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Eur. Phys. J. D 61, 531–550 (2011) DOI: 10.1140/epjd/e2011-10508-7

THE EUROPEAN PHYSICAL JOURNAL D

Colloquium

A survey of existing and proposed classical and quantum approaches to the photon mass G. Spavieri1 , J. Quintero1 , G.T. Gillies2,a , and M. Rodr´ıguez3 1 2 3

Centro de F´ısica Fundamental, Universidad de Los Andes, 5101 Mérida, Venezuela Department of Physics, University of Virginia, Charlottesville, VA 22904-4714, USA Departamento de F´ısica, FACYT, Universidad de Carabobo, Valencia, Venezuela Received 14 September 2010 / Received in final form 20 December 2010 c EDP Sciences, Societ` Published online 25 February 2011 – a Italiana di Fisica, Springer-Verlag 2011 Abstract. Over the past twenty years, there have been several careful experimental, observational and phenomenological investigations aimed at searching for and establishing ever tighter bounds on the possible mass of the photon. There are many fascinating and paradoxical physical implications that would arise from the presence of even a very small value for it, and thus such searches have always been well motivated in terms of the new physics that would result. We provide a brief overview of the theoretical background and classical motivations for this work and the early tests of the exactness of Coulomb’s law that underlie it. We then go on to address the modern situation, in which quantum physics approaches come to attention. Among them we focus especially on the implications that the Aharonov-Bohm and Aharonov-Casher class of effects have on searches for a photon mass. These arise in several different ways and can lead to experiments that might involve the interaction of magnetic dipoles, electric dipoles, or charged particles with suitable potentials. Still other quantum-based approaches employ measurements of the g-factor of the electron. Plausible target sensitivities for limits on the photon mass as sought by the various quantum approaches are in the range of 10−53 to 10−54 g. Possible experimental arrangements for the associated experiments are discussed. We close with an assessment of the state of the art and a prognosis for future work.

1 Introduction One of the central issues and challenges of modern physics involves understanding the origins and properties of elementary particles such as the neutrino, photon, graviton, axion, and so on, which either mediate the fundamental forces of nature or are otherwise involved in explaining them. In particular, physicists seek to understand why they have their specific values of mass, charge and spin, and ultimately, the basis for their existence. Besides gluons, which are not observable as free particles, photons and gravitons are the only known particles whose rest masses may be exactly zero, and the story of the extensive search for their masses has been reviewed by a number of authors, including Goldhaber and Nieto [1,2], and Tu et al. [3]. The search for the mass of the photon has long been of special interest. It began in the time of Newton and Gauss, continued through the eras of Maxwell, Einstein and Proca, and remains an area of very active research today. The notion of a non-vanishing rest mass of the photon, mγ , is often tied conceptually to the possibility of a deviation from exact inverse-square behavior in Coulomb’s a

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law and with long-range, low-frequency deviations from Maxwell’s electrodynamics. This, in turn, has stimulated several increasingly elaborate experimental approaches for determining the value of, or upper bound on, mγ by probing for such departures. Moreover, for the work on the photon mass, history has favored classical methods based mostly on searching for deviations from infinite range electrodynamics, as foreseen by Proca’s equations [4–6] where, as we shall see later, the general Coulomb potential takes on the more specific Yukawa form. Increasingly sensitive null experiments, using essentially static fields and providing ever more stringent bounds on the size of any possible mass, have been developed through several decades. Even so, the search for a finite mγ remains well motivated and quite active, because theories describing such a mass seem well-developed and consistent, even if not considered esthetically appealing by all. In a certain sense, this quest has even penetrated into the undergraduate general physics laboratory, where torsion balance apparatuses for testing Coulomb’s law are found quite frequently. It is not unusual for instructors to motivate the experimental work of their students by pointing out that this class of instruments has been used historically to set very interesting and useful bounds on

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exotic effects like the existence of a non-zero photon rest mass. In general, the prevailing methods for experimental or observational studies in this area include some of the most common classical approaches: (a) bench top Cavendishtype experiments centered on electrical, thermal or dispersive effects and studies of radio waves; (b) solar system tests related to planetary magnetic fields and the solar wind; and (c) cosmic tests related to fields on galactic and larger scales, and the Lakes method. Sometimes, even approaches unique to quantum theory have been considered, e.g., matter waves experiments based on the AharonovBohm (AB) [7] class of effects involving electromagnetic interaction. Interesting though they are, the quantumbased approaches have been of somewhat limited interest because the sensitivities are not yet fully competitive with the traditional classical approaches. Even so, the photon is the quintessential quantum particle. Its story is woven deeply into the fabric of modern physics, and therefore, it is only natural that physicists would seek to develop new quantum-based methods of searching for its rest mass. As a result, this scenario has continued evolving recently for two principal reasons: new quantum effects of the Aharonov-Bohm type have been discovered, and new interferometric techniques have been developed that enhance the observability of those effects. The review component of our paper seeks in part to examine this scenario more carefully, in order to determine if quantumbased searches for the photon mass might become competitive with the classical methods. In dealing with quantum approaches, we will first consider the relevant effects of the AB type that are involved, and then widen the scope to include precision experiments that test quantum electrodynamic predictions, such as the electron g-factor measurement by Van Dyck et al. (VSD) [8]. Also, as a further preview of what is to be discussed, we note that the first approach to the photon mass limit that exploited quantum mechanical phenomena was that developed in 1989 by Boulware and Deser (BD) [9], who used the precise results of phase-shift measurements from the Aharonov-Bohm effect. One year later, Fuchs [10] followed-up on their work with an approach based on the Aharonov-Casher (AC) effect, obtaining a somewhat less sensitive but still very interesting independent upper bound on mγ . The importance of making as many independent measurements of weak effects as possible cannot be overstated, because of the need to be constantly on guard for systematic effects that might be endemic to one class of measurements, but either absent from or weaker in others. With the advent of new interferometric techniques for coherent beams of particles with opposite electromagnetic properties [11–17], and after a revision of the relevant gauge invariance properties, new quantum effects of the AB type also emerged as viable approaches, as proposed and elaborated upon by Sangster et al. [11–13], Dowling et al. [14], Spavieri [18,19], and Tkachuk [20], among others. It somehow seems very fitting that particle/field/potential interferometry can play such a central role in the determination of a fundamental property of the

photon, as all such techniques share a common basis in the general principles of optical interferometry. In a previous overview of studies of Coulomb’s law, Spavieri et al. [21] discussed the potential of some of these quantum effects for achieving better limits on mγ . Indeed, the techniques of Sangster et al. [11–13] have been employed by Rodriguez [22] to improve the results of Fuchs via the AC effect, while Spavieri and Rodriguez [23] used similar techniques applied to the Tkachuk effect and exploited a new version of the AB effect [15–19] to arrive at an improved limit on mγ . Even more recently, the scalar AB effect too has been revived and employed by Neyenhuis et al. [24] for determining a photon mass limit in the same range. Up to the present time, a drawback of most of these studies has been that they are actually realizable only in limited types of experimental arrangements. However, as we shall see, their potential value and utility is nevertheless apparent and provides a clear indication that interesting and competitive mass limits may ultimately be achieved through such routes. Moreover, as discussed in greater detail later, they open the door to additional creative thinking aimed at making further advances in the area of interest. Finally, we step outside of the framework of the AB class of effects and consider a novel approach to the photon mass that further exploits the experiment of Van Dyck et al. (VSD) [8] who measured the electron g-factor and tested the related quantum electrodynamic predictions. Before discussing the experiment of VSD, it is worth recalling some aspects of the recent extended review by Goldhaber and Nieto [2], which also treats the limits on the graviton mass. While our present review focuses largely on the general issue of the photon mass, it also seeks to complement the work of Goldhaber and Nieto [2] in regard to the specific question of quantum approaches to mγ . We believe that, besides the effects of the AB type, it is important to introduce quantum electrodynamic effects into the photon mass scenario, as is possible because of some recent advances in gauge invariance and its violation and restoration, as pointed out by Goldhaber and Nieto in their review [2]. For instance, citing the work of Weinberg [25], Goldhaber and Nieto discuss how a nonvanishing photon mass may lead to the possibility of processes that violate local conservation. Thus, for example, a nonvanishing photon mass might give rise to conflicts with field theories such as quantum electrodynamics. However, Goldhaber and Nieto also mention important recent advances that corroborate the work of Stueckelberg [26], by which full gauge invariance is actually restored. Because of this and for the purpose of discussing other novel quantum approaches to the photon mass limit, we find it useful to consider tests of the electron g-factor and related quantum electrodynamic predictions as well. Therefore, going back to the experiment of Van Dyck, Schwinberg, and Dehmelt on the electron g-factor, we find that a rough estimate of the photon mass limit obtainable in this way, indicates that sensitivities on the order of those provided by the effects of the AB type should be achievable. Moreover, relevant improvements might arise through a unique,

G. Spavieri et al.: A survey of existing and proposed classical and quantum approaches to the photon mass

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dedicated experiment that aims both at the precise measurement of the electron g-factor as well as the photon mass limit. So far, classical approaches that exploit planetary magnetic fields, the solar wind, and fields on galactic scales have provided the most stringent upper bounds on the photon mass. However, the essential elements of those methods, viz., the size of planetary or galactic fields, are imposed by nature and cannot be altered or varied. Thus, from one perspective, there is an intrinsic limit on the sensitivity of such techniques. On the other hand, a potential advantage of methods that exploit quantum effects is that the tests are done in a laboratory environment, where at least some of the most critical parameters can indeed be controlled and sensitivities enhanced. These capabilities, when coupled with many recent burgeoning advances in measurement science and technology, may open the door to improved results (with challenging implications) that might thence emerge from quantum-based approaches to searches for a limit on mγ .

intrinsically high precision involved in experiments based on quantum phenomena, they in turn may provide a suitable scenario for setting improved limits on the photon mass. Most of the results have been obtained by means of “table-top” or “bench-top” experiments. This methodology exploits, as a starting basis, physical effects already well established and verified, such as, the Aharonov-Bohm effect. In light of Proca’s theory, the consequences of a finite mγ on the observables in such cases can be determined. Taking into account the precision of the experiment and the changes of the observables due to the finite mγ , a limit on the photon mass can be established. Moreover, the parameters involved in the experiment can be ideally optimized and, in this way, we can arrive at estimates for improved limits if the experiment were to be performed with the new parameters. Traditionally, proposed improved limits as estimated in this way are of interest if and only if the optimization of the parameters involved is clearly seen to be within the reach of the relevant technologies.

2 Massive electrodynamics and its physical implications

3 Maxwell-Proca theory

In parallel with advances of classical electrodynamics, there have also been notable improvements in the techniques used to test the validity of Coulomb’s law. Many of the early workers may not have realized it, but increases in the precision with which Coulomb’s law could be tested were directly related to concomitant decreases in the upper bound of the photon mass. That inter-relationship was later established theoretically via another advance in classical electrodynamics: the introduction of the Proca equations [4–6]. According to the uncertainty principle, the minimum value of the photon mass should be given by mγ ≈ /(Δt)c2 ≈ 10−66 g, where Δt = 1010 years is taken as the age of the universe. Since it is virtually impossible to conceive an experiment that would allow one to prove that the rest mass of the photon is exactly zero, the best that can be done is to place ever more stringent limits on its value. However, the difficulties in resolving smaller and smaller mass differentials eventually point one towards the physical implications that can be derived from the Proca equations when mγ differs by very small amounts from zero. These include: dependency of the speed of light on wavelength, deviations from exactness in Coulomb’s and Ampere’s laws, existence of longitudinal electromagnetic waves, and addition of a Yukawa component to the dipolar magnetic field. Consequently, these effects open the door to useful experimental approaches and astrophysical/cosmological observations with the object of establishing an upper limit for the photon mass via searches for such violations. During much of the 20th century, the majority of the efforts for establishing limits on mγ relied on classical approaches. However, we will now seek to review how far those and other methods for establishing limits on mγ can be extended to the quantum domain. Because of the

A basic implication of Maxwell’s equations is that electromagnetic radiation travels at the speed of light, c, over a wide range of frequencies. However, besides being described as a wave, light has also been conceived as a particle. Without reviving the atomism of Democritus, who claimed that everything (light included) is reducible to ultimate indivisible particles, we recall that Descartes was perhaps the first to formulate the idea that light consists of swiftly moving particles. Later on, Newton elaborated a corpuscular theory of light that had the immediate merit of explaining polarization. Most likely, Coulomb, Cavendish and their contemporaries, were aware that the photon, like every other corpuscle, could have a rest mass. The term photon for light particles was first used by Lewis in 1926 in an article entitled “The conservation of photons”, although his atomist notion of a photon differed from the actual modern notion. As presently used, the term photon was introduced by Compton at the 5th Solvay Conference in 1927, named “Electrons and Photons”. The photon is characterized by its energy hν, linear momentum hν/c, and spin ±h/2π ≡ ±, where h is the Planck constant and ν is the frequency. Since the development of classical and quantum electrodynamics, the concept of a massless photon has been widely accepted. Even so, many physicists have made notable efforts to determine experimentally whether or not the photon has a residual mass. This posture is due in part to the fact that the effects due to massive photons are readily incorporated in electromagnetism through the Proca equations, which are the simplest relativistic generalization of Maxwell’s equations. 3.1 The Proca equations Electromagnetic phenomena in empty space are characterized by two vector fields, the electric field E(x, t) and

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the magnetic field B(x, t), which obey Maxwell’s equations. Moreover, these fields may be considered to be the classical limit of a quantum description in terms of photons. In Maxwell’s gauge-invariant electromagnetic field theory, the photon rest mass is assumed to be exactly zero. If gauge invariance is abandoned, a mass-related term may be uniquely added to the electromagnetic field’s Lagrangian density [27]

B = ∇ × A,

1 1 L=− Fμν F μν − Jμ Aμ + Aμ Aμ , 16π c 8π

(1)

μ−1 γ is a Aμ and

where typical length related to the photon rest Jμ correspond respectively to the fourmass, vector potential (φ/c, A) and the four-current (cρ, J), where φ is the scalar potential and A the vector potential, and ρ and J are the charge and current densities, respectively. The antisymmetric field tensor Fμν is expressed as: ∂Aν ∂Aμ = − . ∂xμ ∂xν

(2)

(3)

Substituting (2) in (3), yields the Proca wave equation for Aμ : 4π − μ2γ Aμ = − Jμ (4) c where stands for ∇2 − ∂ 2 /∂(ct)2 . In source free space, equation (4) becomes: (5) − μ2γ Aμ = 0 which is essentially a Klein-Gordon equation for the photon. The parameter μγ is related to the photon rest mass, mγ , by means of the expression: mγ =

μγ . c

1 ∂φ = 0. c ∂t

(13)

Proca’s equations provide a complete and self-consistent description of electromagnetic phenomena. Charge conservation is obtained from equations (7) and (10), and the Lorentz condition (13), ∇·J+

∂ρ = 0. ∂t

(14)

Similarly from equations (9), (10), (12) and (13), the equation for the conservation of energy reads ∇·S+

∂w = −J · E ∂t

(15)

where S is the Poynting vector associated with the energy flow density, S=

c E × B + μ2γ φA 4π

(16)

and the energy density of the electromagnetic field w is: w=

1 2 E + B2 + μ2γ φ2 + μ2γ A2 . 8π

(17)

In the approach of Proca even the potentials are physically meaningful, over and beyond their derivatives. In fact, the scalar potential φ and the vector potential A described by the Proca equations become observables because they acquire the energy densities μ2γ φ2 /8π and μ2γ A2 /8π, respectively.

(6) μ−1 γ

4 Early experimental verification of the 1/r2 co- Coulomb law

With this interpretation, the characteristic length incides with the reduced Compton wave length for the photon. This introduces a range of effectiveness in the electromagnetic interaction. Consequently, with massive photons the electromagnetic fields acquire an attenuation that is governed by the term exp(−μγ r). The finite photon mass may be accommodated in a unique manner by changing Maxwell’s equation for Proca’s as follows: ∇ · E = 4πρ − μ2γ φ, 1 ∂B , ∇×E=− c ∂t ∇ · B = 0, 1 ∂E 4π J+ − μ2γ A, ∇×B= c c ∂t

(12)

and the Lorentz condition

The variation of the Lagrangian density (1) with respect to Aμ leads to the Proca equations: 4π ∂Fμν Jμ . + μ2γ Aμ = ∂xν c

(11) 1 ∂A , c ∂t

E = −∇φ −

∇·A+

μ2γ

Fμν

together with

(7) (8) (9) (10)

As has been the case with most of the fundamental laws of physics, Coulomb law emerged from the observations of very basic phenomena. In his research, Coulomb himself was interested in the mutual interaction between electric charges, a topic that was first considered in depth by Priestley [28], but which actually had an empirical beginning in 1755 with some experimental observations of Franklin [29]. Franklin took a small sphere made of cork and placed it inside a charged metallic cup (see Fig. 1) and observed that it did not move, suggesting that it did not feel any interaction. After Franklin communicated his finding to Priestley, the Englishman explained the phenomenon in 1767, providing a line of reasoning analogous to that used by Newton to formulate and enunciate the law of universal


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Wire q Q

d

Mirror

d

Q q

q

Light ray Light source

Fig. 1. Franklin placed a small sphere made of cork inside a metallic cup. When the cup was charged, the sphere did not move, suggesting that there was no interaction between it and the cup. Priestley explained the phenomenon in 1767, providing a line of reasoning analogous to that used by Newton for the law of universal gravitation, implying that the field has an inverse-square dependence on distance. In subsequent tests, instead of a cork sphere inside a metallic cup, others considered a metallic shell enclosed within an outer charged shell or several concentric shells.

gravitation. In fact, Newton demonstrated that the gravitational field inside a material homogeneous spherical shell is null if the field is inversely proportional to the square of the distance r, i.e., if its intensity goes as r−2 . By approximating the cup used by Franklin to a spherical shell, Priestley deduced that the observed phenomenon should be physically analogous and, thus, not only the gravitational force but also the electric force must have a r−2 dependence on distance. All these previous qualitative phenomenological and theoretical insights paved the way to a more consistent quantitative verification of this most basic law of electrical science. This was done by the English physician Robison [30] who in 1769 verified quantitatively that the r−2 dependence was correct. The experiment by Robison was very simple yet elegant: he measured the repulsive force between two charges placed on rods, with the electrical force between them then equilibrated by their mutual gravitational force. By knowing the weight of the rods and by repeating the measurements at different distances, it was possible to calculate the electric force, evaluate its dependence on distance, and thus verify the hypothesized 1/r2 behavior. In fact, Robison concluded that the law must have been of the type F ∝

1 , r2±ε

(18)

where ε represented the limiting size of the deviation from exact inverse-square behavior, i.e., a measure of the precision of the experiment. He found an upper limit of 0.06 for ε in tests of electrical repulsion, i.e., r−2.06 , and for electrical attraction he concluded that the behavior

Fig. 2. The Coulomb torsion balance was similar in principle to the torsion balance used by Cavendish for measurement of the gravitational attraction between masses. The interaction between the charged spheres produced a measurable twist in the torsion fiber, with the apparatus thus rotating until equilibrium is reached. By accurately measuring the torsion angle, Coulomb confirmed the 1/r 2 law with a precision surpassing that of the previous experiments of Robison and Cavendish.

went as r−c where c < 2, thus confirming the expected r−2 dependence for both species of charge. Unfortunately, Robison published his results only in 1801, when Coulomb [31] had already presented his. Before discussing Coulomb’s experiment, we recall the one performed by Cavendish [32] himself, who had previously carried out the seminal laboratory test of Newton’s law of gravity. Inspired by the same idea of his predecessors, he considered a metallic spherical shell enclosed inside another shell composed of two hemispheres that could be opened or closed. In the closed position, the two hemispheres were electrically connected to an electrostatic machine and charged while connected to the internal sphere. The hemispheres were then disconnected from the internal sphere and then opened, and it was verified that they were charged. At that point, by means of an electrometer, he checked that the internal sphere was still uncharged, thus confirming the 1/r2 law, with an uncertainty somewhat less than that of Robison. Less than 1/60 of the charge moved to the inner shell along a thin wire connecting the two spheres. With reference to equation (18), Cavendish obtained1 ε ≤ 0.02. We now turn to the experiment of Coulomb of 1788. He used a torsion balance, as shown in Figure 2, similar to that used by Cavendish for the test of Newton’s law. The interaction between the charged spheres produced a measurable twist in the torsion fiber until equilibrium was reached. By accurately measuring the torsion angle, Coulomb found for equation (18) a value of ε ≤ 0.01, surpassing the precision of the previous experiments of Robison and Cavendish. 1

An improved version of the experiment was later performed by Maxwell [33] who increased the precision of the test and found that the exponent of r in Coulomb’s law could differ from 2 by no more than ε 5 × 10−5 .

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What has thence become known as Coulomb’s law describes the force F between two charges q1 and q2 separated by the distance r, and can be written in scalar form as q1 q2 (19) F =k 2 , r where the value of the constant of proportionality k depends on the units being employed. There are basically two reasons why Coulomb achieved greater success (and recognition) than either Robison or Cavendish: – He performed his tests with combinations of charges of both types, negative and positive, while Cavendish instead used only charges of the same sign. – He published his results immediately, while Robison did not make his findings available until 1801, thirteen years after Coulomb. Moreover, Cavendish did little to disseminate his work, and it was only brought to public attention a century later through citation by Maxwell in his famous essay [33]. The fact that Coulomb published immediately may mean that Coulomb was more aware than his colleagues that he was dealing with a fundamental law and had made a very important discovery.

5 Tests of Coulomb’s law and the null result approach: theory The technique of Cavendish was used in most of the subsequent studies, as it turned out to be potentially the most sensitive. It is an example of a null-result experiment, in the sense that the person doing it had to verify with great precision the absence of the charge, rather than, as in the other tests, having to measure with less precision a nonnull physical quantity. Following Robison and Maxwell and supposing that the exponent in Coulomb’s law is not −2 but −(2 + ε), to first order in ε the electric potential at the point r due to the charge distribution density ρ(r ) is given by ρ (r ) . (20) V (r) = d3 r | r − r |1+ε In the case of exact validity of Coulomb’s law, ε = 0 and +1 d cos θ 1 V (r) = 2πρ = = const., 1/2 2 2 a −1 (r + a − 2ar cos θ ) (21) so that V (r) − V (a) = 0 and the electric field inside the charged spherical shell of radius a vanishes. Thus, for a test of Coulomb’s law, we are interested in the potential induced on a sphere of radius r by a charge uniformly distributed on a concentric sphere of radius a > r. Therefore, to first order in ε, one finds [V (r) − V (a)]/V (a) = εM (a, r),

(22)

where the term M (a, r) turns out to be of the order of unity, so that ε becomes the quotient [V (r) − V (a)]/V (a)

of the measured potential difference V (r) − V (a) and the applied voltage V (a). In the previous sections we introduced Proca’s theory characterized by the photon mass μγ . If μγ = 0, the electric potential of a unit point charge becomes the Yukawa potential U (r) = e−μγ r /r. (23) Therefore, the potential difference appearing in equation (22) may be expressed as a function of μγ . This function was derived by de Broglie [34], who elaborated a simple generalization of Maxwell’s equations for the case of a small nonzero rest mass of the photon, in line with the predictions of Proca’s equations. In the limit μγ a 1, U (r) = 1/r − μγ + 12 μ2γ r and equation (22) yields 1 [V (r) − V (a)]/V (a) = − μ2γ a2 − r2 . 6

(24)

Thus, the potential difference V (r) − V (a) = 0 if Coulomb’s law is invalid. According to Proca’s theory, also if the photon rest mass is not zero the law is invalid and Coulomb’s potential is given by the Yukawa expression (23). However, these effects must be distinguished physically because in fact the one due to a change in the power law (19) could be due for example to the introduction of nonlinear terms in the Maxwell equations (the limits on these terms being much smaller than those on the photon mass). Nevertheless, mathematically a relation can be established between ε and μγ for example comparing (22) with (24). Thus, for direct tests of Coulomb’s law that consist of measuring the static potential difference of charged concentric shells, one may use either equation (22) or equation (24), seeking ε or μγ . However, for testing Coulomb’s law as modified by the Yukawa expression (23), one can also determine μγ with independent, indirect methods that rely on possible variations, due to the presence of the Yukawa potential (23), of the standard fields of massless electrodynamics. An example would be measurements of the Earth’s magnetic field made at either large distances or over long times, where the percentage effect would be much higher. For example one might consider (a) satellite verification that the magnetic field of the earth falls off as 1/r3 out to distances at which the solar wind is appreciable [35,36], (b) observation of the propagation of hydromagnetic waves through the magnetosphere [37], or (c) application of the Schr¨ odinger external field method [38,39], or other methods such as those described below. Any of these three approaches should all give roughly the same limit, μ ≤ 10−11 cm−1 . For the high-frequency direct null test of Coulomb’s law described below, the potential difference becomes V (r, t) − V (a, t) = −μ2γ

Re V0 ei ω t 2 a − r2 , 6

(25)

which is valid when there is no charge on the inner shell. The term dependent on μγ in equation (25) is analogous to that of equation (24).


6 Advanced, direct tests of Coulomb’s law: null experiments In this section we consider Maxwell’s derivation (Eq. (22)) as applied to the simple case of a conducting sphere containing a smaller concentric sphere. The potential of the outer sphere is raised to a value V and the potential difference between them is measured. The actual shape of these conductors should not be relevant because the electric field inside a cavity of any shape vanishes unless Coulomb’s law is violated. Thus, Cochran and Franken [40] could use conducting rectangular boxes (which are easier to manufacture than spheres) to set the limit ε ≤ 10−11 . After the development of the lock-in or phase-sensitive amplifier, new and more sensitive attempts to test Coulomb’s law by the Cochran and Franken method [40] were carried out by Bartlett and Phillips [41]. We also note the experiment of Plimpton and Lawton [42], who in 1936 used alternating potentials to quasi-statically charge the outer sphere. The potential difference between the inner and outer spheres was detected with a resonant-frequency electrometer. The sensitivity was such that a potential difference of 10−6 V was easily observable above the small Brownian fluctuations. With this technique they succeeded in reducing Maxwell’s limit to ε 2 × 10−9 . Another of the classic “null experiments” that tests the exactness of the electrostatic inverse-square law was performed by Bartlett et al. [43] in 1970. In this experiment the outer shell of a spherical capacitor is raised to a potential V with respect to a distant ground and the potential difference V (r) − V (a) of equations (22) and (24) induced between the inner and outer shells is measured. Five concentric spheres were used and a potential difference of 40 kV at 2500 Hz was imposed between the two outer spheres. The result obtained by these authors was ε ≤ 1.3 × 10−13. A comparable result was also found when the frequency was reduced to 250 Hz and the detector was synchronized with the charging current rather than with the charge itself. The best result obtained by improving Cavendish’s technique is still that Williams et al. [44], reported in 1971. These authors used five concentric metallic shells in the form of icosahedra rather than spheres to reduce the errors due to charge dispersion. A high voltage and frequency signal was applied to the external shells and a very sensitive detector checked for any trace of a signal related to variable charge transfer to the internal shell. The detector worked by amplifying the signal of the internal shell and comparing it with an identical reference signal, progressively out of phase with a rhythm of 360 degrees per 1/2 h. Any signal produced by the detector would have been indicative of a violation of the Coulomb’s law. The outer shell, of about 1.5 m in diameter was charged to 10 kV peak-to-peak with a 4 MHz sinusoidal voltage. Centered inside of this charged conducting shell was a smaller conducting shell. Deviations from Coulomb’s law were sought by measuring the line integral of the electric field between these two shells with a detection sensitivity of about 10−12 V peak to peak.

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The null result of this experiment expressed in the form of the photon rest mass squared (Eqs. (25) or (24)) is μ2γ = (1.04 ± 1.2) × 10−19 cm−2 . Expressed as a deviation from Coulomb’s law in the form of equation (22), the result becomes ε = (2.7 ± 3.1) × 10−16 , which extended the validity of Coulomb’s law by two orders of magnitude with respect to the findings of Bartlett, Goldhagen and Phillips. In terms of the Compton wavelength and photon mass, the results are 1.95 × 1010 cm for μ−1 γ and mγ < 1.6 × 10−47 g.

7 Indirect tests of Coulomb’s law geomagnetism measurements After the experiments discussed in the previous sections, subsequent tests of Coulomb’s law employed indirect methods of verification. These are discussed briefly in what follows. 7.1 Geomagnetic and astronomical tests A consequence of Coulomb’s law is that the magnetic field produced by a dipole goes as 1/r3 over distances from its center where the dipole approximation is valid. For the magnetic field of a planet, this distance is equivalent to at least two planetary radii. If the photon rest mass is not zero – which is equivalent to a violation of Coulomb’s law – a Yukawa factor e−r/λC is introduced in the 1/r terms for the electrostatic and magnetostatic potentials. In this case, the magnetic field produced by a dipole no longer goes as 1/r3 but contains corrections related to the = h/mγ c where mγ Compton wavelength λC = 2πμ−1 γ is the photon mass. The value of the standard magnetic dipole moment md , thus becomes 1 2 2 md → md 1 + μγ r + μγ r , (26) 3 and the expression for the magnetic field contains an additional factor Bμγ = −

1 2 2 μ md e−μγ r c 3r γ

(27)

when μγ = 0. In 1968 Goldhaber and Nieto [1] analyzed the best satellite measurements of Earth’s magnetic field and, using equations (26) and (27), they found limits of 5.5 × 1010 cm for λC and 4 × 10−48 g for mγ , the latter corresponding to an equivalent value of ε ≤ 3.37 × 10−17 . In 1975 Davis et al. [45] verified the 1/r3 behavior of the magnetic field of Jupiter (much more intense than that of the earth) from observations made by the Pioneer 10 spacecraft, and were able to improve the precision of the validity of Coulomb’s law, reaching limits of λC ≥ 3.14 × 1011 i.e. μγ ≤ 2 × 10−11 cm−1 or mγ = 8 × 10−49 g.

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Further work on planetary magnetic field-determinations of the photon mass was carried out by Bicknell in 1977 [46]. He extended Schrödinger’s expression for a dipole field [47] to a complete spherical harmonic analysis of a static planetary field. Additional components of the magnetic field were identified as being due to a photon mass. No better limits were set on λC since the most important terms evaluated for the geomagnetic field were of the same order as Schr¨ odinger’s apparent external field. An upper limit for the photon mass was also set by Lowenthal [48] by means of astronomical observations exploiting the gravitational deflection of electromagnetic radiation. This method does not obtain a better upper limit than those mentioned above, but the method is interesting and, since it is not related to the previous ones, it adds to the evidence restricting the magnitude of the photon mass. The question posed by Lowenthal was the following: if general relativity predicts a deflection of starlight by the sun of 1.75 s of arc, how is this deflection altered if the photon has a small rest mass μγ ? Lowenthal shows that the deflection variation is proportional to μ2γ and sets this correction term equal to the difference between the measured deflection angle and the calculated deflection angle for photons of zero rest mass. Taking into account the accuracy of the deflection measurements, an upper limit of mγ < 7 × 10−40 g was reached, which is still a very respectably small quantity. Another interesting approach to the photon mass was provided by a cryogenic Coulomb’s-law or low temperature photon-mass experiment. Modern theories that use the concept of spontaneous symmetry breaking assume that particles, which are massless above a certain critical temperature Tc , acquire mass below this temperature. Within this framework it is natural to speculate that the photon could also be massless above a critical temperature and acquire a rest mass below that temperature. A cryogenic photon-mass experiment was performed by Ryan et al. in 1985 [49]. This test consisted essentially of a null experiment with concentric containers (closed surfaces) similar in a way to the previous direct tests of Coulomb’s law such as the one by William et al. [44]. However, unlike a standard Coulomb’s-law experiment, this method measures the current that flows between two closed surfaces in response to an impressed voltage difference, not the voltage difference itself. The apparatus was immersed in a pool of liquid helium. The result of the experiment indicates that the mass of the photon is mγ ≤ (1.5 ± 1.38) × 10−42 g at 1.36 K. While the sensitivity is not as high as that in some of the previous studies, it is nevertheless quite important because the validity of Coulomb’s law is extended from the standard room temperature on the earth to temperatures typical of a distant interstellar space or the galactic environment. 7.2 The photon mass and the cosmic magnetic vector potential Another novel experimental approach to measurement of the photon mass mγ was based on a toroidal Cavendish

balance which was used to evaluate the product μ2γ A of the photon mass squared (μ−1 γ = /mγ c) and the ambient cosmic vector potential A. This approach, developed by Lakes [50], relies on the fact that the Maxwell-Proca equations modify the standard equations for the curl of B to 1∂E 4π j+ − μ2γ A. c c ∂t Gauge invariance is lost if μγ = 0, since in these equations the potentials themselves have physical significance, not just the usual fields. If a toroid carries an electric current – or it is permanently magnetized with a field B confined within it – the corresponding magnetic vector potential may be represented by a dipole field Ad . If μγ = 0 this dipole field interacts with the ambient cosmic vector potential A to produce a torque τ = Ad × A μ2γ via the energy density of the vector potential. Thus, the method is based on the energy density of A in the presence of mγ , not on measurement of the magnetic field. The experiment does not yield a direct limit for μγ , but rather for A μ2γ . The modified Cavendish balance used to determine the product A μ2γ consisted of a toroid of electrical steel wound with many turns of wire carrying current and supported by water flotation [51]. The ex2 perimental result was A μ2γ < 2 × 10−9 T m/m . If the 12 ambient magnetic vector potential is A ≈ 10 T m due to 10 m. Even with more cluster level fields, then μ−1 γ > 2×10 conservative values for A, the limit set by this experiment improves the precision attained with the Jovian magnetic field technique. The experimental arrangement of Lakes is such that the torque on the torsion balance will vary with the rotation of the earth, making it possible to experimentally detect the variations. However, if the cosmic ambient vector potential were to be fortuitously aligned with Earth’s rotation axis, then this approach would not work. In order to avoid this possibility, Luo et al. [52] carried out an improved version of the experiment in which the torsion balance itself was rotated to ensure efficacy of detection for all possible orientations of the vector potential. They were also able to remove the influences of sidereal environmental disturbances by virtue of a modulation method. The 2 experimental result yielded A μ2γ < 1.1 × 10−11 T m/m . If the ambient cosmic magnetic vector potential is taken to be A ≈ 1012 T m, it yields the new upper limit 11 m and a value for the photon mass of μ−1 γ > 1.66 × 10 ∇×B=

mγ < 1.2 × 10−51 g. After these results were published, Goldhaber and Nieto [53] pointed out that the value of cosmic magnetic vector potential could be zero, or at least very small, in a special region of the universe. If the Earth were located in such a region, the magnitude of A would be small enough to prevent the determination of a limit on mγ . Since an incomplete knowledge of galactic or extragalactic magnetic fields introduces the possibility of inhomogeneities in field and plasma densities, and the torsion balance method imposes a limit on μ2γ A, the problem of incomplete knowledge of A may lead to an inexact limit of μγ .


539

7.3 Other classical approaches Before considering the quantum approaches, it is important to note that there are also some additional interesting classical approaches to measurement of the photon mass. These include studies of the altitude dependence of geomagnetic fields [54], hydromagnetic waves dispersion in the Earth’s magnetosphere [37], dissipation of large scale magnetic fields in the galaxy [55], plasma stability in the galaxy [1], average density of the mass of the galactic disk [56], analysis of the mechanical stability of magnetized gas [57], analysis of the spectral behavior of the dipolar anisotropic background radiation [58], and analysis of the magnetic fields of the solar wind [59]. Interest in this very fundamental quantity has clearly motivated many different and novel approaches to its determination over the years.

Fig. 3. In the magnetic Aharonov-Bohm effect, the charged interfering particles are electrons (e−), which move in parallel on either side of a long, thin solenoid. Since the magnetic field B = ∇ × A is concentrated within the solenoid and is practically zero outside, it does not influence the motion of the electrons. The concentric rings indicate the intensity of the vector potential A, which acts on the wave function of the particles producing an observable phase shift Δφ detected by the interferometer (not shown).

8 The quantum effects of the Aharonov-Bohm type For convenience of the reader, before considering application of the magnetic Aharonov-Bohm (AB) [7] effect, and other related quantum effects of the same type, for establishing limits on the photon mass, we present a brief description of them. The Aharonov-Bohm effect can be represented schematically as shown in Figures 3 and 4. A beam of particles possessing electromagnetic (em) properties encircles an em flux tube. The wave functions of the particles interfere and form an interference pattern visible on a screen. Although the potentials or fields of the em flux tube do not affect the velocity of the particles, the phase φ of the wave function is affected by the potentials or fields of the flux, which produces a phase shift Δφ that leads to an observable displacement of the interference pattern. The type of particles used and the electromagnetic sources of the flux tube (ideally forming a narrow cylindrical singularity) depend on the specific quantum effect involved. We first consider the magnetic AB effect (and the other related effects) in more detail, and then show in the next section they can be exploited to establish limits on the photon mass.

8.1 The magnetic Aharonov-Bohm effect In the magnetic AB effect, the particles are electrons and em “singularity” is the magnetic flux of a thin solenoid, characterized by the vector potential A and the magnetic field B = ∇ × A. In its interaction with the charged particles, the vector potential A acts as an electromagnetic vortex that has the effect of modifying the phase of the wave function of the particles passing on opposite sides of the vortex or singularity. The resulting phase shift Δφ is e e Δφ = A · dx = B·d S, (28) c c

Fig. 4. (Color online) A simplified, unitary scheme for all of the effects of the Aharonov-Bohm type discussed here. A beam of interfering particles approaches a confined electromagnetic flux of density δΦem and splits into the two beams a and b encircling the flux. The beams are then recombined while an interferometer (not shown) measures the phase shift Δφ of the particle wave functions, produced by the interaction with the potentials or fields of δΦem . There is an observable displacement of the interference pattern visible on a screen, which corresponds to the phase shift Δφ.

where δΦem is the flux density and δΦem · d S = B·d S the magnetic flux. Figure 4 shows the essential elements of the AB effect. The magnetic field lines are concentrated inside the solenoid, which is conceptually very long and thin. The beam of charged particles is represented by two straight lines, representing the paths of two electrons on opposite sides of the solenoid. In their swift motion, the particles encircle the solenoid but are not affected by the magnetic field, which is negligible and practically vanishes outside of the very long solenoid. The concentric planar rings picture the intensity of the vortex formed by the vector potential A, which does not affect the particles but

540


instead acts on the phase of the electron wave function, since the phase shift is linked to the enclosed magnetic flux δΦem · d S, in agreement with equation (28). Even when the magnetic field inside the solenoid is zero (i.e., when the current in the solenoid is switched off and B = 0), the interferometric reference pattern of the electron wave functions is being formed and is visible on a screen. If the current in the solenoid is switched on and the magnetic field inside the solenoid is non-zero (i.e.,there is a steady current in the solenoid and B = 0), then the interference pattern is now seen to be displaced by an amount related to the phase shift Δφ (28). Thus, the observable quantity of this quantum effect is the relative displacement of the interference pattern. Experimental verifications of the AB effect were first performed by Chambers [60], who used a Young-type double slit interferometer. Later, Tomonura et al. [61] confirmed the effect using a superconducting magnetic toroid (instead of the solenoid) and an electron interferometer. 8.2 Effects of the Aharonov-Bohm type: unitary view In order to introduce the other effects of the AB type and demonstrate their common characteristics, we present here a unitary description of the electromagnetic interaction involved in these effects. As shown in the previous section, in the AB effect the charged particles interact with the singular magnetic field of a solenoid. In this effect, the interaction momentum between the charge e of the particle and the vector potential is given by Q = ec A. In the Aharonov-Casher (AC) [62] effect the particles are neutral but possess a magnetic dipole moment μm that interacts with the E field of a line of electric charges of density λ. For this effect the interaction momentum is Q = 1c μm ×E. In the Spavieri (S) [15–17] effect for neutral particles possessing an electric dipole moment d, the particles interact with the vector potential A of a semi-infinite magnetic sheet of magnetic density μ and thickness τ . The interaction momentum is Q = 1c (d · ∇)A, which represents the generalization of the AB effect to the electric dipole. In the Tkachuk (T) [20] effect for electric dipoles, the electric dipole d interacts with a radial magnetic field B produced by a line of magnetic dipoles of density μd varying linearly with z. The interaction momentum of this effect is Q = 1c B × d. It would thus appear that the interaction momentum Q differs, depending on the specific type of effect considered. However, a unitary view of these effects may be achieved in the following way. It can be shown [15–19], that, for all the mentioned effects, the Schr¨ odinger equation reads 1 2 (−i∇ − Q) Ψ = EΨ, (29) 2m where Q represents the linear interaction momentum and E is the energy. The solution assumes the form i

Ψ = eiφ Ψ0 = e

Q·dx

where Ψ0 is the solution when Q = 0.

Ψ0 ,

(30)

A common characteristic of these effects is that the beam of particles encloses an electromagnetic singularity, as shown in Figure 4 in analogy with the magnetic AB effect. In order to elucidate this common feature, let us introduce the linear and angular interaction momenta Q and L, and the linear and angular momenta of the electromagnetic fields, Pe and Le , given by 1 Q = ±Pe = ± (31) (E × B) d3 x , 4πc 1 [x × (E × B)] d3 x , (32) L = ±Le = ± 4πc where x is the polar vector indicating the position of the particle and E and B are the interaction fields. For example, in the AB effect, E is the electric field of the electron and B is the magnetic field of the solenoid. It turns out [15–17] that, for each effect, the electromagnetic flux density may be expressed as δΦem = so that the electromagnetic flux is ∇ × Q(x) = L δ(r) r given by the (∇ × Q(x)) · dS = L δ(r) r · dS, where, outside of the singularity on the path of the particle, Q = |L| ∇θ where θ is the polar angle. From the solution (30), the phase shift of these effects may be expressed as [18,19] 1 |L| Δφ = Q (x) · dx = ∇θ·dx 1 |L| L (33) = (∇ × Q (x)) · dS = dθ = 2πn where the minus sign in (31) and (32) applies to the AC effect. Apart from the ± sign, with equation (33) the phase shift Δφ of any of these effects is related in a unitary way in terms of the electromagnetic momentum of the interaction fields Q = ±Pe . For example, if the em momentum Pe of equation (31) is calculated for the AB effect, the expected result, Q = +Pe = eA/c, is found in the Coulomb gauge [18,19]. Always for the AB effect, ∇ ×Q = ec ∇ × A = ec B and from (33) one gets e Δφ = 1 (∇ × Q(x)) · dS = c B · dS in agreement with (28), as expected. Thus, to picture all the mentioned effects of the Aharonov-Bohm type, we appeal to Figure 4, where a beam of particles possessing electromagnetic (em) properties is split into two interfering beams a and b that encircle the em flux tube of density δΦem = ∇ × Q. In complete analogy with the description of the AB effect given above, the wave functions of the particles of all these effects interfere and form an interference pattern visible on a screen. Although the potentials or fields of the em flux tube δΦem do not affect the velocity of the particles, the phase φ of the wave function is affected by the potentials or fields of δΦem , which produces the relative phase shift Δφ = 1 Q · dx that leads to an observable displacement of the interference pattern. Therefore, the generalization suggested in Figure 4 does indeed represent all the effects of the AB type, in close diagrammatic analogy with the usual representation of the magnetic AB effect.

G. Spavieri et al.: A survey of existing and proposed classical and quantum approaches to the photon mass Table 1. Effects of the AB type: field of singularity, particle, and expressions of Q and L. Effect AB AC S T

em singularity ˆ μ B = k2¯ E = 2λ ρρˆ2 ˆ B = k4πμ B = 2q ρρˆ2

Particle

Q (c = 1)

L

e μm d d

eA μm ×E (d · ∇)A B×d

2e¯ μ 2μm λ 2dμτ 4πdμd

The topological properties of the geometric phase are determined by the following relationships, valid for each of the mentioned effects: L = kL = const., Q = |L| ∇θ,

∇ · Q (x) = 0, δ (r) . ∇ × Q (x) = L r

(34)

In Table 1 we summarize all of this and indicate for each effect the field of the electromagnetic singularity, the type of particle, and the corresponding expressions of Q and L. In the effects of the AB type there are no forces acting on the particles in their paths where, outside the singularity, ∇ × Q(x) = 0. In the AB and S effects, the field B is confined and the particles travel in a field-free path (type I effects). In the AC and T effects, the particles travel in the presence of external fields, but still in a force-free path (type II effects).

9 The AB effect and the approach of Boulware-Deser to the photon mass The several effects of the Aharonov-Bohm type as discussed in the literature have been developed assuming electromagnetic interaction of fields of infinite range, i.e. a truly massless photon mass. The possibility that those effects, specifically the Aharonov-Bohm effect itself, might manifest even in finite-range electrodynamics has been discussed by Boulware and Deser [9]. These authors consider the coupling of the photon mass μγ , as predicted by the Proca equation, and calculate the resulting magnetic field B

μ2 Π (μγ ρ) , B = B0 + k γ

(35)

that might be used in a test of the AB effect. The first term, B0 , is the standard magnetic field for zero photon mass – the field confined inside a long solenoid of radius a and with current j – and the second term ΔB =

μ2 Π(μγ ρ) represents a correction due to a nonvanishing k γ rest mass μγ . The function Π(μγ ρ) is expressed in terms of the modified Bessel functions K(μγ ρ) and I(μγ ρ). Thus, on account of expression (28), the photon mass could modify the phase shift Δφ and the corresponding observable displacement of the interference pattern. Taking into account the precision of the experiment and the effect of the extra mass-dependent term, Boulware and Deser compared the theoretical corrections to the flux through an ideal circle of radius ρ ∼ 10 cm with the flux for the massless case.

541

In this way, they were able to predict a nontrivial limit on the range of the transverse photon from a table-top 7 experiment: μ−1 γ > 1.4 × 10 cm, which corresponds to a photon mass limit mγ 2.5 × 10−45 g.

(36)

Because of the findings of Boulware and Deser, it is useful to consider the application of their approach to other quantum effects of the AB type. However, before doing this we describe briefly some of the effects of the AB type that were discovered after the magnetic AB effect for electrons, as introduced above.

10 The photon mass and the effects for dipoles Save for the AB effect itself, the effects of the AB type in general emerge from the particle-dipole interaction with external fields. These are discussed below. 10.1 The Aharonov-Casher effect for magnetic dipoles For the standard Aharonov-Casher (AC) effect [62], the magnetic dipole μm of a neutron interacts with the external electric field E of a line of charges. The photon mass modifies the expression of E and the phase shift (33), analogously to the case of the the approach of Boulware and Deser to the AB effect. For the AC effect the analysis has been performed by Fuchs [10] who points out that, for a neutral particle with a magnetic dipole moment that couples to nongauge fields, no important corrections are expected. By solving Proca’s equations for the AC effect, Fuchs found that the electric field of the line of charges given in Table 1 becomes E = 2μγ λK1 (μγ ρ) ρˆ

(37)

where K1 is the modified Bessel function of second kind. The standard expression is retrieved in the limit μγ → 0. With E given by (37), Q as in Table 1, and Δφ by (33), Fuchs estimated μγ using the parameters of the neutron interferometry experiment by Cimmino et al. [63]. His results show that a Compton wave length of about 10 m or less could be detectable while the corresponding photon mass could be on the order of mγ ∼ 3 × 10−41 g.

(38)

Although an independent method, it nevertheless follows that the AC effect is not as effective as the AB effect for the estimation of μγ . 10.2 The Spavieri and Tkachuk effects for electric dipoles Before discussing the effect of the AB type for electric dipoles suitable for determining the photon mass or its

542


limit, it should be mentioned that the first expression of the phase for the electric dipole that appeared in the literature, was proposed independently by He and McKellar [64], and Wilkens [65,66]. Another expression of the phase for the electric dipole, which in most cases coincides with that of Wilkens et al., has been later derived by Spavieri [15–17]. Discussions on gauge invariance related to the mentioned phases may be found in references [15–19,67,68]. The photon mass limit obtainable from quantum effects of the AB type for electric dipoles has been derived by Spavieri and Rodriguez [23]. These authors note that the interaction term of all the effects for electric dipoles has the same strength [15–17,20] so that, for the purpose of performing a table-top experiment and obtaining the corresponding photon mass limit, it is sufficient to analyze essentially any of these effects. It turns out that it is simpler to consider the Tkachuk effect [20] because the resulting equations for the mass correction possess a symmetry analogous to that of the AB effect. It is important to point out that the phase for electric dipoles of Spavieri [15–17], the one of He and McKellar [64], and Wilkens [65,66], and that of Tkachuk, all coincide in the configuration of sources of the magnetic field presented by Tkachuk in his effect [20]. It would be interesting to discuss the difference between the expressions of the phases of these authors and the implications on gauge invariance. This discussion becomes more appealing and acquires new facets in view of the fact that, in massive electrodynamics with a nonnull photon mass, the difference could be interpreted as a consequence of broken gauge invariance, where the vector potential A assumes a new physical meaning. However, for the purpose of the present paper, which in this section is to determine the photon mass limit exploiting the Tkachuk effect, the discussion would not lead to different physical results because, as we mentioned above, all the different expressions of the phases for the electric dipole coincide in the Tkachuk effect, and the related photon mass or its limit would be the same. Thus, we will defer the discussion on gauge invariance to future contributions. For the Tkachuk effect [20] one may consider a long solenoid with magnetization linear density μd = μd z and a magnetic flux Φ = BS = 4πμd z = πjz a2 , where a is the radius of the solenoid and jz its current density. The resulting vector potential reads A = AAB z, where AAB is the vector potential of the AB effect with μdAB substituted by μd . Thus, the mass treatment for the Tkachuk ˆ can be reduced to effect for the electric dipole d = d k that of Boulware and Deser for the AB effect. With the Tkachuk phase ϕ0 = 4πdμ/c, the relative variation of the phase due to the photon mass found by Spavieri and Rodriguez [23] is Δϕ ja2 = ϕ0 4μ

ρ

μγ ln a

2 μγ ρ

1 2 ρdρ ∼ (μγ ρ) ln 2

2 μγ ρ

.

(39) This result, valid for the Tkachuk effect, can be compared with that of Boulware and Deser derived for the AB effect,

with the result being −1 μ−1 γ = μγBD

or

ϕ0 ϕ0AB

mγ ∼ 3 × 10−41 g.

1/2

∼ 10−4 μ−1 γBD ,

(40) (41)

This represents a range limit of the photon mass 4 orders of magnitude lower than that of BD. As expected, and in agreement with the findings of Fuchs for the AC effect for magnetic dipoles, no improvement for the range μ−1 γ is achieved from a table-top experiment involving electric dipoles because of the lower strength of the em interaction for dipoles.

11 New effects of the Aharonov-Bohm type The seminal work of Aharonov and Bohm led initially to the AB effect for charged particles, and it was only later that other effects were developed associated with neutral particles that have an intrinsic magnetic [62] or electric dipole moment [15–17,20]. For these latter effects, the beam had to be split into two spatially separated beams in order for particles to encircle the singularity (see Fig. 4), and then recombined to form an observable interference pattern. Moreover, the transverse size of the beam, the singularity, the slits of the interferometer and their separation, are all determined by the associated wavelength of the particles. The resulting constraints impose some limitations if these effects are to be exploited for determining the photon mass. Fortunately, research and technological advances have provided improvements in interferometric techniques for beams of coherent superpositions of states of particles with opposite electromagnetic properties. These technological advances, as well as other theoretical developments, have allowed for the realization of more refined and precise experiments involving the mentioned effects and even the discovery of new effects of the AB type. The consequences of these advances may be understood from the conceptual experimental scheme of Figure 5, which shows the coherent superposition of two interfering beams of particles with opposite (±) electromagnetic properties (such as, charge, electric dipole moment, magnetic dipole moment, etc.), not spatially separated, passing on the same side of the singularity, without the need of encircling it. Advanced interferometers are capable of measuring the relative phase shift of the particles with ± electromagnetic properties, which is produced while the particles pass by and interact with the external potentials or fields of the singularity. New effects of the Aharonov-Bohm type make use of a coherent superposition of interfering beams of particles possessing opposite (±) electromagnetic properties. The paths a and b of Figure 4 are now no longer spatially separated and lie on the same side of the electromagnetic flux density δΦem . While interacting with the potentials or fields of δΦem , the phase shift accumulated along the paths is measured by an interferometer. In the figure, we show


543

arrangement of Figure 5. A particle with state + travels on path a and a particle with state − travels on path b. The generic interaction momentum Q has the same strength for the particles possessing opposite (±) electromagnetic properties, but is of opposite sign. The phase difference accumulated along the paths is δφ =

Fig. 5. (Color online) New effects of the Aharonov-Bohm type make use of a coherent superposition of particles possessing opposite (±) electromagnetic properties. The paths a and b of Figure 4 are now no longer spatially separated and lie on the same side of the electromagnetic flux density δΦem . While interacting with the potentials or fields of δΦem , the phase shift accumulated along the paths is measured by an interferometer. In the figure, we show a beam of charged particles passing near a solenoid, where a particle can be either in the charge state + or −. The phases corresponding to the two ± states are oppositely shifted by the vector potential A.

beams of charged particles passing near a solenoid, where a particle can be either in the charge state + or −. The phases corresponding to the two ± states are oppositely shifted by the vector potential A. A coherent superposition of two beams of neutral particles with opposite (±) magnetic dipole moment was used by Sangster et al. [11–13] to test the AC effect, using these new techniques. In that work, the singularity (the line of electric charges producing the external field E) was replaced by the uniform E field of a parallel plate capacitor at high potential difference. The advantages of this arrangement are immediate: the electromagnetic interaction between the magnetic dipole of the particle and the external E field is stronger because the E field of the capacitor can be much more intense than that of the charged wire. Furthermore, the particles can travel inside the capacitor in the presence of the E field for a longer time and over a longer path, which, depending on the details of the interferometric arrangement, can be on the order of meters. Thus, Sangster et al. were able to corroborate the AC effect with high precision and even test the dependence of the quantum effect on the velocity of particles. The possibility of testing the effect for electric dipoles using the new techniques, and the feasibility of realizing a correspondingly precise interferometer, were investigated by Dowling et al. [14], and by Spavieri [15–19]. Furthermore, a generalization of the AB effect for ± charged particles was elaborated by Spavieri [18,19], who also dealt with the problems of gauge and phase invariance involved in these new effects. Let us consider the argument given in references [18,19] about what is being measured by the interferometer in the

1

a

2 Q·d = Q·d, a b

Q·d −

(42)

where the integral 2 a Q·d on path a equals the closed line integral C Q·d, corresponding to the case of particles with the same electromagnetic properties encircling the singularity as in Figure 4. The observable quantity that is actually measured in these effects is the phase shift variation 2 2 Δφ = δ φ − δ φ0 = Q1 · d − Q2 · d, (43) a a where usually Q2 = 0. For example, in the measurement of the AC effect by Sangster et al. [11–13], the phase shift variation δ φ − δ φ0 is obtained by changing the sign of the applied external E field (this implies Q2 (−E) = −Q1 (E) in Eq. (43)). If a phase transformation is performed, the quantity Δφ of equation (43) transforms as 2 (Q1 − ∇χ) · d − (Q2 − ∇χ) · d a a 2 (Q1 − Q2 ) · d = Δφ, (44) = a

2 Δφ =

i.e., the observable Δφ, measured in these quantum effects, is phase invariant [18,19]. Both of the two terms (the integrals) on the rhs of equation (44) are not observable quantities and, taken separately, can be set to zero by a phase transformation. However, their difference Δφ − Δφ is an observable physical quantity that cannot be gaugeor phase-transformed to zero. So far, of all the effects of the AB type, only the AB and AC effects have been corroborated experimentally [61,63]. In particular the AC effect has been corroborated by Sangster et al. [11–13] also with the new techniques, confirming that no phase invariance problems arise for quantum effects involving coherent beams of particles with ± electromagnetic properties. Although the other effects still await experimental realization, it is conceivable that they would be corroborated given the unitary view of all the effects of the AB type presented above. Therefore they can be considered suitable candidates for table-top experiments aimed at determining the photon mass. Moreover, a variant of the Spavieri effect for ± charged particles has been performed already in solid state physics as a test the electron-hole AB effect [69]. Since the electron-hole is a special case of ± charged particles, the experiment corroborates the view that even the effect for ± charged particles is a suitable candidate for a table-top experiment.

544


12 New effects of the AB type and the photon mass

As mentioned above, it has been pointed out [18,19,23] that the observable quantity in the AB effect is actually the phase difference e (48) (A − A0 ) · d Δϕ = c

12.1 The new techniques for the AC effect and the photon mass We recall that in the AC effect the phase shift is given by 1 ΔϕAC = 2 m × E · dx, (45) c where the standard expression of the electric field is modified if the photon mass is non-zero. In the case of the experiment by Sangster et al., the field E has to be calculated solving Proca’s equations for the case of a parallel plate capacitor. The calculations have been performed by Rodriguez [22] who was able to approximate the contribution to the phase shift, Δϕμγ , due to μγ = 0, as: Δϕμγ = ΔϕAC

(μd) 24

2

(46)

where d is the separation of the plates of the capacitor. Taking into account the precision of the experiment of Sangster et al. [11–13] and the value of the parameters that determine ΔϕAC , the following results were obtained [22]: 2 μ−1 γ 1 × 10 m

(47)

corresponding to a photon mass of mγ 5 × 10−42 g. The result (47) represents an improvement upon the result of Fuchs [10] who obtained it from the data of the experiment by Cimmino et al. [63], where a line of charge was used to produce the field E. Thus, result (47), valid for the case of the AC effect, indicates that better results for the photon mass are achievable by means of the new techniques. In a dedicated experiment, where the parameters are optimized, for example, by increasing the length of the capacitor and decreasing the separation d, it is conceivable that an improvement by 1 or 2 orders of magnitude for mγ could be achieved. 12.2 The AB effect for ± charged particles and the photon mass Let us now consider the modified AB effect of Spavieri [18,19] that involves beams of ± charged particles passing on one side of the solenoid. The table-top experiment is still conceptually based on the arrangement of Figure 5 where now the flux tube is no longer a thin solenoid but its radius can be ideally enlarged to a size of >1 m. If the length of the interferometric path is extended to the order of several meters, the effect is magnified by placing a line of solenoids parallel to the path of the particles, instead of using a single much larger solenoid. For this effect, the corrections to B and A due to a nonvanishing photon mass are those calculated by Boulware and Deser in their approach to the magnetic AB effect.

where the integral can be taken over an open path. The considerations on phase (and gauge) invariance presented in the previous section indicate that, in analogy with the AC effect for a coherent superposition of beams of magnetic dipoles of opposite magnetic moments ±μm [11–13] and the effect for electric dipoles of opposite moments ±d [14], an effect of the AB type for a coherent superposition of beams of charged particles with opposite charge ±q is theoretically feasible [18,19]. Depending on the interferometric technique used [11–14], the length of the interferometric path, along which the electromagnetic interaction takes place, can be of the order of a few cm up to a few m. Although the interferometry and other technologies needed for testing the effects of ±q charged particles needs further development, a step in this direction has already been made [18,19] by showing that, at least in principle and as far as gauge invariance requirements are concerned, this measurement is physically feasible. Therefore, using this effect in a table-top experiment analogous to that of BD, we determine its relevance in eventually establishing a bound for the photon mass mγ . In the experimental set ups detecting the traditional AB effect there are limitations imposed by the suitable type of interferometer related to the electron wavelength, the corresponding convenient size of the solenoid or toroid, and the maximum achievable size ρ of the coherent electron beam encircling the magnetic flux [9]. In the analysis made by BD, the radius of the solenoid is a = 0.1 cm, and ρ is taken to be about 10 cm, implying that the electron beam keeps its state of coherence up to a size ρ = 102 a, i.e., about fifty times the solenoid diameter. The advantage of the new approach for the ±q beam of particles is that the dimension of the solenoid has no upper limits and is conditional only on the practical limits of the experimental set up, while the size of the coherent beam of particles plays no important role. In order to calculate the line integral appearing in equation (48) Spavieri and Rodriguez [23] derived an analytical expression for A(x). The result was

μ ρ j a2 1 γ 2 ρ + ln , Aϕ = j (μγ a) 2 ρ 2 2 2 where a is the radius of the solenoid and j its current density. Taking the path of the particles to be along the x axis for a path length 2x with x y, we find ϕ0 = C Amγ = 0 · d −(π/2)a2 j. As done by BD, we neglect the small corrections due to the contribution of the logarithms, so that 2the contribution due to mγ yields A · d = (j/2)(μγ a) yx for the same path length 2x. C Consequently, from equation (48) and reference [23], the observable phase shift variation is Δϕ = 2j(μγ a)2 yx and Δϕ 4 = − μ2γ xy. ϕ0 π

(49)


Following BD [9] we set Δϕ ≥ 2πε = 2π × 10−3 where ε is the precision of the measurement. The value of μγ at which the effect is just observable is fixed by the relation (4/π)μ2γ xy = 2πε/ϕ0 and can be compared with the corresponding one by BD [9]. The question then becomes: what would be the size of the solenoid needed to achieve a photon mass limit of the order provided by the best classical approaches, such as those by Luo et al. [52] and Ryutov [59]? We estimate μγ with respect to μγBD for an ideal experimental set up that, apart from considerations of cost, is realistically within reach of present technology. For a vector potential produced by the large, magnet of a cyclotron-type solenoid (radius a = 5 m and length or height D several times the radius), we estimate ϕ0 /ϕ0BD a2 /(aBD )2 = 52 /(10−3 )2 . For a path of x = 6a = 300ρ at the distance y = 80ρ, Spavieri and Rodriguez [23] obtain μ−1 γ

=

μ−1 γBD

8 ϕ0 xy π ϕ0BD ρ2

1/2

13 106 μ−1 cm, γBD 2 × 10

which would be an improvement of 6 orders of magnitude with respect to the approach of BD based on standard techniques. With their table-top experiment, BD obtained the value μ−1 γBD = 140 km which is equivalent to mγBD = 2.5 × 10−45 g. Therefore, the new limit of the photon mass that appears to be achievable with the proposed new present quantum approach is mγ 2 × 10−51 g

(50)

which would be of the same order of magnitude of that found by Luo et al. [52] and Ryutov [59]. 12.3 The scalar Aharonov-Bohm effect and the photon mass Having exploited the magnetic AB effect in the previous section, we consider now the scalar AB effect. In this effect charged particles do not encircle a singularity, as happens for the previous effects of the AB type discussed above. In fact, for the actual test of the scalar AB effect, a long conducting cylinder of radius R is given a potential V for a time τ while electrons travel inside it. Since no forces act on the charges it is a field-free quantum effect. The standard phase ϕs acquired during the time of interaction is ϕs = 1 eV (t)dt. The feasibility of testing the photon mass with the scalar AB effect has been confirmed by the recent work of Neyenhuis et al. [24], lending support to the quantum approach in general. They consider a table-top experiment equivalent to the scalar AB effect, and project attainment of a photon mass limit of mγ 9 × 10−50 g.

(51)

While reviewing here the original approach by Neyenhuis et al., we will also try to suggest optimized parameters for an improved version of the experiment. If the photon

545

mass does not vanish the potential is modified according to Proca equation. Gauss’ law is modified and the electric potential Φ obeys the equation ∇2 Φ − μ2γ Φ = 0, with the boundary condition that the potential on the cylinder be V . Following reference [24], the solutions for the potential Φ(ρ) in cylindrical coordinates are expressed in terms of the modified Bessel functions of zero order, I0 (μγ ρ) and K0 (μγ ρ) which are regular at the origin and infinite, respectively. The acceptable, approximate solution is μ2γ 2 (52) Φ (ρ) V 1 + ρ − R2 2 where only the first two terms of the expansion of I0 (μγ ρ) have been considered [24]. For two interfering beams of charges passing through separate cylinders, the relative phase shift is 1 δϕs = (53) e [V1 (t) − V2 (t)] dt where V1 (t) and V2 (t) are the potentials applied to cylinders 1 and 2, respectively. In the simple case of one beam travelling inside cylinder 1 and the other travelling outside it (V2 (t) = 0) for a short time interval τ , the phase shift contribution due to μγ , Δϕμγ = δϕ − δϕs is Δϕμγ = −

τ eμ2γ 2 ρ − R2 V 4

(54)

where V = V1 (t) − V2 (t). Interferometric experiments may be performed with a precision of up to 10−4 . Therefore, following the approaches of BD and SR, we set Δϕμγ = ε = 10−4 . Also, we suppose that beam 1 travels nearly at the centre of the cylinder (ρ R) so that πV τ R μ−1 . (55) γ = 2 ε(h/2e) The following values may be used to estimate μ−1 γ : V = 2 7 −15 −2 10 V, h/2e = 2.067 × 10 T m , τ = 5 × 10 s and R = 27 cm. The corresponding range of the photon mass would be 13 μ−1 cm (56) γ = 3.4 × 10 which would lead to an improved photon mass limit of mγ = 9.4 × 10−52 g,

(57)

but one would have to carefully justify the values used above for τ and R, which are both quite large (Spavieri and Gillies [70]). It is interesting to compare the strength of the AB phase of the scalar AB effect with that of the magnetic AB effect. The scalar AB phase may be expressed as eV τ /, while the magnetic AB phase is eAL/(c), and the link between the particle’s classical path is L = τ v with v its speed, assumed to be uniform. According to special relativity, magnetism is a second order effect of electricity,

546


therefore in normal conditions the strength of the coupling eA/c is smaller than the coupling eV . As a consequence of this, the phase variation due to the finite photon mass should be smaller in the magnetic than in the scalar AB effect. In other words, the scalar AB effect should yield a better limit for the photon mass than the magnetic AB effect. However, the above consideration is valid if in the actual experiments we have comparable path lengths, i.e., if τ L/v. In the table-top experiment by SR [23] L is of the order of several meters. Choosing as charged particles heavy ions, for example 133 Cs+ , their speed could be 27 m/s [71]. With this speed and L = 1.35 m for the cylinder length, we get τ = 5 × 10−2 s for the time of flight inside the cylinder. Since τ L/v, the improved result (56) obtained by exploiting the scalar AB effect would appear to be justified. However, the high values chosen for R and L imply that the charged particle beams will have to keep their state of coherence through an extended region of space L = 1.35 m during the interferometric measurement process, while in standard interferometry the path is of the order of at most a few cm. Thus, technological advances are needed in this respect, as also mentioned in the article by SR [23] and the references cited therein. Actually, it is conceivable that one could extend the techniques of references [11–14] to the scalar AB effect for a coherent superposition of beams of charged particles with opposite charge state ±q, as suggested by SR in reference [23]. This might possibly lead to even better limits for the photon mass. In fact, by means of these techniques it is reasonable to expect that the particle paths may be 102 times those considered above. Thus, the time of flight τ becomes 102 times larger. Although the technical details will be given elsewhere, we anticipate that further improvement can be obtained by bending the particle path into a circular one, as in the case of ± charged particles in a cyclotron. In practice, a long conducting cylinder of potential V could be bent to form a hollow toroid inside which the particles would move in a circular path. In this case, τ may conceivably be increased ≈104 times. Thus, we project that, for this table-top experiment, a photon mass limit of the order of mγ 10−54 g could eventually be within the realm of possibility.

13 The electron g factor and the photon mass limit Always for the purpose of establishing an improved photon mass limit, we keep in the quantum domain, but switch to effects other than those of the AB type. According to Dirac, the point electron executes a periodic motion at the speed of light, the Zitterbewegung of Schr¨ odinger, from which the electron spin arises. The spin motion possesses an angular frequency ωs , while the

motion of the electron in an external magnetic field possesses the cyclotron frequency ωc . In a very precise experiment involving the electron magnetic moment from geonium spectra, Van Dyck et al. (VSD) [8] measured the electron g factor, given by g=2

μs ωs =2 , ωc μB

(58)

where μs is the spin magnetic moment and μB the Bohr magneton. In order to perform the measurement, VSD succeeded in confining a single electron in a Penning trap where, in the presence of an external magnetic field B0 , the particle has nearly circular motion at the frequency ωc while its spin precesses at the frequency ωs . The electron motion is brought nearly to rest, although it can oscillate in the axial direction at the frequency ωz in the presence of the electric potential U0 of the trap. In this configuration, VSD determined the observed cyclotron frequency ωc = ωc − δe and the anomaly frequency ωa = ωs − ωc which are related by the small electric shift δe computed using the measured axial frequency and δe = ωz2 /2ωc . The result obtained by these authors, which agrees well with the prediction of quantum electrodynamics, is g/2 = 1.001159652200(40).

(59)

Pushing this value as far as one can and assuming a precision up to the last decimal place, the error involved is Δg/g 10−14 . Our aim here is to investigate how this result is modified in massive electrodynamics where, according to the Proca equations, the electromagnetic fields are modified if the photon possesses a nonvanishing mass mγ . In this case, the resulting g factor may depend on mγ , so that we would have g(mγ ) instead of g. The difference g(mγ ) − g could be evident if it were greater than the experimental error Δg. Therefore, we may assume that g (mγ ) − g ≤ Δg 10−14 ,

(60)

a relation that can be used to set the limit of the photon mass. It is rather cumbersome to solve the Proca equations for the electromagnetic fields of the Penning trap. Nevertheless, without solving the Proca equations, in the following we attempt to provide an idea of the limit of mγ achievable by deriving a very rough estimate of it. Let us first consider the effect of mγ = 0 on the external magnetic field B0 . The spin frequency is related to the field B0 by (1/2)ωs = μs B0 while for the electron of mass me and charge e the cyclotron frequency is ωc = (e/me c)B0 . In short-range massive electrodynamics, the field becomes a function of mγ , i.e., B0 (mγ ) as in the case of the uniform magnetic field of a solenoid discussed by Boulware and Deser for the Aharonov-Bohm effect. The factor g in equation (58) then becomes g = 2μs B0 (mγ )/μB B0 (mγ ) 2μs /μB because the photon mass mγ would affect the spin precession and the cyclotron frequency nearly in the same way. Thus, we do


not expect significant changes for g(mγ ) due to the modification of the field from B0 to B0 (mγ ) and neglect, to a first approximation, the change B0 (mγ ) − B0 . However, if mγ = 0, the usual Coulomb electric potential U of a point charge is modified to the Proca-Yukawa potential U e−μγ r where μγ = mγ c/ = (λC /2π)−1 , λC is the Compton wavelength of the photon, and r is the position of the electrically charged test particle. In fact, according to VSD, in the Penning trap the electron oscillates in the axial direction at the frequency ωz in the presence of an electric potential well of strength U0 , since the frequency ωz is related to U0 by ωz2 = eU0 /me Z02 , where Z0 R is the size of potential well. In these conditions and if mγ = 0, U0 becomes U0 (mγ ) and the axial frequency ωz becomes ωz (mγ ). As a very crude approximation, we may write U0 (mγ ) = U0 e−μγ r where now r R is taken to be of the order of the size R of the trap, so that eU0 e−μγ r = ωz2 e−μγ r me R 2 1 ωz2 1 − μγ r + (μγ r)2 . 2

ωz2 (mγ ) =

(61)

However, this result is speculative because, even assuming that the position U0 (mγ ) = U0 e−μγ r were exact and viable in (61), the correct approach involves calculating the effect of the Yukawa potential U0 e−μγ r at the varying position r of the electron in the Penning trap while it is in motion. Near the sources of U0 at r 0, the Yukawa potential U0 e−μγ r U0 . Thus, assuming oscillatory motion about the average position r = 0, in (61) we can expand e−μγ r up to the second order in μγ r only. At least a hint of what could be the effect of the photon mass on the g factor should be obtainable from our very rough approximation (61), where we first assume that the term in μγ r is dominant, and then consider the more likely case where this term vanishes and the term in (μγ r)2 becomes the dominant one. With the help of equation (58) and the relations ωc = ωc − δe , ωa = ωs − ωc , and δe = ωz2 /2ωc , one gets ωa + ωc ωs =2 , (62) g=2 ωc ωc + ωz2 /2ωc while by equation (61) with (μγ r)2 0, ωa + ωc ωz2 g 1 + μγ r 2 . g (mγ ) 2 ωc + ωz2 /2ωc (1 − μγ r) 2ωc (63) Using the data of VSD, in equation (63) we may take r R = 10−1 m, ωz /ωc = (60 MHz/51 GHz) 1.2 × 10−3 . With μγ = mγ c/, expression (60) and (63) provides the relation that establishes the limit of the photon mass Δg = g (mγ ) − g =

mγ c ωz2 R 2 10−14 . 2ωc

(64)

From equation (64), the sought-for limit would be mγ =

2ωc2 −14 10 0.5 × 10−48 g, cR ωz2

(65)

4 corresponding to the range μ−1 γ = /mγ c 0.7 × 10 km.

547

Comparing these results with the ones obtained by Boulware and Deser with their approach based on the quantum Aharonov-Bohm effect, we see that the limit set by equation (65) would constitute an improvement of about 3-4 orders of magnitude over that obtained by Boulware and Deser. However, it is important to note that in equations (61) and (63) we have assumed the best-case scenario, where terms of the first order in μγ r are relevant. Exact calculations may show that first order terms cancel and only the second order terms (μγ r)2 are important. In this case, we find μ−1 γ 1 km and our projections for the photon mass limit would be reduced by a factor of about 103 , so that we would arrive at roughly the same result as Boulware and Deser (mγ 10−45 g), but at a somewhat better limit than that of Fuchs (mγ 10−41 g), being the results by Boulware and Deser, and Fuchs, obtained exploiting the effects of the AB type. In any case, it might be worthwhile to consider what experimental parameter could be modified if a dedicated experiment employing the g factor measurement were realized for the purpose of improving the limit (65). Tentatively, we may expect that, taking due care, the precision could be improved by perhaps a factor of 10. Then, we could consider modifying the size R and the value of the potential U0 in such a way that the ratio ωc /ωz could be reduced by a factor of 10−2 or 10−3 . With these improvements and when terms of first order in μγ r are relevant, a better mass limit of about mγ 10−54 g

(66)

could arguably be achieved. If only the second order terms (μγ r)2 are important, the mass limit would instead be mγ 10−50 g. Both these limits are on the order of (or slightly better than) the limit proposed as being achievable by Spavieri and Rodriguez with their suggested tabletop approach to the AB effect (mγ 10−51 g). These values for the photon mass limit place this QED approach within the range of those based on the quantum effects of the AB type, and they indicate that different quantum approaches are, at least in principle, more or less comparable. The point is that a very small photon mass limit seems confirmable when the approaches are extended to a wider quantum scenario. Of course, the result (66) applies only for the projections of potential results from a proposed table-top experiment, but if successful, it could be 3 order of magnitudes better than the best limits achieved with the classical approaches of Luo et al., Ryutov and the quantum approaches suggested by Spavieri and Rodriguez (magnetic AB effect) and Spavieri and Gillies [70] (scalar AB effect). Thus, the conclusion we can draw is that quantum approaches, as well as classical, are potentially competitive candidates for setting the limits of the photon mass and should be taken into consideration. A graph showing some of the photon mass limits discussed in this paper, is shown in Figure 6.

548


Fig. 6. (Color online) Graph showing some of the photon mass limits discussed in this paper.

14 Range of applications and conclusions Studies of Coulomb’s law form the foundation of far more than electrostatics alone, because by incorporating special relativity and the principle of superposition, its generalization leads to Maxwell’s equations [72]. The precision of the experimental tests are increasingly stringent and, in recent times, go from measured values of the inversesquare departure parameter ε = (2.7 ± 3.1) × 10−16 for the direct laboratory test of Williams et al. [44] to the very tight limit on the photon mass of m < 1.2 × 10−51 g from the indirect test carried out by Luo et al. [52]. Such studies have provided a very firm foundation for the subsequent work in this field, and help point the way towards ever more interesting investigations still to come. For instance, it is interesting to consider the possibility of space-based (low-earth orbit or otherwise) experimental searches for a non-zero photon mass. In fact, an interesting analog situation is that of measurement of the Newtonian gravitational constant, G, which for centuries now has been the province of terrestrial torsion pendulum experimentation. However, the amelioration in space of the same class of natural and man-made disturbances that plague the precision measurement of G might also produce benefits for measurements of mγ that involve high sensitivity electro-thermo-mechanical apparatus. This might include, for instance, profound reductions in vibrational noise, parasitic electromagnetic fields, and so on. The rather obvious issue is one of cost, as space-based experimentation requires very substantial planning, testing and launch expenditures. As a result, it is likely that work in terrestrial laboratories will continue to set the standards in experimental measurements of mγ for the foreseeable future. Some far-reaching implications of the nonvalidity of Coulomb’s law or, equivalently, the existence of a nonzero photon mass foreseen by Proca’s theory, are the wavelength dependence of the speed of light in free space [73],

deviations from exactness in Ampère law [74], the existence of longitudinal electromagnetic waves [75], and the presence of an additional Yukawa potential [45,76] for magnetic dipole fields as discussed above. Other research hints at the effects that a photon mass may have during early-universe inflation, which might result in magnetic fields on a cosmological scale [77]. There are also investigations of bounds on the photon mass in relationship to thermodynamic phenomena such as blackbody radiation [78]. It is not known what the ultimate impact of a finding of mγ = 0 might be on, e.g., the quest to unify all of the forces, or on our understanding of the behavior and evolution of the early universe. However, it is clear that an unequivocal finding of a non-zero rest mass for the photon would have far-reaching implications for all of modern physics. It is equally clear that the relative simplicity and attractiveness of the theoretical physics for a universe in which the photon rest mass is identically zero cannot be allowed to deflect or deter interest in the basic empirical search for such a quantity, independent of any other motivation that there might be for it. Regarding the limits on the validity of Coulomb’s law with respect to inter-charge distance (or “range”), r, the exactness of the inverse square nature of it has also been verified macroscopically in solar-system measurements (e.g., those involving Jupiter) for distances on the order of r ≈ 106 km. From a microscopic perspective, the limits found with scattering experiments show validity of the law for distances at least down to the atomic (r 10−8 cm = 1 ˚ A) and nuclear (r 10−13 cm) scales, and the cryogenic test extends its validity also to matter at very low temperatures. As per the central theme of this article, we step outside the classical domain and consider quantum approaches to searches for a photon mass. After reviewing the table-top approach of Boulware and Deser [9] to the photon mass,


we verified its applicability to other effects of the AB type. Within the context of the review, we recalled that the interaction momenta Q of all the effects of the AB type arise from the variation of the momentum of the interaction em fields Pe , a property that leads to a unitary view of these effects [15–19,70]. We also considered the new interferometric techniques developed for coherent superposition of beams of interfering particles of opposite electromagnetic properties, and their applications to these quantum effects [11–14,18,19]. We conclude that the effect using beams of charged particles with opposite charge state ±q for the magnetic AB effect, and the effect for charged particles moving in a uniform electric potential (the scalar AB effect), are a good candidates for determining more stringent limits on the photon mass. Using a quantum approach to evaluate the limit of mγ with these effects, and supposing that high precision interferometric technology for coherent beams of particles with opposite electromagnetic properties could be employed, we projected that a bench-top experiment might yield a limit as low as mγ 10−54 g which, if achieved, would be a significant improvement on the limits obtained recently with both classical and quantum approaches. Also the exploitation of quantum electrodynamics effects, such as the very precise electron g-factor experiment [8], seems promising and, in any case, extends the scenario of the quantum approaches. In any event, advances in this area suggest that quantum approaches to the determination of photon mass limits are feasible and may eventually compete with or surpass the traditional classical methods. One can imagine a variety of questions that might follow in the wake of ever higher precision bounds on (or a potential value of) mγ ; for instance, that of the spatial and temporal isotropy of mγ . Given the substantial interest in the possible variation of the fine structure constant with epoch, it might not be unreasonable to expect similar variation in mγ if such a quantity exists. Likewise for spatial isotropy: if the wavelength of photons is subject to gravitational red shift (as in the classic Pound-Rebka experiment), what implications might there be for mγ in similar scenarios? Of course, compiling a long list of such questions is relatively easy, but arriving at non-speculative answers to any of them will certainly be another matter.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

30.

31.

32.

33. This work was supported in part by the CDCHT (Project C1413-06-05-A), ULA, Mèrida, Venezuela. We are grateful to the Postgrado en F´ısica Fundamental (ULA) and to A. Khoudeir and R. Gait´ an for helpful comments and suggestions.

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