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To circumvent some technical difficulties faced by the geometric Lagrangianapproach to the physical degree of freedom count presented in the work of Diaz,Higuita, and Montesinos, J. Math. Phys. 55, 122901 (2014) that prevent itsdirect implementation to field theory, in this paper, we slightly modify thegeometric Lagrangian approach in such a way that its resulting version worksperfectly for field theory (and for particle systems, of course). As inprevious work, the current approach also allows us to directly get theLagrangian constraints, a new Lagrangian formula for the counting of the numberof physical degrees of freedom, the gauge transformations, and the number offirst- and second-class constraints for any action principle based on aLagrangian depending on the fields and their first derivatives withoutperforming any Dirac's canonical analysis. An advantage of this approach overthe previous work is that it also allows us to handle the reducibility of theconstraints and to get the off-shell gauge transformations. The theoreticalframework is illustrated in 3-dimensional generalized general relativity(Palatini and Witten's exotic actions), Chern-Simons theory, $4$-dimensional BFtheory, and $4$-dimensional general relativity given by Palatini's action withcosmological constant.

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