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.
Highlights
Many classical field theories are constructed in such a way that the “most natural” solutions to the equations of motion involve a nonzero field value
In the context of particle physics, the bestknown example is the Higgs field [1,2,3,4]; in the context of condensed-matter physics, this paradigm underlies the modern theory of phase transitions, most notably the Ginzburg-Landau theory of superconductivity [5]
Many nonlinear sigma models can be thought of in this way, if one view the model’s target manifold as being embedded in some higher-dimensional space in which the fields are forced to a nonzero value. Such models have been used in the study of both particle physics [6,7] and in ferromagnetism [8]
Summary
Many classical field theories are constructed in such a way that the “most natural” solutions to the equations of motion involve a nonzero field value. This paradigm has been used to study possible observational signatures of Lorentz symmetry violation Such models include a new vector or tensor field and have equations of motion that are satisfied when the metric is flat and the new tensor field is constant but nonzero. The main purpose of this article is to show that this naïve conjecture is not true in general; in particular, if ψα includes a spacetime vector or tensor field, it may be false In such models, the fields may need to satisfy certain constraints due to the structure of the kinetic terms; adding a new “constraint” to such theories, in the form of a Lagrange multiplier, does not automatically reduce the number of d.o.f. of the theory. All expressions involving repeated indices (either tensor indices or field space indices) can be assumed to obey the Einstein summation convention
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