The polarization of radiation emitted by
distant radio galaxies and quasars offers a
way to search for chiral effects in the propagation
of electromagnetic radiation. Such objects are often elongated in
one direction, so that one may define a position angle 
which
describes the orientation of the object on the sky. Synchrotron
radiation can lead to a significant linear polarization of the source,
and the angle 
of the plane of
polarization may also be measured [1].
(The angle of polarization will typically undergo Faraday rotation, but this
effect can be removed by using the fact that Faraday rotation is
proportional to the square of the wavelength.) One can therefore
study the relative angle 
between the position and
polarization vectors, keeping in mind that this quantity is only defined
modulo 
. It has been found [2, 3] that
is not distributed randomly; there is a large peak at 
, and a smaller enhancement at 
.
Since many of these sources are at significant redshifts, and therefore
very far away, testing whether this relationship is maintained for
distant sources provides constraints on possible chiral
effects on the propagation of light through the universe, which could
rotate away from the intrinsic value (that which would
be measured at the source).
From a field theory point of view, the simplest such chiral effect
arises from a Lagrange density
Here, 
is the field
strength for the electromagnetic field 
, 
is the dual field
strength, and 
is a pseudoscalar field which need not be fundamental
(it can be a function of other fields in the theory). We set c=1
throughout. This Lagrangian is the simplest way to couple a neutral
pseudoscalar to electromagnetism in a parity-invariant way, and often
describes the effective coupling of pseudoscalar particles (such as
pions or axions) to photons.
Our interest here is in the case where varies only very slowly
over extremely large distances. In that case an electromagnetic wave
travelling through the background field will undergo a rotation
in its polarization state which depends on the change in ; such
an effect arises in a variety of contexts [4]-[18].
In the WKB limit where the length scale for variations in is
much larger than the wavelength of the photon, the polarization
angle obeys the simple relation
where 
indicates the change between source and observer.
(Here, and in Eq. (3) below, 
is measured
in radians; elsewhere we measure all angles in degrees;
no confusion should arise.) This effect is independent of
wavelength, and can therefore be distinguished from ordinary
Faraday rotation.
Carroll, Field and Jackiw [10] suggested that observations of
polarized radio sources provide a stringent test of such an effect,
since they afford an opportunity to constrain
over a large interval in space and time (see also
[19, 20]).
The specific model investigated in [10] set 
, where 
is a 4-vector whose expectation
value parameterizes violation of Lorentz invariance (as well as
CPT [21]). It was
hypothesized that there exists a preferred coordinate frame,
close to the background Robertson-Walker frame of our universe, in
which 
. This implies that the predicted
rotation of the polarization angle for a source at redshift z
is given in terms of the timelike component 
and the
spacelike vector 
by
where 
, 
is the angle
between and the direction
toward the source, and r is the proper spacelike distance
traveled. If we take a flat (k=1) universe
as a reasonable approximation, we have
where 
is the Hubble constant today. Regardless of whether or
not one is interested in tests of Lorentz invariance, Eq. (3)
is a useful parameterization of potentially observable chiral
effects.
In [10] it was shown that the radio galaxies at redshift greater
than 0.4, with maximum polarizations greater than 
, were strongly
clustered around , using a sample of
galaxies and redshifts obtained from the literature [2, 3, 22, 23].
Assuming that the timelike component would
be significantly larger than the spacelike part , the limit
was obtained, where 
km/sec/Mpc. Recently, Nodland
and Ralston [24], using the same set of data,
searched for anisotropic effects such as those
that would arise from a nonzero spacelike part in
Eq. (3). Surprisingly,
they claimed to find a significant signal in the data. Given the
fundamental importance of such a result, we have undertaken a
re-examination of the data, and present our results in this paper.
We conclude that the data are most consistent with
no effect, contrary to [24]. Our disagreement stems primarily
from the method used to disentangle the ambiguity in
the quantity , and the use of randomly generated data
for comparison purposes, as will be shown below.
As this manuscript was being completed we received a preprint by Eisenstein and Bunn [25], who come to conclusions similar to those expressed in this paper.