Should Unstable Quantum Field Theories be Lorentz Invariant?

TL;DR

This paper argues that unstable quantum field theories inherently break Lorentz invariance due to their limited applicability in time, implying a preferred frame and questioning the universality of Lorentz symmetry in such contexts.

Contribution

It demonstrates that unstable quantum field theories cannot be Lorentz-invariant and are only valid within a short time regime, establishing a preferred simultaneity hyperplane.

Findings

Unstable theories have a limited regime of applicability.

A preferred simultaneity hyperplane exists for unstable theories.

Current quantization schemes are acceptable regardless of Lorentz invariance.

Abstract

An unstable field theory is what we obtain when we linearise the equations of an interacting field theory near an unstable state. Theories of this kind are adopted to model the onset of spontaneous symmetry breakings, when the fields are sitting on the top of the Mexican hat, and they start to ''roll down'' to the bottom. At present, there exists no rigorous proof that unstable quantum field theories are Lorentz-invariant (in the sense of Wigner's theorem). Here, we show that they shouldn't be. In fact, unstable theories always have a limited regime of applicability, and they are valid only for a very short time. As consequence, there is a preferred simultaneity hyperplane, along which the unstable theory is everywhere applicable, while a generic observer (whose four-velocity is not orthogonal to such hyperplane) must use the full non-linear theory. In summary: the current quantization schemes are ''ok'', independently from whether they lead to a Lorentz-invariant theory.