My book: Generalized Hamiltonian Formalism for Field Theory

Generalized Hamiltonian Formalism for Field Theory

(World Scientific, Singapore, 1995 )

G. SARDANASHVILY

(PDF file)

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 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

1 Geometric Preliminary

1.1 Fibred manifolds

1.2 Jet spaces

1.3 General connections

2 Lagrangian Field Theory

2.1 Lagrangian formalism on fibred manifolds

2.2 De Donder Hamiltonian formalism

2.3 Instantaneous Hamiltonian formalism

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 structure

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 Relativity

4.7 Conservation laws

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