Ch.G. van Weert

Real- and imaginary-time field theory at finite temperature and density

Abstract This report gives a detailed account of relativistic quantum field theory in the grand canonical ensemble. Three approaches are discussed: traditional Euclidean Matsubara, and two recently developed real-time methods, namely, Minkowskian time-path and thermo field dynamics. The first two formulations are derived in a unified manner from the path-integral representation for the contour-ordered generating functional. Fields with spin and gauge fields, in particular, are included. Zero-temperature renormalizability id shown to imply UV finiteness at any temperature and density. Thermo field dynamics, which is basically an operator theory, is presented in a C∗-algebraic context. Relevant parts of the HHW formalism of quantum statistical mechanics, and the Tomita-Takesaki theory are explained. The next chapter contains an analysis of the structure and analytic properties of the self-energy and its relation to the full propagator. In this connection the concept of a statistical quasiparticle is briefly described. This is followed by a discussion of thermal WT identities, Results are applied to discuss transversality of the SU(N) gluon polarization tensor. The final chapter deals with the diagrammatic rules for evaluating the pressure and energy density. The energy-momentum tensor is analyzed as a composite operator, and a renormalized virial theorem is established to provide the link with the thermodynamic potential. The pressure of the SU(N) chromoplasma is calculated up to third order.

Finite-temperature retarded and advanced Green functions

Abstract The relation is established between retarded/advanced (RA) Green functions and the “physical”/“ghost” ones of the standard real-time formalism of finite temperature field theory (RTFT) for arbitrary time-path parameter σ. Causal combinations of RTFT-Green functions are identified with specific RA-Green functions. It is argued that the imaginary part of the two-point RA-Green function is related to a physical reaction rate, and the real part of the three-point function to an effective coupling constant.

Elements of relativistic kinetic theory

The following subjects are treated: scattering matrix, transition rate and cross section in relation to relativistic kinetic theory; unitarity, bilateral normalization and H theorem; covariant one-particle distribution function; relativistic transport equation.

On relativistic kinetic gas theory: VIII. Reciprocal relations for a reactive mixture

Abstract In a reactive system with an affinity much less than kT the scalar transport phenomena of bulk viscosity and mass production may be described by linear laws. It is proved from time (motion) reversal invariance that the cross coefficients Lvr and Lrv which link these two phenomena satisfy the reciprocity relation Lvr = − Lrv. The minus sign is due to the different parity of the two driving thermodynamic forces (the divergence of the velocity field and the affinity) under time reversal.

Maximum entropy principle and relativistic hydrodynamics

Abstract A relativistic theory of hydrodynamics applicable beyond the hydrodynamic regime is developed on the basis of the maximum entropy principle. This allows the construction of a unique statistical operator representing the state of the system as specified by the values of the hydrodynamical densities. Special attention is paid to the thermodynamic limit and the virial theorem which leads to an expression for the pressure in terms of the field-theoretic energymomentum tensor of Coleman and Jackiw. It is argued that outside the hydrodynamic regime the notion of a local Gibbs relation, as usually postulated, must be abandoned in general. In the nontext of the linear approximation, the memory-retaining and non-local generalizations of the relativistic Navier-Stokes equations are derived from the underlying Heisenberg equations of motion. The formal similarity to the Zwanzig-Mori description of non-relativistic fluids is expounded.

Relativistic kinetic theory of quantum systems: III. Transport equation for a neutrino system

Abstract On the basis of the quantum-mechanical equations of motion and the statistical assumption that dynamical correlations present in the initial state may be ignored, a kinetic equation for a dilute neutrino system is derived. If the system is sufficiently uniform in space this equation reduces to the relativistic Boltzmann equation for massless particles with a quantum-mechanical transition probability.

Temperature dependence of the QCD coupling

We present a one-loop calculation of a gauge invariant QCD β-function. Using both momentum and temperature renormalization group equations we investigate the running coupling in the magnetic sector as a function of temperature and momentum scale. At fixed momentum scale we find that, in contrast to λɸ4 or QED, high temperature QCD is strongly coupled, even after renormalization group improvement. However, if the momentum scale is changed simultaneously with temperature in a specified manner, the coupling decreases. We also point out in what regime dimensional reduction occurs. Both the cases Nf smaller and larger than are discussed.

Relativistic kinetic theory of quantum systems

For a dilute and nondegenerate relativistic spin-12 system two kinds of Wigner functions are defined: one has sixteen spinor components and the other four spin components. Their relationship is established. Statistical expressions for the current density, the energy-momentum density and the spin density are obtained in terms of both kinds of Wigner functions. The transformation properties of the latter under Lorentz transformations are discussed.

Relativistic kinetic theory of quantum systems: IV. Transport equation for the neutrino component of an eeνeνe-mixture

Abstract The transport equation for a neutrino system derived previously is extended to include binary collisions involving electrons, positrons and antineutrinos. The transition amplitudes for these collision processes are specified in terms of an effective Hamiltonian density following from the Weinberg-Salam theory. The resulting equation incorporates the reactive processes e e ↔ ν ν and exhibits the electron and positron spins explicitly. It is shown to be form invariant under Lorentz transformations.

Relativistic kinetic theory of quantum systems: II. Neutrino-antineutrino system

Abstract It is shown that the macroscopic current density and energy-momentum density of a neutrino-antineutrino system may be expressed in terms of moments of two scalar Wigner functions, provided that the system is homogeneous on the scale of the de Broglie wavelength of the particles. The equilibrium form of these Wigner functions is established.