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Field theories with global continuous symmetries may admit configurations inwhich time translation invariance is broken by the movement of homogeneousbackground fields evolving along the flat directions implied by the symmetries.In this context, the field fluctuations along the broken symmetry are wellparametrized by a Goldstone boson field that may non-trivially interact withother fields present in the theory. These interactions violate Lorentzinvariance as a result of the broken time translation invariance of thebackground, producing a mixing between the field content and the particlespectrum of the theory. In this article we study the effects of suchinteractions on the low energy dynamics of the Goldstone boson quanta, payingspecial attention to the role of the particle spectrum of the theory. Bystudying the particular case of a canonical two-field model with a mexican-hatpotential, we analyze the derivation of the low energy effective field theoryfor the Goldstone boson, and discuss in detail the distinction betweenintegrating fields v/s integrating quanta, to finally conclude that they areequivalent. In addition, we discuss the implications of our analysis for thestudy of systems where time translation invariance is broken, such as cosmicinflation and time crystals.

On the integration of fields and quanta in time dependent backgrounds