First-Class Constraints, Gauge Transformations, de-Ockhamization, and Triviality: Replies to Critics, Or, How (Not) to Get a Gauge Transformation from a Second-Class Primary Constraint

TL;DR

The paper critically examines the claim that first-class constraints generate gauge transformations, demonstrating that second-class constraints can also generate gauge transformations, challenging traditional distinctions in gauge theory.

Contribution

It provides a novel argument showing second-class primary constraints can generate gauge transformations, undermining the traditional view that only first-class constraints do so.

Findings

Second-class primary constraints can generate gauge transformations.

De-Ockhamization can artificially introduce gauge freedom.

Traditional distinction between first- and second-class constraints is challenged.

Abstract

Recently two pairs of authors have aimed to vindicate the longstanding conventional claim that a first-class constraint generates a gauge transformation in typical gauge theories such as electromagnetism, Yang-Mills and General Relativity, in response to the Lagrangian-equivalent reforming tradition, in particular Pitts, _Annals of Physics_ 2014. Both pairs emphasize the coherence of the extended Hamiltonian formalism against what they take to be core ideas in Pitts 2014, but both overlook Pitts 2014's sensitivity to ways that one might rescue the claim in question, including an additive redefinition of the electrostatic potential. Hence the bulk of the paper is best interpreted as arguing that the longstanding claim about separate first-class constraints is _either false or trivial_ -- de-Ockhamization (using more when less suffices by splitting one quantity into the sum of two) being trivial. Unfortunately section 9 of Pitts 2014, a primarily verbal argument that plays no role in other works, is refuted. Pooley and Wallace's inverse Legendre transformation to de-Ockhamized electromagnetism with an additively redefined electrostatic potential, however, opens the door to a precisely analogous calculation introducing a photon mass, which shows that a _second-class primary_ constraint generates a gauge transformation in the exactly same sense -- a reductio ad absurdum of the claim that a first-class constraint generates a gauge transformation and a second-class constraint does not. Gauge freedom by de-Ockhamization does not require any constraints at all, first-class or second-class, because any dynamical variable in any Lagrangian can be de-Ockhamized into exhibiting trivial additive artificial gauge freedom by splitting one quantity into the sum of two. Physically interesting gauge freedom, however, is typically generated by a tuned sum of first-class constraints.