The Master Field for Rainbow Diagrams and Free Non-Commutative Random Variables

The master field for a subclass of planar diagrams, so called rainbow diagrams, for higher dimensional large N theories is considered. An explicit representation for the master field in terms of noncommutative random variables in the modified interaction representation in the Boltzmannian Fock space is given. A natural interaction in the Boltzmannian Fock space is formulated by means of a rational function of the interaction Lagrangian instead of the ordinary exponential function in the standard Fock space. Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow, e-mail: [email protected] On leave absence from Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow