~ ~ qp
4 il.
VOLUME 43, NUMBER 3 Jur. v 1971
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.'errestria. . ani. '. 'xtraterrestria. '. imits on
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AI.FRED S, GOLDHABKR*
Institute for Theoretica/ Physics, State University of Nezo York at Stony Brook, Stony Brook, Near York 11790
MICHAEL MARTIN NIETO) $
The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
Departmertt of Physics, University of Catiforrtia, Sartta Barbara, Cabforrtia 93106
We give a review of methods used to set a limit on the mass p of the photon. Direct tests for frequency dependence
of the speed of light are discussed, along with more sensitive techniques which test Coulomb's Law and its analog in
magnetostatics. The link between dynamic and static implications of finite p, is deduced from a set of postulates that
make Proca's equations the unique generalization of Maxwell's. We note one hallowed postulate, that of energy con-
servation, which may be tested severely using pulsar signals. We present the merits of the old methods and of possible
new experiments, and discuss other physical implications of finite p, . A simple theorem is proved: For an experiment
confined in dimensions D, effects of finite p, are of order (ttbD)' —there is no "resonance" as the oscillation frequency ~
approaches p, (h, =c=1).The best results from past experiments are (a) terrestrial measurements of c at diRerent fre-
quencies
@&2&(10"g=—7&(10 ' cm '=—10 ' eV;
(b) measurements of radio dispersion in pulsar signals (whistler effect)
@&10 '4 g=—3&(10 ' cm '=—6)&10 "eV;
(c) laboratory tests of Coulomb's law
@&2&(10 47 g—=6)&10 '0 cm i=—10 i4 eV;
(d) limits on a constant "externaV' magnetic Geld at the earth's surface
p, &4)& 10 4' g=—10 "cm '=—3&( 10 "eV.
Observations of the Galactic magnetic field could improve the limit dramatically.
I. INTRODUCTION
One of the great triumphs of classical physics was the
formulation of the Maxwell electromagnetic field
equations. A fundamental prediction of these equations
is that all electromagnetic radiation in vacuum travels
at a constant velocity c. The most recent experiments
have confirmed this prediction with an accuracy near to
one part per million, over a wide range of frequencies
(Froome and Essen, 1969;Taylor, Parker, and Langen-
berg, 1969).
In the context of quantum theory, a relativistic,
quantized electromagnetic field of frequency v is
recognized as an assembly of photon particles with
*Supported in part under the auspices of the United States
Atomic Energy Commission.
f Supported in part by the National Science Foundation.
f Address after September 1, 1971: Department of Physics,
Purdue University, Lafayette, Ind. 47907.
energy hv. These light quanta travel with velocity c,
and hence have zero rest mass. The success of quantum
electrodynamics in predicting experiments to six or
more decimal places has made the massless photon a
tacit axiom of physics. A sign of this is that as late as
1968 the Particle Data Group tables gave experimental
limits on the neutrino masses, but just a zero for the
photon mass (Rosenfeld, et al. 1968) .This is not too sur-
prising since QED is our only "exact" quantum theory.
Nuclear and particle quantum theories do not even
approach such accuracy.
The tacit axiom of masslessness corresponds to the
belief that if the photon has an effective mass p, it does
so only because it is slightly off the mass shell. Using an
uncertainty argument, we would estimate
ts~h/(At)c'=3. 7X10 ' g/T,
where T is the age of the universe in units of 10'0 years.
277
Copyright 19/1 by the American Physical Society
278 RFvrzws oz MoDERx PHvsics ~ JvLv 1971
Alternatively, one could get a similar number, following
de Broglie (1954)' by considering a spherical de Sitter
cosmology. In this model the cosmological constant E
is given by the two equations
wave by
E= Re Eo exp $—i(~t —k.x)],
H= Re Ho exp [—i(&dt —k x)], (2.1)
or
K= 3/(cT) ' K= kPpc/fi]2,
p. = 6"%/Tc'. (1.3)
II. ELECTRODYNAMICS WITH FINITE p
A. Heuristic Discussion
The assertion of a definite nonzero photon mass is
equivalent to the specification of a, free-electromagnetic
'A remarkably similar discussion eras given by Cap {1953).(See also Marochnik, j.968).
Fquations (1.1) and (1.3) give an ultimate limit for a
meaningful experimental measurement of the photon
mass.
Since the time of Cavendish, certain critical physicists
have not been satisfied with speculative assertions on
this subject, and have periodically re-examined the
question (or an equivalent one in the language of their
time) to determine what valid experimental limit could
be placed on the photon mass. In this paper we shall
give a review of methods devised to improve the limit.
In Sec. II, we develop the theory of classical electro-
dynamics from postulates of special relativity, plus the
assumption of a well-defined, locally conserved energy
density associated with electromagnetic fields. We
indicate how this assumption can be tested with pulsar
signals. We proceed in Sec. III to discuss limits that
have been set on the mass by terrestrial methods. These
include determinations of the constancy of the velocity
of light for all wavelengths, and testing the exactness of
Coulomb's Law. The latter method yields the best
laboratory mass limit to date, p(2&& 10 4~ g.
In the next section extraterrestrial methods are
reviewed. The first method is a variation on the terres-
trial velocity of light experiments. Dispersion in the
speed of starlight is inferred from the difference in
arrival times of different colors of light from the same
astronomical event. We then discuss the limits that can
be obtained by studying the effects that a massive
photon would have on the earth's magnetic field. This
yields the lowest limit to date, p(4&10 ' g. Another
technique considered is the study of long period
hydromagnetic waves in plasma. If the photon has a
finite mass, then such waves are damped below a critical
frequency depending on p, and the plasma characteristics.
In the next section the physical eff'ects of longitudinal
photons are derived. We close in Sec. VI with a dis-
cussion of possible future experiments, their efficacy
in improving present limits, and the physical implica-
tions of the results.
E,i= (~/pc) Eoi rest,
Hi —(k/p) XEoi rest,
Hot l = Eo( t rest. (2 4)
If p is much smaller than
l
k, the field of a longitudi-
nal (ll ) photon will be smaller than that of a transverse
(J ) photon by the factor yc/co. Since power absorbed
by electric charges is proportional to E', we infer that
scattering cross sections of longitudinal photons will be
suppressed compared to those of transverse photons by
a factor (pc/~)'; this weak coupling explains how the
longitudinal polarization, if it exists, could have escaped
detection up to the present. The phantom longitudinal
photon is the second consequence of nonzero p.
Finally, we consider the limit of static fields. For
these fields, wehavecu = (k'+ p') 'I'= 0, implying
l
k
l
= ip,
hence, exponential decay of static fields with a range
p '. This behavior is familiar from Yukawa's model for
interaction of nucleons through pion exchange. The
exponential deviation from Coulomb's law, and its
magnetic analog, provide the most sensitive current
test for a photon mass. In the next section we find the
postulates required to link this third effect rigorously
with the previous consequences of finite p.
(~/~) ' —k'= ~' (2 2)
where the last line defines p in units of wavenumber, or
inverse length. Standard arguments (Goldberger and
Watson, 1964) then yield the desired expression for
group velocity of a wave packet
cl k /&u=c
I
kl /(k+& )
—g(~2 ~2g2) 1/2/~ (2 3)
This expression corresponds to a frequency dispersion
of the velocity of light, the first and most direct con-
sequence of a finite photon mass. LNote that here and
in what follows, giving p in units of wavenumber is
using units of c/6. ]
Going to the Lorentz frame in which the photon is at
rest, i.e., k=0, we see that there must be three in-
dependent polarization directions for a massive photon,
since the plane tra, nsverse to k is undefined in this
frame. The argument fails for a massless photon because
it can never have k=0. In the photon rest frame the
electric field energy density E' is proportional to photon
intensity. However, the well-known law of Lorentz
transformations tells us that the fields in a frame with
photon frequency ~ and momentum k will be very
different for photons polarized J or ll to k (Jackson,
1962; unreferenced assertions on electromagnetism in
this paper may be found in Jackson's book):
GoLDHAaER AND NzKTo Limits on the Photon Mass 279
B. Deductive Approach
We adopt the following postulates:
(1) The electromagnetic field is defined through its
action on a test charge q by the Lorentz force law,
F= qPE+ (v/c) XHj. (2 3)
This law determines the behavior of E and H under
Lorentz transformations: they may be identified as
independent components of the antisymmetric 4-tensor
Ii
gaby
If we ignore parity-violating terms as required by
Postulate 3 above, we may write Eq. (2.11) more
simply as
F p(k) = —iD(k)(k Jp —kpJ ), (2.12)
where D is an invariant function of k, and the right-
hand side is the most general antisymmetric tensor built
out of J and its derivatives, i.e.
,
linear in J and an
arbitrary function of k . Thus, the requirements of
Poincare invariance (including parity) are su%cient
to deduce the homogeneous Maxwell equations, which
may be written
(2 6) k e p,)F&'(k) =0, (2.13)
The force law in standard notation becomes
dpp/dT = gs F~p. (2 7)
This latter requirement is applied to assure invariance
of the theory under the transformations of special
relativity. The quantity D p~» must be an invariant
tensor. There are only two possibilities:
D-w~~(x) = D(x) (a-.a~~ g-Cu. )—+D( )~x-~.~, (2 9)
where e is the completely antisymmetric 4-tensor. The
presence of D implies parity violation or magnetic
sources, depending on the point of view. The reason is
that D produces a pseudovector E field, and a vector
H field.
(3) We shall assume there are no magnetic sources
or parity-violating terms in the theory. This eliminates
terms like D.
(4) Finally, we insist that the dependence of the
theory on a small photon mass, p, be such that as p,—+0
there is a smooth transition to the Maxwell theory.
It is easiest to find the consequences of these postulates
in "momentum space". Define (k a 4-vector)
F p(k) = f d4x exp (ik.x)F p(x),
D p),g(k) fd4x exp=(ik.x)D pi„p(x),
J (k) = f d'x exp (ik x)J (x). (2.10)
(2) The electromagnetic field at point x in space-
time is linear in the charge and current densities, and in
the derivatives of these densities, all evaluated at
earlier points x. Further, this linear relationship is
Poincare covariant (translation invariant and Lorentz
covariant):
F p(x) = f d'xD pi, g(x x') B,J)(x—')
+ terms with higher derivatives. (2.8)
and are obviously satisfied by the above form Eq.
(2.12). To state this another way, we have now shown
from invariance requirements alone that the fields
may be derived from a 4-vector potential:
F,&(k) = ilk A—e(k) —keA (k)],
A (k) =D(k)J (k). (2.14)
Next, we study the properties of D(k). Since D is
Lorentz invariant, we shall assume that it is a function
only of the invariant quantity k'—=k k, even for
complex k, giving D(k) =D(k'). This can be proven
from our postulates. ' Let us consider k=0. The condi-
tion D(t(0) = 0 implied by Postulate 2 in turn
implies that if the inverse Fourier transform D(t) =
(2~) 'f d'k exp( ik x)—D(co, k=O) exists, then
D (u, k = 0) is a,nalytic in the upper-half complex ~
plane. Further, the requirement that D is real implies
D(~) =D*(—&o*). Translated into the variable k'=
~'/c' —lr'=aP/c these results imply that D(k') is ana-
lytic in the entire complex k' plane except for the posi-
tive real k' axis, and any discontinuity across this axis is
imaginary. Unless there is a purely local current —current
interaction, D(k') must go to zero as k' goes to infinity.
We exclude the local interaction since it is not present
in the Maxwell theory.
We then may use Cauchy's theorem to write a dis-
persion relation for D by integrating over its imaginary
discontinuity
"dp' fm D(p')
p,' —k'
(2.15)
If Irn D has a delta function, then D has a pole.
Before considering the most general case, let us
specialize by assuming Im D consists of a single delta
function at a particular value p,', giving
( —k'+p')F p ——(4~/c) ( i) (k Jp ——kpJ )
or
Then, the convolution integral Eq. (2.8) becomes
F p=D p/ g( —ik&)Jg
(&+ ') F = (4 /c) (8 J BJ ) . — (2.16)
+ terms with more factors of the 4-vector k. (2.11)
' This can be shown as a trivial example of the discussion in
Streater and Wightman (1964).
280 REVIEWS OF MODERN PHYSieS ~ JVI.Y 1971
F p ——BAp —BpA. (2.17)
Rewriting further gives us the famous Proca equation
(Proca, 1930a, b, c; 1931;1936a, b, c, d, e) for a massive
vector field coupled to a conserved current,
c1 F p+p'Ap (4ir/c)——Jp,
P p ——BAp —BpA. (2, 18)
The whole effect of finite photon mass is to introduce
at each point x a spurious current proportional to the
vector potential and, therefore, a function of the true
current at many earlier points x'. In three-dimensional
notation the massive Maxwell equations become
This may be recognized as the ordinary Maxwell
equation, modified by the addition of p,' to the
D'Alembertian operator. (The free (J =0) solutions of
this equation obey the relation cv/c= (p'+Ir')"'.) We
may rearrange Eq. (2.16) by introducing the vector
potential A satisfying
(~+ti')A = (4ir/c) J,
8 J =0
with
and
P= f d'x(pEM+p „„,),
(dp/«) .«"=t E+(J/c) xH,
(2.23)
(2.24)
the Lorentz force density.
The vector potential is never measured directly, but
it is determined uniquely, and is required for con-
struction of a locally conserved electromagnetic energy
and momentum density.
Let us elevate the principle just mentioned to a
fif th postulate:
(5) There exists a locally conserved energy —momen-
turn density, such that the total energy and momentum
of a system of charges and fields is conserved.
We shall now consider the restrictions implied by this
postulate on Im D(p') .
Clearly, a minimal requirement on Im D(p') is that
it be integrable, i.e., a bounded continuous function
falling faster than 1/ln ti' at high masses, plus a sum of
delta functions and derivatives of delta functions.
Therefore, D(k2) will be a sum of pole terms
fZ d*/(t "—k") }
V E=4vrp —p,'V,
V x E= —(1/c) (BH/Bt),
V H=O
~ x H = (4ir/c) J—ti'A, (2.19)
8a M ——LE'+H'+ ti'(A'+ V') )/8ir,
pEM ——LE x H+ti'VA)/4irc, (2.20)
where the conservation we refer to is the equation
of continuity
(1/c) (BGFM/Bt) +V pEMc =0. (2.21)
When charges and currents are present we obtain
dP/dt=0, (2.22)
with A and V the space and time components of the
4-vector potential 3„.
It is worth noting that the freedom of gauge in-
variance found in conventional electrodynamics is
completely lost here. First of all, the Lorentz gauge
must be used, i.e., 0 A =0. Within that restriction, one
might imagine adding to A a term 8 A, where A is a
scalar function. This does not change F p, of course,
but the Lorentz gauge condition implies A. =0.
Therefore, if -3 is already a solution of the Proca
equation we have the contradictory requirements
C]cj A=O and ( +ti')cj A=O, satisfied only if h. is
constant. Hence, all freedom of gauge change is lost.
It is easy to verify, for free fields, that there exists a
coriserved energy —momentum density (de Broglie,
1957; Bass and Schrodinger, 1955) such that
plus a continuous integral over pole terms (a cut)
( fLd (ti~) /(p~ —k2) ])
plus second or higher order poles Pd/(ti' —k')', etc.).
All these terms can be written as simple poles or limits
of sums of simple poles.
Consider the case of two pole terms (D= di/(pi' —k') +
d&/(ti22 —k')). This leacls to the possibility of arbitrary
free fields with either ~=c(t'ai'+ k') "'or &v =c(p '+ k') "'.
Take the case k=O. One may have an electric field
E=Eo(cos tiict cos ti2ct) with A = Eo(p2 sin pact
p&
' sin tiict). At t=0, both F p and A are zero every-
where, so that any energy density quadratic in F and A
mulct vanish. However, an instant later this is no
longer true. Therefore, there is no conserved electro-
magnetic energy built simply from F and A. For free
fields, a conserved energy density can be constructed
by projecting the parts of E corresponding to each mass
Ei= L(t 2'+ 0)/(t ~' —t i') )E,
E.=I:(t i'+&)/(t i' —t 2'))E (2.25)
With the obvious definitions of AI and A~, etc. , we
get the conserved energy density
8irg= c,kEi'+Hi'+ti, '(Vi'+Ai') )
+cqLE2+Hp+ti2 (V2+A2 )). (2.26)
In the presence of sources, however, our arbitrary but
simple definition of 8 may be seen to fail. For example,
by calculating the potential energy of a charge dis-
tribution and comparing it with the total electro-
magnetic energy E one finds that the two are not equal.
The only way to maintain energy conservation is to
GoLDHABER AND NIETo LAnits on the Photon Muss 281
insist that the fields associated with p~ and p~ are
independent contributors to the energy, even though
there is no general operational distinction, between
them. In particle language, we would say there are two
different photons, though they act on charges in the
same way.
Once this is accepted, it is straightforward to deduce
(d/dt) f d'xg(x) = —I d'xJ (cidiEi+c2AE2). (2.27)
In order that total energy be conserved, this must
balance the effect of the I.orentz force on charges. This
means that
beginning at p,'=0 but very small below p,'=m' and
suppressed at least by 0.', is produced by the dis-
sociation of a virtual photon into three correlated
photons. These cuts are not associated with free
photonlike degrees of freedom and do not violate our
earlier conclusion forbidding a continuous mass photon.
It is amusing to consider in this classical context the
modified electrodynamics of Lee and Wick (1969). In
order to eliminate the small distance divergence in
Maxwell's theory and its quantized version, they
introduce a D with two poles; one at zero mass, and
one at very large mass with
cidi=+ 1, cpl2=+ 1. (2.28) di ———d (2.29)
If G(x) is positive definite, (c,&0), then the residues d,;
must both be positive. This excludes higher order poles,
which are obtained in a limit as simple poles with
residues of both signs approach each other. Another
way to express the difficulty with higher order poles is
to observe that they lead to fields which grow in time,
e.g. , Eot cos pt for a second-order pole at p, . This is a
solution of (C]+p')'E=O. A cut in D(k2) may be
produced as a limit as the number of poles in a certain
interval diverges and the residue d, of each pole goes to
zero. From Eq. (2.28) this means that the coefficient
c; of the corresponding field energy density diverges,
so that in the limit 8(x) is undefined. Thus, it is im-
possible to produce a cut by exciting an infinite number
of photonlike degrees of freedom, and still preserve
energy —momentum conservation: a "continuous-mass"
photon is excluded.
If Postulate (5) holds, we may introduce one or n
new poles in D at a price of the admission of one or
e new photons each with three degrees of freedom. This
would contradict well-k. nown information about black-
body radiation (de Broglie, 1957;Bass and Schrodinger,
1955), and elementary particle reactions (Brodsky and
Drell, 1971) unless either the new photons all have a
mass greater than many GeV, or else their coupling to
charge d; is so small that their degrees of freedom are
not appreciably excited during times of practical
interest. In either case, their existence would have no
significant eRect on a search for eRects of a possible
finite mass of the everyday photon. In fact, there are
known weak cut contributions to D derivable in
quantum electrodynamics and indeed, associated with
new degrees of freedom. For example, at values o
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