Dissipation in the Klein-Gordon Field

The formalism of (±)-frequency parts , previously applied to solution of the D'Alembert equation in the case of the electromagnetic field, is applied to solution of the Klein-Gordon equation for the K-G field in the presence of sources. Retarded and advanced field operators are obtained as solutions, whose frequency parts satisfy a complex inhomogeneous K-G equation. Fourier transforms of these frequency parts are solutions of the central equation, which determines the time dependence of the destruction/creation operators of the field. The retarded field operator is resolved into kinetic and dissipative components. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two—a kinetic and a dissipative component. As in the analogous electromagnetic case the dissipation theorem is derived according to which work done by the dissipative power/force is negative: energy/momentum is dissipated from the sources to the K-G field. Boson quantization conditions are satisfied by the kinetic component but not by the dissipative component of the retarded K-G field.     

Comment on “Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics,” by A. Chubykalo et al.

Fundamental errors in a paper by Chubykalo et al. [2] are highlighted. Contrary to their claim that “… the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously,” it is shown that this instantaneous component is physically irrelevant because it is always canceled by a term contained in the solenoidal component. This result follows directly from the solution of the wave equation that satisfies the solenoidal component. Therefore the subsequent inference of these authors that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one retarded and the other one instantaneous, has no basis. The example given by these authors in which the full electric field of an oscillating charge equals its instantaneous irrotational component on the axis of oscillations is proven to be false.     

Stratified quantization approach to dissipative quantum systems: Derivation of the Hamiltonian and kinetic equations for reduced density matrices

A technique for describing dissipative quantum systems that utilizes a fundamental Hamiltonian, which is composed of intrinsic operators of the system, is presented. The specific system considered is a capacitor (or free particle) that is coupled to a resistor (or dissipative medium). The microscopic mechanisms that lead to dissipation are represented by the standard Hamiltonian. Now dissipation is really a collective phenomenon of entities that comprise a macroscopic or mesoscopic object. Hence operators that describe the collective features of the dissipative medium are utilized to construct the Hamiltonian that represents the coupled resistor and capacitor. Quantization of the spatial gauge function is introduced. The magnetic energy part of the coupled Hamiltonian is written in terms of that quantized gauge function and the current density of the dissipative medium. A detailed derivation of the kinetic equation that represents the capacitor or free particle is presented. The partial spectral densities and functions related to spectral densities, which enter the kinetic equations as coefficients of commutators, are evaluated. Explicit expressions for the nonMarkoffian contribution in terms of products of spectral densities and related functions are given. The influence of all two-time correlation functions are considered. Also stated is a remainder term that is a product of partial spectral densities and which is due to higher order terms in the correlation density matrix. The Markoffian part of the kinetic equation is compared with the Master equation that is obtained using the standard generator in the axiomatic approach. A detailed derivation of the Master equation that represents the dissipative medium is also presented. The dynamical equation for the resistor depends on the spatial wavevector, and the influence of the free particle on the diagonal elements (in wavevector space) is stated.     

Efficient cardiac diffusion tensor MRI by three-dimensional reconstruction of solenoidal tensor fields

Tensor tomography is being investigated as a technique for reconstruction of in vivo diffusion tensor fields that can potentially be used to reduce the number of magnetic resonance imaging (MRI) measurements. Specifically, assessments are being made of the reconstruction of cardiac diffusion tensor fields from 3D Radon planar projections using a filtered backprojection algorithm in order to specify the helical fiber structure of myocardial tissue. Helmholtz type decomposition is proposed for 3D second order tensor fields. Using this decomposition a Fourier projection theorem is formulated in terms of the solenoidal and irrotational components of the tensor field. From the Fourier projection theorem, two sets of Radon directional measurements, one that reconstructs the solenoidal component and one that reconstructs the irrotational component of the tensor field, are prescribed. Based on these observations filtered backprojection reconstruction formulae are given for the reconstruction of a 3D second order tensor field and its solenoidal and irrotational components from Radon projection measurements. Computer simulations demonstrate the validity of the mathematical formulations and demonstrate that a realistic model of the helical fiber structure of the myocardial tissue specifies a diffusion tensor field for which the first principal vector (the vector associated with the maximum eigenvalue) of the solenoidal component accurately approximates the first principal vector of the diffusion tensor. A priori knowledge of this allows the orientation of the myocardial fiber structure to be specified utilizing one half of the number of MRI measurements of a normal diffusion tensor field study.     

Hamiltonian Structure for Dispersive and Dissipative Dynamical Systems

We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, “Brillouin-type,” formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave.     

Sources,potentials and fields in Lorenz and Coulomb gauge: Cancellation of instantaneous interactions for moving point charges

We investigate the coupling of the electromagnetic sources (charge and current densities) to the scalar and vector potentials in classical electrodynamics, using Green function techniques. As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time. Starting from the differential equations that couple the sources to the potentials, we here show in a concise derivation, using the retarded Green function, how the instantaneous interaction cancels in the calculation of the electric field. The time derivative of a specific additional term in the vector potential, present only in Coulomb gauge, yields a supplementary contribution to the electric field which cancels the gradient of the instantaneous Coulomb gauge scalar potential, as required by gauge invariance. This completely eliminates the contribution of the instantaneous interaction from the electric field. It turns out that a careful formulation of the retarded Green function, inspired by field theory, is required in order to correctly treat boundary terms in partial integrations. Finally, compact integral representations are derived for the Liénard–Wiechert potentials (scalar and vector) in Coulomb gauge which manifestly contain two compensating action-at-a-distance terms.     

Planar Yang-Mills theory: Hamiltonian,regulators and mass gap

We carry out the Hamiltonian analysis of non-Abelian gauge theories in (2+1) dimensions in a gauge-invariant matrix parametrization of the fields. A detailed discussion of regularization issues and the construction of the renormalized Laplace operator on the configuration space, which is proportional to the kinetic energy, are given. The origin of the mass gap is analyzed and the lowest eigenstates of the kinetic energy are explicitly obtained; these have zero charge and exhibit a mass gap. The nature of the corrections due to the potential energy, the possibility of an improved perturbation theory and a Schrödinger-like equation for the states are also discussed.     

Excited state mass spectra,decay properties and Regge trajectories of charm and charm-strange mesons

The framework of a phenomenological quark-antiquark potential(Coulomb plus linear confinement)model with a Gaussian wave function is used for detailed study of masses of the ground, orbitally and radially excited states of heavy-light Qq,(Q=c,q=u/d,s) mesons. We incorporate a O(1/m) correction to the potential energy term and relativistic corrections to the kinetic energy term of the Hamiltonian. The spin-hyperfine, spin-orbit and tensor interactions incorporating the effect of mixing are employed to obtain the pseudoscalar, vector, radially and orbitally excited state meson masses. The Regge trajectories in the(J,M~2) and(nr,M~2) planes for heavy-light mesons are investigated with their corresponding parameters. Leptonic and radiative leptonic decay widths and corresponding branching ratios are computed. The mixing parameters are also estimated. Our predictions are in good agreement with experimental results as well as lattice and other theoretical models.     

Hamiltonian approach to 1 + 1 dimensional Yang-Mills theory in Coulomb gauge

We study the Hamiltonian approach to 1 + 1 dimensional Yang-Mills theory in Coulomb gauge, considering both the pure Coulomb gauge and the gauge where in addition the remaining constant gauge field is restricted to the Cartan algebra. We evaluate the corresponding Faddeev-Popov determinants, resolve Gauss’ law and derive the Hamiltonians, which differ in both gauges due to additional zero modes of the Faddeev-Popov kernel in the pure Coulomb gauge. By Gauss’ law the zero modes of the Faddeev-Popov kernel constrain the physical wave functionals to zero colour charge states. We solve the Schrödinger equation in the pure Coulomb gauge and determine the vacuum wave functional. The gluon and ghost propagators and the static colour Coulomb potential are calculated in the first Gribov region as well as in the fundamental modular region, and Gribov copy effects are studied. We explicitly demonstrate that the Dyson-Schwinger equations do not specify the Gribov region while the propagators and vertices do depend on the Gribov region chosen. In this sense, the Dyson-Schwinger equations alone do not provide the full non-abelian quantum gauge theory, but subsidiary conditions must be required. Implications of Gribov copy effects for lattice calculations of the infrared behaviour of gauge-fixed propagators are discussed. We compute the ghost-gluon vertex and provide a sensible truncation of Dyson-Schwinger equations. Approximations of the variational approach to the 3 + 1 dimensional theory are checked by comparison to the 1 + 1 dimensional case.     

Solution of the Dirac equation by separation of variables in spherical coordinates for a large class of non-central electromagnetic potentials

A systematic and intuitive approach for the separation of variables of the three-dimensional Dirac equation in spherical coordinates is presented. Using this approach, we consider coupling of the Dirac spinor to electromagnetic four-vector potential that satisfies the Lorentz gauge. The space components of the potential have angular (non-central) dependence such that the Dirac equation becomes separable in all coordinates. We obtain exact solutions for a class of three-parameter static electromagnetic potential whose time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wave functions are obtained. The Aharonov–Bohm and magnetic monopole potentials are included in these solutions.     

Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics

In this work we substantiate the applying of the Helmholtz vector decomposition theorem (H-theorem) to vector fields in classical electrodynamics. Using the H-theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity v (including a velocity more than c). We show also that the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously.     

Discussion About the Magnetic Field Dimensionality,Invariant Axis Condition,and Coulomb Gauge to Solve the Grad-Shafranov Equation

We discuss the relationship between the Coulomb gauge, the existence of an invariant axis, and the dimensionality (2-D or 2\(\frac {1}{2}\)-D) of the magnetic field in a mathematical-physical formalism that leads us to the Grad-Shafranov (GS) equation. In the literature, we found that a 2-D magnetic structure is used as a prerequisite to derive the GS equation from the Vlasov equation. However, other consulted works are based on a 2\(\frac {1}{2}\)-D (two-and-a-half) magnetic structure as a prerequisite to derive the GS equation from the balance of forces between the pressure gradient and the magnetic force, respectively. We replaced the magnetic vector potential on Ampère’s equation and used the Coulomb gauge to obtain a system of three Poisson equations, one for each component. We also used the same procedure explained above, but without the Coulomb gauge. Comparing z-component in both equation systems, we concluded that there are two possible solutions. We suggest using a 2\(\frac {1}{2}\)-D magnetic field configuration instead of a 2-D, when working with kinetic theory or magnetostatic equilibrium to derive the GS equation. We clarified that there is no relationship between the Coulomb gauge and the magnetic field dimensionality. In this problem, the invariant axis condition is imposed, which means that \(\vec {\nabla }\cdot \vec {A}\) is independent of z, i.e., \(\vec {\nabla }\cdot \vec {A}\) could have any value in which an invariant axis is a sufficient condition to obtain the GS equation.     

Retardation effects in ion-ion collisons

The modification of the two center screened electronic Coulomb potential due to relativistic kinematical effects is investigated in the Coulomb gauge. Both nuclear and electronic charges were approximated by Gaussian distributions. For ion velocitiesv/c =0.1 the effect may roughly be approximated by a 0.1 % increase in the effective strength for the monopole term of the two center potential. Thus for ion kinetic energies not exceeding a few MeV/nucleon this relativistic contribution induces small effects on the binding energy of the 1σ-electrons except for super critical charges.     

Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries

The approximate analytical solutions of the Dirac equation under spin and pseudospin symmetries are examined using a suitable approximation scheme in the framework of parametric Nikiforov-Uvarov method. Because a tensor interaction in the Dirac equation removes the energy degeneracy in the spin and pseudospin doublets that leads to atomic stability, we study the Dirac equation with a Hellmann-like tensor potential newly proposed in this study. The newly proposed tensor potential removes the degeneracy from both the spin symmetry and pseudospin symmetry completely. The proposed tensor potential seems better than the Coulomb and Yukawa-like tensor potentials.     

Exact pseudospin symmetry solution of the Dirac equation for spatially-dependent mass Coulomb potential including a Coulomb-like tensor interaction via asymptotic iteration method

In this Letter, the Dirac equation is exactly solved for spatially-dependent mass Coulomb potential including a Coulomb-like tensor potential under pseudospin symmetry limit by using asymptotic iteration method with arbitrary spin-orbit coupling number κ. The energy eigenvalues and corresponding eigenfunctions are obtained and some numerical results are given.     

Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries

The approximate analytical solutions of the Dirac equation under spin and pseudospin symmetries are examined using a suitable approximation scheme in the framework of parametric Nikiforov-Uvarov method. Because a tensor interaction in the Dirac equation removes the energy degeneracy in the spin and pseudospin doublets that leads to atomic stability, we study the Dirac equation with a Hellmann-like tensor potential newly proposed in this study.The newly proposed tensor potential removes the degeneracy from both the spin symmetry and pseudospin symmetry completely. The proposed tensor potential seems better than the Coulomb and Yukawa-like tensor potentials.     

Bipartite entanglement induced by classically-constrained quantum dissipative dynamics

The properties of some complex many body systems can be modeled by introducing in the dissipative dynamics of each single component a set of kinetic constraints that depend on the state of the neighbor systems. Here, we characterize this kind of dynamics for two quantum systems whose independent dissipative evolutions are defined by a Lindblad equation. The constraints are introduced through a set of projectors that restrict the action of each single dissipative Lindblad channel to the state of the other system. Conditions that guarantee a classical interpretation of the kinetic constraints are found. The generation and evolution of entanglement is studied for two optical qubits systems. Classically constrained dissipation leads to a stationary state whose degree of entanglement depends on the initial state. Nevertheless, independently of the initial conditions, a maximal entangled state is generated when both systems are subjected to the action of local Hamiltonian fields that do not commutate with the constraints. The underlying physical mechanism is analyzed in detail.     

Pseudospin Symmetry in Position-Dependent Mass Dirac-Coulomb Problem by Using Laplace Transform and Convolution Integral

The exact pseudospin symmetry solutions of Dirac equation with position-dependent mass (PDM) Coulomb potential in the presence of Colulomb-like tensor potential are obtained by using Laplace transform (LT) approach. The energy eigenvalue equation of the Dirac particles is found and some numerical results are given. By using Laplace convolution integral, the corresponding radial wave functions are presented in terms of confluent hypergeometric functions.     

Warm Gauge-Flation with General Dissipative Coefficient

In this work, we study the effects of generalized dissipative coefficient on the slow-roll inflation driven by non-Abelian gauge field minimally coupled to gravity. The dynamics of warm intermediate and logamediate inflationary models during weak and strong dissipative regimes is analyzed. In both cases, we explore effective scalar potential, slow-roll parameters, scalar and tensor power spectra, scalar spectral index and tensor to scalar ratio under slow-roll conditions. We conclude that our gauge-flationary model with generalized dissipative coefficient remains consistent with the recent data for dissipative parameter m = 3 and m = 1 for weak and strong dissipative eras, respectively.