Polar Form of the Dirac Operator in Low-Dimensional Space-Times
We express in polar form the Dirac operator as left-multiplication by an assigned matrix: we use this technique to find its right-resolvent for low-dimensional spaces.
math-ph↗
arXiv subjects
Luca Fabbri
Publications and source records attributed to Luca Fabbri.
A covariant approach to the Dirac field in LRS space-times: the case of coplanar frames
We employ the polar decomposition of the Dirac field to describe it as an effective spinorial fluid. We then construct a $(1+1+2)$ covariant formalism for the Dirac field that avoids the introduction of tetrad fields and Clifford matrices. Within this framework, we analyze the conditions under which a self-gravitating Dirac field can be consistently embedded in Locally Rotationally Symmetric (LRS) space-times of types I, II, and III. In accordance with the LRS symmetry requirements, we extend a previous work by assuming that the velocity and spin vector fields of the Dirac field lie in the planes defined pointwise by the generators of the time-like and space-like congruences, which underlie the $(1+1+2)$ decomposition. We present some analytical and numerical solutions to illustrate the applicability of the proposed framework.
gr-qc↗
A note on spinor fields in spherical symmetry
By employing the polar re-formulation, we show that there are no solutions of the Dirac equations in spherical symmetry when the spinor is required to satisfy the same symmetries as the space-time via the Lie derivative.
math-ph↗
Analytic exact solutions to the nonlinear Dirac equation
We present analytic exact solutions to the nonlinear Dirac equation: they display a ring singularity for the Nambu--Jona-Lasinio nonlinearity and a shell singularity for the Soler nonlinearity. For both cases the size of the singular region is of the order of the Compton length.
math-ph↗
Spinors beyond Dirac, Weyl and Majorana: the Flag-Dipoles
We recall the Lounesto classification of 1/2-spin spinor fields, based on the vanishing of spinorial bilinear quantities: the classes are the regular spinor fields (i.e. the Dirac field), as well as singular spinor fields, also known as flag-dipole spinor fields, admitting two limiting sub-classes, given by the dipole spinors (i.e. the Weyl spinor) and the flagpole spinors (i.e. the Majorana spinor). We discuss each class in terms of its representatives, with particular emphasis upon the flag-dipole spinor fields.
math-ph↗
Quantized Dirac Fields in torsionful gravity: cosmological implications and links with the dark universe
We consider a classical field in square torsion theory as a source of torsion for a quantum fermion field in FLRW metric. In the framework of QFT, we obtain vacuum contributions to the energy-momentum tensor and to the axial current that modify the dynamics of the classical field and the field equations as back-reaction. These contributions lead to a modified classical field and therefore to a modified torsion term $L^\mu$ and expectation value of energy-momentum tensor $T^{\mu\nu}$ on the quantum vacuum, altering the field equations in an interative process. We consider the first step of this process and we find that the vacuum condensate could affect the inflationary phase of the Universe. Higher order terms could impact the dark Universe.
hep-th↗
A covariant approach to the Dirac field in LRS space-times
We use the polar decomposition to describe the Dirac field in terms of an effective spinorial fluid. After reformulating all covariant equations in ``spinorial'' signature $(+ -- )$, we develop a $(1+1+2)$ covariant approach for the Dirac field that does not require the use of tetrad fields or Clifford matrices. By identifying the velocity and spin fields as the generators of time-like and space-like congruences, we examine the compatibility of a self-gravitating Dirac field with Locally Rotationally Symmetric space-times of types I, II, and III. We provide illustrative examples to demonstrate the effectiveness of our construction.
gr-qc↗
A Covariant Framework for Generalized Spinor Dual Structures
In this work, we propose a novel framework for defining the dual structure of a spinor. This construction relies on the basis elements of the Clifford algebra, leading to a covariant structure that embeds the dual. The formulation includes free parameters that may be adjusted to meet specific requirements. Remarkably, it enables the explicit construction of representatives for each class within a recently proposed general classification of spinors. In addition to recovering known results, the formalism paves the way for the development of potential new theories in a manifestly covariant setting.
math-ph↗
Dirac Fields in Hydrodynamic Form and their Thermodynamic Formulation
We consider the theory of spinor fields written in polar form and we re-express it in terms of the so-called $1\!+\!1\!+\!2$ covariant splitting: after this is done for the basic kinematic variables, we proceed to decompose the dynamical equations, both for the case of the Dirac differential field equations and for the case of the energy density tensor. As an explicit example of a real physical application we deal with the hydrogen atom, superconductivity and an analogy with the van der Waals gas.
math-ph↗
Madelung Structure of the Dirac Equation
We consider the Dirac equations in polar form proving that they can equivalently be re-configured into a system of equations consisting of derivatives of the velocity density plus the Hamilton-Jacobi equation, giving the momentum in terms of relativistic quantum potentials (i.e. displaying first-order derivatives of the two degrees of freedom of the spinor field): this system is said to have Madelung structure. Conservation laws, second-order equations and multi-valuedness are also discussed.
math-ph↗
Avoiding singularities with propagating torsion
We consider the torsional completion of the theory of gravity in which the torsion is a propagating axial-vector field interacting with spinor fields: we show how this changes the energy conditions leading to singularity formation being avoided.
gr-qc↗
Classical characters of spinor fields in torsion gravity
We consider the problem of having relativistic quantum mechanics re-formulated with hydrodynamic variables, and specifically the problem of deriving the Mathisson-Papapetrou-Dixon equations from the Dirac equation. The problem will be answered on a general manifold with torsion and gravity. We will demonstrate that when plane waves are considered the MPD equations acquire the form given in [1], but we will also see that in such a form the MPD equations become trivial.
gr-qc↗
Unraveling the Physical Meaning Behind Elko's Dual structure
In this work, we shall analyze the necessity of a proper definition of the dual structure for Elko spinors, and singular spinors in general. We examine in detail why the Dirac dual structure fails and it is not functional for these cases, highlighting all the physical consequences of this misdirection. The approach considered here is different from the one usually taken in current literature. We pinpoint the shortcomings that this dual structure brings to particle interpretation, and propagator structure, as well as the implications in non-locality, and the existence of negative energy levels. This investigation furnishes a background for the successful achievements in the field associated with such proper dual formulation.
hep-th↗
How to Achieve High Spatial Resolution in Organic Optobioelectronic Devices?
Light activated local stimulation and sensing of biological cells offers enormous potential for minimally invasive bioelectronic interfaces. Organic semiconductors are a promising material class to achieve this kind of transduction due to their optoelectronic properties and biocompatibility. Here we investigate which material properties are necessary to keep the optical excitation localized. This is critical to single cell transduction with high spatial resolution. As a model system we use organic photocapacitors for cell stimulation made of the small molecule semiconductors H2Pc and PTCDI. We investigate the spatial broadening of the localized optical excitation with photovoltage microscopy measurements. Our experimental data combined with modelling show that resolution losses due to the broadening of the excitation are directly related to the effective diffusion length of charge carriers generated at the heterojunction. With additional transient photovoltage measurements we find that the H2Pc/PTCDI heterojunction offers a small diffusion length of lambda = 1.5 +/- 0.1 um due to the small mobility of charge carriers along the heterojunction. Instead covering the heterojunction with a layer of PEDOT:PSS improves the photocapacitor performance but increases the carrier diffusion length to lambda = 7.0 +/- 0.3 um due to longer lifetime and higher carrier mobility. Furthermore, we introduce electrochemical photocurrent microscopy experiments to demonstrate micrometric resolution with the pn-junction under realistic aqueous operation conditions. This work offers valuable insights into the physical mechanisms governing the excitation and transduction profile and provide design principles for future organic semiconductor junctions, aiming to achieve high efficiency and high spatial resolution.
physics.app-ph↗
Geometry of spinors: doubly-chiral plane-wave expansion
We employ the polar re-formulation of spinor fields to see in a new light their classification into regular and singular spinors, these last also called flag-dipoles, further splitting into the sub-classes of dipoles and flagpoles: in particular, we will study the conditions under which flagpoles may be solutions of the Dirac field equations. We argue for an enlargement of the plane-wave expansion.
math-ph↗
Euler and Pontryagin currents of the Dirac operator
On differential manifolds with spinor structure, it is possible to express the Euler and Pontryagin currents in terms of tensors that also appear as source in the Dirac equation. It is hence possible to tie concepts rooted in geometry and topology to dynamical characters of quantum matter.
math-ph↗
Dirac field, van der Waals gas, Weyssenhoff fluid, Newton particle
We consider the Dirac field in polar formulation, showing that when torsion is taken in effective approximation the theory has the thermodynamic properties of a van der Waals gas, that when the limit of zero chiral angle is taken the theory reduces to that of a Weyssenhoff fluid, and that under the spinless condition it gives the Newtonian particle. This nesting of approximations will allow us to interpret the various objects pertaining to the spinor, with torsion providing a form of negative pressure, and the chiral angle being related to a type of temperature.
physics.gen-ph↗
Polar form of Dirac fields: implementing symmetries via Lie derivative
We consider the Lie derivative along Killing vector fields of the Dirac relativistic spinors: by using the polar decomposition we acquire the mean to study the implementation of symmetries on Dirac fields. Specifically, we will become able to examine under what conditions it is equivalent to impose a symmetry upon a spinor or only upon its observables. For one physical application, we discuss the role of the above analysis for the specific spherical symmetry, obtaining some no-go theorem regarding spinors and discussing the generality of our approach.
math-ph↗