Quantum Foundations

Researchers in quantum foundations claim (D'Ariano, Fuchs, ...): Quantum = probability theory + x and hence: x = Quantum - probability theory Guided by the metaphorical analogy: probability theory / x = flesh / bones we introduce a notion of quantum measurement within x, which, when flesing it with Hilbert spaces, provides orthodox quantum mechanical probability calculus.

We know the mathematical laws of quantum mechanics, but as yet we are not so sure why those laws should be inevitable. In the simpler but related environment of classical inference, we also know the laws (of probability). With better understanding of quantum mechanics as the eventual goal, Kevin Knuth and I have been probing the foundations of inference. The world we wish to infer is a partially-ordered set ('poset') of states, which may as often supposed be exclusive, but need not be (e.g. A might be a requirement for B). In inference, a state of mind about the world degrades from perfect knowledge through logical OR, which allows for uncertain alternatives. We don't need AND, and we don't need NOT; we just need OR. This display of acceptable states of mind is [close to] a mathematical 'lattice'. We find that the OR structure by itself (!) forces the ordinary rules of probability calculus. No other rules are compatible with the structure of a lattice, so the ordinary rules are inevitable. The standard Shannon information/entropy is likewise inevitable. Taking this idea further, the OR of states of mind gives a lattice of 'Questions' that might be useful for automated learning. Disconcertingly, this lattice is very much larger (in class aleph-2), and the natural valuations on it exhibit large range. I will present this extension, and ask whether we can rationally foresee its use in practical application.

Eugene Wigner and Hermann Weyl led the way in applying the theory of group representations to the newly formulated theory of quantum mechanics starting in 1927. My talk will focus, first, on two aspects of this early work. Physicists had long exploited symmetries as a way of simplifying problems within classical physics. Wigner recognized that the theory of group representations would similarly have enormous payoff in quantum mechanics, allowing him to solve problems in atomic spectroscopy ``almost without calculation.'' Here I will describe the novel aspects of symmetry in QM that Wigner clarified in the series of papers leading up to his 1931 textbook (Wigner's theorem, projective representations, etc.). The second aspect is less well-known: Weyl (1927) argued that group theory could also be used to address foundational questions in quantum mechanics, leading to a reformulation of the classical commutation relations and a proposal for quantization. Weyl's program had much less immediate impact, although it led to the Stone-von Neumann theorem and to Mackey's imprimitivity theorem. As a final historical point, I argue that in this early work the theory of group representations was optional (as emphasized by Slater and others) in a sense that it was not in particle physics in the 60s. The closing section of the talk turns to philosophical morals that have been drawn from this historical episode, in particular claims regarding ontic structural realism (French, Ladyman) and the group-theoretic constitution of objects (Castellani).

Quantum foundations in the light of gauge theories We will present the conjecture according to which the fact that q and p cannot be both ``observables'' of the same quantum system indicates that there is a remnant universal symmetry acting on classical states. In order to unpack this claim we will generalize to unconstrained systems the gauge correspondence between properties defined by first-class constraints and gauge symmetries generated by these constraints. As we shall see, this means that the uncertainty principle might be encoded in the very definition of the canonical variables q and p. According to the ontology of quantum objects that stems from this analysis, the quantum-mechanical description of physical objects is complete.

Quantum mechanics is a non-classical probability theory, but hardly the most general one imaginable: any compact convex set can serve as the state space for an abstract probabilistic model (classical models corresponding to simplices). From this altitude, one sees that many phenomena commonly regarded as ``characteristically quantum' are in fact generically ``non-classical'. In this talk, I'll show that almost any non-classical probabilistic theory shares with quantum mechanics a notion of entanglement and, with this, a version of the so-called measurement problem. I'll then discuss what's required for an abstract probabilistic theory to admit a somewhat simplified version of Everett's response to this problem -- an exercise that turns out to be instructive in several ways.

A standard canonical quantization of general relativity yields a time-independent Schroedinger equation whose solutions are static wavefunctions on configuration space. Naively this is in contradiction with the real world where things do change. Broadly speaking, the problem how to reconcile a theory which contains no concept of time with a changing world is called 'the problem of time'. In this seminar we shall study this problem using a reformulation of Newtonian mechanics due to Jacobi (Jacobi's timeless mechanics) which allows one to study the problem of time without all the technical difficulties present in quantized general relativity. We show explicitly that Jacobi's timeless mechanics is a straightforward counterexample to the claim that all first class constraints generate gauge transformations, i.e. physically indistinguishable states. The implications of this is unclear. By making use of deBroglie-Bohm trajectories we derive a necessary and sufficient condition for a time-dependent Schroedinger equation to emerge for subsystems. The importance of 'strong' entanglement between subsystem and environment for the emergence of time is stressed.

The essential ingredients of a quantum theory are usually a Hilbert space of states and an algebra of operators encoding observables. The mathematical operations available with these structures translate fairly well into physical operations (preparation, measurement etc.) in a non-relativistic world. This correspondence weakens in quantum field theory, where the direct operational meaning of the observable algebra structure (encoded usually through commutators) is lost. The situation becomes even worse when we want to give a more dynamical role to spacetime as for example in attempts to formulate a quantum theory of gravity. I argue that a revision of the structures that we think of as fundamental in a quantum theory is in order. I go on to outline a proposal in this direction, based on the so called 'general boundary formulation', emphasizing the operational meaning of the ingredients. If time permits I will also comment on the relation to the framework of algebraic quantum field theory.

Both classical probability theory and quantum theory lend themselves to a Bayesian interpretation where probabilities represent degrees of belief, and where the various rules for combining and updating probabilities are but algorithms for plausible reasoning in the face of uncertainty. I elucidate the differences and commonalities of these two theories, and argue that they are in fact the only two algorithms to satisfy certain basic consistency requirements. In order to arrive at this result I develop an over-arching framework for plausible reasoning that incorporates both classical probability and quantum theory as special cases.

Lee Smolin has argued that one of the barriers to understanding time in a quantum world is our tendency to spatialize time. The question is whether there is anything in physics that could lead us to mathematically characterize time so that it is not just another funny spatial dimension. I will explore the possibility(already considered by Smolin and others) that time may be distinguished from space by what I will call a measure of Booleanity. The Bell-Kochen-Specker Theorem shows that the statistics of quantum systems (unlike that of classical systems) do not in general permit of a Boolean substructure. I will outline reasons for thinking that time is the dimension in which the Booleanity of spacetime (considered as a quantum system) varies, while space is characterized by constant Booleanity. I will not be able to give a mathematically complete characterization of the Booleanity of a region of spacetime, since that would require nothing less than knowing how to quantize spacetime; however, I will argue that something like this is needed if one is to make any sense of an ontological distinction between past, present, and future in terms of modern physics. I will also briefly consider possible objections to this view arising from the relativity of simultaneity, which (on its usual interpretation) apparently places all events on an equal ontological footing. In order to get around this we need a generalized conception of simultaneity that treats Einstein's notion of simultaneity as a special case, and which allows for equivalence classes of spacelike separate events distinguished by covariant quantities such as action, phase, and (as I will argue) any reasonable measure of Booleanity.

Quantum entanglement has two remarkable properties. First, according to Bell\'s theorem, the statistical correlations between entangled quantum systems are inconsistent with any theory of local hidden variables. Second, entanglement is monogamous -- that is, to the degree that A and B are entangled with each other, they cannot be entangled with any other systems. It turns out that these properties are intimately related.

We all know that the EPR argument fails, and we can all provide proofs of one sort or another that it can\'t work. But in spite of this, there\'s something curiously tempting about the reasoning, and the temptation sometimes leads to needless perplexity about other issues. This paper will do two things. It will offer a diagnosis of where the EPR argument goes wrong that shows why we should be suspicious long before we get to Bell-type results, and then use the thought behind this diagnosis to suggest an orientation toward thinking about quantum states. The proposal for understanding states will have some things in common with Bayesian approaches, but will part company with them on some crucial points.

This paper critically examines the view of quantum mechanics that emerged shortly after the introduction of quantum mechanics and that has been widespread ever since. Although N. Bohr, P. A. M. Dirac, and W. Heisenberg advanced this view earlier, it is best exemplified by J. von Neumann’s argument in Mathematical Foundations of Quantum Mechanics (1932) that the transformation of \'a [quantum] state ... under the action of an energy operator . . . is purely causal,\' while, \'on the other hand, the state ... which may measure a [given] quantity ... undergoes in a measurement a non-casual change.\' Accordingly, while the paper discusses all four of these arguments, it will especially focus on that of von Neumann. The paper also offers an alternative, radically noncausal, view of the quantum-mechanical situation and considers the differences between the ensemble and the Bayesian understanding quantum mechanics. It will also discuss the Bayesian approach to quantum information theory in this set of contexts.