Pseudo-Complex Field Theory

Abstract

A new formulation of field theory is presented, based on a pseudo-complex description. An extended group structure is introduced, implying a minimal scalar length, rendering the theory regularized a la Pauli-Villars. Cross sections are calculated for the scattering of an electron at an external Coulomb field and the Compton scattering. Deviations due to a smallest scalar length are determined. The theory also permits a modification of the minimal coupling scheme, resulting in a generalized dispersion relation. A shift of the Greisen-Zatsepin-Kuzmin (GZK) limit of the cosmic ray spectrum is the consequence.Pseudo-Complex Field Theory 1645 energy. The GZK cutoff gives us the opportunity of investigating such high energy events. If not observed, at least we can obtain an upper limit on the smallest length scale l.Consequently, the change in the dispersion relation is visible only at high energies, comparable to the GZK scale. At low energies, the dispersion relation is to very high approximation maintained. One may ask, however, if the smallest length l may also produce deviations at intermediate energies, for example, in the TeV range, accessible to experiment now. In order to be measurable, we look for differences in the cross section of a particular reaction, of the lowest power in l possible.The advantage of the proposed extended field theory is obvious: All symmetries are maintained and thus it permits the calculation of cross sections as we are used to. Still, an invariant length scale appears, rendering the theory regularized and reflecting the deviation of the space-time structure at distances of the order of the Planck length.The main objective of this paper is to formulate the pseudo-complex extension of the standard field theory (SFT). For the extension we propose the name Pseudo-Complex Field Theory (PCFT). First results are reported in Ref. 20.The structure of the paper is as follows: In Sec. 2 the pseudo-complex numbers are introduced and it is shown how to perform calculations, like differentiation and integration. This section serves as a quick reference guide to the reader unfamiliar with the concept of pseudo-complex numbers. In Sec. 3 the pseudo-complex Lorentz and Poincaré groups are discussed. The representations of the pseudocomplex Poincaré group are indicated. Section 4 introduces a modified variational procedure, required in order to obtain a new theory and not two separated old ones. The language is still classical. As examples, scalar and Dirac fields are discussed and an extraction procedure, on how to obtain physical observables, is constructed and at the end formally presented. Section 5 is dedicated to the symmetry properties of the PCFT. Finally, in Sec. 6 the quantization formalism is proposed. In Sec. 7 a couple of cross sections are calculated within the PCFT: (i) the dispersion of a charged particle at a Coulomb field and (ii) the Compton scattering. One could also consider high precision measurements, like the Lamb shift and the magnetic moment of the electron. These, however, re...

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