Abstract
Since Bohmian mechanics is explicitly nonlocal, it is widely believed that it is very hard, if not impossible, to make Bohmian mechanics compatible with relativistic quantum field theory (QFT). I explain, in simple terms, that it is not hard at all to construct a Bohmian theory that lacks Lorentz covariance, but makes the same measurable predictions as relativistic QFT. All one has to do is to construct a Bohmian theory that makes the same measurable predictions as QFT in one Lorentz frame, because then standard relativistic QFT itself guarantees that those predictions are Lorentz invariant. I first explain this in general terms, then I describe a simple Bohmian model that makes the same measurable predictions as the Standard Model of elementary particles, after which I give some hints towards a more fundamental theory beyond standard model. Finally, I present a short story telling how my views of fundamental physics in general, and of Bohmian mechanics in particular, evolved over time.
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Probability is a theoretical tool for making predictions or explanations of the actual occurrences (e.g. the actual number of particles within a given region in 6-dimensional phase space) when some relevant information about the actual system is unknown. As such, probability is a non-ontic entity in both the Bayesian and the frequentist interpretation, while the actual occurrences themselves are ontic. The theoretical frequencies of occurrences computed by probability theory depend on our partial ignorance about the actual system, while the actual frequencies do not depend on our ignorance. Failing to distinguish the former frequencies from the latter frequencies is like failing to distinguish the map from the territory. For a large N, however, owing to the law of large numbers, the two frequencies typically differ only by a negligible amount, which resolves various conceptual confusions in statistical physics. For example, this explains why the “subjective” Gibbs entropy, defined in terms of probabilities depending on our ignorance, in practice often can be treated as being the same as “objective” Gibbs entropy defined by actual frequencies.
In the standard interpretation of QM one often starts from the assumption that \(\psi\) provides a complete description of the system. But \(\psi\) itself is either ontic or non-ontic, and both options are conceptually problematic in a \(\psi\)-complete interpretation. The first option, that \(\psi\) is ontic, is hard to reconcile with the view that \(\psi\) is just a probability amplitude that exhibits an apparent collapse upon measurement. From this point of view, the second option of non-ontic \(\psi\), in which case the “collapse” is nothing but an update of information, seems much more sober. But if \(\psi\) is non-ontic then something else should be ontic, for otherwise one would need to abandon the view that Nature really exists out there, even when we don’t observe it. And yet, if \(\psi\) is both complete and non-ontic, then it is hard to understand what exactly is ontic. This is why the standard interpretation cannot state clearly whether \(\psi\) is ontic or not. A way out of this conundrum is to adopt a non-standard interpretation in which \(\psi\) is not complete, which indeed is the main idea behind the Bohmian interpretation.
Philosophers of physics who find standard QFT textbooks technically formidable may learn a lot of QFT from [12].
QFT interactions are often studied in the interaction picture [13] based on the operator U(t) of unitary evolution, which satisfies the Lorentz non-covariant equation \(H_\mathrm{int}U=i\partial _tU\) with \(H_\mathrm{int}\) being the interaction Hamiltonian. Nevertheless, after a Dyson expansion for U(t), one eventually gets Lorentz invariant Feynman rules for computing matrix elements of the scattering matrix.
For example, if Alice observes that her measuring apparatus shows that the energy of the particle is \(E=7\) MeV, then Bob, who moves with respect to Alice with a relativistic velocity v, will also observe that her apparatus shows \(E=7\) MeV. The result that the Alice’s apparatus shows \(E=7\) MeV can be obtained from calculations in any frame, this is why the result of measurement is Lorentz invariant (and not just covariant). This invariant energy E can be written as \(E=-\eta _{\mu \nu }P^{\mu }U^{\nu }_\mathrm{appar}\), where \(P^{\mu }\) is the covariant 4-momentum of the particle, \(U^{\mu }_\mathrm{appar}\) is the covariant 4-velocity of the measuring apparatus and \(\eta _{\mu \nu }\) is the Minkowski metric with signature \((-+++)\).
More precisely, by “fixed Lorentz frame” in this context one means a fixed slicing of spacetime into space and time. The theory is still covariant under purely spatial rotations and, for that matter, under arbitrary time-independent changes of spatial coordinates.
For some initial conditions \(\mathbf{A}(\mathbf{x},0)\) it may travel through one slit only, but for most initial conditions it travels through both slits.
In BM with point-particle ontology, the two-slit experiment is often presented as an experiment where the Bohmian interpretation is particularly simple. However, this simplicity is misleading and leads to frequent misunderstandings of BM, sometimes even among Bohmian experts. With point-particle ontology it is very simple, indeed, to understand why does particle arrive at only one position at the detection screen. However, this position of a single particle is not what we really see in the experiment. What we really see is a position of a macroscopic pointer associated with the measuring apparatus. The position of the pointer is strongly correlated with the position of the particle, but the explanation of that correlation is not trivial. The explanation requires a use of the theory of quantum measurements, which involves many particles. Hence the explanation of the two-slit experiment in terms of a single particle arriving at a single position at the screen is an over-simplification that misses one of the essential ingredients of BM, namely the theory of quantum measurements. For a formulation of BM that attributes to macroscopic objects (including pointers of measuring apparatuses) a central role see [7].
For philosophers of physics not familiar with the idea of effective field theory I recommend [25].
Of course, since phonons can be derived from the standard model, while the standard model, in this framework, can be derived from a more fundamental theory, one can say that the phonon theory is an effective theory of an effective theory, while the standard model is still more fundamental than the phonon theory. But from the Wilsonian point of view [26,27,28,29,30,31,32], different effective theories, such as the phonon theory and the standard model, are just different coarse grainings of the unknown fundamental theory.
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The author is grateful to T. Jurić for useful comments on the manuscript. This work was supported by the Ministry of Science of the Republic of Croatia.
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Nikolić, H. Relativistic QFT from a Bohmian Perspective: A Proof of Concept. Found Phys 52, 80 (2022). https://doi.org/10.1007/s10701-022-00600-x
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DOI: https://doi.org/10.1007/s10701-022-00600-x