- Perturbative and nonperturbative methods in cosmological problems and astrophysics
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University of Sao Paulo, Brazil
- Developments in noncommutative field theories III: UV-IR mixing
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Syracuse University, USAThe Groenewold-Moyal plane is a noncommutative deformation of the algebra of functions on Rd. Recent work has shown that the diffeomorphism group acts on this plane if its coproduct is deformed. We have argued that such a deformation also deforms the action of the permutation group which defines statistics. A consequence is that the standard Pauli exclusion and hence the commutation/anticommutation rules of creation-annihilation operators of quantum fields are also deformed as we have shown. In this talk, we establish that with these modifications, the UV-IR mixing disappears to all orders in perturbation theory from the S-matrix. This result is in agreement with the earlier results of Oeckl.
- Noncommutativity and deformed symmetries
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SNBNCBS, Kolkata, India
- String field theory and noncommutativity
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SISSA & INFN, Trieste, ItalyI will start with an introduction to SFT stressing in particular the noncommutative aspects. I will show various exact solutions of SFT and their low-energy limit. In the presence of a background B field these solutions reduce to well-known noncommutative field theory solutions. I will present possible applications to gravitational problems.
- Dual families of noncommutative quantum systems
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SNBNCBS, Kolkata, IndiaWe demonstrate how a one parameter family of interacting non-commuting Hamiltonians, which are physically equivalent, can be constructed in non-commutative quantum mechanics. This construction is carried out exactly (to all orders in the non-commutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low energy sector, can be constructed between the interacting commutative and a non-interacting non-commutative Hamiltonian. This approximation holds to order 1/B and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.
- Cosmological singularities from matrices
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University of Kentucky, USA
In string theory space and time are not believed to be fundamental, but derived from some more fundamental structure. For string theories which admit a holographic description we have some hint about these structures - these are field theories of large matrices. I will review some recent developments which show that such theories can lead to spacetimes which appear to have space-like or null singularities. The underlying matrix theory is well defined at these singularities. However the noncommutativity of the matrices become important in this region and this precludes any interpretation in terms of conventional spacetime.
- Quantum black holes: the event horizon as a fuzzy sphere
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NUI, Maynooth, Ireland
- Quantum spacetime and quantum fields
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Rome University, ItalyThe concurrence of the principles of quantum mechanics and of classical general relativity lead to uncertainty relations between the different spacetime coordinates of events, which can be implemented by Poincare covariant commutation relations, thus determining a basic model of quantum spacetime, proposed in [1]. Interaction between quantum fields on quantum spacetime can be formulated in various inequivalent ways [1,2]; merits and defects will be illustrated, with enphasis on the problem of causality and covariance.
[1] Commun. Math. Phys. 172 (1995) 187. [2] hep-th/0105251; Phys. Lett. B 533 (2002) 178; Commun. Math. Phys. 237 (2003) 221; Phys. Rev. D 71 (2005) 025022.
- Geometrical spinoptics and the optical Hall effect
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CPT, France
- Non(anti)commutativity for open superstrings
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SNBNCBS, Kolkata, IndiaNon(anti)commutativity in an open free superstring and also one moving in a background anti-symmetric tensor field is investigated. In both cases, the non(anti)commutativity is shown to be a direct consequence of the non-trivial boundary conditions which, contrary to several approaches, are not treated as constraints. The above non(anti)commutative structures lead to new results in the algebra of super constraints which still remain involutive, indicating the internal consistency of our analysis.
- Non-constant forms of noncommutativity
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ISI, Kolkata, India
- Developments in noncommutative field theories I: Space-time noncommutativity and quantised evolution
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IMSc, India We study the space-time noncommutative quantum theories. Contrary to expectations unitary time evolution can be obtained. We develope formal construction of theory for Moyal plane, a cylinder, R3 and R × S3. In all the later spaces evolution is quantised and leads to changes in energy conservation laws. Further developements will be discussed.
- Exotic Galilean symmetry and noncommutative mechanics, in mathematical and in condensed matter physics
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Universite de Tours, France Around 1995, noncommutative mechanics arose in the representation theory of the planar Galilei group, and, simultaneously and independently, for a semiclassical Bloch electron. The additional term generated in either ways plays a crucial role in understanding the anomalous, the spin, and the optical Hall effects.
- An extra parameter in the theory of gravity
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Perugia University, ItalyA parameter which was introduced by F. J. Barbero and myself in 1996 within the canonical formulation of general relativity, and which appeared to be completely arbitrary, becomes measurable when gravity is coupled to spin-½ fields. I shall summarize the situation and indicate the possible extension to N=1 supergravity.
- A space-time noncommutativity of the Dirac particle as induced by a fifth dimension
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IMSc, IndiaS. K. Bose, A. Gamba and E. C. G. Sudarshan [Phys. Rev. 113 (1959) 1661] have shown that in the extreme relativistic situation the position operators of the Dirac particle do not commute with each other and their commutation relations depend on energy, momentum, spin, etc., whereas the components of position and momentum satisfy the usual commutation relations. To this end, they used a representation of the Dirac Hamiltonian which is appropriate when the mass m is small compared to the momentum p of the particle (c = 1). The transformation they used independently had been given earlier by M. Cini and B. Touschek [Nuovo Cimento 7 (1958) 422]. In this work we attempt to explore the space-time noncommutativity resulting from an analogous treatment of a five-dimensional Dirac Hamiltonian when the extra mass corresponding to the fifth dimension is considered to be small compared to the four dimensional energy of the Dirac particle. This will lead to a space-time noncommutativity for the Dirac particle.
- Higher dimensional quantum Hall droplets
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Lehman College, USAWe present the formulation of quantum Hall effect in higher dimensional spaces, such as CPk and discuss the connection between the lowest Landau level and fuzzy CPk spaces. Further we derive the boundary effective action for the corresponding quantum Hall droplets, which is given by a Wess-Zumino-Witten theory generalized to higher dimensions. In the presence of electromagnetic interactions the bulk effective action is described by a Kahler-Chern-Simons term whose anomaly is cancelled by an appropriate boundary contribution.
- Tachyonic solitons in NCFT, EFT, BCFT and BSFT
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SKKU, Korea We discuss tachyon kinks and vortices in the context of noncommutative field theory and compare with those in Dirac-Born-Infeld type effective field theory, boundary conformal field theory, and boundary string field theory. The obtained codimension-one and two solitons naturally have interpretation as D-branes or DF-composites. In case of D- and DF-strings, cosmological implication is briefly reported.
- Noncommutative gauge theories and Lorentz symmetry
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PU, Chandigarh, India
- Waves on noncommutative spacetimes
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DIAS, Dublin, IrelandWaves on `commutative' spacetimes like Rd are elements of the commutative algebra C0(Rd) of functions on Rd. When C0(Rd) is deformed to a noncommutative algebra A/theta (Rd) with deformation parameter \theta waves being its elements, are no longer complex-valued functions on Rd. Rules for their interpretation, such as measurement of their intensity, thus need to be stated. In this talk I will first address this task and then apply these rules to interference in dimensions less than or equal to 4 and with time-space noncommutativity. I will discuss the cases of deformed interference pattern and no interference at all. As a further application, we discuss the interference of star light due to cosmic strings.
- Noncommutative instantons from twisted conformal symmetries
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Trieste U. & INFN, Naples, ItalyWe construct a five-parameter family of (infinitesimal) gauge-nonequivalent instantons on a noncommutative four sphere. They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the `tangent' to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to its classical value, that is five.
- Infinite conformal symmetry in noncommutative two-dimensional quantum field theory
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INFN & Naples University, Italy I discuss a field theory on the noncommutative plane endowed with the Moyal product. The symmetry is to be understood in the quantum sense, in that while the commutation relations are unchanged and form two copies of the Virasoro algebra, the coaction id deformed by terms epending on the noncommutativity parameter theta. We then quantize the theory. We show that to satisfy the Kac-Moody current algebras (which imply conformal invariance) we must impose a deformation of the commutation relations between the creation and annihilation operators of the theory.
- Noncommutative spacetime and quantum relativistic symmetries
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Wroclaw University, PolandAfter presenting noncommutative space-time as appearing in physical models we consider the deformed Minkowski spaces with constant, linear and quadratic commutators of coordinates. The Hopf-algebraic language will be outlined as describing quantum symmetries. Three classes of twisted Poincare algebras providing the description of deformed spacetime models will be presented. We discuss briefly the decription of dynamics on noncommutative space-time in classical mechanics and field theory.
- Shadows of noncommutative spacetime in cosmology
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INFN, Naples, Italy
- U(1)A anomaly, instantons and 't Hooft vertices in noncommutative SU(3) gauge theory
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Madrid University, SpainThe U(1)A anomaly, the construction of \theta-deformed instantons and the computation of 't Hooft vertices are analysed in noncommutative SU(3) gauge theories.
- Fuzzy spaces and quantum Hall effective actions
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City College of New York, USA We shall briefly discuss how Hall systems can be used to define some unusual fuzzy spaces, such as S3/Z2. We will also consider how functions on a finite dimensional Hilbert space are modified under a shift of background gauge fields. This technique, which is also related to the Seiberg-Witten transformation, will be used to obtain the effective action for quantum Hall systems.
- Glueball mass spectra for supergravity duals of noncommutative gauge theories
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Nihon University, Tokyo, Japan We derive the glueball masses in noncommutative super Yang-Mills theories in four dimensions via the dual supergravity description. The spectrum of glueball masses is discrete due to the noncommutativity and the glueball masses are proportional to the noncommutativity parameter with dimension of length. The mass spectrum in the WKB approximation closely agrees with the mass spectrum in finite temperature Yang-Mills theory.
- Noncommutative Chern-Simons theory on the sphere and the quantum Hall effect
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City College of New York, USAThe noncommutative Chern-Simons action on the sphere is constructed by using a projective representation of the matrix coordinates and a `boundary' term. The resulting theory is shown to reproduce the full Hilbert space of the Haldane-Laughling fractional quantum Hall states on the sphere.
- Fuzzy contact geometry and hydrodynamics of liquid crystals
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University of Rochester, USA We show that the vector fields that preserve a contact structure in three dimensions provide an intermediate case of hydrodynamics in between two and three dimensions: that of certian liquid crystals. The `quantization' of contact manifolds leads to a method of regularization: to approximate the dynamics of liquid crystals by a finite dimensional mechanical system. This is expected to be useful for numerical simulations as well as in formulating the notion of turbulent entropy for liquid crystal dynamics.
- Non(anti)commutative field theories
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La Plata University, Argentina
- Noncommutativity and interactions in quantum Hall systems
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Stellenbosch U., South AfricaWe discuss the role that interactions play in the noncommutative structure that arises when the relative coordinates of two interacting particles are projected onto the lowest Landau level. It is shown that the interactions in general renormalize the noncommutative parameter away from the non-interacting value 1/B. The effective noncommutative parameter is in general also angular-momentum dependent. A heuristic argument, based on the noncommutative coordinates, is given to find the filling fractions at incompressibilty and the results are consistent with known results in the case of singular magnetic fields.
- Super-algebras in the non(anti)commutative superspace
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Seoul National Univ., Korea
- Developments in noncommutative field theories II: Deformed spin and statistics
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IISc, IndiaWe argue that it is possible to write quantum field theories on the noncommutative plane that are (twisted) Lorentz-invariant. Some simple consequences, like the subsequent deformation, the spin-statistics connection are discussed.