A Non-Trivial Zero Length Limit of the Nambu-Goto String

TL;DR

This paper demonstrates that the Nambu-Goto string has a meaningful zero-length limit corresponding to a massless particle with extrinsic curvature, with a consistent quantum description only in eight-dimensional spacetime.

Contribution

It introduces a novel zero-length limit of the Nambu-Goto string leading to a quantum system in eight dimensions with specific constraints.

Findings

Quantum wave function solutions exist in eight dimensions.

The system's constraints enforce zero spin and restrict phase space.

Classically, the system reduces to a point particle with extrinsic curvature.

Abstract

We show that a Nambu-Goto string has a nontrivial zero length limit which corresponds to a massless particle with extrinsic curvature. The system has the set of six first class constraints, which restrict the phase space variables so that the spin vanishes. Upon quantization, we obtain six conditions on the state, which can be represented as a wave function of position coordinates, $x^\mu$, and velocities, $q^\mu$. We have found a wave function $\psi(x,q)$ that turns out to be a general solution of the corresponding system of six differential equations, if the dimensionality of spacetime is eight. Though classically the system is just a point particle with vanishing extrinsic curvature and spin, the quantized system is not trivial, because it is consistent in eight, but not in arbitrary, dimensions.