General Theory of Many-particle Systems

Stephen J. Gustafson, Israel Michael Sigal

Florence A. Aeschlimann

A study was made of the general conditions required to set up a non- relativistic functional theory for systems of particles. Representation of each particle of a system by a space-time function uj(P,T). Formal classification for the particles of the system with respect to indiscernability and the exclusion principle for the fermions. Using the general theory of predictions, the arguments of the functional for predictions are fixed, functional configuration- space is introduced, and the general form of the equation for the predictional elements is determined. Conditions incurred by the limiting cases of the usual wave mechanics for systems and of the single particle in the functional theory of particles. Fundamental principle for the equations of motion concerning the particles of a system: each function uj(P,T) describing a particle of the system obeys a non-linear equation which depends upon the functions u/sub k/(P,T) of the other particles of the system. Brief investigation of the limiting case of classical mechanics. (auth)

K. Kong Wan

A general physical system is a compound system consisting of many constituents. A constituent may be made up of more than one particle, e.g., a molecule is made of atoms which themselves are made up of many particles. In a superconductor at low temperatures the electrons form pairs. 1 Each pair can be treated as a constituent of a superconducting current. For convenience we also call these constituents particles . A general physical system is referred to as a many-particle system. The theory for many-particle systems can be built up from one-particle theory. The situation can be complicated since there are different types of many-particle systems: A compound system may consists of a definite number of particles, i.e., the number of particles in the system remains the same for all time. Such a system may consist of distinguishable particles, or identical, i.e., indistinguishable, particles. 560 A compound system may consists of an indefinite number of particles, i.e., the number of particles in the system may change in time due to interactions. The particles involved may or may not be identical.

Paul C. Martin, Julian Schwinger

Léon Van Hove

Władysław Marcinek

An algebraic formalism for the study of interacting particle systems is developed. Particle processes are described in terms of the category theory. The problem for the unique description of these processes is discussed. Categories relevant for this subject are described. The concept of generalized transmutations of interacting particle systems is considered. The connection with a system with some generalized statistics is explained.