Jacob Barandes

The original version of the de Broglie-Bohm pilot-wave theory, also called Bohmian mechanics, attempted to treat the wave function or pilot wave as a part of the physical ontology of nature. More recent versions of the de Broglie-Bohm theory appearing in the last few decades have tried to regard the pilot wave instead as an aspect of the theory's nomology, or dynamical laws. This paper argues that neither of these views is correct, and that the de Broglie-Bohm pilot wave is best understood as a collection of latent variables in the sense of a hidden Markov model, a construct that was not available when de Broglie and Bohm originally formulated what became their pilot-wave theory. This paper also discusses several other challenges for the ontological view of the pilot wave. One such challenge is due to Foldy-Wouthuysen gauge transformations, which connect up with the Deotto-Ghirardi ambiguity in the de Broglie-Bohm theory. Another challenge arises from the freedom to carry out canonical transformations in the wave function's own notion of phase space, as defined by Strocchi and Heslot.

This paper provides a detailed historical account of early debates over wave-function realism, the modern term for the view that the wave function of quantum theory is physically real. As this paper will show, the idea of physical waves associated with particles had its roots in work by Einstein and de Broglie, who both originally thought of these waves as propagating in three-dimensional physical space. De Broglie quickly turned this wave-particle duality into an early pilot-wave theory, on which a particle's associated “phase wave” piloted or guided the particle along its trajectory. Schrödinger built on de Broglie's phase-wave hypothesis to provide a comprehensive account of the nascent quantum theory. However, Schrödinger's new “undulatory mechanics” came at the cost of replacing de Broglie's phase waves propagating in physical space with a wave function propagating in a system's abstract configuration space. The present work will argue that this move from three-dimensional physical space to a many-dimensional configuration space was a key reason why the founders of quantum theory uniformly abandoned the physical reality of the wave function. This paper will further clarify that de Broglie introduced two distinct pilot-wave theories, and will then argue that it was Bohm's rediscovery of the second of these two pilot-wave theories over two decades later, as well as Bohm's vociferous defense of wave-function realism, that were responsible for resurrecting the idea of an ontological wave function. This idea ended up playing a central role in Everett's development of the many-worlds interpretation.

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of state vectors. Weak values first appeared in the research literature in a pair of papers in 1987 and 1988, and were originally defined as the results of a special kind of experimental protocol involving non-disturbing measurements combined with an explicit form of post-selection. In the years since, subsequent papers on weak values have produced a number of important practical spin-offs, including new methods for signal amplification and quantum-state tomography. The present work is not concerned with those practical spin-offs, but with historical and ongoing attempts to assign weak values a transparent, single-system interpretation, as well as efforts that invoke weak values to make a number of exotic claims about the properties and behavior of individual quantum systems. This paper challenges these interpretational claims by arguing that they involve several forms of fallacious reasoning.

The original version of the de Broglie-Bohm pilot-wave theory, also called Bohmian mechanics, attempted to treat the wave function or pilot wave as a part of the physical ontology of nature. More recent versions of the de Broglie-Bohm theory appearing in the last few decades have tried to regard the pilot wave instead as an aspect of the theory's nomology, or dynamical laws. This paper argues that neither of these views is correct, and that the de Broglie-Bohm pilot wave is best understood as a collection of latent variables in the sense of a hidden Markov model, a construct that was not available when de Broglie and Bohm originally formulated what became their pilot-wave theory. This paper also discusses several other challenges for the ontological view of the pilot wave. One such challenge is due to Foldy-Wouthuysen gauge transformations, which connect up with the Deotto-Ghirardi ambiguity in the de Broglie-Bohm theory. Another challenge arises from the freedom to carry out canonical transformations in the wave function's own notion of phase space, as defined by Strocchi and Heslot.

This paper provides a detailed historical account of early debates over wave-function realism, the modern term for the view that the wave function of quantum theory is physically real. As this paper will show, the idea of physical waves associated with particles had its roots in work by Einstein and de Broglie, who both originally thought of these waves as propagating in three-dimensional physical space. De Broglie quickly turned this wave-particle duality into an early pilot-wave theory, on which a particle's associated “phase wave” piloted or guided the particle along its trajectory. Schrödinger built on de Broglie's phase-wave hypothesis to provide a comprehensive account of the nascent quantum theory. However, Schrödinger's new “undulatory mechanics” came at the cost of replacing de Broglie's phase waves propagating in physical space with a wave function propagating in a system's abstract configuration space. The present work will argue that this move from three-dimensional physical space to a many-dimensional configuration space was a key reason why the founders of quantum theory uniformly abandoned the physical reality of the wave function. This paper will further clarify that de Broglie introduced two distinct pilot-wave theories, and will then argue that it was Bohm's rediscovery of the second of these two pilot-wave theories over two decades later, as well as Bohm's vociferous defense of wave-function realism, that were responsible for resurrecting the idea of an ontological wave function. This idea ended up playing a central role in Everett's development of the many-worlds interpretation.

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of state vectors. Weak values first appeared in the research literature in a pair of papers in 1987 and 1988, and were originally defined as the results of a special kind of experimental protocol involving non-disturbing measurements combined with an explicit form of post-selection. In the years since, subsequent papers on weak values have produced a number of important practical spin-offs, including new methods for signal amplification and quantum-state tomography. The present work is not concerned with those practical spin-offs, but with historical and ongoing attempts to assign weak values a transparent, single-system interpretation, as well as efforts that invoke weak values to make a number of exotic claims about the properties and behavior of individual quantum systems. This paper challenges these interpretational claims by arguing that they involve several forms of fallacious reasoning.

In 1964, Aharonov, Bergmann, and Lebowitz introduced their well-known ‘ABL rule’ with the intention of providing a time-symmetric formalism for computing novel kinds of conditional probabilities in quantum theory. Later papers attached additional significance to the ABL rule, including assertions that it supported violations of the uncertainty principle. The present work challenges these claims, as well as subsequent attempts to salvage the original interpretation of the ABL rule. Taking a broader view, this paper identifies a subtle category error at the heart of the ABL rule that consists of confusing observables that belong to a single system with emergent observables that arise only for physical ensembles. Along the way, this paper points out other problems and fallacious reasoning in the research literature surrounding the ABL rule, including the misuse of post-selection, a reliance on pattern matching to classical formulas, and a posture of ‘measurementism’ that takes experimental data as providing answers to interpretational questions.

Why does quantum theory need the complex numbers? With a view toward answering this question, this paper argues that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. This paper then describes the ‘indivisible interpretation’ of quantum theory, according to which a quantum system can be regarded as an ‘indivisible’ stochastic process unfolding in an old-fashioned configuration space, with wave functions and other exotic Hilbert-space ingredients demoted from having an ontological status. The complex numbers end up being necessary to ensure that the Hilbert-space formalism is indeed a Markovian embedding.

Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of “classical” wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.

According to the stochastic-quantum correspondence, a quantum system can be understood as a stochastic process unfolding in an old-fashioned configuration space based on ordinary notions of probability and ‘indivisible’ stochastic laws, which are a non-Markovian generalization of the laws that describe a textbook stochastic process. The Hilbert spaces of quantum theory and their ingredients, including wave functions, can then be relegated to secondary roles as convenient mathematical appurtenances. In addition to providing an arguably more transparent way to understand and modify quantum theory, this indivisible-stochastic formulation may lead to new possible applications of the theory. This paper initiates a deeper investigation into the conceptual foundations and structure of the stochastic-quantum correspondence, with a particular focus on novel forms of gauge invariance, dynamical symmetries, and Hilbert-space dilations.

It is difficult to extract reliable criteria for causal locality from the limited ingredients found in textbook quantum theory. In the end, Bell humbly warned that his eponymous theorem was based on criteria that “should be viewed with the utmost suspicion.” Remarkably, by stepping outside the wave-function paradigm, one can reformulate quantum theory in terms of old-fashioned configuration spaces together with ‘unistochastic’ laws. These unistochastic laws take the form of directed conditional probabilities, which turn out to provide a hospitable foundation for encoding microphysical causal relationships. This unistochastic reformulation provides quantum theory with a simpler and more transparent axiomatic foundation, plausibly resolves the measurement problem, and deflates various exotic claims about superposition, interference, and entanglement. Making use of this reformulation, this paper introduces a new principle of causal locality that is intended to improve on Bell's criteria, and shows directly that systems that remain at spacelike separation cannot exert causal influences on each other, according to that new principle. These results therefore lead to a general hidden-variables interpretation of quantum theory that is arguably compatible with causal locality.

This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called generalized stochastic systems, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any generalized stochastic system and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasiprobability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any generalized stochastic system on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.

This paper argues that every quantum system can be understood as a sufficiently general kind of stochastic process unfolding in an old-fashioned configuration space according to ordinary notions of probability. This argument is based on an exact correspondence between the class of “indivisible” stochastic processes and quantum theory. This new stochastic-quantum correspondence demotes the wave function from a primary ontological ingredient to a secondary mathematical tool and yields a deflationary account of exotic quantum phenomena, such as interference, decoherence, entanglement, noncommutative observables, and wave-function collapse. At a more practical level, the stochastic-quantum correspondence leads to a novel reconstruction of quantum theory, alongside the Hilbert-space, path-integral, and quasiprobability representations, and also provides a framework for using Hilbert-space methods to formulate highly generic, non-Markovian types of stochastic dynamics, with potential applications throughout the sciences.

We address a long-standing debate over whether classical magnetic forces can do work, ultimately answering the question in the affirmative. In detail, we couple a classical particle with intrinsic spin and elementary dipole moments to the electromagnetic field, derive the appropriate generalization of the Lorentz force law, show that the particle's dipole moments must be collinear with its spin axis, and argue that the magnetic field does mechanical work on the particle's elementary magnetic dipole moment. As consistency checks, we calculate the overall system's energy-momentum and angular momentum, and show that their local conservation equations lead to the same force law and therefore the same conclusions about magnetic forces and work. We also compute the system's Belinfante-Rosenfeld energy-momentum tensor.

Any realist interpretation of quantum theory must grapple with the measurement problem and the status of state-vector collapse. In a no-collapse approach, measurement is typically modeled as a dynamical process involving decoherence. We describe how the minimal modal interpretation closes a gap in this dynamical description, leading to a complete and consistent resolution to the measurement problem and an effective form of state collapse. Our interpretation also provides insight into the indivisible nature of measurement—the fact that you can't stop a measurement part-way through and uncover the underlying 'ontic' dynamics of the system in question. Having discussed the hidden dynamics of a system's ontic state during measurement, we turn to more general forms of open-system dynamics and explore the extent to which the details of the underlying ontic behavior of a system can be described. We construct a space of ontic trajectories and describe obstructions to defining a probability measure on this space.

Wigner's quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the compactness properties of the resulting phase spaces at fixed energy, and show that in order for a classical massless particle to be physically sensible, its phase space must feature a classical-particle counterpart of electromagnetic gauge invariance. By examining the connection between massless and massive particles in the massless limit, we also derive a classical-particle version of the Higgs mechanism.

Standard lore holds that magnetic forces are incapable of doing mechanical work. More precisely, the claim is that whenever it appears that a magnetic force is doing work, the work is actually being done by another force, with the magnetic force serving only as an indirect mediator. However, the most familiar instances of magnetic forces acting in everyday life, such as when bar magnets lift other bar magnets, appear to present manifest evidence of magnetic forces doing work. These sorts of counterexamples are often dismissed as arising from quantum effects that lie outside the classical regime. In this paper, we show that quantum theory is not needed to account for these phenomena, and that classical electromagnetism admits a model of elementary magnetic dipoles on which magnetic forces can indeed do work. In order to develop this model, we revisit the foundational principles of the classical theory of electromagnetism, showcase the importance of constraints from relativity, examine the structure of the multipole expansion, and study the connection between the Lorentz force law and conservation of energy and momentum.

In this paper, we review a general technique for converting the standard Lagrangian description of a classical system into a formulation that puts time on an equal footing with the system's degrees of freedom. We show how the resulting framework anticipates key features of special relativity, including the signature of the Minkowski metric tensor and the special role played by theories that are invariant under a generalized notion of Lorentz transformations. We then use this technique to revisit a classification of classical particle-types that mirrors Wigner's classification of quantum particle-types in terms of irreducible representations of the Poincaré group, including the cases of massive particles, massless particles, and tachyons. Along the way, we see gauge invariance naturally emerge in the context of classical massless particles with nonzero spin, as well as study the massless limit of a massive particle and derive a classical-particle version of the Higgs mechanism.

We summarize a new realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory permits assuming that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the more realistic case of open quantum systems despite the generic development of entanglement. Our interpretation—which we claim is ultimately compatible with Lorentz invariance—reformulates wave-function collapse in terms of an underlying interpolating dynamics, makes it possible to derive the Born rule from deeper principles, and resolves several open questions regarding ontological stability and dynamics.

We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes but leaves the theory’s basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory permits assuming that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the more realistic case of open quantum systems despite the generic development of entanglement. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure—which we call subsystem space—that we believe merits further study. We conclude with a summary, a list of criteria for future work on quantum foundations, and further research directions. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.