Abstract
Under so-called primitive ontology approaches, in fully describing the history of a quantum system, one thereby attributes interesting properties to regions of spacetime. Primitive ontology approaches, which include some varieties of Bohmian mechanics and spontaneous collapse theories, are interesting in part because they hold out the hope that it should not be too difficult to make a connection between models of quantum mechanics and descriptions of histories of ordinary macroscopic bodies. But such approaches are dualistic, positing a quantum state as well as ordinary material degrees of freedom. This paper lays out and compares some options that primitive ontologists have for making sense of the quantum state.
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Notes
Rather than attempting to solve the measurement problem, one can attempt to evade or deflate it by denying that quantum states are in the business of representing how things are in the world. For a sophisticated version of this neo-instrumentalist strategy, see Bub and Pitowsky (2010).
The source of this terminology is Goldstein (1998).
For a survey of these options considered as primitive ontology approaches, see Allori et al. (2008).
Each of the options canvassed below can be found in the literature, except the dispositionalist account.
I.e., we look for the weakest equivalence relation that satisfies these stipulations.
We will see a non-standard approach below, under which inequivalent wave-functions are sometimes taken to correspond to the same quantum state.
Equivalent wave functions will not in general agree about p(q), but they will agree about the integral of p(q) over any region of Q.
For discussion and references, see Callender (2007).
Of course, our real interest is in the quantum state, not the wave-function. In order to see quantum states as analogous to fields, we should first generalize the notion of a field so that it tells us not about the assignment of properties to points of space, but about something like the ratio of integrals of such assignments over regions of space.
For this approach, see Forrest (1988, Chapter 5).
Something along these lines appears to stand behind Bell’s dictum that “No one can understand [Bohmian mechanics] until he is willing to think of ψ as a real objective field rather than just as a ‘probability amplitude’. Even though it propagates not in 3-space but in 3N-space” (Bell 1987, p. 128).
Well-known mathematical difficulties stand in the way of implementing this sort of reciprocity—but in the case of Maxwell’s theory, rigourous models have been constructed for the case of finite charged bodies. See Spohn (2004).
The two wave-functions will differ by a phase factor that depends on x, t, v, and the mass of the particle under consideration. See, e.g., Ballentine (1998, Section 4.3).
Loewer (1996, Section I).
For discussion of this point, see Maudlin (2007, Section 3).
A view of this kind is espoused by Dorr (2009, Finding ordinary objects in some quantum worlds, unpublished manuscript)—whose reasons for rejecting the multi-field approach turn on the subtle question of whether or not there are fundamental non-symmetric relations.
The problem is even more striking on the picture on which the particle configuration variable of the Bohmian theory describes the location of a world-particle in configuration space rather than the locations of many particles in three-space:
the space [wave functions] live in, and (therefore) the space we live in, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out (if space is the right name for it ...) is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three-dimensional space, or in a four-dimensional space-time) is somehow flatly illusory. (Albert 1996, p. 277)
Some readers may well feel that, if true, the claim made by Bell in the passage quoted in fn. 17 above provides ample motivation to prefer the field-in-configuration-space picture to the multi-field-in-three-space picture. Others will be dubious that facts concerning human mathematical processing can carry much weight in the present context.
These are briefly noted in Dürr et al. (1997, Section 12).
Consider, e.g., Lewis’s version of the best-system account (see, e.g., Lewis 1999, pp. 39–43 and 231–236). On this account the laws are the consequences of those generalizations concerning the pattern of instantiation of perfectly natural properties that provide the best (=strongest-simplest) description of the actual world. Because lawhood is, roughly speaking, a global property of systems of axioms, it makes little sense to single out any member of such a system as being more or less necessary than any other.
Consider here the account of Lange (2009), on which the laws are distinguished from accidents via their (collective) exhibition of a certain sort of stability under a certain wide range of counterfactual suppositions. As Lange emphasizes, the set of laws may include a subset distinguished by its (collective) stability under a yet wider range of counterfactual suppositions—such a subset enjoys a form of necessity weaker than metaphysical necessity but somewhat stronger than that it exhibited by the full set of laws. For example: on this account, the symmetry principles of classical mechanics are somewhat more necessary than the other laws of classical mechanics (see Lange 2009, Sections 1.4, 1.9, and 3.5).
It is sometimes suggested that the best-system account of laws be generalized so that the characteristic postulates of statistical mechanics (which are not generalizations, and so do not count as laws under Lewis’s version of the account) will come out as laws at our world (for this suggestion, see Loewer (2001) or Callender (2004)). This leads others to object that it is a weakness of best-system approach that it lends itself so readily to a blurring of the distinction between between laws and initial conditions (see Roberts 2008, Section 1.6).
Other considerations arising out of quantum statistical mechanics suggest a similar sort of picture (see Ruetsche 2011).
For a suggestion along these lines, see Monton (2006).
Contrast with the classical case, where we can say that the bearer of the property is a system of particles and that each property under consideration is an assignment to each particle of a position and a momentum.
Again, on the issue of smoothness, see esp. Berndl et al. (1995, Section 4.4).
For the role that this system played in early criticisms of Bohmian mechanics, see Myrvold (2003). The harmonic oscillator is another simple, charismatic quantum system that could be used to illustrate the following points (it is easier to write down the relevant family of functions for the particle in a box).
Note that this shows that the system is not deterministic if we take Φ and the particle positions to exhaust the instantaneous state and attempt to mimic the dynamics of ordinary Bohmian mechanics. We have seen that ϕ 1 and ϕ 2 determine the same instantaneous dispositional property Φ0. But specifying that at time t 0 the particle is in configuration q 0 and that the dispositional property of the system is Φ0 will not in general suffice to determine the particle’s position at future times: we could choose a Hamiltonian for our system that has ϕ 1 but not ϕ 2 as an eigenstate; then in ordinary Bohmian mechanics, we would find that our initial data (q 0, Φ0) is consistent with the particle remaining eternally in q 0 or with it moving around (depending whether the initial quantum state is given by ϕ 1 or by ϕ 2). Of course, one would hope that determinism would be secured if one were to include one or more ‘time-derivatives of Φ’ among the dynamical variables that describe tinstantaneous states.
Note, though, that a nontrivial superposition of two such solutions would correspond to a quite different dispositional history.
Sheldon Goldstein, private communication (November 2008).
It is important to keep in mind here that it is only when we are treat the free particle in a box or the harmonic oscillator as closed systems that problems arise: if such systems interact with others, then there is no reason to expect the composite system to exhibit difficulties of the kind under discussion. (This point is the analog of the standard Bohmian reply to Einstein’s worry about Bohmian mechanics. Worry—If the quantum state of a particle in a box is an eigenstate of energy, then the particle is stationary—this is unphysical. Reply—Any decent model of the measurement of the momentum of such a system will give predictions agreeing with those of standard quantum mechanics.)
See, e.g., van Fraassen (2007, p. 358).
See Belot (2011, Appendix A).
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Acknowledgements
Earlier versions of this paper were given at conferences at the University at Buffalo in November 2008 and at the University of Western Ontario in April 2009. Thanks to all those present. For helpful discussion and suggestions, thanks to Anonymous1, Anonymous2, Shelly Goldstein, Carl Hoefer, David Malament, Laura Ruetsche, Brad Skow, and Roderich Tumulka.
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Dedicated to the memory of Ayveq, a great pinniped.
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Belot, G. Quantum states for primitive ontologists. Euro Jnl Phil Sci 2, 67–83 (2012). https://doi.org/10.1007/s13194-011-0024-8
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DOI: https://doi.org/10.1007/s13194-011-0024-8