Degrees of freedom in two dimensional string theory

TL;DR

This paper examines the true degrees of freedom in two-dimensional string theory, clarifying classical and quantum aspects, and shows that nonperturbative effects make the geometric entropy finite, implying fewer fundamental degrees of freedom at high energies.

Contribution

It clarifies the nature of degrees of freedom in 2D string theory, emphasizing quantum dispersions and nonperturbative effects on entropy.

Findings

Quantum dispersions are not additional degrees of freedom.

Geometric entropy is finite due to nonperturbative effects.

High-energy degrees of freedom are fewer than expected.

Abstract

We discuss two issues regarding the question of degrees of freedom in two dimensional string theory. The first issue relates to the classical limit of quantum string theory. In the classical theory one requires an infinite number of fields in addition to the collective field to describe ``folds'' on the fermi surface. We argue that in the quantum theory these are not additional degrees of freedom. Rather they represent quantum dispersions of the collective field which are {\em not} suppressed when $\hbar \rightarrow 0$ whenever a fold is present, thus leading to a nontrivial classical limit. The second issue relates to the ultraviolet properties of the geometric entropy. We argue that the geometric entropy is finite in the ultraviolet due to {\em nonperturbative} effects. This indicates that the true degrees of freedom of the two dimensional string at high energies is much smaller than what one naively expects. (Based on talks at Spring Workshop on String theory and Quantum Gravity, ICTP, Trieste, March 1995 and VIIth Regional Conference on Mathematical Physics, Bandar-Anzali, October 1995.)