Dirac quantisation of systems with quadratic constraints

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Abstract
Finite dimensional dynamical systems with one constraint that is quadratic both in canonical momenta and coordinates are considered. The four simplest cases with non-trivial topology are chosen and the Dirac method of quantisation is applied to them. The Wheeler-DeWitt equation is solved by separation of variables. The resulting wavefunctions are interpreted in the regions where they become semiclassical as well as with the help of the DeWitt probability current. The interpretation of initial data for the Wheeler-DeWitt equation is shown to be incompatible with the interpretation of the semiclassical wavepackets in the cases when there is no time function on the configuration spacetime. Thus, the standard way of constructing a unitary theory, with standard time evolution, within the Dirac method, breaks down in a similar way to the reduction one.

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