Aiyalam Balachandran

Dark Radiation from Infrared Gravitons 

The talks first reviews the construction of coherent states from the Weyl algebra for a finite number of degrees of freedom. It then recalls Riesz representation theorem which is important for the theory of inequivalent representations of quantum fields. With these tools in hand, the case of coherent states in QED brought about by infrared radiation is discussed and the proof that these coherent states create inequivalent representations of the Weyl algebra is outlined. A simple proof of the Frohlich et al.result that these inequivalent representations spontaneously break Lorentz invariance is presented. The results are then generalised to QCD. These discussions are then adapted to quantum gravity.There too, infrared gravitons dress the quantum states. That shifts the ADM energy. An estimate of this shift is outlined. It is also argued that boosts are spontaneously broken as in QED or QCD. The final remarks concern superselection rules. If they generate an algebra A, we argue that only their maximal abelian subalgebra define a superselection sector.

Alonso Botero

Multipartite Entanglement

Quantum entanglement remains to this day one of the most intriguing and well-studied properties of quantum physics. However, most of this attention has been concentrated in the study of bipartite entanglement, the structural features of which are pretty much well-understood by now, at least for pure quantum states. In contrast, our understanding of multipartite entanglement, even in the pure-state setting, remains fragmentary. In these lectures we will review some of the main aspects of current interest in the understanding of multipartite entanglement, such as quantification measures and the classification of equivalence classes of states under local operations and classical communication (LOCC). Finally, we discuss some interesting connections that arise between the LOCC classification program, invariant theory, and the representation theory of the symmetric group..

Alexander Cardona

Dirac Structures, Poisson Algebras and Quantization

The main goal of these talks will be to discuss the Poisson algebras of admissible functions associated to Dirac structures on smooth manifolds, and some of their applications to geometric quantization. We will start with the definition and main examples of Dirac structures and then we will show how to associate to them Poisson algebras of functions, and the relation of these Poisson algebras with the so-called Dirac bracket used to describe the dynamics of a constrained system. Then we will come to the problem of quantization of this type of algebras, from the geometric perspective, and the relevance of gerbes -instead of line bundles- in this context.

Bruno Carneiro da Cunha

Isomonodromy, Painlevé Transcendents and Scattering off of Black Holes

We summarise recent developments in obtaining analytical expressions for the scattering coefficients of Kerr and Kerr-de Sitter black holes, using the isomonodromy method. These have a lot in common with the theory of Painlevé transcendents and integrable structures, and physically with recent developments in conformal field theories. After sketching the general method, and consequences for black hole complementarity and unitary evolution, we turn to finding exact scattering coefficients for a conformally coupled field in a black hole background. We are able to derive implicit expressions in terms of Painlevé V and VI tau functions and more amenable analytical expressions in the near-extremal case.

Pramod Padmanabhan

Describing Topological Order Using Lattice Models

The problem of classifying phases of matter at low temperatures is an outstanding problem in condensed matter physics. The mean field theory of order parameters due to Landau-Ginzburg, achieves this for a large variety of systems at higher temperatures but fails at lower temperatures where quantum effects come into play giving rise to a new order named topological order. These phases can be described by exactly solvable lattice models which are described by Hamiltonians made up of commuting projectors. We will illustrate this through the example of the two dimensional toric code model which is based on the discrete group Z_2. The toric code model is the simplest example of a long ranged entangled phase in two dimensions and can be thought of as a particular limit of a lattice gauge theory based on the discrete gauge group Z_2. We will use this fact and construct the transfer matrix of this lattice gauge theory which will then be used to obtain the toric code as a particular limit. The advantage of this approach will be highlighted. We will then generalize this construction by including matter fields to produce more Hamiltonians which are made of commuting projectors. These models exhibit interesting variations of the toric code. Their possible relevance to topological quantum computation will be highlighted.

Juan Manuel Pérez Pardo

Quantum Systems with Boundary. Boundary Control, Topology Change and Edge States.

I will present in a didactic way several features associated to the presence of boundaries in quantum mechanical systems. On one hand I will introduce appropriate mathematical tools to address these problems such as the theory of self-adjoint extensions of differential operators in manifolds with boundary or variational principles. On the other hand I will introduce meaningful examples such as the possibility of controlling the state of a quantum system or even changing its topology by dynamically changing the boundary conditions. Other features intimately related to these like the appearance of edge states will also be discussed.

Aleksandr Pinzul

Spectral Geometry Approach to Foliated Space-Times: IR Regime

Starting with the most general Dirac operator respecting the foliation preserving diffeomorphism symmetry and restricting ourselves to the infrared (IR) limit, i.e. when the operator is still of the first order, we study the natural non-Lorentz invariant coupling of matter to gravity. In particular, we investigate the effect of this coupling on geodesic motion of a test particle. Using the spectral action principle, we obtain the gravitational part of the full action in this limit and speculate on some possible experimental consequences.

Amilcar Queiroz

Entanglement in Fermionic Chains with Finite Range Coupling and with Complex Coupling
(Joint work with: Filiberto Ares, Jose G. Esteve and Fernando Falceto)

Andrés Reyes Lega

Araki's Self-Dual Formalism, Fredholm Modules and Quantum Phase Transitions.

In these lectures I will present several basic examples illustrating cyclic cohomology. I will then discuss Araki's self-dual formalism and explain how it is used in the study of quadratic Hamiltonians. The natural Fredholm module structure present in these models will then be reviewed and used in order to connect cyclic cohomology with previous geometric approaches to quantum criticality.

Joseph Samuel

Classical and Quantum Cloning

Both classical and quantum mechanics are rigidly constrained structures. Classical dynamics respects the symplectic structure of phase space and quantum evolution respects the linear structure of Hilbert space and the projective geometry of ray space. These constraints limit our ability to clone or copy states, as evidenced by the famous ``no cloning theorem’’ in quantum mechanics. I will explore some of the constraints imposed by mechanics on copying states, both in classical and quantum theory. If time permits, I will illustrate the mathematical treatment with a few experimentally realisable sytems.

Supurna Sinha

Coarse Quantum Measurement

We present an analytical study of the Quantum Measurement Process in the context of the Stern Gerlach experiment. We regard the spin of a silver atom as the quantum system and its position as the measuring apparatus. Both the system and the measuring apparatus are treated exactly and quantum mechanically using the unitary evolution via the Schrodinger equation. We invoke the idea that the probes that determine the position of the silver atom always have bounded resources. This results in an apparent non unitary evolution of the spin, so that a pure density matrix appears to evolve to an impure one due to the coarseness of the detection process.

Sachindeo Vaidya

Matrix Yang-Mills and Stratified Boundaries

The talks describe our recent construction of a matrix model for non-Abelian Yang-Mills theory in 3+1 dimensions. This quantum matrix model incorporates locality of the underlying quantum field theory and is free of divergences. It also captures important topological aspects of Yang-Mills theory and it appropriate for both analytical and numerical work. As an application, we show that the gluon spectrum is gapped and estimate some low-lying levels for the cases of 2 and 3 colors. The configuration space of the matrix model is stratified, and we indicate the role of of these stratified boundaries in the structure of inequivalent ground states (or phases) of the Yang-Mills theory. We show that colored states of the theory are necessarily impure, and the implications of this result for the question of confinement.