"Noncommutative Geometry : Physical and Mathematical Aspects Of Quantum Space-Time and Matter"

Venue: S. N. Bose National Centre for Basic Sciences, Kolkata, India

University of Naples Federico II

If a Lie algebra acts on a Poisson manifold via derivations, the problem of deformation quantization of the manifold can be shifted to the existence of a normalized 2-cocycle on the universal enveloping algebra. This process is called Drinfel'd twist deformation quantization. Unlike a general star product, a twist star product inhabits many additional properties which are of particular interest in physics. However, the twisting procedure is limited, i.e. not every deformation quantization is of this form. It is the task of the talk to introduce the notion of Drinfel'd twist deformation quantization and to discuss its strenght and disadvantages together with several examples and counterexamples. The presented results are based on ongoing work of P. Bieliavsky, F.D'Andrea, C. Esposito, S. Waldmann and the speaker.

Universidad de los Andes

The quantum space-time model of Doplicher, Fredenhagen and Roberts (DFR) and in particular its commutative limit and the ensuing divergences for quantum field theories thereon are studied. As a method of renormalization to remove the divergences in question, the application of a Taylor subtraction (as in the BPHZ method for quantum field theory on commutative space-time) is proposed. Specifically, through the calculation of the contributions of fish and sunrise diagrams, we show that the subtraction operator removes divergences that would otherwise appear in the commutative limit. Based on joint work with J.C. Salazar.

King's College London

I will present a noncommutative extension of Palatini-Holst theory on a twist-deformed spacetime, generalizing a model previously proposed in the literature. The twist deformation entails an enlargement of the gauge group, and leads to the introduction of new gravitational degrees of freedom. In particular, the tetrad degrees of freedom must be doubled, thus leading to a bitetrad theory of gravity. The model is shown to exhibit new duality symmetries. I will present the commutative limit of the model, focusing in particular on the role of torsion and non-metricity. The effects of spacetime noncommutativity are taken into account perturbatively, and are computed explicitly in a simple example. Connections with bimetric theories and the role of local conformal invariance in the commutative limit will be briefly discussed.

University of Naples Federico II

I will explain how the graph C*-algebras of a trimmable graph can be decomposed as U(1)-equivariant pullback of two simpler C*-algebras. A main example is given by the algebra of Vaksman-Soibelman quantum sphere, that can be realized as pushout of a lower dimensional quantum sphere and the product of a quantum ball with a circle (we understand the pullback of C*-algebras as pushout of the underlying "noncommutative spaces"). The U(1)-invariant part of this pullback diagram gives a pushout realization of quantum projective spaces that allows to give an explicit description of the K-theory generators. Further examples include quantum lens spaces, one-loop extensions of Cuntz algebras of the Toeplitz algebra. This is a joint work with Francesca Arici, Piotr M. Hajac and Mariusz Tobolski.

Perimeter Institute

I will discuss the following question: Is the structure of the standard model of particle physics based on a non-commutative algebra, or a Jordan algebra?

Max-Planck Institute for Gravitational Physics

I will discuss some of the motivations for considering non-associative differential geometry, and describe some of the progress that has recently been made in this direction.

JNCASR

Abstract to be updated later.

University of Dhaka

The talk is based on the paper J. Math. Phys. 56, 122102 (2015). The work is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions, and starproducts, following a technique developed earlier, viz, using the unitary irreducible representations of the group GNC , which is the three fold central extension of the Abelian group of R4 . These representations have been exhaustively studied in earlier papers. The group GNC is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity?both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.

Queen Mary University of London

I outline a formulation of quantum Riemannian geometry using bimodule connections. I give some examples in the finite case where it is possible for a fixed differential algebra to construct the moduli of all possible quantum Riemannian geometries on it, as a step towards quantum gravity. In particular, this will be solved for the geometry of a square graph, including a reasonable Einstein-Hilbert action for the quantum metric as given by lengths assigned to the edges.

Universidad de Los Andes, Colombia

Universidad de los Andes

A considerable number of mathematical results regarding gapped Hamiltonians of fermions were achieved in recent years. Among the most important are topological protection under small perturbations and the persistence of the spectral gap for interacting fermions. However, information on the classification of the underlying systems in terms of topological indices is lacking. In this presentation I will discuss the first step towards the relation between a sort of topological index ÃÂ, introduced by Araki and Evans, and the Schwinger terms for gapped Hamiltonians of interacting fermion systems on the lattice. Our mathematical framework are Self--Dual CAR Algebras (SdCAR), whose structure contains the information on the symmetries of free fermions embedded in disordered systems, and are also useful to study interacting fermion systems even with supperconductors terms.\par Last but not least, I revisit two of the paradigmatic examples of classical and quantum phase transitions, highlighting the fact that ---in addition to obeying the traditional requirements of the Landau paradigm--- these phase transitions already are of a topological type, and can be characterized in terms of ÃÂ. In fact, Araki and Evans introduced ÃÂ to classify phase transitions of the two--dimensional Ising model. Their results were based on SdCAR and it is known that these to be closely related to the Shale--Stinespring theorem. The latter plays an important role in the study of Fredholm modules in the context of Clifford and CAR algebras. In this way a new approach encompassing examples of both topological as well as more traditional types of phase transitions of simple models, and which is close in spirit to noncommutative geometry, is proposed. Finally, I will comment how to tackle the problem of classification of Gapped Hamiltonians in terms of ÃÂ for the interparticle case. (Join work with N.J.B. Aza and A.F. Reyes-Lega.)

University Of Trieste

Abstract to be updated later.

The Krein's ( one-variable ) Trace Formula was originally motivated by Quantum Mechanics - by certain properties of Scrodinger Oparators . At the sametime , the formula admits a geometric interpretation and hence naturally motivates study of higher-dimensional analogues. Connes' formulation of NCG provides one such possibility , and I shall report some results in that direction.

This workis in collaboration with Arup Chattopadhyay

NISER Bhubaneswar

We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double. This is a joint work with Jyotishman Bhowmick, Arnab Mandal and Adam Skalski.

Saha Institute of Nuclear Physics

Abstract to be updated later.

The Institute Of Mathematical Sciences, Chennai, India

Planck scale acts as a threshold where a new description of spacetime is expected to appear. We consider the invariant Planck scale modified/deformed Poincare algebra proposed by Magueijo and Smolin and study the consequences as the Planck scale correction to the known physics. We get Planck scale modified dispersion relation and an effective invariant ultraviolet energy cut-off (Planck energy). It has been shown by many that there is an intimate connection between such a modified algebra and non commutative phase space. Also a similar cut-off appears in many studies of non commutative geometry. We study various equilibrium thermodynamic properties of blackbody radiation (i.e. a photon gas) and degenerate fermions with such modifications. The energy density, specific heat etc. of the photon gas follows the usual acoustic phonon dynamics as have been well studied by Debye. This is in sync with the expectation of the emergence of the granular structure of spacetime at Planck scale. Other modified thermodynamic quantities like pressure, entropy etc. also get the correction. The usual Stefan- Boltzmann law gets modified. The phase-space measure is also expected to get modified for an exotic spacetime appearing at Planck scale, which in turn leads to the modification of Planck energy density distribution and the Wien?s displacement law. We also found that the non-perturbative nature of the thermodynamic quantities in the SR limit (for both the case with ultraviolet cut-off and the modified measure case), due to nonanalyticity of the leading term, is a general feature of the theory accompanied with an ultraviolet energy cut-off. Due to such modifications, the energy momentum tensor TÎ¼Î½ gets modified and which leads to possible modification in case of the physics of Big Bang and the age of the known Universe. The dynamics of the compact Stellar objects such as white dwarfs too gets modified.

Copernicus Center for Interdisciplinary Studies

While studying the causal structure of the space of states on noncommutative Lorentzian spectral triples (as defined by N.Franco and M.Eckstein), we have proposed and studied a natural extension of the standard causal precedence relation $J^+$ onto the space $\mathscr{P}(\mathcal{M})$ of all Borel probability measures on a given spacetime $\mathcal{M}$. Using the tools from the optimal transport theory, one can utilize thus obtained notion of `causality between measures' to model a causal time-evolution of a spatially distributed physical entity in a globally hyperbolic spacetime. In my talk, after briefly presenting the abovementioned extension of $J^+$ and explaining what it means for a measure to evolve causally, I will discuss how this formalism can be extended to encompass many-particle systems. Thus obtained `$N$-particle causality theory', even though commutative, might offer a good starting point towards the study of the causal structure of Lorentzian spectral triples modelling many-particle quantum systems.

Universit? "D'Annunzio" di Chieti-Pescara

The extension of noncocommutative geometry to the semi-riemannian context has attracted a considerable and longstanding attention. This has led in particular to a sound notion of (globally hyperbolic) Lorentzian Spectral Triples (LSTs) and to the study of their causal and metric properties. However, some technical difficulties still affect the general construction, especially concerning the (lorentzian) distance formula. In this talk we propose a solution to these problems and test it in the case of the (lorentzian) Moyal Plane, providing some new explicit results on causality relations and distances of a large class of states.

Indian Statistical Institute

We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold and prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including Connes-Landi deformations of spectral triples on Rieffel-deformation of a compact manifold equipped with a free isometric toral action, the matrix geometry on the fuzzy sphere and quantum Heisenberg manifolds. As an application, we compute the Ricci and scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative 2-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

University of Gdansk

The spectral action principle put forward by Ali Chamseddine and Alain Connes offers an overarching rule for modelling the dynamics of fields. The primary interest in the spectral action, and its physical applications, is connected with almost commutative geometries, where one has at disposal the powerful tools of pseudo-differential calculus and heat kernel expansion. However, beyond the almost commutative realm the spectral action is still poorly understood. In my talk I will summarise the recent progress in the domain of spectral action in full glory of noncommutative geometry. The talk is based on the book "Spectral Action in Noncommutative Geometry" co-authored with Bruno Iochum.

Universidad Carlos III de Madrid

The development of quantum information processing and quantum computation goes hand in hand with the ability of dressing and manipulating quantum systems. Quantum Control Theory has provided a successful framework, both theoretical and experimental, to design and develop the control of such systems. In particular, to finite dimensional quantum systems or finite dimensional approximations to them. A concrete example is provided by the so called superconducting circuits, which are a promising technology to achieve the goal of quantum computation if one is able to improve the coherence times and reduce the errors committed during the manipulation. One possibility in order to achieve this is to go beyond the limitations of finite dimensional Quantum Control and to address the problem of control in infinite dimensional quantum systems. I will introduce notions of controllability well-suited for the infinite-dimensional setting and prove that certain families of quantum systems associated with superconducting circuits are controllable.

Universit? di Napoli Federico II

I analyse the localization of states and the role of observers in a noncommutative space with kappa-Poincare symmetry, and noncommutative coordinates of the kappa-Minkowski kind. The tools used mirror the ones of usual quantum mechanics.

Indian Institute of Science, Bangalore

Abstract to be updated later.

Syracuse University, NY, USA

Any three-dimensional manifold can be built up from the so-called prime manifolds. Quantum gravity on such spatial slices describes topological excitations called quantum geons. It is shown that associated spacetimes can be made noncommutative by generalising the Moyal star product. Diffeomorphisms also act consistently when the Drinnfeld twist is applied. These constructions extend the Moyal star product , which are based on spacetime translations , to more general abelian diffeomorphisms. Associated quantum fields can be constructed from those appropriate in the absence of the star product. Extended structures like knots exhibit similar features as are also discussed.

Chennai Mathematical Institute, Chennai

Quantum theory on manifolds with boundary present novel features due to boundary conditions. There are edge states localised at the boundary. These are useful in scale symmetry breaking. This has several applications like fractional QHE, topological insulators etc. They also play interesting role in counting microscopic states in blackhole geometry. But when geometry is described by noncomutative algebra the boundary issue becomes more complicated. We describe procedure which has proper continuum limit along with their boundary conditions. We also discuss modifications it causes in the applications.

IISc, Bangalore

One of the most celebrated discoveries of twentieth century is the existence of limiting mass of white dwarfs, which is one of the three compact objects formed once nuclear burning stops inside the star. On approaching this limiting mass 1.4 solar mass, called Chandrasekhar mass-limit, a white dwarf is believed to sparks off with an explosion called type Ia supernova, which is considered to be a standard candle. However, observations of several over-luminous, peculiar type Ia supernovae indicate that the Chandrasekhar-limit to be significantly larger. By considering noncommutativity of components of position and momentum variables, hence uncertainly in their measurements, at the quantum scales, we show that the mass of white dwarfs could be significantly super-handrasekhar and arrive at a new mass limit of about 2.5 solar mass. This provides an explanation for the origin of over-luminous peculiar type Ia supernovae. The idea of noncommutativity, apart from Heisenberg's uncertainly principle, is there for quite sometime, without any observational proof however. Our finding offers a plausible astrophysical evidence of noncommutativity, arguing for a possible second standard candle, which has many far reaching implications.

ISI, Kolkata

We'll give a brief outline of the theory of quantum isometry groups for noncommutative manifolds given by spectral triples, , which is a generalization of the group of Riemannian isometries of classical Riemannian manifolds. We'll devote some time to discuss the special and interesting case of classical smooth manifolds and discuss the background and very brief sketch of proof of the following result: a compact connected smooth manifold cannot have a genuine quantum symmetry given by a faithful, smooth co-action (to be explained in the talk) of a compact quantum group. However, if one drops the condition of compactness of the quantum group or the smoothness of the manifold, there are examples of genuine quantum group co-actions.

Jagiellonian University & Warsaw Institute of Mathematics, Poland

In the construction of spectral triples the real structure and the first-order condition assured that the Dirac operators were noncommutative counterparts of first-order differential operators. I will present a variations of the condition with examples that illustrate the applications of twisted first order condition and relate the construction to twisted and modular spectral triples.

Universita di Roma, Italy

Perturbative algebraic quantum field theory (pAQFT) is an approach to QFT in which the algebra of interacting observales is constructed perturbatively by a deformation of the product of classical obseravbles. We adapt its methods to a class of non-local QFTs that arise from QFT on DFR quantum spacetime with a suitably defined interaction. As an application, we analyze the limit, for vanishing Planck length, of the S-matrix to all perturbative orders, and argue that the result has potentially interesting cosmological consequences. (Joint work with S. Doplicher, N. Pinamonti.)

SNBNCBS

In this talk, we present the results of our investigations of a possible link between noncommutativity and dissipation. Our instincts were motivated by an analysis of tâ Hooft in which a new scheme of quantization was proposed as an attempt to understand Planck scale physics where the existence of a possible underlying relation between quantization and dissipation of information was pointed out. Here, we propose a new approach to the problem of quantization of the damped harmonic oscillator. To start with, we adopt the standard method of doubling the degrees of freedom of the system (Bateman form) and then by introducing some new parameters we get a generalized coupled set of equations from the Bateman form. Using the corresponding time-independent Lagrangian, quantum effects on a pair of Bateman oscillators embedded in an ambient noncommutative space (Moyal plane) is analyzed using both path integral and canonical quantization schemes within the framework of Hilbert-Schmidt operator formulation. We then discuss the results obtained and their implications . (Work done under the supervision of Prof. Biswajit Chakraborty and in collaboration with Partha Nandi).

SNBNCBS

Abstract to be updated later.

IMSC, Chennai

The connection between coarse-graining of measurement and emergence of classicality has been investigated for some time, if not well understood. Recently in (PRL 112, 010402, (2014)) it was pointed out that coarse-graining measurements can lead to non-violation of Bell-type inequalities by a state which would violate it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type inequality, this is not reflected at the state level. Under such measurements the post-measurement states can be non-classical (in the quantum optical sense) and in certain cases coarse-graning can lead to an increase in this non-classicality with respect to the coarse-graining parameter. While there is no universal way to quantify non-classicality, we do so using well understood notions in quantum optics such as the negativity of the Wigner function and the singular nature of the Gluaber-Sudharshan P distribution. [Joint work with Madhav Krishnan V and Tanmoy Biswas]