Scientific Series

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. We review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. With the above general considerations, we derive concretely the path integral formulations for GR from the canonical theory. We also show that there in principle exists a spin-foam model consistent with the canonical theory of GR.

The asymptotic formula for the Ponzano-Regge model amplitude is given for non-tardis triangulations of handlebodies in the limit of large boundary spins. The formula produces a sum over all possible immersions of the boundary triangulation in three dimensional Euclidean space weighted by the cosine of the Regge action evaluated on these immersions. Furthermore the asymptotic scaling registers the existence of flexible immersions.

Scalar field models of early universe inflation are effective field theories, typically valid only up to some UV energy scale, and receive corrections through higher dimensional operators due to the UV physics. Corrections to the tree level inflationary potential by these operators can ruin an otherwise suitable model of inflation. In this talk, I will consider higher dimensional kinetic operators, and the corrections that they give to the dynamics of the inflaton field. In particular, I will show how inflationary solutions exist even when the higher dimensional operators are important and not tuned to be negligible. I will then show that these solutions, which include the usual slow roll inflationary solutions, are attractors in phase space. I will end by speculating on the role of the corrections from these higher dimensional operators in alleviating the homogeneous initial conditions problem for inflation.

When a pair of particles is produced close to threshold, they may form a bound state if the potential between them is attractive. Can we use such bound states to obtain information about new colored particles at the LHC? I will discuss the relevant issues using examples from the MSSM and other beyond the standard model scenarios.

In the context of AdS/CFT correspondence the AdS_3/CFT_2 instance of the duality stands apart from other well studied cases, like AdS_5/CFT_4 or AdS_4/CFT_3. One of the reasons is that the CFT side of this duality is not a theory of matrices but rather a two dimensional orbifold based on the group of permutations. In this talk we will discuss some aspects of this theory. In particular a diagrammatic language, akin to Feynman diagrams used for gauge theories, will be developed. Moreover, we will compute a large set of protected quantities in a certain symmetric product orbifold CFT, and show that these are elegantly given in terms of Hurwitz numbers.

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. Inspired by the success of the statistical study of classical constraint optimization problems, we study random ensembles of the QMA$_1$-complete quantum satisfiability (QSAT) problem introduced by Bravyi. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are product satisfiable, entangled satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for a fixed interaction graph exhibit the same dimension of the satisfying manifold. This establishes the generic QSAT decision problem as equivalent to a purely graph theoretic property and that the hardest typical instances are likely to be localized in a bounded range of clause density. Based on papers: C.R. Laumann, R. Moessner, A. Scardicchio, and S.L. Sondhi. Phase transitions and random quantum satisfiability. QIC 10 (1/2), (2009). arXiv:0903.1904 C.R. Laumann, A.M. Lauchli, R. Moessner, A. Scardicchio, and S.L. Sondhi. On product, generic and random generic quantum satisfiability. arXiv:0910.2058

In quantum field theory it is possible to create negative local energy densities. This would violate the Generalized Second Law (GSL) unless there is some sort of energy condition requiring the negative energy to be counterbalanced by positive energy. TO explore what this energy condition is, I will assume that the GSL holds in semiclassical gravity for all future causal horizons. From CPT symmetry it follows that the time-reverse of the GSL, properly understood, holds for all past causal horizons. These two conditions together then imply that the Averaged Null Energy Condition (ANEC) holds on any null line, i.e. a complete achronal lightlike null geodesic. In curved spacetimes, the ANEC can be violated on general geodesics. But even if the ANEC only holds on null lines, theorems by Sorkin, Penrose and Woolgar, and by Graham and Olum imply that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. These results can thus be seen as consequences of the GSL. However, these theorems don't apply when gravitational fluctuations are taken into account. In that case, the GSL argument suggests a modification to the ANEC which may make these theorems applicable to perturbative quantum gravity.

The scaling analysis in the large spin limit of Feynman amplitudes for the Bosonic colored group field theory are considered in any dimension starting with dimension 4. By an explicit integration of two colors, we show that the model is positive. This formulation could be useful for the constructive analysis of this type of models.

Spin foam models aim at defining non-perturbative and background independent amplitudes for quantum gravity. In this work, I argue that the dynamics and the geometric properties of spin foam models can be nicely studied using recursion relations. In 3d gravity and in the 4d Ooguri model, the topological invariance leads to recursion relations for the amplitudes. I also derive recursions from the action of holonomy operators on spin network functionals. Their geometric content is discussed in terms of elementary moves on simplices, and is related to the classical constraints of the underlying theories. Interestingly, our recurrence relations apply to any SU(2) invariant symbol. Another method is considered for non-topological objects, and applied to the 10j-symbol which defines the Barrett-Crane spin foam model.

Supersymmetry is a leading candidate for physics Beyond the Standard Model. However, a tree level in the Minimal Supersymmetric Standard Model the Higgs boson should be lighter than the Z boson. LEP did not discover the Higgs boson, so typically large radiative corrections are required to push the Higgs above the LEP lower limit, leading to fine tuning issues. In this talk I will describe how to avoid limits from the searches at LEP and discuss a potential early signal of a 90 GeV Higgs at the LHC.