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Perturbation theory has long been the method of choice in addressing the

A genuine testing of the U(1)

Some of the contributions to this present special issue focus on just such nonperturbative approaches to pure Yang-Mills theory, QCD, and Yang-Mills thermodynamics. Namely, the work of H. Reinhardt et al. investigates the nonperturbative vacuum functional in the Hamiltonian formulation of SU(2) and SU(3) QCD in Weyl-Coulomb gauge both at finite and at zero temperature and for both situations, pure Yang-Mills and fluctuating quark fields. The task is to solve the functional Schrödinger equation with a duly gauge-fixed Hamiltonian for the ground-state functional. This is done in an approximate form in terms of minimizing energy in a well-motivated Gaussian-variational ansatz (and cubic plus quartic generalizations thereof) for the ground-state functional. The associated kernels can be expressed as solutions to Dyson-Schwinger and to gap equations which can be solved subject to simplifying assumptions and boundary (horizon) conditions to implement confinement (a highly nonperturbative feat). Interesting quantities to compute are the Wilson and Coulomb string tension. One then is led to ask about the field configurations that mainly induce them. This is done in comparison of the present continuum approach to lattice results in discussing the role of center-vortex loops and the magnetic monopoles that associate with them. Finite temperature is introduced by compactification of the

There are two interesting contributions in this special issue, one by S. J. Brodsky and the other one by H. G. Dosch, who base their nonperturbative approach to QCD on the so-called AdS-QCD conjecture which is believed to be related to the AdS-CFT holography correspondence, conjectured by J. Maldacena to hold between 4D SU_{5}). In particular, S. J. Brodsky first introduces a mass scale into the QCD light-front Hamiltonian by adding a term proportional to the special conformal operator which then gives rise to a confining potential. This does not break conformal invariance on the level of the QCD action. This somewhat ad hoc procedure can be shown to follow from the conjectured duality between light-front QCD and AdS_{5} if one introduces the AdS_{5} action by the dilaton along the fifth dimension. Generalizing this by appealing to the full superconformal algebra, Brodsky obtains, among interesting insights about the light-front vacuum structure, a unified hadron spectroscopy for mesons, baryons, and tetraquarks with supersymmetric relations between meson and baryon masses. H. G. Dosch (in collaboration with S. J. Brodsky and G. de Teramont) makes the case for supersymmetry across the light hadron spectrum very explicit in appealing to the superconformal algebra to fix the form of the light-front potential. On the other hand, heavy-quark symmetry seems to ensure the survival of supersymmetry even though conformal symmetry is strongly broken in the heavy-quark limit.

A long-standing question about the asymptotic behavior of perturbative expansions, here in QED, is pursued by I. Huet et al. There are good, classical arguments that this expansion does not converge. However, in particular, the leading large-

P. Mathieu discusses the possibility of performing exact nonperturbative computations of functional integrals related to the partition function and observables in 3D U(1) Chern-Simons theory thanks to the Deligne cohomology classes of its fiber bundles.

H. Weigel reports about his development of an efficient method, based on scattering data (spectral method) about a scalar field configuration of given topology (static solution to the classical equation of motion), to compute its one-loop effective potential (or vacuum polarization energy, VPE) in a 1 + 1D

T. Krajewski presents a beautiful discussion on an extension of validity of the Wigner semicircle law for the probability distribution of eigenvalues of

The contribution by J. L. Rubin is a bit off the main theme of the special issue, yet highly interesting: insights derived from relativistic positioning systems for the structure of space-time. Namely, he shows how causal axiomatics and certain local 1D and 2D projective structures attached to emitters are sufficient to deduce the 4D projective structure of space-time. This allows, e.g., for a modification of Newton’s force law on large distances.

M. Faber in his paper proposes a model of the electron along the lines of a Skyrme-model-like construction (but with a different potential term). The basic quantity, a spatial dreibein, parameterizes unit quaternions, that is, group elements of SU(2) with nontrivial winding on its group manifold

The paper by T. Grandou et al. addresses an effective formulation of QCD which exhibits locality in fermionic Green’s functions upon a sophisticated and highly nonperturbative functional integration (actually, functional differentiation) of gluonic gauge-field fluctuations, using the Halpern quadrature of the QCD partition function for the gauge field

Finally, there are two papers, one by I. Bischer and one by S. Hahn and R. Hofmann, which explore consequences of deconfining SU(2) Yang-Mills thermodynamics. The former contribution addresses certain (nonperturbative) radiative corrections to the pressure, arising from massive quasi-particle fluctuations. It is shown that fixed-order dihedral diagrams exhibit a high-temperature dependence starkly exceeding the Stefan-Boltzmann behavior. However, it is demonstrated that an all-order resummation cures this apparent problem and leads to well-bounded, purely imaginary contributions at leading order (implying that these radiative corrections do not admit a thermodynamical interpretation). The second paper investigates the consequences of the postulate that an SU(2) rather than a U(1) gauge principle governs thermal photon gases, e.g., the Cosmic Microwave Background (CMB), for the high-

All papers of this special issue have undergone peer review by one or two high-calibre referees. A substantial fraction of manuscripts submitted had to be rejected.