Propagators of Charged Particles in External Active Media
In this chapter, we give different representations of the charged particle propagators in an external active environment that will be needed for the analysis of quantum processes. The transformation from one representation to another are provided which ca
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Propagators of Charged Particles in External Active Media
In this chapter, we give different representations of the charged particle propagators in an external active environment that will be needed for the analysis of quantum processes. The transformation from one representation to another are provided which can be useful from a methodological point of view. The exact propagator for an electron in a constant uniform magnetic field as the sum over Landau levels is obtained by the direct derivation by standard methods of quantum field theory from exact solutions of the Dirac equation in the magnetic field. In this chapter, we use the notation for the 4-vectors and their components: X μ = (t, x, y, z); digital indices are used to enumerate the various 4-vectors. Throughout the chapter, all the masses squared are assumed to have small negative imaginary parts, m 2 → m 2 − i.
3.1 Propagators of Charged Particles in a Magnetic Field The magnetic field influence on the particle properties is determined by the specific charge, i. e. by the particle charge and mass ratio. Hence, the charged fermion which is the most sensitive to the external field influence is the electron. The calculations of specific physical phenomena in strong external field are based on the application of Feynman diagram technique generalization. It consists in the following procedure: in initial and final states the electron is described by the exact solution of the Dirac equation in the external field, and internal electron lines in quantum processes correspond to exact propagators that are constructed on the basis of these solutions. The expression for the exact electron propagator in the constant uniform magnetic field was obtained by J. Schwinger [1] in the Fock proper-time formalism [2]; see e.g. [3]. There are another propagator representations given in a number of works. Thus, in Refs. [4, 5] the case was considered of superstrong field and the contribution of the ground Landau level to the electron propagator was obtained. In Ref. [6], see also Ref. [7], the propagator was transformed from the form of Ref. [1] into the sum over Landau levels. Also in Ref. [7] the electron propagator decomposition over the power series of the magnetic field strength was given. A. Kuznetsov and N. Mikheev, Electroweak Processes in External Active Media, Springer Tracts in Modern Physics 252, DOI: 10.1007/978-3-642-36226-2_3, © Springer-Verlag Berlin Heidelberg 2013
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3 Propagators of Charged Particles in External Active Media
In our opinion, it is quite important to know different representations of the electron propagator in the external magnetic field and the conditions of their applicability. There were some examples where misunderstanding of such conditions has led to erroneous papers. Thus, in Refs. [8, 9] the self-energy operator of a neutrino in the magnetic field was calculated by the analysis of the one-loop diagram ν → e− W + → ν. The authors of the paper restricted themselves by consideration of the ground Landau level contribution to the
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