Degrees of freedom in two dimensional string theory

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 limit one requires an infinite number of fields w_j(x, t) in addition to the standard collective field whenever a "fold" forms on the fermi surface of the underlying fermionic field theory. We show that in the quantum theory these are not additional degrees of freedom, but represent quantum dispersions of the collective field. These dispersions become O(1) rather than O precisely after fold formation, thus giving additional classical quantities and 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 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.