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arXiv:math-ph/0502038 · Stochastic Analysis: Classical and Quantum

Micro-Macro Duality in Quantum Physics

Izumi Ojima

Published Oct 1, 2005• 38 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/9789812701541_0012

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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.
After the mathematical notion of "Micro-Macro Duality" for understanding mutual relations between microsopic quantum systems (Micro) and their macroscopic manifestations (Macro) is explained on the basis of the notion of sectors and order parameters, a general mathematical scheme is proposed for detecting the state-structure inside of a sector through measurement processes of a maximal abelian subalgebra of the algebra of observables. For this purpose, the Kac-Takesaki operators controlling group duality play essential roles, which naturally leads to the composite system of the observed system and the measuring system formulated by a crossed product. This construction of composite systems will be shown to make it possible for the Micro to be reconstructed from its observational data as Macro in the light of the Takesaki duality for crossed products.
A unified scheme for quantum measurement processes is formulated on the basis of micro-macro duality as a mathematical expression of the general idea of quantum-classical correspondence. In this formulation, we can naturally accommodate the amplification processes necessary for magnifying quantum state changes at the microscopic end of the probe system into the macroscopically visible motion of the measuring pointer. Its essence is exemplified and examined in the concrete model of the Stern-Gerlach experiment for spin measurement, where the Helgason duality controlling the Radon transform is seen to play essential roles.
On the basis of the mathematical notion of “micro-macro duality” for understanding mutual relations between microsopic quantum systems (micro) and their macroscopic manifestations (macro) in terms of the notion of sectors and order parameters, a general mathematical scheme is proposed for detecting the state-structure inside of a sector through measurement processes of a maximal abelian subalgebra of the algebra of observables. For this purpose, the Kac-Takesaki operators controlling group duality play essential roles, which naturally leads to the composite system of the observed system and the measuring system formulated by a crossed product. This construction of composite systems will be shown to make it possible for the micro to be reconstructed from its observational data as macro in the light of the Takesaki duality for crossed products.
Traditional quantum theory has a very rigid structure, making it difficult to accommodate new properties emerging from novel systems. This book presents a flexible and unified theory for physical systems, from micro and macro quantum to classical. This is achieved by incorporating superselection rules and maximal symmetric operators into the theory. The resulting theory is applicable to classical, microscopic quantum and non-orthodox mixed quantum systems of which macroscopic quantum systems are examples. A unified formalism also greatly facilitates the discussion of interactions between these systems. A scheme of quantization by parts is introduced, based on the mathematics of selfadjoint and maximal symmetric extensions of symmetric operators, to describe point interactions. The results are applied to treat superconducting quantum circuits in various configurations.This book also discusses various topics of interest such as the asymptotic treatment of quantum state preparation and quantum measurement, local observables and local values, Schrodinger's cat states in superconducting systems, and a path space formulation of quantum mechanics.This self-contained book is complete with a review of relevant geometric and operator theories, for example, vector fields and operators, symmetric operators and their maximal symmetric extensions, direct integrals of Hilbert spaces and operators.
Philosophical debate on the measurement problem of quantum mechanics has, for the most part, been confined to the non-relativistic version of the theory. Quantizing quantum field theory, or making quantum mechanics relativistic, yields a conceptual framework capable of dealing with the creation and annihilation of an indefinite number of particles in interaction with fields, i.e. quantum systems with an infinite number of degrees of freedom. I want to show that a solution to the standard measurement problem is available if we exploit the properties of the infinite quantum models available in this broader conceptual framework. There is a qualitative difference between quantum systems with an infinite number of degrees of freedom and quantum systems with a finite number of degrees of freedom. In the infinite case, there exist many unitarily inequivalent irreducible Hilbert space representations of the algebra of observables of the system. In the finite case, there is only one such representation.