Fermionic Equations Research Articles

Relativistic free fermions in an elastic medium with screw dislocations

We study the Dirac equation for free fermions in an elastic medium with a linear topological defect or distortion analogous to a screw dislocation. We obtain its wave functions for free particle and the corresponding energy spectrum, as well as the fermionic current density. Furthermore, we provide the energy–momentum density for four well-known forms: Einstein, Landau–Lifshitz, Papapetrou and Bergmann–Thomson.

Quantum BGK Model near a Global Fermi--Dirac Distribution

In this paper, we consider the existence and asymptotic behavior of the fermionic quantum BGK model, which is a relaxation model of the quantum Boltzmann equation for fermions. More precisely, we e...

Einstein\u2013Cartan\u2013Dirac gravity with U(1) symmetry breaking

Einstein–Cartan theory is an extension of the standard formulation of General Relativity where torsion (the antisymmetric part of the affine connection) is non-vanishing. Just as the space-time metric is sourced by the stress-energy tensor of the matter fields, torsion is sourced via the spin density tensor, whose physical effects become relevant at very high spin densities. In this work we introduce an extension of the Einstein–Cartan–Dirac theory with an electromagnetic (Maxwell) contribution minimally coupled to torsion. This contribution breaks the U(1) gauge symmetry, which is suggested by the possibility of a torsion-induced phase transition in the early Universe, yielding new physics in extreme (spin) density regimes. We obtain the generalized gravitational, electromagnetic and fermionic field equations for this theory, estimate the strength of the corrections, and discuss the corresponding phenomenology. In particular, we briefly address some astrophysical considerations regarding the relevance of the effects which might take place inside ultra-dense neutron stars with strong magnetic fields (magnetars).

Bilinearization and Soliton Solutions of a Supersymmetric Multicomponent Coupled Dispersionless Integrable System

We present a supersymmetric generalization of a multicomponent coupled dispersionless (SUSY-MCD) integrable system. Expanding the superfields of the SUSY-MCD system, we obtain the component fermionic and bosonic held equations. We propose a bilinear form of the SUSY-MCD system and find soliton solutions of it.

Light composite fermions from holography

Motivated by Beyond the Standard Model theories of composite fermions or top partners, we propose a holographic mechanism that generates light baryonic states in a strongly coupled gauge theory. The starting point are the fermionic fluctuations of massive probe branes embedded into AdS5× S5. We first consider the D3/probe D7-brane system. We derive in detail the fermionic fluctuation equations and show the supersym- metric degeneracy of the mesinos with the mesons. Here we view the fermionic mesinos as potential realizations of composite fermions or top partners. We then add higher dimension operators and study their impact on the mesino spectrum. In particular we show that the ground state mesino mass can be pushed to an arbitrarily light value by a suitable choice of the coupling of the higher dimension operator, g. No matter the value of g, the masses of higher excited states never fall below the mass of the ground state at g = 0. We also present similar results for the supersymmetric D3/D3 and D3/D5 systems.

A classification of Markovian fermionic Gaussian master equations

We introduce a classification scheme for the generators of open fermionic Gaussian dynamics. We simultaneously partition the dynamics along the following four lines: (1) unitary versus non-unitary, (2) active versus passive, (3) state-dependent versus state-independent, and (4) single-mode versus multi-mode. We find that only nine of these 16 types of dynamics are possible. Using this partition we discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, we show that completely positive dynamics must be either unitary (and so can be implemented without an quantized environment) or active (and so must involve particle exchange with an environment).

Quantum fields in the background spacetime of a polymeric loop black hole

The description of black holes in loop quantum gravity is a hard and tricky task. In this article, we focus on a minisuperspace approach based on a polymerization procedure. We consider the resulting effective metric and study the propagation of quantum fields in this background. The cross sections for scalar particles and fermions are explicitly calculated. The Teukolsky–Chandrasekhar procedure used to derived the fermionic radial equation of motion for usual spacetimes is entirely generalized to a much larger class. The resulting radial equation can be used in quite a lot of other contexts.

The self-coupled Einstein–Cartan–Dirac equations in terms of Dirac bilinears

In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann–Cartan spacetime with torsion; the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein–Cartan–Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit rational expression of the four-vector potential of the gauge field in terms of the spinors. Substitution of this expression into Maxwell’s equations yields the bilinear form of the self-interactive Maxwell–Dirac equations. In the present (purely gravitational) case, the inversion process yields two rational four-vector expressions in terms of Dirac bilinears, which act as gravitational analogues of the electromagnetic vector potential. These ‘potentials’ also appear as irreducible summand components of the connection, along with a traceless residual term of mixed symmetry. When taking the torsion field equation into account, the residual term can be written as a function of the object of anholonomity. Using the local tetrad frame associated with observers co-moving with the Dirac matter, a generic vierbein frame can described in terms of four Dirac bilinear vector fields, normalized by a scalar and pseudoscalar field. A corollary of this is that in regions where the Dirac field is non-vanishing, the self-coupled Einstein–Cartan–Dirac equations can in principle be expressed in terms of Dirac bilinears only.

Stationary Solutions of the Second-Order Equation for Fermions in Kerr–Newman Space-Time

When using the quantum mechanical second-order equation with the effective potential of the Kerr-Newman (KN) field for fermions, results were obtained that qualitatively differ from results obtained when using the Dirac equation. In presence of two event horizons, existence of degenerate stationary bound states was proved for charged and uncharged fermions with square integrable wave functions vanishing on event horizons. The fermions in such states are localized near the event horizons with the maxima of probability densities away from the event horizons by fractions of the Compton wave length of fermions versus the values of coupling constants, the values of angular and orbital momenta $j,l$ and the value of the azimuthal quantum number $m_{\varphi}$. In the case of extreme KN fields, absence of stationary bound states of fermions was shown for any values of coupling constants. Existence of discrete energy spectra was shown for charged and uncharged fermions in the field of naked KN singularity at definite values of physical parameters. The KN naked singularity poses no threat to cosmic censorship because of the regular behavior of the effective potentials of the KN field in quantum mechanics with the second-order equation.

Second-Order Equations for Fermions on Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman Space–Times

We obtain relativistic self-adjoint second-order equations for fermions in Schwarzschild, Reissner–Nordstrom, Kerr, and Kerr–Newman gravitational and electromagnetic fields. Second-order equations with effective potentials and spinor wave functions extend opportunities for obtaining regular solutions of quantum mechanics equations for spin-1/2 particles.

Equilibration in fermionic systems

The time evolution of a finite fermion system towards local statistical equilibrium is investigated using analytical solutions of a nonlinear partial differential equation that had been derived earlier from the Boltzmann collision term. The solutions of this fermionic diffusion equation are rederived in closed form, evaluated exactly for simplified initial conditions, and applied to hadron systems at low energies in the MeV-range, as well as to quark systems at relativistic energies in the TeV-range where antiparticle production is abundant. Conservation laws for particle number including created antiparticles, and for the energy are discussed.

Consistent relativistic chiral kinetic theory: A derivation from on-shell effective field theory

We formulate the on-shell effective field theory (OSEFT) in an arbitrary frame and study its reparametrization invariance (RI), which ensures that it is respectful of Lorentz symmetry. In this formulation the OSEFT Lagrangian looks formally equivalent to the sum over light-like velocities of soft collinear effective field theory in the Abelian limit, differences remain in the scale of the gauge fields involved in the two effective theories. We then use the OSEFT Lagrangian expanded in powers of the on-shell energy to derive how the classical transport equations for charged massless fermions are corrected by quantum effects, as derived from quantum field theory. We provide a formulation in a full covariant way, and explain how the consistent form of the chiral anomaly equation can be recovered from our results. We also show how the side jump transformation of the distribution function associated with massless charged fermions can be derived from the RI transformation rules of the OSEFT quantum fields. Finally, we discuss differences in our results with others found in the literature.

Stationary Solutions of Second-Order Equations for Point Fermions in the Schwarzschild Gravitational Field

When using a second-order Schr\"odinger-type equation with the effective potential of the Schwarzschild field, existence of a stationary state of half-spin particles with energy $E=0$ is proved. For each of the values of quantum numbers $j,l$, the physically meaningful energy $E=0$ (the binding energy is $E_{b}=mc^{2}$) is implemented at the value of the gravitational coupling constant $\alpha\geq\alpha_{min}$. The particles with $E=0$ are, with the overwhelming probability, at some distance from the event horizon within the range from zero to several fractions of Compton wavelength of a fermion depending on value of the gravitational coupling constants and values $j,l$. In this paper, similar solutions of the second-order equation are announced for bound states of fermions in the Reissner-Nordstr\"om, Kerr, Kerr-Newman fields. Atomic-type systems: collapsars with fermions in bound states are proposed as particles of dark matter.

Stationary Solutions of Second-Order Equations for Fermions in Reissner–Nordström Space-Time

Existence of degenerate stationary bound states with square integrable radial wave functions was proved when second-order equations are used with the effective potential of the Reissner-Nordstr\"{o}m (RN) field with two event horizons for charged and uncharged fermions. The fermions in such states are localized near event horizons within the ranges from zero to several fractions of Compton wave length of fermions versus the values of gravitational and electromagnetic coupling constants and the values of angular and orbital momenta $j,l$. In case of extreme RN fields, absence of stationary bound states of fermions with the energies of $E<mc^{2}$ is shown for solutions of the second-order equation for any value of gravitational and electromagnetic coupling constants. Existence of the discrete energy spectrum is shown for the naked RN singularity due to solution of the second-order equation at definite values of physical parameters. The discrete spectrum exists for both charged and uncharged fermions. The naked RN singularity in quantum mechanics with the second-order equation for half-spin particles poses no threat to cosmic censorship since it is covered with an infinitely large potential barrier. Electrically neutral systems of atomic type (RN collapsars with the definite number of fermions in degenerate bound states) are proposed to consider as particles of dark matter.

Composite fermions and their pair states in a strongly-coupled Fermi liquid

Our goal is to understand the phenomena arising in optical lattice fermions at low temperature in an external magnetic field. Varying the field, the attraction between any two fermions can be made arbitrarily strong, where composite bosons form via so-called Feshbach resonances. By setting up strong-coupling equations for fermions, we find that in spatial dimension $d>2$ they couple to bosons which dress up fermions and lead to new massive composite fermions. At low enough temperature, we obtain the critical temperature at which composite bosons undergo the Bose-Einstein condensate (BEC), leading to BEC-dressing massive fermions. These form tightly bound pair states which are new bosonic quasi-particles producing a BEC-type condensate. A quantum critical point is found and the formation of condensates of complex quasi-particles is speculated over.

Non-Relativistic Limit of Neutron Beta-Decay Cross-Section in the Presence of Strong Magnetic Field

One of the most important reactions of the URCA that lead to the cooling of a neutron star, is neutron beta-decay ( ). In this research, the energy spectra and wave functions of massive fermions taking into account the Anomalous Magnetic Moment (AMM) in the presence of a strong changed magnetic field are calculated. For this purpose, the Dirac-Pauli equation for charged and neutral fermions is solved by Perturbation and Frobenius series method, respectively. The results of the Frobenius series method are in good agreement with the results of Nikiforov-Uvarov method (NU). In continuous, using the calculated wave functions, the general relation of neutron decay cross-section in the non-relativistic limit has been obtained. This relation has been derived by the four-fermion Lagrangian within the framework of the standard model of weak interactions. These calculations from the perspective of nuclear astrophysics can be important.

Self-Adjoint Second-Order Equations for Fermions and the Foldy–Wouthuysen Representation

Self-adjoint second-order equations with spinor wave functions in an external electromagnetic field are obtained for fermions. The relationship is found between these equations and the Dirac equations in the Foldy–Wouthuysen representation. Closed chiral symmetric equations are written for left and right spinors by using the Dirac matrices in chiral representation. A closed chiral symmetric Foldy–Wouthuysen equation is obtained for an arbitrary electromagnetic field. There are no summands mixing left and right spinors in these equations irrespective of the presence or absence of fermion mass.

Electronic Properties of Curved Few-Layers Graphene: A Geometrical Approach

We show the presence of non-relativistic Lévy-Leblond fermions in flat three- and four-layers graphene with AB stacking, extending the results obtained in Cariglia et al. 2017 for bilayer graphene. When the layer is curved we obtain a set of equations for Galilean fermions that are a variation of those of Lévy-Leblond with a well defined combination of pseudospin, and that admit Lévy-Leblond spinors as solutions in an approriate limit. The local energy of such Galilean fermions is sensitive to the intrinsic curvature of the surface. We discuss the relationship between two-dimensional pseudospin, labelling layer degrees of freedom, and the different energy bands. For Lévy-Leblond fermions, an interpretation is given in terms of massless fermions in an effective 4D spacetime, and in this case the pseudospin is related to four dimensional chirality. A non-zero energy band gap between conduction and valence electronic bands is obtained for surfaces with positive curvature.

Coupled kinetic equations for fermions and bosons in the relaxation-time approximation

Kinetic equations for quarks and gluons are solved numerically in the relaxation time approximation for the case of one-dimensional boost-invariant geometry. Quarks are massive and described by the Fermi-Dirac statistics, while gluons are massless and obey Bose-Einstein statistics. The conservation laws for the baryon number, energy, and momentum lead to two Landau matching conditions which specify the coupling between the quark and gluon sectors and determine the proper-time dependence of the effective temperature and baryon chemical potential of the system. The numerical results illustrate how a non-equlibrium mixture of quarks and gluons approaches hydrodynamic regime described by the Navier-Stokes equations with appropriate forms of the kinetic coefficients. The shear viscosity of a mixture is the sum of the shear viscosities of quark and gluon components, while the bulk viscosity is given by the formula known for a gas of quarks, however, with the thermodynamic variables characterising the mixture. Thus, we find that massless gluons contribute in a non-trivial way to the bulk viscosity of a mixture, provided quarks are massive. We further observe the hydrodynamization effect which takes place earlier in the shear sector than in the bulk one. The numerical studies of the ratio of the longitudinal and transverse pressures show, to a good approximation, that it depends on the ratio of the relaxation and proper times only. This behaviour is connected with the existence of an attractor solution for conformal systems.

Field equations from Killing spinors

From the Killing spinor equation and the equations satisfied by their bilinears, we deduce some well-known bosonic and fermionic field equations of mathematical physics. Aside from the trivially satisfied Dirac equation, these relativistic wave equations in curved spacetimes, respectively, are Klein-Gordon, Maxwell, Proca, Duffin-Kemmer-Petiau, Kähler, twistor, and Rarita-Schwinger equations. This result shows that, besides being special kinds of Dirac fermions, Killing fermions can be regarded as physically fundamental. For the Maxwell case, the problem of motion is analysed in a reverse manner with respect to the studies of Einstein-Groemer-Infeld-Hoffmann and Jean Marie Souriau. In the analysis of the gravitino field, a generalised 3-ψ rule is found which is termed the vanishing trace constraint.