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  • Open Access

Reflection positivity in Euclidean formulations of relativistic quantum mechanics of particles

Gohin Shaikh Samad*

W. N. Polyzou†

  • *Contact author: ssamad@mcneese.edu
  • †Contact author: polyzou@uiowa.edu
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Phys. Rev. C 112, 064917 – Published 24 December, 2025

DOI: https://doi.org/10.1103/2cf4-qx1h

Abstract

Background: Phenomenological relativistic models of strongly interacting systems have proved to be useful for modeling few-nucleon and few-quark systems at the few-GeV scale. They have the advantage that relativistic invariance is exact; however, satisfying cluster properties requires a complicated construction which has never been used in applications. In addition, there is no straightforward connection to local field theories.

Purpose: To explore an alternative approach for constructing relativistic quantum mechanical models that addresses these two deficiencies. This work examines the general structure of the dynamical input to these models.

Method: Relativistic models that address these two deficiencies can be formulated using reflection positive Euclidean kernels. Relaxing the locality requirement of the axioms of Euclidean field theories leads to a weaker form of reflection positivity. Constraints imposed by the spectral condition and cluster properties are used to determine the structure of a class of kernels satisfying this weaker form of reflection positivity.

Results: The resulting kernels lead to a representation of the quantum mechanical Hilbert space in terms of Euclidean variables that has a positive norm, a set of self-adjoint Poincaré generators with a Hamiltonian satisfying cluster properties and a spectral condition. Relativistic dynamical calculations can be performed directly in the Euclidean representation without analytic continuation. Schwinger functions of a local field theory also satisfy the weakened form of reflection positivity, which provides the desired connection with local field theories.

Conclusion: This work provides the structure of a large class of reflection positive kernels that can be used to test this Euclidean formulation of relativistic quantum mechanics. What is still missing is a dynamical principle that generates model kernels with the desired properties.

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Phys. Rev. C 112, 064917– Published 24 December, 2025
  • Received 4 July 2025
  • Revised 24 October 2025
  • Accepted 13 November 2025
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