Ultrarelativistic limit

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).

As we have mentioned above it is possible to evaluate the electromagnetic lepton pair production in the limiting case of infinite Lorentz factors y. One interesting aspect among others is that peripheral heavy-ion collisions at ultrarelativistic energies offer... 

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. 

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... 

In addition, such a power series expansion is, however, only permitted for analytic, i.e., holomorphic functions and must never be extended beyond a singular point. Since the square root occurring in the relativistic energy-momentum relation Ep of Eq. (11.11) possesses branching points at X = p/nteC = i, any series expansion of Ep around the static nonrela-tivistic limit T = 0 is only related to the exact expression for Ep for non-ultrarelativistic values of the momentum, i.e., t < 1. This is most easily seen by rewriting Ep as... 

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