For general interacting quantum field theories, one must choose among in-
equivalent possibilities for representations of the canonical commutation rela-
tions, finding one on which the operators of the interacting field theory are
well-defined. This makes interacting quantum field theory a much more com-
plex subject than free field theory and is the source of well known difficulties
with infinities that appear when standard calculational methods are applied.
A proper definition of an interacting quantum field theory generally requires
introducing cutoffs that make the number of degrees of freedom finite so that
standard properties used in the finite dimensional case still hold, then studying
what happens as the cutoffs are removed, trying to find a physically sensible
limit (“renormalization”).
The reader is warned that this chapter is of a much sketchier nature than
earlier ones, intended only to indicate some outlines of how certain foundational
ideas about representation theory and quantization developed for the finite di-
mensional case apply to quantum field theory. This material will not play a
significant role in later chapters.
39.1 Inequivalent irreducible representations
In our discussion of quantization, an important part of this story was the Stone-
von Neumann theorem, which says that the Heisenberg group has only one in-
teresting irreducible representation, up to unitary equivalence (the Schr ̈odinger
representation). In infinite dimensions, this is no longer true: there will be an
infinite number of inequivalent irreducible representations, with no known com-
plete classification of the possibilities. Before one can even begin to compute
things like expectation values of observables, one needs to find an appropriate
choice of representation, adding a new layer of difficulty to the problem that
goes beyond that of just increasing the number of degrees of freedom.
To get some idea of how the Stone-von Neumann theorem can fail, one can
consider the Bargmann-Fock quantization of the harmonic oscillator degrees of
freedom and the coherent states (see section 23.2)
|α〉=D(α)| 0 〉=eαa†−αa
| 0 〉whereD(α) is a unitary operator. These satisfy
|〈α| 0 〉|^2 =e−|α|2Each choice ofαgives a different, unitarily equivalent usingD(α), representation
of the Heisenberg group. This is on the space spanned by
D(α)(a†)n| 0 〉= (a(α)†)n|α〉where
a(α)†=D(α)a†D−^1 (α)