Scientific Series

I'll discuss a reformulation of twistor-string theory as a heterotic string. This clarifies why conformal supergravity arises and provides a link between the Berkovits and Witten pictures. The talk is based on arXiv:0708:2276 with Lionel Mason.

In 3d quantum gravity, Planck's constant, the Planck length and the cosmological constant control the lack of (co)-commutativity of quantities like angular momenta, momenta and postion coordinates. I will explain this statement, using the quantum groups which arise in the 3d quantum gravity but avoiding technical details. The non-commutative structures in 3d quantum gravity are quite different from those in the deformed version of special relativity desribed by the kappa-Poincare group, but can be related to the latter by an operation called semi-dualisation. I will explain this operation, and make comments on its possible physical significance. The talk is based on joint work with Shahn Majid.

Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a classical curved spacetime, as described by general relativity. QFTCS has been applied to describe such important and interesting phenomena as particle creation by black holes and perturbations in the early universe associated with inflation. However, by the mid-1970\'s, it became clear from phenomena such as the Unruh effect that \'particles\' cannot be a fundamental notion in QFTCS. By the mid-1980\'s it was understood how to give a mathematically rigorous formulation of the theory of a free quantum field in curved spacetime. During the past decade, major progress has been made in providing a completely mathematically satisfactory formulation of renormalization in interacting QFTCS, thereby overcoming the difficulties caused by the absence of Poincare symmetry as well as the lack of a preferred vacuum state and a fundamental notion of \'particles\'. This talk will describe these developments and some of the insights that have thereby been attained.

The non-Gaussianity of the primordial cosmological perturbations will be strongly constrained by future observations like Planck. It will provide us with important information about the early universe and will be used to discriminate among models. I will review how different models of the early universe can generate different amount and shapes of non-Gaussianity.

I will discuss a solution generating technique that allows to generate stationary axisymmetric solutions of five-dimensional gravity, starting from static ones. This technique can be used to add angular momentum to static configurations. It can also be used to add KK-monopole charge to asymptotically flat five-dimensional solutions, thus generating geometries that interpolate between five-dimensional and four-dimensional solutions.

In this talk we discuss how large classes of classical spin models, such as the Ising and Potts models on arbitrary lattices, can be mapped to the graph state formalism. In particular, we show how the partition function of a spin model can be written as the overlap between a graph state and a complete product state. Here the graph state encodes the interaction pattern of the spin model---i.e., the lattice on which the model is defined---whereas the product state depends only on the couplings of the model, i.e., the interaction strengths. As main examples, we find that the 1D Ising model corresponds to the 1D cluster state, the 2D Ising model without external field is mapped to Kitaev's toric code state, and the 2D Ising model with external field corresponds to the 2D cluster state---but the mappings are completely general in that arbitrary graphs, and also q-state models can be treated. These mappings allow one to make connections between concepts in (classical) statistical mechanics and quantum information theory and to obtain a cross-fertilization between both fields. As a main application, we will prove that the classical Ising model on a 2D square lattice (with external field) is a "complete model", in the sense that the partition function of any other spin model---i.e., for q-state spins on arbitrary lattices---can be obtained as a special instance of the (q=2) 2D Ising partition function with suitably tuned (complex) couplings. This result is obtained by invoking the above mappings from spin models to graph states, and the property that the 2D cluster states are universal resource states for one-way quantum computation. Joint work with Wolfgang Duer and Hans Briegel, see PRL/ 98 117207 (2007)/ and quant-ph/0708.2275. For related work, see also S. Bravyi and R. Raussendorf, quant-ph/0610162.

I will discuss an alternative approach to simulating Hamiltonian flows with a quantum computer. A Hamiltonian system is a continuous time dynamical system represented as a flow of points in phase space. An alternative dynamical system, first introduced by Poincare, is defined in terms of an area preserving map. The dynamics is not continuous but discrete and successive dynamical states are labeled by integers rather than a continuous time variable. Discrete unitary maps are naturally adapted to the quantum computing paradigm. Grover's algorithm, for example, is an iterated unitary map. In this talk I will discuss examples of nonlinear dynamical maps which are well adapted to simple ion trap quantum computers, including a transverse field Ising map, a non linear rotor map and a Jahn-Teller map. I will show how a good understanding of the quantum phase transitions and entanglement exhibited in these models can be gained by first describing the classical bifurcation structure of fixed points.

In this talk, I will describe recent work in string phenomenology from the perspective of computational algebraic geometry. I will begin by reviewing some of the long-standing issues in heterotic model building and the goal of producing realistic particle physics from string theory. This goal can be approached by creating a large class of heterotic models which can be algorithmically scanned for physical suitability. I will outline a well-defined set of heterotic compactifications over complete intersection Calabi-Yau manifolds using the monad construction of vector bundles. Further, I will describe how a combination of analytic methods and computer algebra can provide efficient techniques for proving stability and calculating particle spectra.

The Origin of the Large Scale Structure is one of the key issue in Cosmology. A plausible assumption is that structures grow via gravitational amplification and collapse of density fluctuations that are small at early times. The growth history of cosmological fluctuations is a fundamental observable which helps in hunting for evidences of new physics, currently missing from our picture of the universe, but potentially crucial to explain its past, present and future history. I'll show how we investigated if the gradual growth of structures observed over a period of nearly 9 billion years can be used to discriminate between different gravitational models. I'll also discuss how the measurement of the cosmic growth rate provides an alternative independent probe to understand the origin of the accelerated expansion of the universe.

An ingredient in recent discussions of curvature singularity avoidance in quantum gravity is the "inverse scale factor" operator and its generalizations. I describe a general lattice origin of this idea, and show how it applies to the Coulomb singularity in quantum mechanics, and more generally to lattice formulations of quantum gravity. The example also demonstrates that a lattice discretized Schrodinger or Wheeler-DeWitt equation is computationally equivalent to the so called "polymer" quantization derived from loop quantum gravity.

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.

In nearly every quantum algorithm which exponentially outperforms the best classical algorithm the quantum Fourier transform plays a central role. Recently, however, cracks in the quantum Fourier transform paradigm have begun to emerge. In this talk I will discuss one such development which arises in a new efficient quantum algorithm for the Heisenberg hidden subgroup problem. In particular I will show how considerations of symmetry for this hidden subgroup problem lead naturally to a different transform than the quantum Fourier transform, the Clebsch-Gordan transform over the Heisenberg group. Clebsch-Gordan transforms over finite groups thus appear to be an important new tool for those attempting to find new quantum algorithms. [Part of this work was done in collaboration with Andrew Childs (Caltech) and Wim van Dam (UCSB)]