Abstract. These lecture notes review the foundations and some applications of lightcone quantization. First I explain how to choose a time in special relativity. Inclusion of Poincaré invariance naturally leads to Dirac's forms of relativistic dynamics. Among these, the front form, being the basis for light-cone quantization, is my main focus. I explain a few of its peculiar features such as boost and Galilei invariance or separation of relative and center-of-mass motion. Combining light-cone dynamics and field quantization results in light-cone quantum field theory. As the latter represents a first-order system, quantization is somewhat nonstandard. I address this issue using Schwinger's quantum action principle, the method of Faddeev and Jackiw, and the functional Schrödinger picture. A finite-volume formulation, discretized light-cone quantization, is analysed in detail. I point out some problems with causality, which are absent in infinite volume. Finally, the triviality of the light-cone vacuum is established. Coming to applications, I introduce the notion of light-cone wave functions as the solutions of the light-cone Schrödinger equation. I discuss some examples, among them nonrelativistic Coulomb systems and model field theories in two dimensions. Vacuum properties (like chiral condensates) are reconstructed from the particle spectrum obtained by solving the light-cone Schrödinger equation. In a last application, I make contact with phenomenology by calculating the pion wave function within the Nambu and Jona-Lasinio model. I am thus able to predict a number of observables like the pion charge and core radius, the r.m.s. transverse momentum, the pion structure function and the pion distribution amplitude. The latter turns out to be the asymptotic one.

We derive effective actions for SU (2) Polyakov loops using inverse Monte Carlo techniques. In a first approach, we determine the effective couplings by requiring that the effective ensemble reproduces the single-site distribution of the Polyakov loops. The latter is flat below the critical temperature implying that the (untraced) Polyakov loop is distributed uniformly over its target space, the SU (2) group manifold. This allows for an analytic determination of the Binder cumulant and the distribution of the meanfield, which turns out to be approximately Gaussian. In a second approach, we employ novel lattice Schwinger-Dyson equations which reflect the SU (2) × SU (2) invariance of the functional Haar measure. Expanding the effective action in terms of SU (2) group characters makes the numerics sufficiently stable so that we are able to extract a total number of 14 couplings. The resulting action is short-ranged and reproduces the Yang-Mills correlators very well.

We investigate the mesonic light-front bound-state equations of the 't Hooft and Schwinger model in the two-particle, i.e. valence sector, for small fermion mass. We perform a high precision determination of the mass and light-cone wave function of the lowest lying meson by combining fermion mass perturbation theory with a variational approach. All calculations are done entirely in the fermionic representation without using any bosonization scheme. In a step-by-step procedure we enlarge the space of variational parameters. We achieve good convergence so that the calculation of the meson mass squared can be extended to third order in the fermion mass. Within a numerical treatment we include higher Fock states up to six particles. Our results are consistent with all previous numerical investigations, in particular lattice calculations. For the massive Schwinger model, we find a small discrepancy ( < ∼ 2%) in comparison with known results. Possible resolutions of this discrepancy are discussed. PACS number(s):where all masses are measured in units of the basic scale µ 0 = e/ √ π, the mass of the boson in the massless Schwinger model, which is thus represented by the '1' in (1.1). The first order coefficient M 1 was obtained analytically in [12] via bosonization, M 1 = 2e γ = 3.56215 . (1.2) with γ = 0.577216 being Euler's constant. Shortly afterwards Bergknoff, using light-cone Hamiltonian techniques (see below), found a value [13] M 1 = 2π/ √ 3 = 3.62760 , (1.3)

We perform a lattice analysis of the Faddeev-Niemi effective action conjectured to describe the low-energy sector of SU (2) Yang-Mills theory. To this end we generate an ensemble of unit vector fields ('color spins') n from the Wilson action. The ensemble does not show long-range order but exhibits a mass gap of the order of 1 GeV. From the distribution of color spins we reconstruct approximate effective actions by means of exact lattice Schwinger-Dyson and Ward identities ('inverse Monte Carlo'). We show that the generated ensemble cannot be recovered from a Faddeev-Niemi action, modified in a minimal way by adding an explicit symmetry-breaking term to avoid the appearance of Goldstone modes.

We study effective Polyakov loop models for SU(N) Yang-Mills theories at finite temperature. In particular effective models for SU(3) YM with an additional adjoint Polyakov loop potential are considered. The rich phase structure including a center and anti-center directed phase is reproduced with an effective model utilizing the inverse Monte-Carlo method. The demon method as a possibility to obtain the effective models' couplings is compared to the method of Schwinger-Dyson equations. Thermalization effects of microcanonical and canonical demon method are analyzed. Finally the elaborate canonical demon method is applied to the finite temperature SU(4) YM phase transition.

The light-cone wave function of the pion is calculated within the Nambu-JonaLasinio model. The result is used to derive the pion electromagnetic form factor, charge radius, structure function, π-γ-transition form factor and distribution amplitude.

This contribution presents an overview of fundamental QED processes in the presence of an external field produced by an ultra-intense laser. The discussion focusses on the basic intensity effects on vacuum polarisation and the prospects for their observation. Some historical remarks are added where appropriate.

The standard approach to the infra-red problem is to sum over degenerate final states to remove soft divergences (Bloch-Nordsiech), and over both initial and final states for collinear divergences (Lee-Nauenberg). We show that this division is inconsistent, and further that the Lee-Nauenberg recipe leads to ill-defined results in a variety of theories. We then argue that infra-red divergences are due to using unphysical variables to describe in and out states and introduce gauge invariant, physical 'dressed' states. Perturbative and non-perturbative properties of physical electron and quark states are then discussed. It is shown in SU(2) lattice simulations that the overlap between the ground state and our dressed state increases as we approach finer lattices.