Terrestrial and Extraterrestrial Limits on The Photon Mass

~ ~ qp 4 il. VOLUME 43, NUMBER 3 Jur. v 1971 '. .'errestria. . ani. '. 'xtraterrestria. '. imits on '. . '. xe '. . ~oton '. V. :ass 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