Abstract
Lounesto’s classification of spinors is a comprehensive and exhaustive algorithm that, based on the bilinears covariants, discloses the possibility of a large variety of spinors, comprising regular and singular spinors and their unexpected applications in physics and including the cases of Dirac, Weyl, and Majorana as very particular spinor fields. In this paper we pose the problem of an analogous classification in the framework of second quantization. We first discuss in general the nature of the problem. Then we start the analysis of two basic bilinear covariants, the scalar and pseudoscalar, in the second quantized setup, with expressions applicable to the quantum field theory extended to all types of spinors. One can see that an ampler set of possibilities opens up with respect to the classical case. A quantum reconstruction algorithm is also proposed. The Feynman propagator is extended for spinors in all classes.
Highlights
A more viable approach seems to be the perturbative one
We start the analysis of two basic bilinear covariants, the scalar and pseudoscalar, in the second quantized setup, with expressions applicable to the quantum field theory extended to all types of spinors
In the sections that follow, to start with, we will construct the pseudoscalar and the scalar bilinear covariants, constructed out of such arbitrary second quantized free spinor fields at different points, and analyze them. In this way we come across the bilinears of the expansion coefficients, which are ordinary classical spinors to which the Lounesto classification can be applied. This leads to a new puzzling aspect: the expansion coefficients are classical spinors, but they depend on the momentum, not on the coordinates
Summary
A more viable approach seems to be the perturbative one. From a perturbative point of view one of the basic ingredients of a quantum field theory are the propagators. In order to evaluate a free propagator we need an expansion of the spinor field in plane waves, which in turn requires that the spinor satisfies the Klein–Gordon (KG) equation. In the sections that follow, to start with, we will construct the pseudoscalar and the scalar bilinear covariants, constructed out of such arbitrary second quantized free spinor fields at different points, and analyze them. In this way we come across the bilinears of the expansion coefficients, which are ordinary classical spinors to which the Lounesto classification can be applied. In the sections that follow, we analyze the pseudoscalar and the scalar bilinear covariants, constructed upon arbitrary second quantized spinor fields.
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