How to Observe Quantum Fieldsand Recover Them From Observational Data? — Takesaki Duality as a Micro-Macro Duality
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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.
KeywordsIntrinsic Dynamic Quantum Observable Microscopic Quantum Maximal Abelian Subalgebra Invariant Haar Measure
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- 1.I. Ojima, Micro-macro duality in quantum physics, in: Proc. Intern. Conf. on Stochastic Analysis, Classical and Quantum, World Scientific, 2005, pp. 143–161.Google Scholar
- 2.I. Ojima, A unified scheme for generalized sectors based on selection criteria — Order parameters of symmetries and of thermality and physical meanings of adjunctions, Open Sys. Information Dyn. 10, 235 (2003); I. Ojima, Temperature as order parameter of broken scale invariance, Publ. RIMS 40, 731 (2004).Google Scholar
- 3.J. Dixmier, C*-Algebras, North-Holland, 1977; G. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, 1979.Google Scholar
- 4.J. Dixmier, Von Neumann Algebras, North-Holland, 1981.Google Scholar
- 5.M. Ozawa, Quantum measuring processes of continuous observables, J. Math. Phys. 25,79 (1984); M. Ozawa, Publ. RIMS, Kyoto Univ. 21, 279 (1985); M. Ozawa, Ann. Phys. (N.Y.) 259, 121 (1997).Google Scholar
- 6.M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.Google Scholar
- 8.Y. Nakagami and M. Takesaki, Lecture Notes in Mathematics 731, Springer, 1979.Google Scholar
- 9.M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer, 1992.Google Scholar
- 12.M. Takesaki, Theory of Operator Algebras II, Springer-Verlag, 2003.Google Scholar
- 13.M. A. Rieffel, Induced representations of C*-algebras, Adv. Math. 13, 176 (1974); M. A. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure and Appl. Alg. 5, 51 (1974).Google Scholar
- 14.I. Ojima, in preparation.Google Scholar