Local nets and self-adjoint extensions of quantum field operators

Borchers, H. J.; Yngvason, Jakob

doi:10.1007/bf00401649

Cited by 3 publications

(4 citation statements)

References 5 publications

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“…The authors thank C. Capoferri, C. Dappiaggi, S. Mazzucchi and N. Pinamonti for useful discussions. We are grateful to C. Fewster for helpful comments and for pointing out to us reference [3] and to the anonymous referees for their the helpful suggestions.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…If π ω (a) is essentially selfadjoint for every ω, then no issue pops out and we can conclude that the algebraic observable a has a definite meaning as a quantum observable also in the Hilbert space formalism. 3 . However, we cannot a priori exclude the possibility that for some (possibly unnormalized) state ω and some a = a * ∈ A, π ω (a) may admit different selfadjoint extensions.…”

Section: Issuementioning

confidence: 99%

“…Already at the level of selfadjoint observables, commutativity of symmetric operators (say π ω (a 1 ) and π ω (a 2 )) on a dense invariant domain of essential selfadjointness (D ω ) does not imply the much more physically meaningful commutativity of their respective PVMs (as proved by Nelson [24]) and the existence of a joint PVM. These aspects have been investigated in [3], where a stronger positivity condition for the state ω has been imposed to ensure the existence of self-adjoint extensions for the operators π ω (a 1 ), . .…”

Section: Open Issuesmentioning

confidence: 99%

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The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

Drago

Moretti

2020

Lett Math Phys

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We address some usually overlooked issues concerning the use of * -algebras in quantum theory and their physical interpretation. If A is a * -algebra describing a quantum system and ω : A → C a state, we focus in particular on the interpretation of ω(a) as expectation value for an algebraic observable a = a * ∈ A, studying the problem of finding a probability measure reproducing the moments {ω(a n )} n∈N . This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π ω (a) in the GNS representation of ω and thus it has important consequences for the interpretation of a as an observable. We provide physical examples (also from QFT) where the moment problem for {ω(a n )} n∈N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences {ω b (a n )} n∈N , being b ∈ A and ω b (·) := ω(b * · b). Letting µ

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Section: Acknowledgmentsmentioning

confidence: 99%

Section: Issuementioning

confidence: 99%

Section: Open Issuesmentioning

confidence: 99%

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The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

Drago

Moretti

2020

Lett Math Phys

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“…Any realistic experiment involves devices with finite ranges of possible values, thus any outcome should be predictable with arbitrary precision by a theory dealing with bounded operators only, like in the AQFT framework. 10 In the discussion of connection to the Wightman axioms, one asks two questions: whether, starting with a Wightman field smeared out with a compactly supported test function, one can find a self-adjoint bounded operator, and whether, starting with a net of algebras of bounded operators, one can obtain Wightman fields by a limiting process shrinking the regions to points (such questions were studied, e.g., in [324][325][326][327][328][329][330]).…”

Section: Algebraic Qftmentioning

confidence: 99%

Snowmass White Paper: The Quest to Define QFT

Dedushenko¹

2022

Preprint

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This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions that exist in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians.

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Snowmass white paper: The quest to define QFT

Dedushenko

2023

Int. J. Mod. Phys. A

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This paper provides a review of the literature on rigorous definitions and constructions in quantum field theory (QFT), spanning the period of seven decades. Compared with the ideas and constructions found in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians.

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Local nets and self-adjoint extensions of quantum field operators

Cited by 3 publications

References 5 publications

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

Snowmass White Paper: The Quest to Define QFT

Snowmass white paper: The quest to define QFT

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