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What about infrared divergences?
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Renormalization theory deals with the regularization of ultraviolet
divergences, occuring at very high but unobservable energies.
In contrast, infrared divergences arise if there are problems at
very low energies. They are not cured by renormalization and need
completely different techniques.
Theories without massless particles have no infrared problems at all,
since at low energies only few particles can coexist. Indeed,
the sum of the rest masses of physical particles is bounded by the
total energy of the system.
In QED one has infrared problems because the photon is massless,
so a bound on the sum of the rest masses does not limit the number of
possible photons. indeed, a closer calculations shows that there
may be an arbitrary number of very low energy ('soft') photons.
One can handle the situation in some approximation by giving the
photon a tiny mass mu. But this is an _additional_ parameter, quite
different from the renormalization scale M. And the renormalized
theory at finite mu depends on mu (so that one needs to take
in the end the limit mu to 0 to get physically correct results),
while it is still independent of M.
A better way to handle the infrared divergences is to avoid them
completely by using coherent states. These sum the contributions of
arbitrarily many soft photons in a coherent way. See
N. Papanicolaou
Infrared problems in quantum electrodynamics
Phys. Reports 24 (1976), 229-313.
See also:
G. Scharf
On a construction of the S -matrix in QED
Il Nuovo Cimento A 74 (1983), 302-324.
and
H.P. Stapp
Exact solution of the infrared problem
Phys. Rev. D 28, 1386-1418 (1983)
For nonabelian gauge theories, see also
Thomas Appelquist and J. Carazzone,
Infrared singularities and massive fields,
Phys. Rev. D 11, 2856-2861 (1975)
and
D.A. Forde,
Infrared Finite Amplitudes
Ph.D. thesis, Durham, UK, 2004.
http://www.ippp.dur.ac.uk/Research/Theses/forde.ps.gz
The QED Hamiltonian is the limit Lambda-->inf of the family of
Hamiltonians H_Lambda that include a finite cutoff energy Lambda
and counterterms depending on Lambda in the way prescribed by
renormalization theory.
From a functional aanalysis point of view, the infrared problem is not
a problem of this Hamiltonian but of the wave functions to which it is
applied. The QED Hamiltonian is an unbounded operator, and unbounded
operators are always defined only on their domain, which is a proper
subspace of the Hilbert space defined by the physical inner product.
The massless Fock states are not in the domain of the QED Hamiltonian.
This gives divergences as for all other operators applied outside
of their domain.
On the other hand, photon coherent states are in the domain of the QED
Hamiltonian, and - as the many applications in quantum optics show -
do not suffer from infrared problems.