The Levy-Leblond equation

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) 

This is an equation for a 4-component spinor field o analogous to the Dirac 4-component spinor with an upper component po and a lower component xoi both of which are two-component spinors. 

Here we do not follow the original derivation of Levy-Leblond [16], but we derive the LLE as the non-relativistic limit (nrl) of the Dirac equation (DE) [12], and do this at the time-independent level. We hence start by 

The decomposition of the eigenvalue W into a rest mass contribution mc and a remainder E is only meaningful if IT 0 and IE] C mc, i.e. for a state with its energy close to the onset of the positive-energy continuum, in particular for a bound positive-energy state. If we were interested in a state near the negative-energy continuum, we would define E as VT - - mc vide infra). 

We know that for a positive-energy bound state (or an unbound state with W close to me ) the lower component is (except possibly very close to a nucleus, for atoms depending on the sign of k) smaller than the upper component by a factor of 0(c ) (hence the alternative names large and small components for p and respectively - which are only valid for this kind of state). In order to deal with quantities of the same order of magnitude we introduce [33] 

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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... 

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 Levy-Leblond equation one gets a different limit of G z) for 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. 

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... 

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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