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Classical Topology and Quantum States
Classical Topology and Quantum States
Date: 23 May 2011, 03:46

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The topology of the configuration space for many classical systems of physical importance, such as the free particle and the harmonic oscillator in R-", happens to be very simple. The conventional exposition of classical and quantum mechanics generally concentrates on such systems, explains their quantization and goes on to show that there is a representation of wave functions as functions on the configuration space Q. However, already in 1931, Dirac understood in his work on magnetic monopoles that there are problems of interest where it is not natural to regard wave functions as functions on Q. Subsequent developments, especially in recent years, have made it clear that there are numerous important physical theories, all distinguished by a nontrivial topology of Q, where too it is not appropriate to regard wave functions as functions on Q. Rather they are to be thought of as special kinds of functions on a principal fibre bundle over Q or as sections of an associated vector bundle.

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