Algebraic Conformal Quantum Field Theory in Perspective

Rehren, Karl-Henning

doi:10.1007/978-3-319-21353-8_8

Cited by 14 publications

(18 citation statements)

References 79 publications

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“…The category C should be interpreted as the category of "spacetimes" of interest, and the orthogonality relation ⊥ encodes the commutative behavior of certain observables. Our construction is very flexible and in particular it reveals an operadic structure underlying various kinds of quantum field theories, including Haag-Kastler theories on Minkowski spacetime [HK64], locally covariant theories on all Lorentzian spacetimes [BFV03,FV15], chiral conformal theories [Kaw15, Reh15,BDH15] and also Euclidean theories [Sch99]. Each of these models is obtained by a different choice of category C and orthogonality relation ⊥, however the operadic structure is formally the same.…”

Section: Quantum Field Theory Operadsmentioning

confidence: 99%

“…▽ Remark 3.23. Mimicking the previous examples, one can introduce further interesting orthogonal categories which give rise to algebraic quantum field theories on a fixed spacetime [HK64], chiral conformal quantum field theories [Kaw15, Reh15,BDH15] and Euclidean quantum field theories [Sch99]. For the latter two scenarios, the orthogonality relation is determined by disjointness instead of causal disjointness.…”

Section: (I) Given Any Orthogonality Relationmentioning

confidence: 99%

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Operads for algebraic quantum field theory

Benini

Schenkel

Woike

2020

Commun. Contemp. Math.

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We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions, such as timeslicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen's universal algebra. Report no.: ZMP-HH/17-26, Hamburger Beiträge zur Mathematik Nr. 682Keywords: algebraic quantum field theory, locally covariant quantum field theory, colored operads, change of color adjunctions, Fredenhagen's universal algebra MSC 2010: 81Txx, 18D50

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Section: Quantum Field Theory Operadsmentioning

confidence: 99%

Section: (I) Given Any Orthogonality Relationmentioning

confidence: 99%

Operads for algebraic quantum field theory

Benini

Schenkel

Woike

2020

Commun. Contemp. Math.

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“…In the later constructions, this symmetry between wedges and their 3 Although we will mostly be working with Minkowski space here, it should be noted that similar families of regions can also be defined in other situations: On the one-dimensional line, the half lines (a, ∞) and (−∞, a), a ∈ R, have the same properties as the wedges in Minkowski space (see also the discussion in Section 3.5). Also the family of all intervals on a circle, of prominent importance in chiral conformal field theory [Reh15], shares many properties with the family of wedges on Minkowski space, see for example [Lon08].…”

Section: )mentioning

confidence: 99%

Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

Lechner

2015

Advances in Algebraic Quantum Field Theory

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Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.

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“…This structure is in fact very well known in local quantum physics. One of the highlights of algebraic quantum field theory is the study of superselection sectors initiated by Doplicher, Haag and Roberts [19,20], see also this volume [24,56]. This leads in a natural way to a fusion category as above.…”

Section: Introductionmentioning

confidence: 99%

Kitaev’s Quantum Double Model from a Local Quantum Physics Point of View

Naaijkens¹

2015

Advances in Algebraic Quantum Field Theory

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A prominent example of a topologically ordered system is Kitaev's quantum double model D(G) for finite groups G (which in particular includes G = Z2, the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of D(G), and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups G is more complicated. We outline how one could use amplimorphisms, that is, morphisms A → Mn(A) to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.

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Algebraic Conformal Quantum Field Theory in Perspective

Cited by 14 publications

References 79 publications

Operads for algebraic quantum field theory

Operads for algebraic quantum field theory

Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

Kitaev’s Quantum Double Model from a Local Quantum Physics Point of View

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