Abstract:

Bounded Symmetric Domains are domains with a rich group of automorphisms. There is a ternary product structure uniquely associated with the domain.

The ball of all possible velocities is a Bounded Symmetric Domain with respect to the Lorenz group. The generators of this group can be identified with relativistic fields, for instance with the electromagnetic field. Under this representation the formulae for non-commutation of rotations and boosts is complicated. However, replacing the velocity by the so-called symmetric velocity, that is its relativistic half, simplifies the above non-commutation formulae. Conformal maps on these velocities represent the Lorentz group. The generators of this presentation lead to the spin triple product, which is related to the Geometric product (also called the Clifford product). The latter simplifies greatly the Maxwell equation, the Dirac equaion and other equations of physics.

The measuring process defines the geometry of the state space of a Quantum system. This geometry leads to a ternary product, converting the state space into a Bounded Symmetric Domain.