Quantum Gravity

It is a standard axiom of quantum mechanics that the Hamiltonian H must be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary. In this talk we examine an alternative formulation of quantum mechanics in which the conventional requirement of Hermiticity is replaced by the more general and physical condition of space- time reflection (PT) symmetry. We show that if the PT symmetry of H is unbroken, Then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are $H=p^2+ix^3$ and $H=p^2-x^4$. Amazingly, the energy levels of these Hamiltonians are all real and positive despite the ``wrong'' sign in the $x^4$ potential! We show that such PT-symmetric Hamiltonians specify physically acceptable quantum-mechanical theories in which the norms of states are positive and time evolution is unitary. To do so we demonstrate that a Hamiltonian that has an unbroken PT symmetry also possesses a new physical symmetry that we call C. Using C, we construct an inner product whose associated norm is positive definite. The result is a new class of consistent complex quantum theories. In effect, we have extended and generalized quantum mechanics into the complex domain. We then discuss PT-symmetric quantum field theories. PT-symmetric scalar field-theoretic Hamiltonians corresponding to the above quantum-mechanical Hamiltonains have interaction terms $ig\phi^3$ and $-g\phi^4$. The latter Theory is interesting because (1) it is asymptotically free and (2) the expectation value of $\phi$ is nonzero. (Thus, such a theory might be useful in describing the Higgs sector.) PT symmetry resolves the long-standing problem of ghosts in the Lee model. When the renormalized coupling constant in this model increases past a critical value, the Hamiltonian ceases to be Hermitian and a negative-norm ghost state appears. At this transition the Hamiltonian becomes PT-symmetric, and the ghost is a physical particle. PT-symmetric QED and the PT-symmetric massive Thirring model will also be discussed. Finally, we mention recent papers which suggest that PT-symmetry may provide insight into cosmological problems.

Space-time measurements and gravitational experiments are made by the mutual relations between objects, fields, particles etc... Any operationally meaningful assertion about spacetime is therefore intrinsic to the degrees of freedom of the matter (i.e. non-gravitational) fields and concepts such as ``locality'' and ``proximity'' should, at least in principle, be definible entirely within the dynamics of the matter fields. We propose to consider the regions of space just as general ``subsystems''. By writing the Hilbert space of the matter fields as a generic tensor product of subsystems we analyse the evolution of a state vector on an information theoretical basis and discuss general principles to recover a posteriori the usual space-time relations. We apply such principles to generic interacting second quantized models with a finite number of fermionic degrees of freedom. Finally, we discuss the possible role of gravity in this framework.

We calculate analytically the highly damped quasinormal mode spectra of generic single-horizon black holes using the rigorous WKB techniques of Andersson and Howls. We thereby provide a firm foundation for previous analysis, and point out some of their possible limitations. The numerical coefficient in the real part of the highly damped frequency is generically determined by the behavior of coupling of the perturbation to the gravitational field near the origin, as expressed in tortoise coordinates. This fact makes it difficult to understand how the (in)famous ln(3) could be related to the quantum gravitational microstates near the horizon.

The phenomenology of quantum gravity can be examined even though the underlying theory is not yet fully understood. Effective extensions of the standard model allow us to study specific features, such as the existence of extra dimensions or a minimal length scale. I will talk about some applications of this approach which can be used to make predictions for particle- and astrophysics, and fill in some blanks in the puzzle of quantum gravity. A central point of this investigations is the physics of black holes. I will comment on possible ways to proceed and on the missing pieces I find most important to look for.

Observers agree that a citizen of Ohio had much larger voting power than a citizen of Texas or California in the recent US presidential election. Why is it so? A brief introduction to the theory of voting will be provided. We analyze the voting power of a member of a voting body, or of a person which elects his representative, who will take part in the voting on her behalf. The notion of voting power is illustrated by examples of the systems of voting in the European Council. We propose a representative voting system based on the square root law of Penrose. Using statistical approach and considering fictitious countries with randomly chosen populations we study the problem of selecting an optimal quota.

A joint Guelph-Waterloo Gravity Group/Perimeter Institute Seminar --------------------------------------------------------------------------- Observational evidence suggests that the large scale dynamics of the universe is presently dominated by dark energy, meaning a non-luminous cosmological constituent with a negative value of the pressure to density ratio w, which would be unstable if purely fluid, but could be stable if effectively solid with sufficient rigidity. It was suggested by Bucher and Spergel that such a solid constituent might be constituted by an effectively cold (meaning approximately static) distribution of cosmic strings with w=-1/3, or membranes with the observationally favoured value w=-2/3, but it was not established whether the rigidity in such models actually would be sufficient for stabilisation. For cases (exemplified by an approximately O(3) symmetric scalar field model) in which the number of membranes meeting at a junction is even (though not if it is odd) it is easy to obtain an explicit evaluation of the rigidity to density ratio, which is shown to 3/15 in both string and membrane cases, and it is confirmed that this is indeed sufficient for stabilisation.