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After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), $ 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications.The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered. 1. I N T R O D U C T I O NH E recent rapid growth of physics is producing not only an ever-growing spate of papers-often repeating what has been published independently T elsewhere-but also an increasing degree of specialization. Different branches of physics often use different terms for the same concept, and it takes a long time before new techniques filter through from one branch of physics to another. This is a great pity since in theoretical physics, for instance, techniques which are useful in one field are often also applicable to other fields. Two such techniques will be discussed in the present paper. T h e first-and main-one is the use of the so-called density matrix which was introduced by von Neumann (1927 a, b, c ; see also Dirac 1929, and von Neumann 1932, and was until recently used mainly in statistical mechanics. T h e other one is the use of the so-called Green functions which were originally used in field theory and which PO

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