Abstract
We study the vacuum structure of a class of Lorentz invariant field theories where the vacuum expectation values are not constant but are (phase) modulated. The vacua are classified into spatial, temporal, and light-like modulation types according to the patterns of spontaneous breaking of translational symmetry. The conditions for having temporal or light-like modulated vacua imply severe constraints on the models. We utilize the notion of generalized Nambu-Goldstone modes which appear in the modulated vacua. Finally, we examine fluctuation modes around these vacua and discuss their dynamics and the absence of ghosts.
Highlights
Finding the vacuum structure is one of the most fundamental issues in the course of understanding quantum field theories
For models that consist of a canonical kinetic term with a potential V, this is given by a constant field configuration determined by a minimum of V
The constant vacuum expectation value (VEV) trivially satisfies the equation of motion and the constant field configurations automatically yield the minimum energy since varying a field costs gradient energy
Summary
Finding the vacuum structure is one of the most fundamental issues in the course of understanding quantum field theories. [32,33], we realized such a possibility in a simple Lorentz-invariant model with higher-derivative terms, by studying the vacuum structure possessing a nonconstant VEV.. The generalized NG modes correspond to the flat direction of the “potential” whose quadratic kinetic term disappears in the Lagrangian for the case of spatial modulation [32]. This turns out not to be the case for temporal or lightlike modulation. Such a time dependence is not usually called a temporal modulation Note, that this is not the case for Lorentz-invariant (relativistic) theories discussed in this paper.
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