Topological field theory and the quantum double of SU(2)

Bais, F.A.; Muller, N. M.

doi:10.1016/s0550-3213(98)00572-0

Cited by 54 publications

(112 citation statements)

References 30 publications

(84 reference statements)

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“…In fact, most systems which can be described by means of two-dimensional conformal field theory have this property (for reviews, see for instance [19,20,21]). Examples in (2+1) dimensions are the discrete gauge theories we will treat in this paper, but also (2+1)-dimensional gravity [22,23] and certain fractional quantum Hall systems [24,25]. In two spatial dimensions, the exchanges of a system of n particles are governed by the braid group B n .…”

Section: Definitionmentioning

confidence: 99%

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Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Bais¹,

Schroers²,

Slingerland³

2003

J. High Energy Phys.

112

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Gauge theories in 2+1 dimensions whose gauge symmetry is spontaneously broken to a finite group enjoy a quantum group symmetry which includes the residual gauge symmetry. This symmetry provides a framework in which fundamental excitations (electric charges) and topological excitations (magnetic fluxes) can be treated on equal footing. In order to study symmetry breaking by both electric and magnetic condensates we develop a theory of symmetry breaking which is applicable to models whose symmetry is described by a quantum group (quasitriangular Hopf algebra). Using this general framework we investigate the symmetry breaking and confinement phenomena which occur in (2+1)-dimensional gauge theories. Confinement of particles is linked to the formation of string-like defects. Symmetry breaking by an electric condensate leads to magnetic confinement and vice-versa. We illustrate the general formalism with examples where the symmetry is broken by electric, magnetic and dyonic condensates.The irreps of our transformation group algebras are then described by the following theorem, which is a simple consequence of theorem 3.9 in [29]

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Section: Definitionmentioning

confidence: 99%

“…the quantum group theoretical framework of [22,23]. A generalization to weak quasi-Hopf algebras would bring any physical system which has a description in terms of Chern-Simons theory or rational conformal field theory within the reach of our methods.…”

Section: Jhep05(2003)068mentioning

confidence: 99%

Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Bais¹,

Schroers²,

Slingerland³

2003

J. High Energy Phys.

112

View full text Add to dashboard Cite

show abstract

“…We refer the reader to [2] for a discussion and to [3] for further geometrical background. Particles are introduced by marking points on the surface Σ and coupling the particle's phase space to the phase space of the theory.…”

Section: Einstein Gravity As Chern-simons Theorymentioning

confidence: 99%

“…with a suitable P The coupling of the particle degrees of freedom to the gauge field via minimal coupling is described in [3]. The upshot is that we specify the kinematic state of each particle in terms of representative iso(2, 1) elementsξ (i) = M i J 0 + s i P 0 and Poincaré elements (a i , L i ).…”

Section: Einstein Gravity As Chern-simons Theorymentioning

confidence: 99%

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Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

2002

Self Cite

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Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we show that the gravitational interactions deform the Poincaré symmetry of flat spacetime to a quantum group symmetry. The relevant quantum group is the quantum double of the universal cover of the (2+1)-dimensional Lorentz group, or Lorentz double for short. We construct the Hilbert space of two gravitating particles and use the universal R-matrix of the Lorentz double to derive a general expression for the scattering cross section of gravitating particles with spin. In appropriate limits our formula reproduces the semi-classical scattering formulae found by 't Hooft, Deser, Jackiw and de Sousa Gerbert.

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Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Fields

Glazebrook

Marcianò

2022

Fortschritte der Physik

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We introduce novel methods for implementing generic quantum information within a scale-free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis too develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.

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Topological field theory and the quantum double of SU(2)

Cited by 54 publications

References 30 publications

Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

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