We obtain a generalization of the relativistic diffusion of Schay and Dudley for particles with spin. The diffusion equation is a classical version of an equation for the Wigner function of an elementary particle. The elementary particle is described by a unitary irreducible representation of the Poincare group realized in the Hilbert space of wave functions in the momentum space. The arbitrariness of the Wigner rotation appears as a gauge freedom of the diffusion equation. The spin is described as a connection of a fiber bundle over the momentum hyperbolic space (the mass-shell). Motion in an electromagnetic field, transport equations and equilibrium states are discussed.
We carry out a constructive review of non-standard solutions of relativistic wave equations. Such solutions are obtained via splitting of relativistic wave equations written in spinor form. All these solutions are also solutions of the Dirac equation and are non-standard because they involve higher-order spinors. The main finding is that non-standard solutions describe decaying states.
We obtain a limit when mass tends to zero of the relativistic diffusion of Schay and Dudley. The diffusion process has the log-normal distribution. We discuss Langevin stochastic differential equations leading to an equilibrium distribution.We show that for the Juttner equilibrium distribution the relativistic diffusion is a linear approximation to the Kompaneetz equation describing a photon diffusion in an electron gas.The stochastic equation corresponding to the Juttner distribution is explicitly soluble. We relate the relativistic diffusion to imaginary time quantum mechanics. Some astrophy…
We show that the relativistic dynamics in a Gaussian random electromagnetic field can be approximated by the relativistic diffusion of Schay and Dudley. Lorentz invariant dynamics in the proper time leads to the diffusion in the proper time. The dynamics in the laboratory time gives the diffusive transport equation corresponding to the Juettner equilibrium at the inverse temperature β^{-1}=mc^{2}. The diffusion constant is expressed by the field strength correlation function (Kubo's formula).
We discuss relativistic dynamics in a random electromagnetic field which can be considered as a high temperature limit of the quantum electromagnetic field in a heat bath (cavity) moving with a uniform velocity w. We derive diffusion approximation for the particle's dynamics generalizing the diffusion of Schay and Dudley. It is shown that the Juttner distribution is the equilibrium state of the diffusion.
We study a relativistic diffusion equation on the Riemannian phase space defined by Franchi and Le Jan. We discuss stochastic Ito (Langevin) differential equations (defining the diffusion) as a perturbation by noise of the geodesic equation. We show that the expectation value of the angular momentum and the energy grow exponentially fast. We discuss drifts leading to an equilibrium. It is shown that the diffusion process corresponding to the Juettner or quantum equilibrium distributions has a bounded expectation value of angular momentum and energy. The energy and the angular momentum tend exp…
We obtain a non-linear generalization of the relativistic diffusion of particles with spin. We discuss diffusion equations whose non-linearity is a consequence of quantum statistics. We show that the assumptions of the relativistic invariance and an interpretation of the solution as a probability distribution substantially restrict the class of admissible non-linear diffusion equations. We consider relativistic invariant as well as covariant frame dependent diffusion equations with a drift. In the latter case we show that there can exist stationary solutions of the diffusion equation besides t…
We discuss transport equations resulting from relativistic diffusions in the proper time. We show that a solution of the transport equation can be obtained from the solution of the diffusion equation by means of an integration over the proper time. We study the stochastic processes solving the relativistic diffusion equation and the relativistic transport equation. We show that the relativistic transport equation for massive particles in the light cone coordinates and for massless particles in spatial momentum coordinates are related to the (generalized) Bessel diffusion which has an analytic …
We consider Wong equations for a particle with a continuous mass spectrum in a random Yang-Mills field approximating the quantum field at finite temperature. We show that particle time evolution can be approximated by a relativistic diffusion. Kubo's generator of the relativistic diffusion is defined as an expectation value of the square of the Liouville operator.
Elementary particles, i.e. the basic constituents of nature, are characterized by quantum recurrences in time. The flow of time of every physical system can be therefore decomposed in elementary cycles of time. This allows us to enforce the local nature of relativistic time, yielding interesting unified descriptions of fundamental aspects of modern physics, as shown in recent publications and reviewed in [arXiv:1305.2802]. Every particle can be regarded as a reference clock with time resolution of the order of its Compton time, typically many orders of magnitude more accurate than the atomic c…
In this paper, based on the principles of gauge/gravity duality and considering the so called \textit{hydrodynamic} limit we compute various charge transport properties for a class of strongly coupled non relativistic CFTs corresponding to $ z=2 $ fixed point whose dual gravitational counter part could be realized as the consistent truncation of certain non relativistic $ Dp $ branes in the non extremal limit. From our analysis we note that unlike the case for the AdS black branes, the charge diffusion constant in the non relativistic background scales differently with the temperature. This sh…
We extend some results of group representation theory and von Neumann algebras to the quaternionic Hilbert space case, proving the double commutant theorem (whose quaternionic proof requires a different procedure) and extend to the quaternionic case a result concerning the classification of irreducible von Neumann algebras. Secondly, we consider elementary relativistic systems in Wigner's view defined as a locally-faithful irreducible strongly-continuous unitary representation of Poincaré group in quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the syst…
In this paper, we discuss dissipation process of the binary mixture gas in the thermally relativistic flow \textcolor{red}{by focusing on the characteristics of the diffusion flux}. As an analytical object, we consider the relativistic rarefied-shock layer problem around the triangle prism. Numerical results of the diffusion flux are compared with the Navier-Stokes-Fourier (NSF) order approximation of the diffusion flux, which is calculated using the diffusion and thermal-diffusion coefficients by Kox \textit{et al}. [Physica A, 84, 1, pp.165-174 (1976)]. In the case of the uniform flow with t…
It is shown that the use of extended sets of irreducible representations of the Lorentz group opens new possibilities for the theory of relativistic wave equations from the point of view of the space-time description of both the internal structure and the isospin degrees of freedom of elementary particles. The approach developed in this work also makes it possible to apply the methods of the theory of relativistic wave equations in superstring and gauge models of fundamental interactions.
We review the General Relativistic model of a (quasi) point-like particle represented by a massive shell of electrically charged matter, which displays an ADM mass M equal to the electric charge |Q| in the small-volume limit. We employ the Israel-Darboux's junction equations to explicitly derive this result, and then study the modifications introduced by the existence of a minimum length scale lambda. For lambda of the order of the Planck length (or larger), we find that the ADM mass becomes equal to the bare mass m_0 of the shell, like it occurs for the neutral case.
We conjecture that the random walk and the corresponding diffusion in the relativistic velocity space is an adequate method for describing the acceleration process in relativistic jets. Considering a simple toy model, the main features of diffusion in the velocity space are demonstrated in both non-relativistic and relativistic regimes.
The spectral dimension $d_s$ for high energies is calculated using the Relativistic Schrödinger Equation Analytically Continued (RSEAC) instead of the so-called Telegraph's Equation (TE), in both ultraviolet (UV) and infrared (IR) regimens. Regarding the TE, the recent literature presents difficulties related to its stochastic derivation and interpretation, advocating the use of the RSEAC to properly describe the relativistic diffusion phenomena. Taking into account that the Lorentz symmetry is broken in UV regime at Lifshitz point, we show that there exists a degeneracy in very high energies,…
The spin-3/2 elementary particle, known as Rarita-Schwinger (RS) fermion, is described by a vector-spinor field ψ_{μα}, whose number of components is larger than its independent degrees of freedom (DOF). Thus the RS equations contain nontrivial constraints to eliminate the redundant DOF. Consequently the standard procedure adopted in realizing relativistic spin-1/2 quasi-particle is not capable of creating the RS fermion in condensed matter systems. In this work, we propose a generic method to construct a Hamiltonian which implicitly contains the RS constraints, thus includes the eigenstates a…
We show that a short-time regime, in which a deviation from the exponential decay law occurs, exists also in the framework of a superrenormalizable relativistic quantum field theory. This, in turn, implies the possibility of a quantum Zeno effect also for elementary decays. The attention is then focused on the typical order of magnitude of strong decay rates of mesons: for these particles, strong deviations from the exponential decay law are present during a period of time comparable with their mean life time. As a concrete example, the case of the $ρ$ meson is studied.
