Abstract
Inspired by the appearance of split-complex structures in the dimensional reduction of string theory, and in the theories emerging as byproducts, we study the hyper-complex formulation of Abelian gauge field theories, by incorporating a new complex unit to the usual complex one. The hypercomplex version of the traditional Mexican hat potential associated with the $U(1)$ gauge field theory, corresponds to a {\it hybrid} potential with two real components, and with $U(1)\times SO(1,1)$ as symmetry group. Each component corresponds to a deformation of the hat potential, with the appearance of a new degenerate vacuum. Hypercomplex electrodynamics will show novel properties, such as the spontaneous symmetry breaking scenarios with running masses for the vectorial and scalar Higgs fields, and the Aharonov-Bohm type strings defects as exact solutions; these topological defects may be detected only by quantum interference of charged particles through gauge invariant loop integrals. In a particular limit, the {\it hyperbolic} electrodynamics does not admit topological defects associated with continuous symmetries
Highlights
Explorations involving hypercomplex structures have appeared recently in the literature, for example, in the dimensional reduction of M-theory over a Calabi-Yau-3 fold, where a five-dimensional N = 2 supergravity theory emerges, it turns out that the hyperbolic representation based on para- or split-complex numbers is the most natural way to formulate the scalar fields of the five-dimensional universal multiplet, gaining insight in the understanding of the string theory landscape [1, 2]
In [10] the requirement of hermiticity on the Poincare mass operator defined on the commutative ring of the hyperbolic numbers H, leads to a decomposition of the corresponding hyperbolic Hilbert space into a direct product of the Lorentz group related to the spacetime symmetries, and the hyperbolic unitary group SU (4, H), which is considered as an internal symmetry of the relativistic quantum state; the hyperbolic unitary group is equivalent to the group SU (4, C) × SU (4, C) of the Pati-Salam model [11]
One of the motivations of the present work is to explore the realizations of the hyperbolic symmetries as an internal gauge symmetry in classical gauge field theories; we determine the effects of the incorporation of those symmetries on the geometry and topology of the vacuum manifolds, and the subsequent effect on the formation of topological defects
Summary
In [12] the hyperbolic Klein-Gordon equation for fermions and bosons is considered as a para-complex extension of groups and algebras formulated in terms of the product of ordinary complex and hyperbolic unit; this implies the existence of hyperbolic complex gauge transformations, and the possibility of new interactions; certainly there is not currently experimental indications of them, either evidence against If these new interactions are effectively absence, it is of interest to understand why the hyperbolic complex counterparts for the other interactions there no exist in nature at presently known energies, in spite of the consistence of hyperbolic extensions from the theoretical point of view. In general the classical and quantum descriptions of noncompact σ models show problems such as the unitarity of the S matrix, and the spontaneous symmetry breaking realizations
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
