# A New Geometric Proposal for the Hamiltonian

Description of Classical
Field Theories ^{†}^{†}thanks: The paper is in final form and will not be published elsewhere.

###### Abstract

We consider the geometric formulation of the Hamiltonian formalism for field theory in terms of Hamiltonian connections and multisymplectic forms. In this framework the covariant Hamilton equations for Mechanics and field theory are defined in terms of multisymplectic –forms, where is the dimension of the basis manifold, together with connections on the configuration bundle. We provide a new geometric Hamiltonian description of field theory, based on the introduction of a suitable composite fibered bundle which plays the role of an extended configuration bundle. Instead of fibrations over an –dimensional base manifold , we consider fibrations over a line bundle fibered over . The concepts of extended Legendre bundle, Hamiltonian connection, Hamiltonian form and covariant Hamilton equations are introduced and put in relation with the corresponding standard concepts in the polymomentum approach to field theory.

Key words: fiber bundles, jets, connections, Hamilton equations.

2000 MSC: 53C05,58A20,70H05,37J05.

## 1 Introduction

A geometric formulation of the Hamiltonian formalism for field theory in terms of Hamiltonian connections and multisymplectic forms was developed in [19, 20, 21]. We recall that, in this framework, the covariant Hamilton equations for Mechanics and field theory are defined in terms of multisymplectic –forms, where is the dimension of the basis manifold, together with connections on the configuration bundle.

We provide here a new geometric Hamiltonian description of field theory, based on the introduction of a suitable composite fibered bundle which plays the role of an extended configuration bundle. One of the main features of this approach is that one can describe the polymomenta and other objects appearing in the polymomentum formulation of field theory (see e.g [3, 8, 9, 10, 14, 16] and references therein) in terms of differential forms with values in the vertical tangent bundle of an appropriate line bundle . The introduction of the line bundle can be here understood as a suitable way of describing the gauge character appearing in the Hamiltonian formalism for field theory (see [11] for a nice introduction to this topic). Instead of bundles over an –dimensional base manifold , we consider fibrations over a line bundle fibered over ; the concepts of event bundle, configuration bundle and Legendre bundle are then introduced following the analogous setting introduced in [19, 20, 21] for Mechanics and for the polymomentum approach to field theory. Moreover, Hamiltonian connections, Hamiltonian forms and covariant Hamilton equations can be suitably described in this framework. This new approach takes into account the existence of more than one independent variable in field theory, but enables us to keep as far as possible most of the nice features of time–dependent Hamiltonian Mechanics.

Already in the seventies, Kijowski stressed the prominent role of the symplectic structures in field theories [10, 11, 13, 14]. Our approach can provide a suitable geometric interpretation of the canonical theory of gravity and gravitational energy, as presented in [12], where the local line bundle coordinate plays the role of a parameter and enables one to consider the gravitational energy as a ‘gravitational charge’.

In Section 2 we state the general framework of composite fiber bundles, their jet prolongations and composite connections. Section 3 contains the main result of this note, i.e Theorem 1, which relates the abstract Hamiltonian dynamics introduced here with the standard Hamilton–De Donder equations (see [16] for a detailed review on the topic and recent developments). Furthermore, since it stresses the underlying algebraic structure of field theory, this ‘extended’ approach turns out to be very promising, e.g via the application of some results concerned with a new –theory for vector bundles carrying this special kind of multisymplectic structure (see [24]). However, this topic will be developed elsewhere.

## 2 Jets of fibered manifolds and connections

The general framework is a fibered bundle , with and and, for , its jet manifold . We recall the natural fiber bundles , , , and, among these, the affine fiber bundles . We denote by the vector subbundle of the tangent bundle formed by vectors on which are vertical with respect to the fibering (see e.g. [22]).

Greek indices run from to and they label base coordinates, while Latin indices run from to and label fibre coordinates, unless otherwise specified. We denote multi–indices of dimension by boldface Greek letters such as , with , ; by an abuse of notation, we denote with the multi–index such that , if , , if . We also set . The charts induced on are denoted by , with ; in particular, we set . The local bases of vector fields and –forms on induced by the coordinates above are denoted by and , respectively.

The contact maps on jet spaces [17] induce the natural complementary fibered morphisms over the affine fiber bundle

(1) |

with coordinate expressions, for , given by

(2) |

and the natural fibered splitting [17, 19, 22]

(3) |

Let us consider the following dual exact sequences of vector bundles over :

(4) |

###### Definition 1

A connection on the fiber bundle is defined by the dual linear bundle morphisms over

(5) |

which split the exact sequences (4).

###### Remark 1

Let be the local components of the connection . The above linear morphisms over yield uniquely a horizontal tangent–valued –form on , which we denote by and which projects over the soldering form on . Dually, a connection on can be also represented by the vertical–valued –form (see [19]). Taking this into account, the canonical splitting (3) provides the horizontal splitting .

###### Proposition 1

In the following a relevant role is played by the composition of fiber bundles

(6) |

where , and are fiber bundles. The above composition was introduced under the name of composite fiber bundle in [6, 18, 20] and shown to be useful for physical applications, e.g for the description of mechanical systems with time–dependent parameters. We recall some structural properties of composite fiber bundles [19].

###### Proposition 2

Given a composite fiber bundle (6), let be a global section of the fiber bundle . Then the restriction of the fiber bundle to is a subbundle of the fiber bundle .

###### Proposition 3

Given a section of the fiber bundle and a section of the fiber bundle their composition . Conversely, every section of the fiber bundle is the composition of the section and some section of over the closed submanifold . of is a section of the composite bundle

### 2.1 Connections on composite bundles

We shall be concerned here with the description of connections on composite fiber bundles. We will follow the notation and main results stated in [19]; see also [2].

We shall denote by , and , the jet manifolds of the fiber bundles , and respectively.

Let be a connection on the composite bundle projectable over a connection on , i.e such

###### Remark 2

Let be a section of . Every connection induces the pull–back connection on the subbundle . We recall that the composite connection

We have the following exact sequences of vector bundles over a composite bundle :

(7) |

where and are the vertical tangent and cotangent bundles to the bundle .

###### Remark 3

Every connection on provides the dual splittings

(8) |

of the above exact sequences.

By means of these splittings we can construct the vertical covariant differential on the composite bundle , i.e the first order differential operator

(9) |

The restriction of , induced by a section of , coincides with the covariant differential on relative to the pull–back connection [19].

## 3 Hamiltonian formalism for field theory

We recall now that the covariant Hamiltonian field theory can be conveniently formulated in terms of Hamiltonian connections and Hamiltonian forms [6, 20]. Here we shall construct a Hamiltonian formalism for field theory as a theory on the composite bundle , with a line bundle having local fibered coordinates .

Let us now consider the extended Legendre bundle . There exists the canonical isomorphism

(10) |

###### Definition 2

We call the fiber bundle the abstract event space of the field theory. The configuration space of the field theory is then the first order jet manifold .

The abstract Legendre bundle of the field theory is the fiber bundle .

Let now be a connection on and be a connection on . We have the following non–canonical isomorphism

(11) |

In this perspective, we consider the canonical bundle monomorphism over providing the tangent–valued Liouville form on , i.e

(12) |

the coordinate expression of which is

(13) |

where are generators of vertical –forms (i.e contact forms) on and “” is the isomorphism defined by (11).

The polysymplectic form on is then intrinsically defined by

where is an arbitrary –form on ; its coordinate expression is given by

(14) |

###### Remark 4

Let be the first order jet manifold of the extended Legendre bundle . By Proposition 1 a connection on the extended Legendre bundle is in one–to–one correspondence with global sections of the affine bundle .

###### Definition 3

A connection on the extended Legendre bundle is said to be a Hamiltonian connection iff the exterior form is closed.

As a straightforward application of the relative Poincaré lemma we have then [19] the following.

###### Proposition 4

Let be a Hamiltonian connection on and be an open subset of . Locally, we have

(15) |

where .

###### Definition 4

The local mapping is called a Hamiltonian. The form on the extended Legendre bundle is called a Hamiltonian form.

Every Hamiltonian form admits a Hamiltonian connection such that

(16) |

Let now set . Then the Hamiltonian form is the Poincaré–Cartan form of the Lagrangian on , with values in .

###### Definition 5

The Hamilton operator for is defined as the Euler–Lagrange operator associated with , namely:

(17) |

The kernel of the Hamilton operator (17), i.e the Euler–Lagrange equations for , is an affine closed embedded subbundle of , locally given by the equations

(18) |

###### Definition 6

The kernel of the Hamilton operator defines the covariant Hamilton equations (18) on the extended Legendre bundle .

###### Remark 5

Notice that a global section of is a Hamiltonian connection satisfying relation (16).

In the sequel we state the main result of this note, which points out the relation with the standard polysymplectic approach (for a review of the topic see e.g [3, 8, 9, 10, 14, 16] and references quoted therein). The basic idea is that the present geometric formulation can be interpreted as a suitable generalization to field theory of the so–called homogeneous formalism for Mechanics.

Let be the vertical covariant differential (see (9)) relative to the connection on the abstract Legendre bundle .

###### Definition 7

We define the abstract covariant Hamilton equations to be the kernel of the first order differential operator .

###### Lemma 1

Let be a Hamiltonian connection on . Let and be connections on and , respectively. Let and be sections of the bundles and , respectively.

Then the standard Hamiltonian connection on turns out to be the pull–back connection induced on the subbundle by the section of .

Proof. The abstract Legendre bundle is in fact a composite bundle , so that it is possible to apply the results concerning connections on composite bundles recalled in Subsection 2.1, for any connection . Our claim then follows for any section of the composite bundle of the type , since the extended Legendre bundle can be also seen as the composite bundle .

We can then state our main result as follows.

###### Theorem 1

Let be the covariant differential on the subbundle relative to the pull–back connection . The kernel of coincides with the Hamilton–De Donder equations of the standard polysymplectic approach to field theories.

###### Remark 6

Our approach provides a suitable geometric interpretation of the canonical theory of gravity and gravitational energy, as presented in [12], where plays the role of a parameter and enables one to consider the gravitational energy as a ‘gravitational charge’. This topic is currently under investigation and it will be developed in a separate forthcoming paper.

## 4 Acknowledgments

One of the authors (M. P.) is grateful to J. Kijowski for helpful comments concerning his lectures on Canonical Gravity held in Levoča, August 2000. Thanks are due to R. Vitolo for useful remarks. M. P. and E. W. also acknowledge the kind invitations at the Department of Mathematics E. De Giorgi of the University of Lecce, October 2000 and August–September 2001. This paper has been written within the GNFM–INdAM research project Formalismo Hamiltoniano in teoria dei campi and the University of Torino project Giovani Ricercatori 2001.

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