Quantum Foundations

If spacetime is "quantized" (discrete), then any equation of motion compatible with the Lorentz transformations is necessarily non-local. I will present evidence that this sort of nonlocality survives on length scales much greater than Planckian, yielding for example a nonlocal effective wave-equation for a scalar field propagating on an underlying causal set. Nonlocality of our effective field theories may thus provide a characteristic signature of quantum gravity.

In stochastic treatments of the ERRB set-up, it is equivalent to impose Bell\'s inequalities, a local causality condition, or a certain \"non-contextual hidden variables\" condition. But these conditions are violated by quantum mechanics. On the other hand, it is possible to view quantum mechanics as part of \"quantum measure theory\", a generalization of probability measure theory that allows pair wise interferences between histories whilst banning higher order interference. In this setting, is may be possible find quantum analogues of the three stochastic conditions. Following this line of inquiry, we will see that quantum measure theory allows no stronger violations of Bell\'s inequalities than does standard quantum theory. We also gain some insights into how to define causality in quantum theory.

The Everett (many-worlds) interpretation has made great progress over the past 20-30 years, largely due to the role of decoherence in providing a solution to the preferred basis problem. This makes it a serious candidate for a realist solution to the measurement problem. A remaining objection to the Everett interpretation (and one that is often considered fatal) is that that interpretation cannot make adequate sense of quantum probabilities. Dvaid Deutsch and David Wallace have argued that, by applying decision theory to the case of a rational agent who believes in the many-worlds interpretation, we can prove that such agents _act as if_ the theory predicted objective probabilities in the sense of fundamental indeterminism, or ignorance of initial conditions. I raise the issue of whether or not this, if true, is all that the many-worlds theorist needs from \'probability\'. I first suggest a reason for thinking that the answer might be \'no\': the reason is that knowing how to act on the assumption that a given theory is true is prima facie irrelevant to the question of whether we have any reason to believe the theory in the first place. I then go on to offer a solution to this problem, drawing on resources from Bayesian confirmation theory. My conclusion is that the problem of probability in the Everett interpretation has been solved.

Newton\'s first law of motion - and the very meaning of inertia - has been described as either completely obvious (D\'Alembert) or a \'logician\'s nightmare\' (ex-editor of the American Journal of Physics). Sometimes the simplest things in physics are the most subtle. The first law will be described in historical context, explaining a connection with the ancient Greeks’ distinction between natural and violent motion and with Descartes\' natural philosophy. You will also learn why it still requires careful handling and what it tells us about time in physics. \'Time and Motion\', Harvey Brown, time, motion, relative, Copernicus, Ptolemy, Galileo, Copernicanism, Descartes, inertia, Newton, standard of time, Fitzgerald, duration, inertial frame

I will discuss various different ways of quantifying the differences between two quantum observables (POVMs). Each of these approaches gives rise to a notion of approximately measuring one observable by means of measuring some other observable. This will be illustrated in the case of position and momentum by studying the question which POVMs on phase space can reasonably be said to represent a joint approximate determination of these observables. A new, universally valid trade-off relation for the associated inaccuracies will be rigorously formulated. I will sketch the proof which is an adaptation of some interesting techniques and properties of covariant phase space observables used recently by R Werner in a related project. Recommended reading (optional): quant-ph/0405184 (R Werner), quant-ph/0609185 (PB et al), and also for further background information quant-ph/0309091 (M Hall), quant-ph/0310070 (M Ozawa), quant-ph/9803051 (DM Appleby).

Quantum mechanics is a non-classical probability calculus -- but hardly the most general one imaginable. In this talk, I'll discuss some familiar non-classical properties of quantum-probabilistic models that turn out to be features of {em all} non-classical models. These include a generic no-cloning theorem obtained in recent work with Howard Barnum, Jon Barrett and Matt Leifer.

My field is the foundations of quantum mechanics, in particular Bohmian mechanics, a non-relativistic theory that is empirically equivalent to standard quantum mechanics while solving all of its paradoxes in an elegant and simple way, essentially by assuming that particles have trajectories. Bohmian mechanics possesses a straightforward generalization to relativistic space-time, be it flat or curved, if one assumption against the spirit of relativity is granted: the existence of a "time foliation", i.e., a physical object mathematically represented by a slicing of space-time into spacelike 3-surfaces, which evolves according to a Lorentz-invariant law. On the basis of this kind of theory, describing particles in a background 4-geometry, I propose an extension in which the space-time geometry is dynamically generated, as in general relativity. Whether my model is empirically equivalent to any known type of quantum gravity I don't know. In this model, there is a Lorentzian metric on configuration-space-time, evolving according to the higher-dimensional analog of the Einstein field equation. The 4-metric is obtained from the configuration-space-time metric and the actual particle configuration. Thus, this Bohm-like model generates (up to diffeomorphisms) a 4-metric and particle world lines from a given wave function.

How should we think about quantum computing? The usual answer to this question is based on ideas inspired by computer science, such as qubits, quantum gates, and quantum circuits. In this talk I will explain an alternate geometric approach to quantum computation. In the geometric approach, an optimal quantum computation corresponds to "free falling" along the minimal geodesics of a certain Riemannian manifold. This reformulation opens up the possibility of using tools from geometry to understand the strengths and weaknesses of quantum computation, and perhaps to understand what makes certain physical operations difficult (or easy) to synthesize.

In this talk, I will outline a quantum generalization of causal networks that are used to analyze complex probabilistic inference problems involving large numbers of correlated random variables. I will review the framework of classical causal networks and the graph theoretical constructions that are abstracted from them, including entailed conditional independence, d-separation and Markov equivalence. I will show how to generalize the definition of causal networks to the quantum case, such that the same graph theoretic constructions apply, and give an explicit representation of the states supported on the graph as the Gibbs states of certain classes of Hamiltonians.

Not only general relativity but also quantum theory plays important roles in current cosmology. Quantum fluctuations of matter fields are supposed to have provided the initial seeds of all the structure of the current universe, and quantum gravity is assumed to have been essential in the earliest stages. Both issues are not fully understood, although several heuristic effects have been discussed. In this talk, implications of an effective framework taking into account the coupling of matter and gravity are discussed. This touches on interpretational issues of quantum mechanics, cosmological observations and properties of quantum gravity.