Symmetries and physical functions in general gauge theory

TL;DR

This paper analyzes the symmetry structure of general gauge theories, linking gauge transformations with the constraint structure, and provides a systematic method to describe all gauge symmetries and their relation to physical functions.

Contribution

It introduces a constructive procedure to solve the symmetry equation using an orthogonal basis for constraints, revealing the full gauge structure of singular theories.

Findings

The gauge charge includes both first- and second-class constraints.

A systematic method to describe all gauge transformations is developed.

Proof of the equivalence of gauge invariance and Dirac's physicality condition.

Abstract

The aim of the present article is to describe the symmetry structure of a general gauge (singular) theory, and, in particular, to relate the structure of gauge transformations with the constraint structure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry structure of a theory action can be completely revealed by solving the so-called symmetry equation. We develop a corresponding constructive procedure of solving the symmetry equation with the help of a special orthogonal basis for the constraints. Thus, we succeed in describing all the gauge transformations of a given action. We find the gauge charge as a decomposition in the orthogonal constraint basis. Thus, we establish a relation between the constraint structure of a theory and the structure of its gauge transformations. In particular, we demonstrate that, in the general case, the gauge charge cannot be constructed with the help of some complete set of first-class constraints alone, because the charge decomposition also contains second-class constraints. The above-mentioned procedure of solving the symmetry equation allows us to describe the structure of an arbitrary symmetry for a general singular action. Finally, using the revealed structure of an arbitrary gauge symmetry, we give a rigorous proof of the equivalence of two definitions of physicality condition in gauge theories: one of them states that physical functions are gauge-invariant on the extremals, and the other requires that physical functions commute with FCC (the Dirac conjecture).