Constraints and degrees of freedom in Lorentz-violating field theories

TL;DR

This paper examines the relationship between potential-based and Lagrange multiplier-based Lorentz-violating vector field theories, revealing conditions under which degrees of freedom are reduced, thus clarifying the structure of such models.

Contribution

It demonstrates that a Lagrange multiplier reduces degrees of freedom only when the vacuum manifold function commutes with primary constraints in vector models.

Findings

Lagrange multipliers do not always eliminate degrees of freedom in vector models.

The reduction of degrees of freedom depends on the commutation relation between the vacuum function and primary constraints.

Provides insight into the structure of Lorentz-violating field theories with vector fields.

Abstract

Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the na\"ive expectation that a Lagrange multiplier "kills off" one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints.