The relativistic phase space and Newman-Penrose basis

Abstract

We define a complex relativistic phase space which is the space $\mathbb{C}^4$ equipped with the Minkowski metric and with a geometric tri-product on it. The geometric tri-product is similar to the triple product of the bounded symmetric domain of type IV in Cartan's classification, called the spin domain. We show that there are to types of tripotents-the basic elements of the tri-product in the relativistic phase space. We construct a spectral decomposition for elements of this space. A description of compatibility of element of the relativistic phase space is given. We show that the relativistic phase space has two natural bases consisting of compatible tripotents. The fist one is the natural basis for four-vectors and the second one is the Newman-Penrose basis. The second one determine Dirac bi-spinors on the phase space. Thus, the relativistic phase space has similar features to the quantum mechanical state space.