We consider the theory of spinor fields written in polar form, that is the form in which the spinor components are given in terms of a module times a complex unitary phase respecting Lorentz covariance. In this formalism, spinors can be treated in their most general mathematical form, without the need to restrict them to plane waves. As a consequence, calculations of scattering amplitudes can be performed by employing the most general fermion propagator, and not only the free propagator usually employed in QFT. In this article, we use these quantities to perform calculations in two notable processes, the electron-positron and Compton scatterings. We show that although the methodology differs from the one used in QFT, the final results in the two examples turn out to give no correction as predicted by QFT.
One of the most important mathematical tools necessary for Quantum Field Theory calculations is the field propagator. Applications are always done in terms of plane waves and although this has furnished many magnificent results, one may still be allowed to wonder what is the form of the most general propagator that can be written. In the present paper, by exploiting what is called polar form, we find the most general propagator in the case of spinors, whether regular or singular, and we give a general discussion in the case of vectors.
The commonly-known Chern-Simons extension of Einstein gravitational theory is written in terms of a square-curvature term added to the linear-curvature Hilbert Lagrangian. In a recent paper, we constructed two Chern-Simons extensions according to whether they consisted of a square-curvature term added to the square-curvature Stelle Lagrangian or of one linear-curvature term added to the linear-curvature Hilbert Lagrangian [Ref. 4]. The former extension gives rise to the topological extension of the re-normalizable gravity, the latter extension gives rise to the topological extension of the least-order gravity. This last theory will be written here in its torsional completion. Then a consequence for cosmology and particle physics will be addressed.
In this paper, we perform the polar analysis of the spinorial fields, starting from the regular cases and up to the singular cases: we will give for the first time the polar form of the spinorial field equations for the singular cases constituted by the flag-dipole spinor fields. Comments on the role of further spinor sub-classes containing Majorana and Weyl spinors will be sketched.
We write the most general parity-even re-normalizable Chern-Simons term for massive axial-vector propagating torsion fields. After obtaining the most comprehensive action, we perform the causal structure analysis to see what self-interaction term must be suppressed. In view of such a restriction for the Lagrangian, we will obtain the field equations, investigating some of their properties.
The usual Chern-Simons extension of Einstein gravity theory consists in adding a squared Riemann contribution to the Hilbert Lagrangian, which means that a square-curvature term is added to the linear-curvature leading term governing the dynamics of the gravitational field. However, in such a way the Lagrangian consists of two terms with a different number of curvatures, and therefore not homogeneous. To develop a homogeneous Chern-Simons correction to Einstein gravity we may, on the one hand, use the above-mentioned square-curvature contribution as the correction for the most general square-curvature Lagrangian, or on the other hand, find some linear-curvature correction to the Hilbert Lagrangian. In the first case, we will present the most general square-curvature leading term, which is in fact the already-known re-normalizable Stelle Lagrangian. In the second case, the topological current has to be an axial-vector built only in terms of gravitational degrees of freedom and with a unitary mass dimension, and we will display such an object. The comparison of the two theories will eventually be commented.
Spinor fields are considered in a generally covariant environment where they can be written in the polar form. The polar form is the one in which spinorial fields are expressed as a module times the exponential of a complex pseudo-phase, and in this form the full spinorial field theory can in turn be expressed by employing only real tensorial quantities. Such a reformulation makes it possible to emphasize properties of the spinorial field theory, and this would enrich our understanding in ways that have never been followed up until this moment.
In a series of recent papers, we have introduced an object that was constructed on the connection but which was proven to be a tensor: this object, thus called tensorial connection, has been defined and some of its properties have been given. In the present paper, we intend to present all the results found so far, complementing them with some new ones, in a systematic and organic manner.
In this paper, we consider the theory of ELKO written in their polar form, in which the spinorial components are converted into products of a real module times a complex unitary phase while the covariance under spin transformations is still maintained: we derive an intriguing conclusion about the structure of ELKO in their polar decomposition when seen from the perspective of a new type of adjunction procedure defined for ELKO themselves. General comments will be given in the end.
Spinor fields are written in polar form so as to compute their tensorial connection, an object that contains the same information of the connection but which is also proven to be a real tensor. From this, one can still compute the Riemann curvature, encoding the information about gravity. But even in absence of gravity, when the Riemann curvature vanishes, it may still be possible that the tensorial connection remains different from zero, and this can have effects on matter. This is shown with examples in the two known integrable cases: the hydrogen atom and the harmonic oscillator. The fact that a spinor can feel effects due to sourceless actions is already known in electrodynamics as the Aharonov-Bohm phenomenon. A parallel between the electrodynamics case and the situation encountered here will be drawn. Some ideas about relativistic effects and their role for general treatments of quantum field theories are also underlined.
