Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space

TL;DR

This paper explores the representations of the Canonical group acting on noncommuting extended phase space, aiming to incorporate a broader symmetry framework for particle physics beyond the Poincare group.

Contribution

It introduces the study of unitary irreducible representations of the Canonical group C(1,3), unifying quantum space and dynamical symmetries, extending prior models like the Poincare group.

Findings

Representations include a product with U(1,3) that models hadron dynamics.

Discrete series representations form infinite ladders with U(3) or C(2) sub-representations.

The framework potentially encompasses a spectrum of particle states with extended symmetries.

Abstract

The unitary irreducible representations of the covering group of the Poincare group P define the framework for much of particle physics on the physical Minkowski space P/L, where L is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a SU(3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} are the time, energy, position, and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the Canonical group C(1,3) = U(1,3) *s H(1,3) and in this theory the non-commuting space Q= C(1,3)/ SU(1,3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the Canonical group, T^2 + E^2 - Q^2/c^2-P^2/b^2 +(2h I/bc)(Y/bc -2) where {T,E,Q,P,I,Y} are the generators of the algebra of Os(1,3). The idea is to study the representations of the Canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the Canonical group contain a direct product term that is a representation of U(1,3) that Kalman has studied as a dynamical group for hadrons. The U(1,3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of U(3) (finite dimensional) or C(2) (with degenerate U(1)* SU(2) finite dimensional representations) corresponding to the rest or null frames.