Polysymplectic Hamiltonian Field Theory

TL;DR

This paper develops a covariant polysymplectic Hamiltonian formalism for field theories, establishing its relation to Lagrangian theory, analyzing constraints, and exploring quantization methods, thus extending Hamiltonian approaches to a covariant setting.

Contribution

It introduces a polysymplectic Hamiltonian formalism for field theories, generalizing the Hamiltonian approach to be covariant and applicable to both regular and singular Lagrangians.

Findings

Polysymplectic formalism is equivalent to Lagrangian theory for hyperregular Lagrangians.

Non-regular Lagrangians lead to constraints and require multiple Hamiltonians.

The formalism applies to quadratic systems like Yang-Mills gauge theory.

Abstract

Applied to field theory, the familiar symplectic technique leads to instantaneous Hamiltonian formalism on an infinite-dimensional phase space. A true Hamiltonian partner of first order Lagrangian theory on fibre bundles $Y\to X$ is covariant Hamiltonian formalism in different variants, where momenta correspond to derivatives of fields relative to all coordinates on $X$. We follow polysymplectic (PS) Hamiltonian formalism on a Legendre bundle over $Y$ provided with a polysymplectic $TX$-valued form. If $X=\mathbb R$, this is a case of time-dependent non-relativistic mechanics. PS Hamiltonian formalism is equivalent to the Lagrangian one if Lagrangians are hyperregular. A non-regular Lagrangian however leads to constraints and requires a set of associated Hamiltonians. We state comprehensive relations between Lagrangian and PS Hamiltonian theories in a case of semiregular and almost regular Lagrangians. Quadratic Lagrangian and PS Hamiltonian systems, e.g. Yang - Mills gauge theory are studied in detail. Quantum PS Hamiltonian field theory can be developed in the frameworks both of familiar functional integral quantization and quantization of the PS bracket.