No-Go theorems and quantization
Keywords:Quantization, Groenewold van-Hove theorem, entanglement, contextuality,
AbstractIn this review, we would like to highlight the three known no-go theorems in quantum physics in relation to the process
of quantization that maps classical observables to quantum ones. The quantization approach considered is a mixture of
Isham’s group-theoretic quantization and geometric quantization with special emphasis on underlying compact phase
space geometry of spheres. The first is Groenewold-van Hove theorem that states the obstruction of quantizing the full
algebra of observables and in the sphere case, only limited to the spin observables plus the constant functions. The
other two are theorems of Bell and Kochen-Specker stating that the only hidden variable theories allowed by quantum
physics are nonlocal and contextual ones. We give simple examples of these no-go theorems and indicate some
interesting problems arising from them for the field of quantization.
N.P. Landsman, “Between classical and quantum”, (arXiv: quant-ph/0506082 v2) to appear in Handbook of Philosophy of Science Vol. 2: Philosophy of Physics, (eds.) J. Earman & J. Butterfield, Elsevier.
G. Auletta, Foundations and Interpretation of Quantum Mechanics: In the Light of a Critical-Historical Analysis of the Problems and of a Synthesis of the Results, Rev. Edition, World Scientific, Singapore, 2001.
H.J. Groenewold, Physica 12, (1946) 405.
L. van Hove, Proc. Roy. Acad. Sci. Belgium 26, (1951) 1.
J.S. Bell, Physica 1, (1964) 195.
S. Kochen & E. Specker, J. Math. Mech. 17, (1967) 59-87.
P.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, London, 1958.
N.E. Hurt, Geometric Quantization in Action, D. Reidel, Dordrecht, 1983.
N.M.J. Woodhouse, Geometric Quantization, 2nd Edition, Clarendon Press, Oxford, 1992.
C.J. Isham, “Topological and global aspects of quantum theory” in Relativity, Groups and Topology II,
(eds.) B.S. deWitt & R. Stora, North-Holland, Amsterdam, 1984.
G.W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, Benjamin/Cummings, Reading, 1978.
S.T. Ali & M. Englis, Rev. Math. Phys. 17, (2005) 391.
R.P. Feynman & A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965.
J. Glimm & A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987.
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz & D. Sternheimer, Ann. Phys. 110, (1978) 61, 111.
M.A. Rieffel, Comm. Math. Phys. 122, (1989) 531.
G. Lusztig, Introduction to Quantum Groups, Birkhauser, Boston, 1993.
N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer-Verlag, Berlin, 1998.
M.J. Gotay & H. Grundling, Rep. Math. Phys. 40, (1997) 107.
M.J. Gotay, H. Grundling & C.A. Hurst, Trans. Amer. Math. Soc. 348, (1995) 1579.
M.J. Gotay, H. Grundling & G.T. Tuynman, J. Nonlinear Sci. 6 (1996) 469.
M.J. Gotay, “Obstructions to Quantization”, (arXiv: math-ph/9809011).
M.J. Gotay, “On full quantization of the torus” in Quantization, Coherent States and Complex Structures, (eds.) J.-P. Antoine et al., Plenum, New York, 1995.
H. Zainuddin, “Groenwold-van Hove Problem and Group-Theoretic Quantization of Hall Systems on Torus”, in Prosiding Persidangan Fizik Kebangsan, PERFIK 2000, 4-5 September 2000, Port Dickson, Universiti
Kebangsaan Malaysia, 2000.
H. Zainuddin, Phys. Rev. D 40, (1989) 636.
M.M. Sinolecka, K. Zyczkowski & M. Kus, Acta Phys. Pol. B 33, (2002) 2081.
Toh Sing Poh & H. Zainuddin, “Understanding Basic Idea of Contextuality Through the Quantum Mechanical Interpretation of Schutte’s Tautology” to appear in Proc. of ITMA S&T 2005, ITMA, UPM, 2005.
H. Zainuddin, M. Zainy, J. Hassan, F.P. Zen, M.Y. Sulaiman & Z.A. Hassan, “Qubit Geometry, Phase Spaces and Quantization” to appear in Proceedings of Conference on Advances in Theoretical Sciences,
International Advanced Technology Congress 2005 (CD Proceedings), 6-8 Dec, 2005, IOI Marriott Hotel, Putrajaya