On a Symmetry Argument for the Guidance Equation in Bohmian Mechanics

Abstract

Bohmian mechanics faces an underdetermination problem: when it comes to solving the measurement problem, alternatives to the Bohmian guidance equation work just as well as the official guidance equation. Dürr, Goldstein, and Zanghì have argued that of the candidate guidance equations, the official guidance equation is the simplest Galilean‐invariant candidate. This symmetry argument—if it worked—would solve the underdetermination problem. But the argument does not work. It fails because it rests on assumptions about how Galilean transformations (especially boosts) act on the wavefunction that are (in this context) unwarranted. My discussion has larger morals about the physical significance of certain mathematical results (like, for example, Wigner’s theorem) in non‐orthodox interpretations of quantum mechanics.

Acknowledgements

Thanks to two anonymous referees, to Bryan Roberts, and to David Baker, Gordon Belot, Laura Ruetsche, and everyone else at the University of Michigan who read this paper.

Notes

[1] This is the form of the guidance equation preferred by (among others) Bell (1987b) and Dürr, Goldstein, and Zanghì (1992). There are alternative, mathematically equivalent forms of the equation—Bohm 1952, for example, prefers to write it as a second‐order differential equation similar in form to Newton’s second law. Although these forms of the guidance equation are mathematically equivalent there may be conceptual differences between them, and these differences may suggest different strategies for justifying the equation. But in this paper I will focus only on the first‐order formulation of the equation.

[2] As is well known, because the word ‘measurement’ appears in the formulation of orthodox theory it is not clear just what predictions the theory does make. (The word ‘measurement’ does not appear in the formulation of Bohmian mechanics.) So it is not clear just what it takes for an interpretation of (or replacement for, or completion of) orthodox quantum mechanics to match that theory’s predictions. I am going to set this problem aside.

[3] This sketch of the Bohmian solution to the measurement problem resembles the one given in Bell 1987a. But the full story of how Bohmian mechanics solves the measurement problem is much more complicated. For example, I ignore the fact that due to the pervasiveness of quantum entanglement particles almost never have their own wavefunctions. Bohmians must explain why we are entitled to treat them as if they do. I also say nothing about why Bohmians are entitled to assume that the frequency with which electrons appear in some region S near the source is given by . For the full story, including how the theory deals with measurements of things other than position, see Dürr, Goldstein, and Zanghì 1992. Although what I say is oversimplified in many ways, it is accurate enough to let us see the role the guidance equation plays.

[4] In slightly more detail, to say that the guidance equation preserves probability is to say that

where ρ(x,t) = |ψ(x,t)|² and v(x,t) is the velocity field given by the guidance equation. Inspection shows that if we changed the guidance equation so that

where v is a solution to the original guidance equation and w(x,t) is divergence‐free, then the continuity equation is still true of . So the guidance equation is not the unique equation that, together with |ψ|², satisfies the continuity equation; it is not the unique equation that ‘preserves probability’. (An extended discussion of alternative Bohm‐like theories may be found in Deotto and Ghirardi 1998.)

[5] They are not empirically equivalent if we give up the assumption that the frequency with which electrons begin in some region S near the source is given by —that is, if we give up the quantum equilibrium hypothesis. But I am looking for a justification of the guidance equation that holds even under this hypothesis.

[6] Whether the first law follows from the second law by setting F = 0 is a matter of dispute (see, for example, Earman and Friedman 1973; an interpretation of the first law that makes it a consequence of the second appears on page 337). But since I am using Newton’s laws just to illustrate the type of symmetry argument I am interested in, it is safe to assume that this understanding of the first law is correct.

[7] I am discussing this example as an aid to understanding the structure of the symmetry argument for the guidance equation. So do not think that I take this motivation for seeking a justification for the law of inertia seriously. It may be that we should reject the presupposition that there is a distinction to be made between forced motion and natural motion.

[8] Let’s not worry about how we could have grounds for this assumption in a context in which we do not know what the correct law of inertia is.

[9] Some equations using higher derivatives can be ruled out; the equation γ″ = γ″¢‴ is equivalent to γ″ = k + γ′ where k is some constant, and (as we will see) that equation is not Galilean invariant. I am not sure how far this line of thought extends. But for my purposes, it is not important.

[10] It is not clear to me that Dürr et. al. offer the symmetry argument as an argument for preferring Bohmian mechanics to its Bohm‐like rivals. But Wallace (2008, section 6.5) sees this as a use to which the argument could be put.

[11] And together with the operators that represent spatial translations, rotations, and time translations, they will form a projective representation of the Galilean group. But this further constraint will not matter for what follows.

[12] The fact that boosting by v is equivalent to boosting twice by v/2 can be used to establish that the transformations representing boosts must be unitary.

[13] As above, if S is a (measurable) region of space, then the probability that the particle will be found in S is , where χ_S is 1 on S and 0 elsewhere. The function χ_S is just a projection operator on our Hilbert space, so this integral is just the inner product .

[14] See, for example, Jordan 1969 (ch. 7) or Ballentine 1998 (ch. 3) for more details.

[15] Holland (1993, 122–124) appears to give this justification for (B).

[16] Maudlin (2007, 3169) claims that The Wavefunction is an objectively real field that does not transform like a scalar field under boosts. But his grounds for this are that it cannot transform that way if the guidance equation is to be Galilean invariant. Obviously, this is of no help in the present context.

[17] David Albert uses similar premises to argue that time reversal should not change the magnetic field in classical electromagnetism; see Albert 2000, ch. 1.

[18] The argument here that Bohmian mechanics is not Galilean invariant rests on an assumption about how the wavefunction is to be interpreted. One might try to construct an argument that Bohmian mechanics is not Galilean invariant that does not rest on any assumptions like that. The argument in Valentini 1997 that I mentioned above is an argument of this form. Since my primary interest here is in what the wavefunction represents in Bohmian mechanics, I will not pursue this kind of argument here.

[19] Allori et al. 2008 defend this interpretation of the wavefunction and then appeal to the guidance equation to justify the transformation properties of the wavefunction. Of course, once again, if one proceeds in this way then one cannot turn around and give the symmetry argument for the guidance equation.

[20] Insofar as the orthodox theory itself has a clear meaning, of course.