© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:quant-ph/9704031

The Geometry of Consistency: Decohering Histories in Generalized Quantum Theory

D. A. Craig

Published Apr 17, 1997• 8 citations

Abstract

The geometry of decoherence in generalized "consistent histories" quantum theory is explored, revealing properties of the theory that are independent of any particular application of it. It is shown how the decoherence functional of a closed quantum system may be regarded as an Hermitian form on the space of linear operators on the Hilbert space of the system. This identification makes manifest a number of structural properties of decoherence functionals. For example, a bound on the maximum number of histories in a consistent set is determined. When the decoherence functional is positive -- as in conventional quantum mechanics -- it defines a semi-inner product on the space of history operators. This shows that consistent sets of histories are just orthogonal sets in this inner product. It further implies the existence in general of Cauchy-Schwarz and triangle inequalities for positive decoherence functionals. The geometrical significance of the ILS theorem classifying all possible decoherence functionals is illuminated, and a version of the ILS theorem for decoherence functionals on class operators is given. The class of history operators consistent according to a given decoherence functional is found, and, conversely, it is shown how to construct the decoherence functionals according to which a given set of histories is consistent. More generally, the "geometric" point of view here developed supplies a powerful unified language with which to solve problems in generalized quantum theory.

DOI: 10.48550/arxiv.quant-ph/9704031

Related papers

Completeness of Decoherence Functionals

S. Schreckenberg

The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham it is natural to identify the space \\UP of propositions about histories with an orthoalgebra or lattice. When \\UP is given by the lattice of projectors \\PV in some Hilbert space \\V, consistent sets correspond to certain partitions of the unit operator in \\V into mutually orthogonal projectors \\{\\a_1,\\a_2,\\ldots\\}, such that the function d(\\a,\\a) is a probability distribution on the boolean algebra generated by \\{\\a_1,\\a_2,\\ldots\\}. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where \\V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(\\a) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(\\a,\\a)=p(\\a) holds.

•Jan 1, 1995•9 citations

Completeness of decoherence functionals

Stephan Schreckenberg

The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.

arXiv:gr-qc/9502036•Journal of Mathematical Physics•Sep 1, 1995•9 citations

Symmetries of decoherence functionals

S. Schreckenberg

The basic ingredients of the “consistent histories” approach to quantum theory are a space 𝒰𝒫 of “history propositions” and a space 𝒟 of “decoherence functionals.” In this article we consider such history quantum theories in the case where 𝒰𝒫 is given by the set of projectors 𝒫(𝒱) on some Hilbert space 𝒱. Using an analog of Wigner’s theorem in the context of history quantum theories proven earlier, we develop the notion of a “symmetry of a decoherence functional” and prove that all such symmetries form a group which we call “the symmetry group of a decoherence functional.” We calculate for the case of standard quantum mechanics—when looked at from the perspective of the history program—some of these symmetries explicitly and relate them to some discussions that have appeared previously.

arXiv:gr-qc/9607050•Journal of Mathematical Physics•Feb 1, 1997•15 citations

Quantum Histories and Their Implications

Adrian Kent

Classical mechanics and standard Copenhagen quantum mechanics respect subspace implications. For example, if a particle is confined in a particular region $R$ of space, then in these theories we can deduce that it is confined in regions containing $R$. However, subspace implications are generally violated by versions of quantum theory that assign probabilities to histories, such as the consistent histories approach. I define here a new criterion, ordered consistency, which refines the criterion of consistency and has the property that inferences made by ordered consistent sets do not violate subspace relations. This raises the question: do the operators defining our observations form an ordered consistent history? If so, ordered consistency defines a version of quantum theory with greater predictive power than the consistent histories formalism. If not, and our observations are defined by a non-ordered consistent quantum history, then subspace implications are not generally valid.

arXiv:gr-qc/9607073•Jul 30, 1996•9 citations

Quantum prediction algorithms

Adrian Kent, Jim McElwaine

The consistent histories formulation of the quantum theory of a closed system with a pure initial state defines an infinite number of incompatible consistent sets, each of which gives a possible description of the physics. We investigate the possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set. We explain the problems which arise, set out some possible algorithms, and explain their properties with the aid of simple models. Though the discussion is framed in the language of the consistent histories approach, it is intended to highlight the difficulty in making any interpretation of quantum theory based on decoherence into a mathematically precise theory.

arXiv:gr-qc/9610028•Physical Review A•Mar 1, 1997•19 citations

Symmetry and history quantum theory: An analog of Wigner’s theorem

S. Schreckenberg

The basic ingredients of the ‘‘consistent histories’’ approach to quantum theory are a space 𝒰𝒫 of ‘‘history propositions’’ and a space 𝒟 of ‘‘decoherence functionals.’’ In this article we consider such history quantum theories in the case where 𝒰𝒫 is given by the set of projectors 𝒫(𝒱) on some Hilbert space 𝒱. We define the notion of a ‘‘physical symmetry of a history quantum theory’’ (PSHQT) and specify such objects exhaustively with the aid of an analog of Wigner’s theorem. In order to prove this theorem we investigate the structure of 𝒟, define the notion of an ‘‘elementary decoherence functional,’’ and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT ‘‘physically equivalent’’ and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.

arXiv:gr-qc/9607051•Journal of Mathematical Physics•Dec 1, 1996•14 citations

The geometry of decoherence in generalized "consistent histories" quantum theory is explored, revealing properties of the theory that are independent of any particular application of it. It is shown how the decoherence functional of a closed quantum system may be regarded as an Hermitian form on the space of linear operators on the Hilbert space of the system. This identification makes manifest a number of structural properties of decoherence functionals. For example, a bound on the maximum number of histories in a consistent set is determined. When the decoherence functional is positive -- as in conventional quantum mechanics -- it defines a semi-inner product on the space of history operators. This shows that consistent sets of histories are just orthogonal sets in this inner product. It further implies the existence in general of Cauchy-Schwarz and triangle inequalities for positive decoherence functionals. The geometrical significance of the ILS theorem classifying all possible decoherence functionals is illuminated, and a version of the ILS theorem for decoherence functionals on class operators is given. The class of history operators consistent according to a given decoherence functional is found, and, conversely, it is shown how to construct the decoherence functionals according to which a given set of histories is consistent. More generally, the "geometric" point of view here developed supplies a powerful unified language with which to solve problems in generalized quantum theory.

The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham it is natural to identify the space \\UP of propositions about histories with an orthoalgebra or lattice. When \\UP is given by the lattice of projectors \\PV in some Hilbert space \\V, consistent sets correspond to certain partitions of the unit operator in \\V into mutually orthogonal projectors \\{\\a_1,\\a_2,\\ldots\\}, such that the function d(\\a,\\a) is a probability distribution on the boolean algebra generated by \\{\\a_1,\\a_2,\\ldots\\}. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where \\V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(\\a) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(\\a,\\a)=p(\\a) holds.

The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.

The basic ingredients of the “consistent histories” approach to quantum theory are a space 𝒰𝒫 of “history propositions” and a space 𝒟 of “decoherence functionals.” In this article we consider such history quantum theories in the case where 𝒰𝒫 is given by the set of projectors 𝒫(𝒱) on some Hilbert space 𝒱. Using an analog of Wigner’s theorem in the context of history quantum theories proven earlier, we develop the notion of a “symmetry of a decoherence functional” and prove that all such symmetries form a group which we call “the symmetry group of a decoherence functional.” We calculate for the case of standard quantum mechanics—when looked at from the perspective of the history program—some of these symmetries explicitly and relate them to some discussions that have appeared previously.

Classical mechanics and standard Copenhagen quantum mechanics respect subspace implications. For example, if a particle is confined in a particular region

R

of space, then in these theories we can deduce that it is confined in regions containing

R

. However, subspace implications are generally violated by versions of quantum theory that assign probabilities to histories, such as the consistent histories approach. I define here a new criterion, ordered consistency, which refines the criterion of consistency and has the property that inferences made by ordered consistent sets do not violate subspace relations. This raises the question: do the operators defining our observations form an ordered consistent history? If so, ordered consistency defines a version of quantum theory with greater predictive power than the consistent histories formalism. If not, and our observations are defined by a non-ordered consistent quantum history, then subspace implications are not generally valid.

The consistent histories formulation of the quantum theory of a closed system with a pure initial state defines an infinite number of incompatible consistent sets, each of which gives a possible description of the physics. We investigate the possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set. We explain the problems which arise, set out some possible algorithms, and explain their properties with the aid of simple models. Though the discussion is framed in the language of the consistent histories approach, it is intended to highlight the difficulty in making any interpretation of quantum theory based on decoherence into a mathematically precise theory.

The basic ingredients of the ‘‘consistent histories’’ approach to quantum theory are a space 𝒰𝒫 of ‘‘history propositions’’ and a space 𝒟 of ‘‘decoherence functionals.’’ In this article we consider such history quantum theories in the case where 𝒰𝒫 is given by the set of projectors 𝒫(𝒱) on some Hilbert space 𝒱. We define the notion of a ‘‘physical symmetry of a history quantum theory’’ (PSHQT) and specify such objects exhaustively with the aid of an analog of Wigner’s theorem. In order to prove this theorem we investigate the structure of 𝒟, define the notion of an ‘‘elementary decoherence functional,’’ and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT ‘‘physically equivalent’’ and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.