Quantum field theory of a hyper-complex scalar field on a commutative ring

Abstract

Inspired by the structural unification of unitary groups (quantum field theory) with orthogonal groups (relativity) proposed recently through a non-division algebra, we construct a hypercomplex field theory with an internal symmetry that unifies the U(1) compact gauge group with the SO(1,1) noncompact gauge group, using the commutative ring of hypercomplex numbers. From the quantum field theory point of view, the hypercomplex field encodes two charged bosons with opposite charge, and corresponds thus to a neutral compound boson. Furthermore, normal or- dering of operators is not required for controling the vacuum divergences; in an analogy with SUSY, the theory under study contains U(1) boson particles and their hyperbolic SO(1,1) boson partners, whose contributions to the vacuum energy cancel out exactly to a zero value. In fact the present scheme allows us to compare finite measuments of squeezed boson-number statistics obtained with and without normal ordering. Additionally we discuss on the potential applications of the squeezed boson states constructed on the commutative ring, in quantum teleportation and in related areas.