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数理解析研究所講究録

Generalized sectors and macroscopic order parameters (Mathematical Study of Quantum Dynamical Systems and Its Application to Quantum Computer)

Izumi Ojima

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equi- librium and non-equilibrium states (see [1] for details).This is achieved by the help of $carrow q$ and $qarrow c$ channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the su- perselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters.

Related papers

A Unified Scheme for Generalized Sectors Based on Selection Criteria: Order Parameters of Symmetries and of Thermality and Physical Meanings of Adjunctions

Izumi Ojima

A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to some basic aspects in measurement processes. This is achieved by the help of c → q and q → c channels: the former determines the states to be selected and to be parameterized by the order parameters, and the latter provides the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parameterized by the variables belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.

arXiv:math-ph/0303009•Open Systems & Information Dynamics•Sep 1, 2003•34 citations

A Unified Scheme for Generalized Sectors based on Selection Criteria. I. Thermal situations, unbroken symmetries and criteria as classifying categorical adjunctions

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A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to the some basic aspects in measurement processes. This is achieved by the help of \textit{c} $\to$ \textit{q} and \textit{q} $\to$ \textit{c} channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the superselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown, as will be reported in a succeeding paper.

arXiv:math-ph/0206034•Jun 20, 2002•0 citations

Entanglement entropy and superselection sectors. Part I. Global symmetries

Horacio Casini, Marina Huerta, Javier M. Magán +1 more

Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.

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The Modular Superselection Framework: A Formal Proposal for Quantum Sector Structures

Lazzari, Giancarlo

This work introduces the Modular Superselection Framework, a mathematical and conceptual proposal based on the assumption that a quantum system may be structured into modular sectors labeled by a conserved quantity defined modulo an integer MMM. In this framework, observable states must remain within a single sector, and transitions between sectors are restricted by the modular selection rule ΔQ≡0(modM)\Delta Q \equiv 0 \pmod{M}ΔQ≡0(modM). The work develops this idea axiomatically, presenting formal definitions, structural propositions, and examples illustrating its internal coherence. Unlike standard superselection rules derived from symmetry groups or environmental decoherence, the modular architecture is introduced as a primitive structural feature of Hilbert space. Its goal is not to describe existing physics, but to provide a mathematically consistent, falsifiable, and expandable theoretical scaffold for future investigations in quantum foundations, mathematical physics, and the study of constrained quantum systems.

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Generalized sectors and macroscopic order parameters (Mathematical Study of Quantum Dynamical Systems and Its Application to Quantum Computer)

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数理解析研究所講究録•Jan 1, 2004•0 citations

Micro-Macro Duality in Quantum Physics

Izumi Ojima

Micro-Macro Duality means here the universal mutual relations between the microscopic quantum world and various macroscopic classical levels, which can be formulated mathematically as categorical adjunctions. It underlies a unified scheme for generalized sectors based upon selection criteria proposed by myself in 2003 to control different branches of physics from a unified viewpoint, which has played essential roles in extending the Doplicher-Haag-Roberts superselection theory to various situations with spontaneously as well as explicitly broken symmetries. Along this line of thought, the state correlations between a system and a measuring apparatus necessary for measurements can canonically be formulated within the context of group duality; the obtained measurement scheme is not restricted to the quantum mechanical situations with finite number of particles but can safely be applied to quantum field theory with infinite degrees of freedom whose local subalgebras are given by type III von Neumann algebras.

arXiv:math-ph/0502038•Stochastic Analysis: Classical and Quantum•Oct 1, 2005•38 citations
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equi- librium and non-equilibrium states (see [1] for details).This is achieved by the help of carrowqcarrow qcarrowq and qarrowcqarrow cqarrowc channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the su- perselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters.
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to some basic aspects in measurement processes. This is achieved by the help of c → q and q → c channels: the former determines the states to be selected and to be parameterized by the order parameters, and the latter provides the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parameterized by the variables belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to the some basic aspects in measurement processes. This is achieved by the help of \textit{c} →\to→ \textit{q} and \textit{q} →\to→ \textit{c} channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the superselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown, as will be reported in a succeeding paper.
Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.
This work introduces the Modular Superselection Framework, a mathematical and conceptual proposal based on the assumption that a quantum system may be structured into modular sectors labeled by a conserved quantity defined modulo an integer MMM. In this framework, observable states must remain within a single sector, and transitions between sectors are restricted by the modular selection rule ΔQ≡0(modM)\Delta Q \equiv 0 \pmod{M}ΔQ≡0(modM). The work develops this idea axiomatically, presenting formal definitions, structural propositions, and examples illustrating its internal coherence. Unlike standard superselection rules derived from symmetry groups or environmental decoherence, the modular architecture is introduced as a primitive structural feature of Hilbert space. Its goal is not to describe existing physics, but to provide a mathematically consistent, falsifiable, and expandable theoretical scaffold for future investigations in quantum foundations, mathematical physics, and the study of constrained quantum systems.
Micro-Macro Duality means here the universal mutual relations between the microscopic quantum world and various macroscopic classical levels, which can be formulated mathematically as categorical adjunctions. It underlies a unified scheme for generalized sectors based upon selection criteria proposed by myself in 2003 to control different branches of physics from a unified viewpoint, which has played essential roles in extending the Doplicher-Haag-Roberts superselection theory to various situations with spontaneously as well as explicitly broken symmetries. Along this line of thought, the state correlations between a system and a measuring apparatus necessary for measurements can canonically be formulated within the context of group duality; the obtained measurement scheme is not restricted to the quantum mechanical situations with finite number of particles but can safely be applied to quantum field theory with infinite degrees of freedom whose local subalgebras are given by type III von Neumann algebras.