Abstract
We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins.
Highlights
Since the original Stueckelberg’s work [1], the idea remains attractive for decades concerning inclusion of auxiliary fields into the action in such a way that modified theory becomes gauge invariant while it is still equivalent to the original one
The Stueckelberg gauge symmetry of the original fields is assumed to remain the same as for the gauge invariant part of the action, while the transformations of Stueckelberg fields are chosen to compensate the non-invariance of the rest part
We propose a systematic way for inclusion of Stueckelberg fields such that guarantees equivalence of the resulting gauge theory to the original system
Summary
Since the original Stueckelberg’s work [1], the idea remains attractive for decades concerning inclusion of auxiliary fields into the action in such a way that modified theory becomes gauge invariant while it is still equivalent to the original one. The procedure of the article [8] allows one to iteratively include Stueckelberg fields for any field theory proceeding from the involutive closure of the original Lagrangian equations, and it is proven to be unobstructed. This procedure implies inclusion of independent consequences into the involutive closure of Lagrangian equations Given this starting point, one arrives at the irreducible Stueckelberg gauge symmetry. We consider inclusion of Stueckelberg fields proceeding from the involutive closure which involves a reducible set of consequences of Lagrangian equations. To exemplify the general procedure, we consider the third order involutive closure of the Proca equations when the original equations are complemented, besides the first order consequence, by the antisymmetric combinations of the derivatives of the Lagrangian equations This leads to inclusion, besides the usual Stueckelberg scalar, of the Stueckelberg field, being the second rank antisymmetric tensor. The results and further perspectives are discussed in the Conclusion
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