Introduction to Algebraic Quantum Field Theory

Horuzhy, S. S.

doi:10.1007/978-94-009-1179-6

Cited by 54 publications

(45 citation statements)

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“…This type of quantum theory predicts the notorious EPR correlations [10][Chapter 9]. Relativistic quantum field theory (in the so called algebraic approach [11], [7], [1]) is based on probability theory of the form (P(N ), φ), where P(N ) is the projection lattice of a type III von Neumann algebra and φ is a normal state on N . Quantum field theory predicts an abundance of correlations between observables that are localized in spacelike separated spacetime regions.…”

Section: Common Cause Completability and The Common Cause Principlementioning

confidence: 99%

Common cause completability of non classical probability spaces

Gyenis¹,

Rédei²

2016

Belgr Philosop Ann

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We prove that under some technical assumptions on a general, non-classical probability space, the probability space is extendible into a larger probability space that is common cause closed in the sense of containing a common cause of every correlation between elements in the space. It is argued that the philosophical significance of this common cause completability result is that it allows the defence of the Common Cause Principle against certain attempts of falsification. Some open problems concerning possible strengthening of the common cause completability result are formulated. Main resultIn this paper we prove a new result on the problem of common cause completability of non-classical probability spaces. A non-classical (also called: general) probability space is a pair (L, φ), where L is an orthocomplemented, orthomodular, non-distributive lattice and φ : L → [0, 1] is a countably additive probability measure. Taking L to be a distributive lattice (Boolean algebra), one recovers classical probability theory; taking L to be the projection lattice of a von Neumann algebra, one obtains quantum probability theory. A general probability space (L, φ) is called common cause completable if it can be embedded into a larger general probability space which is common cause complete (closed) in the sense of containing a common cause of every correlation in it. Our main result (Proposition 4.2.3) states that under some technical conditions on the lattice L a general probability space (L, φ) is common cause completable. This result utilizes earlier results on common cause closedness of general probability theories ( The main conceptual-philosophical significance of the common cause extendability result proved in this paper is that it allows one to deflect the arrow of falsification directed against the Common Cause Principle for a very large and abstract class of probability theories. This will be discussed in section 5 in the more general context of how one can assess the status of the Common Cause Principle, viewed as a general metaphysical claim about the causal structure of the world. Further sections of the paper are organized as follows: Section 2 fixes some notation and recalls some facts from lattice theory and general probability spaces needed to formulate the main result. Section 3

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Section: Common Cause Completability and The Common Cause Principlementioning

confidence: 99%

Common cause completability of non classical probability spaces

Gyenis¹,

Rédei²

2016

Belgr Philosop Ann

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“…Readers interested in more in depth treatments can consult any number of good texts (e.g. Haag 1992, Horuzhy 1990, and Bratelli and Robinson 1987.…”

Section: The Apparatusmentioning

confidence: 99%

Some Puzzles and Unresolved Issues About Quantum Entanglement

Earman

2014

Erkenn

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Schrödinger (1935) averred that entanglement is the characteristic trait of quantum mechanics. The …rst part of this paper is simultaneously an exploration of Schrödinger's claim and an investigation into the distinction between mere entanglement and genuine quantum entanglement. The typical discussion of these matters in the philosophical literature neglects the structure of the algebra of observables, implicitly assuming a tensor product structure of the simple Type I factor algebras used in ordinary QM. This limitation is overcome by adopting the algebraic approach to quantum physics, which allows a uniform treatment of ordinary QM, relativistic QFT, and quantum statistical mechanics. The algebraic apparatus helps to distinguish several di¤erent criteria of quantum entanglement and to prove results about the relation of quantum entanglement to two additional ways of characterizing the classical vs. quantum divide, viz. abelian vs. non-abelian algebras of observables, and the ability vs. inability to interrogate the system without disturbing it. Schrödinger's claim is reassessed in the light of this discussion. The second part of the paper deals with the relativity-to-ambiguity threat: the entanglement of a state on a system algebra is entanglement of the state relative to a decomposition of the system algebra into subsystem algebras; a state may be entangled with respect to one decomposition but not another; hence, unless there is some principled way to choose a decomposition, entanglement is a radically ambiguous notion. The problem is illustrated in terms a Realist vs. Pragmatist debate, the former claiming that the decomposition must correspond to real as opposed to virtual subsystems, while the latter claims that the real vs. virtual distinction is bogus and that practical considerations can steer the choice of decomposition. This debate is applied to the fraught problem of measuring entanglement for indistinguishable particles. The paper ends with some (intentionally in ‡ammatory) remarks about claims in the philosophical literature that entanglement undermines the separability or independence of subsystems while promoting holism.

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“…Algebraic quantum field theory exists in two versions: the Haag-Araki theory which uses von Neumann algebras on a Hilbert space, and the HaagKastler theory which uses abstract C*-algebras (cf. [9]). Here we examine no-cloning theorem in the Haag-Kastler theory.…”

Section: Preliminarymentioning

confidence: 99%

“…The following assumptions are made in the Haag-Kastler theory ( [8], cf. [9]). Relativistic covariance Let g = (Λ, a) denote a Poincaré transformation x ∈ M → Λx+a ∈ M, where a ∈ M is the amount of space-time translation and Λ is a Lorentz transformation.…”

Section: Preliminarymentioning

confidence: 99%

Imperfect Cloning Operations in Algebraic Quantum Theory

Kitajima

2014

Found Phys

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No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal ǫ-imperfect cloning operation which tolerates a finite loss ǫ of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal ǫ-imperfect cloning operation in the case where the loss of fidelity is less than 1/4. Therefore in this case no universal ǫ-imperfect cloning operation is possible in algebraic quantum theory.

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Introduction to Algebraic Quantum Field Theory

Cited by 54 publications

References 0 publications

Common cause completability of non classical probability spaces

Common cause completability of non classical probability spaces

Some Puzzles and Unresolved Issues About Quantum Entanglement

Imperfect Cloning Operations in Algebraic Quantum Theory

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