I will present an efficient quantum algorithm for an additive
approximation of the famous Tutte polynomial of any planar graph at
any point. The Tutte polynomial captures an extremely wide range of
interesting combinatorial properties of graphs, including the
partition function of the q-state Potts model. This provides a new
class of quantum complete problems.
Our methods generalize the recent AJL algorithm for the approximation
of the Jones polynomial; instead of using unitary representations, we
allow non-unitarity, which seems counter intuitive in the quantum
world. Significant contribution of this is a proof that non-unitary
operators can be used for universal quantum computation.
A relativistic modified gravity (MOG) leads to a self-consistent, stable gravity theory that can describe the solar system, galaxy and clusters of galaxies data and cosmology without dark matter. A review is given of fits to galaxy rotation curves, mass profiles of X-ray clusters and weak and strong lensing of galaxy clusters including the bullet cluster E10657-56. MOG can explain the CMB power spectrum and the observed acceleration of the expansion of the universe.
We show that the recently suggested Euclidean quantum gravity density matrix of the Universe which generates the set of inflationary universes is, in fact, the density matrix of the microcanonical ensemble in {\em Lorentzian} quantum cosmology. This ensemble corresponds to the uniform (weight one) distribution over an entire phase space of {\em true physical variables}, but in terms of the observable spacetime geometry it is peaked about complex saddle-points of the Lorentzian path integral, given by the recently obtained Euclidean gravitational instantons belonging to the bounded range of the cosmological constant. It is shown that the conformal rotation, designed to solve the problem of unboundedness of the Euclidean gravitational action, is, in fact, the manifestation of correct integration range over the lapse and shift functions in Lorentzian quantum gravity, enforcing the Hamiltonian and momentum constraints. Inflationary cosmologies generated by the gravitational instantons at late stages of expansion incorporate cosmological acceleration phenomenon whose low-energy scale can be attained by invoking the ideas of dynamically evolving compactified extra dimensions. Thus, together with the bounded range of the early cosmological constant, this initial state of the Universe simultaneously suggests the dynamical restriction mechanism for the landscape of string vacua and the solution of the dark energy problem.
I describe how vacuum energy can be controlled by a symmetry principle that necessitates a ghost sector. I argue that the implied instability of Minkowski spacetime is natural and consistent with experience if gravity is fundamentally Lorentz-violating, and describe attempts to construct such exotic dynamics. I briefly discuss the more robust experimental/observational signatures of such a scenario.
Motivated by the severity of the bounds on Lorentz violation in the presence of ordinary gravity, we study frameworks in which Lorentz violation does not affect the spacetime geometry. We show that there are at least two inequivalent classes of spontaneous Lorentz breaking that even in the presence of gravity result in Minkowski space. The first one generically corresponds to the condensation of tensor fields with tachyonic mass, which in turn is related to ghost-condensation. In the second class, realized in the DGP model or in theories of massive gravity, spontaneous Lorentz breaking is induced by the expectation value of sources. The generalization to de-Sitter space is also discussed.
In this talk, I summarize a current status of the DGP braneworld emphasizing the theoretical consistency of the model.First I review the behaviour of the linearized gravity and show the existence of the ghost. Then I discuss the issue of the non-linearity of gravity in this model.