Geometric Lagrangian approach to the physical degree of freedom count in field theory

TL;DR

This paper introduces a modified geometric Lagrangian method that efficiently counts physical degrees of freedom, identifies constraints, and determines gauge transformations in field theories without canonical analysis.

Contribution

It presents a new version of the geometric Lagrangian approach that handles reducibility and off-shell gauge transformations in field theories, improving upon previous methods.

Findings

Successfully applied to 3D generalized gravity and Chern-Simons theory.

Provides explicit formulas for constraints and degrees of freedom.

Avoids Dirac's canonical analysis, simplifying the process.

Abstract

To circumvent some technical difficulties faced by the geometric Lagrangian approach to the physical degree of freedom count presented in the work of Diaz, Higuita, and Montesinos, J. Math. Phys. 55, 122901 (2014) that prevent its direct implementation to field theory, in this paper, we slightly modify the geometric Lagrangian approach in such a way that its resulting version works perfectly for field theory (and for particle systems, of course). As in previous work, the current approach also allows us to directly get the Lagrangian constraints, a new Lagrangian formula for the counting of the number of physical degrees of freedom, the gauge transformations, and the number of first- and second-class constraints for any action principle based on a Lagrangian depending on the fields and their first derivatives without performing any Dirac's canonical analysis. An advantage of this approach over the previous work is that it also allows us to handle the reducibility of the constraints and to get the off-shell gauge transformations. The theoretical framework is illustrated in 3-dimensional generalized general relativity (Palatini and Witten's exotic actions), Chern-Simons theory, $4$-dimensional BF theory, and $4$-dimensional general relativity given by Palatini's action with cosmological constant.