  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.
