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S8g. Nonrelativistic quantum field theory
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The right way to understand relativistic QFT is to regard it as
a limit of nonlocal nonrelativistic quantum field theory.
The latter is much better behaved.
Interacting QFT in 3+1 dimensions exists, however, as a rigorous
mathematical theory in the nonrelativistic case, since there only
finite renormalizations are needed and no infinities occur.
In this context, Feynman-Dyson perturbation theory can be given a
rigorous meaning. Note that nonrelativistic QFT is nonlocal
because of the Coulomb potential interaction.
Interacting QFT based on Feynman-Dyson perturbation theory
in 3+1 dimensions exists as a rigorous mathematical theory
in the relativistic case, as a limit of smeared, nonrelativistic
theories. This is done for Phi^4 theory in all details in
Salmhofer's book. For technical reasons, one gets the results
however only in a very weak topology corresponding to power series
in the coupling constant, rather than as true functions of the
coupling constants. Thus perturbative relativistic QFT is rigorously
established in 4D while nonperturbative relativistic QFT in 4D
is still elusive.
However, the infinities that plague 4D relativistic QFT are already
present in 3D, and there rigorous construction have been given.
Exactly the same kind of renormalization tricks are used in 3D.
Thus our present lack of understanding cannot be blamed on
renormalization, but has to do with the difficulty of getting
the hard analytical estimates needed to justify the constructions.