Lorentz Symmetries of a Doubly Hyperbolic Phase Space

The Einstein addition law of velocities implies a hyperbolic geometry for relativistic velocity and momentum space. The simplest model of an open, expanding universe implies a hyperbolic geometry for position space. It is natural to investigate the kinematics of a phase space combining hyperbolic geometries in both the velocity-momentum manifold H(3)$_{\mbox{vel}}$ and the position manifold H(3)$_{\mbox{pos}}$. Each of these sustains its own Lorentz subgroup, L$_{\mbox{vel}}$ = O(1,3)$_{\mbox{vel}}$ and L$_{\mbox{pos}}$ = O(1,3)$_{\mbox{pos}}$. These form a direct product group L$^2$ = L$_{\mbox{vel}} \times$ L$_{\mbox{pos}}$, a 12-parameter group, represented by 8 $\times 8$ matrices. Among its operators are a subgroup L$_{\mbox{boost}}$ of Lorentz velocity boosts that operate on the elements of L$_{\mbox{vel}}$ by Einstein addition and on those of L$_{\mbox{pos}}$ by the Lorentz transformation. There is also a conjugate subgroup L$_{\mbox{shift}}$ of hyperbolic translational shifts that operate on the elements of L$_{\mbox{pos}}$ translationally, and on those of L$_{\mbox{vel}}$ to describe the Hubble effect of distance on velocity vectors. The structure, symmetries, Lie algebra and important operators and quantum numbers of the resulting representation of L$^2$ will be reported. (See also F.T. Smith, Ann. Fond. L. de Broglie, 30, 179 (2005).)