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Particles, fields, and the measurement of electron spin

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Abstract

This article compares treatments of the Stern–Gerlach experiment across different physical theories, building up to a novel analysis of electron spin measurement in the context of classical Dirac field theory. Modeling the electron as a classical rigid body or point particle, we can explain why the entire electron is always found at just one location on the detector (uniqueness) but we cannot explain why there are only two locations where the electron is ever found (discreteness). Using non-relativistic or relativistic quantum mechanics, we can explain both uniqueness and discreteness. Moving to more fundamental physics, both features can be explained within a quantum theory of the Dirac field. In a classical theory of the Dirac field, the rotating charge of the electron can split into two pieces that each hit the detector at a different location. In this classical context, we can explain a feature of electron spin that is often described as distinctively quantum (discreteness) but we cannot explain another feature that could be explained within any of the other theories (uniqueness).

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Notes

  1. Stern–Gerlach spin measurements play a central role in textbook treatments of the philosophy of quantum physics, such as Albert (1992), Lewis (2016), Norsen (2017), Maudlin (2019), Barrett (2020).

  2. Stern and Gerlach saw their experiment as demonstrating “directional quantization,” though this was not initially applied to electron spin as that idea had not yet entered quantum theory (Gerlach and Stern 1922; Weinert 1995; Sauer 2016; Schmidt-Böcking et al. 2016). Now, it is common to present the experiment as showing that electron spin is quantized. I will use the term “discrete” instead of “quantized” because “quantized” suggests “quantum” and, as we will see, the outcomes of Stern–Gerlach experiments can be discrete in theories that are not quantum.

  3. For philosophical discussion and comparison of rigid body classical mechanics (used in Sect. 2), point particle classical mechanics (used in Sect. 3), and continuum mechanics (used in Sect. 6), see Wilson (1998, (2013).

  4. We will not examine relativistic classical theories of a point electron in this article (see Wen et al. 2016; Barandes 2019a, b and references therein).

  5. Here and throughout I adopt Gaussian cgs units.

  6. For the definition of magnetic moment in terms of current density, see Jackson (1999, sec. 5.6).

  7. Keeping the electron’s angular momentum and magnetic moment fixed, the electron will have to rotate faster the smaller it is. For a sufficiently small radius, its edges would have to move faster than the speed of light (Tomonaga 1997, p. 35; Griffiths 2005, problem 4.25; Rohrlich 2007, p. 127; Sebens 2019). Let us assume that the electron is large enough that it does not have to rotate so rapidly.

  8. See Uhlenbeck (1976, p. 47), Pais (1989, p. 39), Jackson (1999, p. 187), Griffiths (2013, problem 5.58).

  9. See Sebens (2019).

  10. This expression for the magnetic field follows (Griffiths 2005, p. 181).

  11. See Feynman et al. (1964, sec. 15.6), Lange (2002, ch. 5).

  12. See Feynman et al. (1964, sec. 34.3), Landau and Lifshitz (1971, sec. 45), Alstrøm et al. (1982), Griffiths (2005, sec. 4.4.2).

  13. The spherical electron of Sect. 2 had a magnetic moment as well, but it was not intrinsic. It was the result of flowing charge. Similarly, the spherical electron had angular momentum that was the result of flowing mass.

  14. Although we can derive the additional term in the force law that must be postulated and the equation for the torque that must be added by analyzing the theory of electromagnetism with rigid bodies, these features must be included as part of the fundamental dynamical laws in this theory of electromagnetism with point particles.

  15. Griffiths (2013, p. 378) considers this modification to the Lorentz force law to incorporate intrinsic magnetic moment and remarks: “I don’t know whether a consistent theory can be constructed in this way, but in any event it is not classical electrodynamics, which is predicated on Ampère’s assumption that all magnetic phenomena are due to electric charges in motion, and point magnetic dipoles must be interpreted as the limits of tiny current loops.” Barandes (2019a, (2019b) analyzes such modifications to the force law, considering intrinsic magnetic dipole moments, intrinsic electric dipole moments, and also other multipole moments.

  16. For a point electron, this potential energy cannot be derived from the total energy in the electromagnetic field (as in Sect. 2). Instead, this potential energy can be calculated by considering the work that must be done to rotate a point magnetic dipole in an external magnetic field (Griffiths 2013, problem 6.21) or the work required to bring the dipole to a given location from infinity (Good and Nelson 1971, p. 227). Or, it can be calculated by asking what potential would generate the additional term in the modified Lorentz force law (19) when one takes its negative gradient (Jackson 1999, sec. 5.7).

  17. For a general derivation from the theory of electromagnetism with rigid bodies of the law for torque that should be included in a theory with point bodies that possess intrinsic magnetic moments and angular momenta, see Jackson (1999, sec. 5.7).

  18. When the electron’s location is not relevant, its spin state is sometimes just specified by two complex numbers. We will not analyze the electron using that simplified representation in this article.

  19. See Bjorken and Drell (1964, eq. 1.34), Berestetskii et al. (1971, eq. 33.7), Dürr and Lazarovici (2020, eq. 1.28).

  20. This approximation is used in Griffiths (2005, example 4.4), Ballentine  (2014, sec. 9.1), Dürr and Lazarovici (2020, sec. 1.7.1).

  21. See Platt (1992).

  22. Similar examples of time evolution for gaussian wave packets appear in Griffiths (2005, problem 2.43), Ballentine (2014, problem 5.8).

  23. This effect is discussed in Griffiths (2005, example 4.3), Ballentine (2014, sec. 12.1). Larmor precession is often ignored in simplified and idealized presentations of Stern–Gerlach experiments. This precession is particularly relevant in repeated series of Stern–Gerlach setups, such as two-path experiments. For example, one often assumes that if an x-spin up electron passes through the Stern–Gerlach magnetic field and then the two wave packets that diverge are brought back together, a subsequent x-spin measurement will show that the electron is x-spin up (when in actuality the electron’s spin has been rotated by the first magnetic field). This kind of simplified treatment appears in Albert (1992, pp. 7–11), Maudlin (2019, pp. 22–25), Barrett (2020, ch. 2).

  24. For detailed introductions to these interpretations of quantum mechanics, see the references in footnote 1.

  25. Albert (1992, ch. 7), Bohm and Hiley (1993, ch. 10), Dürr and Teufel (2009, sec. 8.4), Norsen (2014, p. 346), for example, do not attribute an intrinsic magnetic moment to the electron itself.

  26. This point is explained clearly in relation to collapse theories by Maudlin (2019, pp. 101–102) and in relation to Bohmian mechanics by Norsen (2014).

  27. Bjorken and Drell (1964, p. 11) explain the interaction term as an operator version of the classical potential energy of a point charge in a static electromagnetic field (Feynman et al. 1964, sec. 15.6). This raises a number of questions, as we noted earlier (in Sect. 2) that potential energies are insufficient for achieving conservation of energy in classical electromagnetism and, putting that aside, the additional potential energy associated with the electron’s intrinsic magnetic moment (20) has not been included.

  28. See Bjorken and Drell (1964, sec. 1.4), Berestetskii et al. (1971, sec. 33), Bohm and Hiley (1993, sec. 10.4), Ryder (1996, sec. 2.6), Nowakowski (1999).

  29. This assumption focuses our attention on positive energy modes, allowing us to set aside questions regarding the interpretation of negative energy modes (sometimes addressed by introducing an infinite “sea” of negative energy electrons—the Dirac sea).

  30. By setting \(\phi \) equal to zero we are choosing to ignore the electric field that would be present in the frame where the electron begins at rest (a choice that was explained in Sect. 2).

  31. The state in (41) also appears in Sebens (2019, eq. 36). Note that by taking a non-relativistic approximation we have ended up with a wave function that is not normalized.

  32. See Pauli (1925, (1946), Tomonaga (1997, ch. 2), Morrison (2007), Giulini (2008), Regt and Henk (2017, sec. 7.4).

  33. See, for example, Gross (1966), Glassey and Strauss (1979), Flato et al. (1987).

  34. I have argued elsewhere that we should modify equations like (45) and (46) so that the negative frequency modes of the Dirac field are associated with negative charge and positive energy (Sebens 2020). However, we don’t have to be worry about that here because the negative frequency modes are not important in the non-relativistic approximation that we are using (Bjorken and Drell 1964, p. 10).

  35. For an examination and defense of the idea that forces can be exerted upon fields, see Sebens (2018).

  36. Consider, for example, working backwards through the proofs in Jackson (1999, sec. 6.7), Griffiths (2013, sec. 8.2).

  37. For philosophical discussion of self-interaction, see Lange (2002), Frisch (2005), Earman (2011), Lazarovici (2018), Maudlin (2018), Hartenstein and Hubert (forthcoming).

  38. Hypothetical forces holding the electron together have been called “Poincaré stresses.” See Feynman et al. (1964, ch. 28), Rohrlich (1973), Pearle (1982), Schwinger (1983), Jackson (1999, ch. 16), Rohrlich (2007, sec. 6.3), Griffiths (2012, sec. 5).

  39. The electron’s contribution to the electromagnetic field is also relevant for a precise calculation of the total angular momentum of the electron itself and the electromagnetic field that surrounds it, as there is angular momentum in both the Dirac and electromagnetic fields (Sebens 2019).

  40. This assumption is part of our non-relativistic approximation (Bjorken and Drell 1964, p. 39; Sebens 2019, sec. 5).

  41. This current density is discussed in Ohanian (1986, sec. 4), Sebens (2019, eq. 31).

  42. This section has focused on magnetic moment and the flow of charge. Angular momentum results from the flow of relativistic mass, or you could say the flow of energy (as relativistic mass is proportional to energy). This flow is analyzed in Ohanian (1986), Sebens (2019).

  43. See footnote 15 for Griffiths’ statement of this criticism.

  44. Depending on the way the particle approach is executed, positrons may either be viewed as fundamental particles or as holes in the Dirac Sea. Both options have been explored by scholars working to extend Bohmian mechanics to quantum field theory, the former by Dürr et al. (2004, (2005) and the latter by Bohm himself (Bohm 1953, p. 275; Bohm and Hiley 1993, ch. 12) and more recently by Colin and Struyve (2007), Deckert et al. (2019).

  45. For more on the particle and field approaches to quantum field theory, see the references in Sebens (2019, footnote 2).

  46. Wave functionals over Grassmann-valued fields are used in Floreanini and Jackiw (1988), Jackiw (1990), Hatfield (1992), Valentini (1992, (1996). The issues mentioned above are discussed in Struyve (2010, sec. 9.2), Struyve (2011, sec. 3.3), Sebens (2020, appendix A).

  47. For discussion of the various difficulties involved in extending the interpretations of quantum mechanics from Sect. 4 to relativistic quantum field theory, see Wallace (2008), Barrett (2014), Maudlin (2019, ch. 7), Dürr and Lazarovici (2020, ch. 11–12).

  48. See Wallace (2008, (2018, (2020), Wallace (2012, sec. 1.7).

  49. See Struyve (2010), Struyve (2011), Tumulka (2018) and references therein.

  50. See also Kaloyerou (1994, sec. 4).

  51. This table is modeled on Feynman’s table distinguishing the behavior of bullets, water waves, and electrons (Feynman 1967, figure 31).

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Acknowledgements

Thank you to Valia Allori, Jacob Barandes, Jeffrey Barrett, Mario Hubert, and the anonymous reviewers for helpful feedback and discussion.

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    Charles T. Sebens

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Sebens, C.T. Particles, fields, and the measurement of electron spin. Synthese 198, 11943–11975 (2021). https://doi.org/10.1007/s11229-020-02843-5

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