arXiv subjects

A. P. Balachandran

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

At least 127 records · Page 7Linked to original sources

arXiv1998

Properties of Quantum Hall Skyrmions from Anomalies

It is well known that the Fractional Quantum Hall Effect (FQHE) may be effectively represented by a Chern-Simons theory. In order to incorporate QH Skyrmions, we couple this theory to the topological spin current, and include the Hopf term. The cancellation of anomalies for chiral edge states, and the proviso that Skyrmions may be created and destroyed at the edge, fixes the coefficients of these new terms. Consequently, the charge and the spin of the Skyrmion are uniquely determined. For those two quantities we find the values $eνN_{Sky}$ and $νN_{Sky}/2$, respectively, where $e$ is electron charge, $ν$ is the filling fraction and $N_{Sky}$ is the Skyrmion winding number. We also add terms to the action so that the classical spin fluctuations in the bulk satisfy the standard equations of a ferromagnet, with spin waves that propagate with the classical drift velocity of the electron.

cond-mat.mes-hall↗

arXiv1998

Skyrmions, Spectral Flow and Parity Doubles

It is well-known that the winding number of the Skyrmion can be identified as the baryon number. We show in this paper that this result can also be established using the Atiyah-Singer index theorem and spectral flow arguments. We argue that this proof suggests that there are light quarks moving in the field of the Skyrmion. We then show that if these light degrees of freedom are averaged out, the low energy excitations of the Skyrmion are in fact spinorial. A natural consequence of our approach is the prediction of a $(1/2)^{-}$ state and its excitations in addition to the nucleon and delta. Using the recent numerical evidence for the existence of Skyrmions with discrete spatial symmetries, we further suggest that the the low energy spectrum of many light nuclei may possess a parity doublet structure arising from a subtle topological interaction between the slow Skyrmion and the fast quarks. We also present tentative experimental evidence supporting our arguments.

hep-th↗

arXiv1997

Anomalous Defects and Their Quantized Transverse Conductivities

Using a description of defects in solids in terms of three-dimensional gravity, we study the propagation of electrons in the background of disclinations and screw dislocations. We study the situations where there are bound states that are effectively localized on the defect and hence can be described in terms of an effective 1+1 dimensional field theory for the low energy excitations. In the case of screw dislocations, we find that these excitations are chiral and can be described by an effective field theory of chiral fermions. Fermions of both chirality occur even for a given direction of the magnetic field. The ``net'' chirality of the system however is not always the same for a given direction of the magnetic field, but changes from one sign of the chirality through zero to the other sign as the Fermi momentum or the magnitude of the magnetic flux is varied. On coupling to an external electromagnetic field, the latter becomes anomalous, and predicts novel conduction properties for these materials.

hep-th↗

arXiv1997

Bringing Up a Quantum Baby

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 vectors in the domain of the Hamiltonian: The ``probability densities'' (hermitean forms) ψ^\dagger χfor ψ,χin this domain generate an algebra from which the classical configuration space with its topology (and with further refinements of the axiom, its C^K and C^infinity structures) can be reconstructed using Gel'fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the configuration space 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↗

arXiv1997

Emergent Chiral Symmetry: Parity and Time Reversal Doubles

There are numerous examples of approximately degenerate states of opposite parity in molecular physics. Theory indicates that these doubles can occur in molecules that are reflection-asymmetric. Such parity doubles occur in nuclear physics as well, among nuclei with odd A $\sim$ 219-229. We have also suggested elsewhere that such doubles occur in particle physics for baryons made up of `cbu' and `cbd' quarks. In this article, we discuss the theoretical foundations of these doubles in detail, demonstrating their emergence as a surprisingly subtle consequence of the Born-Oppenheimer approximation, and emphasizing their bundle-theoretic and topological underpinnings. Starting with certain ``low energy'' effective theories in which classical symmetries like parity and time reversal are anomalously broken on quantization, we show how these symmetries can be restored by judicious inclusion of ``high-energy'' degrees of freedom. This mechanism of restoring the symmetry naturally leads to the aforementioned doublet structure. A novel by-product of this mechanism is the emergence of an approximate symmetry (corresponding to the approximate degeneracy of the doubles) at low energies which is not evident in the full Hamiltonian. We also discuss the implications of this mechanism for Skyrmion physics, monopoles, anomalies and quantum gravity.

hep-th↗

arXiv1996

Lattice Gauge Fields and Noncommutative Geometry

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified spacetime or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD theta term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.

hep-lat↗

arXiv1996

Parity Doubles in Quark Physics

There are numerous examples of very nearly degenerate states of opposite parity in molecular physics. The ammonia maser is based on one such double. Theory shows that these parity doubles can occur if the nuclear shape in the molecule is reflection-asymmetric because the time scales of the shape and the electronic cloud are well-separated. Parity doubles occur in nuclear physics as well for odd $A \sim 219-229$. We discuss the theoretical foundation of these doubles and on that basis suggest that parity doubles should occur in particle physics too. In particular they should occur among baryons composed of $cbu$ and $cbd$ quarks.

hep-ph↗

arXiv1996

Edge States and Entanglement Entropy

It is known that gauge fields defined on manifolds with spatial boundaries support states localized at the boundaries. In this paper, we demonstrate how coarse-graining over these states can lead to an entanglement entropy. In particular, we show that the entanglement entropy of the ground state for the quantum Hall effect on a disk exhibits an approximate ``area " law.

hep-th↗

arXiv1995

Edge States in Canonical Gravity

It is well-known that gauge fields defined on manifolds with spatial boundaries support states localized at the boundary. In this talk, we show how similar states arise in canonical gravity abd discuss their physical relevance using their analogy to quantum Hall effect.

gr-qc↗

arXiv1995

Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold $M$ can be reconstructed from the commutative C*-algebra $\cc(M)$ of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on $M$. The latter also serve to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C$^*$-algebra $\ca $. This fact makes any poset a genuine `noncommutative' (`quantum') space, in the sense that the algebra of its `continuous functions' is a noncommutative C$^*$-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using $\ca$.

hep-th↗

arXiv1995

Chern-Simons Duality and the Quantum Hall Effect

In previous work on the quantum Hall effect on an annulus, we used $O(d,d;{\bf Z})$ duality transformations on the action describing edge excitations to generate the Haldane hierarchy of Hall conductivities. Here we generate the corresponding hierarchy of ``bulk actions'' which are associated with Chern-Simons (CS) theories, the connection between the bulk and edge arising from the requirement of anomaly cancellation. We also find a duality transformation for the CS theory exactly analogous to the $R\rightarrow \frac{1}{R}$ duality of the scalar field theory at the edge.

hep-th↗

arXiv1995

TOPOLOGY CHANGE AND QUANTUM PHYSICS

The role of topology in elementary quantum physics is discussed in detail. It is argued that attributes of classical spatial topology emerge from properties of state vectors with suitably smooth time evolution. Equivalently, they emerge from considerations on the domain of the quantum Hamiltonian, this domain being often specified by boundary conditions in elementary quantum physics. Several examples are presented where classical topology is changed by smoothly altering the boundary conditions. When the parameters labelling the latter are treated as quantum variables, quantum states need not give a well-defined classical topology, instead they can give a quantum superposition of such topologies. An existing argument of Sorkin based on the spin-statistics connection and indicating the necessity of topology change in quantum gravity is recalled. It is suggested therefrom and our results here that Einstein gravity and its minor variants are effective theories of a deeper description with additional novel degrees of freedom. Other reasons for suspecting such a microstructure are also summarized.

gr-qc↗

arXiv1994

Edge states in Gravity and Black Hole Physics

We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.

gr-qc↗

arXiv1994

Edge States in Gauge Theories: Theory, Interpretations and Predictions

Gauge theories on manifolds with spatial boundaries are studied. It is shown that observables localized at the boundaries (edge observables) can occur in such models irrespective of the dimensionality of spacetime. The intimate connection of these observables to charge fractionation, vertex operators and topological field theories is described. The edge observables, however, may or may not exist as well-defined operators in a fully quantized theory depending on the boundary conditions imposed on the fields and their momenta. The latter are obtained by requiring the Hamiltonian of the theory to be self-adjoint and positive definite. We show that these boundary conditions can also have nice physical interpretations in terms of certain experimental parameters such as the penetration depth of the electromagnetic field in a surrounding superconducting medium. The dependence of the spectrum on one such parameter is explicitly exhibited for the Higgs model on a spatial disc in its London limit. It should be possible to test such dependences experimentally, the above Higgs model for example being a model for a superconductor. Boundary conditions for the 3+1 dimensional $BF$ system confined to a spatial ball are studied. Their physical meaning is clarified and their influence on the edge states of this system (known to exist under certain conditions) is discussed. It is pointed out that edge states occur for topological solitons of gauge theories such as the 't Hooft-Polyakov monopoles.

hep-th↗

arXiv1994

Duality and the Fractional Quantum Hall Effect

The edge states of a sample displaying the quantum Hall effect (QHE) can be described by a 1+1 dimensional (conformal) field theory of $d$ massless scalar fields taking values on a $d$-dimensional torus. It is known from the work of Naculich, Frohlich et al.\@ and others that the requirement of chirality of currents in this \underline{scalar} field theory implies the Schwinger anomaly in the presence of an electric field, the anomaly coefficient being related in a specific way to Hall conducvivity. The latter can take only certain restricted values with odd denominators if the theory admits fermionic states. We show that the duality symmetry under the $O(d,d;{\bf Z})$ group of the free theory transforms the Hall conductivity in a well-defined way and relates integer and fractional QHE's. This means, in particular, that the edge spectra for dually related Hall conductivities are identical, a prediction which may be experimentally testable. We also show that Haldane's hierarchy as well as certain of Jain's fractions can be reproduced from the Laughlin fractions using the duality transformations. We thus find a framework for a unified description of the QHE's occurring at different fractions. We also give a derivation of the wave functions for fractions in Haldane's hierarchy.

hep-th↗

arXiv1994

Discrete Time from Quantum Physics

't Hooft has recently developed a discretisation of (2+1) gravity which has a multiple-valued Hamiltonian and which therefore admits quantum time evolution only in discrete steps. In this paper, we describe several models in the continuum with single-valued equations of motion in classical physics, but with multiple-valued Hamiltonians. Their time displacements in quantum theory are therefore obliged to be discrete. Classical models on smooth spatial manifolds are also constructed with the property that spatial displacements can be implemented only in discrete steps in quantum theory. All these models show that quantization can profoundly affect classical topology.

hep-th↗

arXiv1994

Finite Quantum Physics and Noncommutative Geometry

Conventional discrete approximations of a manifold do not preserve its nontrivial topological features. In this article we describe an approximation scheme due to Sorkin which reproduces physically important aspects of manifold topology with striking fidelity. The approximating topological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold $M$, continuous probability densities generate the commutative C*-algebra $\cc(M)$ of continuous functions on $M$. It has a fundamental physical significance, containing the information to reconstruct the topology of $M$, and serving to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C*-algebra $\ca $. As noncommutative geometries are based on noncommutative C*-algebras, we therefore have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. Various methods for doing quantum physics using $\ca $ are explored. Particular attention is paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics.

hep-th↗

arXiv1993

Finite Approximations to Quantum Physics: Quantum Points and their Bundles

There exists a physically well motivated method for approximating manifolds by certain topological spaces with a finite or a countable set of points. These spaces, which are partially ordered sets (posets) have the power to effectively reproduce important topological features of continuum physics like winding numbers and fractional statistics, and that too often with just a few points. In this work, we develop the essential tools for doing quantum physics on posets. The poset approach to covering space quantization, soliton physics, gauge theories and the Dirac equation are discussed with emphasis on physically important topological aspects. These ideas are illustrated by simple examples like the covering space quantization of a particle on a circle, and the sine-Gordon solitons.

hep-th↗