomentum Hamiltonian forms always exist. Moreover, the following example shows that the complete family of associated multimomentum Hamiltonian forms may exist if even a Lagrangian density is neither semiregular nor almost regular.

88

CHAPTER 3.


Example. Let Y be the bundle R3 → R2 coordinatized by (x1 , x2 , y). The jet manifold J 1 Y and the Legendre bundle Π over Y are coordinatized by (x1 , x2 , y, y1 , y2 ) and (x1 , x2 , y, p1 , p2 ) respectively. Let us consider the Lagrangian density 1 1 L = [ (y1 )3 + (y2 )2 ]ω. 3 2 The associated Legendre morphism reads b = (y )2 , p1 ◦ L 1 2 b p ◦ L = y2 .

(3.72)

The corresponding constraint space Q is given by the coordinate relation p1 ≥ 0. It is not even a submanifold of Π. There exist two associated multimomentum Hamiltonian forms 3 2 1 H+ = pλ dy ∧ ωλ − [ (p1 ) 2 + (p2 )2 ]ω, 3 2 3 2 1 H− = pλ dy ∧ ωλ − [− (p1 ) 2 + (p2 )2 ]ω 3 2

on Q which correspond to different solutions y1 =

q

p1

and q

y1 = − p1 of Eq.(3.72). One can verify that the multimomentum Hamiltonian forms H+ and H− constitute the complete family. 2 Note that, if the imbedded constraint space Q is not an open subbundle of the Legendre bundle Π, it can not be provided with the multisymplectic structure. Roughly speaking, not all canonical momenta pλi live on such a constraint space in general. Therefore, to consider solutions of the Hamilton equations even on Q, one must set up a multimomentum Hamiltonian form and the corresponding Hamilton equations at least on some open neighborhood of the constraint space Q.

3.6. CAUCHY PROBLEM

89

Another way is to construct the De Donder-Hamilton type equations on the imbedded constraint space Q itself. Let an almost regular Lagrangian density L be also semiregular. Let HQ be the restriction of the associated multimomentum Hamiltonian forms to the constraint space Q. In virtue of Proposition 3.22, it is uniquely defined. For sections r of the fibred manifold Q → X, we can write the equations r∗ (ucdHQ ) = 0

(3.73)

where u is an arbitrary vertical vector field on Q → X. Since dHQ 6= dH |Q , these equations fail to be equivalent to the Hamilton equations restricted to the constraint space Q. At the same time, we have b (J 1 Y ) = H (Q) Ξ L Q

and so, Eqs.(3.73) are equivalent to the De Donder-Hamilton equations (2.23). Thus, the feature of the multimomentum Hamiltonian approach to constraint field systems is clarified. In the framework of this approach, field equations are set up at least on an open neighborhood of the Lagrangian constraint space though only their solutions living on the constraint space are considered. Whereas in the framework of the De Donder Hamiltonian formalism and, in fact, the Lagrangian formalism, field equations themselves are defined only on the constraint space.

3.6

Cauchy problem

This Section covers the Cauchy problem for the Euler-Lagrange equations and the Hamilton equations. The key point consists in the fact that the Hamilton equations must be modified in order to formulate the Cauchy task. We observe that, although the first order Lagrangian equations are equivalent to the second order ones, the Cauchy tasks for them fail to be equivalent. At the same time, if a Lagrangian density is regular, the Cauchy problem for the second order Euler-Lagrange equations is equivalent to the Cauchy problem for the Hamilton equations. The system of the first order Euler-Lagrange equations (2.15a) and (2.15b) has the standard form λ (x, φ)∂λ φb = fa (x, φ) Sab

(3.74)

90

CHAPTER 3.


for the Cauchy problem or, to be more precise, for the general Cauchy problem since λ the coefficients Sab depend on the variable functions φ in general. Here, φb are the collective notation of field functions si and siλ . However, the characteristic form λ det(Sab cλ ),

cλ ∈ R,

(3.75)

of the first order Euler-Lagrange equations is singular, that is, it is not different from 0 because of Eqs.(2.15a). If the index i takes the single value, the second order Euler-Lagrange equations (2.16) also have the standard form for the Cauchy problem. In general case, one can reduce the Cauchy problem for these equations to the standard form as follows. Let us examine the extension of the Euler-Lagrange operator (2.11) to the repeated jet manifold J 1 J 1 Y which is given by the horizontal exterior form EL00 = (∂i − ∂bλ ∂iλ )Ldy i ∧ ω,

(3.76)

i i ∂bλ = ∂λ + y(λ) ∂i + yµλ ∂iµ ,

on J 1 J 1 Y → Y . Note that the exterior form (3.76) differs from the exterior form (2.10) as follows: i ΛL = EL00 + (y(λ) − yλi )dπiλ ∧ ω.

We single out a local coordinate x1 and consider local sections s of the fibred jet manifold J 1 Y → X which satisfy the conditions ∂1 si = si1 , ∂1 siλ = ∂λ si1 ,

λ 6= 1.

(3.77)

It should be noted that the first order jet prolongations J 1 s of such sections s do not take their values into the sesquiholonomic jet manifold Jb2 Y in general. Let us consider the system of the first order differential equations (3.77) and the first order differential equations EL00 ◦ J 1 s = 0, i i (yλi , y(λ) , yµλ ) ◦ J 1 s = (siλ , ∂λ si , ∂λ siµ ),

for sections s of the fibred jet manifold J 1 Y → X. The latter take the form ∂i L − (∂λ + ∂λ sj ∂j + ∂λ sjµ ∂jµ )∂iλ L = 0.

(3.78)

3.6. CAUCHY PROBLEM

91

This system has the standard fashion (3.74) for the Cauchy problem on the initial conditions si (x) = φi (x), siλ (x) = ∂λ φi (x), si1 (x) = φi1 (x)

λ 6= 1, (3.79)

on some local hypersurface of X transversal to coordinate lines x1 . Its characteristic form (3.75) is defined only by the Hessian (2.14). In virtue of the well-known theorem,33 if local functions (si (x), siλ (x)) are solutions of the Cauchy problem for Eqs.(3.77) and (3.78) [in analytic functions or functions of class C 2 ], they satisfy still Eqs.(2.15a). It follows that they take their values into the second order jet manifold J 2 Y and si (x) are solutions of the second order Euler-Lagrange equations (2.16). We thus observe that, on solutions of the Cauchy problem for the second order Euler-Lagrange equations (2.16), Eqs.(2.15a) with λ 6= 1 are the corollaries of Eqs.(3.77) and the initial conditions (3.79). In other words, the system of first order Euler-Lagrange equations is overdefined on solutions of the Cauchy problem. Lemma 3.27. The characteristic form (3.75) for the system of Eqs.(3.77) and (3.78) is singular iff there exists a collection of real numbers cj such that, whenever bµ ∈ R, ∂jµ ∂iλ Lcj bµ = 0.

(3.80)

2 Outline of proof: The conditions (3.80) are equivalent to the conditions that λ columns of the matrix Sab bλ corresponding to Eqs.(3.78) are linearly dependent. 2 In particular, if the determinant of the Hessian (2.14) of a Lagrangian density L is different from zero, the characteristic form (3.75) for the system of Eqs.(3.77) and (3.78) is not singular. It means that the Cauchy problem for these equations may be formulated. Let us turn now to the Cauchy problem for the Hamilton equations. The Hamilton equations (3.35a) and (3.35b) have the standard form (3.74) for the Cauchy problem, but one faces the same difficulties as for the Caushy problem of the first order Euler-Lagrange equations (2.15a) and (2.15b). Their characteristic form (3.75) fails to be different from 0. These difficulties are overcomed in the same way as in case of the second order Euler-Lagrange equations.

92

CHAPTER 3.


Let us single out a local coordinate x1 and replace Eqs.(3.35a) by the equations ∂1 ri = ∂1i H, ∂b1 ∂λi H = ∂bλ ∂1i H, λ 6= 1, i b ∂µ = ∂µ + ∂µ r ∂i + ∂µ riλ ∂λi .

(3.81)

The system of Eqs.(3.81) and (3.35b) have the standard form for the Cauchy problem on the initial conditions ri (x0 ) = φi (x0 ), riµ (x0 ) = φµi (x0 ), ∂λ ri = ∂λi H, λ 6= 1,

(3.82)

on some local hypersurface S of X transversal to coordinate lines x1 . Proposition 3.28. If ri are solutions of the Cauchy problem for Eqs.(3.81) and (3.35b) [in functions of class C 2 ] on the initial conditions (3.82), they satisfy all Eqs.(3.35a). 2 Outline of proof: The proof shadows the standard one.33 If x is a point of an open neighborhood of the point x0 ∈ S, we have the relations i

r (x) =

Zx1

∂1i Hds + φi (x0 ),

0 i

∂λ r (x) =

Zx1

∂λ ∂1i Hds + ∂λ φi (x0 )

0

=

Zx1

∂1 ∂λi Hds + ∂λi H(x0 ) = ∂λi H(x).

λ 6= 1.

0

This statement can be extended to analytical functions. 2 It follows that, in order to formulate the Cauchy problem for the Hamilton equations in the multimomentum Hamiltonian formalism, one should single a one of coordinates and consider the system of equations (3.81) and (3.35b). It is readily observed that, if a Lagrangian density is regular, the system of equations (3.79) and (2.15b) is equivalent to the system of equations (3.81) and (3.35b) for the Cauchy task of the second order Euler-Lagrange equations.

3.7. ISOMULTISYMPLECTIC STRUCTUTE

3.7

93

Isomultisymplectic structute

The canonical multisymplectic structure defined by the generalized Liouville form θ [Eq.(3.6)] and the multisymplectic form Ω [Eq.(3.7)] is by no means the unique multisymplectic structure on the Legendre bundle Π [Eq.(3.2)] over a fibred manifold Y → X. This Section exemplifies its following deformation. Let φ = φλµ (x)dxµ ⊗ ∂λ be a tangent-valued 1-form on X corresponding to some isomorphism of the tangent bundle T X of X. We shall denote by the same index φ pullback onto the fibred Legendre manifold Π → X. Given the generalized Liouville form θ [Eq.(3.6)] and the multisymplectic form Ω [Eq.(3.7)], let us consider their deformations θφ = θcφ = −φλµ (x)pµi dy i ∧ ω ⊗ ∂λ

(3.83)

Ωφ = Ωcφ = φλµ (x)dpµi ∧ dy i ∧ ω ⊗ ∂λ

(3.84)

and

respectively. In comparison with the canonical forms θ and Ω, the forms θφ and Ωφ set up another multisymplectic structure on the Legendre bundle Π. We shall call it the isomultysimplectic structure in the spirit of Santilli’s modification of the Poisson bracket in mechanics.46 In particular, Proposition 3.3 is extended to the forms θφ and Ωφ as follows. Proposition 3.29. For each exterior 1-form σ on X, the generalized Liouville form (3.83) and the isomultisymplectic form (3.84) obey the same relations d(Ωφ cσ) = 0, Ωφ cσ = −d(θφ cσ) as the relations (3.10). 2 Building on the forms θφ and Ωφ , one can develop the multimomentum isoHamiltonian formalism by analogy with the multimomentum Hamiltonian formalism from previous Sections. Definition 3.30. We shall say that a jet field γ on the fibred Legendre manifold Π → X is an iso-Hamiltonian jet field if the exterior form γcΩφ is closed. 2

94

CHAPTER 3.


Definition 3.31. An exterior n-form Hφ on the Legendre manifold Π is called the iso-Hamiltonian form if, on an open neighborhood of each point of Π, there exists an iso-Hamiltonian jet field γ satisfying the equation γcΩφ = dHφ . 2 The following assertion shows that the sets of iso-Hamiltonian jet fields and iso-Hamiltonian forms on Π are not empty. e its lift (3.12) onto Proposition 3.32. Let Γ be a connection on Y → X and Γ Π → X. We have the iso-Hamiltonian form

HΓφ = φλµ (pµi dy i ∧ ωλ − Γiλ pλi ω) and the associated iso-Hamiltonian jet field e+ γ=Γ

1 −1 µ α β ν (φ )λ (−∂α φαν − K α βα φβν + Kνβ φα )pi dxλ ⊗ ∂µi . n

2 In case of iso-Hamiltonian forms, also Proposition 3.9 remains true. As an immediate consequence of Proposition 3.32 and the analogue of Proposition 3.9, we get the following corollary. Corollary 3.33. Every iso-Hamiltonian form is given by the expression f ω = φλ pµ dy i ∧ ω − H ω Hφ = HφΓ + H λ φ Γ µ i

(3.85)

where Γ is a connection on Y → X. 2 By analogy with the multimomentum Hamiltonian formalism, one can introduce the Hamilton operator and obtain the Hamilton equations associated with the isoHamiltonian form (3.85). For sections r of the fibred Legendre manifold Π → X, these equations read φλµ ∂λ ri = ∂µi Hφ , ∂λ (φλµ riµ ) = −∂i Hφ . Every iso-Hamiltonian form Hφ (3.85) yields the associated momentum morc given by the coordinate expression phism H φ c = (φ−1 )λ ∂ i H . yµi ◦ H φ µ λ φ

3.7. ISOMULTISYMPLECTIC STRUCTUTE

95

However, the Lagrangian counterpart of an iso-Hamiltonian form is another Lepagean equivalent φλµ πiµ dy i ∧ ωλ − (φλµ πiµ yλi − L)ω, but not the Poincaré-Cartan form.

96

CHAPTER 3.


Chapter 4 Hamiltonian Field Theory With the multimomentum Hamiltonoan formalism, the tools are now at hand to canonically analize constraint field systems on the covariant finite-dimensional level. Maintaining covariance has the principal advantages of describing field theories, for any space-time splitting shades the covariant picture of field constraints. The Lagrangian densities of field models are almost always semiregular and almost regular that enables one to utilize the relations stated above between Lagrangian and Hamiltonian formalisms. Moreover, the Lagrangian densities of all fundamental field theories are quadratic and affine in the derivatives of field functions. Gauge theory exemplifies the degenerate quadratic Lagrangian density, whereas gravity and fermion fields are described by the affine ones. In first Section of this Chapter, we shall spell out Lagrangian and Hamiltonian systems in case of affine and almost regular quadratic Lagrangian densities. The goal is the general procedure of describing constraint field systems. The most of contemporary field models are concerned with gauge theory. We observe that several attributes of gauge theory such as gauge freedom and gauge conditions are the common attributes of systems with degenerate quadratic and affine Lagrangian densities, without appealing to any symmetry group.

4.1

Constraint field systems

This Section presents the complete families of multimomentum Hamiltonian forms associated with affine and almost regular quadratic Lagrangian densities. The key ingredient in our consideration is splitting of the configuration space in the dynamic sector and the gauge sector. The latter consists with the kernal of the Legendre morphism. As an immediate consequence of this splitting, a part of the Hamilton equations comes to the gauge-type conditions independent of canonical momenta. 97

98

CHAPTER 4. HAMILTONIAN FIELD THEORY

Different associated multimomentum Hamiltonian forms are responsible for different such conditions. Given a fibred manifold Y → X, let us consider a quadratic Lagrangian density L = Lω, 1 i j λ i L = aλµ ij (y)yλ yµ + bi (y)yλ + c(y), 2

(4.1)

where a, b and c are local functions on Y with the corresponding transformation laws. The associated Legendre morphism reads b = aλµ y j + bλ . pλi ◦ L ij µ i

(4.2)

Lemma 4.1. The Lagrangian density (4.1) is semiregular. 2 Outline of proof: If q ∈ Q, the system of linear algebraic equations (4.2) for yµi , by definition, have solutions, and these solutions form an affine space. 2 The Legendre morphism (4.2) is an affine morphism over Y . It implies the corresponding linear morphism L : T ∗ X ⊗ V Y → Π, Y

pλi

j ◦ L = aλµ ij y µ ,

over Y where y jµ are bundle coordinates of the vector bundle (1.23). Almost all quadratic Lagrangian densities of field models take the particular form 1 L = aλµ y i y j + c, (4.3) 2 ij λ µ y iµ = yµi − Γiµ , where Γ is a certain connection on Y → X and yµi → y iµ is the corresponding covariant differential (1.64). This fashion is equivalent to condition that the Lgrangian constraint space Q defined by the Legendre morphism (4.2) contains the image of b ) of the Legendre bundle Π → Y . Let us set Y under the canonical zero section 0(Y b =L b −1 (0(Y b )). Ker L

It is an affine subbundle of the jet bundle J 1 Y → Y which is modelled on the vector bundle −1

b )). Ker L = L (0(Y

4.1. CONSTRAINT FIELD SYSTEMS

99

b Then, there exists a connection Γ on Y → X which takes its values into Ker L: b Γ : Y → Ker L, λµ j aij Γµ + bλi = 0.

(4.4) (4.5)

With this connection, the Lagrangian density (4.1) can be brought into the form (4.3). For instance, if the Lagrangian density (4.1) is regular, the connection (4.4) is the unique solution of the algebraic equations (4.5). Proposition 4.2. Let L be an almost regular quadratic Lagrangian density b ) ⊂ Q. Then, there exists a linear pullback-valued horizontal 1-form such that 0(Y σ : Π → T ∗ X ⊗ V Y, y iλ ◦ σ =

(4.6)

Y ij µ σλµ pj ,

on the Legendre bundle Π → Y such that L ◦ σ ◦ iQ = i Q , jk λ k aλµ ij σµα = δα δi ,

(4.7)

where iQ denotes the natural imbedding of Q into Π. 2 Outline of proof: Since ImL = Q, every global linear section of the bundle L : T ∗X ⊗ V Y → Q Y

is extended to the morphism (4.6). 2 If the Lagrangian density (4.1) is regular, the form (4.6) is determined uniquely by the algebraic equations (4.7). The connection (4.4) and the form (4.6) play the prominent role in our construction. Since L and σ are linear morphisms, their composition L ◦ σ is a surjective submersion of Π onto Q. It follows that σ = σ ◦ L ◦ σ,

(4.8)

100


and the jet bundle J 1 Y → Y has the splitting b ⊕ Imσ, J 1 Y = Ker L

(4.9)

Y αµ j α ik α ik j (aαµ yλi = [yλi − σλα kj yµ + bk )] + [σλα (akj yµ + bk )].

Moreover, there exists the form σ [Eq.(4.6)] such that also the Legendre bundle meets the splitting Π = Ker σ ⊕ Q.

(4.10)

Y

Given the form σ [Eq.(4.6)] and the connection Γ [Eq.(4.4)], let us consider the affine momentum morphism b + σ, Φ=Γ ij µ pj , Φiλ = Γiλ (y) + σλµ

(4.11)

b is the pullback (3.13) of Γ onto Π. It is easy to see that this momentum where Γ b is the Legendre morphism (4.2). Conversely, morphism satisfies Eq.(3.48) where L every affine momentum morphism satisfying the condition (3.48) is of the type (4.11). Building on the condition (4.8), one can check that the multimomentum Hamiltonian form HLΦ [Eq.(3.47)] corresponding to Φ [Eq.(4.11)] is associated with the Lagrangian density (4.1). It is given by the expression

1 1 ij λ µ H = pλi dy i ∧ ωλ − [Γiλ (pλi − bλi ) + σλµ pi pj − c]ω. 2 2

(4.12)

This expression consists with the canonical splitting (3.26) of H where ΓH = Γ. Note that, on Ker σ, the multimomentum Hamiltonian form (4.12) becomes affine in canonical momenta as it was pointed out in Section 3.5. We claim that the multimomentum Hamiltonian forms (4.12) where Γ are connections (4.4) constitute the complete family. Given the multimomentum Hamiltonian form (4.12), let us consider the Hamilton equations (3.35a) for sections r of the fibred Legendre manifold Π → X. They read b + σ) ◦ r, J 1 s = (Γ

or ij µ ∇λ ri = σλµ rj

s = πΠY ◦ r,

(4.13)

4.1. CONSTRAINT FIELD SYSTEMS

101

where ∇λ ri = ∂λ ri − (Γ ◦ s)iλ is the covariant derivative (1.65). With splitting (4.9), we have the following surjections b S := pr 1 : J 1 Y → Ker L, α j ik (aαµ S : yλi → yλi − σλα kj yµ + bk ),

and F := pr 2 : J 1 Y → Imσ, b : y i → σ ik (aαµ y j + bα ). F =σ◦L k λα kj µ λ With respect to these surjections, the Hamilton equations (4.13) break into two parts S ◦ J 1 s = Γ ◦ s, j α ik ∇λ ri = σλα (aαµ kj ∂µ r + bk ).

(4.14)

F ◦ J 1 s = σ ◦ r, ik j α ik α σλα (aαµ kj ∂µ r + bk ) = σλα rk .

(4.15)

and

The Hamilton equations (4.14) are independent of canonical momenta rkα and make the sense of gauge-type conditions. By analogy with gauge theory, one can think of the preimage F −1 (y) of every point y ∈ Imσ as being a gauge-type class. Then, Eq.(4.14) makes the sense of a gauge-type condition. We shall say that this condition is universal if it singles out one representative of every gauge-type class. It is readily observed that Eq.(4.14) is the universal gauge-type condition on sections γ of the jet bundle J 1 Π → Π when the algebraic Hamilton equations (3.31a) and (3.31b) are considered, otherwise on sections r of the Legendre manifold Π → X. At the same time, one can conclude that it is one or another condition for the quantity ik j α S ◦ J 1 s = ∂λ ri − σλα (aαµ kj ∂µ r + bk )

which may supplement the underdetermined Euler-Lagrange equations for the degenerate Lagrangian density (4.1). Moreover, for every section s of the fibred manifold Y → X (in particular, every solution of the Euler-Lagrange equations), there exists a connection Γ [Eq.(4.4)]

102


such that the gauge-type conditions (4.14) holds. Indeed, let Γ0 be a connection whose integral section is s. Set Γ = S ◦ Γ0 , i ik 0j α Γ = Γ0 λ − σλα (aαµ kj Γ µ + bk ). In this case, the momentum morphism (4.11) satisfies Eq.(3.66) b ◦ J 1 s = J 1 s. Φ◦L

It follows that the multimomentum Hamiltonian forms (4.12) constitute really the complete family. It must be noted that the multimomentum Hamiltonian forms from this family differ from each other only in connections Γ [Eq.(4.4)] which imply the different gauge-type conditions (4.14). The complete family of the multimomentum Hamiltonian forms (4.12) is by no means minimal. It may be often minimized by choice of a certain subset of connections (4.4). Let us turn now to an affine Lagrangian density L = Lω, L = bλi (y)yλi + c(y).

(4.16)

It is almost regular and semiregular. The corresponding Legendre morphism b = bλ (y) pλi ◦ L i

(4.17)

is a bundle over the imbedded constraint submanifold Q = b(Y ) of the Legendre manifold Π determind by the section b of the Legendre bundle Π → Y . b the associated moLet Γ be a connection on the fibred manifold Y → X and Γ b is the mentum morphism. This momentum morphism satisfies Eq.(3.48) where L Legendre morphism (4.17). Therefore, let us consider the multimomentum Hamiltonian form (3.47) H = HΓ + L ◦ Γ = pλi dy i ∧ ωλ − (pλi − bλi )Γiλ ω + cω,

(4.18)

b It is easy to see that this multimomentum Hamiltonian form is corresponding to Γ. associated with the affine Lagrangian density (4.16). Note that this multimomentum Hamiltonian form is affine in canonical momenta everywhere on Π. The corresponding momentum morphism reads c = Γi yλi ◦ H λ

(4.19)

4.2. HAMILTONIAN GAUGE THEORY

103

and so, the Hamilton equations (3.35a) reduce to the gauge-type condition ∂λ ri = Γiλ . Their solution is an integral section of the connection Γ. Conversely, for each section s of the fibred manifold Y → X, there exists the connection Γ on Y which has the integral section s. Then, the corresponding momentum morphism (4.19) obeys the condition c◦L b ◦ J 1 s = J 1 s. H

It follows that the multimomentum Hamiltonian forms (4.18) constitute the complete family. Thus, the universal procedure is now at hand to describe constraint field theories including gauge theory and gravitation theory.

4.2

Hamiltonian gauge theory

Gauge theory, in conjuction with the mechanism of spontaneous symmetry breaking, makes the adequate picture of fundamental interactions where principal connections model the mediators of interaction possessing a certain group of symmetries. This Section covers the gauge theory with unbroken symmetries. Gauge theory of principal connections is described by the degenerate quadratic Lagrangian density, and its multimomentum Hamiltonian formulation shadows the general procedure for degenerate quadratic models from previous Section. The feature of gauge theory consists in the fact that splittings (4.9) and (4.10) of configuration and phase spaces of fields are canonical. In the rest of this Chapter, a base manifold X is proposed to be oriented. We call it the world manifold. By gµν [or g µν ] is meant a nondegenerate fibre metric in the tangent [or cotangent] bundle of X. We denote g = det(gµν ). Structure groups of bundles throughout are supposed to be finite-dimensional real connected Lie groups. Let πP : P → X be a principal bundle with a structure Lie group G which acts freely and transitively on P on the right: rg : p 7→ pg,

(4.20)

104


p ∈ P,

g ∈ G.

Remark. In other words, the principal bundle P → X is the general affine bundle modelled P ×G → P X

on the right on the trivial group bundle X × G. The principal bundle P is also the general affine bundle modelled on the left on the associated group bundle Pe with the standard fibre G on which the structure group G acts by the adjoint representation. The corresponding fibre-preserving morphism reads e p) 7→ pp e ∈ P. Pe × P 3 (p,

Note that the standard fibre of the group bundle Pe is the group G, while that of the principal bundle P is the group space of G on which the structure group G acts on the left. 2 A principal bundle P → X with a structure Lie group G possesses the canonical trivial vertical splitting (1.8) α : V P → P × gl , pr 2 ◦ α ◦ em = Jm ,

(4.21)

where gl is the left Lie algebra of left-invariant vector fields on the group G, {Jm } is a basis for gl and em denote the corresponding fundamental vector fields on the principal bundle P . The general approach to connections as jet fields is suitable to formulate the classical concept of principal connections.15,36 If P → X is a principle bundle with a structure group G, the exact sequence (1.6a) implies the exact sequence 0 → V G P ,→ T G P → T X → 0 X

(4.22)

where T G P = T P/G, V G P = V P/G are the quotients of the tangent bundle T P and the vertical tangent bundle V P of P respectively by the tangent prolongation of the canonical action (4.20) of G on P


105

on the right. The bundle V G P → X is called the adjoint bundle. Its standard fibre is the right Lie algebra gr of the right-invariant vector fields on the group G. The group G acts on this standard fibre by the adjoint representation. A principal connection A on a principal bundle P → X is defined to be a Gequivariant global jet field on P such that, whenever the canonical morphism (4.20), the following diagram commutes: A

P −→ J 1 P J 1 rg

rg

?

?

P −→ J 1 P A

We have A ◦ rg = J 1 rg ◦ A, g ∈ G, λ m A = dx ⊗ [∂λ + Aλ (p)em ], m −1 Am λ (pg) = Aλ (p)adg (em ).

p ∈ P,

A principal connection A determines splitting T X ,→ T G P X

of the exact sequence (4.22). Let A be a principal connection on P → X. Building on the complementary morphism (1.28) and the canonical vertical splitting (4.21), one recovers the familiar connection form A = α ◦ θ1 ◦ A on the principal bundle P . Relative to a bundle atlas ΨP = {Uξ , ψξP } of P , we obtain the local connection 1-forms λ Aξ = zξ∗ A = −Am λ (x)dx ⊗ Jm .

(4.23)

Here, {zξ } is the family of local sections of P → X associated with an atlas ΨP : (pr 2 ◦ ψξP ◦ zξ )(x) = 1G ,

x ∈ Uξ ,

where 1G is meant the unit element of the group G.

106


Because of the manifest G-equivariance of principal connections, there is the 1:1 correspondence between the principal connections on a principal bundle P → X and the global sections of the quotient C := J 1 P/G → X

(4.24)

of the jet bundle J 1 P → P by the first order jet prolongations J 1 rg of the canonical morphisms (4.20). We shall call C → X the principal connection bundle. It is an affine bundle modelled on the vector bundle C = T ∗X ⊗ V GP

(4.25)

and so, there is the canonical vertical splitting V C = C × C. Remark. The bundle C [Eq.(4.24)] fails to be a bundle with a structure group. The jet prolongation J 1 P × J 1 (X × G) → J 1 P of the canonical action (4.20) brings the fibred jet manifold J 1 P → X into the general affine bundle modelled on the right on the group bundle J 1 (X × G) = G × (T ∗ X × gl ) over X. However, the latter fails to be generally a trivial bundle because of T ∗ X. Therefore, J 1 P → X is not a principal bundle. At the same time, the bundle J 1P = C × P → C X

is the G-principal bundle over C. 2 Given a bundle atlas ΨP of P , the principal connection bundle C is provided with the fibred coordinates (xλ , kµm ) so that (kµm ◦ A)(x) = Am µ (x) are coefficients of the local connection 1-form (4.23). In gauge theory, global sections of the principal connection bundle (4.24) are treated the gauge potentials. Their finite-dimensional configuration space is the first order jet manifold J 1 C of C. It is endowed with the adapted coordinates m (xλ , kµm , kµλ ).

(4.26)


107

The affine jet bundle J 1 C → C is modelled on the vector bundle T ∗ X ⊗(C × T ∗ X ⊗ V G P ). C

Lemma 4.3. There exists the canonical splitting 2

J 1 C = C+ ⊕ C− = (J 2 P/G) ⊕(∧ T ∗ X ⊗ V G P ) C

C

(4.27)

C

over C where 2

C− = C × ∧ T ∗ X ⊗ V G P X

and C+ → C is the affine bundle modelled on the vector bundle 2

C + = ∨ T ∗ X ⊗ V G P. C

2 Outline of proof: To perform the splitting (4.27), one can use the splitting (1.45) and the canonical isomorphism of Jb2 P/G to J 1 C. 2 Relative the coordinates (4.26), the splitting (4.27) reads 1 m 1 m m m n l m m n l kµλ = (kµλ + kλµ + cm nl kλ kµ ) + (kµλ − kλµ − cnl kλ kµ ) 2 2 where ckmn are structure constants of the Lie algebra gr with respect to its basis {Im }. There are the corresponding canonical surjections S := pr 1 : J 1 C → C+ , m m m n l Sµλ = kµλ + kλµ + cm nl kλ kµ , and F := pr 2 : J 1 C → C− 1 m λ F = Fλµ dx ∧ dxµ ⊗ Im , 2 m m m n l Fλµ = kµλ − kλµ − cm nl kλ kµ .

108


For every principal connection A, we observe that F ◦ J 1 A = F, 1 m λ dx ∧ dxµ ⊗ Im , F = Fλµ 2 m m m n k = ∂λ Am Fλµ µ − ∂µ Aλ − cnk Aλ Aµ , is the familiar strength of the gauge potential A. On the configuration space (4.27), the conventional Yang-Mills Lagrangian density LY M of gauge potentials is given by the expression LY M =

1 G λµ βν m n q a g g Fλβ Fµν | g | ω 4ε2 mn

(4.28)

where aG is a nondegenerate G-invariant metric in the Lie algebra gr and ε is a coupling constant. This Lagrangian density is almost regular and semiregular. Remark. Note that the Lagrangian density (4.28) is the unique gauge invariant quadratic Lagrangian density on J 1 C. In gauge theory several types of gauge transformations are considered.37,57 Here, we are concerned only with so-called principal morphisms. Given a principal bundle P → X, by a principal morphism is meant its G-equivariant isomorphism ΦP over X together with the first order jet prolongations J 1 ΦP . Whenever g ∈ G, we have rg ◦ ΦP = Φ P ◦ r g . Every such isomorphism ΦP is brought into the form ΦP (p) = pfs (p),

p ∈ P,

(4.29)

where fs is a G-valued equivariant function on P : fs (qg) = g −1 fs (q)g,

g ∈ G.

There is the 1:1 correspondense between these functions and the global section s of the group bundle Pe : s(π(p))p = pfs (p). Principal morphisms ΦP constitute the gauge group which is isomorphic to the group of global sections of the P -associated group bundle Pe . The Sobolev completion of the gauge group is a Banach Lie group. Its Lie algebra in turn is the Sobolev completion of the algebra of generators of infinitesimal principal morphsms. These generators are represented by the corresponding vertical vector fields on a P -associated bundle


109

which carries representation of the gauge group. We call them principal vector fields. Note that the jet lift u [Eq.(1.25)] of a principal vector field u on a P -associated bundle is a principal vector field on the fibred jet manifold J 1 Y → X. Then, it is readily observed that a Lagrangian density L on the configuration space J 1 Y is gauge invariant iff, whenever principal vector field u on Y → X, Lu L = 0 where L denotes the Lie derivative of exterior forms. For instance, local principal vector fields on the principal connection bundle C read l n µ u = (∂µ αµ + cm nl kµ α )∂m

(4.30)

where αm (x) are local real functions on X. 2 The finite-dimensional phase space of gauge potentials is the Legendre bundle πΠC : Π → C, n

Π = ∧ T ∗ X ⊗ T X ⊗[C × C]∗ , C

C

over the principal connection bundle C [Eq.(4.24)]. It is endowed with the canonical coordinates (xλ , kµm , pµλ m ). The Legendre bundle Π over C as like as the jet bundle J 1 C → C admits the canonical splitting Π = Π + ⊕ Π− ,

(4.31)

C

1 µλ 1 µλ (µλ) λµ λµ + p[µλ] pµλ m = (pm + pm )) + (pm − pm ). m = pm 2 2 The Legendre morphism associated with the Lagrangian density (4.28) takes the form b p(µλ) ◦L Y M = 0, m

(4.32a) q

−2 G λα µβ n b p[µλ] m ◦ LY M = ε amn g g Fαβ | g |.

A glance at this morphism shows that b Ker L Y M = C+

(4.32b)

110


and 1 b Q=L Y M (J C) = Π− .

It follows that splittings (4.27) and (4.31) are similar to the splittings (4.9) and (4.10) in general case of a degenerate Lagrangian density. Thus, to construct the complete family of multimomentum Hamiltonian forms associated with the Yang-Mills Lagrangian density (4.28), we can follow the standard procedure for degenerate Lagrangian densities from previous Section. In accordance with this procedure, let us consider connections on the principal b connection bundle C which take their values into Ker L: S : C → C+ , n l m m − Sλµ − cm Sµλ nl kλ kµ = 0.

(4.33)

Lemma 4.4. Given a symmetric linear connection K ∗ [Eq.(1.67)] on the cotangent bundle T ∗ X of X, every principal connection B on a principal bundle P gives rise to the connection SB : C → C+ , SB ◦ B = S ◦ J 1 B, on the principal connection bundle C. Relative the coordinates (4.26), the connection SB reads 1 m n l m m SB m µλ = [cnl kλ kµ + ∂µ Aλ + ∂λ Aµ 2 n l n l β m m −cm nl (kµ Aλ + kλ Aµ )] − K µλ (Aβ − kβ ).

(4.34)

2 The proof is straightforward. Given a connection (4.33), one can check directly that the multimomentum Hamiltonian form m µλ m f H = pµλ m dkµ ∧ ωλ − pm Sµλ ω − HY M ω, 2

f H YM =

ε mn [µλ] [νβ] a gµν gλβ pm pn | g |−1/2 , 4 G

is associated with the Lagrangian density LY M .

(4.35)


111

Remark. In contrast with the Lagrangian density LY M , the multimomentum Hamiltonian forms (4.35) are not gauge invariant, otherwise their restriction 1 m n l m f HQ = pµλ m (dkµ ∧ ωλ − cnl kλ kµ ω) − HY M ω 2 to the constraint space Q. By gauge transformations in multimomentum canonical variables are meant isomorphisms of the Legendre bundle Π over C which are induced by principal morphisms of the principal connection bundle C. The corresponding principal vector fields on Π → X read µλ n m l l n µ u = (∂µ αµ + cm nl kµ α )∂m − cnm pl α ∂µλ .

2 We can justify that the multimomentum Hamiltonian forms (4.35) constitute the complete family. Moreover, to get the complete family, it suffices to take the subset of connections (4.34). Let K ∗ be a fixed symmetric linear connection (1.67) on the cotangent bundle ∗ T X of X. In virtue of Lemma 4.4, every principal connection B on P gives rise to the connection SB [Eq.(4.34)]. We shall denote by HB the multimomentum Hamiltonian form (4.35) where S is the connection SB [Eq.(4.34)]. Given HB , the corresponding Hamilton equations for sections r of the Legendre bundle Π → X are µλ (λν) ∂λ rm = −cnlm rνl rn[µν] + cnml Bνl rn(µν) − K µ λν rm , m m m ∂λ rµ + ∂µ rλ = 2SB (µλ)

(4.36) (4.37)

and plus Eq.(4.32b). The Hamilton equations (4.37) and (4.32b) are similar to Eqs.(4.14) and (4.15) respectively. The Hamilton equations (4.32b) and (4.36) restricted to the constraint space (4.32a) are precisely the Yang-Mills equations for a gauge potential A = πΠC ◦ r. Whenever HB , these equations are the same. They exemplify the De DonderHamilton equations (3.73). Different multimomentum Hamiltonian forms HB lead to different Eqs.(4.37). The Hamilton equation (4.37) is independent of canonical momenta and it is just the gauge-type condition (4.14) SB ◦ A = S ◦ J 1 A.

112


A glance at this condition shows that, given a solution A of the Yang-Mills equations, there always exists a multimomentum Hamiltonian form HB which obeys the condition (3.66) 1 1 c ◦L b H B Y M ◦ J A = J A.

For instance, this is HB=A . It follows that the Hamiltonian forms HB constitute really the complete family. It must be noted that the gauge-type condition (4.37) differs from the familiar gauge condition in gauge theory. The latter singles out a representative of each gauge coset (with 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 A0 which also is 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 a system of the Yang-Mills equations and one or another 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, the system of these gauge conditions is not complete. In gauge theory, this lack is not essential since one think of all gauge conjugate potentials as being physically equivalent, otherwise in case of other constraint field theories, e.g. the Proca field. In the framework of the multimomentum 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 to which this solution satisfies. Given a principal connection B, the gauge-type condition (4.37) is universal on solutions of the algebraic Hamilton equations, but it is rather rigid on solutions of the differential Hamilton equations. In gauge theory where gauge conjugate solutions are treated physically equivalent, one may replace Eq.(4.37) by one or another condition for 1 m m m n l (S ◦ J 1 A)m µλ = (∂λ Aµ + ∂µ Aλ + cnl Aλ Aµ ) 2 which can supplement the Yang-Mills equations. In particular, m g µλ (S ◦ J 1 A)m µλ = α (x)

recovers the familiar generalized Lorentz gauge condition.

4.3. ELECTROMAGNETIC FIELDS

4.3

113

Electromagnetic fields

As a test case of Hamiltonian gauge theory, let us consider electromagnetic fields on a 4-dimensional world manifold X 4 . In accordance with the gauge approach, electromagnetic potentials are identified with principal connections on a principal bundle P → X 4 with the structure group U (1). In this case, the adjoint bundle is isomorphic to the trivial linear bundle V G P = X 4 × R. The corresponding principal connection bundle C [Eq.(4.24)] is an affine bundle modelled on the cotangent bundle T ∗ X of X 4 . It is coordinatized by (xλ , kµ ). The finite-dimensional configuration space of electromagnetic potentials is the affine jet bundle J 1 C → C modelled on the pullback tensor bundle 2

C := ⊗ T ∗ X × C → C. X

Its canonical splitting (4.27) reads 2

J 1 C = C+ ⊕(∧ T ∗ X × C) C

(4.38)

X

where C+ → C is the affine bundle modelled on the pullback symmetric tensor bundle 2

C + = ∨ T ∗ X × C. X

Relative to the adapted coordinates (xλ , kµ , kµ,λ ) of J 1 C, the splitting (4.38) is expressed as 1 kµλ = (Sµλ + Fµλ ) = k(µλ) + k[µλ] . 2 For any section A of C, we observe that Fµλ = Fµλ ◦ J 1 A = ∂λ Aµ − ∂µ Aλ is the familiar strength of an electromagnetic field. For the sake of simplicity, we further let X 4 be the Minkowski space with the Minkowski metric η = (+, − − −).

114


On the configuration space (4.38), the conventional Lagrangian density of an electromagnetic field is written LE = −

1 λµ βν η η Fλβ Fµν ω. 16π

(4.39)

The finite-dimensional phase space of electromagnetic potentials is the Legendre bundle 4

Π = (∧ T ∗ X ⊗ T X ⊗ T X) × C X

endowed with the canonical coordinates (xλ , kµ , pµλ ). With respect to these coordinates, the Legendre morphism associated with the Lagrangian density (4.39) is written b = 0, p(µλ) ◦ L E

(4.40a)

b = − 1 η λα η µβ F . p[µλ] ◦ L αβ E 4π

(4.40b)

The multimomentum Hamiltonian forms f ω, HB = pµλ dkµ ∧ ωλ − pµλ SBµλ ω − H E 1 SBµλ = (∂µ Bλ + ∂λ Bµ ), 2 f HE = −πηµν ηλβ p[µλ] p[νβ] ,

(4.41)

parameterized by electromagnetic potentials B are associated with the Lagrangian density (4.39) and constitute the complete family. Given the multimomentum Hamiltonian form HB [Eq.(4.41)], the corresponding Hamilton equations are ∂λ rµλ = 0 ∂λ rµ + ∂µ rλ = ∂λ Bµ + ∂µ Bλ

(4.42) (4.43)

and plus Eq.(4.40b). On the constraint space (4.40a), Eqs.(4.40b) and (4.42) come to the Maxwell equations without matter sources. At the same time, Eq.(4.43) independent on canonical momenta plays the role of a gauge-type condition. Although its solution πΠC ◦ r = B.

4.4. PROCA FIELDS

115

is by no means unique, this gauge-type condition, B being fixed, is rather rigid. Since gauge conjugate electromagnetic potentials are treated physically equivalent, one may replace Eq.(4.43) by one or another condition for the quantity ∂λ rµ + ∂µ rλ . In particular, the familiar generalized Lorentz gauge condition η µλ ∂µ rλ = α(x) is recovered. In gauge theory, the gauge conditions which supplement the Euler-Lagrange equations are well-known, otherwise in general case of constraint field systems.

4.4

Proca fields

The model of massive vector Proca fields exemplifies a constraint field theory which shadows the electromagnetic theory, but without gauge invariance. Note that at present, the Proca field model is not actual, for vector mesons appeared to be composite particles and the Higgs mechanism of generating a mass is dominant in contemporary particle physics. The vector Proca fields are represented by sections of the cotangent bundle T ∗ X → X 4 , whereas electromagnetic potentials are sections of the affine bundle modelled on T ∗ X. The finite-dimensional configuration space of Proca fields is the affine jet bundle J 1T ∗X → T ∗X modelled on the pullback tensor bundle 2

⊗ T ∗ X × T X → T X.

(4.44)

It is provided with the adapted coordinates (xλ , kµ , kµ,λ ) where kµ = x˙ µ are the familiar induced coordinates of T ∗ X. Let X 4 be the Minkowski space. The Lagrangian density of Proca fields is viewed as the electromagnetic Lagrangian density (4.39) minus the massive-type term: LP = LE −

1 2 µλ m η kµ kλ ω. 8π

(4.45)

116


It is semiregular and almost regular. The finite-dimensional phase space of Proca fields is the Legendre bundle 4

Π = ∧ T ∗X ⊗ T X ⊗ T X × T ∗X endowed with the canonical coordinates (xλ , kµ , pµλ ). With respect to these coordinates, the Legendre morphism associated with the Lagrangian density (4.45) takes the form b = 0, p(µλ) ◦ L P b =− p[µλ] ◦ L P

(4.46a)

1 λα µβ η η Fαβ . 4π

(4.46b)

b . We have It is viewed exactly as the Legendre morphism L E 2

b = ∨ T ∗X × T ∗X Ker L P

and 4

2

Q = ∧ T ∗X ⊗ (∧ T X), ∗ T X

p(µλ) = 0. Following the general procedure of describing constraint field systems from Section 4.1, let us consider the form σ [Eq.(4.6)]: k µλ ◦ σ = −2πηµν ηλβ p[νβ] where k µλ are the bundle coordinates of the bundle (4.44). Since 2

Imσ = ∧ T ∗ X × T ∗ X and 4

2

Ker σ = ∧ T ∗ X ⊗ (∨ T X) × T ∗ X, one can perform the corresponding splitting (4.9) of the configuration space 2

2

J 1T ∗X = ∨ T ∗X ⊕ ∧ T ∗ X, ∗ T X

1 kµλ = (Sµλ + Fµλ ) = k(µλ) + k[µλ] 2

(4.47)

4.4. PROCA FIELDS

117

and the splitting (4.10) of the phase space 4

2

Π = [∧ T ∗ X ⊗ (∨ T X)] ⊕ Q, ∗ ∗ p

µλ

=p

(µλ)

T X [µλ]

+p

T X

.

Following Section 4.1, let us consider connections on the cotangent bundle T ∗ X b . Every such a connection is expressed as taking their values into Ker L P S = φµλ dxµ ⊗ ∂ λ where φ is a symmetric soldering form on T ∗ X, for K = 0 in the Minkowski space. By analogy with case of the electormagnetic field, it suffies to take the connections 1 SB = dxλ ⊗ [∂λ + (∂µ Bλ + ∂λ Bµ )∂µ ] 2 where B is a section of T ∗ X. Then one can justify that the multimomentum Hamiltonian forms f ω, HB = pµλ dkµ ∧ ωλ − pµλ SBµλ ω − H P 1 f =H f + H m2 η µν kµ kν , P E 8π

are associated with the Lagrangian density LP [Eq.(4.39)] and constitute the complete family. Given the multimomentum Hamiltonian form HB , the corresponding Hamilton equations are 1 2 µν m η rν , 4π ∂λ rµ + ∂µ rλ = ∂λ Bµ + ∂µ Bλ ∂λ rµλ = −

(4.48) (4.49)

and plus Eq.(4.46b). On the constraint space (4.46a), Eqs.(4.46b) and (4.48) come to the Euler-Lagrange equations which are supplemented by the gauge-type condition (4.49). In view of the splitting (4.47), this supplementary condition is universal on solutions of the algebraic Hamilton equations for HB , but is rather rigid on solutions of the differential Hamilton equations. At the same time, one may replace Eq.(4.49) by one or another condition for the quantity ∂λ rµ + ∂µ rλ , e.g., the generalized Lorentz gauge condition η µλ ∂µ rλ = α(x).

118


But, in contrast with case of the electromagnetic field, no such condition exausts all physically non-equivalent solutions of the Euler-Lagrange equations for Proca fields. For instance, the Lorentz gauge condition η µλ ∂µ rλ = 0 is compatible with the wave solutions. Note that, in another model, massive vector fields represented by sections of the cotangent bundle T ∗ X may be described also by the hyperregular Lagrangian density 1 L = η µν (η λα kµλ kνα − m2 kµ kν )ω 2 on the configuration space J 1 T ∗ X. In this case, they make the sense of matter fields.

4.5

Matter fields

In gauge theory, matter fields possessing only internal symmetries, i.e., scalar fields are described by sections of a vector bundle associated with a principal bundle P . One calls it the matter bundle. Let Y → X be a bundle associated with a principal bundle P → X. The structure group G of P acts freely on the standard fibre V of Y on the left. The total space of the P -associated bundle Y , by definition, is the quotient Y = (P × V )/G of the product P × V by identification of its elements (qg × gv) for all g ∈ G. The P -associated bundle Y is provided with an atlas Ψ = {Uξ , ψξ } associated with an atlas ΨP = {Uξ , zξ } of the principal bundle P as follows: ψξ−1 (x × V ) = [zξ (x)]V (V ),

x ∈ Uξ ,

where by [p]V is meant the restriction of the canonical map [P ]V : P × V → Y

4.5. MATTER FIELDS

119

to the subset p × V . Remark. For each P associated bundle Y , there exists the fibre-preserving representation morphism e y) 7→ py e ∈Y Pe × Y 3 (p,

where Pe is the P -associated group bundle. Building on this representation morphism, one can induce principal morphisms of Y : Φs : Y 3 y 7→ (s ◦ π)(y)y ∈ Y where s is a global section of Pe . The corresponding principal vertical vector fields on the P -associated vector bundle Y → X read u = αm (x)Im i j y j ∂i where Im are generators of the structure group G acting on V and αm (x) are arbitrary local functions on X. 2 Every principal connection A on a principal bundle P yields the associated connection Γ on a P -associated bundle Y → X such that the diagram J 1 P × V −→ J 1 Y A×Id V

6

6Γ

P × V −→ Y is commutative. We call it the associated principal connection. Relative to the associated atlases Ψ of Y and ΨP of P , this connection is written i j Γ = dxλ ⊗ [∂λ + Am µ (x)Im j y ∂i ]

(4.50)

where Am µ (x) are coefficients of the local connection 1-form (4.23). The curvature (1.59) of the connection (4.50) reads i m Rλµ = Fλµ Im i j y i .

Turning now to the model of scalar matter fields, we let Y → X be a P -associated matter bundle. It is assumed to be provided with a G-invariant fibre metric aY . Because of the canonical vertical splitting (1.9), the metric aY is also a fibre metric in the vertical tangent bundle V Y → X. A linear connection Γ on a matter bundle Y is supposed to be an associated principal connection (4.50).

120


The finite-dimensional configuration space of matter fields is the jet manifold J Y provided with the adapted coordinates (xλ , y i , yλi ). Relative to these coordinates, the Lagrangian density of scalar matter fields in the presence of a background gauge field, i.e., a connection Γ [Eq.(4.50)] on Y reads q 1 (4.51) LM = aYij [g µν (yµi − Γiµ )(yνj − Γjν ) − m2 y i y j ] | g |ω. 2 It is hyperregular. The finite-dimensional phase space of matter fields is the Legendre bundle 1

n

Π = (∧ T ∗ X ⊗ T X ⊗ Y ∗ ) × Y where by Y ∗ is meant the vector bundle dual to Y → X. It is provided with the canonical coordinates (xλ , y i , pλi ). The unique multimomentum Hamiltonan form on Π associated with the Lagrangian density (4.51) reads i j f HM = pλi dy i ∧ ωλ − pλi Am (4.52) λ Im j y ω − HM , q f = 1 (aij g pµ pν | g |−1 +m2 aY y i y j ) | g |, H M µν i j ij 2 Y where aY is the fibre metric in Y ∗ dual to aY . The corresponding Hamilton equations for sections r of the Legendre manifold Π → X take the form ij µ i j −1/2 ∂λ ri = Am , λ Im j r + aY gλµ rj | g |

q

i λ 2 Y j ∂λ riλ = −Am | g |. λ Im j rj − m aij r

They are equivalent to the Euler-Lagrange equations corresponding to the Lagrangian density (4.51). In case of unbroken symmetries, the total configuration space of gauge potentials and matter fields is the direct product J 1 Y × J 1 C.

(4.53)

X

Accordingly, the total Lagrangian density describing matter fields in the presence of dynamic gauge potentials is the sum of the Lagrangian density (4.28) and the Lagrangian density (4.51) where we must set up Γiλ = kλm Im i j y j .

(4.54)

The associated multimomentum Hamiltonian forms are the sum of the multimomentum Hamiltonian forms (4.35) where S = SB [Eq.(4.34)] and the multimomentum Hamiltonian form (4.52) where the connection Γ is given by the expression (4.54). In this case, the Hamilton equation (4.36) contains the familiar Nöether current λ Jm = pλi Im i j y j

as a matter source of gauge potentials.

4.6. HAMILTON EQUATIONS OF GENERAL RELATIVITY

4.6

121

Hamilton equations of General Relativity

The contemporary concept of gravitation interactions is based on the gauge gravitation theory with two types of gravitational fields.24,39,43,50 These are tetrad gravitational fields and Lorentz gauge potentials. They correspond to different matter sources: the energy-momentum tensor and the spin current of matter. At present, all Lagrangian densities of classical and quantum gravitation theories are expressed in these variables. They are of the first order with respect to these fields. Only General Relativity without spin matter sources utilizes traditionally the HilbertEinstein Lagrangian density LHE which is of the second order with respect to a pseudo-Riemannian metric. One can reduce its order by means of the Palatini variables when the Levi-Civita connection is regarded on the same footing as a pseudo-Riemannian metric. This Section is devoted to the so-called affine-metric gravitation theory when gravitational variables are both pseudo-Riemannian metrics g on a world manifold X 4 and linear connections K [Eq.(1.67)] on the tangent bundle of X 4 . We call them a world metric and a world connection respectively. Given a world metric, every world connection meets the well-known decomposition in the Cristoffel symbols, contorsion and the nonmetricity term. We here are not concerned with the matter sources of a general linear connection, for they, except fermion fields, are non-Lagrangian and hypothetical as a rule. In this Section, X is an oriented 4-dimensional world manifold which obeys the well-known topological conditions in order that a gravitational field exists on X 4 . Let LX → X 4 be the principal bundle of linear frames in the tangent spaces to X 4 . The structure group of LX is the group GL4 = GL+ (4, R) of general linear transfromations of the standard fibre R4 of the tangent bundle T X. The world connections are associated with principal connections on the principal bundle LX → X 4 . Hence, there is the 1:1 correspondence between the world connections and the global sections of the principal connection bundle C = J 1 LX/GL4 .

(4.55)

Therefore, we can apply the standard scheme of gauge theory in order to describe the configuration and phase spaces of world connections. There is the 1:1 correspondence between the world metrics g on X 4 and the global sections of the bundle Σg of pseudo-Riemannian bilinear forms in tangent

122


spaces to X 4 . This bundle is associated with the GL4 -principal bundle LX. The 2-fold covering of the bundle Σg is the quotient bundle Σ = LX/SO(3, 1) where by SO(3, 1) is meant the connected Lorentz group.50 Thus, the total configuration space of the affine-metric gravitational variables is represented by the product of the corresponding jet manifolds: J 1 C × J 1 Σg .

(4.56)

X4

Given a holonomic bundle atlas of LX associated with induced coordinates of T X and T ∗ X, this configuration space is provided with the adapted coordinates (xµ , g αβ , k α βµ , g αβ λ , k α βµλ ). Also the total phase space Π of the affine-metric gravity is the product of the Legendre bundles over the above-mentioned bundles C and Σg . It is coordinatized by the corresponding canonical coordinates (xµ , g αβ , k α βµ , pαβ λ , pα βµλ ). On the configuration space (4.56), the Hilbert-Einstein Lagrangian density of General Relativity reads LHE = −

1 βλ α √ g F βαλ −gω, 2κ

(4.57)

F α βνλ = k α βλν − k α βνλ + k α εν k ε βλ − k α ελ k ε βν . It is affine in connection velocities k α βµλ and, moreover, it is independent of metric velocities g αβ λ at all. Therefore, one can follow the general procedure of analyzing constraint field theories from Section 4.1. The corresponding Legendre morphism is given by the expressions b pαβ λ ◦ L HE = 0, βνλ b pα βνλ ◦ L = HE = πα

√ 1 ν βλ (δα g − δαλ g βν ) −g. 2κ

(4.58)

These relations define the constraint space of General Relativity in multimomentum canonical variables. Building on all the set of connections on the bundle C × Σg , one can construct the complete family of multimomentum Hamiltonian forms (4.18) associated with

4.6. HAMILTON EQUATIONS OF GENERAL RELATIVITY

123

the Lagrangian density (4.57). To minimize it this complete family, we consider the following subset of these connections. Let K be a world connection and 1 SK α βνλ = [k α εν k ε βλ − k α ελ k ε βν + ∂λ K α βν + ∂ν K α βλ 2 −2K ε (νλ) (K α βε − k α βε ) + K ε βλ k α εν + K ε βν k α ελ −K α ελ k ε βν − K α εν k ε βλ ] the corresponding connection (4.34) on the bundle C [Eq.(4.55)]. Let K 0 be another symmetric world connection which induces an associated principal connection on the bundle Σg . On the bundle C × Σg , we consider the following connection α

β

Γαβ λ = −K 0 ελ g εβ − K 0 ελ g αε , Γα βνλ = SK α βνλ − Rα βνλ

(4.59)

where R is the curvature of the connection K. The corresponding multimomentum Hamiltonian form (4.18) is given by the expression HHE = (pαβ λ dg αβ + pα βνλ dk α βν ) ∧ ωλ − HHE ω, α

HHE = −pαβ λ (K 0 ελ g εβ + K 0 −Rα βνλ (pα βνλ − πα βνλ ).

β

ελ g

αε

) + pα βνλ Γα βνλ (4.60)

It is associated with the Lagrangian density LHE . We shall justify that the multimomentum Hamiltonian forms (4.60) parameterized by all the world connections K and K 0 constitute the complete family. Given the multimomentum Hamiltonian form HHE [Eq.(4.60)], the corresponding covariant Hamilton equations for General Relativity read α

β

∂λ g αβ + K 0 ελ g εβ + K 0 ελ g αε = 0, ∂λ k α βν = Γα βνλ − Rα βνλ , ε ε ∂λ pαβ λ = pεβ σ K 0 ασ + pεα σ K 0 βσ √ 1 1 + (Rαβ − gαβ R) −g, κ 2 βνλ ε[νγ] β ∂λ pα = −pα k εγ + pε β[νγ] k ε αγ −pα βεγ K ν (εγ) − pα ε(νγ) K β εγ + pε β(νγ) K ε αγ .

(4.61a) (4.61b)

(4.61c)

(4.61d)

The Hamilton equations (4.61a) and (4.61b) are independent of canonical momenta and so, reduce to the gauge-type condition (4.19). In accordance with the canonical

124


splitting (4.27) of J 1 C, the gauge-type condition (4.61b) breaks into two parts F α βνλ = Rα βνλ , ∂ν (K α βλ − k α βλ ) + ∂λ (K α βν − k α βν ) −2K ε (νλ) (K α βε − k α βε ) + K ε βλ k α εν + K ε βν k α ελ −K α ελ k ε βν − K α εν k ε βλ = 0.

(4.62)

(4.63)

It is readily observed that, for a given world metric g and a world connection k, there always exist the world connections K 0 and K such that the gauge-type conditions (4.61a), (4.62) and (4.63) hold (e.g. K 0 is the Levi-Civita connection of g and K = k). It follows that the multimomentum Hamiltonian forms (4.60) consitute really the complete family. At the same time, one can think of connections K and K 0 as being variable parameters expressed into dynamical variables g and k and so, can substitute them into other equations. Being restricted to the constraint space (4.58), the Hamilton equations (4.61c) and (4.61d) comes to √ 1 1 (Rαβ − gαβ R) −g = 0, κ √ 2 √ √ Dα ( −gg νβ ) − δαν Dλ ( −gg λβ ) + −g[g νβ (k λ αλ − k λ λα ) +g λβ (k ν λα − k ν αλ ) + δαν g λβ (k µ µλ − k µ λµ )] = 0

(4.64) (4.65)

where Dλ g αβ = ∂λ g αβ + k α µλ g µβ + k β µλ g αµ . Substituting Eq.(4.62) into Eq.(4.64), we obtain the Einstein equations 1 Fαβ − gαβ F = 0. 2

(4.66)

It is easy to see that Eqs.(4.65) and (4.66) are the familiar equations of gravitation theory phrased in terms of the nonsymmetric Palatini variables. In particular, the former is the equation for the torsion and the nonmetricity term of the connection k α βν . In the absence of matter sources, it admits the well-known solution 1 k α βν = {α βν } − δνα Vβ , 2 βγ βγ Dα g = Vα g , where Vα is an arbitrary covector field corresponding to the well-known projective freedom.45

4.7. CONSERVATION LAWS

4.7

125

Conservation laws

Unless n = 1, the physical meaning of a multimomentum Hamiltonian form is not evident. To make it more understood, let us explore the covariany Hamiltonian formulation of the energy-momentum conservation law. In the framerwork of the multimomentum Hamiltonian formalism, we get the fundamental identity whose restriction to the Lagrangian constraint space reproduces the familiar energy-momentum conservation law, without appealing to any symmetry condition. Let H be a Hamiltonian form on the Legendre bundle Π [Eq.(3.2)] over a fibred manifold Y → X. Let r be a section of of the fibred Legendre manifold Π → X and (y i (x), pλi (x)) its local components. Given a connection Γ on Y → X, we consider the following T ∗ X-valued (n − 1)-form on X: TΓ (r) = −(ΓcH) ◦ r, f )]dxµ ⊗ ω , TΓ (r) = [pλi (yµi − Γiµ ) − δµλ (pαi (yαi − Γiα ) − H λ Γ

(4.67)

f is the Hamiltonian density in the splitting (3.22) of H with respect to the where H Γ connection Γ. Let

τ = τ λ ∂λ be a vector field on X. Given a connection Γ on Y → X, it gives rise to the vector field τeΓ = τ λ ∂λ + τ λ Γiλ ∂i + (−τ µ pλj ∂i Γjµ − pλi ∂µ τ µ + pµi ∂µ τ λ )∂λi f on a section on the Legendre manifold Π. Let us calculate the Lie derivative LeτΓ H Γ r. We have f ) ◦ r = pλ Ri + d[τ µ T λ (r)ω ] − (τe cE ) ◦ r (LeτΓ H Γ Γ µ λ ΓV H i λµ

(4.68)

where R is the curvature (1.59) of the connection Γ, EH is the Hamilton operator (3.28) and τeΓV is the vertical part of the canonical horizontal splitting (1.33) of the vector field τeV on Π over J 1 Π. If r is a solution of the Hamilton equations, the equality (4.68) comes to the identity d i TΓ λ µ (r) = pλi Rλµ . dxλ On solutions of the Hamilton equations, the form (4.67) reads f − (∂µ + Γiµ ∂i − ∂i Γjµ pλj ∂λi )H Γ

(4.69)

µ f − δ λ (pα ∂ i H f f TΓ (r) = [pλi ∂µi H Γ µ i α Γ − HΓ )]dx ⊗ ωλ .

(4.70)

126


Lemma 4.5. The identity (4.69) do not depend upon choice of the connection Γ on the fibred manifold Y → X. 2 Outline of proof: Substitute relations Γ = Γ0 + σ, TΓ λ µ = −pλµ σµi + TΓ0 λµ and Eq.(1.63) into Eq.(4.69). 2 Lemma 4.5 and the fact that the identity (4.69) holds on no additional conditions for a multimomentum Hamiltonian form make this identity fundamental. For instance, if X = R and Γ is the trivial connection corresponding to the splitting (3.38), then f dt TΓ (r) = H 0 f is a Hamiltonian, and the identity (4.69) consists with the familiar energy where H 0 conservation law f f ∂H dH 0 0 = dt ∂t

in analytical mechanics. Unless n = 1, the identity (4.69) can not be regarded directly as the energymomentum conservation law. To clarify its physical meaning, let us turn to the Lagrangian formalism. Let ΞL be the Poincaré-Cartan form (2.6) associated with a Lagrangian density L on the jet manifold J 1 Y of Y . Let s be a section of the fibred jet manifold J 1 Y → X. Given a connection Γ on Y → X, we introduce the T ∗ X-valued (n − 1)-form TΓ (s) = −(ΓcΞL ) ◦ s = [πiλ (siµ − Γiµ ) − δµλ L]dxµ ⊗ ωλ

(4.71)

on X. One can think of this form as being the canonical energy-momentum tensor of fields s with respect to the connection Γ on Y . If the fibration Y → X is trivial, one can choose the trivial connection Γ = θX where θX is the pullback-valued form (1.17) on J 1 Y . In this case, the form (4.71) is viewed as the familiar canonical energy-momentum tensor. For the sake of simplicity, we shall denote TΓH (s) by T .


127

Remark. The form (4.71) appears in the framework of the following general construction. Let u = uµ ∂µ + ui ∂i be a vector field on a fibred manifold Y → X and u its jet lift (1.25) on the fibred jet manifold J 1 Y → X. Given a Lagrangian density L on J 1 Y , let us compute the Lie derivative Lu L. Using the canonical horizontal splitting (1.38) of u determined over J 2 Y , we get Lu L = [∂bλ (πiλ (ui − uµ yµi ) + uλ L) + (ui − uµ yµi )δi L]ω

(4.72)

where δi = ∂i − ∂bλ ∂iλ is the variational derivative (2.11) and ∂bλ is the total derivative (1.36). For instance, if L is a gauge invariant Lagrangian density of some gauge model, the Lie derivative (4.72) for every principal vector field u vanishes: Lu L = 0. On solutions of the Euler-Lagrange equations, this identity comes to the conservation law d (ui πiλ ) = 0. λ dx In particular, if u is the principal vector field (4.30), this conservation law comes to the familiar Nöether identities. Now, let τ be a vector field on X and τΓ = τ µ (∂µ + Γiµ ∂i ) its horizontal lift on the fibred manifold Y by a connection Γ on Y → X. For every solution s of the first order Euler-Lagrange equations, we have Lτ Γ L = −

d µ λ [τ TΓ µ (s)]ω dxλ

(4.73)

where TΓ (s) is given by the expression (4.71). In comparison with Eq.(4.69), the equality (4.73) is not viewed as a fundamental identity, for the Lie derivatives Lτ Γ L have not the standard form for all Lagrangian densities. The Hamiltonian counterpart of Eq.(4.73) is the universal identity (4.69) restricted to solutions living on the Lagrangian constraint space specified for each Lagrangian density. 2

128


Let a multimomentum Hamiltonian form H be associated with a semiregular Lagrangian density L. Lemma 4.6. Let r be a solution of the Hamilton equations for H which lives on the Lagrangian constraint space Q and s the associated solution of the first order Euler-Lagrange equations for L so that they satisfy the conditions (3.68). Then, we have TΓ (r) = TΓ (s). 2 The proof is straightforward. It follows that the form (4.70) may be treated as a Hamiltonian canonical energymomentum tensor wth respect to a background connection Γ on the fibred manifold Y → X. However, the examples below will show that, in field models, the identity (4.69) is precisely the energy-momentum conservation law for the metric energymomentum tensor, not the canonical one. In the Lagrangian formalism with a Lagrangian density L, the metric energymomentum tensor is defined to be √

−gtαβ = 2

∂L . ∂g αβ

In case of a background world metric g, this object is well-behaved, otherwise in the gauge gravitation theory. In the framework of the multimomentum Hamiltonian formalism, one can introduce the similar tensor √

−gtH αβ = 2

∂H . ∂gαβ

(4.74)

If a multimomentum Hamiltonian form H is associated with a semiregular Lagrangian density L, the relations (3.44b) and (3.62) result in the equalities tαβ (xλ , y i , ∂λi H(q)) = g αµ g βν tµν (xλ , y i , ∂λi H(q)) = tH αβ (q), tαβ (z) = g αµ g βν tµν (z) = tH αβ (xλ , y i , πiλ (z)) where q ∈ Q and c ◦ L(z) b H = z.

In view of these equalities, we can think of the tensor (4.74) restricted to the Lagrangian constraint space Q as being the Hamiltonian metric energy-momentum


129

tensor. On Q, the tensor (4.74) does not depend upon choice of a multimomentum Hamiltonian form H associated with L. Therefore, we shall denote it by the common symbol t. In the presence of a background world metric g, the identity (4.69) takes the form √ f = d T λ + pλ R i tλ α {α λµ } −g + (Γiµ ∂i − ∂i Γjµ pλj ∂λi )H Γ Γ µ i λµ dxλ

(4.75)

where d = ∂λ + ∂λ y i ∂i + ∂λ pµi ∂µi λ dx and by {α λµ } are meant the Cristoffel symbols of the world metric g. For instance, let us examine matter fields in the presence of a background gauge potential A which are described by the multimomentum Hamiltonian form (4.52). In this case, we have the equality √ λ ν −1 tλ µ −g = T λ µ = [aij Y gµν pi pj (−g) √ 1 α ν −1 −δµλ (aij + m2 aYij y i y j )] −g Y gαν pi pj (−g) 2 and the gauge invariance condition f = 0. Im j i pλj ∂λi H

The identity (4.75) then takes the form of the energy-momentum conservation law √ m −g∇λ tλ µ = −pλi Fλµ Im i j y j where ∇λ is the covariant derivative relative to the Levi-Civita connection and F is the strength of the background gauge potential A. Let us consider gauge potentials A described by the complete family of the multimomentum Hamiltonian forms (4.35) where S = SB are the connections (4.34). On the solution A = B, the curvature of the connection SB is reduced to 1 q n m m β m β m Rλαµ = (∂λ Fαµ − cm qn kλ Fαµ − { µλ }Fβα − { αλ }Fµβ ). 2 Set S

λ

µ

=

m f p[αλ] m ∂αµ HY M

ε2 [αλ] [βν] = √ amn G gµν gαβ pm pn . 2 −g

130


In virtue of Eqs.(4.32a), (4.32b) and (4.36), we obtain the relations m λ β λ p[λα] m Rλαµ = ∂λ S µ − { µλ }S β , α β αλ n f ∂nβ Γm αµ pm ∂βλ HY M = { αµ }S β ,

√ f tλ µ −g = 2S λ µ − δµλ H YM and √ tλ µ −g = T λ µ + S λ µ . The identity (4.75) then takes the form of the energy-momentum conservation law ∇λ tλ µ = 0. In particular, the familiar energy-momentum conservation law for electromagnetic field is reproduced.

Chapter 5 Field Systems on Composite Manifolds It is the multimomentum Hamiltonian formalism that enables one to describe dynamic systems on composite manifolds. In analytical mechanics, these are systems with variable parameters, e.g., the Berry oscillator.52 In field theory, composite manifolds (1.3) Y →Σ→X characterize spontaneous symmetry breaking, particularly, in the gauge gravitation theory.49,50 Application of composite manifolds to field theory is based on the following speculations. Given a global section h of the fibred manifold Σ → X, the restriction Yh of the fibred manifold Y → Σ to h(X) ⊂ Σ is a fibred submanifold of Y → X. Moreover, there is the 1:1 correspondence between the global sections sh of Yh → X and the global sections of the composite manifold Y which are projected onto the section h. Therefore, one can say that sections sh of the portion Yh → X describe fields in the presence of a background parameter field h, whereas sections of the composite manifold Y describe all the pairs (sh , h). The total configuration space of these pairs thus is the jet manifold J 1 Y of the composite manifold Y , and their total phase space is the Legendre bundle Π over Y . This approach is especially advantageous when, for different global sections h and h0 of Σ, the fibred manifolds Yh and Yh0 are not equivalent. In analytical mechanics, the fields h play the role of variable parameters. In field theory, they are exemplified by Higgs fields, e.g., a gravitational field. 131

132

5.1

CHAPTER 5. FIELD SYSTEMS ON COMPOSITE MANIFOLDS

Geometry of composite manifolds

Let π := πΣX ◦ πY Σ : Y → Σ → X

(5.1)

be a composite manifold (1.3) provided with the fibred coordinates (xλ , σ m , y i ) (1.4) where (xµ , σ m ) are fibred coordinates of the fibred manifold Σ → X. We further suppose that the fibred manifold YΣ := Y → Σ is a bundle, and the coordinates (1.4) are bundle coordinates of YΣ . Note that, even if the fibred manifold Σ → X also is a bundle, the composite manifold (5.1) fails to be a bundle in general. The following assertion is one of the cornerstones of field theory on composite manifolds. Proposition 5.1. Let Y be the composite manifold (5.1). Given a section h of the fibred manifold Σ → X and a global section sΣ of the bundle Y → Σ, their composition s = sΣ ◦ h

(5.2)

is a section of the composite manifold Y . Conversely, if the bundle Y → Σ has a global section, every global section s of the fibred manifold Y → X is represented by some composition (5.2). 2 Outline of proof: Given a global section s of Y → X, we set h = πY Σ ◦ s. Then, sΣ is an extension of the local section h(X) → s(X) of the bundle YΣ over the closed imbedded submanifold h(X) ⊂ Σ. 2 Corollary 5.2. Given a global section h of the fibred manifold Σ → X, the restriction Yh = h∗ YΣ

(5.3)

of the bundle YΣ to h(X) is a fibred imbedded submanifold ih : Yh ,→ Y

(5.4)

5.1. GEOMETRY OF COMPOSITE MANIFOLDS

133

of the fibred manifold Y → X. In virtue of Proposition 5.1, there is the 1:1 correspondence between the global sections sh of Yh and the global sections (5.2) of the composite manifold Y which cover the section h. 2 In view of Proposition 5.1 and Corollary 5.2, one can think of sections sh of the bundle Yh → X as being fields in the presence of a background parameter field h, whereas the sections (5.2) of the composite manifold (5.1) can describe all pairs (sh , h). Given the composite manifold (5.1), let J 1 Σ, J 1 YΣ and J 1 Y be the first order jet manifolds of Σ → X, Y → Σ and Y → X respectively which are endowed with the corresponding adapted coordinates (1.68): (xλ , σ m , σλm ), i (xλ , σ m , y i , yeλi , ym ), λ m i m i (x , σ , y , σλ , yλ ). Corollary 5.3. There exists the canonical surjection ρ : J 1 Σ × J 1 YΣ → J 1 Y,

Σ 1 1 ρ(jx h, jh(x) sΣ ) = jx1 (sΣ i m yλi ◦ ρ = ym σλ + yeλi ,

◦ h),

(5.5)

where sΣ and h are sections of the bundle Y → Σ and the fibred manifold Σ → X respectively. 2 In particular, let AΣ = dxλ ⊗ (∂λ + Aeiλ ∂i ) + dσ m ⊗ (∂m + Aim ∂i )

(5.6)

be a connection on the bundle Y → Σ and Γ = dxλ ⊗ (∂λ + Γm λ ∂m ) a connection on the fibred manifold Σ → X. Building on the morphism (5.5), one can construct the connection i m ei A = dxλ ⊗ [∂λ + Γm λ ∂m + (Am Γλ + Aλ )∂i ]

on the composite manifold Y in accordance with the commutative diagram J 1 Σ × J 1 YΣ

ρ

−→ J 1 Y

Σ

Γ×AΣ

6A

6

Σ × YΣ

←−

πY Σ ×Id Y

Y

(5.7)

134


We call A [Eq.(5.7)] the composite connection because of the term λ Aim Γm λ dx ⊗ ∂i .

(5.8)

Example. Set Y =E×Σ→Σ→X where E → X is a fibred manifold provided with coordinates (xλ , y i ). Let Γ and Γ0 be connections on the fibred manifolds Σ → X and E → X respectively. We have the pullback connection i

AΣ = dxλ ⊗ (∂λ + Γ0 λ ∂i ) + dσ m ⊗ ∂m

(5.9)

on the bundle Y → E. Then, the corrresponding composit connection (5.7) on Y → X is precisely the direct product connection Γ0 × Γ. 2 The composite connections (5.7) are by no means the unique type of connections on a composite manifold. We consider them because of the following property. Let h be a global section of the fibred manifold Σ. The injection ih [Eq.(5.4)] gives rise to the injection J 1 ih : J 1 Yh → J 1 Y. Let h be an integral section of the connection Γ on Σ → X, that is, Γ ◦ h = J 1 h, ∂µ hm = Γm µ. In this case, the composite connection (5.7) on Y is reducible to the connection Ah = dxλ ⊗ [∂λ + (Aim ∂λ hm + (Ae ◦ h)iλ )∂i ]

(5.10)

on the fibred submanifold Yh (5.3) of Y → X in accordance with the commutative diagram J 1i

J 1 Yh −→h J 1 Y Ah

6A

6

Yh ,→ Y ih

Given a composite manifold Y [Eq.(5.1)], the exact sequences 0 → V YΣ ,→ V Y → Y × V Σ → 0,

(5.11)

0 → Y × V ∗ Σ ,→ V ∗ Y → V ∗ YΣ → 0

(5.12)

Σ

Σ

over Y take place, besides the familiar exact sequences (1.6a) and (1.6b). Every connection (5.6) on the bundle YΣ determines:

5.1. GEOMETRY OF COMPOSITE MANIFOLDS

135

• the horizontal splitting V Y = V YΣ ⊕(Y × V Σ), Y

(5.13)

Σ

y˙ i ∂i + σ˙ m ∂m = (y˙ i − Aim σ˙ m )∂i + σ˙ m (∂m + Aim ∂i ), of the vertical tangent bundle V Y of Y → X and, consequently, the splitting of the exact sequence (5.11); • the dual horizontal splitting V ∗ Y = V ∗ YΣ ⊕(Y × V ∗ Σ), Y

(5.14)

Σ

y˙ i dy i + σ˙ m dσ m = y˙ i (dy i − Aim dσ m ) + (σ˙ m + Aim y˙ i )dσ m , of the vertical cotangent bundle V ∗ Y of Y → X and, consequently, the splitting of the exact sequence (5.12). It is readily observed that the splitting (5.13) is uniquely characterized by the form ω ∧ AΣ = ω ∧ dσ m ⊗ (∂m + Aim ∂i ).

(5.15)

So different connections AΣ on YΣ can determine the same splitting (5.13). Building on the horizontal splitting (5.13) relative to a connection AΣ on Y → Σ, one can constract the following first order differential operator on the composite manifold Y : f = pr ◦ D : J 1 Y → T ∗ X ⊗ V Y → T ∗ X ⊗ V Y , D A Σ 1 f D

Y m i i i = dx ⊗ [yλ − Aλ − Am (σλ − dxλ ⊗ (yλi − Aeiλ − Aim σλm )∂i , λ

Y

Γm λ )]∂i

= (5.16)

where DA is the covariant differential (1.64) relative to the composite connection A f the which is composition of AΣ and some connection Γ on Σ → X. We shall call D vertical covariant differential. Let h be an integral section of the connection Γ and Yh the portion of YΣ over h(X). It is readily observed that the vertical covariant differential (5.16) restricted to J 1 Yh ⊂ J 1 Y comes to the familiar covariant differential for the connection Ah [Eq.(5.10)] on the portion Yh → X [Eq.(5.3)].

136


Thus, it is the vertical covariant differential (5.16) that we may utilize in order to construct a Lagrangian density (2.2) n

e D

L : J 1 Y → T ∗ X ⊗ V YΣ → ∧ T ∗ X

(5.17)

Y

for fields on a composite manifold. It should be noted that such a Lagrangian density is never regular because of the constraint conditions µ . πiµ Aim = πm

5.2

Hamiltonian systems on composite manifolds

In this Section, by Y throughout is meant a composite manifold (5.1). The major feature of Hamiltonian systems on a composite manifold Y lies in the following. The horizontal splitting (5.14) yields immediately the corresponding splitting of the Legendre bundle Π over the composite manifold Y . As a consequence, the Hamilton equations (3.35a) for sections h of the fibred manifold Σ → X reduce to the gauge-type conditions independent of momenta. Thereby, these sections play the role of parameter fields. Let Y be a composite manifold (5.1). The Legendre bundle Π over Y is endowed with the canonical coordinates (xλ , σ m , y i , pλm , pλi ). Let AΣ be a connection (5.6) on the bundle Y → Σ. With a connection AΣ , the splitting n

Π = ∧ T ∗ X ⊗ T X ⊗[V ∗ YΣ ⊕(Y × V ∗ Σ)] Y

Y

Y

(5.18)

Σ

of the Legendre bundle Π is performed as an immediate consequence of the splitting (5.14). We call this the horizontal splitting of Π. Given the horizontal splitting (5.18), the Legendre bundle Π is provided with the coordinates pλi = pλi , pλm = pλm + Aim pλi

(5.19)

which are compatible with this splitting. These coordinates however fail to be canonical. Let h be a global section of the fibred manifold Σ → X. Given the horizontal splitting (5.18), the submanifold {σ = h(x), pλm = 0}

(5.20)

5.2. HAMILTONIAN SYSTEMS ON COMPOSITE MANIFOLDS

137

of the Legendre bundle Π over Y is isomorphic to the Legendre bundle Πh over the portion Yh [Eq.(5.3)] of YΣ . But, only with a connection on the bundle Y → Σ, the particular Hamiltonian system in the presence of the background parameter field h can be inserted into the total Hamiltonian system characterized by the Legendre bundle Π over Y . Let the composite manifold Y be provided with the composite connection (5.7) determined by connections AΣ on Y → Σ and Γ on Σ → X. Relative to the coordinates (5.19) compatible with the horizontal splitting (5.18), every multimomentum Hamiltonian form on the Legendre bundle Π over Y as a section of the bundle Z → Π [Eq.(2.25)] can be given by the expression H = (pλi dy i + pλm dσ m ) ∧ ωλ − f µ m i µ µ [pλi Aeiλ + pλm Γm λ + H(x , σ , y , pm , pi )]ω

(5.21)

where λ i λ m pλi Aeiλ + pλm Γm λ = pi Aλ + pm Γλ .

The corresponding Hamilton equations are mf f ∂λ pλi = −pλj [∂i Aejλ + ∂i Ajm (Γm λ + ∂λ H)] − ∂i H,

(5.22a)

mf i f ∂λ y i = Aeiλ + Aim (Γm λ + ∂λ H) + ∂λ H, f f − pλ ∂ Γm − ∂ H, ∂λ pλm = −pλi [∂m Aeiλ + ∂m Ain (Γnλ + ∂λn H)] m n m λ mf ∂λ σ m = Γm λ + ∂λ H

(5.22b) (5.22c) (5.22d)

and plus constraint conditions. Let the multimomentum Hamiltonian form (5.21) be associated with a Lagrangian density (5.17) which contains the velocities σµm only inside the vertical covariant f appears independent of the differential (5.16). Then, the Hamiltonian density Hω momenta pµm and the Lagrangian constraints read pµm = 0.

(5.23)

In this case, Eq.(5.22d) comes to the gauge-type condition ∂λ σ m = Γm λ

(5.24)

independent of momenta. Substitution of Eqs.(5.22a) - (5.22c) into Eq.(5.23) results in the equation i i f = (∂ + Aj ∂ )H f pλi [Rλm + Γnλ Rnm + ∂j Aim ∂λj H] m m j

(5.25)

138


where R is the curvature of the connection AΣ . It is the condition which the connection AΣ must meet in order that the Hamilton equations have a solution living on the constraint space. In particular, if AΣ is the pullback connection (5.9), then the constraint condition (5.23) and the gauge-type condition (5.25) take the form pµm = 0

(5.26)

f = 0. ∂m H

(5.27)

and

For instance, the gravitational Hamilton equations (4.61a), (4.64) and the first from Eqs.(4.58) exemplify Eqs.(5.24), (5.27) and (5.26) respectively. In Section 5.5, gravity will be described as the Hamiltonian system on a composite manifold. Let us consider now a Hamiltonian system in the presence of a background parameter field h(x). Substituting Eq.(5.22d) into Eqs.(5.22a) - (5.22b) and restricting them to the submanifold (5.20), we obtain the equations f ∂λ pλi = −pλj ∂i [(Ae ◦ h)jλ + Ajm ∂λ hm ] − ∂i H, f ∂λ y i = (Ae ◦ h)iλ + Aim ∂λ hm + ∂λi H

(5.28)

for sections of the Legendre manifold Πh → X of the bundle Yh endowed with the connection (5.10). Equations (5.28) are the Hamilton equations corresponding to the multimomentum Hamiltonian form f µ , hm (x), y i , pµ , pµ = 0)]ω Hh = pλi dy i ∧ ωλ − [pλi Ah iλ + H(x i m

(5.29)

on Πh which is induced by the multimomentum Hamiltonian form (5.21) on Π. It follows that the Hamiltonian description of the total system of pairs (sh , h) is compatible with the particular Hamiltonian model of fields sh in the presence of a background parameter field h.

5.3

Classical Berry’s oscillator

As a test case, let us examine the one-dimensional classical oscillator whose frequency is a fixed function h(t) of time t ∈ R. In accordance with the adiabatic hypothesis, there exists a coordinate Y such that this ascillator differs from the standard one only in the kinetic energy. Let us consider the composite manifold Y := R × R+ × R → R+ × R → R

(5.30)

5.3. CLASSICAL BERRY’S OSCILLATOR

139

provided with the fibred coordinates (t, σ > 0, y). Let Π be the Legendre bundle over Y [Eq.(5.30)]. It is coordinatized by (t, σ, y, pσ , py ). In accordance with the adiabatic hypothesis, the above-mentioned oscillator is described by the multimomentum Hamiltonian form H = pσ dσ + py dy − Hdt, 1 H = pσ Γσt dt + (σ 2 p2y + y 2 ), 2 where the associated connection

(5.31)

ΓH = dt ⊗ (∂t + Γσt ∂s i)

(5.32)

on Y → X is the composite connection (5.7) which is yielded by some linear connection Γ = dt ⊗ (∂t + Γt σ∂σ ) on the parameter bundle Σ = R+ × R → R and by the trivial connection AΣ = dt ⊗ ∂t + dσ ⊗ ∂σ on the bundle Y → Σ. Note that one can not bring the connection Γ into the trivial one by coordinate transformations since the coordinate σ of Σ is fixed by the frequency function h(t). Let us consider the canonical transformation y = σ 0 y, 1 py = p0 y , σ 1 0 0 y p y. σ It should be noted that, on quantum level, it fails to be transformation of physical equivalence. Relative to the new coordinates pσ = p0 σ −

(t, σ, y 0 , p0 σ , p0 y ),

140


the connecion (5.32) is written ΓH = dt ⊗ [∂t + Γσt (∂σ − y 0 ∂y0 )] and the multimomentum Hamiltonian form (5.31) reads H = p0 σ dσ + p0 y dy 0 − Hdt, 1 2 2 H = p0 σ Γσt − p0 y y 0 Γt + (p0 y + σ 2 y 0 ). 2 The corresponding Hamilton equations (5.22a) - (5.22d) take the form 1 0 σ p Γ − σ2y0, σ y t 1 ∂t y 0 = − y 0 Γσt + p0 y , σ 2 0 ∂t p σ = −p0 σ Γt − σy 0 , ∂t σ = Γσt .

∂t p0 y =

(5.33a) (5.33b) (5.33c) (5.33d)

In the context of the physical task which we discuss, the Hamilton equation (5.33c) for the canonical momentum p0 σ is not valuable, while Eq.(5.33d) selects a multimomentum Hamiltonian form (5.31) which describes the oscillator with the fixed frequency function σ = h(t). Substitution of Eq.(5.33c) into Eqs.(5.33a) and (5.33b) lead to the Hamilton equations (5.28) corresponding to the multimomentum Hamiltonian form (5.29) H = p0 y dy 0 − Hdt, 1 1 2 2 H = − ∂t σy 0 p0 y + (p0 y + σ 2 y 0 ), σ 2 of the well-known Berry’s oscillator40,44 when the connection (5.10) is the Berry connection Ah = dt ⊗ (∂t −

5.4

1 ∂t σy 0 ∂y0 ). σ

Higgs fields

Contemporary field models are almost always theories with broken symmetries. Spontaneous symmetry breaking is quantum phenomenon. In algebraic quantum field theory, it is characterized by nonequivalent Gaussian states of quantum field algebras. In classical field theory, spontaneous symmetry breaking is modelled by

5.4. HIGGS FIELDS

141

classical Higgs fields. In geometric terms, the necessary condition for spontaneous symmetry breaking consists in reduction of a structure group G of a principal bundle P to its closed subgroup K of exact symmetries43,50,60 . We assume that a world manifold X meets the requisite topological conditions in order that this reduction takes place. Classical Higgs fields are represented by global sections of the bundle P/K → X which is the quotient of P by the canonical action (4.20) of the subgroup K ⊂ G on P on the right. It will be called the Higgs bundle. Let πP : P → X be a principal bundle with a structure Lie group G and K its closed subgroup. The Higgs bundle P/K → X is the P -associated bundle Σ = P/K = (P × G/K)/G with the standard fibre G/K on which the structure group G acts on the left. There is the composite fibration πΣX ◦ πP Σ : P → P/K → X

(5.34)

where PΣ = P → P/K is the principal bundle with the structure group K. It should be noted that the composite manifold (5.34) fails to be a principal bundle since the canonical action (4.20) of the structure group G on P violates the composite fibration (5.34), while the action of K ⊂ G on P is not transitive. Let the structure group G of a principal bundle P be reducible to its closed subgroup K. It means existence of a reduced principal subbundle Ph of P whose structure group is K. By the well-known theorem, there is the 1:1 correspondence between the global sections h of the Higgs bundle P/K → X [if they exist] and the reduced K-principal subbundles Ph of P which consists with the restrictions of the K-principal bundle PΣ to h(X): πP Σ (Ph ) = (h ◦ πP )(Ph ). Recall the following facts. • Every principal connection Ah on a reduced subbundle Ph gives rise to a principal connection on P .

142


• Conversely, a principal connection A on P is reducible to a principal connection on the reduced subbundle Ph iff the global section h of the bundle P/K → X is an integral section of the connection A. • Every principal connection AΣ on the K-principal bundle P → P/K, whenever h, induces a principal connection on the reduced subbundle Ph of P . One says that sections sh of a vector bundle Yh → X with a standard fibre V describe matter fields in the presence of a Higgs field h if Yh is associated with the reduced subbundle Ph of the principal bundle P , that is, Yh = (Ph × V )/K.

(5.35)

Matter fields sh in the presence of different Higgs fields h and h0 are represented by sections of the matter bundles Yh and Yh0 associated with different reduced subbundles Ph and Ph0 of P . We shall restrict our consideration to the case when the standard fibre V carries only represenation of the exact symmetry subgroup K ⊂ G and so, there is no canonical isomorphism between the bundles Yh and Yh0 . [For instance, spontaneous symmetry breaking in gravitation theory where Dirac fermion fields possess only exact Lorentz symmetries belongs to this type.] In this case, a connection on Yh is assumed to be associated with a principal connection on the reduced subbundle Ph . When extended to a principal connection on P , this principal connection however fails to be reducible to a connection on another reduced subbundle Ph0 6=h . It follows that a matter field possessing only exact symmetries must be regarded only in a pair (sh , h) together with a certain Higgs field h. In particular, they fail to constitute a linear space. To describe these pairs for different h as sections of the same fibred manifold, composite the manifolds have been suggested.49 Lemma 5.4. Given the composite manifold (5.34), the canonical morphism (5.5) results in the surjection J 1 PΣ /K × J 1 Σ → J 1 P/K

(5.36)

Σ

over J 1 Σ. Let AΣ be a principal connection on PΣ and Γ be a connection on Σ. It is readily observed that the corresponding composite connection (5.7) on the composite manifold (5.34) is equivariant under the canonical action of K on P . If the connection Γ has an integral global section h of Σ, the composite connection (5.7) is reducible to the connection (5.10) which is a principal connection on Ph . 2 Outline of proof: If h is an integral section of Γ, the connection (5.10) consists with the connection on Ph induced by the principal connection AΣ on PΣ . 2

5.4. HIGGS FIELDS

143

Let us consider the composite manifold Y = (P × V )/K → P/K → X

(5.37)

where the bundle YΣ = (P × V )/K → P/K is associated with the K principal bundle P → P/K. Given a reduced subbundle Ph of P , the associated bundle (5.35) is isomorphic to the portion (5.3) of YΣ over h(X). The composite manifold (5.37) can be provided with the composite connection (5.7) where: (i) the connection AΣ on the bundle Y → P/K is associated with a principal connection on the K-principal bundle PΣ and (ii) the connection Γ on Σ is associated with a principal connection on some reduced subbundle Ph of P . In virtue of Lemma 5.4, such composite connection A is reducible to the connection (5.10) on the bundle Yh [Eq.(5.35)] as a subbundle of the composite manifold Y . This connection appears to be associated with a principal connection on Ph . Thus, they are sections of the composite manifold (5.37) which describe all the above-mentioned pairs (sh , h) of matter fields and Higgs fields. Their configuration space J 1 Y has the following composite fibration J 1 Y −→ J 1 Σ ?

?

Y −→ Σ −→ X Accordingly, the pairs (Ah , h) of reducible connections Ah and Higgs fields h can be represented by sections of the composite manifold CK := (J 1 P )/K → J 1 (P/K) → P/K → X

(5.38)

which cover jet prolongations J 1 h of global sections h of the Higgs bundle P/K → X. Moreover, in the context of the composite manifold (5.38), we can still describe deviations from a Higgs field h by sections σ of the jet manifold J 1 (P/K) → X which are projected onto the section h, but do not consist with J 1 h: σ m (x) = hm (x), m σm λ 6= ∂λ h . The jet manifold J 1 CK has the composite fibration J 1 CK −→ J 1 J 1 Σ ?

?

CK −→ J 1 Σ −→ Σ −→ X

(5.39)

144


The total configuration space of matter fields, gauge potentials, Higgs fields and their deviations (5.39) reads J 1 Y × J 1 CK . J 1Σ

Remark. The total space (P ×V )/K of the composite manifold (5.37) possesses the structure of the bundle associated with the G-principal bundle P . Its standard fibre is (G × V )/K on which the structure group G of P (and its subgroup K) acts by the law G 3 g : (G × V )/K → (gG × V )/K. However, it differs from thatof the structure group K of the bundle PΣ by the law K 3 g : (G × V )/K → (Gg −1 × V )/K. If the standard fibre V of the bundle YΣ carriers representation of the whole group G, these two actions are equivalent, otherwise in general case. Therefore, we are not concerned with the P -associated bundle structure of Y [Eq.(5.37)]. 2

5.5

Gauge gravitation theory

Gravitation theory is just the gauge theory with spontaneous symmetry breaking of the above-mentioned type.24,48,50 In gravitation theory, spontaneous symmetry breaking of world symmetries is established by the equivalence principle reformulated in the terms of Klein-Chern geometries of invariants. In Einstein’s General Relativity, the equivalence principle is called to provide transition to Special Relativity with respect to some reference frames. In the spirit of F.Klein’s Erlanger program, the Minkowski space geometry can be characterized as geometry of Lorentz invariants. The geometric equivalence principle then postulates existence of reference frames with respect to wich Lorentz invariants can be defined everywhere on a world manifold X 4 . This principle has the adequate mathematical formulation in terms of fibre bundles. Let LX be the principal bundle of linear frames in tangent spaces to X 4 . The geometric equivalence principle requires that its structure group GL4 = GL+ (4, R) is reduced to the connected Lorentz group L = SO(3, 1).

5.5. GAUGE GRAVITATION THEORY

145

It means that there is given a reduced subbundle Lh X of LX whose structure group is L. Then, they are atlases of Lh X extended to atlases of LX with respect to which Lorentz invariants can be defined. In other words, the geometric equivalence principle provides a world manifold with the so-called L-structure.58 From the physical point of view, it singles out the Lorentz group as the exact symmetry subgroup of world symmetries broken spontaneously.24 The associated classical Higgs field is a tetrad gravitational field. Remark. By X 4 is further meant an oriented world manifold which satisfies the well-known global topological conditions in order that gravitational fields, spacetime structure and spinor structure can exist. To summarize these conditions, we assume that X 4 is not compact and the linear frame bundle LX over X 4 is trivial.50 2 In accordance with the well-known theorem, there is the 1:1 correspondence between the reduced L-principal subbundles Lh X of LX and the tetrad gravitational fields h which, by definition, are represented by global sections of the quotient bundle Σ := LX/L → X 4

(5.40)

with the standard fibre GL4 /L. Given a tetrad field h, let Ψh = {zξh } be an atlas of the linear frame bundle LX such that the local sections zξh of LX take their values into the reduced subbundle Lh X. Relative to Ψh , the pseudoRiemannian metric g associated with a gravitational field h comes to the Minkowski metric and exemplifies the Lorentz invariant defined in accordance with the geometric equivalence principle. With respect to an atlas Ψh and a holonomic atlas ΨT = {ψξT } of LX, a tetrad field h can be represented by a family of GL4 -valued tetrad functions hξ = ψξT ◦ zξh , h = πP Σ ◦ zξh , which determine the atlas transformation dxλ = hλa (x)ha

(5.41)

146


between the fibre bases {dxλ } and {ha } for T ∗ X associated with atlases ΨT and Ψh respectively. Remark. For the first time, the conception of a graviton as a Goldstone particle corresponding to violation of Lorentz symmetries in a curved space-time had been advanced in mid 60s by Heisenberg and Ivanenko in discussion on cosmological and vacuum asymmetries. This idea was revived in connection with constructing the induced representations of the group GL4 23,41,42 and then in the framework of the approach to gravitation theory as a nonlinear σ-model.43 In geometric terms, the fact that a pseudo-Riemannian metric is exactly a Higgs field has been pointed out by A.Trautman61 and by us.47 To justify it, the modified geometric formulation of the equivalence principle has been suggested.24 2 In the gauge gravitation theory, dynamic gravitational variables are pairs of tetrad gravitational fields h and gauge gravitational potentials Ah identified with principal connections on Ph . Following previous Section, one can describe these pairs (h, Ah ) by sections of the bundle (5.38) where P = LX and K = L. The corresponding configuration space is the jet manifold J 1 CK of CK . The Legendre bundle 4

Π = ∧ T ∗ X 4 ⊗ T X 4 ⊗ V ∗ CK . CK

(5.42)

CK

over CK plays the role of a phase space of the gauge gravitation theory. The bundle CK is endowed with the local fibred coordinates λ (xµ , σaλ , k ab λ = −k ba λ , σaµ )

where λ (xµ , σaλ , σaµ )

are coordinates of the jet bundle J 1 Σ. Given a section s of CK , we have familiar tetrad functions and Lorentz gauge potentials (σaλ ◦ s)(x) = hλa (x), (k ab λ ◦ s)(x) = Aab λ (x) respectively. The jet manifold J 1 CK of CK is provided with the adapted coordinates λ λ λ (xµ , σaλ , k ab λ = −k ba λ , σaµ = σa(µ) , k ab µλ , σaµν ).

(5.43)

5.5. GAUGE GRAVITATION THEORY

147

The associated coordinates of the Legendre manifold (5.42) are λ λµ aνµ (xµ , σaλ , k ab λ , σaν , paµ λ , pab , pλ )

where (xµ , σaλ , paµ λ ) are coordinates of the Legendre manifold of the bundle Σ. For the sake of simplicity, we here consider the Hilbert-Einstein Lagrangian density of classical gravity. In the coordinates (5.43), it reads LHE = −

1 ab F µλ σaµ σbλ σ −1 ω, 2κ

(5.44)

F ab µλ = k ab λµ − k ab µλ + k a cµ k cb λ − k a cλ k cb µ , σ = det(σaµ ). b The corresponding Legendre morphism L HE is given by the coordinate expressions

pab (λµ) = 0, paµ λ = 0, aνµ pλ = 0, pab [λµ] = πab [λµ] =

−1 [µ λ] σ σ . κσ a b

(5.45)

We construct the complete family of multimomentum Hamiltonian forms associated with the affine Lagrangian density (5.44). Let K be a world connection associated with a principal connection B on the linear frame bundle LX. To minimize the complete family, we consider the following connections on the bundle CK : Γλaµ = B b aµ σbλ − K λ νµ σaν , Γλaνµ = ∂µ B d aν σdλ − ∂µ K λ βν σaβ λ β λ +B d aµ (σdν − Γλdν ) − K λ βµ (σaν − Γβaν ) + K β νµ (σaβ − Γλaβ )

+B d aν Γλdµ − K λ βν Γβaµ , 1 Γab λµ = [k a cλ k cb µ − k a cµ k cb λ + ∂λ B ab µ + ∂µ B ab λ 2 −B b cµ k ac λ − B b cλ k ac µ − B a cµ k cb λ − B a cλ k cb µ ] 1 +K ν λµ k ab ν − K ν (λµ) B ab ν − Rab λµ , 2 where R is the curvature of the connection B.

148


The complete family of multimomentum Hamiltonian forms associated with the Lagrangian density (5.44) consists of the forms given by the coordinate expressions aνµ λ λ HHE = (pab λµ dk ab λ + paµ λ dσa + pλ dσaν ) ∧ ωµ − HHE ω, aνµ λ λ HHE = (pab λµ Γab λµ + paµ λ Γaµ + pλ Γaνµ ) + 1 ab R λµ (pab [λµ] − πab λµ ). 2 The Hamilton equations corresponding to such a multimomentum Hamiltonian form read

F ab µλ = Rab µλ , ∂µ k ab λ + ∂λ k ab µ = 2Γab (µλ) , ∂µ σaλ = Γλaµ ,

(5.46a) (5.46b) (5.46c)

λ ∂µ σaν = Γλaνµ , ∂HHE ∂µ pac λµ = − ac , ∂k λ ∂HHE ∂µ paµ , λ = − ∂σaλ

(5.46d) (5.46e) (5.46f)

plus the equations which are reduced to the trivial identities on the constraint space (5.45). The equations (5.46a) - (5.46d) make the sense of gauge-type conditions. The equation (5.46d) has the solution λ σaµ = ∂ν σaλ

which shows that the Hilbert-Einstein Lagrangian density of classical gravity fails to describe deviations (5.39) of a Higgs gravitational field. The gauge condition (5.46b) has the solution k(x) = B. It follows that the forms HHE really constitute the complete family of multimomentum Hamiltonian forms associated with the Hilbert-Enstein Lagrangian density (5.44). On the constraint space, Eqs.(5.46e) and (5.46f) are brought to the form ∂µ πac λµ = 2k b cµ πab λµ + πac βγ Γλ βγ , N cb βµ ∂λa πcb βµ = 0.

(5.47a) (5.47b)

The equation (5.47a) shows that k(x) is the Levi-Civita connection for the tetrad field h(x). Substitution of Eqs.(5.46a) into Eqs.(5.47b), leads to the familiar Einstein equations.

5.6. FERMION FIELDS

5.6

149

Fermion fields

The underlying physical reason of spontaneous symmetry breaking in gravitation theory lies in existence of Dirac fermion matter possesing only exact Lorentz symmetries. Although several spinor models of fermion matter have been suggested, all observable fermions are apparently Dirac fermions. We describe Dirac fermion fields as follows. Given a Minkowski space M with the Minkowski metric η, let AM = ⊕ M n , n

M 0 = R,

n

M n>0 = ⊗ M,

be the tensor algebra modelled on M . The complexified quotient of this algebra by the two-sided ideal generated by elements e ⊗ e0 + e0 ⊗ e − 2η(e, e0 ) ∈ AM ,

e ∈ M,

constitutes the complex Clifford algebra C1,3 . A spinor space V is defined to be a linear space of some minimal left ideal of C1,3 on which this algebra acts on the left. We then have the representation γ : M ⊗ V → V,

(5.48)

of elements of the Minkowski space M ⊂ C1,3 by γ-matrices on V : γ(ea ⊗ y A vA ) = γ aA B y B vA , where {e0 ...e3 } is a fixed basis for M , vA is a basis for V , and γ a are Dirac’s matrices on V .6 Let us consider the transformations preserving the representation (5.48). These are pairs (l, ls ) of Lorentz transformations l of the Minkowski space M and invertible elements ls of C1,3 such that lM = ls M ls−1 , γ(lM ⊗ ls V ) = ls γ(M ⊗ V ). Elements ls form the Clifford group whose action on M however is not effective. We here restrict ourselves to its spinor subgroup Ls = SL(2, C), L = Ls /Z2 , whose generators act on V by the representation 1 Iab = [γa , γb ]. 4

150


Remark. Since (Iab )+ 6= −Iab , the standard Hamiltonian metric in the spinor space V fails to be Ls -invariant. At the same time, we have (Lab )+ γ 0 = −γ 0 Iab . Hence, the Ls -invariant spinor metric in V is defined to be8 a(v, v) = v + γ 0 v. 2 Let us consider a bundle of complex Clifford algebras C3,1 over X 4 together with its spinor subbundle SM → X 4 and the subbundle YM → X 4 of Minkowski spaces of generating elements of C3,1 . To describe Dirac fermion fields on a world manifold, one must require that the bundle of Minkowski spaces YM is isomorphic to the cotangent bundle T ∗ X of the world manifold X 4 . It takes place if only the structure group of the principal linear frame bundle LX is reducible to the Lorentz group L and LX contains a reduced L-principal subbundle Lh X such that YM = (Lh X × M )/L = T ∗ X. In this case, the spinor bundle SM is associated with the Ls -principal lift Ph of Lh X: r : Ph → Lh X = Ph /Z2 , SM = (Ph × V )/Ls = Sh .

(5.49)

With the corresponding tetrad field h, one can define the representation γh : T ∗ X ⊗ Sh = (Ph × (M ⊗ V ))/Ls → (Ph × γ(M ⊗ V ))/Ls = Sh

(5.50)

of cotangent vectors to the world manifold X 4 by Dirac’s γ-matrices on elements of the spinor bundle Sh [Eq.(5.49)]. Relative to an atlas {zξ } of the Ls -principal lift Ph of Lh X and the associated atlas {zξh = r ◦ zξ } of Lh X, the morphism (5.50) is written γh (ha ⊗ y A vA (x)) = γ aA B y B vA (x)

5.6. FERMION FIELDS

151

where {ha } and {vA (x)} are the associated bases for fibres Tx∗ X of T ∗ X and fibres Vx of Sh respectively. As a shorthand, we can write b a = γ (ha ) = γ a . h h b λ = γ (dxλ ) = hλ (x)γ a . dx h a

We shall say that, given the representation (5.50), sections of the spinor bundle Sh describe fermion fields in the presence of the tetrad gravitational field h. Let Ah be a principal connection on the spinor bundle Sh [Eq.(5.49)] and D : J 1 Sh → T ∗ X ⊗ V S h , Sh

D=

(yλA

−A

Sh

ab

A B λ λ (x)Iab B y )dx

⊗ ∂A ,

the corresponding covariant differential (1.64). Given the representation (5.50), one can construct the first order differential Dirac operator Dh = γh ◦ D : J 1 Sh → T ∗ X ⊗ V Sh → V Sh ,

(5.51)

Sh

y˙ A ◦ Dh = hλa (x)γ aA B (yλB − Aab λ Iab A B y B ), on Sh . Here, γh is the pullback γh : T ∗ X ⊗ V Sh → V Sh , a

Sh A

Sh

γh (h ⊗ y˙ ∂A ) = γ aA B y˙ B ∂B , over Sh of the bundle morphism (5.50) owing to the canonical splitting (1.9) V S h = S h × Sh . On sections sh of Sh , we recover the familiar expression b λ ∇ s = hλ (x)h b a ∇ s = hλ (x)γ a ∇ s Dh ◦ sh = dx λ h λ h λ h a a

for the Dirac operator in the presence of a background tetrad gravitational field h. Dirac fermion fields in the presence of a background tetrad field h exemplify the constraint field theory with an affine Lagrangian density on the configuration space J 1 (Sh ⊕ Sh∗ ) provided with the adapted coordinates + ). (xµ , y A , yA+ , yµA , yAµ

152


On the configuration space J 1 Y , the Lagrangian density of Dirac fermion fields reads i + L = { [yA+ (γ 0 γ µ )A B (yµB − AB Cµ y C ) − (yAµ − A+C Aµ yC+ )(γ 0 γ µ )A B y B ] 2 −myA+ (γ 0 )A B y B }h−1 ω, (5.52) γ µ = hµa (x)γ a ,

h = det(hµa ),

where AA Bµ = Aab µ (x)Iab A B is a principal connection on the principal spinor bundle Ph . Note that one can always choose an atlas Ψh such that the operator γ 0 is globally defined. We can follow the standard procedure from Section 4.1 in order to construct the complete family of associated multimomentum Hamiltonian forms associated with the affine Lagrangian density (5.52). The Legendre bundle 4

Π = ∧ T ∗ X ⊗ T X ⊗ ∗ [(Sh ⊕ Sh∗ ) × (Sh∗ ⊕ Sh )] Sh ⊕Sh

over the spinor bundle Sh ⊕ Sh∗ is provided with the canonical coordinates (xµ , y A , yA+ , pµA , pµA + ). Relative to these coordinates, the Legendre morphism associated with the Lagrangian density (5.52) is written i pµA = πAµ = yB+ (γ 0 γ µ )B A h−1 , 2 i µA µA p+ = π+ = − (γ 0 γ µ )A B y B h−1 . 2

(5.53)

It defines the constraint subspace of the Legendre bundle Π. Given a soldering form S = S A Bµ (x)y B dxµ ⊗ ∂A on the bundle Sh , let us consider the connection A + S on Sh . The multimomentum Hamiltonian forms (4.18) associated with the Lagrangian density (5.52) read + HS = (pµA dy A + pµA + dyA ) ∧ ωµ − HS ω, + 0 A B −1 HS = pµA AA Bµ y B + yB+ A+B Aµ pµA + + myA (γ ) B y h µA (pµA − πAµ )S A Bµ y B + yB+ S +B Aµ (pµA + − π+ ).

(5.54) +

5.7. FERMION-GRAVITATION COMPLEX

153

The corresponding Hamilton equations for a section r of the fibred Legendre manifold Π → X take the form ∂µ yA+ = yB+ (A+B Aµ + S +B Aµ ), ∂µ pµA = −pµB AB Aµ − (pµB − πBµ )S B Aµ − i (myB+ (γ 0 )B A + yB+ S +B Cµ (γ 0 γ µ )C A )h−1 2

(5.55a)

(5.55b)

plus the equations for the components y A and pµA + . The equation (5.55a) and the A similar equation for y imply that y is an integral section for the connection A + S on the spinor bundle Sh . It follows that the multimomentum Hamiltonian forms (5.54) constitute the complete family. On the constraint space (5.53), Eq.(5.55b) reads i ∂µ πAµ = −πBµ AB Aµ − (myB+ (γ 0 )B A + yB+ S +B Cµ (γ 0 γ µ )C A )h−1 . 2

(5.56)

Substituting Eq.(5.55a) into Eq.(5.56), we obtain the familiar Dirac equation in the presence of a tetrad gravitational field h.

5.7

Fermion-gravitation complex

For different tetrad fields h and h0 , Dirac fermion fields are described by sections of spinor bundles Sh and Sh0 associated with Ls -principal lifts Ph and Ph0 of different reduced L-principal subbundles of LX. Therefore, the representations γh and γh0 [Eq.(5.50)] are not equivalent as follows. For two arbitrary elements q ∈ Ph and q 0 ∈ Ph0 over the same point x ∈ X, there is an element g ∈ GL4 so that rq 0 = (rg ◦ r)q. Let Tx∗ be the cotangent space to X 4 at x ∈ X 4 . Since Tx∗ = [rq]M M = ([rq 0 ]M ◦ g −1 )M, we can write γh : Tx∗ ⊗ Vx = [rq]M M ⊗ [q]V V → ([q]V ◦ γ)(M ⊗ V ), γh0 : Tx∗ ⊗ V 0 x = ([rq]M ◦ g)M ⊗ [q 0 ]V V → ([q 0 ]V ◦ γ)(gM ⊗ V ). If g ∈ GL4 \ L, there is no isomorphism lV of the spinor space V such that γ(gM ⊗ lV V ) = lV γ(M ⊗ V ).

154


It follows that a Dirac fermion field must be regarded only in a pair together with a certain tetrad gravitational field. These pairs constitute the so-called fermiongravitation complex.41,48,50 In virtue of Corollary 5.2, there is the 1:1 correspondence between the above-mentioned pairs and sections of the corresponding composite spinor bundle S → Σ → X4 where Σ is the Higgs bundle (5.40). In gravitation theory, we have the composite manifold πΣX ◦ πP Σ : LX → Σ → X 4

(5.57)

where Σ is the Higgs bundle (5.40) and LXΣ := LX → Σ is the L-principal bundle. Building on the double universal covering of the group GL4 , one can perform the Ls -principal lift PΣ of LXΣ such that PΣ /Ls = Σ, LXΣ = r(PΣ ) = PΣ /Z2 . In particular, there is imbedding of the Ls -principal bundle Ph → X 4 onto the portion of PΣ over h(X 4 ). Let us consider the composite spinor bundle S := πΣX ◦ πSΣ : (PΣ × V )/Ls → Σ → X 4

(5.58)

where SΣ := S → Σ is the spinor bundle associated with the Ls principal bundle PΣ . It is readily observed that, given a global section h of the Higgs bundle Σ → X 4 , the restriction S → Σ to h(X 4 ) consists with the spinor bundle Sh Eq.(5.49)] whose sections describe Dirac fermion fields in the presence of the background tetrad field h. Let us provide the principal bundle LX with a holonomic atlas {Uξ , ψξT } and the principal bundles PΣ and LXΣ with associated atlases {U , zs } and {U , z = r ◦ zs }


155

respectively. Relative to these atlases, the composite spinor bundle (5.58) is endowed with the fibred coordinates (xλ , σaµ , y A ) where (xλ , σaµ ) are fibred coordinates of the Higgs bundle Σ → X which is coordinatized by matrix components σaµ of the group elements GL4 3 (ψξT ◦ z )(σ) : R4 → R4 , σ ∈ U , πΣX (σ) ∈ Uξ . Given a section h of Σ → X 4 , we have zξh (x) = (z ◦ h)(x), (σaλ ◦ h)(x) = hλa (x), h(x) ∈ U , x ∈ Uξ , where hλa (x) are the tetrad functions (5.41). The jet manifolds J 1 Σ, J 1 SΣ and J 1 S of the bundles Σ, SΣ and S respectively are provided with the adapted coordinates µ (xλ , σaµ , σaλ ), λ µ A A Aa (x , σa , y , yeλ , y µ ), µ (xλ , σaµ , y A , σaλ , yλA ).

Let us consider the bundle of Minkowski spaces (LX × M )/L → Σ associated with the L-principal bundle LXΣ . It is isomorphic to the pullback Σ × T ∗ X which we denote by the same symbol T ∗ X. Building on the morphism (3.2), one can define the bundle morphism γΣ : T ∗ X ⊗ SΣ = (PΣ × (M ⊗ V ))/Ls → Σ

(PΣ × γ(M ⊗ V ))/Ls = SΣ

(5.59)

over Σ. In coordinate terms, this morphism reads b λ = γ (dxλ ) = σ λ γ a dx Σ a

where dxλ is the basis for the fibre of T ∗ X over σ ∈ Σ. Owing to the canonical vertical splitting V S Σ = SΣ × SΣ , Σ

156


the morphism (5.59) implies the corresponding morphism γΣ : T ∗ X ⊗ V SΣ → V SΣ .

(5.60)

S

We use this morphism in order to construct the total Dirac operator on the composite spinor bundle (5.58). Let Ba a µ Ae = dxλ ⊗ (∂λ + AeB λ ∂B ) + dσa ⊗ (∂µ + A µ ∂B )

(5.61)

be a connection on the bundle SΣ . This determines the horizontal splitting (5.13) of the vertical tangent bundle V SΣ . Lemma 5.5. There exist the canonical form (5.15) on SΣ given by the expression ω ∧ ⊗[∂µC + AB C µ ∂B ],

(5.62)

abc B A AB C µ = A µ Iab A y , 1 Aabcµ = − (η ca σµb − η cb σµa ), 2

(5.63)

and, consequently, the canonical horizontal splitting (5.13). 2 Outline of proof: The form (5.62) correspond to the canonical invariant connection on the bundle GL4 → GL4 /L. 2 Composition of the morphism (5.60) and the vertical covariant differential (5.16) for the connection (5.61) results in the first order differential operator f : J 1S → T ∗X ⊗ V S → V S , D = γΣ ◦ D Σ Σ

(5.64)

S

Ba µ y˙ A ◦ D = σaλ γ aA B (yλB − AeB λ − A µ σaλ ),

on S where AB aµ is given by the expression (5.63). In what follows, one can treat this morphism as the total Dirac operator. • For each section h of Σ, when restricted to the submanifold h(X 4 ) ⊂ Σ for a section h of Σ, the morphism (5.59) comes to the morphism γh [Eq.(5.50)].


157

• For each section h of Σ, the fibred jet manifold J 1 Sh → X 4 is a fibred submanifold of J 1 S given by the coordinate relations σaµ = hµa (x), µ σaλ = ∂λ hµa (x). As an immediate consequence of these assertions, we have Proposition 5.6. For every tetrad field h, the restriction of D to J 1 Sh ⊂ J 1 S comes to the Dirac operator Dh [Eq.(5.51)] in the presence of a principal connection Ba µ Ah = dxλ ⊗ [∂λ + ((Ae ◦ h)B λ + A µ ∂λ ha )∂B ].

2 The total configuration space of the fermion-gravitation complex is the product J 1 S × J 1 CK . J 1Σ

Its total phase space Π admits the splitting (5.18): µ λ B Bc µ aλ µ ω ⊗ ∂λ ⊗ [pλB dy B + paλ µ dσa ] = ω ⊗ ∂λ ⊗ [pB (dy − A µ dσc ) + pµ dσa ]

where aλ Ba λ paλ µ = pµ + A µ pB

and AB cµ is given by Eq.(5.63). The corresponding constraint conditions paλ µ = 0 modify the constraint condition (5.45) paλ µ = 0 of gravity without matter. We here are not concerned with dynamics of the fermion-gravitation complex, for it involves generally Lagrangian densities quadratic in tetrad fields and gauge gravitation potentials and needs a lot of technical computations.

158


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