On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces

Abstract:

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the (Formula presented.) quaplectic group that is given by the semi-direct product of (Formula presented.) with the (Formula presented.) (noncommutative) Weyl–Heisenberg group corresponding to (Formula presented.) fiber coordinates and momenta (Formula presented.) ; (Formula presented.). This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks (Formula presented.) ; (Formula presented.). We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting (Formula presented.) operator variables (associated to an 8D curved phase space) to the canonical (Formula presented.) operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the (Formula presented.) limit, the embedding functions (Formula presented.) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric (Formula presented.), the fiber metric of the vertical space (Formula presented.), and the nonlinear connection (Formula presented.) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

Año de publicación:

2023

Keywords:

• Born reciprocal relativity
• Finsler Geometry
• Yang Algebra
• Phase spaces

Fuente:

scopus