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arXiv:1504.03041 · Brazilian Journal of Physics

On the Symplectic Dirac Equation

R. G. G. Amorim, S. C. Ulhoa, Edilberto O. Silva

Published Sep 22, 2015• 5 citations
arXivHTMLPDF
DOI
10.1007/s13538-015-0368-1

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"Symplectic unitary representations for the Poincar ́ e group are studied. The formalism is based on the noncommutative structure of the star–product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quan- tum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. In this context, we derive the Klein–Gordon and Dirac equations. As an application, we study the Dirac equation with electromagnetic interaction in phase space."
A carefully motivated symmetric variant of the Poisson bracket in ordinary (not Grassmann) phase space variables is shown to satisfy identities which are in algebraic correspondence with the anticommutation postulates for quantized Fermion systems. "Symplecticity" in terms of this symmetric Poisson bracket implies generalized Hamilton's equations that can only be of Schroedinger type (e.g., the Dirac equation but not the Klein-Gordon or Maxwell equations). This restriction also excludes the old "four-Fermion" theory of beta decay.
An approach is formulated to the problem of obtaining approximate solutions to many-body quantum mechanics. The starting point is the representation of quantum mechanics as Hamiltonian mechanics on a symplectic manifold (phase space). It is shown that Dirac's variation of an action integral provides a natural mechanism for constraining the dynamics to symplectic submanifolds and gives rise to a hierarchy of approximate many-body theories of which Hartree-Fock, random-phase approximation, time-dependent Hartree-Fock, and the double commutator equations of motion formalism are special cases.