The covariance problem and the Hamiltonian formalism in quantum mechanics

Giuseppe Antonio NISTICO'

Wanda Hanus

Olímpia Lombardi, Mario Castagnino, Juan Sebastián Ardenghi

Manuel Gadella, José M. Gracia-Bondı́a, Luis Miguel Nieto +1 more

The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.

U. Kasper, Eckhard Kreisel, H.‐J. Treder

Rafael Ferraro, Claudia Yastremiz, Mario Castagnino

We show that it is possible to formulate the canonical Hamiltonian formalism for the relativistic particle, in a covariant fashion and without some well-known problems, when we introduce the concept of a reference system to establish the notions of time and space. Starting from the observer-dependent Hamiltonian formalism, we obtain the result that the wave function describing the quantum behavior of the particle must be observer dependent and we give the corresponding wave equation.