Abstract:
Dynamical quasiparticle properties are determined from lattice QCD along the line of the Peshier model for the running strong coupling constant in case of three light flavors. By separating time-like and space-like quantities in the number density and energy density the effective degrees of freedom in the gluon and quark sector may be specified from the time-like densities. The space-like parts of the energy densities are identified with interaction energy (or potential energy) densities. By using the time-like parton densities (or scalar densities) as independent degrees of freedom variations of the potential energy densities with respect to the time-like gluon and/or fermion densities lead to effective mean-fields for time-like gluons and quarks as well as to effective gluon-gluon, quark-gluon and quark-quark (quark-antiquark) interactions. The latter dynamical quantities are found to be approximately independent on the quark chemical potential and thus well suited for an inplementation in off-shell parton transport approaches. Results from the dynamical quasiparticle model (DQPM) in case of two dynamical light quark flavors are compared to lattice QCD calculations for the net quark density as well as for the 'back-to-back' differential dilepton production rate by $q-{\bar q}$ annihilation. The DQPM is found to pass the independent tests.

Abstract:
The production and propagation of mesons ($\pi, \eta, \rho, \omega, \Phi, K, \bar{K}, J/\Psi$) in proton-nucleus and nucleus-nucleus collisions from 1 - 200 GeV/u is studied within the covariant transport approach HSD, which explicitly allows to investigate selfenergy effects of the hadrons at finite baryon density. Whereas the experimental pion and $K^+$ spectra can be described without introducing any selfenergies for the mesons, the $K^-$ yield in Ni + Ni collisions is underestimated by a factor of 5 - 7 at 1.66 and 1.85 GeV/u. However, introducing density dependent antikaon masses in line with effective chiral Lagrangians a satisfactory agreement with the data is achieved. A dropping of the $\rho$-meson mass with baryon density, as suggested by QCD sumrule studies, is proposed to explain the dilepton spectra for S + Au and Pb + Au at SPS energies, which indicates independently that a partial restoration of chiral symmetry might be found already in the present experiments.

Abstract:
This review provides a written version of the lectures presented at the Schladming Winter School 2008, Austria, on 'Nonequilibrium Aspects of Quantum Field Theory'. In particular, it shows the way from quantum-field theory - in two-particle irreducible approximation - to the Kadanoff-Baym (KB) equations and various approximations schemes of the KB equations in phase space. This ultimately leads to the formulation of an off-shell transport theory that well incorporates the underlying quantum physics. Remarkably, these transport equations may be solved within a testparticle representation that allows to study non-equilibrium quantum systems in the weak and strong coupling regime. Actual applications to dilepton production in heavy-ion reactions are presented in comparison with available data. The approach, furthermore, allows to address the hadronization process from partonic to hadronic degrees of freedom.

Abstract:
The production of antibaryons is calculated in a microscopic transport approach employing multiple meson fusion reactions according to detailed balance relations with respect to baryon-antibaryon annihilation. It is found that the abundancies of antiprotons as observed from peripheral to central collisions of $Pb + Pb$ at the SPS and $Au + Au$ at the AGS can approximately be described on the basis of multiple interactions of 'formed' hadronic states which drive the system to chemical equilibrium by flavor exchange or quark rearrangement reactions.

Abstract:
The production and decay of vector mesons ($\rho, \omega$) in pA and AA reactions is studied with particular emphasis on their in-medium spectral functions. It is explored within transport calculations if hadronic in-medium decays like $\pi^+\pi^-$ or $\pi^0 \gamma$ might provide complementary information to their dilepton ($e^+e^-$) decays. Whereas the $\pi^+ \pi^-$ signal from the $\rho$-meson is found to be strongly distorted by pion rescattering, the $\omega$-meson Dalitz decay to $\pi^0 \gamma$ appears promising even for more heavy nuclei in $\gamma A$ and pA reactions. Furthermore, the influence of nucleon and kaon/antikaon potentials on the $K^\pm$ yields and spectra in pA collisions is calculated and compared to the recent data from the ANKE Collaboration.

Abstract:
We briefly review the dilepton experiments at BEVALAC/SIS and SPS energies for p+p, p+A and A+A collisions as well as our present understanding of the data within transport theoretical simulations. Since dileptons from p+A and A+A collisions in particular probe the in-medium spectral functions of vector mesons, a novel semiclassical off-shell transport approach is introduced on the basis of the Kadanoff-Baym equations that describes the dynamical evolution of broad hadron spectral functions. The implications of the off-shell dynamics -- relative to the conventional on-shell transport dynamics -- are discussed for proton spectra, high energy $\gamma$, pion, kaon and antikaon production from GANIL to AGS energies in comparison to experimental data.

Abstract:
The nuclear equation of state (EoS) is investigated by flow phenomena in relativistic heavy-ion collisions, both in transverse and radial direction, in comparison to experimental data from 150 A MeV to 11 A GeV. To this aim the collective dynamics of the nucleus-nucleus collision is described within a transport model of the coupled channel RBUU type. There are two factors which dominantly determine the baryon flow at these energies: the momentum dependence of the scalar ($U_S$) and vector potentials ($U_\mu$) for baryons and the resonance/string degrees of freedom for energetic hadron excitations. We fix the explicit momentum dependence of the nucleon-meson couplings by the nucleon optical potential up to 1 GeV and extrapolate to higher energy. When assuming the optical potential to vanish identically for $E_{kin} \geq 3.5$ GeV we simultaneously describe the sideward flow data of the PLASTIC BALL, FOPI, EoS and E877 collaborations, the elliptic flow data of the E895 and E877 collaborations and approximately the rapidity and transverse mass distribution of protons at AGS energies without employing any {\it explicit} assumption on a phase transition in the EoS. However, the gradual change from hadronic to string degrees of freedom with increasing bombarding energy can be viewed as a transition from {\it hadronic} to {\it string} matter, i.e. a dissolution of hadrons at high energy density.

Abstract:
The properties of $\pi, \eta, K^+$ and $K^-$ mesons are studied in nuclear reactions from SIS to SPS energies within the covariant transport approach HSD in comparison to the experimental data. Whereas the pion, $\eta$ and kaon abundancies and spectra indicate little or vanishing selfe-nergies for these mesons in the medium, antikaons (as well as antiprotons) are found to experience strong attractive potentials in nucleus-nucleus collisions at SIS energies. However, even when including these potentials the $K^+$ and $K^-$ spectra at AGS energies are noticeably underestimated showing an experimental excess of strangeness that points towards a nonhadronic phase in these reactions. On the other hand the $K^+, K^-$ production at SPS energies is well described by the hadronic approach without incorporating any parton degrees of freedom.

Abstract:
The production and decay of vector mesons ($\rho, \omega$) in $pA$ reactions at COSY energies is studied with particular emphasis on their in-medium spectral functions. It is explored within transport calculations, if hadronic in-medium decays like $\pi^+\pi^-$ or $\pi^0 \gamma$ might provide complementary information to their dilepton ($e^+e^-$) decays. Whereas the $\pi^+ \pi^-$ signal from the $\rho$-meson is found to be strongly distorted by pion rescattering, the $\omega$-meson Dalitz decay to $\pi^0 \gamma$ appears promising even for more heavy nuclei. The perspectives of scalar meson ($f_0, a_0$) production in $pp$ reactions are investigated within a boson-exchange model indicating that the $f_0$-meson might hardly be detected in these collisions in the $K \bar{K}$ or $\pi \pi$ decay channels whereas the exclusive channel $pp \to d a_0^+$ looks very promising.

Abstract:
In this study it is demonstrated that a simple picture of the QCD gluon liquid emerges in the dynamical quasiparticle model that specifies the active degrees of freedom in the time-like sector and yields a potential energy density in the space-like sector. By using the time-like gluon density (or scalar gluon density) as an independent degree of freedom - instead of the temperature $T$ as a Lagrange parameter - variations of the potential energy density lead to effective mean-fields for time-like gluons and an effective gluon-gluon interaction strength at low density. The latter yields a simple dynamical picture for the gluon fusion to color neutral glueballs when approaching the phase boundary from a temperature higher than $T_c$ and paves the way for an off-shell transport theoretical description of the parton dynamics.