Skip to main content
SpringerLink
Log in

Eliminating Electron Self-repulsion

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Problems of self-interaction arise in both classical and quantum field theories. To understand how such problems are to be addressed in a quantum theory of the Dirac and electromagnetic fields (quantum electrodynamics), we can start by analyzing a classical theory of these fields. In such a classical field theory, the electron has a spread-out distribution of charge that avoids some of the problems of self-interaction facing point charge models. However, there remains the problem that the electron will experience self-repulsion. This self-repulsion cannot be eliminated within classical field theory without also losing Coulomb interactions between distinct particles. But, electron self-repulsion can be eliminated from quantum electrodynamics in the Coulomb gauge by fully normal-ordering the Coulomb term in the Hamiltonian. After normal-ordering, the Coulomb term contains pieces describing attraction and repulsion between distinct particles and also pieces describing particle creation and annihilation, but no pieces describing self-repulsion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Early models of the electron posited forces holding the electron together, now known as “Poincaré stresses” [1, ch. 28, 2,3,4,5, 6, ch. 16, 7, sec. 5]. However, no such forces appear in quantum electrodynamics or the standard model. Thus, for our purposes here in understanding the relation between classical and quantum field theories, we need not consider such forces.

  2. The Hartree-Fock method for approximating the ground state energy of an atom or molecule includes contributions from Coulomb repulsion between distinct electrons but explicitly excludes self-repulsion, and a more recent method for calculating ground state energies, called “density functional theory,” requires a self-interaction correction that corrects for the initial inclusion of electron self-repulsion [8, p. 436, 9, 10, sec. 8.3, 11, p. 559, 12].

  3. Barut and his collaborators have studied such a theory of interacting classical interacting Dirac and electromagnetic fields, considering it to be an alternative to standard quantum electrodynamics. They have shown that a number of important phenomena can be explained without quantum physics [13, 14]. These authors retain self-interaction effects, including self-repulsion. In the context of a discussion of the hydrogen atom, Barut [13, p. 39] explains why we do not observe electron self-repulsion (even though he thinks that it does in fact occur) as follows: “in a static situation, the interaction potential between electron and the proton is just Ze/r [where \(Z=1\) and r is the distance from the proton], hence we expect that the static self-field of the electron should have no effect - it is already taken into account by the physical charge and mass of the electron.” I do not see how one could explain the absence of any observed self-repulsion by simply redefining the mass and/or charge of the electron. The classical Coulomb energy associated with electron self-repulsion (8) would vary depending on how compactly the electron’s charge is spread.

  4. See [16, ch. 4].

  5. Given that the electron will be modeled classically in the next section as a lump of energy and charge in the classical Dirac field, one might wonder whether it makes sense to think of the above density of force as acting on a field (the Dirac field). For defense of the idea that forces act on both the electromagnetic and Dirac fields, see [28, 29].

  6. Although I will not do so here, there are reasons (coming from quantum field theory) to treat the components of the Dirac field \(\psi \) at a given location as anticommuting Grassmann numbers instead of complex numbers (see [15] and references therein).

  7. These two interpretations of the Dirac equation and their relation to quantum field theory are discussed in [15].

  8. In [31] I propose changing the charge and current densities so that the (free) classical Dirac field describes both negatively charged electrons and positively charged positrons. For our purposes here, where we are concerned first and foremost with electrons, we can stick with the standard charge and current densities. However, ultimately it would be better to extend the treatment in [31] to include interactions with the electromagnetic field.

  9. In [32] I discuss the interpretation of different terms in this Hamiltonian, arguing that the first two terms give the energy of the electromagnetic field and the remainder gives the energy of the Dirac field (with the final term being a potential energy of the Dirac field).

  10. For presentations of quantum electrodynamics in the Coulomb gauge (including discussion of the Hamiltonian for interacting Dirac and electromagnetic fields), see [33, sec. 15.2 and 17.9, 34, sec. 5.2 and 8.1, 35, sec. 8.3, 36, sec. 6.4]. To limit the scope of this paper, I will work entirely within the Coulomb gauge and not ask how the points made here might be extended to other gauges.

  11. Greiner and Reinhardt [39, sec. 8.6] do not explicitly write down the normal-ordered Coulomb term, but they do seem to think that the entire Hamiltonian (in both spinor and scalar quantum electrodynamics) should be fully normal-ordered. Greiner and Reinhardt [39, p. 238] explain that normal-ordering removes an undesirable kind of self-interaction, writing that “the prescription of normal-ordering of the interaction operator eliminates the interaction of a particle with itself at the same point x.” However, they do not directly discuss self-repulsion.

  12. There are a number of competing proposals as to the nature of these quantum states. Elsewhere, I have argued that states in quantum field theory should be viewed as wave functionals assigning quantum amplitudes to classical field configurations [15].

  13. See [35, eq. 8.3.12, 36, sec. 6.4, 40].

  14. When Bjorken and Drell [33, sec. 17.9] use the Coulomb term in deriving the Feynman rules for quantum electrodynamics, they only consider its contribution to diagrams involving interactions between distinct particles. Thus, although they do not explicitly normal order the entire term, they treat it as if it was normal-ordered.

  15. See, e.g., [17, sec. 8b, 33, p. 90, 41, p. 58, 36, ch. 5].

  16. See the discussion of Haag’s theorem in [15, sec. 4.3, 42, sec. 3, 43, sec. 3].

References

  1. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, Vol. II, Addison-Wesley Publishing Company, Boston (1964)

  2. Rohrlich, F.: The electron: development of the first elementary particle theory. In: Mehra, J. (ed.) The Physicist’s Conception of Nature, pp. 331–369. D. Reidel Publishing Company, Dordrecht (1973)

    Chapter  Google Scholar 

  3. Rohrlich, F.: (2007) Classical Charged Particles, 3rd edn. World Scientific, Singapore

  4. Pearle, P.: Classical electron models. In: Teplitz, D. (ed.) Electromagnetism: Paths to Research, pp. 211–295. Plenum Press, New York (1982)

  5. Schwinger, J.: Electromagnetic mass revisited. Found. Phys. 13(3), 373–383 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  6. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999)

  7. Griffiths, D.J.: Resource letter EM-1: electromagnetic momentum. Am. J. Phys. 80, 7–18 (2012)

    Article  ADS  Google Scholar 

  8. Blinder, S.M.: Basic concepts of self-consistent-field theory. Am. J. Phys. 33(6), 431–443 (1965)

    Article  ADS  Google Scholar 

  9. Perdew, J.P., Zunger, A.: Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23(10), 5048–5079 (1981)

    Article  ADS  Google Scholar 

  10. Parr, R.G., Yang, W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press, Oxford (1989)

  11. Levine, I.N.: Quantum Chemistry. 7th edn. Pearson, London (2014)

  12. Sebens, C.T.: Electron charge density: a clue from quantum chemistry for quantum foundations. Found. Phys. 51, 75 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Barut, A.O.: The Schrödinger and the Dirac equation—linear nonlinear and integrodifferential. In: De Filippo, S., Marinaro, M., Marmo, G., Vilasi, G. (eds.) Geometrical and Algebraic Aspects of Nonlinear Field Theory, pp. 37–51. Elsevier, New York (1989)

    Chapter  Google Scholar 

  14. Barut, A.O.: Foundations of self-field quantumelectrodynamics. In: Barut, A.O. (ed.) New Frontiers in Quantum Electrodynamics and Quantum Optics, pp. 345–365. Plenum Press, New York (1991)

    Google Scholar 

  15. Sebens, C.T.: The fundamentality of fields. Synthese 200(5), 380 (2022)

    Article  MathSciNet  Google Scholar 

  16. Lange, M.: An Introduction to the Philosophy of Physics: Locality, Energy, Fields, and Mass. Blackwell, Oxford (2002)

    Google Scholar 

  17. Schweber, S.S.: Introduction to Relativistic Quantum Field Theory. Harper & Row, Manhattan (1961)

    MATH  Google Scholar 

  18. Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  19. Greiner, W., Reinhardt, J.: Quantum Electrodynamics, 3rd edn. Springer, New York (2003)

  20. Schwartz, M.D.: Quantum Field Theory and the Standard Model. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  21. Maudlin, T.: Ontological clarity via canonical presentation: electromagnetism and the Aharonov-Bohm effect. Entropy 20(6), 465 (2018)

    Article  ADS  Google Scholar 

  22. Frisch, M.: Inconsistency in classical electrodynamics. Philos. Sci. 71(4), 525–549 (2004)

    Article  MathSciNet  Google Scholar 

  23. Frisch, M.: Inconsistency, Asymmetry, and Non-locality: A Philosophical Investigation of Classical Electrodynamics. Oxford University Press, Oxford (2005)

    Book  Google Scholar 

  24. Frisch, M.: Conceptual problems in classical electrodynamics. Philos. Sci. 75(1), 93–105 (2008)

    Article  MathSciNet  Google Scholar 

  25. Frisch, M.: Philosophical issues in electromagnetism. Philos. Compass. 4(1), 255–270 (2009)

    Google Scholar 

  26. Belot, G.: Is classical electrodynamics an inconsistent theory? Can. J. Philos. 37(2), 263–282 (2007)

    Article  Google Scholar 

  27. Lazarovici, D.: Against fields. Eur. J. Philos. Sci. 8(2), 145–170 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sebens, C.T.: Forces on fields. Stud. Hist. Philos. Mod. Phys. 63, 1–11 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Sebens, C.T.: Particles, fields, and the measurement of electron spin. Synthese 198(12), 11943–11975 (2021)

    Article  MathSciNet  Google Scholar 

  30. Duncan, A.: The Conceptual Framework of Quantum Field Theory. Oxford University Press, Oxford (2012)

    Book  MATH  Google Scholar 

  31. Sebens, C.T.: Putting positrons into classical Dirac field theory. Stud. Hist. Philos. Mod. Phys. 70, 8–18 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  32. Sebens, C.T.: The disappearance and reappearance of potential energy in classical and quantum electrodynamics (2021). arXiv:2112.14643

  33. Bjorken, J.D., Drell, S.D.: Relativistic Quantum Fields. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  34. Hatfield, B.: Quantum field theory of point particles and strings. Front. Phys. 75 (1992)

  35. Weinberg, S.: The Quantum Theory of Fields, Volume 1: Foundations. Cambridge University Press, Cambridge (1995)

  36. Tong, D.: Lectures on quantum field theory (2007). http://www.damtp.cam.ac.uk/user/tong/qft.html

  37. Sebens, C.T.: How electrons spin. Stud. Hist. Philos. Mod. Phys. 68, 40–50 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Sebens, C.T.: Possibility of small electron states. Phys. Rev. A 102, 052225 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  39. Greiner, W., Reinhardt, J.: Field Quantization, Springer, New York (1996)

  40. Kay, B.S.: Quantum electrostatics, Gauss’s law, and a product picture for quantum electrodynamics; or, the temporal gauge revised (2020). arXiv:2003.07473

  41. Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Westview Press, Boulder (1995)

    Google Scholar 

  42. Earman, J., Fraser, D.: Haag’s theorem and its implications for the foundations of quantum field theory. Erkenntnis 64, 305–344 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fraser, D.: The fate of ‘particles’ in quantum field theories with interactions. Stud. Hist. Philos. Mod. Phys. 39(4), 841–859 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thank you to Jacob Barandes, Maaneli Derakhshani, Michael Miller, Logan McCarty, Simon Streib, and anonymous reviewers for helpful feedback and discussion.

Author information

Authors and Affiliations

  1. Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA

    Charles T. Sebens

Authors
  1. Charles T. Sebens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles T. Sebens.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sebens, C.T. Eliminating Electron Self-repulsion. Found Phys 53, 65 (2023). https://doi.org/10.1007/s10701-023-00702-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-023-00702-0

Keywords

Search

Navigation