Ultrarelativistic States

Of course, the Dirac operator for H-like ions has continuum states as well, including ultrarelativistic ones. One can therefore neither expect that all eigenstates are analytic in c , nor that the entire Dirac operator allows an expansion in powers of c. This can at best be the case for the projection of D to positive-energy non-ultrarelativistic states. The paradigm, on which the Foldy-Wouthuysen transformation is based, to construct a Hamiltonian, related to the Dirac operator by a unitary transformation, in an expansion in c ... 

We arrive so at the SE for the lower component in a potential (and also an energy) with opposite sign, i.e. the SE for a positron. If we want to consider an ultrarelativistic state, we do not get a nonrelativistic limit at all. 

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. 

A convergent expansion in powers of c is only possible if we restrict the domain of H to non-ultrarelativistic states, for which... 

Note that eigenstates of D lare also eigenstates of p, i.e. for a free particle p is a constant of motion. Even for a free particle an expansion of the transformed Hamiltonian (after removal of the rest energy) is only possible in a very limited sense. There is divergence for ultrarelativistic states. The nrl is obviously... 

Abstract. The Coulomb interaction which occurs in the final state between two particles with opposite charges allows for creation of the bound state of these particles. In the case when particles are generated with large momentum in lab frame, the Lorentz factors of the bound state will also be much larger than one. The relativistic velocity of the atoms provides the oppotrunity to observe bound states of (-n+fx ), (7r+7r ) and (7x+K ) with a lifetime as short as 10-16 s, and to measure their parameters. The ultrarelativistic positronium atoms (.4oe) allow us to observe the effect of superpenetration in matter, to study the effects caused by the formation time of A e. from virtual e+e pairs and to investigate the process of transformation of two virtual particles into the bound state. 

At present, the ultrarelativistic A2e have been only observed in the process in which a time-like photon 7 is converted into a bound state of the electron and positron (Fig. 6(c)). If in the radiative process (Fig. 6(a)) where a, b are the particles and 7 is the photon... 

In this case the probability of the passage of an atom through a layer of matter becomes greater than the one that follows from the usual exponential dependence. This phenomenon, superpenetration of ultrarelativistic A.2e, allows for measurement of the time of conversion of a non-stationary state of e+e, formed in the target, to stationary states and to verify the form of the Lorentz transformations for the time [8]. The theory of superpenetration has been formulated in [9,10,11]. A quantitative calculation shows that even for a film thickness L = 2.5A the deviation from an exponential absorption law reaches 100%. 

Figure 1.10 Spectrum of positrons in the pi/2 (m = —1/2) state at b — 530 fm calculated at the ultrarelativistic limit (solid curve), with the finite-element method (short dashes) and with perturbation theory (longer dashes).

Fortunately, if we use DPT, the Brown-Ravenhall disease does not show up at all, because we start from an n-electron state in the nrl, and there is no chance for unphysical positronic or ultrarelativistic components to mix in. This is easily seen in the quasidegenerate formalism, where we insist on relating the upper and lower components via the correct X-operator for electrons, which clearly takes care that we only move in the world of... 

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