Abstract
Dirac’s formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant, and we calculate the Feynman propagator using the extended phase space description. We show that the Feynman propagator’s quantum phase is given by the boundary term of the canonical transformation of the extended phase space. We propose a new canonical transformation within the extended phase space that leads to a Lewis invariant generalization, and we sketch some possible applications.
Highlights
Time-dependent Hamiltonian systems are broadly used in physics, both in classical and quantum mechanics, with many applications [1,2,3]
We have shown that the Dirac’s formalism for the system given by the action (4) in the extended phase space gives rise to the standard Hamiltonian analysis with Hamiltonian H and coordinates (q, p)
It is worth mentioning that the gauge fixing process is mainly used within the path integral formulation of quantum mechanics, while the additional constraint procedure is commonly used in the canonical quantization scheme in order to derive the Dirac brackets and the reduced phase space
Summary
Time-dependent Hamiltonian systems are broadly used in physics, both in classical and quantum mechanics, with many applications [1,2,3]. The analysis given in [19,20] by Struckmeier lacks some results that might be useful for the quantization of such systems It is not clear what commutation relation should be employed in order to canonically quantize the system or whether there is a more general transformation to that used in [19] leading to the exact invariant of the system. It is broadly used in particle physics as well as in field theory physics since it provides the tools to quantize constrained systems [21,22] Another advantage of using Dirac’s method for time-dependent Hamiltonian systems in the extended phase space is that it paves the way to the path integral analysis of these systems.
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