Generalized Langevin Equation and Anomaly

Abstract

In quantum field theories, a physical observable is expressed as a function or functional of field configurations. Therefore, once the Langevin equations for the field configurations are given, we can derive a Langevin equation for a physical observable itself, which we call a generalized Langevin equation (Namiki, Ohba and Tanaka 1986; Namiki, Ohba, Tanaka and Yanga 1987. See also Caracciolo, Ren and Wu 1985; Aldazabal et al. 1983). This derivation is based on the so-called Ito calculus, explained in Appendix B. In this chapter, we derive the generalized Langevin equations for some physical systems, and then apply them to anomaly problems.