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Electromagnetism as Quantum Physics

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Abstract

One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function. Here I examine whether Maxwell’s equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon’s quantum wave function. I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good in Phys Rev 105(6):1914–1919, 1957). This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations. The fact that such a quantum interpretation of Maxwell’s equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon.

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Notes

  1. More accurately: Maxwell’s equations and the Lorentz force law giving the force exerted upon a charged body were already special relativistic, though the reaction of bodies to forces from the electromagnetic field must be understood properly (as the rate of change of relativistic momentum) if the theory as a whole is to be relativistic.

  2. See Bjorken and Drell [10, Sect. 1.3].

  3. The Dirac equation is treated as part of a classical field theory in, e.g., Hatfield [34], Valentini [69, Ch. 4], Peskin and Schroeder [55], Ryder [60, Sect. 4.3], Greiner and Reinhardt [31, Ch. 5], Sebens [63] and it is treated as part of a quantum particle theory in, e.g., Frenkel [27], Dirac [18], Schweber [61], Bjorken and Drell [10], Thaller [68]. There is much to be said about how one moves from either one of these options to a full quantum field theory. However, as the focus of this paper is on the physics of the photon, I will not say much about that here. Let me just mention that at some point along the road to quantum field theory, one must shift to thinking of Dirac’s equation as part of a theory of both electrons and positrons.

  4. Classical Dirac field theory has not proved useful as a theory of macroscopic physics. This is because, unlike classical electromagnetism, classical Dirac field theory does not emerge as a classical limit of quantum field theory [20, p. 221]. However, the theory is still of interest to physics because, like classical electromagnetism, classical Dirac field theory serves an important foundational role in a field approach to quantum field theory (as mentioned above).

  5. The question of whether we should take a field or a particle approach to various quantum field theories has been a popular topic of discussion in the philosophy of physics literature. See, e.g., Malament [46], Halvorson and Clifton [33], Fraser [26], Baker [4, 5], Struyve [67], Wallace [71].

  6. The relativistic Bohmian dynamics for the electron are presented in Bohm [12], Bohm and Hiley [13, Sect. 12.2], Holland [37, Sect. 12.2].

  7. This field approach dates back to Bohm [11]. For some discussion as to why the field approach has been taken instead of the particle approach, see Bohm and Hiley [13, Ch. 11], Holland [36], Holland [37, Sect. 12.6], Struyve [67], Flack and Hiley [24].

  8. Einstein’s hope for what looks like a Bohmian theory of the photon has been mentioned by Holland [37, p. 538], Kiessling and Tahvildar-Zadeh [41]. See also Pais [53, pp. 440–443], Wigner [75, p. 262].

  9. Lindell [45, Ch. 1] discusses the mathematical properties of complex vectors.

  10. Kiessling and Tahvildar-Zadeh [41, Appendix A] present a historical case as to why \(\vec {F}\) should be called the “Weber vector” (after Heinrich Martin Weber) and not the “Riemann-Silberstein vector” (a name introduced by Bialynicki-Birula).

  11. The idea that \(\vec {F}\) might be interpreted as a wave function for the photon is explored in: Rumer [59], Archibald [3], Good [28, 29], Good and Nelson [30, Ch. 11], Moses [50], Kobe [42], Esposito [23], Holland [38], Raymer and Smith [57], Keller [40], Cugnon [17], Chandrasekar [16], Norsen [52, p. 117], Kiessling and Tahvildar-Zadeh [41], and a number of publications by Bialynicki-Birula and Bialynicka-Birula, including [7,8,9]. The earliest sources for this quantum interpretation of \(\vec {F}\) appear to be Rumer [59] and Majorana (who worked on this between 1928 and 1932; see [49]). For more on the role of the Weber vector in classical electromagnetism, see Weber and Riemann [72], Silberstein [65, 66], Hestenes [35], Landau and Lifshitz [43, Sect. 25], Dresden [19, Sect. 16.II], Riesz [58].

  12. Philosophers of physics have wondered why the magnetic field should be flipped under time reversal and also why quantum wave functions should be complex conjugated under time reversal [2, 14]. Thinking of the Weber vector as a quantum wave function suggests that these issues may be connected: flipping the sign of \(\vec {B}\) is the same operation as complex conjugating \(\vec {F}\).

  13. These matrices appear in a number of the references mentioned in footnote 11, though note that in Good [28] the definition differs in sign.

  14. One might wonder why the putative higher-spin photon wave function F (or \(\phi \) in Sect. 4) has fewer components than the lower-spin electron wave function \(\psi \). The reason for this is that \(\psi \) has enough degrees of freedom to describe two different spin-1 / 2 particles—the electron and the positron.

  15. One could alternatively seek alignment by introducing a wave function for many electrons (obeying a multi-particle Dirac equation) and comparing this to F. However, such a multi-particle wave function would be defined over configuration space, unlike F which is defined over physical space.

  16. The idea that a weak electromagnetic field might actually be a quantum wave function is provocative and its metaphysical implications are well worth exploring further, as there has been much discussion of the extent to which the quantum wave function resembles a classical field in debates about the ontological status of the wave function (e.g., [51]).

  17. Holland [37, pp. 540–541, 545] sees the fact that the number of photons can be indefinite as an obstacle to finding a Bohmian theory in which photons follow definite trajectories (to be contrasted with a Bohmian approach in which the electromagnetic field has a definite configuration). But, it is possible to have a Bohmian theory in which there is a true number of particles even though the quantum state is in a superposition of different numbers of particles. See, for example, the way particle creation and annihilation is handled in Dürr et al. [21].

  18. See Akhiezer and Berestetskii [1, Sect. 1.2].

  19. It follows from (15) that only complex-valued functions \(\vec {{\widetilde{E}}}(\vec {k})\) and \(\vec {{\widetilde{B}}}(\vec {k})\) satisfying \(\vec {{\widetilde{E}}}^*\!\!(\vec {k})=\vec {{\widetilde{E}}}(-\vec {k})\) and \(\vec {{\widetilde{B}}}^*\!\!(\vec {k})=\vec {{\widetilde{B}}}(-\vec {k})\) can correspond to real-valued electric and magnetic fields. Because \(\vec {F}(\vec {x})\) is complex-valued, \(\vec {{\widetilde{F}}}(\vec {k})\) does not have such a symmetry.

  20. This condition is discussed further in Good and Nelson [30, pp. 583–584].

  21. This expression for total photon number appears in Landau and Peierls [44], Good [28], Zel’dovich [76], Bialynicki-Birula [8, pp. 318], Sebens [62].

  22. The move from the first line of (21) to the second makes use of the symmetry noted in footnote 19.

  23. On this picture: Integrating the Weber vector’s magnitude squared (over \(8 \pi \)) in either position or momentum space yields the expectation value of the energy, not the total energy.

  24. This name comes up in Mandel and Wolf [47, Sect. 12.11.5], Bialynicki-Birula [8], Keller [40].

  25. Good [28] does not restrict his attention to states of the electromagnetic field describing a single photon. He thus treats \(\phi ^\dagger \phi \) as a number density, not a probability density—see (27).

  26. Pauli [54, p. 191], Akhiezer and Berestetskii [1, Eq. 1.6], Mandel and Wolf [47, p. 637] introduce somewhat similar photon wave functions which also involve removing an energy weighting. Akhiezer and Berestetskii caution against Fourier transforming their momentum-space photon wave function and interpreting the result as a position-space photon wave function (see also [54]). Their concern is that “the presence of a photon can be established only as a result of its interaction with charges” and that the force a charge will feel at a given point is determined by the electric and magnetic fields at that point. Since a wave function like \(\phi \) at a point \(\vec {x}\) is dependent on the electric and magnetic fields at distant points—see (24)—they argue that it will not give the right interactions. They conclude that “the concept of probability density for the localization of a photon does not exist.” This argument is a bad mix of classical and quantum ideas. The interaction is modeled classically and yet quantum probabilities are expected to emerge. When the electromagnetic field of a single photon hits the screen at the end of a double-slit experiment like the one depicted in Fig. 1, its interaction with the screen is not a matter of weak fields exerting weak forces all over the screen. The photon is found at just one location. Via (27), the position-space photon wave function tells us how likely each location is.

  27. Because \(\vec {\phi }\) at \(\vec {x}\) is not fixed by \(\vec {E}\) and \(\vec {B}\) at \(\vec {x}\), one might conclude that \(\vec {\phi }\) is not a “local” field. If the electromagnetic field is fundamental and the photon wave function is defined from it, I think it would be reasonable to classify \(\vec {\phi }\) as non-local. However, one could invert (24) and express the electromagnetic field non-locally in terms of the photon wave function. If we take the wave function to be fundamental and the electromagnetic field to be defined from it, then it is the electromagnetic field which should be deemed non-local. Not knowing how to settle such a question of fundamentality at this point, I think it is best to just keep in mind that switching from one field to the other is not a local maneuver.

  28. The idea that probability density is proportional to \(F^\dagger F\) has been proposed by Archibald [3], Wesley [73], Esposito [23].

  29. See Bohm [12], Bohm and Hiley [13, Sect. 12.2], Holland [37, Sect. 12.2].

  30. See, e.g., Bjorken and Drell [10, Sect. 1.3 and 2.2], Schweber [61, Sect. 4c].

  31. Kiessling and Tahvildar-Zadeh [41] have recognized that the probability density and probability current defined from the Weber vector, (29) and (30), do not transform together as a four-vector and, because of this defect, concluded that the Weber vector is unacceptable as a wave function for the photon.

  32. The fact that the energy and momentum of the electromagnetic field do not form a four-vector is well-known. The energy and momentum form a four-by-four tensor (the energy-momentum tensor) when combined with the momentum flux density of the field.

  33. At least, neither is acceptably relativistic when paired with its natural probability density and current: (27) and (28) or (29) and (30).

  34. Of course, this pathology could be easily remedied by imagining space to be finite and imposing periodic boundary conditions.

  35. For derivation of the way electromagnetic waves transform under Lorentz boosts, see discussions of the relativistic doppler effect in, e.g., Einstein [22, Sect.  7], Griffiths [32, prob.  12.47], Pollack and Stump [56, pp.  467–468]. Alternatively, one can calculate the way these waves transform using the general transformation properties of the Weber vector [43, Eq.  25.5] and [30, Sect.  35].

  36. It is interesting to think about where problem with relativity arises if one approaches the situation from the perspective of the many-worlds interpretation. The basic physics just includes a wave function obeying the wave equation (25), and all of that is entirely relativistic. The claims about probability density and probability current, like those in (27) and (28), are not fundamental posits of the theory. They should be derived from the dynamics of the wave function, along with some (hopefully true) assumptions about probability and/or rationality (see, e.g., [6, 15, 64, 70]). If these derived probabilities don’t transform properly, that seems problematic—even if the fundamental dynamics are perfectly relativistic.

  37. See, e.g., Messiah [48, pp. 884–888], Itzykson and Zuber [39, Sect. 2-1-1], Hatfield [34, Sects. 3.1 and 3.4], Ryder [60, Sect. 2.2].

  38. Bohm and Hiley [13, p. 235] use a case in which the electric and magnetic fields are parallel and we boost along the axis they pick out to argue that the two above methods will not generally agree for the guidance equation derived from the Weber vector (33).

  39. According to (33), a photon in (47) will not be moving at all (because there is no energy flow). Thus, from the perspective of an observer moving with velocity \(u{\hat{z}}\), it should be moving backwards with velocity \(-u{\hat{z}}\). However, if you apply (33) to (49) you get a different photon velocity.

  40. Alternatively, instead of looking within the equations of classical electromagnetism, one could attempt to find a relativistic quantum particle theory for the photon by proposing new equations (see, e.g., [41]). Then, one would need to either integrate these equations into a non-standard way of understanding our existing quantum field theory for the photon, or, put forward and defend an appropriately modified quantum field theory based on this new quantum theory of the photon.

  41. There is reasonably wide consensus that such an interpretation is unavailable, though the reasons given for this conclusion are varied. See the references in Kiessling and Tahvildar-Zadeh [41].

  42. Bohm and Hiley [13] treat bosons as fields and fermions as particles in their chapters on extending Bohmian mechanics to relativistic quantum field theory.

  43. From a field perspective, Sect. 4 shows that you can make electromagnetism look very similar to classical Dirac field theory by rewriting the equations of electromagnetism in terms of \(\phi \) (which does not have to be interpreted as a quantum wave function; it can be seen instead as an unusual way of writing the state of the classical electromagnetic field).

  44. Along similar lines, others have noted that the fact that we cannot interpret the Klein–Gordon field as a single-particle relativistic wave function (discussed in Sect. 5) suggests that we ought to interpret both the Klein–Gordon and Dirac equations as giving the dynamics for classical fields when we are using them to build quantum field theories. See Ryder [60, Ch. 4], cf. Fleming and Butterfield [25, Sect. 3].

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Acknowledgements

Thank you to Steve Carlip, Sheldon Goldstein, Chris Hitchcock, Michael Kiessling, Matthias Lienert, Ward Struyve, A. Shadi Tahvildar-Zadeh, Roderich Tumulka, David Wallace, and an anonymous referee for helpful feedback and discussion. This project was supported in part by funding from the President’s Research Fellowships in the Humanities, University of California (for research conducted while at the University of California, San Diego).

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Sebens, C.T. Electromagnetism as Quantum Physics. Found Phys 49, 365–389 (2019). https://doi.org/10.1007/s10701-019-00253-3

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