Chapter 39

Quantization of Infinite

dimensional Phase Spaces

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 di-
mensional 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 dimen-
sional phase spaces, the symplectic or orthogonal groups acting on these spaces
will be infinite dimensional. In this chapter we’ll 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.
The most important difference in the infinite dimensional case is that the
Stone-von Neumann theorem and its analog for Clifford algebras no longer hold.
One no longer has a unique (up to unitary equivalence) representation of the
canonical commutation (or anticommutation) relations. It turns out that only
for a restricted sort of infinite dimensional symplectic or orthogonal group does
one recover the Stone-von Neumann uniqueness of the finite dimensional case,
and even then new phenomena appear. The arbitrary constants found in the
definition of the moment map now cannot be ignored, but may appear in com-
mutation relations, leading to something called an “anomaly”.
Physically, new phenomena due to an infinite number of degrees of freedom
can have their origin in the degrees of freedom occurring at arbitrarily short
distances (“ultraviolet divergences”), but also can be due to degrees of freedom
corresponding to large distances. In the application of quantum field theories to
the study of condensed matter systems it is the second of these that is relevant,
since the atomic scale provides a cutoff distance scale below which there are no
degrees of freedom.