Covariant quantization of membrane dynamics

A Lorentz covariant quantization of membrane dynamics is defined, which also leaves unbroken the full three dimensional diffeomorphism invariance of the membrane. This makes it possible to understand the reductions to string theory directly in terms of the Poisson brackets and constraints of the theories. Two approaches to the covariant quantization 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 all physical states are “zero energy.” A covariant matrix formulation may be defined, but it is not known if the full diffeomorphism invariance of the membrane may be consistently imposed. The problem is the non-area-preserving diffeomorphisms: they are realized nonlinearly in the classical theory, but in the quantum theory they do not seem to have a consistent implementation for finite N. Finally, an approach to a genuinely background independent formulation of matrix dynamics is briefly described.