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I will identify six \'problems of time\' that arise in connection with quantum gravity and review the extent to which some of them can be regarded as solved, highlighting the very different aspects that they assume depending on one\'s starting point: Hamiltonian vs. path-integral, discrete vs continuous.

There is now a great deal of evidence confirming the existence of a very hot and dense early stage of the universe. Much of this data comes from a detailed study of the cosmic microwave background (CMB) - radiation from the early universe that was most recently measured by NASA\'s WMAP satellite. But the information presents new puzzles for scientists. One of the most blatant examples is an apparent paradox related to the second law of thermodynamics. Although some have argued that the hypothesis of inflationary cosmology solves some of the puzzles, profound issues remain. In this talk, Professor Penrose will describe a very different proposal, one that suggests a succession of universes prior to our own. He will also present a recent analysis of the CMB data that has a profound bearing on these issues.

Exactly half a century after Minkowski’s justly famous lecture, Dirac’s efforts to quantize gravity led him “to doubt how fundamental the four-dimensional requirement in physics is”. Dirac does not appear to have explored this doubt further, but I shall argue that it needs to be considered seriously. The fact is that Einstein and Minkowski fused space and time into a four-dimensional continuum but never directly posed the two most fundamental questions in dynamics: What is time? What is motion? It was an historical accident that Einstein attempted to implement Mach’s principle after he had created special relativity; otherwise he would have been forced to address these questions, which have never been properly considered. I shall show how they can be answered and suggest that: 1) time and space are utterly different; 2) the dynamical law of the universe may define absolute simultaneity in a manner that is still consistent with local validity of Minkowski’s marvellous notion of spacetime.

The evidence for the big bang is now overwhelming. However, the basic question of what caused the bang remains open. One possibility is that time somehow \'emerged,\' placing the universe in an inflationary state. Another, perhaps more conservative possibility, is that the big bang was a violent event in a pre-existing universe. I will describe model calculations employing the AdS/CFT correspondence which show how this is possible, and which point to a new explanation for the origin of large scale structure in the universe.

In recent work with Bob Coecke and others, we have developed a categorical axiomatization of quantum mechanics. This analyzes the main structural features of quantum mechanics into simple and general elements, which admit an elegant diagrammatic representation. This enables an illuminating and effective analysis of quantum information protocols and computational structures. One aspect which is brought to light in this analysis is that protocols such as teleportation use entanglement to achieve a logical information flow which has an apparent retro-causal (or even `backward-in-time\') component. However, there is also a physical or operational description of the same systems, based on an abstract structural characterization of quantum measurements, which is entirely causally consistent. The systematic relationship between these two descriptions suggests a novel perspective on the flow of time and information in the quantum realm.

In recent years there has been a growing awareness that studies on quantum foundations have close relationships with other fields such as probability and information theory. In this talk I give another example of how such interdisciplinary work can be fruitful, by applying some of the lessons from quantum mechanics, in particular from Bell\'s theorem, to a debate on the philosophical foundations of decision theory. I argue that the basic assumptions of the popular causal decision theory -- which was developed partly in response to a puzzle proposed by the physicist William Newcomb and published by the philosopher Robert Nozick -- are analogous to the basic assumptions of a local hidden-variables theory in the context of Bell\'s theorem. Both have too strong a prejudice about the causal structure of the world: there are possible games the world can pose such that an agent who operates by those theories is constrained to choose losing strategies no matter what evidence he or she acquires.

I\'ll sketch of a proposal for unifying classical and quantum probability, arguing first for the need to recognize a measure over phase space as a component of classical theories (indeed, of any theory satisfying certain constraints and capable of generating predictions for open systems) and then showing how to use that measure to define objective chances. Time permitting, I\'ll briefly address questions about the nature and interpretation of the measure.

As is well known, time-energy uncertainty generically manifests itself in the short time behavior of a system weakly coupled to a bath, in the energy non-conservation of the interaction term (H_I does not commute with H_0). Similarly, the monotonic evolution of the system density operator to its equilibrium value which is a universal property of quantum dynamical semigroups (Spohn\'s theorem), e.g., systems with Lindbladian evolution, is in general violated at short (non-Markovian) timescales. For example, frequent, brief non-demolition measurements of the energy states of a two level system (TLS) coupled to a bath, disturbs the thermal equilibrium between them, despite leaving the system and bath states separately unperturbed. For sufficiently short intervals between measurements (Zeno regime) the system and bath heat up immediately following the measurement. It is also possible to have net cooling in an intermediate (anti-Zeno-like) regime. The evolution of the system state away from its equilibrium value, not only violates the Markovian-dynamics version of the 2nd law (Spohn\'s theorem), but also Lindblad\'s theorem on which it rests, which is valid for any evolution described by a completely positive map. This does not imply that the evolution is not completely-positive, but rather that it is not a well-defined map at allthe evolution of the state of the system is not determined by this state alone (nor even together with the reduced state of the bath), but rather by the full joint system-bath state (this indeterminacy was shown previously, by Buzek et al., for special cleverly constructed joint states). Ref: N. Erez, G. Gordon, M. Nest & G. Kurizki, Nature 452, 724 (2008)

In reference [1] R. D. Sorkin investigated a formulation of quantum mechanics as a generalized measure theory. Quantum mechanics computes probabilities from the absolute squares of complex amplitudes, and the resulting interference violates the (Kolmogorov) sum rule expressing the additivity of probabilities of mutually exclusive events.However, there is a higher order sum rule that quantum mechanics does obey, involving the probabilities of three mutually exclusive possibilities. We could imagine a yet more general theory by assuming that itviolates the next higher sum rule.An experiment is in progress in our laboratory which sets out to test the validity of this second sum rule by measuring the interference patterns produced by three slits and all the possible combinations of those slits being open or closed. We use either attenuated laser light or a heralded single photon source (using parametric down conversion) combined with single photon counting to confirm the single photon character of the measured light. We will show results that bound the possible violation of the second sum rule and will point out ways toobtain a tighter experimental bound.[1] R. D. Sorkin, Quantum Mechanics as Quantum Measure Theory,Mod. Phys. Lett. A 9, 3119 (1994).