Quantum Mechanics and Fundamentals of Quantum Field Theory.- Path Integral Quantization.- Broken Symmetry and Phase Transition.- Some Applications - Warming Ups.- Superconductivity.- Quantum Hall Liquids and Chern--Simons Gauge Field.- Appendix.

Why is quantum field theory of condensed matter physics necessary? Condensed matter physics deals with a wide variety of topics, ranging from gas to liquids and solids, as well as plasma, where owing to the inter­ play between the motions of a tremendous number of electrons and nuclei, rich varieties of physical phenomena occur. Quantum field theory is the most appropriate "language", to describe systems with such a large number of de­ grees of freedom, and therefore its importance for condensed matter physics is obvious. Indeed, up to now, quantum field theory has been succesfully ap­ plied to many different topics in condensed matter physics. Recently, quan­ tum field theory has become more and more important in research on the electronic properties of condensed systems, which is the main topic of the present volume. Up to now, the motion of electrons in solids has been successfully de­ scribed by focusing on one electron and replacing the Coulomb interaction of all the other electrons by a mean field potential. This method is called mean field theory, which made important contributions to the explanantion of the electronic structure in solids, and led to the classification of insulators, semiconductors and metals in terms of the band theory. It might be said that also the present achievements in the field of semiconductor technology rely on these foundations. In the mean field approximation, effects that arise due to the correlation of the motions of many particles, cannot be described.