In this paper, we extend the method of Kadanoff-Baym equations for open quantum systems to arbitrary kinds of systems and heat baths, either fermionic or bosonic. This includes three spacial dimensions and different potentials for the system-bath interaction or external traps. We study the quantum-mechanical formation of bound states in one and also in three dimensions with the full Kadanoff-Baym equations and compare them to more simplified approaches with and without memory effects. An in-depth examination of the thermodynamics of open systems is performed, showing perfect equilibration of the system's degrees of freedom along with a comprehensive investigation of the influence of the heat bath on the system's wave functions. The formation time, decay time and regeneration of bound states and their dependence on the temperature and coupling strength is explored We evaluate the non-equilibrium Kadanoff-Baym equations for the system particles, assuming that interactions are elastic two-particle collisions with the heat-bath particles. Finally, we describe in detail the method used to numerically solve the corresponding spatially heterogeneous integro-differential equations for the set of one-particle Green's functions.

We consider a problem borrowed from heavy-ion collisions, namely, the formation of bound states, as for example the deuteron, in the nonrelativistic regime by using the techniques of Lindblad dynamics considering an open quantum system. We show that the position-space Lindblad equation can be reformulated in terms of a diffusion-advection equation with sources and therefore provides a hydrodynamical formulation of a dissipative quantum master equation. Making use of this advanced machinery and these insights, we describe the possible formation of a bound state, which is realized by a P & ouml;schl-Teller-like potential, of a particle in interaction with a heat bath in a one-dimensional setting and discuss general aspects of open quantum systems, such as decoherence, entropy production, purity, thermalization time, and thermalization. Finally, we also show an example of a much deeper potential, where we allow for three bound states, just in the spirit of quarkonia.

The Lindblad master equation is a frequently used Markovian approach to describe open quantum systems in terms of the temporal evolution of a reduced density matrix. Here, the thermal environment is traced out to obtain an expression to describe the evolution of what is called a system: one particle or a chain of interacting particles, which is/are surrounded by a thermal heat bath. In this work, we investigate the formation of non-relativistic bound states, involving the Pöschl-Teller potential, to discuss the formation time and the thermal equilibrium, applying scales from nuclear physics. This problem is borrowed from the field of heavy-ion collisions, where the deuteron is a probe which is measured at temperature regimes around the freeze out temperature, while the deuteron itself has a binding energy which is much lower. This is known and often described as a “snowball in hell". We use a reformulated Lindblad equation, in terms of a conservative diffusion–advection equation with sources and therefore provide a hydrodynamical formulation of a dissipative quantum master equation.

The invariant mass spectra of dileptons radiated from the fireballs formed in high-energy heavy-ion collisions have been successfully used to investigate the properties of hot and dense QCD matter. Using a realistic model for the in-medium electromagnetic spectral function, we predict polarization observables and compare them to experiment. This allows, for the first time, independent tests of the longitudinal and transverse components of the virtual photon's selfenergy. While the low- and high-mass regions exhibit the expected limits of transverse and unpolarized photons, respectively, baryon-driven medium effects in the ρ-meson mass region create a marked longitudinal polarization that transits into a largely unpolarized emission from the quark-gluon plasma, thus providing a sensitive test of microscopic emission processes in QCD matter. Applications to available data from the HADES and NA60 experiments at SIS and SPS energies, respectively, are consistent with our predictions and set the stage for quantitative polarization studies at FAIR and collider energies.

This report summarizes the work of the EMMI Rapid Reaction Task Force on “Real and Virtual Photon Production at Ultra-Low Transverse Momentum and Low Mass at the LHC”. We provide an overview of the soft-photon puzzle, i.e., of the long-standing discrepancy between experimental data and predictions based on Low’s soft-photon theorem, also referred to as “anomalous” soft photon production, and we review the current theoretical understanding of soft radiation and soft theorems. We also focus on low-mass dileptons as a tool for determining the electrical conductivity of the medium produced in high-energy nucleus–nucleus collisions. We discuss how both topics can be addressed with the planned ALICE 3 detector at the LHC.

Multi-differential measurements of dilepton spectra serve as a unique tool to characterize the properties of matter in the interior of the hot and dense fireball created in heavy-ion collisions. An important property of virtual photons is their spin polarization defined in the rest frame of the virtual photon with respect to a chosen quantization axis. Microscopic calculations of in-medium electromagnetic spectral functions have mostly focused on integrated yields which are proportional to the sum of the longitudinal and transverse components of the virtual photon’s self-energy, while photon polarization results from the difference of these components. As the processes that drive the medium effects in the spectral function change with invariant mass and momentum, this becomes a powerful tool for studying the medium composition. We present the polarization observables of thermal virtual photons as a function of mass and momentum and confront the results with existing measurements from HADES and NA60.

We study the temporal evolution of quantum mechanical fermionic particles exhibiting one bound state within a one-dimensional attractive square-well potential in a heat bath of bosonic particles. For this open quantum system we formulate the non-equilibrium Kadanoff-Baym equations for the system particles by taking the interactions to be elastic 2-2 scatterings with the heat-bath particles. The corresponding spatially imhomogeneous integro-differential equations for the one-particle Greens's function are solved numerically. We demonstrate how the system particles equilibrate and thermalize with the heat bath and how the off-diagonal elements of the density matrix, expressed in the one-particle energy eigenbasis, decohere, so that only the diagonal entries, i.e. the occupation numbers, survive. In addition, the time evolution of the (retarded) Green's function also determines the spectral properties of the various one-particle quantum states.

The diffusion of heavy quarks through the quark–gluon plasma (QGP) as produced in high-energy heavy-ion collisions has long been recognized as an excellent probe of its transport properties. In addition, the experimentally observed heavy-flavor hadrons carry valuable information about the hadronization process of the transported quarks. Here we review recent progress in the theoretical developments of heavy-quark interactions in the QGP and how they relate to the nonperturbative hadronization process, and discuss the recent status of the pertinent phenomenology in heavy-ion collisions at the RHIC and the LHC. The interactions of heavy quarks in the QGP also constitute a central building block in the description of the heavy quarkonia which controls their transport parameters as well. We will thus focus on theoretical approaches that aim for a unified description of open and hidden heavy-flavor particles in medium, and discuss how they can be constrained by lattice-QCD “data” and utilized to deduce fundamental properties of the microscopic interactions and emerging spectral properties of the strongly coupled QGP.

• the gravitational red shift of spectral lines. This has also been predicted by Einstein in his first papers, and the famous “Einstein tower” in Potsdam was an observatory dedicated explicitly to confirm the gravitational red-shift by spectroscopy of the light of the Sun. The accuracy for this has not been sufficient at the time. The effect of gravitational red (or in this case rather blue shift) has then been successfully observed for the first time using γ -rays and the Mössbauer effect in the gravitational field of the Earth.

We study the temporal formation of quantum-mechanical bound states within a one-dimensional attractive square-well potential by first solving the time-independent Schrodinger equation and then studying a time -dependent system with an external time-dependent potential. For this we introduce Gaussian potentials with different spatial and temporal extensions and generalize this description also for subsequent pulses and for random noisy potentials. Our main goal is to study the timescales in which the bound state is populated and depopulated. Particularly, we clarify a likely connection between the uncertainty relation for energy and time and the transition time between different energy eigenstates. We demonstrate that the formation of states is not delayed due to the uncertainty relation but follows the pulse shape of the perturbation. In addition we investigate the (non-)applicability of first-order perturbation theory on the considered quantum system.

In this study an effective description in the 2PI effective-action formalism for systems of quarks and mesons in and out of equilibrium within a numerical approach is developed, allowing to approximate the complexity of QCD by taking only the lightest and most relevant degrees of freedom into account. In particular the temporarily building up of fluctuations of the net-baryon number encoded by the fourth-order cumulant (or the rescaled curtosis) for lower momenta is being demonstrated when the phase transition occurs near the critical point, or even stronger when the phase transition is of first order, although the initial system is prepared with purely Gaussian fluctuations in the net baryon number. This is the result of the evolving slow and critical order parameter, i.e., the σ-field. On the other hand, depending on the speed of the (Hubble-)expansion scale, the final dissipative evolution due to the collisions among the mesons, the quarks and anti-quarks and the order field weakens the final fluctuations considerably.

In this comment it is shown that the argument for a unique determination of the electromagnetic potentials in classical electrodynamics by Davis (2020 Eur. J. Phys. 41 045202) is flawed. To the contrary the ‘gauge freedom’ of the electromagnetic potentials has proven to be one of the most important properties in the development of modern physics, where local gauge invariance with its extension to non-abelian gauge groups is a key feature in the formulation of the standard model of elementary particles in terms of a relativistic quantum field theory.

Exciting new scientific opportunities are presented for the PANDA detector at the High Energy Storage Ring in the redefined $$\bar{\text {p}} \text {p}(A)$$ collider mode, HESR-C, at the Facility for Antiproton and Ion Research (FAIR) in Europe. The high luminosity, $$L \sim 10^{31}$$ cm$$^{-2}$$ s$$^{-1}$$, and a wide range of intermediate and high energies, $$\sqrt{s_{\text {NN}}}$$ up to 30 GeV for $$\bar{\text {p}} \text {p}(A)$$ collisions will allow to explore a wide range of exciting topics in QCD, including the study of the production of excited open charm and bottom states, nuclear bound states containing heavy (anti)quarks, the interplay of hard and soft physics in the dilepton production, and the exploration of the regime where gluons—but not quarks—experience strong interaction.

This paper is a write-up of the ideas that were presented, developed and discussed at the third International Workshop on QCD Challenges from pp to A–A, which took place in August 2019 in Lund, Sweden (Workshop link: https://indico.lucas.lu.se/event/1214/ ). The goal of the workshop was to focus on some of the open questions in the field and try to come up with concrete suggestions for how to make progress on both the experimental and theoretical sides. The paper gives a brief introduction to each topic and then summarizes the primary results.

Exciting new scientific opportunities are presented for the PANDA detector at the High Energy Storage Ring in the redefined $$\overline{\text {p}} \text {p}(A)$$ collider mode, HESR-C, at the Facility for Antiproton and Ion Research (FAIR) in Europe. The high luminosity, $$L \sim 10^{31}$$ cm$$^{-2}$$ s$$^{-1}$$, and a wide range of intermediate and high energies, $$\sqrt{s_{\text {NN}}}$$ up to 30 GeV for $$\overline{\text {p}} \text {p}(A)$$ collisions will allow to explore a wide range of exciting topics in QCD, including the study of the production of excited open charm and bottom states, nuclear bound states containing heavy (anti)quarks, the interplay of hard and soft physics in the dilepton production, probing short-range correlations in nuclei, and the exploration of the early, complete $$\overline{\text {p}}$$-p- annihilation phase, where an initially pure Yang–Mills gluon plasma is formed.

We compare the thermal escape rates of a Brownian particle, initially trapped into one of the two wells of an asymmetric double-well potential, for thermal Markovian and non-Markovian noise. The Markovian treatment of this problem goes originally back to the studies of Kramers in 1940 and is therefore often referred to as “Kramers’ escape rate problem”. We solve the generalized Langevin equation for the trajectories of the particles numerically and analytically for both limiting cases, Markovian and non-Markovian thermal noise. We compute the escape rate and work out the fundamental differences arising from finite correlation times of the thermal noise.

The direct-current-conducting infinitely long wire is often discussed in the context of relativistic electrodynamics. It is of course a completely academic discussion since for the typical household currents the drift velocity of the electrons in the wire, making up the conduction current, is tiny (of the order O (1mm/s)!). Nevertheless it is unfortunately only quite confusingly discussed in the literature. So here is my attempt for a more consistent description using a naive classical model of a metal as consisting of a continuum of effectively positive bound charges (consisting of atoms and the bound electrons making up the lattice forming the metal, in the following called “ions”) and negative freely moving conduction electrons, treated as a freely moving fluid subject to some friction when moving against the positive charged rigid background. The confusion starts with the fact that usually it is not carefully discussed in which reference frame the wire is uncharged. It is not the rest frame of the wire (i.e., the rest frame of the ions) but the rest frame of the conduction electrons [Pet85]. The qualitative argument is simple: We consider a straight wire with a constant current. In the rest frame of the conduction electrons there is a current due to the moving positively charged background and a corresponding magnetic field. The charges within the positive background are however bound and can be considered not to move due to the electromagnetic field. Thus in this reference frame the charge density vanishes everywhere within the wire as if there were no current at all since the freely movable conduction electrons are at rest, and there is thus no net force acting on them and thus in this reference frame no charge separation occurs. Since the charge density (times c ) and the current density form a four-vector, consequently in the rest frame of the wire (i.e., the rest frame of the ions) there must be a non-vanishing charge density within the wire due to the Lorentz-transformation properties of vector components. This is also easily explained dynamically: In this reference frame the conduction electrons move along the wire with constant velocity, and thus a magnetic field is present, which causes a radial force on the conduction electrons, which consequently arrange such that an electric field is built up which exactly compensates this magnetic force. This is, of course, nothing else than the “self-induced” Hall effect. This is taken into account automatically when the correct relativistic version of Ohm’s Law is considered, which is sometimes approximated with the simple non-relativistic form, consequently leading to non-covariant approximations of the fields.

The phenomenon of Bose-Einstein condensation is investigated in the context of the color-glass-condensate description of the initial state of ultrarelativistic heavy-ion collisions. For the first time, in this paper, we study the influence of particle-number changing 2↔3 processes on the transient formation of a Bose-Einstein condensate within an isotropic system of scalar bosons by including 2↔3 interactions of massive bosons with constant and isotropic cross sections, following a Boltzmann equation. The one-particle distribution function is decomposed in a condensate part and a nonzero momentum part of excited modes, leading to coupled integro-differential equations for the time evolution of the condensate and phase-space distribution function, which are then solved numerically. Our simulations converge to the expected equilibrium state, and only for σ23/σ22≪1, we find that a Bose-Einstein condensate emerges and decays within a finite lifetime in contrast to the case where only binary scattering processes are taken into account, and the condensate is stable due to particle-number conservation. Our calculations demonstrate that Bose-Einstein condensates in the very early stage of heavy-ion collisions are highly unlikely, if inelastic collisions are significantly participating in the dynamical gluonic evolution.

We present a study of the elliptic flow and $$R_{AA}$$ of $$\text {D}$$ and $$\bar{\text {D}}$$ mesons in Au+Au collisions at FAIR energies. We propagate the charm quarks and the $$\text {D}$$ mesons following a previously applied Langevin dynamics. The evolution of the background medium is modeled in two different ways: (I) we use the UrQMD hydrodynamics + Boltzmann transport hybrid approach including a phase transition to QGP and (II) with the coarse-graining approach employing also an equation of state with QGP. The latter approach has previously been used to describe di-lepton data at various energies very successfully. This comparison allows us to explore the effects of partial thermalization and viscous effects on the charm propagation. We explore the centrality dependencies of the collisions, the variation of the decoupling temperature and various hadronization parameters. We find that the initial partonic phase is responsible for the creation of most of the $$\text {D}/\bar{\text {D}}$$ mesons elliptic flow and that the subsequent hadronic interactions seem to play only a minor role. This indicates that $$\text {D}/\bar{\text {D}}$$ mesons elliptic flow is a smoking gun for a partonic phase at FAIR energies. However, the results suggest that the magnitude and the details of the elliptic flow strongly depend on the dynamics of the medium and on the hadronization procedure, which is related to the medium properties as well. Therefore, even at FAIR energies the charm quark might constitute a very useful tool to probe the quark–gluon plasma and investigate its physics.