Analytic representation of the square-root operator

Abstract

In this paper, we use the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the square-root energy operator of relativistic quantum theory, which is valid for all values of the spin. We focus on the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, in order to understand how to interpret the operator. Our general representation is uniquely determined by the Green's function for the corresponding Schrödinger equation. We find that, in general, the operator has a representation as a nonlocal composite of (at least) three singularities. In the standard interpretation, the particle component has two negative parts and one (hard core) positive part, while the antiparticle component has two positive parts and one (hard core) negative part. This effect is confined within a Compton wavelength such that, at the point of singularity, they cancel each other providing a finite result. Furthermore, the operator looks like the identity outside a few Compton wavelengths (cutoff). To our knowledge, this is the first example of a physically relevant operator with these properties.When the magnetic field is constant, we obtain an additional singularity, which could be interpreted as particle absorption and emission. The physical picture that emerges is that, in addition to the confined singularities and the additional attractive (repulsive) term, the effective mass of the composite acquires an oscillatory behavior.We also derive an alternate relationship between the Dirac equation (with minimal coupling) and the square-root equation that is much closer than the one obtained via the FoldyWouthuysen method, in that there is no change in the wave function. This is accomplished by considering the scalar potential to be a part of the mass. This approach leads to a new KleinGordon equation and a new square-root equation, both of which have the same eigenvalues and eigenfunctions as the Dirac equation. Finally, we develop a perturbation theory that allows us to extend the range of our theory to include suitable spacetime-dependent potentials.2

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

“…The example of the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, is presented in details in order to understand how to interpret the operator. Note that corresponding results for the RCQM can be found from the FW canonical field theory results [31] with the help of our transformation (see the section 9 in [1]). Using the momentum space representation, the authors of [32] presented an analytical treatment of the one-dimensional spinless Salpeter equation with a Coulomb interaction.…”

“…An extension to confining potentials, which are not in L p (R n ), is also presented. In the paper [31], the authors used the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the square-root energy operator γ 0 − → p 2 + m 2 + V of FW canonical field theory, which is valid for all values of the spin. The example of the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, is presented in details in order to understand how to interpret the operator.…”

“…The transition to the RCQM is given by the transformation (30), (31). Thus, in the quantum-mechanical representation the seven Γ matrices (43) (in the terms of standard Γμ 2N matrices (34), (35)) have the form (38) together with …”

“…This second set is given by the generators p 0 , p from (76) together with the orbital parts of the generators j n , j 0 from (76), (77), respectively. In another way this set follows from the set (27) after the transformation (30), (31).…”

“…Dirac's doubts were overcome in [6,7,14]. Today the significance of the Schrödinger-Foldy equation is confirmed by more then a hundred publications about FW (see, e. g., [31,45,50,[115][116][117][118][119][120][121][122][123][124][125][126][127][128] and the references therein) and the spinless Salpeter equations [8, 13, 20-30, 32, 33, 35, 36], which have wide-range application in contemporary theoretical physics.…”