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S6b. The formal functional integral approach to QFT
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On a purely formal level (i.e., with power series in place of actual
numbers), 4D QFT is very alive and useful. It is now almost
always based upon functional integrals.
The path integral is discussed e.g., in Weinberg I, Chapter 9, or
Peskin/Schroeder, also Chapter 9. As one can see there, the
path integral formalism involves no operators at all, only classical
(commuting or anticommuting) fields.
The quantities obtained in the expansion of the path integral in
powers of hbar are time-ordered vacuum expectation values.
Since the original ordering in a time-ordered vacuum expectation value
is immaterial (apart from a sign for fermions), the same must be the
case for the path integral itself, which explains why the fields
in the path integral are classical (i.e., commute or anticommute
at all arguments).
The main strength of the path integral approach is precisely that
it avoids quantum operators and replaces all operator arguments by
averages over classical paths. (The main weakness is that this
averaging process is logically ill-defined.
There exists no prescription how the limit in the ``definition'' of
the path integral is to be taken to yield (in theory - independent
of the difficulty of computing them) numbers that have the properties
commonly ascribed to the path integral.)
The older canonical quantization approach was fraught with difficulties
because of inconsistencies in the operator approach.
For example, the canonical commutation rules (CCR) are
valid only in the free case, and no one knows how they should
be in the interacting case - though one knows that (anti)commutators
must still vanish at spacelike related arguments.
Moreover, the renormalization program plays havoc with operators.
Unfortunately, this means that dynamical isssues and bound states
questions, which are comparatively easy to handle in an operator
framework, become almost intractable in the path integral approach.
However, as Weinberg stresses in his QFT book, an understanding of
the relation between path integral and canonical quantization is
essential to get the properties of the latter correct in cases like
the nonlinear sigma model.