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The fundamentality of fields

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Abstract

There is debate as to whether quantum field theory is, at bottom, a quantum theory of fields or particles. One can take a field approach to the theory, using wave functionals over field configurations, or a particle approach, using wave functions over particle configurations. This article argues for a field approach, presenting three advantages over a particle approach: (1) particle wave functions are not available for photons, (2) a classical field model of the electron gives a superior account of both spin and self-interaction as compared to a classical particle model, and (3) the space of field wave functionals appears to be larger than the space of particle wave functions. The article also describes two important tasks facing proponents of a field approach: (1) legitimize or excise the use of Grassmann numbers for fermionic field values and in wave functional amplitudes, and (2) describe how quantum fields give rise to particle-like behavior.

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Notes

  1. In Sebens (2019c) I give a non-technical introduction this debate.

  2. In his recent book Foundations of Modern Physics, Weinberg (2021, pp. 251–252) writes: “Even though it is not generally useful to do so, we can also introduce wave functions for fields – they are functionals of the field, quantities that depend on the value taken by the field at every point in space, equal to the component of the state vector in a basis labeled by these field values.”

  3. Hobson (2013), for example, argues that fields are more fundamental than particles without mentioning wave functionals.

  4. Notable omissions include particle localizability, renormalization, spontaneous symmetry breaking, and the Unruh effect (see Barrett, 2002; Lupher, 2010; Earman, 2011b; Ruetsche, 2011; Myrvold, 2015; Baker, 2016; Wallace, 2021; 2022 and references therein).

  5. This kind of particle approach is described in Schweber (1961, Sects. 6f, 6h, and 7c); Dürr et al. (2004, 2005); Tumulka (2018). Although the focus here is on wave functions that assign amplitudes to different particle arrangements at a single time, some have proposed (for better harmony with special relativity) using multi-time wave functions where there is a separate time coordinate for each particle position (Lienert et al., 2017; Lienert et al., 2020, ch. 4).

  6. Some authors present the particle approach for momenta as a potentially viable option (at least in the absence of interactions), but challenge the idea that Fourier transforming yields a relativistically acceptable representation in terms of positions. See Teller (1995, pp. 48–56, 85–91); Myrvold (2015).

  7. See Bjorken & Drell (1964, ch. 3).

  8. These are controversial assumptions. For more on why such assumptions might be made and on the consequences that follow from them, see the references in footnote 41.

  9. Detailed technical introductions to the field approach are given in Jackiw (1987, 1990); Floreanini & Jackiw (1988); Hatfield (1992); Bohm & Hiley (1993, ch. 11); Holland (1993a, Sect. 12.4); Kiefer & Wipf (1994); Kaloyerou (1994, 1996); Huang (2008, pp. 29–33). The field approach is also discussed in Valentini (1992, 1996); Huggett (2000); Wallace (2001, 2006, 2021); Baker (2009, 2016); Struyve (2010, 2011); Myrvold (2015); Dürr & Lazarovici (2020, Sect. 11.2). For an introduction to the field approach aimed at a general audience, see Carroll (2019, ch. 12).

  10. There are mathematical issues regarding the definition of a measure over the (infinite-dimensional) space of possible field configuration—a measure that is necessary for a mathematically rigorous account as to how the amplitude-squared of the wave functional serves as a probability density (see Struyve, 2010, Sect. 2.2.2).

  11. This classical theory is discussed in Barut (1964); Bjorken & Drell (1965, Sect. 15.2); Doughty (1990, Sect. 20.9); Greiner & Reinhardt (1996, Sect. 5.1); Hatfield (1992, Sect. 8.1); Sebens (2021a).

  12. See Bohm (1952, appendix A); Hatfield (1992, Sect. 10.2); Kaloyerou (1994, 1996); Struyve (2010, Sect. 4.2); Flack & Hiley (2016).

  13. See Floreanini & Jackiw (1988); Jackiw (1990); Valentini (1992, 1996); Hatfield (1992); Kiefer & Wipf (1994).

  14. See Hatfield (1992); Peskin & Schroeder (1995, Sect. 9.5); Ryder (1996, Sect. 6.7); Greiner & Reinhardt (1996, Sect. 12.8); Zee (2010, Sect. 11.5); Duncan (2012, Sect. 10.3.2); Schwartz (2014, Sect. 14.6).

  15. The indices on \(\psi \) are dropped in (11) and some other equations. One could write \(\psi ^\dagger \psi \) as \(\sum _{i=1}^4 \psi _i^*\psi _i\).

  16. The difficulties involved in constructing a relativistic quantum theory for the photon are reviewed in Bohm et al. (1987, Sect. II.5.2); Holland (1993b); Holland (1993a, Sect. 12.6); Kiessling & Tahvildar-Zadeh (2018); Valentini (2020, Sect. 7.4).

  17. See Rumer (1930); Mignani et al. (1974).

  18. Similar proposals appear in Akhiezer & Berestetskii (1965, Eq. 1.6); Pauli (1980, p. 191); Mandel & Wolf (1995, p. 637).

  19. Recent proposals for such a theory have been given in Kiessling & Tahvildar-Zadeh (2018); Bialynicki-Birula & Bialynicka-Birula (2019); Hawton (2019, 2021).

  20. Berestetskii et al. (1982, p. 14) write that “the coordinate wave function of the photon cannot be interpreted as the probability amplitude of its spatial localization.” (See also Akhiezer & Berestetskii, 1965, Sect. 2.2.)

  21. Such mixed approaches are considered in Bohm et al. (1987); Bohm & Hiley (1993); Kaloyerou (1994, p. 293); Kaloyerou (1996, p. 155).

  22. Lazarovici (2018) advocates this kind of approach.

  23. Kaloyerou (1996, p. 155) gives a different argument for consistency in the approaches used for bosons and fermions (in the context of seeking a Bohmian quantum field theory):

    “A criterion that has been introduced by Bohm, regarded as preliminary by the present author, is that where the classical limit of the equation of motion of the field is a wave equation, then the entity can be consistently regarded as an objectively existing field, but where the classical limit is a particle equation, then the entity must be regarded as an objectively existing particle. The former is the case for bosons, such as the electromagnetic field and the mesons, and the latter for fermions. The problem with this criteria is that the field ontology of bosons is in direct conflict with that of fermions when it is recalled that some bosons are fermion composites (e.g., mesons are quark-antiquark pairs) and quarks are fermions. It seems likely instead that fermions and bosons should have the same ontology.”

  24. See Good (1957); Sebens (2020b, Sect. 6).

  25. This total energy can be calculated from the 00 component of either the canonical or the symmetrized energy-momentum tensor for the free Dirac field. See Heitler (1954, p. 419); Schweber (1961, p. 219); Sebens (2020b, Eq. 3).

  26. Here we are discussing a fully classical theory where the electron is modeled as a point particle that has an intrinsic “spin” angular momentum and an intrinsic “spin” magnetic moment. In a Bohmian version of quantum mechanics or quantum field theory, you might include a point electron with these properties or without them (Holland, 1993a, ch. 9; Bohm & Hiley, 1993, ch. 10).

  27. Barandes (2019, 2021) develops a detailed classical relativistic theory of particles with intrinsic properties (like angular momenta and magnetic moments) interacting with the electromagnetic field.

  28. These densities can be modified so that negative-frequency modes carry positive charge (as would be appropriate for representing positrons; see Sebens 2020b), but we will not need to introduce that complication here as we are focused on electron spin.

  29. This expansion of the current density is discussed in Gordon (1928); Frenkel (1934, pp. 321–322); Huang (1952, p. 479); Ohanian (1986); Sebens (2019b, 2020a).

  30. The question as to whether it is only moving charges that produce magnetic fields has recently been discussed by Fahy & O’Sullivan (2022); Griffiths (2022).

  31. Another common objection is that the electron’s gyromagnetic ratio does not match the classical prediction. But, that classical prediction assumes that mass and charge rotate at the same rate—which will not be the case for the mass and charge of the Dirac field (Sebens, 2019b).

  32. Although I hope that we can find a subluminal velocity of energy flow, I do not think this is a necessary condition for the picture of electron spin outlined here to be viable. It may be better to focus on the densities of energy and momentum, recognizing that the above-defined velocity of energy flow is not always well-behaved.

  33. See Takabayasi (1957, Sect. 2b); Bohm & Hiley (1993, Sect. 10.4); Holland (1993a, Sect. 12.2); Sebens (2019b, 2020a).

  34. For philosophical discussion of self-interaction in classical electrodynamics, see Lange (2002); Frisch (2005); Earman (2011a); Maudlin (2018, Sect. 4); Lazarovici (2018); Hartenstein & Hubert (2021). For discussion in physics, see Pearle (1982); Jackson (1999); Rohrlich (2007); Baez (2021).

  35. There is much that could be said about the strengths and weaknesses of action-at-a-distance formulations of classical electrodynamics. Briefly, note that such theories are non-local and violate both energy and momentum conservation (though there are ways of understanding locality and conservation that allow one to contest these apparent defects—see Lazarovici, 2018).

  36. See also Blum & Joas (2016); Blum (2017, Sects. 2.5.2 and 3.1).

  37. See Seben (2022b, Sect. 2.2).

  38. Thank you to David Baker for clarifying this point in correspondence.

  39. Difficulties related to functional integration were mentioned earlier in footnote 10.

  40. See also Jackiw (1990, p. 88).

  41. The reasons for introducing a high-momentum cutoff and a finite spatial region—and the costs that come with doing so—are discussed in Wallace (2006; 2021); Duncan (2012, Sect. 10.5); Baker (2016); Deckert et al. (2020, Sect. 2).

  42. If the above kind of strategy works for introducing particle wave functions in interacting theories, there might be a way of combining the space of particle wave functions from the free theory with the various spaces used for different interacting theories to get a large space of states (that could perhaps be as big as the space of wave functionals).

  43. The problems with Grassmann numbers have led some to conclude that a field approach for fermions is either unavailable or unattractive. See Bohm et al. (1987, p. 374); Dürr & Lazarovici (2020, p. 202); Struyve (2010, 2011); Wallace (2021, Sect. 9.2).

  44. Because it is ultimately the quantum field theory that needs to have a precise formulation, one might be willing to tolerate problems with energy and charge in the pre-quantization classical field theory so long as they do not deeply damage the post-quantization quantum field theory. I would prefer, if possible, to start with a clear and consistent classical field theory.

  45. See Floreanini & Jackiw (1988); Jackiw (1990); Hatfield (1992, Sect. 10.3); Kiefer & Wipf (1994).

  46. At least, we can define a measure here as easily as in the bosonic case. That being said, there are challenges there (see footnote 10).

  47. See Bohm et al. (1987, pp. 363–373); Bohm & Hiley (1993, ch. 11); Kaloyerou (1994, Sect. 4); Valentini (1992, Sect. 4.1; 1996, pp. 54–55); Sebens (2021c).

  48. One might also wish to derive some quantum theory for the photon, but (as was discussed in Sect. 4.1) we have no theory like relativistic electron quantum mechanics for the photon—so the goalposts will look different for the photon.

  49. See Desclaux (2002).

  50. See Bjorken & Drell (1964, Sect. 1.4); Berestetskii et al. (1971, Sect. 33); Bohm & Hiley (1993, Sect. 10.4); Ryder (1996, Sect. 2.6); Nowakowski (1999).

  51. The details of this project will depend on one’s preferred strategy for making the laws and ontology of quantum theories precise. On the many-worlds interpretation, the task is as described above. In an interpretation that includes some form of wave function collapse, one would have to propose a theory of wave functional collapse in quantum field theory and show that the collapse of the wave functional induces a satisfactory collapse of the particle wave function. In a Bohmian field approach to quantum field theory where one supplements the wave functional with an actual field state evolving by a new equation of motion, one would have to show that the evolution of that field state leads to unique outcomes in quantum measurements. One would not expect to (and would not need to) recover the point particles of elementary Bohmian quantum mechanics from the fields posited in the kind of Bohmian quantum field theory just described.

  52. See Blum (2017).

  53. See Lange (2002, ch. 7).

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Acknowledgements

Thank you to David Baker, Jacob Barandes, Jeffrey Barrett, Sean Carroll, Eddy Keming Chen, Maaneli Derakhshani, Benjamin Feintzeig, Mario Hubert, Dustin Lazarovici, Logan McCarty, Tushar Menon, David Mwakima, Ward Struyve, Roderich Tumulka, Jim Weatherall, and anonymous reviewers for helpful feedback and discussion.

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    Charles T. Sebens

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Sebens, C.T. The fundamentality of fields. Synthese 200, 380 (2022). https://doi.org/10.1007/s11229-022-03844-2

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