Particles in quantum electrodynamics
------------------------------------
Modern (renormalized) QED was created in 1943-1947 by Feynman,
Tomonaga and Schwinger, who received the 1965 Nobel prize for this
achievement. In 1951, Dyson analyzed their achievements, and related
the action-based formulations of QED by Feynman, Tomonaga and Schwinger
to the Hamiltonian form, in which the meaning of the QED formalism could
be interpreted along traditional lines.
Modern textbooks of quantum field theory tend to emphasize the
formalistic part that leads most quickly to computable results,
at the cost of failing to create a clear picture of the physical
content of the theory as it appears from a quantum mechanical point
of view. Therefore the following facts are not as well-known as they
deserve to be.
The conventional textbook treatment of QED involves meaningless
(''divergent'') intermediate expressions in the ultraviolet (UV)
and infrared (IR) limit. The UV divergences are rendered finite by a
procedure called renormalization, while the IR divergences are
controlled by a summation over infinitely many soft photons.
Both aspects invite the critique of those who want to understand
physics in terms of well-defined intermediate concepts only.
It is not so well known that both kinds of divergences can be avoided
completely by a careful approach to QED. In particular, the results of
G. Scharf,
Finite Quantum Electrodynamics: The Causal Approach, 2nd ed.,
Springer, New York 1995
for the UV regime, together with the results of
P.P. Kulish and L.D. Faddeev,
Asymptotic conditions and infrared divergences in quantum
electrodynamics,
Theoretical and Mathematical Physics 4 (1970), 745-757
for the IR regime, prove the existence of an exact S-matrix for QED
in the sense of a series
S(alpha) = sum_k alpha^k S_k
in powers of the physical fine structure constant, with well-defined,
exactly representable operator distributions S_k, defined as quadratic
forms on the space of asymptotic states. Explicit formulas for the S_k
are known for small k, and can in principle be generated for all k,
using diagrammatic techniques. For larger k, they can be written as a
sum of a finite but exponentially large number of terms, each one
involving integrations over more and more particle momenta.
However, since alpha is small, the first few terms are sufficient to
predict physical phenomena with astonishingly high accuracy.
The only aspect that is currently not known is which of the uncountably
many existing operator-valued functions S(alpha) that have the same
asymptotic series represents the true S-matrix of QED.
For those who don't have access to Scharf's book on QED,
let me recommend the paper
DR Grigore
Gauge invariance of quantum electrodynamics in the causal approach
to renormalization theory
http://arxiv.org/pdf/hep-th/9911214
It contains in 35 pages (including references and an exposition of
all concepts not familiar from standard QFT textbooks) a complete
proof of renormalizability of scalar and spinor QED (without taking
the infrared limit), based on the causal approach, without
encountering a single UV divergence anywhere on the road.
The physical, renormalized Hamiltonian is defined perturbatively
via Dyson's intermediate representation, or in more modern terminology,
via similarity renormalization. See
http://arxiv.org/pdf/hep-th/9806097 (and ref. 5 there).
A nonperturbative definition is not known at present. In particular,
no explicit form of the physical QED Hamiltonian can currently be
given, though approximations to any order are in principle computable.
This is typically done under the name NRQED. For Lamb shift
calculations in NRQED, see
T. Kinoshita and M. Nio,
Radiative corrections to the muonium hyperfine structure:
The alpha^2(Z alpha) correction,
Phys. Rev. D 53, 4909-4929 (1996).
The physical vacuum is the ground state of the renormalized QED
Hamiltonian in the rest frame of the system. The Poincare group
preserves the vacuum state, and its generators map it to zero.
In particular, the ground state energy is exactly zero, and
the vacuum is dynamically inert,
(The idea of a dynamical vacuum with virtual particle pairs popping in
and out of existence for very short times -- which often appears in
loose talk about the subject -- comes from thinking in terms of bare,
nonphysical particles.)
The physical (''dressed'') electron is the lowest eigenstate with
fermion number 1 of the renormalized QED Hamiltonian. Its deviation
from pointlikeness is described by its nontrivial form factor,
which gives rise to the anomalous magnetic moment and the Lamb shift.
The physical particles are the objects that are related by the
S-matrix elements computed in all treatments of QED. They are
represented by _external_ lines in Feynman diagrams with
_renormalized_ (''fully resummed'') propagators satisfying the
Dyson equation.
On the other hand, lines in Feynman diagrams corresponding to free
propagators correspond to free, pointlike (''bare'') particles,
which are physically meaningless and serve only to illustrate
the perturbative apparatus in terms of which the traditional
covariant calculations are done.