Dirac spinor


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Dirac spinor

[di′rak ′spin·ər]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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where the [??] is the modified Dirac spinor, [m.sub.0] is mass of the Dirac particle, [[bar.[sigma]].sup.[mu]](x) are the space-time-dependent Dirac matrices, and [[GAMMA].sub.[mu]](x) are the spin affine connection for spin-1/2 particle [69].
The massive Dirac spinor [PSI] is taken in the Weyl representation; that is, it is described by a pair of two-component spinors, [[phi].sup.A] and [[bar.[chi]].sub.A'], with opposite chirality.
These two waves are a Dirac spinor satisfying the Dirac Equation.
Provided a certain transformational condition is met [i.e., the condition given in equation (28) of [14]], it [[psi]] can be the typical Dirac spinor.
Under the infinitesimal Lorentz transformation [delta][x.sup.[mu]] = [[omega].sup.[mu]] [x.sup.v], the Dirac spinor [PSI] = ([[PSI].sub.1], [[PSI].sub.2], [[PSI].sub.3], [[PSI].sub.4]) transforms as follows:
Also, the relationship between the Weitzenbock spin connection, [mathematical expression not reproducible], and general relativity (Lorentz) spin connection, [mathematical expression not reproducible], becomes as [mathematical expression not reproducible] As the teleparallel gravity is characterized by the vanishing Weitzenbock spin connection (i.e., [mathematical expression not reproducible] [43,44,48], the covariant derivative of the Dirac spinor, [D.sub.[mu]][psi], and its adjoint, [D.sub.[mu]] [bar.[psi]], can be expressed, respectively, as [48, 53]
[f.sub.1], [f.sub.2], [g.sub.1], and [g.sub.2] are the four components of Dirac spinor in the Newman-Penrose formalism.
where N is the normalization constant; on the other hand, the lower component of the Dirac spinor can be calculated from (11) as
In Section 3, we search the bound state solutions for upper and lower component of the Dirac spinor separately and give the normalization constant.
Using the creation and annihilation operators and the raising and lowering operators [[sigma].sup.[+ or -]] = (1/2)([[sigma].sup.x] [+ or -] i[[sigma].sup.y]) Dirac Spinor one can rewrite the previous equation as
In the latter case, the variables nj[pi] (here [pi] denotes parity and it takes the values [+ or -] 1) is an equivalent notation for a relativistic configuration because l = j [+ or -] 1/2 and the numerical parity of the l-value of a Dirac spinor upper part defines the single particle's parity.