Alon E. Faraggi, Marco Matone
We show that if space is compact, then time cannot be defined by Jacobi's theorem in the quantum Hamilton--Jacobi theory. The key point is that time parameterisation t-t_0=\partial_E {S}_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. This provides a stringent connection between the quantum HJ theory and the Copenhagen interpretation. Together with tunnelling and the energy quantization theorem for confining potentials, formulated in quantum HJ theory, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similarly to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limit case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time leading to trajectories in quantum HJ theory having the probabilistic interpretation of the Copenhagen School.
View original:
http://arxiv.org/abs/1211.0798
No comments:
Post a Comment