Natural Science Forum / Physics / Research / August 2008

What you find there is of course more about "scalar electrodynamics
with a massive photon", i.e., it's a massive vector field coupled in
the minimal way to a scalar field. This theory is used as the most
simple effective model for the charged pions and rho-mesons and then
assuming "vector-meson dominance" for the electromagnetism of pions,
leading to a nice description of the em. form factor of pions etc.
Nowadays this is mainly of interest because of ultra-relativistic
heavy-ion physics and the measurement of lepton-antilepton pairs (e^+
e^- or mu^+ mu^-) pairs emerging from the hot and dense fireball
created in such collisions. The standard review about this is

R. Rapp, J. Wambach, in Adv. Nucl. Phys 25, 1 (2000)
http://arxiv.org/abs/hep-ph/9909229

The famous vector-dominance model mentioned above goes originally back
to J. J. Sakurai. The proof of renormalizability and the explicitly
gauge-invariant formulation can be found in

Kroll, N. M., Lee, T. D., and Zumino, B., Phys. Rev. 157, 1376 (1967)
http://link.aps.org/abstract/PR/v157/i5/p1376

Of course you can use the model as one for electrodynamics with a
massive photon. Instead of coupling a scalar field you couple a Dirac
field in the usual minimal way.

The auxiliary (real) scalar field is a ghost in the sense that it
decouples from the rest of the world, i.e., it does not interact with
any of the observable particles. I called the Stueckelberg ghost in
the preprint. Its purpose is to make the theory manifestly gauge
invariant. To keep the gauge invariance of course the coupling to the
matter fields (i.e., the pion field in my paper or an electron field
for massive QED), one has to couple the massive vector field to
conserved currents and then the demand to build an (at least
superficially) Dyson-renormalizable theory leaves you with the
minimal-coupling scheme.

The rest is then straight forward: You fix the gauge in the
path-integral formula for the generating functional of (connected)
Green's functions, thereby introducing another pair of ghosts
(analogous to the Faddeev-Popov ghosts in the non-abelian case, in
the context of QED or more generally abelian gauge theories also
known as the Feynman ghosts) which are Grassmann-valued, however
scalar fields. In the gauge choosen in the paper (again strictly
analogous to 't Hooft's R_xi gauge for Higgsed non-abelian theories),
the Stueckelberg and Feynman ghosts stay free fields, i.e., they do
not couple to all the other fields and do thus not lead to observable
particles. The particle content of the theory becomes most obvious in
the socalled "unitary gauge", which is given for xi->\infty. In this
limit the Stueckelberg and Feynman ghost together with the scalar
(i.e. unphysical!) part (four-longitudinal component) of the vector
field masses become infinite and thus decouple. You are left with the
three physical (four-transverse) components of the vector field and
the matter fields (i.e., pions in the model discussed in the paper or
electrons for the massive-QED model). The rest of the paper deals
with the proof that all this holds true to any order of perturbation
theory and that the model is not only superficially but really Dyson
renormalizable, making use of the Ward-Takahashi identities from the
gauge invariance of the model. This proof, much simplified compared
to the Kroll-Lee-Zumino one) is very much the same as for QED (or
non-abelian theories in background-field gauge).

The Stueckelberg and Feynman ghosts become only relevant in the
finite-temperature QFT of this model. In the vacuum you don't need to
consider the ghosts since they do not interact. So your Feynman rules
deal only with massive vector fields and the matter fields since the
non-interacting ghosts do not appear in the S-matrix (neither in
loops for radiation corrections). At finite temperature the ghosts
however provide the correct counting of field-degrees of freedom in
the ideal-gas limit (i.e., for the non-interacting model):

You start with a four-component vector field of which the scalar piece
has to be separated out (this is bourne out of the formal
representation theory of the Poincare group, as you can find in
Weinberg's quantum theory of fields or my QFT script

http://nucleus.physik.uni-giessen.de/~hees/publ/lect.pdf
).

In the here discussed model that's done as follows:

First you add the Stueckelberg ghost, making together five bosonic
degrees of freedom. As you can read off the free propagators of these
fields, you get three fields of mass m (the mass of the physical
vector mesons) and 2 of squared mass xi m^2. That's another way to
see that these must not become physical since the appearance of xi
indicates that these fields are gauge dependent. However, you also
have to introduce the Feynman ghosts which can be seen as two scalar
Grassmann fields, i.e., scalar fields with Fermi statistics which
again indicates from another point of view that these fields must
never be coupled to the rest of the particles since they cannot
describe physical particles either, because this would violate the
spin-statistics theorem which states that in a local, micro-causal
theory with stable vacuum state scalar fields (more generally, any
field with a integer-valued spin) must be quantized with commutation
rules, i.e., must be bosons.

Now comes the magic: if you calculate the partition sum (e.g., with
the path-integral formalism), the two Feynman ghosts which also have
a squared mass xi m^2. Thus in the partition sum the contribution
from the Feynman ghosts precisely cancels the contributions from the
Stuecelberg ghost and the unphysical scalar (four-longitudinal)
degree of freedom of the vector field, leaving you with the partition
sum of three bosonic degrees of freedom of squared mass m^2 from the
vector fields as it must be. The partition sum becomes independent of
the gauge parameter as it must be since it must be a
gauge-independent quantity since one can derive observable
thermodynamic quantities from it.

Again, this is very parallel to the thermodynamics of massless photons
(Planck's radiation law derived in covariant Lorenz (or Feynman)
gauges): Also there you start with a four-component vector field, of
which now two components are unphysical since a massless vector
particles has only two physical degrees of freedom.

You can see the reason in two ways:

(a) In classical electrodynamics the Lorenz gauge is not sufficient to
fix the gauge of the free em. field, but there is a restricted gauge
freedom left, namely you can add the four-gradient of the scalar
field which obeys the wave equation, and you can fix this in the way
to make the time component of the vector potential vanish (the
socalled radiation gauge). The remaining three-vector potential has
to be three-transverse due to the Lorenz-gauge condition with
vanishing time component of the vector field, i.e., only the two
three-transverse three-vector components are physical. Now you can
build a unique Hamiltonian with these two physical degrees of freedom
and quantize the fields canonically (which is a pretty handwaving way
anyway).

(b) You look at the representation theory of the Poincare group.
Again, it turns out that a massless vector field has only two degrees
of freedom. The most natural way is to introduce the helicity as
intrinsic quantum number, and for a massless vector field this must
have the values +1 or -1 (corresponding to right- and left-circular
polarized em. waves). It also turns out that the only way to
represent massless vector particles with a local field is as a gauge
field, i.e., in the massless case one has to use (at least an
abelian) gauge theory if one doesn't get trouble with the basic
principles of local QFT.

Coming back to the thermodynamics of the free photon field: You start
with the four degrees of freedom of the vector potential. Then you
have to fix the gauge and therefore introduce the two Feynman ghosts
which (in this abelian case) turn out to be not interacting. These
two Feynman ghosts cancel the two (four-longitudinal and time-like)
unphysical degrees of freedom of the vector field in the partion sum,
leaving you with two bosonic degrees of freedom and a
gauge-independent partition sum as it must be.

--
Hendrik van Hees                        Institut für Theoretische Physik
Phone:  +49 641 99-33342                Justus-Liebig-Universität Gießen
Fax:    +49 641 99-33309                D-35392 Gie=DFen
http://theory.gsi.de/~vanhees/faq/