Symmetries in the hyperbolic Hilbert space

TL;DR

This paper explores the symmetries of a hyperbolic Hilbert space using Clifford algebra and hyperbolic numbers, revealing a decomposition into Lorentz and internal symmetries related to the Pati-Salam model.

Contribution

It introduces a mathematical framework for hyperbolic Hilbert spaces that decomposes sixteen-dimensional symmetry into familiar Lorentz and internal symmetries.

Findings

Decomposition of hyperbolic Hilbert space symmetry into Lorentz and SU(4,H) groups.

Connection of the internal symmetry to the Pati-Salam model.

Representation of the Poincare mass operator in this framework.

Abstract

The Cl(3,0) Clifford algebra is represented with the commutative ring of hyperbolic numbers H. The canonical form of the Poincare mass operator defined in this vector space corresponds to a sixteen-dimensional structure. This conflicts with the natural perception of a four-dimensional space-time. The assumption that the generalized mass operator is an hermitian observable forms the basis of a mathematical model that decomposes the full sixteen-dimensional symmetry of the hyperbolic Hilbert space. The result is a direct product of the Lorentz group, the four-dimensional space-time, and the hyperbolic unitary group SU(4,H), which is considered as the internal symmetry of the relativistic quantum state. The internal symmetry is equivalent to the original form of the Pati-Salam model.