for different values ofj. Note that we are using normal ordered operators
here, which is necessary since in the limit as one or both cutoffs are removed,
H 1 becomes infinite dimensional, and only the normal ordered version of the
Hamiltonian used here is well-defined (the non-normal ordered version will differ
by an infinite sum of^12 s).
Everything in this section has a straightforward analog describing a multi-
particle system of fermionic particles with energy-momentum relation given by
the free particle Schr ̈odinger equation. The annihilation and creation operators
will be the fermionic ones, satisfying the canonical anticommutation relations
[aFpj,aF†pk]+=δjkimplying that states will have occupation numbersnpj = 0,1, automatically
implementing the Pauli principle.
36.3 Continuum formalism
The use of cutoffs allows for a finite dimensional phase space and makes it pos-
sible to straightforwardly use the Bargmann-Fock quantization method. Such
cutoffs however introduce very significant problems, by making unavailable some
of the continuum symmetries and mathematical structures that we would like
to exploit. In particular, the use of an infrared cutoff (periodic boundary con-
ditions) makes the momentum space a discrete set of points, and this set of
points will not have the same symmetries as the usual continuous momentum
space (for instance in three dimensions it will not carry an action of the rotation
groupSO(3)). In our study of quantum field theory we would like to exploit the
action of space-time symmetry groups on the state space of the theory, so need a
formalism that preserves such symmetries. In this section we will outline such a
formalism, without attempting a detailed rigorous version. One reason for this
choice is that for the case of physically interesting interacting quantum field
theories this continuum formalism is inadequate, since a rigorous definition will
require first defining a finite, cutoff version, then using renormalization group
methods to analyze the very non-trivial continuum limit.
If we try and work directly with the infinite dimensional space of solutions of
the free Schr ̈odinger equation, for the three forms of the Fock space construction
discussed in section 36.1.1 we find:
- The occupation number construction of Fock space is not available (since
it requires a discrete basis). - For the Bargmann-Fock holomorphic function state space and inner prod-
uct on it, one needs to make sense of holomorphic functions on an infinite
dimensional space, as well as the Gaussian measure on this space. See
section 36.6 for references that discuss this. - For the symmetric tensor product representation, one needs to make sense
of symmetric tensor products of infinite dimensional Hilbert spacesH 1