Constructing Quantum Mechanics from a Clifford substructure of the relativistic point particle

Abstract

We show that the quantized free relativistic point particle can be understood as a string in a Clifford space which generates the space-time coordinates through its inner product. The generating algebra is preserved by a unitary symmetry which becomes the symmetry of the quantum states. We start by resolving the space-time canonical variables of the point particle into inner products of Weyl spinors with components in a Clifford algebra. Next, we show that a system of N particles has a U(N) symmetry that mixes the Clifford coordinates and momenta belonging to different particles. The inner products of these variables are assembled into Hermitian matrices X and P which are employed in defining a general unitarily invariant dynamical system. When X and P commute, this system can be gauged back into the original system of independent particles. When they do not commute, the system becomes irreducible and infinite and generates a space-time canonical system formally identical to Matrix Mechanics. The continuum limit is identified as a particular parametrization of a relativistic string in Clifford space.