Point Particle with Extrinsic Curvature as a Boundary of a Nambu-Goto String: Classical and Quantum Model

TL;DR

This paper models a string as a point particle with extrinsic curvature, deriving its classical equations and quantum spin properties, linking string dynamics to Dirac operators and spin-1/2 states.

Contribution

It introduces a generalized Howe-Tucker action for a rigid particle approximation of a string, analyzing its classical and quantum properties, including spin representation via Dirac matrices.

Findings

The algebra of Dirac brackets includes the spin tensor.

Quantized operators correspond to Dirac matrices projected orthogonally to momentum.

States satisfy the Dirac equation with an effective mass, exhibiting spin-1/2 characteristics.

Abstract

It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe-Tucker action for such a "rigid particle" and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli-Lubanski vector composed with such projected Dirac matrices is equal to the Pauli-Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.