FROM LOOPS TO STATES IN 2-DIMENSIONAL QUANTUM-GRAVITY

We investigate macroscopic loop amplitudes (at genus zero) using the matrix model and the Liouville theories of two-dimensional quantum gravity. Some loop amplitudes, interpreted as wave functions of local operators, satisfy a linear differential equation known as the Wheeler-deWitt equation. Moreover, from the properties of the loop amplitudes an inner-product space structure on the space of wave functions emerges naturally. In the course of our analysis we resolve several apparent discrepancies between the matrix model and Liouville theory. Macroscopic loops provide a natural ultraviolet cutoff on the theory, rendering universal analytic terms in the coupling constants. They contain more information than the local operators and should be regarded as fundamental.