Reflection positivity in Euclidean formulations of relativistic quantum mechanics of particles

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. 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. 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. 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. 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.