Let us discuss these equations in the Dirac theory.
Passing from the Schroodinger to Heisenberg picture, the time derivative of an operator a is i[??]a = [a, H], and for the expectation values of basic operators of the Dirac theory we obtain the equations
Many people noticed that in the Dirac theory it is possible to construct other operators that obey reasonable equations; see [16, 80].
The Dirac theory
allows us to think of the complex 4-spinors [[psi].sub.a] at each point as indicating the local direction of the local current of the particle corresponding to it.
This equation is describes a particle of spin-1/2 such as the electron in the Dirac theory
with q-deformed hyperbolic Scarf potential including a tensor coupling.
Here Dyson elaborates on his ground-breaking articles, "The radiation theories of Tomonaga, Schwinger and Feynman" and "The S matrix in quantum electrodynamics," working through the Dirac theory
, scattering problems and born approximation, field theory, examples of quantized field theories, free particle scattering problems, the general theory of free particle scattering, and scattering by a static potential with a comparison to experimental results.
Wouthuysen, "On the dirac theory of spin 1/2 particles and its non-relativistic limit," Physical Review A: Atomic, Molecular and Optical Physics, vol.
Ribeiro, "Exact Foldy-Wouthuysen transformation for a Dirac theory with the complete set of," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol.
So the onset radius for electron-position pair production is an important parameter in the Dirac theory
of the electron.
As the Dirac theory
has a vacuum including Zitterbewegung oscillations between positive and negative energy states and perfectly explains an interaction of a Dirac particle with an atomic structure of the materials, it gives a reasonable cosmological solutions for the early-time inflation and late-time acceleration of the universe .
The interaction involves both the charge of the particle and its magnetic moment resulting from its spin magnetic moment (SMM) derived from the Dirac theory
and the quanta of the 4-vector electromagnetic field are spin 1 photons.
For k = 1 the expression for [E.sub.4] is equal to the Darwin term of the Dirac theory
. In the Dirac theory
the Darwin term has to be introduced separately for l = 0 states, whereas in our model [E.sub.4] already provides for l = 0 through the degeneracy (l = 0,1) associated with the k = 1 level.