methods can be used to compute a power-series approximation in the small pa-
rameter. This is an important topic in physics, covered in detail in the standard
textbooks.
The standard approach to quantization of infinite dimensional systems is
to begin with “regularization”, somehow modifying the system to only have a
finite dimensional phase space, for instance by introducing cutoffs that make
the possible momenta discrete and finite. One quantizes this theory by taking
the state space and canonical commutation relations to be the unique ones for
the Heisenberg Lie algebra, somehow dealing with the calculational difficulties
in the interacting case (non-quadratic Hamiltonian).
One then tries to take a limit that recovers the infinite dimensional system.
Such a limit will generally be quite singular, leading to an infinite result, and
the process of manipulating these potential infinities is called “renormalization”.
Techniques for taking limits of this kind in a manner that leads to a consistent
and physically sensible result typically take up a large part of standard quantum
field theory textbooks. For many theories, no appropriate such techniques are
known, and conjecturally none are possible. For others there is good evidence
that such a limit can be successfully taken, but the details of how to do this
remain unknown, with for instance a $1 million Millenium Prize offered for
showing rigorously this is possible in the case of Yang-Mills gauge theory (the
Hamiltonian in this case will be discussed in chapter 46).
39.6 For further reading
Berezin’sThe Method of Second Quantization[6] develops in detail the infinite
dimensional version of the Bargmann-Fock construction, both in the bosonic
and fermionic cases. Infinite dimensional versions of the metaplectic and spinor
representations are given there in terms of operators defined by integral kernels.
For a discussion of the infinite dimensional Weyl and Clifford algebras, together
with a realization of their automorphism groupsSpresandOres(and the corre-
sponding Lie algebras) in terms of annihilation and creation operators acting on
the infinite dimensional metaplectic and spinor representations, see [64]. The
book [70] contains an extensive discussion of the groupsSpresandOresand the
infinite dimensional version of their metaplectic and spinor representations. It
emphasizes the origin of novel infinite dimensional phenomena in the geometry
of the complex structures used in infinite dimensional examples.
The use of Bogoliubov transformations in the theories of superfluidity and
superconductivity is a standard topic in quantum field theory textbooks that
emphasize condensed matter applications, see for example [53]. The book [11]
discusses in detail the occurrence of inequivalent representations of the commu-
tation relations in various physical systems.
For a discussion of “Haag’s theorem”, which can be interpreted as showing
that to describe an interacting quantum field theory, one must use a represen-
tation of the canonical commutation relations inequivalent to the one for free
field theory, see [19].