Stochastic Quantization Method of Fermion Fields

Recently, Parisi and Wu l ) proposed a stochastic method of quantizing boson fields. In this method, an extra fifth time t is introduced and the evolution of classical boson fields with respect to the time t is described by the Langevin equation. Then, it can be proved that, in the t-infinity limit, correlation functions in the stochastic process are identical with the corresponding' ones which are given in the usual path integral form of Euclidean field theories. It is essential for this equivalence between the stochastic process and the path integral that the probability distribution has the unique equilibrium when t goes to infinity. The approach to the equilibrium is realized by the damping factor coming from the drift force proportional to Euclidean Klein-Gordon operator (0 + m ). It is the purpose of this paper to investigate whether and how the stochastic quantization can be extended to fermion. At the first sight, some difficulties may arise in its extension: (j) There is no classical analogue because of the anti-commuting nature of fermion fields. (ij) What is the origin of the damping factor in fermionic case where the classical equation of motion is described by a first differential equation (Dirac equation) ? The point (i) leads us to use Grassmann numbers, which seem to be incompatible with the concept of probability. *) However, we can safely construct the probabilistic interpretation even in this case. As for the point (ij), the mass term of fermion field gives the damping factor, when the Langevin equations are set up properly.