Matrix Mechanics of the Relativistic Point Particle and String in Clifford Space

Abstract

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two N × N Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw–Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the ${SL(2.\mathbb{C})}$ charge and space-time momentum of the quantized string satisfy the Poincaré algebra.