Soft theorems for boosts and other time symmetries

TL;DR

This paper derives soft theorems related to spontaneously broken time symmetries, such as Lorentz boosts, involving unequal-time correlation functions with a soft Goldstone, with applications to superfluids and inflation.

Contribution

It introduces new soft theorems for broken time symmetries involving unequal-time correlators and explores their implications for observable quantities.

Findings

Soft theorems involve unequal-time correlators with soft Goldstone insertions.

Explicit checks performed for superfluid and inflationary effective theories.

Unequal-time identities reveal information not accessible through equal-time correlators.

Abstract

We derive soft theorems for theories in which time symmetries -- symmetries that involve the transformation of time, an example of which are Lorentz boosts -- are spontaneously broken. The soft theorems involve unequal-time correlation functions with the insertion of a soft Goldstone in the far past. Explicit checks are provided for several examples, including the effective theory of a relativistic superfluid and the effective field theory of inflation. We discuss how in certain cases these unequal-time identities capture information at the level of observables that cannot be seen purely in terms of equal-time correlators of the field alone. We also discuss when it is possible to phrase these soft theorems as identities involving equal-time correlators.