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A. P. Balachandran

Publications and source records attributed to A. P. Balachandran.

At least 109 records · Page 6Linked to original sources
arXiv2002

The Star Product on the Fuzzy Supersphere

The fuzzy supersphere $S_F^{(2,2)}$ is a finite-dimensional matrix approximation to the supersphere $S^{(2,2)}$ incorporating supersymmetry exactly. Here the star-product of functions on $S_F^{(2,2)}$ is obtained by utilizing the OSp(2,1) coherent states. We check its graded commutative limit to $S^{(2,2)}$ and extend it to fuzzy versions of sections of bundles using the methods of [1]. A brief discussion of the geometric structure of our star-product completes our work.

hep-th↗
arXiv2002

Non-abelian Topological Strings and Metastable States in Linear Sigma Model

Non-abelian (NA) topological string defects exist in QCD with the spontaneous chiral symmetry breaking $U(N_f)_L \times U(N_f)_R \to U(N_f)_V$. Anomaly effects lead to domain walls connected to these strings. In the framework of linear sigma model we find the configuration of these defects. Also we show that in this model metastable CP violating vacuua exist. Strings as well as extended regions of metastable vacuua may form during the chiral transition which could have interesting effects.

hep-ph↗
arXiv2002

Quantum Spacetimes in the Year 1

We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $ CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics.

hep-th↗
arXiv2002

Continuous Time-Dependent Measurements: Quantum Anti-Zeno Paradox with Applications

We derive differential equations for the modified Feynman propagator and for the density operator describing time-dependent measurements or histories continuous in time. We obtain an exact series solution and discuss its applications. Suppose the system is initially in a state with density operator $ρ(0)$ and the projection operator $E(t) = U(t) E U^\dagger(t)$ is measured continuously from $t = 0$ to $T$, where $E$ is a projector obeying $Eρ(0) E = ρ(0)$ and $U(t)$ a unitary operator obeying $U(0) = 1$ and some smoothness conditions in $t$. Then the probability of always finding $E(t) = 1$ from $t = 0$ to $T$ is unity. Generically $E(T) \neq E$ and the watched system is sure to change its state, which is the anti-Zeno paradox noted by us recently. Our results valid for projectors of arbitrary rank generalize those obtained by Anandan and Aharonov for projectors of unit rank.

quant-ph↗
arXiv2002

Topological String Defect Formation During the Chiral Phase Transition

We extend and generalize the seminal work of Brandenberger, Huang and Zhang on the formation of strings during chiral phase transitions(berger) and discuss the formation of abelian and non-abelian topological strings during such transitions in the early Universe and in the high energy heavy-ion collisions. Chiral symmetry as well as deconfinement are restored in the core of these defects. Formation of a dense network of string defects is likely to play an important role in the dynamics following the chiral phase transition. We speculate that such a network can give rise to non-azimuthal distribution of transverse energy in heavy-ion collisions.

hep-ph↗
arXiv2001

Fuzzy Complex Projective Spaces and their Star-products

We derive an explicit expression for an associative *-product on fuzzy complex projective spaces. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on fuzzy complex projective spaces, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on ordinary projective space. Derivatives on the fuzzy projective space are also expressed as matrix commutators.

hep-th↗
arXiv2001

Fuzzy CP2

Regularization of quantum field theories (QFT's) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a ``fuzzy manifold''. Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem. In this paper, we report on our work on the ``fuzzification'' of the four-dimensional CP2 and its QFT's. CP2 is not spin, but spin${}_c$. Its Dirac operator has many unique features. They are explained and their fuzzy versions are described.

hep-th↗
arXiv2001

A Novel Spin-Statistics Theorem in (2 + 1)d Chern-Simons Gravity

It has been known for some time that topological geons in quantum gravity may lead to a complete violation of the canonical spin-statistics relation : there may exist no connection between spin and statistics for a pair of geons. We present an algebraic description of quantum gravity in (2 + 1)d based on the first order formalism of general relativity and show that, although the usual spin-statistics theorem is not valid, statistics is completely determined by spin. Hence, a new spin-statistics theorem can be formulated.

hep-th↗
arXiv2000

Fuzzy Actions and their Continuum Limits

Previously matrix model actions for ``fuzzy'' fields have been proposed using non-commutative geometry. They retained ``topological'' properties extremely well, being capable of describing instantons, $θ$--states, the chiral anomaly, and even chiral fermions with no ``doubling''. Here, we demonstrate that the standard scalar and spinor actions on a $d$--dimensional manifold are recovered from such actions in the limit of large matrices if their normalizations are correctly scaled as the limit is taken.

hep-th↗
arXiv2000

Fermion doubling problem and noncommutative geometry II

In our previous paper (hep-th/9911087), we proposed a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its cartesian products. In this paper after a review of that work, we bring out its relationship to the Ginsparg-Wilson approach.

hep-th↗
arXiv2000

Classical Topology and Quantum States

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its $C^K-$ and $C^\infty-$ structures) can be reconstructed using Gel'fand - Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.

quant-ph↗
arXiv2000

Instantons and Chiral Anomaly in Fuzzy Physics

In continuum physics, there are important topological aspects like instantons, theta-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we present discrete representations of theta-terms and topological susceptibility for gauge theories and derive axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions is bounded by the modulus of the instanton number as in the continuum.

hep-th↗
arXiv1999

Monopoles and Solitons in Fuzzy Physics

Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy sigma-model action for the two-sphere fulfilling a fuzzy Belavin-Polyakov bound is also put forth.

hep-th↗
arXiv1999

Quantum Topology Change in (2 + 1)d

The topology of orientable (2 + 1)d spacetimes can be captured by certain lumps of non-trivial topology called topological geons. They are the topological analogues of conventional solitons. We give a description of topological geons where the degrees of freedom related to topology are separated from the complete theory that contains metric (dynamical) degrees of freedom. The formalism also allows us to investigate processes of quantum topology change. They correspond to creation and annihilation of quantum geons. Selection rules for such processes are derived.

hep-th↗
arXiv1999

A Quantum Anti-Zeno Paradox

We establish an exact differential equation for the operator describing time-dependent measurements continuous in time and obtain a series solution. Suppose the projection operator $E(t) = U(t) E U^\dagger(t)$ is measured continuously from t = 0 to T, where E is a projector leaving the initial state unchanged and U(t) a unitary operator obeying U(0) = 1 and some smoothness conditions in t. We prove that the probability of always finding E(t) = 1 from t = 0 to T is unity. If $U(t) \neq 1$, the watched kettle is sure to `boil'.

quant-ph↗
arXiv1999

The Spin-Statistics Connection in Quantum Gravity

It is well-known that is spite of sharing some properties with conventional particles, topological geons in general violate the spin-statistics theorem. On the other hand, it is generally believed that in quantum gravity theories allowing for topology change, using pair creation and annihilation of geons, one should be able to recover this theorem. In this paper, we take an alternative route, and use an algebraic formalism developed in previous work. We give a description of topological geons where an algebra of "observables" is identified and quantized. Different irreducible representations of this algebra correspond to different kinds of geons, and are labeled by a non-abelian "charge" and "magnetic flux". We then find that the usual spin-statistics theorem is indeed violated, but a new spin-statistics relation arises, when we assume that the fluxes are superselected. This assumption can be proved if all observables are local, as is generally the case in physical theories. Finally, we also show how our approach fits into conventional formulations of quantum gravity.

hep-th↗
arXiv1998

Current Oscillations, Interacting Hall Discs and Boundary CFTs

In this paper, we discuss the behavior of conformal field theories interacting at a single point. The edge states of the quantum Hall effect (QHE) system give rise to a particular representation of a chiral Kac-Moody current algebra. We show that in the case of QHE systems interacting at one point we obtain a ``twisted'' representation of the current algebra. The condition for stationarity of currents is the same as the classical Kirchoff's law applied to the currents at the interaction point. We find that in the case of two discs touching at one point, since the currents are chiral, they are not stationary and one obtains current oscillations between the two discs. We determine the frequency of these oscillations in terms of an effective parameter characterizing the interaction. The chiral conformal field theories can be represented in terms of bosonic Lagrangians with a boundary interaction. We discuss how these one point interactions can be represented as boundary conditions on fields, and how the requirement of chirality leads to restrictions on the interactions described by these Lagrangians. By gauging these models we find that the theory is naturally coupled to a Chern-Simons gauge theory in 2+1 dimensions, and this coupling is completely determined by the requirement of anomaly cancellation.

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