Fluctuations in the relativistic quantum plasma

The fluctuations of various parameters of the relativistic quantum plasma are studied with the use of the covariant Wigner function techniques. The fluctuations of the covariant Wigner function of the ideal Fermi gas at thermal equilibrium are calculated. The result is a function of the one particle Wigner function and the anticommutator of the Dirac fields only. As a consequence second order correlation functions of the four-current and the momentum-energy tensor are obtained and analyzed. On the basis of the fluctuation-dissipation theorem, the polarization tensors at first order in e² of the magnetized and the non-magnetized electron gas are derived from the four-current fluctuations of the ideal plasma. Expressions are given for the magnetized vacuum polarization in two representations, integral and discrete sums over Landau levels, for arbitrary values of the photon four-momentum. The dispersion relations of the magnetized gas are studied in the limit of low frequencies and wavenumbers. It is shown that owing to relativistic quantum effects, the characteristic frequencies of the plasma modes (plasma and Larmor frequencies) split into effective transverse and longitudinal frequencies. The existence of an acoustic mode (zero sound) for the one-component plasma is also shown.     

Covariant Wigner function approach to the relativistic charged gas in a strong magnetic field: I. Electron gas in thermal equilibrium

The relativistic quantum electron gas embedded in a strong magnetic field is studied by calculating its covariant Wigner function in thermal equilibrium. Previous results obtained earlier by Canuto and Chiu are then recovered in a unified way. The polarization tensor is calculated with the use of a covariant quantum BGK equation. Also the lifetime of the neutron in such a medium is calculated for the sake of illustration of the usefulness of the covariant Wigner function.     

Relativistic quantum kinetic analysis of a pion-nucleon system

A relativistic plasma of nucleons interacting through pions via the usual isospin-invariant Yukawa coupling is analyzed in the framework of the covariant Wigner function technique. The method is manifestly covariant and the temperature effects are considered. The relativistic quantum BBGKY hierarchy for the pion-nucleon system is derived. By generalizing the Bogolioubov analysis of the classical BBGKY hierarchy a non-perturbative renormalizable method is elaborated which allows the solution of the kinetic problem in form of power series of two cluster parameters which measure the importance of correlations. In the lowest order of the cluster expansion (Hartree approximation or zero-order approximation) the quasi-nucleon Fock space is introduced, the fermion Wigner function in the thermodynamic equilibrium is obtained and the vacuum effects are renormalized. In this approximation the plasma behaves as a perfect Fermi gas of nucleons and antinucleons, but there exists an abnormal configuration with a uniform pion condensate which is unstable. In the next approximation (quadratic in the small parameters) the quasi-pion dispersion relation is obtained and the vacuum polarization tensor is renormalized. The quasi-pion rest-mass spectra (“plasma frequency”) and the effective-coupling behaviour as functions of the thermodynamic state are given. By estimating the size of the cluster parameters the self-consistency of the approximation scheme is proved. The quasi-pion Fock space is introduced and the quasi-pion equilibrium Wigner function is obtained. From these results the problem of the higher-order corrections to the Hartree thermodynamics is outlined.     

Fluid moment hierarchy equations derived from quantum kinetic theory

A set of quantum hydrodynamic equations are derived from the moments of the electrostatic mean-field Wigner kinetic equation. No assumptions are made on the particular local equilibrium or on the statistical ensemble wave functions. Quantum diffraction effects appear explicitly only in the transport equation for the heat flux triad, which is the third-order moment of the Wigner pseudo-distribution. The general linear dispersion relation is derived, from which a quantum modified Bohm-Gross relation is recovered in the long wave-length limit. Nonlinear, traveling wave solutions are numerically found in the one-dimensional case. The results shed light on the relation between quantum kinetic theory, the Bohm-de Broglie-Madelung eikonal approach, and quantum fluid transport around given equilibrium distribution functions.     

Quantum phase distribution and the number—phase Wigner function of the generalized squeezed vacuum states associated with solvable quantum systems

The quantum phase properties of the generalized squeezed vacuum states associated with solvable quantum systems are studied by using the Pegg-Barnett formalism.Then,two nonclassical features,i.e.,squeezing in the number and phase operators,as well as the number-phase Wigner function of the generalized squeezed states are investigated.Due to some actual physical situations,the present approach is applied to two classes of generalized squeezed states:solvable quantum systems with discrete spectra and nonlinear squeezed states with particular nonlinear functions.Finally,the time evolution of the nonclassical properties of the considered systems has been numerically investigated.     

Self-Averaging of Wigner Transforms in Random Media

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved.     

Moyal Quantization and Wave Equations

We develop the Moyal formalism in the context of covariant wave equations, such as Dirac's or Proca's wave equations, using the Moyal formalism for the canonical or Wigner representation and the relation between covariant and canonical representations.     

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say , of the w^*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincaré, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincaré symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Jüttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.     

Covariant Relativistic Statistical Mechanics of Many Particles

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of event linking (concatenations) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view.     

Quantum transport theory for abelian plasmas

The gauge invariant relativistic quantum equations of motion for the fermion and photon Wigner operators are derived from QED. In the mean field (Hartree) approximation, we extract the generalized quantum Vlasov and mass-shell constraint equations for fermions. In addition, a complete spinor decomposition is performed. A systematic method for computing quantum corrections to all orders in h is developed. First order quantum (spin) corrections are computed explicitly. Finally, the relations between gauge dependent and independent definitions of the photon Wigner function and their corresponding transport equations are discussed.     

Quantization effects in the plasma universe

It is suggested that a unification of the morphology of the solar system, anomalous intrinsic red shifts of quasars and galaxies, the structure of the hydrogen atom, the Einstein equations of general relativity, and Maxwell's equations can be accomplished by a basic consideration of the minimum-action states of cosmic and/or virtual vacuum field plasmas. A formalism of planetary formation theory leads naturally to a generalization which describes relativistic gravitational field theory in terms of a `pregeometry'. A virtual plasma associated with the vacuum state is postulated. It is demonstrated that the relaxed state of the virtual plasma underlies Einstein's field equation and predicts the proper form for the effective gravitational potential generated by the Schwarzschild solution of those equations. A further extension of the theory demonstrates that it also predicts the structure of the hydrogen atom described in terms of the Schrodinger equation of quantum mechanics. These concepts are applied in an attempt to explain the quantized anomalous red shifts in related galaxies as observed by H. Arp and J.H. Sulentic (1985). A possible unified field theory is suggested based on the above-mentioned concepts     

Wigner Measures in Noncommutative Quantum Mechanics

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.     

Non-existence of Goldstone bosons with non-zero helicity

In a local relativistic quantum field theory a conserved covariant tensor current may lead to a spontaneously broken symmetry if it generates zero mass states from the vacuum (Goldstone theorem). Here it is shown that in addition it is necessary that these massless states have helicity zero if the underlaying state space has a positive metric.     

Non-linear Liouville and Shrödinger equations in phase space

Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schrödinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.     

Self-consistent hartree description of finite nuclei in a relativistic quantum field theory

Relativistic Hartree equations for spherical nuclei are derived from a relativistic nuclear quantum field theory using a coordinate-space Green function approach. The renormalizable field theory lagrangian includes the interaction of nucleons with σ, ω, ρ and π mesons and the photon. The Hartree equations represent the “mean-field” approximation for a finite nuclear system. Coupling constants and the σ-meson mass are determined from the properties of nuclear matter and the rms charge radius in ⁴⁰Ca, and pionic contributions are absent for static, closed-shell nuclei. Calculated charge densities, neutron densities, rms radii, and single-nucleon energy levels throughout the periodic table are compared with data and with results of non-relativistic calculations. Relativistic Hartree results agree with experiment at a level comparable to that of the most sophisticated non-relativistic calculations to date. It is shown that the Lorentz covariance of the relativistic formalism leads naturally to density-dependent interactions between nucleons. Furthermore, non-relativistic reduction reveals non-central and non-local aspects inherent in the Hartree formalism. The success of this simple relativistic Hartree approach is attributed to these features of the interaction.     

Thermodynamic Potential of Relativistic Electron Plasma

Within the real time formalism of quantum field theory at finite temperatures based on the closed-time-path Green's function approach, a closed analytical expression of the thermodynamic potential of a relativistic electron plasma is derived under the random phase approximation by summing up all ring diagrams. The result is natural extension to the relativistic case sf our previous formula derived in the case of a Coulomb gas.     

Relativistic Linear Theory in the Absence of External Fields

The dielectric tensor for a multi-component, homogeneous, field-free relativistic plasma is derived in manifestly covariant form. From the dielectric tensor, linear dispersion relations are obtained explicitly when each component of the plasma is isotropic in its rest frame. If the components are relativistic Maxwellians, these dispersion relations are expressible in terms of the relativistic plasma dispersion function. Special attention is given to the Weible and two-stream instabilities and to the normal modes of a quiescent, hot electron gas. For the last case the dispersion relations are solved numerically and compared against computer simulation data. An appendix applies the formalism to cold plasmas.     

Non-linear relativistic diffusions

We obtain a non-linear generalization of the relativistic diffusion. 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 the equilibrium solution corresponding to the quantum or Tsallis distributions. We define the relative entropy as a function of the diffusion probability and prove that it is monotonically decreasing in time when the diffusion tends to equilibrium. We discuss its relation to the thermodynamic behavior of diffusing particles.