An Introduction to Algebraic Quantum Field Theory

Fredenhagen, Klaus

doi:10.1007/978-3-319-21353-8_1

Cited by 6 publications

(9 citation statements)

References 75 publications

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“…Indeed, we have long known how to translate causal relations into properties of a quantum operator algebra [18]. Bekenstein's seminal work, as generalized by Jacobson, Fischler-Susskind, and Bousso, completes the process of translation.…”

Section: The Fischler-susskind Boundmentioning

confidence: 99%

Cosmological Implications of the Bekenstein Bound

Banks¹,

Fischler

2019

Jacob Bekenstein

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Section: The Fischler-susskind Boundmentioning

confidence: 99%

Cosmological Implications of the Bekenstein Bound

Banks¹,

Fischler

2019

Jacob Bekenstein

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“…Hence we are in the familiar setting of local quantum physics, where we have a net Λ → A(Λ) of observables associated to bounded regions (cf. [24]). It is not difficult to see that this net is local, that is,…”

Section: Topological Ordermentioning

confidence: 99%

“…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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“…Our analysis will follow closely the script of Klaus Fredenhagen. 53 As a starting point, we show that, whenever we represent a * -algebra on a Hilbert space via linear operators, we can automatically construct several states: Lemma 4.3. Let A be any topological * -algebra with an identity element and H a Hilbert space with scalar product (·, ·), such that there exists a faithful strongly continuous representation π : A → L(D), where D is a dense subspace of H, L(D) is the space of continuous linear operators on D and where π(a * ) = π(a) * for all a ∈ A.…”

Section: Algebraic States and Hilbert Space Representationsmentioning

confidence: 99%

“…Other relevant sources, whose presentation is often complementary to ours are Refs. 5,7,18,44,53, as well as the book written by Wald. 124 This review is organized as follows: In Section 2 we discuss the key geometrical concepts which lie at the heart of the construction of quantum field theory on curved backgrounds, particularly the notion of globally hyperbolic spacetimes.…”

Section: Introductionmentioning

confidence: 99%

Quantum Field Theory on Curved Backgrounds — A Primer

Benini

Dappiaggi

Hack

2013

Int. J. Mod. Phys. A

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Goal of this review 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 to the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.Keywords: quantum field theory on curved backgrounds, algebraic quantum field theory PACS numbers: 04.62.+v, 11.10.Cd or lightlike vector depending whether its length g(v p , v p ) is smaller, greater or equal to 0 respectively. In this way, as in Minkowski spacetime, we divide T p M into two regions, the set of spacelike vectors, and the two-folded light cone stemming from 0, the origin of the vector space T p M . For each point p ∈ M we have the freedom to designate each of the folds of the light cone of T p M as the set of future-directed and of pastdirected vectors respectively. If we can smoothly specify at each point which one of the two cones is the future one, we say that (M, g) is time orientable. 123 This is equivalent to the existence of a global vector field on M which is timelike everywhere. Henceforth we will only consider pairs (M, g) enjoying this property and, moreover, we shall assume that a time orientation has been fixed. This allows to introduce J ± M even when (M, g) is not isometric to (R 4 , η). To be precise, in the first place one has to define timelike, lightlike and spacelike curves: A piecewise smooth curve γ : I → M , I ∈ [0, 1], is timelike (lightlike or spacelike) if, for every t ∈ I, the vector tangent to the curve at γ(t) is timelike (respectively lightlike or spacelike). A curve is called causal if it is nowhere spacelike. For causal curves, one can also specify the direction according to the time orientation of (M, g): A causal curve γ : I → M is future-(past-) directed if for each t ∈ I each vector tangent to γ at γ(t) lies in the future (respectively past) fold of T γ(t) M . Given these preliminaries, for any p ∈ M we call causal future of p the set J + M (p) of points q ∈ M which can be reached by a future-directed causal curve stemming from p. Replacing future-directed causal curves with past-directed ones, we define also the causal past of p, J − M (p). Notice that, if we allow only timelike (in place of causal) curves, we replace J ± M (p) with I ± M (p), namely the chronological future (+) an...

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An Introduction to Algebraic Quantum Field Theory

Cited by 6 publications

References 75 publications

Cosmological Implications of the Bekenstein Bound

Cosmological Implications of the Bekenstein Bound

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

Quantum Field Theory on Curved Backgrounds — A Primer

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