On gauge theories and redundant degrees of freedom

Question

Given an action or Lagrangian with the additional information that it is a gauge system, how do we know this field has how many physical or redundant degrees of freedom? Is there any systematic method to derive it?

Answer 1

1. That a Lagrangian/action/Hamiltonian is a gauge theory is not "additional information" - whether or not the solutions to the equations of motion are underdetermined / the action possess an infinite-dimensional symmetry group are observable facts about the action itself needing no additional input.

2. The theory of constrained Hamiltonian systems provides a well-defined (if sometimes difficult to carry out explicitly) algorithm to deal with such systems. See e.g. this answer of mine for a discussion of how to turn a given Lagrangian with gauge symmetries into a constrained Hamiltonian system. The essence is that the Hamiltonian theory describing a general physical theory consists of

   • a "naive" Hamiltonian $H(p,q)$ on a phase space of dimension $2n$ ("$2n$ degrees of freedom")
   • a number $F$ of first-class constraints $\phi_i(p,q) = 0$ that Poisson-commute with each other and the Hamiltonian (if $F=0$, the theory has no gauge symmetries)
   • a number $S$ of second-class constraints $\chi_i(p,q) = 0$ that have non-zero Poisson bracket with at least one other constraint

   and the physical d.o.f. are the dimension of the reduced phase space you get when you take the constraint surface $\Sigma$ (defined by the submanifold of the phase space where all the constraints are simultaneously fulfilled) and then quotient out the remaining gauge symmetries generated by the $F$ first-class constraints (mathematically this is symplectic or Marsden-Weinstein reduction). In this geometric language it is then clear that this reduced phase space has dimension $2N - 2F - S$ (if one assumes a certain niceness/regularity of the constraints).

For a full exposition of the theory of constrained Hamiltonian systems with a particular eye towards their quantization, see "Quantization of Gauge Systems" by Hennaux and Bunster.