Lattice QCD at Finite Temperature and Density

A general introduction into the subject aimed at a general theoretical physics audience. We introduce the sign problem posed by finite density lattice QCD, and we discuss the main methods proposed to circumvent it, with emphasis on the imaginary chemical potential approach. The interrelation between Taylor expansion and analytic continuation from imaginary chemical potential is discussed in detail. The main applications to the calculation of the critical line, and to the thermodynamics of the hot and normal phase are reviewed.. 1. Outline This is a general introduction into the subject aimed at a general theoretical physics audience. Several recent results on the behavior of the cold and hot phases, as well as of the critical line have been included, however no attempt was made at being exhaustive: further material is to be found in the contributions by A. Nakamura and Ph. De Forcrand, as well as in introductory 1 and topical reviews 2. This writeup is organized as follows: we give a first introduction and physical motivation. Next, we briefly review field theory thermodynamics, and the lattice approach. Technical difficulties (the sign problem) are introduced there, while the three main methods to circumvent them are discussed in the following Section. The following section discusses in some detail the interrelation between the Taylor expansion and the imaginary chemical potential approach. Section VI, VII, VIII give an overview of the results on the critical line, hot and cold phases respectively. A short summary concludes the paper.