Quantization of Infinite dimensional Phase Spaces

Abstract

While finite dimensional Lie groups and their representations are rather well-understood mathematical objects, this is not at all true for infinite dimensional Lie groups, where only a fragmentary such understanding is available. In earlier chapters, we have studied in detail what happens when quantizing a finite dimensional phase space, bosonic or fermionic. In these cases, a finite dimensional symplectic or orthogonal group acts and quantization uses a representation of these groups. For the case of quantum field theories with their infinite dimensional phase spaces, the symplectic or orthogonal groups acting on these spaces will be infinite dimensional. In this chapter, we will consider some of the new phenomena that arise when one looks for infinite dimensional analogs of the role these groups and their representations play in quantum theory in the finite dimensional case.