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A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures. The talk is based on work carried out in collaboration with D.W. Hook and L.P. Hughston.

We consider pure three dimensional quantum gravity with a negative cosmological constant. The torus partition function can be computed exactly as a sum over geometries, including all known quantum corrections. The answer provides important clues about the structure of quantum gravity; in particular, in order for the theory to be a proper quantum mechanical system some extra ingredients are needed beyond the usual real geometries considered in general relativity. One possiblity is that complex geometries need to be included; this leads to holomorphically factorized partition functions. These partition functions provide a wealth of information about black hole microphysics. For example, the Hawking-page phase transition can be studied exactly; it is a phase transition of the type described by Lee and Yang, which is associated with a condensation of zeros in the complex temperature plane.

I discuss the status of Quantum Gravity Phenomenology, focusing separately on the 3 key areas: ability to discover, ability to constrain, and ability to falsify. And I stress the importance of adopting carefully taylored test theories as a remedy to difficulties encountered when comparing experimental evidence to theory evidence.

I discuss how physics beyond the Planck scale and before inflation might leave an imprint on the primordial spectrum. There are interesting limitations connected with the information paradox that suggests unexpected new ways to test ideas on quantum gravity.

We use the example of inflationary physics to discuss the possibility that short distance physics might be imprinted on long-distance observables. In particular, we focus on issues involving decoupling in field theory.

What if the second law of thermodynamics, in the hierarchy of physical laws, were at the same level as the fundamental laws of mechanics, such as the great conservation principles? What if entropy were an intrinsic property of matter at the same level as energy is universally understood to be? What if irreversibility were an intrinsic feature of the microscopic dynamical law of all physical objects, including an individual qubit or qudit? This talk will show how positive answers to these questions need not contradict any of the known results of quantum mechanics. We construct a logically consistent, mathematically sound and definite, physically intriguing, non-relativistic and non-statistical quantum theory, in which the second law of thermodynamics is embedded as a fundamental microscopical law. The theory hinges upon a nonlinear extension of unitary Hamiltonian dynamics which for uncorrelated and noninteracting systems reduces to the usual Schroedinger equation for the zero entropy states, but in general generates a group (not a semi group) of irreversible time evolutions, where the non-Hamiltonian entropy generating term in the evolution equation attracts the state towards the direction of maximal entropy increase. Various examples and features of this highly non-conventional dynamical theory are discussed. References available at http://www.quantumthermodynamics.org/