Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Continuous-time histories: observables, probabilities, phase space structure and the classical limit

Charis Anastopoulos

Published Jan 1, 2008• 18 citations
OpenAlex

Abstract

In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the analogue of the Weyl group) and the implementation of a histories Wigner-Weyl transform. The latter enables us to write quantum theory solely in terms of phase space quantities. These results enable the implementation of an algorithm for identifying the classical (stochastic) limit of a general quantum system.

Related papers

Continuous-time histories: Observables, probabilities, phase space structure and the classical limit

Charis Anastopoulos

The continuous-time histories program stems from the consistent histories approach to quantum theory and aims to provide a fully covariant formalism for quantum mechanics. In this paper we examine some structural points of the formalism. We demonstrate a general construction of history Hilbert spaces and identify a large class of time-averaged observables. We pay particular attention to the construction of the decoherence functional (the object that encodes probability information) in the continuous-time limit and its relation to the temporal structure of the theory. Phase space observables are introduced, through the study of general representations of the history group, which is the analog of the canonical group in the formalism. We can also define a closed-time-path (CTP) generating functional for each observable, which encodes the information of its correlation functions. The phase space version of the CTP generating functional leads to the implementation of Wigner–Weyl transforms, that gives a description of quantum theory solely in terms of phase space histories. These results allow the identification of an algorithm for going to the classical (stochastic) limit for a generic quantum system.

arXiv:quant-ph/0008052•Journal of Mathematical Physics•Aug 1, 2001•21 citations

Quantum correlation functions and the classical limit

Charis Anastopoulos

We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, and quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of the back-reaction of quantum fields in spacetime geometry.

arXiv:gr-qc/0011111•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 24, 2001•15 citations

Phase space representations and perturbation theory for continuous-time histories

Aidan Burch

We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.

arXiv:quant-ph/0701186•Journal of Mathematical Physics•Jul 1, 2007•1 citation

Continuous Time in Consistent Histories

Ntina Savvidou

We discuss the case of histories labelled by a continuous time parameter in the {\em History Projection Operator} consistent-histories quantum theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time averages of the energy. We define the action operator for the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. We show that the action operator is the generator of two types of time transformations that may be related to the two laws of time-evolution of the standard quantum theory: the `state-vector reduction' and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories; we demonstrate how the requirement of the temporal logic structure of the theory is sufficient for the definition of classical histories. Furthermore, we show the relation of the action operator to the decoherence functional which describes the dynamics of the system. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

arXiv:gr-qc/9912076•Dec 17, 1999•13 citations

Quantum fields in nonstatic background: A histories perspective

Charis Anastopoulos

For a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time-dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous-time histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time-averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closed-time-path generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding n-point functions.

arXiv:gr-qc/9903026•Journal of Mathematical Physics•Feb 1, 2000•13 citations

Aspects of Time in Quantum Theory

James M. Yearsley

We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.

arXiv:1110.5790•Oct 1, 2011•4 citations
In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the analogue of the Weyl group) and the implementation of a histories Wigner-Weyl transform. The latter enables us to write quantum theory solely in terms of phase space quantities. These results enable the implementation of an algorithm for identifying the classical (stochastic) limit of a general quantum system.
The continuous-time histories program stems from the consistent histories approach to quantum theory and aims to provide a fully covariant formalism for quantum mechanics. In this paper we examine some structural points of the formalism. We demonstrate a general construction of history Hilbert spaces and identify a large class of time-averaged observables. We pay particular attention to the construction of the decoherence functional (the object that encodes probability information) in the continuous-time limit and its relation to the temporal structure of the theory. Phase space observables are introduced, through the study of general representations of the history group, which is the analog of the canonical group in the formalism. We can also define a closed-time-path (CTP) generating functional for each observable, which encodes the information of its correlation functions. The phase space version of the CTP generating functional leads to the implementation of Wigner–Weyl transforms, that gives a description of quantum theory solely in terms of phase space histories. These results allow the identification of an algorithm for going to the classical (stochastic) limit for a generic quantum system.
We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, and quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of the back-reaction of quantum fields in spacetime geometry.
We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.
We discuss the case of histories labelled by a continuous time parameter in the {\em History Projection Operator} consistent-histories quantum theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time averages of the energy. We define the action operator for the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. We show that the action operator is the generator of two types of time transformations that may be related to the two laws of time-evolution of the standard quantum theory: the `state-vector reduction' and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories; we demonstrate how the requirement of the temporal logic structure of the theory is sufficient for the definition of classical histories. Furthermore, we show the relation of the action operator to the decoherence functional which describes the dynamics of the system. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.
For a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time-dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous-time histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time-averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closed-time-path generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding n-point functions.
We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.