Dimensional reduction of the Dirac theory

Abstract

We perform a reduction from three to two spatial dimensions of the physics of a spin-1/2 fermion coupled to the electromagnetic field, by applying Hadamard's method of descent. We consider first the free case, in which motion is determined by the Dirac equation, and then the coupling with a dynamical electromagnetic field, governed by the Dirac-Maxwell equations. We find that invariance along one spatial direction splits the free Dirac equation in two decoupled theories. On the other hand, a dimensional reduction in the presence of an electromagnetic field provides a more complicated theory in 2+1 dimensions, in which the method of descent is extended by using the covariant derivative. Equations simplify, but decoupling between different physical sectors occurs only if specific classes of solutions are considered.

Dimensional reduction of the Dirac equation in arbitrary spatial dimensions

“…i.e., by putting the gamma matrices inside the box and the chiral matrix outside, (85) the latter is diagonal: by adopting such representations, the decoupled Dirac theories obtained by performing dimensional reduction on the Dirac equation on an even-dimensional spacetime are mapped into each other via reflection of one spatial coordinate. This generalizes the results already obtained in [17] for the Dirac equations in (3 + 1)-and (2 + 1)-dimensional spacetimes. A further descent along the last spatial coordinate finally yields two equivalent theories.…”

“…Carrying forward this line of research, here we will discuss the dimensional reduction for the free Dirac equation in d dimensions, with d ≥ 2 (hereafter, d denotes the total number of coordinates, including the temporal one). We will generalize and complete the results found in [17] on the Dirac equation in a (3+1)-and a (2+1)-dimensional spacetime, by showing that such a procedure is indeed compatible with the whole hierarchy of Dirac equations in all dimensions d, in the following sense: In this sense, when "descending" along the family of Dirac equations with different space dimensionality, a modulo 2 periodicity is observed. This simple and elegant result can be obtained either via a more "algebraic" approach, or in a "constructive" way, that is, by constructing and working into suitable representations in which the picture underlined above is particularly simple.…”

“…This is the hard case, including as a notable instance the reduction of the ordinary Dirac equation from 4 to 3 spacetime dimensions that has been investigated in [17]. The treatment we will give here of the even-to-odd descent is a generalization of that special case.…”

“…The restricted covariance of the decompositions (33) as well as of the corresponding block structures (39)-( 40) is ensured by the fact that κ 2 −1 is a scalar with respect to the (2 − 1)-dimensional restricted Lorentz group (17), that is, κ 2 −1 commutes with all its generators i 2 γ μ γ ν . Therefore the x 2 −1 -independent wavefunctions ψ ± depend on the same coordinates, and have the same number of components and Lorentz algebra as a (2 − 1)-dimensional spinor field.…”

“…[16] for another approach, based on boundary conditions). More recently, it has been applied to a Dirac particle, both free and minimally coupled to the electromagnetic field [17]; remarkably, in the interacting case the descent prescription can be "upgraded" to match a gauge-covariance request.…”