Euclidean formulations of relativistic quantum mechanics - part 2

Wayne Polyzou, PhD; Department of Physics and Astronomy, University of Iowa

I summarize progress on a line of research that Fritz Coester and I started many years ago. One of the challenges in formulating relativistic quantum mechanical models is the difficulty in simultaneously satisfying cluster properties and Poincare invariance. Because of this we began exploring alternate formulations of relativistic quantum theories where cluster properties was easy to satisfy. One approach that we explored was based on a subset of the Euclidean axioms of quantum field theory, where the microscopic locality axiom is discarded. The resulting theory still has a Hilbert space structure, a unitary representation of the Poincar\'e Lie algebra, a spectral condition and satisfies cluster properties. The research has led to a number of interesting and sometimes surprising results. One surprise is that Hilbert space representation and a self-adjoint representation of the Poincar\'e Lie Algebra can be formulated in the Euclidean representation of the Hilbert space without the need for analytic continuation. Another both practical and formal surprise is that it is possible to perform time-dependent scattering calculations in the Euclidean framework, using a Euclidean generalization of Haag Ruelle scattering with some results from formal scattering theory. We were able to emonstrate that polynomials in the 4 dimensional Euclidean Laplacian were complete on the Euclidean representation of the physical Hilbert, which was needed to formulate scattering with strong limits. We were able to construct Euclidean representations of all positive mass positive energy irreducible representation of the Poincare group and develop an understanding of the structure of a large class of distributions that are consistent with the axioms. This is still a work in progress; what is still missing is a dynamical principle that generates the input.

Zoom link to live session: https://uiowa.zoom.us/j/99570315915