The God has created a man in order that he creates that the God fails to do

## Sunday, 11 December 2011

### Covariant (polysymplectic) Hamiltonian field theory (from my Scientific Biography)

Classical field theory is formulated as Lagrangian theory. All fundamental field equations are Euler – Lagrange equations derived from some Lagrangian. At the same time, classical autonomous (conservative) mechanics admits both Lagrangian and Hamiltonian formulations, which, however, are not equivalent. A Hamiltonian formulation of mechanics is based on symplectic geometry which a phase space is provided with. Naturally, a long time ago there arose a question about a Hamiltonian formulation of field theory. However, a straightforward application of symplectic Hamiltonian formalism to field theory, when canonical momenta are correspondent to derivatives of field functions only with respect to time, leads to an infinite-dimensional phase space, where  canonical variables are functions in any given instant. The Hamilton equations on such a phase space are not familiar differential equations and, in no way, comparable to the Euler - Lagrange equations of field theory. Such a symplectic Hamiltonian construction is utilized exclusively in quantum field theory to obtain the commutation relations of quantum field operators.

At the same time, a finite-dimensional phase space can be obtained if one considers canonical momenta correspondent to derivatives of field functions relative to all space-time coordinates. Such an approach is called the covariant Hamiltonian field theory. Its different variants are considered. These are polysymplectic, multisymplectic, k-symplectic Hamiltonian theories in accordance with a choice of a phase space and entered structure on it, generalizing symplectic geometry. In 1990, this question attracted attention of my student and collaborator Oleg Zakharov, who, in 1992, published an article in Journal of Mathematical Physics. However, he met a problem of constructing Hamilton equations of fields similar to the Euler – Lagrange ones. I built these equations, and then close interested in this topic.

We restricted our consideration to first order field theory, and developed polysymplectic Hamiltonian formalism on fibre bundles which, in the case of fibre bundles over the temporal axis X=R, led to non-autonomous Hamiltonian mechanics with the usual canonical variables. We constructed a globally defined polysymplectic form on a phase space, developed polysymplectic Hamiltonian formalism and, given a Lagrangian, built the associated Hamiltonians. The main results were presented in [65,66] in 1992 and, in 1993, we published already quite detailed theory . The main problem was that Lagrangian and Hamiltonian formalisms on fibre bundles are not equivalent, unless only a Lagrangian is hyperregular, i.e., when the Legendre map of a configuration space to a phase space is a diffeomorphism. In a general case, one and the same Lagrangian is associated to different Hamiltonians, or no one. The comprehensive relationship between Lagrangian and polysymplectic Hamiltonian formalisms can be given in the case of the so-called semiregular and almost regular Lagrangians. The basic theorems are presented in papers [67,69] and books [10,11], and the final theory was published in the book  in 1997 and in the  article  in 1999.

Polysymplectic Hamiltonian formalism was considered in application to the basic field models, all of which are almost regular. We studied a possibility of quantization of fields in covariant canonical variables [70,114]. However, a question remains still open because additional gauge symmetries, arising in field models in covariant canonical variables, are not studied till now.

Reference:

G.Sardanashvily, My Scientific Biography

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