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The non-representational character of explanation at the heart of modern science

In today’s blog we talk to Dr. Jeffrey Bub, PhD in mathematical physics at the University of London, and a Distinguished University Professor in the Department of Philosophy, the Institute for Physical Science and Technology, and the Center for Quantum Information and Computer Science, at the University of Maryland at College Park.

David: Hello Jeffrey. At a colloquium I attended some 18 years ago, you raised the possibility that Boolean logic may not apply to quantum systems. I found this unpalatable at the time, and despite some effort to come to terms with this over the years, still place it somewhere between sophistry and circular reasoning, today. Is there an argument that might help me see the light? How do you view the idea of accommodating non-Boolean logic with epistemology?

Jeffrey: First of all, the claim is not that we need to change our logic in order to make sense of quantum mechanics. But I need to go into some background history to answer your question fully.

Fundamentally, quantum mechanics replaces the commutative algebra of physical quantities of a classical system with a noncommutative algebra of “observables." This is an extraordinary move, quite unprecedented in the history of physics, and arguably requires us to re-think what counts as an acceptable explanation in physics. To understand what noncommutativity involves, it’s helpful to think of two-valued observables. These represent properties of a quantum system (for example, the property that the energy of the system lies in a certain range of values, with the two values of the observable representing “yes" or “no"), or propositions (the proposition asserting that the value of the energy lies in a certain range, with the two values representing “true" or “false"). The two-valued observables of a classical system form a Boolean algebra.

We all, in a loose sense, understand the concept of a Boolean algebra, even if the term is unfamiliar. It’s simply a formalization of the way in which we commonly think of properties or propositions fitting together when we combine them with connectives “and,” “or,” “not," so that it’s possible to imagine a classical or common-sense "state of reality" in which every relevant proposition is assigned a truth value, either “true" or “false,” consistently with the connectives. (For example, if p is true and q is true then “p and q” is true, and so on.) George Boole characterized this algebraic structure he identified in 1847 as capturing "the conditions of possible experience.” Every Boolean algebra is isomorphic to a set of subsets of a set, so you can think of a Boolean algebra, if you like, as just an algebra of sets under the operations of intersection, union, and complement, which correspond to “and, “or,” and “not,” in that order.

To say that the algebra of observables of a quantum system is noncommutative is formally equivalent to saying that the sub-algebra of two-valued observables representing properties or propositions is non-Boolean. Hilbert space formalizes this non-Booleanity in a particular way. The Boolean algebra of classical mechanics is replaced by a collection of Boolean algebras, one for each set of commuting two-valued observables. The interconnections of commuting and noncommuting observables preclude the possibility of embedding the whole collection into one inclusive Boolean algebra, so you can’t assign truth-values consistently to the propositions about observable values in all these Boolean algebras. Putting it differently, there are Boolean algebras in the collection of Boolean algebras of a quantum system — for example, the Boolean algebras for position and momentum, or for spin components in different directions — that don’t fit together into a single Boolean algebra, unlike the corresponding collection for a classical system.

Bohr did not refer to Boolean algebras, but the concept is simply a precise way of codifying a significant aspect of what Bohr meant by his constant insistence that "the account of all evidence must be expressed in classical terms," that’s to say, "unambiguous language with suitable application of the terminology of classical physics," for the simple reason, as he put it, that we need to be able "to tell others what we have done and what we have learned." Formally speaking, the significance of “classical" here as being able "to tell others what we have done and what we have learned” is that the events in question conform to Boole’s "conditions of possible experience" and fit together as a Boolean algebra.

In the non-Boolean theory of quantum mechanics, probabilities arise via the Born rule as a feature of the geometry of Hilbert space, related to the angle between rays in Hilbert space representing “pure" quantum states. As von Neumann put it, the quantum probabilities are "sui generis” and "uniquely given from the start" as a feature of the geometry of Hilbert space — quite different from the way probabilities are introduced in a classical theory, as measures over classical states or points in a phase space. These quantum probabilities can’t be understood as quantifying ignorance about the pre-measurement value of an observable, as in a Boolean theory. From this perspective, quantum mechanics is a new sort of non-representational theory for an irreducibly indeterministic universe, with a new type of nonlocal probabilistic correlation for “entangled" quantum states of separated systems, where the correlated events are intrinsically random, not merely apparently random like coin tosses.

On this view, quantum mechanics does not provide a representational explanation of events as a classical theory does, loosely a picture of what’s going on. Noncommutativity or non-Booleanity makes quantum mechanics quite unlike any theory we have dealt with before, and there is no reason, apart from tradition, to assume that a noncommutative or non-Boolean theory should provide the sort of explanation we are familiar with in a theory that is commutative or Boolean at the fundamental level. A representational theory proposes a primitive ontology, perhaps of particles or fields of a certain sort, and a dynamics that describes how things change over time. The non-Boolean physics of quantum mechanics does not provide a representational explanation of phenomena. Rather, a quantum mechanical explanation involves showing how a Boolean output (a measurement outcome) is obtained from a Boolean input (a state preparation) via a non-Boolean link. In this sense, a quantum mechanical explanation is "operational” rather than “representational,” but this is not simply a matter of convenience or philosophical persuasion. We adopt quantum mechanics — the theoretical formalism of the non-Boolean link between Boolean input and output — for empirical reasons, the failure of classical physics to explain certain phenomena, and because there is no satisfactory representational theory of such phenomena.

This is a very different way of thinking about quantum mechanics than the familiar textbook formulation of the theory as a wave theory suggests. Heisenberg developed the idea of a noncommutative quantum mechanics in collaboration with Born and Jordan in 1925. As Heisenberg put it in a letter to Pauli in 1925: "All of my meagre efforts go toward killing off and suitably replacing the concept of the orbital paths that one cannot observe." The orbital paths Heisenberg was referring to were the "stationary states" of electrons revolving around a central nucleus in Bohr’s theory of the atom. Bohr associated electrons jumping between these fixed orbits with photons emitted or absorbed by an atom, with discrete frequencies corresponding to the observed lines or gaps in atomic spectra. Heisenberg thought Bohr’s orbits, which conflicted with classical electrodynamics as well as classical mechanics, were unphysical. His aim was to get rid of the orbits by replacing classical mechanics with “a theoretical quantum mechanics … in which only relations between observable quantities occur.” He accomplished this by replacing quantities associated with a single orbit with arrays of quantities associated with transitions between orbits, representing the observable features of atomic spectra instead of the unobservable orbits. Multiplication of these transition arrays turned out to be noncommutative: AB ≠ BA. It subsequently became apparent, with Heisenberg’s interpretation of his commutation relations in 1927 and the Dirac-Jordan transformation theory, that systems in this noncommutative mechanics can’t have definite values for all physical quantities simultaneously. In particular, an electron can’t have definite position and momentum values and so can’t have a well-defined orbit in an atom.

Schrödinger published a wave-mechanical version of the theory in 1926 that kept the orbits and explained their quantization as a wave phenomenon, and he subsequently proved the empirical equivalence of wave mechanics and Heisenberg’s matrix mechanics for experiments relevant at the time. The general theoretical question of equivalence was first addressed by Jordan and Dirac in their transformation theory and definitively settled by John von Neumann in a series of papers in 1927, in which he reformulated quantum mechanics as a theory of “observables” represented by operators and states represented by rays in Hilbert space. Physicists, not surprisingly, found wave mechanics more familiar and intuitively appealing than Heisenberg’s formulation of quantum mechanics. The wave theory seemed to provide an explanation of observed phenomena in terms of causal processes evolving continuously in space and time, as opposed to Heisenberg’s derivation of irreducible transition probabilities from features of a noncommutative algebraic structure. As Schrödinger initially saw it, the wave theory had the advantage of “Anschaulichkeit," usually translated as “clarity" or “visualizability," here in the specific sense of representability as a causal picture of events unfolding continuously in space and time.

Heisenberg disagreed. As he put it in a letter to Pauli in 1926: "The more I think about the physical portion of Schrödinger’s theory, the more repulsive [abscheulich] I find it. . . . What Schrödinger writes about the Anschaulichkeit of his theory 'is probably not quite right,' in other words it’s crap [Mist].” Schrödinger expressed similar views about Heisenberg’s quantum mechanics. The dispute about “Anschaulichkeit" is the root of the debate about the “completeneness’ of quantum mechanics, which morphed into a debate about “realism” (really, “representationalism”) between an orthodox camp, represented primarily by the diverse but related views of Bohr, Heisenberg, Pauli, and Rosenfeld, dubbed the Copenhagen interpretation by Heisenberg, and the dissidents, represented by Schrödinger with Einstein as a powerful ally, and later by Bohm, Everett, Bell, and others. As I see it, Schrödinger’s wave formulation of quantum mechanics, while mathematically equivalent to the Heisenberg-Born-Jordan formulation, is misleading in terms of the conceptual picture it presents, and the source of the conceptually puzzling features of the theory.

David: Does the concept of information give us a roadmap into understanding quantum physics?

Jeffrey: Yes, I think it does. On the view of quantum mechanics I outlined, quantum mechanics is about probability. That is, the theory is essentially a new framework for representing probabilities, including new sorts of probabilistic correlations that arise for entangled states and have no causal explanation, as Bell showed — correlations that are not possible (without assuming instantaneous action at a distance) in a Boolean or set-theoretical theory, where probabilities are represented as measures over classical states (essentially, points in a set, phase space). In quantum mechanics, probabilities are, as von Neumann put it, "sui generis" and "uniquely given from the start’” as a feature of the geometry of Hilbert space. The central interpretative question for quantum mechanics as a non-Boolean theory is how we should understand these "sui generis" probabilities, since they can't be interpreted as measures of ignorance about quantum properties associated with the actual values of observables prior to measurement. As I see it, information theory is a branch of the mathematical theory of probability, an application of probability theory to a certain domain of phenomena. So understanding quantum information as a new type of information, not encompassed by Shannon’s Boolean theory, is a consequence of understanding Hilbert space as the kinematic framework for the physics of an indeterministic universe, just as Minkowski space provides the kinematic framework for the physics of a non-Newtonian, relativistic universe. In special relativity, the geometry of Minkowski space imposes spatio-temporal constraints on events to which the relativistic dynamics is required to conform. In quantum mechanics, the non-Boolean projective geometry of Hilbert space imposes objective kinematic (i.e., pre-dynamic) probabilistic constraints on correlations between events to which a quantum dynamics of matter and fields is required to conform. In this non-Boolean theory, new sorts of nonlocal probabilistic correlations are possible for “entangled" quantum states of separated systems, where the correlated events are intrinsically random, not merely apparently random like coin tosses. In special relativity, Lorentz contraction is a kinematic effect of the spatio-temporal constraints on events imposed by the geometry of Minkowski space. In quantum mechanics, the loss of information in a quantum measurement — Bohr’s "irreducible and uncontrollable disturbance” — is a kinematic (i.e., pre-dynamic) effect of any process of gaining information of the relevant sort, irrespective of the dynamical processes involved in the measurement process, given the objective probabilistic constraints on correlations between events imposed by the geometry of Hilbert space. Today, the significant results in the foundations of quantum mechanics are all associated with research in quantum information.

David: Recent work suggests that degrees of entanglement may explain space fundamentally. From the perspective of someone that has explored entanglement for decades, what do you make of this project? Jeffrey: I think this is a fascinating avenue to explore, especially with the aim of understanding something new about quantum gravity. My colleague, Brian Swingle, at the University of Maryland works on this, but I’m not familiar enough with this research to say anything insightful. David: You worked with David Bohm who interacted with Einstein. I’m not an historian, but my intuition suggests that Einstein supported Bohm’s ideas mostly because they provided ammunition against the Copenhagen cult. Do you know the extent to which Einstein was sympathetic towards pilot wave theory? Jeffrey: There is a very good recent biography of Bohm by Olival Freire in which he writes about Bohm’s relation with Einstein in some detail. You’re right that Einstein regarded Bohm as an ally in his rejection of the Copenhagen interpretation. But he didn’t like Bohm’s hidden variable theory, perhaps because it is nonlocal in the sense, as Bell put it in his review article on hidden variable disproofs, that in Bohm’s theory "an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece." As Einstein wrote in a letter to Max Born in 1952: “Have you noticed that Bohm believes (as de Broglie did, by the way, 25 years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me.” I don’t think that Einstein was bothered by the determinism, per se, as Pauli pointed out in a letter to Born, but by the necessity to introduce instantaneous action at a distance to get a deterministic (Boolean) theory to produce the quantum statistics.

David: Thank you Professor!

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