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The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics

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Abstract

In electrostatics, we can use either potential energy or field energy to ensure conservation of energy. In electrodynamics, the former option is unavailable. To ensure conservation of energy, we must attribute energy to the electromagnetic field and, in particular, to electromagnetic radiation. If we adopt the standard energy density for the electromagnetic field, then potential energy seems to disappear. However, a closer look at electrodynamics shows that this conclusion actually depends on the kind of matter being considered. Although we cannot get by without attributing energy to the electromagnetic field, matter may still have electromagnetic potential energy. Indeed, if we take the matter to be represented by the Dirac field (in a classical precursor to quantum electrodynamics), then it will possess potential energy (as can be seen by examining the symmetric energy-momentum tensor of the Dirac field). Thus, potential energy reappears. Upon field quantization, the potential energy of the Dirac field becomes an interaction term in the Hamiltonian operator of quantum electrodynamics.

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Notes

  1. See [1, ch. 27]; [2, ch. 5]; [3, Sect. 2.4.4].

  2. Sir John Herschel and Ernst Mach regarded potential energy as unreal energy and reacted to the above problem by downgrading conservation of energy, with Mach describing it as a mere “housekeeping principle” [14, p. 191].

  3. Lange [2, ch. 5] discusses this idea and its relation to field energy.

  4. By contrast, kinetic energy appears to be intrinsic (though that is not to say that it appears to be the only kind of energy that is intrinsic). The kinetic energy of a point particle in a particular frame is fixed by its mass and velocity in that frame. Thus, whether a particle’s kinetic energy is intrinsic turns on whether its mass and velocity are intrinsic properties. There are reasons why one might question whether the velocity of a particle is an intrinsic property of that particle. First, there has been philosophical debate as to whether a body’s velocity is an intrinsic property of that body at a particular instant in time [15,16,17]. Still, velocity might be an intrinsic property of a body even if it requires considering a time interval and thus is not an intrinsic property of that body at a moment. Second, one could argue that the velocity of a particle is not intrinsic because it depends on the relation of that body to the background spacetime in which it is moving. For our purposes here, let us put this concern aside and count velocity as intrinsic-enough to support a useful distinction between a particle’s intrinsic kinetic energy and any extrinsic potential energy it may have.

  5. In Sect. 5, will see that if charged matter is modeled by the classical Dirac field then its energy density depends on the electromagnetic field and is thus not entirely intrinsic. We can identify a potential energy density that captures the dependence on the state of the electromagnetic field. Although it is not apparent from just studying interactions between the Dirac and electromagnetic fields, the energy density of the electromagnetic field is arguably also not entirely intrinsic because, within general relativity, it depends on the gravitational field (or, you might say, on spacetime structure) [18]. That context is beyond the scope of this article, but I would be inclined to say that in general relativity the electromagnetic field has a gravitational potential energy density.

  6. Noting that neutral wires (with \(\rho ^q=0\)) can carry non-zero currents, Griffiths [3, p. 357] suggests in a footnote that we introduce separate charge densities and velocity fields for positive and negative charges, writing the current as \(\vec {J} = \rho ^q_+ \vec {v}^{\,q}_+ + \rho ^q_- \vec {v}^{\,q}_-\). This is a good idea, but to keep things simple I have left this complication out of the above equations.

  7. These two energy densities for electrostatics are discussed in, e.g., [1, Sect. 8.5]; [19]; [3, Sect. 2.4]; [20, Sect. 3.6].

  8. See [2]; [4, Sect. 4.2] for discussion of this argument and a similar argument from conservation of momentum.

  9. For discussion of these candidate energy densities for Newtonian gravity, see [22, Sect. 1.2]; [19]; [23, box 13.4]; [24]; [25, Sect. 3.1]. One might hope that general relativity would give us the benefit of hindsight, telling us which of the available energy densities accurately describes the distribution of gravitational energy in the Newtonian limit. Unfortunately, the status of gravitational energy in general relativity is a thorny issue that remains unsettled. There are multiple candidate energy-momentum tensors that can be constructed and existing proposals disagree on the energy density in the Newtonian limit [19, 26]; [25, Sect. 5.1].

  10. This perspective on potential energy pairs well with approaches to electromagnetism where one attempts to remove the electromagnetic field and have particles interact directly with one another across spatiotemporal gaps (such as Wheeler-Feynman electrodynamics). See [27]; [22, p. 80]; [2, ch. 5]; [4, Sect. 4.2].

  11. See [1, ch. 27]; [28, Sect. 31]; [29, ch. 5]; [30, Sect. 6.7]; [2, ch. 5]; [31, ch. 3]; [20, Sect. 15.4]; [3, Sect. 8.1.2].

  12. My notation in this explanation of Lagrangian densities and energy-momentum tensors matches [30, ch. 12], including the use of a \((+,-,-,-)\) signature for the metric.

  13. This Lagrangian density is discussed in [32, p. 102]; [33, ch. 8]; [34, p. 349]; [30, Sect. 12.7]; [35, Sect. 4.9]; [20, ch. 24].

  14. See [36].

  15. The canonical energy-momentum tensor for the Dirac and electromagnetic fields will contain this \(\rho ^q \phi \) potential energy density, though that will not be the full potential energy density of matter. An explanation as to why the other term does not lead to a violation of conservation of energy will be given near the end of Sect. 5.

  16. See also [2, box 8.3].

  17. See [40]; [41]; [42, appendix 1]; [32, ch. 3]; [43]; [33, Sect. 9.5]; [21, Sect. 19.5]; [34, Sect. 7.4]; [44, pp. 47–49]; [45, ch. 7]; [46].

  18. Soper [33, p. 121] remarks on a few “nice things” that happen when one moves from the canonical to the symmetric electromagnetic energy-momentum tensor, including the fact that (for a particular charged fluid model of matter) the total energy-momentum tensor for matter and field can be written as the sum of a separate matter energy-momentum tensor and field energy-momentum tensor such that “one may speak of the energy of matter ... and the energy of the electromagnetic field ... without needing an interaction energy like [\(\rho ^q \phi \)].”

  19. Here I follow moves made in [47, Sect. 15.2]; [48, Sect. 8.1].

  20. Because the Dirac field operators anticommute in quantum field theory, the classical Dirac field is sometimes treated as Grassmann-valued instead of complex-valued. For more on this issue, see [49, appendix A]; [12, Sect. 5.1] and references therein.

  21. See [21, pp. 444, 453]; [50, p. 221].

  22. Note that for the theory of interacting scalar and electromagnetic fields that is the classical precursor to scalar quantum electrodynamics (as contrasted with spinor quantum electrodynamics, which will be the quantum successor of the classical field theory examined here), the Lagrangian density does not fit the general form in (27) as the interaction is different. (See [51, eq. 9.11] for the Lagrangian density of scalar quantum electrodynamics.)

  23. These two Lagrangian densities differ by a four-divergence and yield the same field equations [32, pp. 102–103]; [52, p. 221]; [44, pp. 117–122]. The first is used in [42, 47, 48, 53] and the second is used in [21, 54,55,56].

  24. See [53, pp. 132–134]; [32, pp. 144]; [55, pp. 275–276]; [21, p. 451].

  25. To be precise, two of Maxwell’s equations are derived from the Lagrangian density and two are automatically satisfied by the use of scalar and vector potentials satisfying (24) and (25).

  26. See [54, p. 1442]; [36, pp. 477–478]; [21, Sect. 20.9]; [44, p. 151]; [57, p. 57].

  27. For discussion of the canonical energy-momentum tensor generated from \(\mathscr {L}_m\) in (43), see [54, p. 1444]; [42, p. 170, footnote 1]; [21, pp. 440–441]; [44, pp. 121–122].

  28. This energy density is discussed in [42, p. 171]; [53, pp. 132, 419]; [47, eq. 15.13]; [48, eq. 8.7]; [58].

  29. Discussing interactions between the electromagnetic field and matter in general, Konopinski [37, pp. 419, 425] identifies this kind of term as an interaction energy density.

  30. See [47, eq. 15.14]; [36, eq. 3.6]; [48, eq. 8.9].

  31. This expression (49) appears in [47, Sect. 15.2]; [34, Sect. 8.3]; [58, p. 983]; [59, Sect. 6.4] and a similar expression appears in [48, Sect. 8.1].

  32. The symmetric energy-momentum tensor for interacting Dirac and electromagnetic fields is given in [54, p. 1444]; [42, Sect. 21]; [53, p. 419]; [43, 60].

  33. This point has been noted for the free Dirac field in [42, p. 171]; [53, p. 419]; [55, p. 219]; [49, footnote 8].

  34. There is an alternative picture of potential energy available. Focusing on the first line of (52), we might interpret \(e \phi \psi ^\dagger \psi = - \rho ^q \phi \) as the potential energy of the Dirac field (which disappears in the temporal gauge) and the rest of that line as the non-potential energy of the Dirac field. On that interpretation, the Dirac field’s non-potential energy density depends on the field’s rate of change—unlike the energy density of the electromagnetic field (23).

  35. Feynman et al. [61, Sects. 21.1–21.3] include a discussion of two different kinds of momentum for classical and quantum particles that may be helpful in assessing whether (A) or (A) + (B) should be regarded as the kinetic energy density of the Dirac field. Leclerc [58, p. 983] takes the integral of \(T_m^{00}\) to give the total kinetic energy of the Dirac field.

  36. If we take one of the two options where the Dirac field’s potential energy (B) is part of its kinetic energy density, then the kinetic energy of the Dirac field will not be an intrinsic property of the field (contra footnote 4).

  37. Such states are discussed in [62,63,64,65,66].

  38. This Hamiltonian is given in [47, eq. 15.14]; [36, eq. 3.6]; [48, eq. 8.9]; [59, Sect. 6.4]. A variant is given in [69, eq. 15].

  39. To generate the standard radiative interaction vertex, the last term in (57) must be combined with part of the \(\frac{\widehat{E}^2}{8 \pi }\) term. Tong [59, Sect. 6.4.1] gives an alternative set of Feynman rules for quantum electrodynamics where the last term in (57) alone generates a non-standard radiative interaction vertex (and where there is also another kind of interaction).

  40. For a mathematical description of this vector potential and the Aharonov–Bohm effect, see [56, Sect. 3.4].

  41. In this classical field theory context, the interference pattern will be visible in the charge density of the Dirac field.

  42. See [2, 70,71,72,73] and references therein.

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Acknowledgements

Thank you to Jacob Barandes, Sean Carroll, Davison Soper, and, especially, Logan McCarty and the anonymous reviewers for helpful feedback and discussion.

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    Charles T. Sebens

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Appendices

Appendix

A Comparing Candidate Potential Energy Densities

In the Dirac field’s energy density (52), should we take (B) or (A) + (B) to be the potential energy density of the Dirac field? Given the clarification in Sect. 2 that we are understanding potential energy to be energy of one entity that is dependent on the state of some other entity (in this case, energy of the Dirac field that is dependent on the state of the electromagnetic field), one might first think only (B) should be classified as potential energy density. However, (B)’s dependence on the state of the electromagnetic field cannot be separated out from (A) + (B)’s dependence in a gauge-invariant way. So, one could argue that (A) + (B) should be regarded as the potential energy density.

To illustrate the difference between the two options, let us suppose that we have a Dirac field wave packet traveling forward in the x direction in empty space (where there is no external electromagnetic field). If we set aside the electromagnetic field created by the wave packet itself, then we can set the vector and scalar potentials to zero everywhere and have the Dirac field’s potential energy density (B) vanish everywhere. However, if we perform a gauge transformation (45) with \(\alpha \) equal to some positive constant C times x, the vector potential becomes \(\vec {A} = (C,0,0)\) (depicted in Fig. 3). This gauge transformation will change the Dirac field by an x-dependent phase factor of \(e^{- i \frac{e C}{\hbar c} x}\), which leaves the charge and current densities unaltered. The gauge transformation lowers (A) and increases (B), leaving (A) + (B) unchanged at every point. To see that (B) is positive, note that the wave packet carries negative charge and is moving forward in the x direction, so the current density \(\vec {J}\) points in the negative x direction (opposite the vector potential). If (B) is the potential energy, then the wave packet has no potential energy in the first gauge and positive potential energy in the second gauge. The potential energy is gauge-dependent, just as it was in Newtonian gravity and electrostatics. On the other hand, if (A) + (B) is the potential energy then the packet (oddly) already possessed potential energy in the first gauge (where the potentials were zero everywhere) and carries the same potential energy in the second gauge.

Fig. 3
figure 3

The first image shows the charge density of a wave packet in the Dirac field moving to the right in free space through a vector potential that is everywhere zero. The second image shows the same situation in a different gauge, with arrows depicting a constant vector potential pointing to the right at every point in space

Full size image

Moving to a more complex case, let us consider the Aharonov–Bohm effect (in the context of classical field theory, not quantum physics). Wallace [70] has recognized the importance of the fact that gauge transformations affect both the matter field and the electromagnetic vector potential in the Aharonov–Bohm setup (though his focus is on a charged scalar field, not the Dirac spinor field). From his analysis of the Aharonov–Bohm effect, Wallace concludes that the matter field and the vector potential are “interlinked,” jointly representing the physical state of a single entity. Although I have not collapsed the Dirac and electromagnetic fields into a single entity, I agree that, in an important way, the two fields are interlinked: the Dirac field has a potential energy density that depends on the physical state of the electromagnetic field. But, is that potential energy density given by (B) or (A) + (B)?

Fig. 4
figure 4

This image depicts charge densities for two wave packets that are traveling above and below a long solenoid (shown as a white circle), through a vector potential. As time goes on, the wave packets will move toward the detection screen on the right while spreading and interfering

Full size image

The Aharonov–Bohm setup is illustrated in Fig. 4. Let us suppose that the Dirac field is in a (classical) superposition of a wave packet that passes above the solenoid and a wave packet that passes below the solenoid. Assuming the solenoid is long and carefully shielded, these wave packets pass through regions where the electromagnetic field is essentially zero. However the vector potential is not negligible. In the most natural choice of gauge, the vector potential circles the solenoid as depicted in Fig. 4.Footnote 40 When the wave packets hit the screen, the interference patternFootnote 41 will depend on the strength of the magnetic field in the solenoid. Seeking to avoid non-local interaction between the solenoid and the matter field, this shift in the interference pattern (observed in quantum experiments) has been taken as evidence that the scalar and vector potentials are more fundamental than electric and magnetic fields. I will not attempt to sort out issues of locality here, or questions about the nature of the electromagnetic field.Footnote 42 Instead, I want to use this experiment to better understand our options for isolating the potential energy density of the Dirac field.

If the potential energy density is (B) and we adopt the choice of gauge in Fig. 4, the wave packet that passes underneath the solenoid will acquire a positive potential energy because its current points opposite the vector potential. Accompanying that potential energy will be a shift in the phase of the wave packet from what it would have been had the packet been moving at the same velocity in the absence of a vector potential (just as in the case from Fig. 3). The wave packet that passes above the solenoid will acquire a negative potential energy and an opposite shift in the phase. These phase shifts are responsible for the interference pattern. In a different gauge, the potential energies of the wave packets would be different and so would the phase shifts. But, the interference pattern would be the same. Thus, taking (B) to be the potential energy density, we can attribute the shift in the interference pattern to a difference in potential energies between the wave packets. Although we can push that bump in the rug around through gauge transformations, it cannot be eliminated. By contrast, if we take (A) + (B) to be the potential energy density, then the potential energies of the two wave packets would be just the same as if they were propagating in free space. The predicted interference pattern is the same, but it cannot be understood as the result of the different wave packets having different potential energies.

I do not have a definitive argument to offer in favor of taking (B) to be the potential energy density, but it is the option that I lean towards and thus I have chosen to treat (B) as the potential energy density of the Dirac field in the main text of the article. Taking (B) to be the potential energy density gives an attractive understanding of the two cases described above. This potential energy density is gauge-dependent, but that feature is shared with the potential energy densities used in Newtonian gravity and electrostatics (see Sect. 3).

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Sebens, C.T. The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics. Found Phys 52, 113 (2022). https://doi.org/10.1007/s10701-022-00630-5

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