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Lattice Quantum Chromodynamics (LQCD) is the most versatile and powerful method to investigate non-perturbative effects in strong interaction physics. In view of the much improved statistical accuracy the control of systematic uncertainties became the most important issue in recent years. The most severe problem is the implementation of chiral symmetry. The BGR collaboration works with two different Dirac operators, both having good chiral properties but are suited best for the analysis of different aspects of QCD: Chirally Improved (CI) fermions are especially well suited for the investigation of excited hadrons while Fixed-Point (FP) fermions are ideal for studies in the so-called e- and δ-regime.

In the last chapter we started the construction of QCD on the lattice. We discussed a particularly important feature of QCD, namely the self-interaction of gluons. This self-interaction makes pure gluodynamics, i.e., QCD without quarks, an interesting, highly nontrivial theory. Pure gluodynamics shows color confinement, an important property of full QCD. Since gluodynamics is much easier to handle than QCD with quarks, it is an important subject for studies of confinement and its underlying mechanisms.

Chiral symmetry and its spontaneous breaking are central properties of QCD with important phenomenological implications. They explain why pions have unexpectedly small masses and why we do not see degenerate masses for parity partners in the baryon sector. This important role of chiral symmetry and the mechanism of its breaking require that a reasonable lattice version of QCD has to implement chiral symmetry properly.

In our first look at QCD on the lattice in Chap.2 we have already announced that our formulation of lattice fermions has not reached its final form. In particular we still need to take into account the correct statistics. In the first section of this chapter we will show that the required Fermi statistics can be implemented by using anti-commuting numbers for the quark fields, so-called Grassmann numbers. We will discuss the rules for calculating with Grassmann numbers, and we will derive a few key formulas for Gaussian integrals with Grassmann numbers.

A basic simulation of pure SU(3) gauge theory is something that by now can be done already on a modern PC and certainly is a pedagogically valuable exercise. This section provides the techniques necessary for such a calculation. It also serves as an introduction to Monte Carlo simulations, although for the sake of having an easy-to-follow presentation, we concentrate on the simplest algorithms.

In Chap. 1 we derived the lattice path integral for a scalar field theory using the canonical approach for the quantization of the system. Subsequently, we changed our point of view and adopted the lattice path integral itself as a method of quantization. The steps involved are a lattice discretization of the classical Euclidean action and the construction of the measure for the integration over “all configurations” of the classical fields. In this chapter we begin with the implementation of these two steps for quantum chromodynamics.

The implementation of chiral symmetry on the lattice discussed in Chap. 7 was an unsolved puzzle for many years. The problem has continuously inspired the lattice community and led to a variety of formulations for fermions on the lattice. So far we have focused on two kinds of fermions in our presentation, Wilson-type and overlap fermions. Wilson fermions provide a simple robust and easy-to-use discretization, while overlap fermions implement chiral symmetry in the most transparent way.

When introducing the QCD action on the lattice we had to discretize the derivative terms that show up in the continuum action. It was pointed out then that any discretization, e.g., symmetric differences for the first derivative in the fermion action, gives rise to discretization effects. Typically the discretization effects are of (a) for fermions and of (a 2) for the gauge fields. These disappear only in the continuum limit when the lattice spacing it a is sent to zero. Performing the continuum limit is, however, a nontrivial task. As one decreases a, the number of lattice points has to increase, such that the physical volume remains constant (ideally one would first send the number of lattice points to infinity before sending a to zero). Thus in a numerical simulation one always works with finite a and the discretization errors have to be dealt with, e.g., by including them in the extrapolation to vanishing a.

Since the big bang the universe has cooled down substantially. Still, understanding QCD at high temperature and at high matter density is important for various reasons. One of them is obviously a better understanding of quarkand gluon matter shortly after the big bang and of the condensation of hadrons in the cooling process. Many objects in the universe like neutron stars supposedly have high enough density or temperature to expect that hadronic matter behaves differently compared to usual atoms. On earth very energetic collisions of heavy ions provide also possibilities to study matter under extreme conditions. Thus studying QCD at high temperature and density from first principles is a challenging task for the lattice approach.

The basic tool for quantizing fields on the lattice is the Euclidean path integral. Our first chapter is dedicated to the introduction of the path integral formalism and to its interpretation. In order to develop the idea without getting lost in technicalities we introduce the path integral for the simplest case, a scalar field theory. We derive and discuss the two key equations of lattice field theory.

We review the techniques of lattice QCD calculations for excited hadrons with light quarks and outline the future challenges that are faced in calculations with fully dynamical fermions.

QED_2 with mass and flavor has in common many features with QCD, and thus is an interesting toy model relevant for four dimensional physics. The model is constructed using Euclidean path integrals and mass perturbation series. The vacuum functional is carefully decomposed into clustering states being the analogue of the theta-vacuum of QCD. Finally the clustering theory can be mapped onto a generalized Sine-Gordon model. Having at hand this bosonized version, several lessons on the theta-vacuum, the U(1)-problem and Witten-Veneziano-type formulas will be drawn. This sheds light on the corresponding structures of QCD.