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 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
