**Bounded Symmetric Domains as an expression for symmetry in physics**

**Abstract:**

In the talk it will be presented how the Bounded symmetric Domains and the algebraic structure associated with them provide a model for different areas in Physics.

It will be shown that the Bounded Symmetric Domains occur from fundamental laws of Relativity and Quantum Mechanics and that the geometry of these domains determines the evolution of systems.

It will be shown how the commutation relations of the Poincare group define a triple product defining the geometry of the spin domain -Bounded Symmetric Domain of type 4 in Cartan's classification. Also will be shown how this domain can be used to represent both spin 1 and spin 1/2 particles.

We will shown that from symmetry, following from the Principle of Relativity, alone we can infer that there are only two possibilities for space time transformations between inertial systems: the Galilean transformations or the Lorentz transformations. The approach was presented in the book "Physical Applications of Homogenous Balls".

Similarly, the symmetry following from the General Principle of Relativity leads to a description of the transformations between accelerated systems without assuming the clock hypothesis.