# Relativistic invariance in Euclidean formulations of quantum mechanics

## TL;DR

This paper explores how Euclidean formulations of quantum mechanics can maintain relativistic invariance, providing explicit Poincaré generators and verifying key mathematical properties for hadronic systems.

## Contribution

It derives explicit representations of Poincaré generators in Euclidean space-time for all positive-mass positive-energy irreducible representations, ensuring relativistic invariance.

## Key findings

- Explicit Poincaré generators in Euclidean variables derived
- Hermiticity and self-adjointness of generators established
- Reflection positivity of kernels verified

## Abstract

Relativistic invariance in Euclidean formulations of quantum mechanics is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. Euclidean formulations of relativistic quantum mechanics have some computational advantages. In the Euclidean representation the physical Hilbert space inner product is expressed in terms of Euclidean space-time variables with no need for any analytic continuation. The identification of the complex Euclidean group with the complex Poincar\'e group relates the infinitesimal generators of both groups. In this work explicit representations of the Poincar\'e generators in Euclidean space-time variables for all positive-mass positive-energy irreducible representations of the Poincar\'e group are derived. The commutation relations are checked, both hermiticity and self-adjointness are established, and reflection positivity of the kernels is verified.