A Note on Reflection Positivity and the Umezawa-Kamefuchi-Kallen-Lehmann Representation of Two Point Correlation Functions

TL;DR

This paper demonstrates that lattice field theories satisfying certain symmetry and positivity conditions allow for a specific spectral representation of two-point functions, and discusses an example where these conditions are violated.

Contribution

It establishes the necessity and sufficiency of reflection positivity and related conditions for the Umezawa-Kamefuchi-Kallen-Lehmann representation in lattice models.

Findings

Models satisfying (A1)-(A4) admit the spectral representation.

Positivity of spectral density is necessary for these models.

Overlap scalar boson violates reflection positivity.

Abstract

It will be proved that a model of lattice field theories which satisfies (A1) Hermiticity, (A2) translational invariance, (A3) reflection positivity, and (A4) polynomial boundedness of correlations, permits the Umezaa-Kamefuchi-Kallen-Lehmann representation of two point correlation functions with positive spectral density function. Then, we will also argue that positivity of spectral density functions is necessary for a lattice theory to satisfy conditions (A1) - (A4). As an example, a lattice overlap scalar boson model will be discussed. We will find that the overlap scalar boson violates the reflection positivity.