Levy-Leblond equation

The limit c —can now be taken provided that (1) V is everywhere non-singular, which is true for finite nuclei [42] but not point nuclei, and that (2) E < c which is true for the (shifted) positive-energy solutions only. With this procedure all relativistic effects are eliminated and one obtains the four-component non-relativistic Levy-Leblond equation [34,43]... 

These operators may be contrasted with the property operators obtained at the non-relativistic level. The Levy-Leblond equation given in (29) is consistent with the rigorous non-relativistic limit in that only scalar potentials are included. We now follow the standard, pragmatic approach and consider the operator form... 

Earlier we mentioned briefly that the electron spin is perfectly consistent with the non-relativistic four-component Levy-Leblond theory [44,45]. The EC type interaction does not manifest in Dirac or Levy-Leblond theory. We shall show that on reducing the four-component Levy-Leblond equation into a two-component form the EC contribution arises naturally. A non-relativistic electron in an electromagnetic radiation field is described by the Levy-Leblond equation given by... 

Fig. 4. The same as in figures 1 and 3, but the values of L and S are not equal to the ones of the exact solutions. In the case of the Dirac equation L = S = i and in the case of the Levy-Leblond equation L = S = y have been taken. The saddlepoint coordinates (tXg, Pq are equal to (114,99) in the Dirac case and (66,84) in the Levy-Leblond case. The cross-sections of the energy surface by the planes a = p (broad solid line), and a = Uq (broken line) are plotted versus P, while the cross-sections by theplane P = Po (broken line with dots) and by the surface P = Pmax (thin solid line) areplotted versus a, in D2 (Dirac) and in L2 (Levy-Leblond). The scale o/P is shown in the horizontal axes. The scale of a has been chosen so that the curve for which P = Po und the onefor which a = Oq match at the saddle point The scale of a may be obtained by adding Oq - Po to the values o/P displayed in the axes of D2 and L2.

The Dirac and the Levy-Leblond equations establish relationships between the large and the small components of the wavefunctions. If these relationships are to be fulfdled by the functions derived from a variational procedure, the basis sets for the large and for the small components have to be constructed accordingly. In particular, the relation... 

Another important feature of the Dirac-Pauli representation is its natural adaptation to the non-relativistic limit. If V —E l << x( then Eq. (2) transforms directly to its non-relativistic counterpart known as the Levy-Leblond equation ... 

No such singularities arise if one uses the direct perturbation theory (DPT) [12, 13, 15], which starts directly from the Dirac equation, with the Levy-Leblond equation [16] as zeroth-order (non-relativistic) approximation. Unfortunately, most practical applications of DPT were so far limited to the leading order, which is particularly easily implemented. This has sometimes led to the unjustified identification of DPT with its lowest order. Higher orders of DPT are straightforward, but have only occasionally been evaluated [17, 18, 19]. Even an infinite-order treatment of DPT is possible [12, 20], where one starts with a non-relativistic calculation and improves it iteratively towards the relativistic result. 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

The Levy-Leblond equation is hardly mentioned in textbooks. One of the rare counterexamples is that of Greiner [37]. 

One should, nevertheless, be aware, that it has been essential for the proof of the holomorphicity of G z) = G z+mc ) to consider this resolvent for values of 2 in the neighborhood of the energies of the bound positive-energy states, i.e. for 2 positronic states, i.e. for 2 = 2 — mc, and no limit at all for ultrarelativistic states. 

We now make a change of the metric (15) in order to prepare the construction of the Levy-Leblond equation. 

The Pauli equation is equivalent to the Levy-Leblond equation in a magnetic field, in the same sense as the Schrodinger equation is equivalent to the Levy-Leblond in the absence of a magnetic field. 

For H-like ions we get the zeroth order upper and lower radial components g(j r) and fo r) from the radial Levy-Leblond equation. We have given them for the Is, 2ps, and Is, 2pi states of H-like ions in section (2.4). From these we obtain immediately 2 as... 

Obviously (460) is the counterpart of the Levy-Leblond equation (21) for two particles. In terms of etc. it reads... 

The nonrelativistic Schrodinger equation can be obtained by applying the analogous elimination of the small component transformation on the Levy-Leblond equation [65,66]. The 4-component L6vy-Leblond equation... 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

The implication of the above arguments is that we have taken the maximum of the second part of Fq and the minimum of the first part. Fq is called the Levy-Leblond functional because the solutions of the Levy-Leblond equation make this functional stationary. The Lagrange multiplier is simply the nonrelativistic energy. 

This is similar to the Levy-Leblond equation, in that there is a zero in the metric and the negative-energy solutions would be pathological. From the second row in the matrix equation we have... 

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