Abstract
As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents.
Highlights
Symmetries play a crucial role in the development and analysis of physical theories.Beyond the paradigmatic examples in the classical field theory of Einstein’s equations of general relativity, or the Yang–Mills equation for gauge fields, symmetries are used, for instance, to determine the form of the Lagrangian function within the Lagrangian formulation theory
We first show how to formulate the theory within the multisymplectic Hamiltonian formalism constructing the presymplectic structure on the space of solutions of the theory, we show that the dynamics of the theory lies entirely in the kernel of a presymplectic manifold since the presymplectic constraint algorithm (PCA) leads to a final manifold in which the Hamiltonian identically vanishes
The case of gauge theories, i.e., theories such that the space of solutions carries a presymplectic structure with a non-trivial characteristic distribution, was dealt by constructing a symplectic regularization of it using the coisotropic embedding theorem
Summary
Symmetries play a crucial role in the development and analysis of physical theories. Beyond the paradigmatic examples in the classical field theory of Einstein’s equations of general relativity, or the Yang–Mills equation for gauge fields, symmetries are used, for instance, to determine the form of the Lagrangian (or respectively of the Hamiltonian) function within the Lagrangian (respectively, Hamiltonian) formulation theory (an interesting example, in this perspective, is given by Utiyama’s theorem). Symmetries are used to uncover significant global properties of a theory itself, such as the existence of different phases or sectors, or even as an effective tool to analyze its quantum aspects (ranging from the study of the properties of quantum states, to the study of the renormalizability or to obtain definite predictions when anomalies arise) In most cases, these symmetries of interest can be termed “geometrical”, as they emerge from underlying geometrical structures (consider, for instance, geometrical symmetries of the space–time of the theory). Other examples of geometrical symmetries include conformal invariance, which is instrumental in the analysis of quantum field theories in two dimensions, or the group of causal diffeomorphisms of a space–time, in the case of theories of gravitation There is another use of symmetries that runs parallel to the development of the previous physical motivations.
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