New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split

TL;DR

This paper introduces a new way to treat auxiliary variables in gauge theories, including general relativity, by considering them as cyclic coordinates with natural boundary conditions, leading to more relational actions.

Contribution

It proposes a novel formalism for gauge theories that treats auxiliary variables as cyclic coordinates, enhancing the relational character of the actions and extending existing principles.

Findings

New formalism for auxiliary variables as cyclic coordinates

Actions become more manifestly relational

Potential for generalizations in gauge theory

Abstract

I show how there is an ambiguity in how one treats auxiliary variables in gauge theories including general relativity cast as 3 + 1 geometrodynamics. Auxiliary variables may be treated pre-variationally as multiplier coordinates or as the velocities corresponding to cyclic coordinates. The latter treatment works through the physical meaninglessness of auxiliary variables' values applying also to the end points (or end spatial hypersurfaces) of the variation, so that these are free rather than fixed. [This is also known as variation with natural boundary conditions.] Further principles of dynamics workings such as Routhian reduction and the Dirac procedure are shown to have parallel counterparts for this new formalism. One advantage of the new scheme is that the corresponding actions are more manifestly relational. While the electric potential is usually regarded as a multiplier coordinate and Arnowitt, Deser and Misner have regarded the lapse and shift likewise, this paper's scheme considers new {\it flux}, {\it instant} and {\it grid} variables whose corresponding velocities are, respectively, the abovementioned previously used variables. This paper's way of thinking about gauge theory furthermore admits interesting generalizations, which shall be provided in a second paper.