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Strong Time Operators in Algebraic Quantum Mechanics and Quantum Field Theory

Asao Arai

Published Apr 1, 2010• 1 citation
OpenAlex

Abstract

Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described. As for the facts presented in QFT, they have essentially been established in a previous paper (A. Arai, Rev. Math. Phys. 17(2005), 1071–1109).

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Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},
Abstract Arguments by Sorkin (Impossible measurements on quantum fields. In: Directions in general relativity: proceedings of the 1993 International Symposium, Maryland, vol 2, pp 293–305, 1993) and Borsten et al. (Phys Rev D 104(2), 2021. https://doi.org/10.1103/PhysRevD.104.025012 ) establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which unitary operation is performed in a spacelike separated region. Sorkin [1] labels such scenarios ‘impossible measurements’. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use Lüders’ rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the ‘impossible measurements’ problem. Two of the approaches are: a measurement theory based on detector models proposed in Polo-Gómez et al. (Phys Rev D, 2022. https://doi.org/10.1103/physrevd.105.065003 ) and a measurement framework for algebraic QFT proposed in Fewster and Verch (Commun Math Phys 378(2):851–889, 2020). Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT. These morals are about the role that dynamics plays in eliminating ‘impossible measurements’, the abandonment of the operational interpretation of local algebras A(O){\mathcal {A}}(O)A(O) A ( O ) as representing possible operations carried out in region O , and the interpretation of state update rules. Finally, we examine the form that the ‘impossible measurements’ problem takes in histories-based approaches and we discuss the remaining challenges.
The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems. We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math. Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math. Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math. Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math. Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems. We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography. We review the connection with modular theory. The latter provides a way to extend the classical notions to the quantum domain. We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003]. Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model. The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.