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A Case for an Empirically Demonstrable Notion of the Vacuum in Quantum Electrodynamics Independent of Dynamical Fluctuations

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Abstract

A re-evaluation of the notion of vacuum in quantum electrodynamics is presented, focusing on the vacuum of the quantized electromagnetic field. In contrast to the ‘nothingness’ associated to the idea of classical vacuum, subtle aspects are found in relation to the vacuum of the quantized electromagnetic field both at theoretical and experimental levels. These are not the usually called vacuum effects. The view defended here is that the so-called vacuum effects are not due to the ground state of the quantized electromagnetic field. Nevertheless it is possible to maintain an empirically demonstrable notion of vacuum state that is consistent with the interpretation of the formalism of the theory.

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Notes

  1. The Casimir effect consists in the existence of an attractive force between two conducting plates located face to face in a spatial vacuum. According to Casimir’s standard interpretation this dynamical effect is due to the fact that the plates are supposed to change the boundary conditions of the quantized electromagnetic field ground state (vacuum state), and that this will result in an attractive force between the plates (see Sect. 3.2).

  2. In Sect. 3.3 I will make the case for an interpretation of the variance in terms of a statistical spread (distribution) in the results of independent measurements made on identically prepared systems. This interpretation is made within the broader framework of the ensemble interpretation of quantum mechanics. This meaning of variance is to be distinguished from the standard and confusing term ‘fluctuations’ associated with an intuitive idea of actual fluctuations of the field related to the so-called zero-point energy.

  3. See, e.g., Bogoliubov and Shirkov (1959).

  4. Schweber (1961, 245–251).

  5. Mandl and Shaw (1984, 89).

  6. Schweber (1961, 251), Källén (1972, 42).

  7. Jauch and Rohrlich (1976, 47).

  8. Dirac (1927).

  9. Barut (1964, 138), Bogoliubov and Shirkov (1959, 84).

  10. The use of perturbative methods has a long history in celestial mechanics. One example is the development of an analytical perturbation theory for the three-body problem: the Sun-Earth-Moon system (Hoskin and Taton 1995, 89–107). From the planets, perturbative methods went to the planetary models of atoms, being a calculational tool present in the so-called old quantum theory (Darrigol 1992, 129 and 171). Also it became fundamental in the creation of matrix mechanics, as it was from the perturbative study of the anharmonic oscillator that Werner Heisenberg developed his quantum-theoretical approach (Darrigol 1992, 266–267; Paul 2007, 4–5). Soon after, Heisenberg and Max Born put together a perturbation theory within the formalism of quantum mechanics recently developed (van der Waerden 1967, 43–50; see also Lacki 1998).

  11. Veltman (1994, 62–67).

  12. Dyson (1952a, 81).

  13. Ibid.

  14. See also Falkenburg (2007, 129–131).

  15. Dyson (1952a, 94).

  16. Pais (1986, 374–376).

  17. Schweber (1994, 527–544).

  18. Quoted in Schweber (1994, 565).

  19. Anselmi (2003, 311).

  20. Dyson (1952b, 631).

  21. Quoted in Schewber (1994, 565).

  22. Even if strict mathematical proof of the divergence of the S-matrix does not exist, further strong evidence in favour of Dyson’s claim has been given in the last decades (see e.g. Aramaki 1989, 91–92; West 2000, 180–181; Jentschura 2004, 86–112; Caliceti et al. 2007, 5–6).

  23. Dyson (1952a, 79).

  24. Schweber (1994, 565).

  25. Anselmi (2003, 311).

  26. Ibid.

  27. This brings the philosophical question of what to make of a ‘theory’ that seems to provide only an approximate scheme for calculations. As Meinard Kuhlmann remarked (in a broader context), quantum electrodynamics seems more like “a set of formal strategies and mathematical tools than a closed theory” (Kuhlmann 2006). This is an important question, but it would go beyond the scope of this paper to address it here.

  28. Considering quantum electrodynamics on a lattice the situation is not different. It is usually considered that the lattice regularization is non-perturbative because from the start the space–time lattice implies an energy–momentum cutoff to all orders of the perturbative calculation. However in lattice quantum electrodynamics we still have a divergent S-matrix, and it is this that makes the theory intrinsically approximate (see, e.g., Montvay and Münster 1997).

  29. Schweber (1961, 322).

  30. Redhead (1982, 86), Schweber (1961, 655).

  31. Scharf (1995, 314–318).

  32. Aitchison (1985, 342–345), Milonni (1994, 79–111).

  33. See, e.g., Milonni (1994, 115–138), Zinkernagel (1998, 48–60).

  34. The main difference between Zinkernagel’s views and mine, regarding the Casimir effect, is that I consider that within standard quantum electrodynamics there are sound approaches that show that the Casimir effect is not a vacuum effect. In this way I will not consider for example Schwinger’s source theory (see e.g. Rugh et al. 1999). I will frame my discussion within standard quantum electrodynamics by considering Milonni’s approach as Zinkernagel did. This does not mean that there are no other standard quantum electrodynamics calculations which enable to calculate the Casimir effect without reference to the vacuum state (see, e.g., Jaffe 2003, 2005; Graham et al. 2004). Simply, choosing Milonni’s work makes it easier to see the difference between my views and Zinkernagel’s.

  35. See, e.g., Teller (1995, 129–131).

  36. Casimir (1948).

  37. Milonni (1994, 44). From a classical perspective the series expansion of a classical electromagnetic wave seems to be unproblematic; we can regard it as rigorous for an electromagnetic field in a cavity and for the case of, for example, a radiation field one can always consider the possibility of taking the spatial dimensions to infinity and the wave to be represented by a Fourier integral (see, e.g., Schwartz 1987, 225–26). In the case of the quantized field the situation is trickier, since we have to consider interpretation issues. In particular when adopting a Bohrian definition of quantum phenomena, what we call the quantized electromagnetic field cannot be considered by abstracting away the experimental setup where it is manifested (see, e.g., Howard 2004). In this case, for example for experiments of quantum optics, the box quantization can be seen as an abstract implementation of the contextual definition of quantum field taking into account the actual experimental ‘boundary conditions’ due to the experimental material systems that enable the quantum phenomena to appear, i.e. there really is no quantum field without material objects. I do not develop this line of inquiry in this work. I am solely interested in showing that, within quantum electrodynamics, there is a consistent and meaningful definition of vacuum state with experimental relevance.

  38. See, e.g., Graham et al. (2004).

  39. Milonni and Shih (1992, 4241).

  40. Saunders (2002, 23).

  41. Ibid., 19.

  42. Milonni and Shih (1992, 4241).

  43. Ibid., 4243.

  44. Milonni (1994, 138).

  45. Ibid., 52.

  46. Ibid.

  47. Ibid., 53.

  48. Ibid., 52.

  49. Ibid., 53.

  50. Ibid.

  51. See, e.g., Rugh and Zinkernagel (2002, 675).

  52. This, as mentioned, is the main difference between Zinkernagel’s views on the Casimir effect and my own. Zinkernagel considers that Milonni’s work is not sound and he explores approaches that go beyond standard quantum electrodynamics to show that the Casimir effect is not a vacuum effect. My view is that Milonni’s work is sound and shows that we do not have to see the Casimir effect as a vacuum effect (see also Jaffe 2003, 2005, Graham et al. 2004).

  53. See, e.g., Milonni (1988, 106).

  54. See, e.g., Milonni (1984, 342), (1988, 106), (1984, 50–54).

  55. The only experimental consequence of the vacuum, according to the view being presented here, is the non-vanishing variance of the quantized electromagnetic field when in this state. To measure the field we need a charged ‘probe’ (i.e. charged matter). This fits nicely with Milonni’s ‘formal result’, pointing to the need to always consider charged matter together with the electromagnetic field.

  56. See, e.g., Sakurai (1967, 32–33), Aitchison (1985, 246–247).

  57. Aitchison (1985, 347).

  58. Rugh and Zinkernagel (2002, 673).

  59. Isham (1995, 80–81), Peres (1995, 24–26), Ballentine (1998, 225–227), Falkenburg (2007, 205–207).

  60. Isham (1995, 80).

  61. Peres (1995, 24–25).

  62. Leonhardt (1997, 23, 47 and 84–88).

  63. Gerry and Knight (2005, 167–168).

  64. Vogel et al. (2001, 225).

  65. Leonhardt (1997, 99).

  66. Ibid., 40.

  67. Ibid., 46–47.

  68. Ibid., 23.

  69. In squeezed states, of two operators satisfying a commutation relation (like for example the quadrature operators, which have their standard deviation related by an Heisenberg uncertainty relation), the variance related to one of the operators is experimentally ‘squeezed’ while increasing at the same time the other operator’s variance, in a way that is according to the uncertainty relation (see, e.g., Gerry and Knight 2005).

  70. See, e.g., Breitenbach et al. (1997), Bertet et al. (2002).

  71. As mentioned in footnote 37, there are interpretations of the quantum formalism according to which the quantum field is only meaningful in the context of a material experimental setup. However, even when adopting this type of interpretation, we can talk about a quantum field in a 0-photon state in the same sense that we talk, for example, about a quantum field in a 1-photon state, without having to consider that the experimental results related to the 1-photon state need an account in terms of ‘physical effect’ due to the experimental setup.

  72. Vogel et al. (2001, 169–190).

  73. Milonni (1994, 125).

  74. We must recall that this variance is related with measurements made on identically prepared systems not one individual system. As mentioned, there is a tendency in the literature to refer to the non-vanishing variance as ‘fluctuations’ of the vacuum state (Sakurai 1967, 32–33; Aitchison 1985, 346–247). This view is misleading since the non-zero variance of the ground state of the quantized electromagnetic field cannot be related to a fluctuation in time: “there is no time evolution of this vacuum state” (Rugh and Zinkernagel 2002, 673). Considering measurements made on equally prepared systems, they will show fluctuations in the results of the successive observations—according to the interpretation of the theory followed here (Isham 1995, 80–81; Peres 1995, 24–26; Ballentine 1998, 225–227; Falkenburg 2007, 205–207). The non-vanishing variance is not a temporal property of one single system. We observe a statistical fluctuation on the results of measurements made on equally prepared systems, not a temporal fluctuation of the same system.

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    Mario Bacelar Valente

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Valente, M.B. A Case for an Empirically Demonstrable Notion of the Vacuum in Quantum Electrodynamics Independent of Dynamical Fluctuations. J Gen Philos Sci 42, 241–261 (2011). https://doi.org/10.1007/s10838-011-9162-0

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