Symmetries in the hyperbolic Hilbert space

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.