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What is It Like to be a Relativistic GRW Theory? Or: Quantum Mechanics and Relativity, Still in Conflict After All These Years

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Abstract

The violation of Bell’s inequality has shown that quantum theory and relativity are in tension: reality is nonlocal. Nonetheless, many have argued that GRW-type theories are to be preferred to pilot-wave theories as they are more compatible with relativity: while relativistic pilot-wave theories require a preferred slicing of space-time, foliation-free relativistic GRW-type theories have been proposed. In this paper I discuss various meanings of ‘relativistic invariance,’ and I show how GRW-type theories, while being more relativistic in one sense, are less relativistic in another. If so, the initial claim that GRW-type theories have a greater compatibility with relativity is unwarranted: both type of theories violate relativity, one way or another.

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Notes

  1. For sake of precision, however, notice that the theory originally proposed by Bohm (1952) is arguably not the same theory as the one developed by Dürr, Goldstein and Zanghí (1992), and now known as Bohmian mechanics. In fact, both are theories of particles; however, in Bohm’s theory is a first-order theory in which the wavefunction is considered a real physical field in space, and there is a quantum potential, while nothing of the sort is present in Bohmian mechanics (see Dürr et al. for a comparison). Nonetheless, it is common practice among physicists interested in foundations and philosophers of physics to use the locutions ‘Bohmian mechanics’ and ‘de Broglie-Bohm pilot-wave theory’ interchangeably (see e.g., Bricmont 2016b, Norsen 2016). In the following, I will not consider Bohm’s theory.

  2. Also in this case, it has been argued that Everett did not endorse a many-worlds interpretation of his formulation of quantum mechanics (see Barrett and Byrne 2012). Nonetheless, it seems common practice among philosophers of physics and physicists to understand ‘Everettian mechanics’ to denote the many-worlds theory.

  3. For other attempts, see Bassi and Ghirardi (2020) and references therein.

  4. For more on this theory, see Wallace (2012). See also Barrett (2018), Vaidman (2014), and references therein. For an assessment of its compatibility with relativity, see Myrvold (2021).

  5. See, e.g., Goldstein et al. (2011), Vaidman (2016). However, see also Norsen (2016). For an explicitly nonlocal many-worlds theory within the primitive ontology framework see Allori et al. (2014). In addition, let me notice that, if one takes Bell’s theorem as a general argument for nonlocality, every theory will have to be nonlocal, including the many-worlds theory.

  6. Among these theories one also finds the so-called modal interpretation. Since it has been argued (e.g., Myrvold 2002a) that they suffer from the same difficulties as the pilot-wave theory, in this paper I will only discuss the latter because it is simpler. For additional discussion on the modal interpretation and its compatibility with relativity, see also Myrvold (2009, 2021).

  7. Goldstein (2017) and references therein.

  8. See, e.g., Allori et al. (2008).

  9. Allori et al. (2008).

  10. Allori et al. (2011). Another example of a Bohmian theory of this type is BQFT (Bohmian quantum field theory), which has deterministically evolving fields (Bohm 1952, Struyve and Westman 2006).

  11. Nelson (1985). Otherwise, in Bell-type quantum field theory, a stochastic evolution of the particles allows for creation and annihilation (see Bell 1986, Dürr et al. 2004; 2005; nonetheless, one could arguably describe particle creations and annihilations even with a deterministic dynamics, see Colin 2003, Colin and Struyve 2007; see also Nikolić 2010; for discussion, see also Oldofredi 2020).

  12. There are two new constants of nature, the localization accuracy \(d={10}^{-7}\) m, and the localization frequency \(f={10}^{-16} {s}^{-1}\), so that microscopic systems localize on average every hundred million years, while macroscopic systems every \({10}^{-7}\) seconds. There is a more general class of GRW-like theories, namely theories in which the wavefunction spontaneously collapses, which goes under the name of CSL, continuous spontaneous localization, which is an extension of the GRW logic (see Bassi and Ghirardi 2020). I will include these theories under the label ‘GRW-type theories.’

  13. See Allori et al. (2008), Allori (2013a). See also Allori (2019a) and references therein. For criticism, see e.g., Belot (2012), Ney and Phillips (2013), Albert (2015), Lewis (2015), Myrvold (2015, 2019), Wallace (2020), Egg (2021), Egg and Saatsi (2021), and references therein.

  14. Bell (1987).

  15. The matter density function has been introduced in Benatti et al. (1995). It is considered as a possible primitive ontology explicitly in Allori et al. (2008).

  16. Allori et al. (2008, 2014); Allori (2020a).

  17. Notice that a stochastically evolving primitive ontology (a ‘stochastic primitive ontology’) has been combined with a deterministically evolving wavefunction (a ‘deterministic wavefunction’), like in Sf, stochastic mechanics or certain Bell-type quantum field theories. Also, a ‘stochastic primitive ontology’ has been paired with a stochastically evolving wavefunction (a ‘stochastic wavefunction’), as for instance in all GRW-type theories. Instead, a deterministically evolving primitive ontology (a ‘deterministic primitive ontology’) has only been successfully combined with a deterministically evolving wavefunction (a ‘deterministic wavefunction’), as in the original de Broglie-Bohm pilot-wave theory.

  18. In this reconstruction I follow Goldstein et al. (2011), Maudlin (2011), Norsen (2016), Bricmont (2016a). I assume that this reconstruction is correct, as my main goal in this paper is to show that, even granting that Bell has shown that reality is nonlocal, it is still problematical to think that GRW theories are more compatible with relativity than the pilot-wave theory. Indeed, most people who argue that GRW theories are more compatible with relativity than pilot-wave theories accept this reconstruction.

  19. Einstein, Podolsky and Rosen (1935). See also Eisntein (1948).

  20. Think for instance about the objections to Newton’s theory of gravitation. Newton agreed it was a problem and replied that his theory was incomplete, and that a future, better, theory should make this action at a distance go away (see Norsen 2011).

  21. This is due to the relativity of simultaneity: since two spacelike separated observers, namely observers such that a signal connecting them would have to propagate faster than light, will disagree on the temporal ordering of events, to avoid causal chains to go backwards one requires physical influences to travel slower than the speed of light.

  22. When Alice measures spin up in one direction, Bob will measure spin down in that direction.

  23. It is interesting to notice, as emphasized by Norsen (2011), that EPR’s conclusion that quantum theory is incomplete is similar in spirit to Newton’s reply to the objection that his theory requires action at a distance (see footnote 21).

  24. Bell (1990).

  25. For more on Bell’s notion of local causality and on local beables, see Norsen (2011).

  26. However, see Allori (2021) for a distinction.

  27. Notice, in passing, that this requirement seems to rule out the many-worlds theory: since it is only about the evolution of the wavefunction, it does not have any local beables.

  28. Aspect et al. (1982).

  29. For discussions of Bell’s proof in general, see e.g., the contributions to Gao and Bell (2016).

  30. Bell (1987).

  31. See Caluser et al. (1985). See also Norsen (2009), Bricmont (2016a), Chen (2021) for further criticisms. For a defense of superdeterminism, see Hossenfelder and Palmer (2020) while for a recent discussion, see Baas and Le Bihan (forthcoming).

  32. For more on retrocausality in quantum mechanics, see e.g., Freidman (2019) and references therein.

  33. At some point, this is what Bell seemed to have argued, at least according to Norsen (2011).

  34. Bell (1987).

  35. Following Bell, see Tumulka (2006). See also e.g., Ghirardi (2012), Myrvold (2002, 2016, 2021), and references therein.

  36. Some have also maintained that there is the additional problem that a relativistic quantum theory needs to be a field theory (Malament 1996; see also Myrvold 2021). I will not consider this difficulty in this paper because, as I will be evident in the text, the point I wish to make arises independently of this issue.

  37. See Leinert et al. (2020), and references therein.

  38. Notice that this type of theories is nonlocal regardless of the ontological status of the wavefunction. Nonlocality is particularly explicit if one thinks of the wavefunction as a field in configuration space (see e.g., Albert 2015, Ney 2020), or as a part of the structure of the law of interaction among of the particles (Goldstein and Zanghì 2013, Allori 2020b). It is less explicit if one considers the wavefunction to be a multi-filed in three-dimensional space (Forrest 1988; Belot 2012, Hubert and Romano 2017), or tentatively eliminated (Norsen 2015) but the nonlocality of the interaction is still there.

  39. Berndl et al. (1996), Dürr et al. (1999).

  40. See also Maudlin (1996).

  41. Technically, the worry is that one could make anything relativistic invariant (or invariant with respect to any transformation) by adding suitable structure. Take a non-Lorenz invariant theory with primitive ontology \(P\) and law \(L\). That means that the ‘trajectories’ of \(P\) (i.e., their worldlines), when transformed according to the Lorentz group, are no longer solutions (that is, they are no longer possible states of affairs of the world). However, we can always add something to the primitive ontology, so that \(P?=(P,X)\), in such a way that by stipulation the new law \(L?\) would transform solutions into solutions under the Lorentz group. Such theory would be Lorentz invariant, but not genuinely so (see Bell 1987, Berndl et al. 1996).

  42. Dürr et al. (2013).

  43. Tumulka (2007).

  44. This is because the flashes are constructed here in generations, and the distribution of a flash depends upon which of the other flashes belong to the same or the previous generation.

  45. Other relativistic pilot-wave theories have been proposed, but are not viable, as they have no equivariant measure and therefore they have predictions which are inconsistent with quantum mechanics. They all use the multi-time formalism and Lorentz invariant equations for the wavefunction. For instance, in one proposal (Berndl et al. 1996; Dewdney and Horton 2001; Nikolić 2005) a current vector defined by the wavefunction generates spatiotemporal paths parametrized by a common parameter. Moreover, another proposal is Lorentz invariant without a foliation, as it uses the light-cone structure as simultaneity structure (Goldstein and Tumulka 2001, Tumulka 2007) but it has a backward microscopic arrow of time (a theories defined on the past light-cone would have been local). This theory does not describe interactions and has no equivariant measure. However, it constitutes an example on how one can use the spacetime structure to achieve nonlocality, by allowing for retrocausality.

  46. This can be seen as the non-relativistic shadow of Lorentz invariance, because it is equivalent to assuming that absolute simultaneity plays no role in the theory: if it would, then the two times in two distant systems could not be shifted independently.

  47. See also Maudlin (2011).

  48. Some other attempts to relativistic GRW-type theories have been proposed which do not use the multi-time wavefunction but still have a Lorentz invariant stochastic evolution (see for instance, Bedingham 2011, Pearle 2015). This is of no consequence for the main conclusion of this paper, as I will show in Sect. 6, because the stochasticity of the evolution of the wavefunction, regardless of how this stochasticity is implemented, is the relevant ingredient.

  49. Tumulka (2006a). See also Tumulka (2020) for a theory with interaction.

  50. For completeness, let me mention other proposals for relativistic spontaneous collapse theories. Some early proposals are found in Pearle (1990); Ghirardi, Grassi and Pearle (1990) but have been recognized to be problematical (see Bassi and Ghirardi 2020). Dowker and Henson (2004) construct a spontaneous collapse theory which is very similar to rGRWm, but it is defined on a lattice spacetime. It is a theory with a primitive ontology of field values at the lattice sites, and it is Lorentz invariant in the ‘correct lattice sense.’ Dove and Squires (1995) extend a previous proposal by Hellwig and Kraus (1970) developed in the context of ordinary quantum theory to a flash GRW theory. In this model, the wavefunction collapses along the past light-cone of the spacetime point at which a measurement takes place. However, the theory involves retrocausation, as explained in Tumulka (2009). Also, they only propose a Lorentz-invariant collapse rule for the wavefunction given the flashes, but no distribution law for the flashes. For more discussion on these theories, see Ghirardi and Bassi (2020), Tumulka (2006a,b, 2007).

  51. Norsen (2010a).

  52. It seems important to notice that the situation may be made less severe in this respect by considering the wavefunction as a multi-field in three-dimensional space, or not as part of the ontology of matter (for instance, one could consider the wavefunction part of the law of these theories). Similarly, Rovelli’s relational interpretation (1996) or Bub and Pitowsky information-theoretic account of quantum theory (2010), seem to be in a better position in this respect (even if by considering the wavefunction as epistemic they run into other problems, see e.g. Norsen 2016). Nonetheless, in addition to the ontology being local, one would also have to face the fact that the interaction is nonlocal, and this is what Bell has shown. Therefore, the advantage of these formulations seems minimal in this respect.

  53. Norsen, p.c.

  54. That is, no ‘spooky action at a distance’ (Einstein, Born 1971).

  55. This idea has been criticized by Maudlin (2011).

  56. Dürr et al. (2013).

  57. Maudlin (1996); see also Maudlin (2011). This understanding of ‘spirit of relativity’ is shared also by Tumulka (see, e.g., 2006, 2009) and Myrvold (see, e.g., 2016, 2021).

  58. A remark on the meaning of ‘foliation-free theory.’ This terminology should be taken to mean that there is no preferred foliation with a dynamical role. That is, the dynamical laws do not involve any such structure. In particular, as mentioned earlier, the fact that one could extract a preferred foliation from the wavefunction, does not imply that every quantum theory, in virtue of having a wavefunction, will also have a preferred foliation, because such foliation could be not dynamical.

  59. See e.g., Dürr et al. (2016) on this point.

  60. Also, as mentioned already, some relativistic GRW-type theories do not use this formalism, see Bedingham (2011), Pearle (2015).

  61. Tumulka (2009).

  62. In GRWf and GRWm, as well as in rGRWf and rGRWm, the flashes are distributed as dictated by the collapses of the wavefunction and the matter density is defined as.

    \(m\left(x,t\right)=\sum _{i=1}^{N}{m}_{i}\int d{q}_{1}\dots d{q}_{N}{\delta }^{3}\left({q}_{i}-x\right){\left|{\psi }_{t}\left({q}_{1}\dots {q}_{N},t\right)\right|}^{2},\)where \(N\) is the number of ‘particles,’ and \({m}_{i}\) are their masses.

  63. They are doubly deterministic in the sense that both the particles and the wavefunction evolve deterministically (respectively according to the guidance equation and the Schrödinger equation).

  64. As discussed in Goldstein and Tumulka (2001), and in Tumulka (2007), a theory using past light-cones is local.

  65. See Allori (2020a) and references therein.

  66. The first to propose to rewrite the locality condition in terms of two different conditions was Jarrett (1984), and he distinguished between ‘simple locality’ and ‘completeness.’ His conclusions have been criticized especially by Norsen (2009) and Maudlin (2011). The terminology ‘parameter independence’ and ‘outcome independence’ is by Shimony (1984) and seem to better characterize the content of these conditions.

  67. See, e.g., the first part of the proof in Butterfield et al. (1993) as reported in Norsen (2010b).

  68. Jarrett (1984), Shimony (1984); see also, e.g., Redhead (1987), Cushing and McMullin (1989), Butterfield (1992), Ghirardi (2012), Myrvold (2016), Bassi and Ghirardi (2020), and references therein.

  69. Maudlin (2011), Goldstein p.c., Norsen p.c.

  70. Assuming there is no retrocausation. See also Ghirardi et al. (1993).

  71. Lewis (2020) and Esfeld (2020), for instance, have argued that their primitive ontology is not as familiar as a particle ontology, regardless of the status of the wavefunction.

  72. See also Lewis (2003), McQuinn (2015) and references therein for more.

  73. For a small macroscopic object made of roughly \({10}^{19}\) ‘particles,’ if each ‘particle’ undergoes a collapse every \({10}^{16}\) seconds, there is a flash roughly once every \({10}^{-3}\) seconds, and nothing at all in between.

  74. See Allori (2020a) for more, and Esfeld (2020) for a defense of the flash ontology.

  75. However, within the primitive ontology framework some have maintained that fields should be seen as part of the law too, just as the wavefunction (Allori 2015). If so, the objection seems less severe.

  76. However, Allori (2018) has argued that this is still possible.

  77. See e.g., Monton (2002, 2006, 2013), Allori et al. (2008, 2011), Albert (2013, 2015), Allori (2013b), Emery (2017), Maudlin (2019). See replies in Albert (2015), Ney (2021) and references therein. Notice however, that these replies aim at recovering the appearances of macroscopic three-dimensional objects from the wavefunction, so they do not dispute the unfamiliarity of their ontology.

  78. At east most of them, if one does not consider, for instance, Bell-type quantum field theories.

  79. This is in a 1926 letter to Born; in Born and Einstein (1971).

  80. For propensities see Popper (1957). For more discussion, see also, among others, Frigg and Hoefer (2007), Dorato and Esfeld (2010), Esfeld and Gisin (2010), Monton (2013), Egg and Esfeld (2015), and Esfeld et al. (2017).

  81. See also Ghirardi and Grassi (1996), Esfeld and Gisin (2014), Myrvold (2016).

  82. Goldstein (2012), Lazarovoci and Reicher (2015), Wilhelm (2019), Bricmont (2020).

  83. However, see Allori (2019b).

  84. See also Einstein’s distinction between constructive and principle theories (1919). Constructive theories are theories in which the macroscopic phenomena are accounted for in terms of the microscopic dynamics, while principle theories constrain the phenomena with principles which govern what can and cannot happen. Flores (1999) has argued that constructive theories are compatible with Salmon’s mechanistic explanation (1987), principle theories can be tied to unification approaches such as the ones of Friedman (1974), Kitcher (1989). See also Felline (2011).

  85. The current models of explanatory unification generalize the idea that explanations are arguments (developed by the deductive-nomological model), and therefore are committed to the so-called ‘expectability thesis:’ a unifying explanation must show how the explanandum is to be expected from the explanans. With stochastic laws, events with a low probability of happening cannot be expected. One possible reply is to use van Camp’s proposal (2011) that principle theories unify in the sense that they provide the conceptual schema necessary to talk about explanation within a theory in the first place. The idea is that principle theories play a distinctive conceptual role as necessary preconditions for explanation (see also DiSalle 2006).

  86. Tumulka (2006b) similarly argues that the direction of influence is indeterminate.

  87. See e.g., Lewis (2021).

  88. See Myrvold (2002) for the original argument. See also Myrvold (2016, 2019, 2021).

  89. Myrvold (2002).

  90. See e.g., Maudlin (2011), Bedingham et al. (2013), Bassi and Ghirardi (2020).

  91. This experiment has been first proposed by Einstein; see Norsen (2005).

  92. Bricmont (2016b).

  93. Maudlin (2011), Bricmont (2016b).

  94. Norsen p.c.

  95. I am grateful to an anonymous reviewer for pressing me on this point.

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Acknowledgements

I am particularly grateful to Jean Bricmont for discussing these matters with me at length. I also wish to thank Travis Norsen and Roderich Tumulka for their helpful insights on some issues considered in this paper.

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    Valia Allori

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Allori, V. What is It Like to be a Relativistic GRW Theory? Or: Quantum Mechanics and Relativity, Still in Conflict After All These Years. Found Phys 52, 79 (2022). https://doi.org/10.1007/s10701-022-00595-5

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