Stochastic field theory, holomorphic quantum mechanics, and supersymmetry

Abstract

Holomorphic quantum mechanics are studied from the point of view of stochastic quantization in Minkowski space which involves the introduction of two stochastic fields, one in the external space and the other in the internal space. The equilibrium condition is given by Z2 symmetry between the external and internal fields. In the nonequilibrium case, N=2 Wess–Zumino quantum fields are arrived at giving rise to supersymmetry. This helps to define the supercharge operator Q when the Hamiltonian is given by H=Q2 and an index theorem is derived for an interacting case when the superpotential is given by V(φ)=λφn, φ being complex with n>2. It is found that the vacuum is degenerate and is in conformity with the result obtained by Jaffe, Lesniewski, and Lewenstein [Ann. Phys. 178, 313 (1987)] in the two-dimensional N=2 Wess–Zumino quantum field model.