Energy Quantisation and Time Parameterisation

Abstract

We show that if space is compact, then trajectories cannot be defined in the framework of quantum Hamilton--Jacobi equation. The starting point is the simple observation that when the energy is quantized it is not possible to make variations with respect to the energy, and the time parameterisation t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. It should be stressed that this follows directly form the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunneling and the energy quantization theorem for confining potentials, formulated in the framework of quantum HJ equation, 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 limiting 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 parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.

11 pages. The main addition concerns a discussion on the variational principle in the case of discrete energy spectra (Jacobi's Theorem). References added

References in corpus (10)

• Bohmian Mechanics and Quantum Field Theory
• Reconciling Semiclassical and Bohmian Mechanics: I. Stationary states
• Top Quark Mass in Exophobic Pati--Salam Heterotic String Model
• Classical and quantum massive cosmology for the open FRW universe
• Reconstructing the primordial power spectrum from the CMB
• Higher genus superstring amplitudes from the geometry of moduli spaces
• Superstring measure and non-renormalization of the three-point amplitude
• Quantum Hamilton-Jacobi Theory
• Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian
• Modular Invariant Regularization of String Determinants and the Serre GAGA Principle

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• The Möbius Symmetry of Quantum Mechanics
• Universe as Klein-Gordon Eigenstates