A Note on Background (In)dependence

TL;DR

This paper distinguishes between fluctuating and non-fluctuating modes in quantum systems, emphasizing the background dependence of non-fluctuating modes and arguing for the finiteness of string perturbation expansions around these backgrounds.

Contribution

It clarifies the role of background-dependent non-fluctuating modes in 2D gravity and string theory, highlighting their superselection nature and implications for perturbation theory.

Findings

Non-fluctuating modes are superselection parameters.

The theory's background dependence is limited to non-fluctuating modes.

Perturbation expansion around these backgrounds is finite.

Abstract

In general quantum systems there are two kinds of spacetime modes, those that fluctuate and those that do not. Fluctuating modes have normalizable wavefunctions. In the context of 2D gravity and ``non-critical'' string theory these are called macroscopic states. The theory is independent of the initial Euclidean background values of these modes. Non-fluctuating modes have non-normalizable wavefunctions and correspond to microscopic states. The theory depends on the background value of these non-fluctuating modes, at least to all orders in perturbation theory. They are superselection parameters and should not be minimized over. Such superselection parameters are well known in field theory. Examples in string theory include the couplings $t_k$ (including the cosmological constant) in the matrix models and the mass of the two-dimensional Euclidean black hole. We use our analysis to argue for the finiteness of the string perturbation expansion around these backgrounds.