Numerical simulations in quantum field theory of elementary particles

Quantum field theory of particles like electrons or quarks, which are the elementary constituents of matter, describes the interactions between these particles, in particular their frequent transmutations due to quantum fluctuations. Many experimental data on the structure of matter (e.g., of proton), and its properties observed by means of the high energy particle accelerators, can be explained by such a theory. In many cases massive numerical simulations of this theory formulated on a finite space-time lattice are required, however. We explain the basic physical principles of the lattice formulation of quantum field theory and its mathematical realization. For finite values of the lattice spacing and finite volumes, the theory can be simulated numerically by means of the Monte Carlo methods. The process of approaching the continuous space-time, using the statistical theory of critical phenomena, is understood in principle, but it requires immense computational resources and elaborate algorithms. We introduce some of these algorithms and give examples of current achievements like an explanation of the confinement of quarks, the fact that quarks cannot be observed experimentally as individual free particles.