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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Definition of a class of transformations leading to equivalent Hamiltonian representations of the Dirac equation

J. F. Roux, Phan Van Loc

Published Sep 1, 1975• 3 citations
Publisher
DOI
10.1007/bf02732241

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A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics

Manuel Gadella, F. Gómez-Cubillo

Foundations of Physics•Jun 1, 2002•55 citations
We define and study the probability current and the Hamiltonian operator for a fully general set of Dirac matrices in a flat spacetime with affine coordinates, by using the Bargmann-Pauli hermitizing matrix. We find that with some weak conditions on the affine coordinates, the current, as well as the spectrum of the Dirac Hamiltonian, thus all of quantum mechanics, are independent of that set. These results allow us to show that the tensor Dirac theory, which transforms the wave function as a spacetime vector and the set of Dirac matrices as a third-order affine tensor, is physically equivalent to the genuine Dirac theory, based on the spinor transformation. The tensor Dirac equation extends immediately to general coordinate systems, thus to non-inertial (e.g. rotating) coordinate systems.
We define and study the probability current and the Hamiltonian operator for a fully general set of Dirac matrices in a flat spacetime with affine coordinates, by using the Bargmann-Pauli hermitizing matrix. We find that with some weak conditions on the affine coordinates, the current, as well as the spectrum of the Dirac Hamiltonian, thus all of quantum mechanics, are independent of that set. These results allow us to show that the tensor Dirac theory, which transforms the wave function as a spacetime vector and the set of Dirac matrices as a third-order affine tensor, is physically equivalent to the genuine Dirac theory, based on the spinor transformation. The tensor Dirac equation extends immediately to general coordinate systems, thus to non-inertial (e.g. rotating) coordinate systems.
A nonrelativistic reduction of the second-order Dirac equation is investigated. An equation K2+[Π0; K−Π̂]=m2+Π̂2 is obtained for a ‘‘kinetic Hamiltonian’’ K such that Φ obeys a nonrelativistic Schrödinger-type equation Π0Φ=KΦ in addition to obeying the second-order Dirac equation. Non-self-adjoint terms in K are removed by a change of representation from Φ to ΦT ≡PΦ, and a closed expression P≡[ 1/2 +(K† −Π̂)(K−Π̂)/2(m)2]1/2 is obtained for P. Similar results are obtained for antiparticle states, and a type of ‘‘factorization theorem’’ for the second-order Dirac equation is obtained in which the second-order Dirac equation is replaced by a pair of uncoupled first-order equations. Since nonperturbative techniques are used throughout, the factorization theorem holds independently of the question of convergence of the Foldy–Wouthuysen series and is valid whenever the equation K2+[Π0; K−Π̂]=m2+Π̂2 admits a solution for K. The connection with the usual linear Dirac equation is investigated, and closed expressions are described for a transformation to a representation of that equation which is ‘‘even’’ in the language of Foldy and Wouthuysen. The transformation described is equivalent to two consecutive ‘‘odd’’ unitary transformations.