# Examples of reflection positive Euclidean field theories

## TL;DR

This paper investigates reflection positivity in various Euclidean scalar field theories, highlighting the importance of the analytical structure of Schwinger functions and exploring conditions for reflection positivity in non-integer dimensions and non-local theories.

## Contribution

It demonstrates when reflection positivity holds or fails in scalar field theories, including higher derivative and non-integer dimensional cases, and constructs non-local reflection positive theories from interacting fields.

## Key findings

- Reflection positivity fails for higher derivative scalar fields.
- Reflection positivity holds for scalar fields on non-integer dimensional spaces within certain ranges.
- Non-local reflection positive Euclidean theories can be constructed from interacting local field theories.

## Abstract

The requirement of reflection positivity(RP) for Euclidean field theories is considered. This is done for the cases of a scalar field, a higher derivative scalar field theory and the scalar field theory defined on a non-integer dimensional space(NIDS). It is shown that regarding RP, the analytical structure of the corresponding Schwinger functions plays an important role. For the higher derivative scalar field theory RP does not hold. However for the scalar field theory on a NIDS, RP holds in a certain range of dimensions where the corresponding Minkowskian field is defined on a Hilbert space with a positive definite scalar product that provides a unitary representation of the Poincar\'e group. In addition, and motivated by the last example, it is shown that, under certain conditions, one can construct non-local reflection positive Euclidean field theories starting from the corrected two point functions of interacting local field theories.