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

After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the d e f o r m e d quaplectic group that is given by the semi-direct product of U (1 , 3) with the d e f o r m e d (noncommutative) Weyl–Heisenberg group corresponding to n o n c o m m u t a t i v e fiber coordinates and momenta [ X a , X b ] ≠ 0 ; [ P a , P b ] ≠ 0 . 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 [ X a 1 a 2 ⋯ a n , X b 1 b 2 ⋯ b n ] ≠ 0 ; [ P a 1 a 2 ⋯ a n , P b 1 b 2 ⋯ b n ] ≠ 0 . 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 x μ , p μ operator variables (associated to an 8D curved phase space) to the canonical Y A , Π A operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the c l a s s i c a l limit, the embedding functions Y A (x , p) , Π A (x , p) 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 g μ ν (x , p) , the fiber metric of the vertical space h a b (x , p) , and the nonlinear connection N a μ (x , p) 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. [ABSTRACT FROM AUTHOR]