Abstract
The aim of this chapter is to examine the observables of a quantum system, described on the Hilbert space H, by means of elementary results from the theory of von Neumann algebras. von Neumann algebras will be used as a tool to formalize superselection rules.
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- 1.
According to (3)(d) Exercise 6.13, this is equivalent to requiring \({\mathfrak R} \vee {\mathfrak R}' = {\mathfrak B}({\mathsf H})\).
- 2.
Considering algebraic states instead of normal states defines C ∗ -independence, a notion eligible for generic C ∗-algebras as well.
References
H. Araki, Mathematical Theory of Quantum Fields (Oxford University Press, Oxford, 2009)
O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vols. I, II, 2nd edn. (Springer, Berlin, 2002)
A. Dvurecenskij, Gleason’s Theorem and Its Applications (Kluwer Academic Publishers, Dordrecht, 1992)
G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley-Interscience, New York, 1972)
R. Haag, Local Quantum Physics (Second Revised and Enlarged Edition) (Springer, Berlin, 1996)
J. Hamhalter, Quantum Measure Theory (Springer, Berlin, 2003)
H. Halvorson with an appendix by M. Müger, Algebraic Quantum Field Theory. Philosophy of Physics (Handbook of the Philosophy of Science), vol. 2, ed. by J. Butterfield, J. Earman (North Holland, Amsterdam, 2006)
J.M. Jauch, B. Misra, Supersymmetries and essential observables. Helv. Phys. Acta 34, 699–709 (1961)
R. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Graduate Studies in Mathematics, vols. I,II,III,IV (AMS, Providence, 1997)
K. Landsman, Foundations of Quantum Theory (Springer, New York, 2017)
V. Moretti, Spectral Theory and Quantum Mechanics, 2nd edn. (Springer, Cham, 2018)
M. Redéi, Quantum Logic in Algebraic Approach (Kluwer Academic Publishers, Dordrecht, 1998)
W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New Delhi, 1986)
S.J. Summers, On the Independence of Local Algebras in Quantum Field Theory. Rev. Math. Phys. 02(02), 201–247 (1990)
M. Takesaki, Theory of Operator Algebras I, II, III (Springer, Berlin, 2002–2010)
A.S. Wightman, Superselection rules; old and new. Nuovo Cimento B 110, 751–769 (1995)
J. Yngvason, The role of type III factors in quantum field heory. Rep. Math. Phys. 55(1), 135–147 (2005)
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Moretti, V. (2019). von Neumann Algebras of Observables and Superselection Rules. In: Fundamental Mathematical Structures of Quantum Theory. Springer, Cham. https://doi.org/10.1007/978-3-030-18346-2_6
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