Note on background (in)dependence

In general quantum systems, there are two kinds of spacetime modes: those that fluctuate and those that do not. Fluctuating modes have normalizable wave functions. In the context of two-dimensional gravity and "noncritical" string theory these are called macroscopic states. The theory is independent of the initial Euclidean background values of these modes. Nonfluctuating modes have nonnormalizable wave functions and correspond to microscopic states. The theory depends on the background values of these nonfluctuating 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.