Abstract
Symmetries are introduced into the convexity approach to physics. This allows one to make connections between classical and quantum theories by exploiting the properties of quantum mechanics on phase space. The measurement problem is discussed and many of the known no-go theorems are shown not to apply. Finally, hidden variable theories exhibiting these physical symmetries are shown to have a certain required group structure, if they exist at all.
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References
R. Abraham, J. E. Mardsen, and T. Ratiu,Manifolds, Tensor Analysis, and Applications (Addison-Wesley, Reading, Massachusetts, 1983).
E. Beltrametti, “Trends in foundations of physics,” inStructures and Norms in Science, J. van Benthem, M. L. Dalla Chiara, and D. Mundici, eds. (Kluwer Academic, Dordrecht, 1996).
E. G. Beltrametti and S. Bugajski, “A classical extension of quantum mechanics,”J. Phys. A: Math. Gen. 28, 3329–3343 (1995).
J. A. Brooke and F. E. Schroeck, Jr., “Localization of the photon on phase space,”J. Math. Phys. 37, 5958–5986 (1996).
P. Busch, “A characterization of informationally complete sets of physical quantities,”Int. J. Theor. Phys. 30, 1217–1227 (1991).
P. Busch, G. Cassinelli, and P. J. Lahti, “Probability structures for quantum states,”Rev. Math. Phys. 7, 1105–1121 (1195).
P. Busch, M. Grabowski, and P. J. Lahti,Operational Quantum Physics (Springer, Berlin, 1995).
P. Busch, P. J. Lahti, and P. Mittelstaedt,The Quantum Theory of Measurement, 2nd edn. (Springer, Berlin, 1996).
D. Foulis and F. E. Schroeck, Jr., “Stochastic quantum, mechanics viewed from the language of manuals,”Found. Phys. 20, 823–858 (1990).
D. R. Grigore and O. T. Popp, “The complete classification of generalized homogeneous symplectic manifolds,”J. Math. Phys. 30, 2476–2483 (1989).
V. Guillemin and S. Sternberg,Symplectic Techniques in Physics (Cambridge University Press, New York, 1991).
D. M. Healy Jr. and F. E. Schroeck, Jr., “On informational completeness of covariant localization observables and Wigner coefficients,”J. Math. Phys. 36, 453–507 (1995).
G. C. Hegerfeldt, “Remark on causality and particle localization,”Phys. Rev. D 10, 3320–3321 (1974).
G. W. Mackey, “Imprimitivity for representations of locally compact groups,”Proc. Natl. Acad. Sci. USA 35, 537–545 (1949).
G. W. Mackey,Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963).
T. D. Newton and E. P. Wigner, “Localized states for elementary systems,”Rev. Mod. Phys. 21, 400–406 (1949).
E. Prugovecki, “Information theoretical aspects of quantum measurements,”Int. J. Theor. Phys. 16, 321–331 (1977).
F. E. Schroeck Jr.,Quantum Mechanics on Phase Space (Kluwer Academic, Dordrecht, 1996).
V. S. Varadarajan,Geometry of Quantum Theory, Vol. I (Van Nostrand, Princeton, 1968).
A. S. Wightman, “On the localizability of quantum mechanical systems”Rev. Mod. Phys. 34, 845–872 (1962).
N. Woodhouse,Geometric Quantization (Oxford University Press, Oxford, 1980; 2nd edn., 1992).
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Schroeck, F.E. Symmetry in quantum theory: Implications for the convexity formalism, the measurement problem, and hidden variables. Found Phys 27, 1375–1396 (1997). https://doi.org/10.1007/BF02551518
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DOI: https://doi.org/10.1007/BF02551518