An application of the theory of moments to Euclidean relativistic quantum mechanical scattering

An application of the theory of moments to Euclidean relativistic quantum mechanical scattering
Gordon J. Aiello

2017

Department of Mathematics (Applied Mathematical and Computational Sciences program), The University of Iowa, Iowa City, IA 52242, UNITED STATES OF AMERICA.

ABSTRACT

One recipe for mathematically formulating a relativistic quantum mechanical scattering theory utilizes a two-Hilbert space approach, denoted by H and H0, upon each of which a unitary representation of the Poincaré Lie group is given. Physically speaking, H models a complicated interacting system of particles one wishes to understand, and H0 an associated simpler (i.e., free/noninteracting) structure one uses to construct “asymptotic boundary conditions” on so-called scattering states in H. Simply put, H0 is an attempted idealization of H one hopes to realize in the large time limits ? → ±∞.

The above considerations lead to the study of the existence of strong limits of operators of the form ??????−??0?, where ? and ?0 are the self-adjoint generators of the time translation subgroup of the unitary representations of the Poincaré group on H and H0, and ? is a contrived mapping from H0 into H that provides the internal structure of the scattering asymptotes.

The existence of said limits in the context of Euclidean quantum theories (satisfying precepts known as the Osterwalder-Schrader axioms) depends on the choice of ? and leads to a marvelous connection between this formalism and a beautiful area of classical mathematical analysis known as the Stieltjes moment problem, which concerns the relationship between numerical sequences {??}?=0 and the existence/uniqueness of measures ?(?) on the half-line satisfying ?? = ∫0 ????(?).

 



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