The K\"all\'en-Lehmann representation for Lorentz-violating field theory

TL;DR

This paper extends the K"allén-Lehmann representation to Lorentz-violating field theories with scalar and fermion models, deriving explicit propagator forms, commutators, and spectral sum rules.

Contribution

It provides the first detailed derivation of the K"allén-Lehmann representation in Lorentz-violating models involving scalars and fermions.

Findings

Exact propagator forms derived for Lorentz-violating models

Explicit expressions for equal-time field commutators

Sum rules established for spectral density functions

Abstract

We consider field-theoretic models, one consisting purely of scalars, the other also involving fermions, that couple to a set of constant background coupling coefficients transforming as a symmetric observer Lorentz two-tensor. We show that the exact propagators can be cast in the form of a K\"all\'en-Lehmann representation. We work out the resulting form of the Feynman propagators and the equal-time field commutators, and derive sum rules for the spectral density functions.