Matrix mechanics of the relativistic point particle and string in Clifford space

TL;DR

This paper develops a matrix mechanics framework for relativistic particles and strings in Clifford space, utilizing spinor and algebraic structures to describe their dynamics and symmetries.

Contribution

It introduces a novel matrix mechanics approach for particles and strings in Clifford space, incorporating spinor variables and algebraic symmetries.

Findings

Clifford spinors form a canonical system with Poisson brackets.

Inner products form Hermitian matrices transforming under U(N).

Quantized string charges satisfy the Poincare algebra.

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 NxN 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.C) charge and space-time momentum of the quantized string satisfy the Poincare algebra.