Hyperbolic complex structures in physics

Abstract

A brief introduction to the parabolic and hyperbolic complex number systems is given. Parabolic and hyperbolic phase transformations are shown to be equivalent to two-dimensional Galilean and Lorentzian relativity transformations, respectively. Basic properties and definitions for the hyperbolic complex numbers are given, and they are then applied to special relativistic physics and the Dirac equation in 1+1 dimensions. Turning to string theory, it is briefly shown that for Minkowski signature the string world sheet possesses an integrable almost hyperbolic complex structure, with a metric that is generalized Hermitian. Next, the hyperbolic complex numbers are applied to the formalism of Dirac spinors in 3+1 dimensions. It is shown that a four-component Dirac spinor is equivalent to a two-component hyperbolic complex spinor, that the Lorentz group is equivalent to a generalized SU(2), and that the Dirac adjoint corresponds to a generalized Hermitian adjoint. A complete formalism is presented for the hyperbolic complex two-component spinors. Lastly, it is shown that the operations of C, P, and T on Dirac spinors are closely related to the three types of complex conjugation that exist when both hyperbolic and ordinary imaginary units are present.