(Semi)-nonrelativistic limits of the Dirac equation with external time-dependent electromagnetic field
The authors start with the Dirac equation describing a fast electron in an electromagnetic field: \[ i\gamma^\mu\partial_\mu- M+ g\gamma^\mu A_\mu)\psi= 0,\quad \mu= 1,2,3,4, \] and \(\psi\) is a 4-spinorfield vector, \(\psi(t,x)\in \mathbb{C}^4\). \(A_\mu\) denotes the time dependent electromagnetic vector potential. The authors study a semi-nonrelativistic approximation (semi-means within the first-order term of a ``small'' parameter \(\varepsilon= V/c\)) as velocity of light compared to reference velocity approaches infinity. In the proposed scaling the magnetic field appears as a relativistic effect which is not present when the Dirac equation is replaced by the Schrödinger equation (zeroth order approximation). The first-order approximation yields the Pauli equation for the 4-spinor. The authors organize their approach as follows: After a discussion of the pseudodifferential projector operators uniformly bounded in \(\varepsilon\) derive the nonrelativistic limit of the Dirac equation. Then they rescale time which implies subtracting the rest energy from electron and positron components. This permits the authors to avoid making smallness assumption on the initial data. In the next section, they obtain the crucial results for ``positron - small component'' obtained by adding rest energy in the transformation rescaling the time variable. However, more regularity of data and electromagnetic potential is required then for the nonrelativistic limit, and a smallness assumption has to be made for the initial positron component. The approximation of Dirac equation by Pauli equation (within \(\varepsilon^2\)) is derived in the last section of the paper.
regularity of data and electromagnetic potential
scientific article; zbMATH DE number 1294588