Abstract
Englert et al. (Zeitschrift für Naturforschung, 47a, 1175–1186, 1992) claim that, in certain circumstances, the Bohmian trajectory of a test particle does not match the reports of which-path detectors, concluding that the Bohmian trajectories are not real, but “surrealistic.” However, Hiley and Callaghan (Physica Scripta, 74, 336–348, 2006) argue that, if Bohm’s interpretation is correctly applied, no such mismatch is ever sanctioned. Unfortunately, the debate was never settled since nobody showed where the source of disagreement resided. In this paper, I reassess the debate over such “surrealistic” trajectories and I derive both a necessary and a sufficient condition for there to be a mismatch between the Bohmian trajectories and the reports of which-path detectors. I conclude that the mismatch is possible as a matter of principle, but can be ruled out in practice. I explore in depth the philosophical consequences of such mismatch arguing that it does not render realism about the Bohmian trajectories untenable. In addition, I show that the opposing conclusion of Hiley and Callaghan is due to the fact that they assume a set of trajectories that are incompatible with the postulates of Bohmian mechanics.
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Notes
If a system of particles with spin is considered, the right-hand side of (2) involves the appropriate scalar product among spinor wave functions. If, on the contrary, the wave function is just a scalar, the factor Ψ* both in the numerator and denominator of the right-hand side of this expression cancel out, leading to the usual, simpler formula: \( {\mathbf{v}}_k=\frac{\hslash }{m_k}\mathrm{Im}\left(\frac{\nabla_k\Psi}{\Psi}\right) \).
See Bohm (1952).
Here we have Bell on this matter: “[Bohmian] Particles are not attributed angular momenta, energies, etc., but only positions as functions of time. Peculiar ‘measurement’ results for angular momenta, energies, and so on, emerge as pointer positions in appropriate experimental setups.” (Bell 1990, 39)
The support of a function is the set of points of its domain in which the function has a non-zero value.
It is worth noting that underlying the bounce of the test particle there is a change in the wave packet that effectively guides the particle. If the test particle initially follows path 1 [2], its movement is determined by the wave packet ψ 1 [ψ 2] until it reaches the interference area; however, when it abandons the interference area, it is no longer carried by ψ 1 [ψ 2] but by ψ 2 [ψ 1].
Measuring the energy state of the cavity—that is, ascertaining whether there is a tell-tale photon inside the cavity or not—requires a rather complicated process. This process is typically known as the «reading process». It involves sending a second atom through the apparatus, the so-called reader atom, which must be suitably excited so that it will absorb the tell-tale photon, if one is present. A field that is only able to ionise the reader atom if it has absorbed the tell-tale photon is previously set up at the outset of the cavity. Finally, the ionising of the reader atom triggers an irreversible process of amplification.
As is obvious, in ESSW’s experiment, P is not a particle but a whole atom. In this case, the wave functions |ψ 1〉 and |ψ 2〉 are meant to represent the centre of mass of the atom.
According to many proposals in the literature (including that of ESSW), the test particle is not really a single particle, but an atom, with many internal degrees of freedom. Typically, the interaction with the which-path detector involves an energy exchange that results in some de-excitation of these internal degrees of freedom. If there is only one detector, to be fully rigorous, the internal degrees of freedom need to be taken into account and instead of (15), the complete wave function after the interaction is \( 1/\sqrt{2}\left({\psi}_1{\eta}_e\ {\varphi}_{\mathrm{No}}+{\psi}_2{\eta}_{ue}{\varphi}_{\mathrm{Yes}}\right) \), where η e represents the internal degrees of freedom in their initial excited state and η ue is the wave function corresponding to the internal degrees of freedom having lost some of the energy of excitation. Now, if we consider a symmetric setup with which-path detectors placed in both arms of the interferometer, after interaction with the apparatuses, the factor η ue appears in both branches of the wave function and can be factored out, thereby becoming irrelevant. For this reason, in what follows, I omit from my treatment any reference to the internal degrees of freedom of the test particle. For simplicity, I only consider the spatial part of the wave function, omitting reference to the spin.
Aharonov and Vaidman (1996) first discuss the surrealistic trajectories effect by means of a very abstract model. It is no worth discussing this model here which I think is not problematic. Later on, however, the authors maintain that the surrealistic trajectories effect can be exemplified if a delayed bubble chamber is used as a which-path detector (see Aharonov and Vaidman 1996, 153). The key idea is that the bubbles develop so slowly that no macroscopically discernible bubble has yet formed when the test particle is within the interference area. Hiley and Callaghan (2006, 342ff) rightly show that this latter device does not meet (SURR1). To see that this is the case, we need to realise that the operation of the chamber involves two distinct processes: in order for a bubble to develop, first, the test particle must collide with a molecule of the chamber and ionise it; next, in virtue of a complicated chemical process of amplification, a bubble will grow wherever the ionised molecule happens to be. This second process of bubble formation is the one that takes place very slowly in a delayed bubble chamber. However, the very first process of ionisation already involves a violation of (SURR1), regardless of the speed of bubble formation. Let M be the electron ejected from a molecule of the chamber because of the collision with the test particle. Before such collision occurs, M is located in the interior of the molecule; after the collision, however, the probability of finding M in the interior of the molecule is nearly zero. Therefore, the wave functions representing M both before and after the ionization process do not significantly overlap, violating (SURR1).
These authors include some numerical calculations of the Bohmian trajectories, as implied by their model, showing that some of them do not cross the plane of symmetry of the interferometer.
For a definition of the Hermite polynomials and a discussion of the relation between those polynomials and the solutions of the harmonic oscillator, see Cohen-Tannoudji et al. (1977, 530ff.).
This is also the case of the proposal in Brown et al. (1995).
For a realistic treatment of spin in Bohmian mechanics, see Dewdney et al. (1987).
Personal communication with Basil Hiley (October 28, 2008).
The degree of discomfort may depend on which interpretation of Bohmian mechanics are you endorsing. If you are a friend of the quantum potential approach—like Hiley is—and therefore you believe that energy is a real property of Bohmian particles, you will surely feel more discomfort than will a Bohmian minimalist who thinks that Bohmian particles only have positions.
See Popper (1968, p.98).
For more details about the functioning of the micromaser cavity, see the original ESSW paper and appropriate references therein.
I characterise Standard Quantum Mechanics as including (i) the eigenvalue-eigenstate link, according to which a particle has a given property if and only if its quantum state is an eigenstate of the corresponding operator, and (ii) the assumption that wave function collapse occurs upon measurement.
This conclusion reminds Wheeler’s well-known dictum regarding delayed choice experiments: “the past has no existence except as it is recorded in the present.” (Wheeler 1978, 194)
I will discuss in a while Hiley’s motivation to do so.
Hiley assumes, sometimes, that the additional axis represents the cavity’s energy of excitation and, at some other times, that it represents the internal energy of excitation of the test particle.
For instance, the dynamics of a Bohmian system can be represented in phase space instead of configuration space. Given that the momenta of the particles is p(q, t) = ∇S(q, t), where S is the phase of the wave function according to the decomposition (6), the probability density of the representative point of the system in phase space is f(q, p, t) = R2(q, t)δ[p − ∇S(q, t)].
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Acknowledgements
I would like to thank, first, Basil Hiley for terrific discussions even if we did not finally agree on the issue of the occurrence of the surrealistic trajectories. I am also greatly indebted to two anonymous referees that contributed to improve the manuscript with very pertinent remarks. Finally, I thank Carl Hoefer for useful comments on various stages of the manuscript. Research towards this paper was funded by the Spanish Ministry of Economy and Competitiveness through research projects FFI2011-29834-C03-03, CSD2009-00056 and FFI2012-37354.
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Solé, A. Surrealistic Bohmian trajectories appraised. Euro Jnl Phil Sci 7, 467–492 (2017). https://doi.org/10.1007/s13194-017-0170-8
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DOI: https://doi.org/10.1007/s13194-017-0170-8