scholarly journals ON OCNEANU'S THEORY OF ASYMPTOTIC INCLUSIONS FOR SUBFACTORS, TOPOLOGICAL QUANTUM FIELD THEORIES AND QUANTUM DOUBLES

1995 ◽  
Vol 06 (02) ◽  
pp. 205-228 ◽  
Author(s):  
DAVID E. EVANS ◽  
YASUYUKI KAWAHIGASHI
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Finite Depth ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Natural Basis ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Topological Quantum

A fully detailed account of Ocneanu's theorem is given that the Hilbert space associated to the two-dimensional torus in a Turaev-Viro type (2+1)-dimensional topological quantum field theory arising from a finite depth subfactor N⊂M has a natural basis labeled by certain M∞- M∞ bimodules of the asymptotic inclusion M∨(M'∩M∞)⊂M∞, and moreover that all these bimodules are given by the basic construction from M∨(M'∩M∞)⊂M∞ if the fusion graph is connected. This Hilbert space is an analogue of the space of conformal blocks in conformal field theory. It is also shown that after passing to the asymptotic inclusions we have S- and T-matrices, analogues of the Verlinde identity and Vafa's result on roots of unity. It is explained that the asymptotic inclusions can be regarded as a subfactor analogue of the quantum double construction of Drinfel'd. These claims were announced by Λ. Ocneanu in several talks, but he has not published his proofs, so details are given here along the lines outlined in his talks.

scholarly journals FROM SUBFACTORS TO 3-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND BACK: A detailed account of Ocneanu’s theory

1995 ◽  
Vol 06 (04) ◽  
pp. 537-558 ◽  
Author(s):  
DAVID E. EVANS ◽  
YASUYUKI KAWAHIGASHI
Keyword(s):  
Field Theory ◽  
Finite Depth ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Frobenius Reciprocity ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Full Proof

A full proof of Ocneanu’s theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum field theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

scholarly journals QUANTUM FIELD THEORY ON CURVED BACKGROUNDS — A PRIMER

2013 ◽  
Vol 28 (17) ◽  
pp. 1330023 ◽  
Author(s):  
MARCO BENINI ◽  
CLAUDIO DAPPIAGGI ◽  
THOMAS-PAUL HACK
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Degrees Of Freedom ◽  
Algebraic Approach ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
System A ◽  
Step Procedure ◽  
Free Fields

Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

scholarly journals Quantum Field Theory of Open Spin Networks and New Spin Foam Models

2003 ◽  
Vol 18 (supp02) ◽  
pp. 83-96 ◽  
Author(s):  
A. Miković
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vector Bundles ◽  
Tensor Category ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Spin Networks ◽  
Spin Foam Models ◽  
Topological Quantum ◽  
Spin Foam

We describe how a spin-foam state sum model can be reformulated as a quantum field theory of spin networks, such that the Feynman diagrams of that field theory are the spin-foam amplitudes. In the case of open spin networks, we obtain a new type of state-sum models, which we call the matter spin foam models. In this type of state-sum models, one labels both the faces and the edges of the dual two-complex for a manifold triangulation with the simple objects from a tensor category. In the case of Lie groups, such a model corresponds to a quantization of a theory whose fields are the principal bundle connection and the sections of the associated vector bundles. We briefly discuss the relevance of the matter spin foam models for quantum gravity and for topological quantum field theories.

scholarly journals M$_k$ models: the field theory connection

2017 ◽  
Vol 3 (1) ◽  
Author(s):  
Thessa Fokkema ◽  
Kareljan Schoutens
Keyword(s):  
Field Theory ◽  
Finite Size ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Minimal Models ◽  
Size Spectra ◽  
Conformal Field ◽  
Lattice Fermions ◽  
Clustering Property

The M_kk models for 1D lattice fermions are characterised by {\cal N}=2 supersymmetry and by an order-kk clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of {\cal N}=2 supersymmetric conformal field theory (CFT) - we analyse finite size spectra on open chains with a variety of supersymmetry preserving boundary conditions. Specific staggering perturbations lead to a gapped regime corresponding to massive {\cal N}=2 supersymmetric QFT with Chebyshev superpotentials. At ‘extreme staggering’ we uncover a simple physical picture with degenerate supersymmetric vacua and mobile kinks. We connect this kink-picture to the Chebyshev QFTs and use it to derive novel CFT character formulas. For clarity the focus in this paper is on the simplest models, M_11, M_22 and M_33.

scholarly journals Global anomalies on the Hilbert space

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Diego Delmastro ◽  
Davide Gaiotto ◽  
Jaume Gomis
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Symmetry Algebra ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Strongly Coupled ◽  
Topological Quantum ◽  
Anomalous Theory

Abstract We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology “layers” that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2 + 1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.

scholarly journals Locality and General Vacua in Quantum Field Theory

2021 ◽  
Author(s):  
Daniele Colosi ◽  
◽  
Robert Oeckl ◽  
◽  
◽  
...  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Vacuum State ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Algebraic Quantum ◽  
Particle Pairs ◽  
The One ◽  
Gns Construction

We extend the framework of general boundary quantum field theory (GBQFT) to achieve a fully local description of realistic quantum field theories. This requires the quantization of non-Kähler polarizations which occur generically on timelike hypersurfaces in Lorentzian spacetimes as has been shown recently. We achieve this in two ways: On the one hand we replace Hilbert space states by observables localized on hypersurfaces, in the spirit of algebraic quantum field theory. On the other hand we apply the GNS construction to twisted star-structures to obtain Hilbert spaces, motivated by the notion of reflection positivity of the Euclidean approach to quantum field theory. As one consequence, the well-known representation of a vacuum state in terms of a sea of particle pairs in the Hilbert space of another vacuum admits a vast generalization to non-Kähler vacua, particularly relevant on timelike hypersurfaces.

Quantum field theory in phase space

2019 ◽  
Vol 34 (08) ◽  
pp. 1950037 ◽  
Author(s):  
R. G. G. Amorim ◽  
F. C. Khanna ◽  
A. P. C. Malbouisson ◽  
J. M. C. Malbouisson ◽  
A. E. Santana
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Phase Space ◽  
Relativistic Quantum ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Feynman Rules ◽  
Klein Gordon ◽  
Quantization Rules

The tilde conjugation rule in thermofield dynamics, equivalent to the modular conjugation in a [Formula: see text]-algebra, is used to develop unitary representations of the Poincaré group, where the Hilbert space has the phase space content, a symplectic Hilbert space. The state is described by a quasi-amplitude of probability, which is a sort of wave function in phase space, associated with the Wigner function. The quantum field theory in phase space is then constructed, including the quantization rules for the Klein–Gordon and the Dirac fields, the derivation of the electrodynamics in phase space and elements of a relativistic quantum kinetic theory. Towards a physical interpretation of the theory, propagators are associated with the corresponding Wigner functions. The Feynman rules follow accordingly with vertices similar to those of usual non-Abelian quantum field theories.

On 3-dimensional homotopy quantum field theory II: The surgery approach

2014 ◽  
Vol 25 (04) ◽  
pp. 1450027 ◽  
Author(s):  
Vladimir Turaev ◽  
Alexis Virelizier
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.

ON THE SYMMETRIES OF TOPOLOGICAL QUANTUM FIELD THEORIES

1995 ◽  
Vol 10 (31) ◽  
pp. 4483-4499 ◽  
Author(s):  
LAURENT BAULIEU
Keyword(s):  
Field Theory ◽  
Gauge Symmetry ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

We display properties of the general formalism which associates with any given gauge symmetry a topological action and a system of topological BRST and anti-BRST equations. We emphasize the distinction between the antighosts of the geometrical BRST equations and the antighosts occurring in field theory. We propose a transmutation mechanism between these objects. We illustrate our general presentation by examples.

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