COVARIANT QUANTIZATION OF MEMBRANE DYNAMICS

A Lorentz covariant quantization of membrane dynamics is defined, whic h also leaves unbroken the full three dimensional diffeomorphism invar iance of the membrane; This makes it possible to understand the reduct ions to string theory directly in terms of the Poisson brackets and co nstraints of the theories. Two approaches to the covariant quantizatio n are studied, Dirac quantization and a quantization based on matrices , which play a role in recent work on M theory. In both approaches the dynamics is generated by a Hamiltonian constraint, which means that a ll physical states are ''zero energy.'' A covariant matrix formulation may be defined, but it is not known if the full diffeomorphism invari ance of the membrane may be consistently imposed. The problem is the n on-area-preserving diffeomorphisms: they are realized nonlinearly in t he classical theory, but in the quantum theory they do not seem to hav e a consistent implementation for finite N. Finally, an approach to a genuinely background independent formulation of matrix dynamics is bri efly described.