Instantons with noise: Equations for two-dimensional models.

Abstract

We discuss a generalization of (Euclidean) quantum mechanics on a manifold to infinite-dimensional complex manifolds. For configuration spaces characterized by a topological charge we obtain first-order partial differential equations, which are random perturbations of instanton equations. The complex structure of the manifold is essential for Euclidean invariance and for the Markov property, which enable a construction of relativistic quantum fields. Semiclassical estimates for the randomly perturbed dynamical system show the relevance of classical approximations to quantum models. The solution of a stochastic equation determines a nonlinear transformation of the (free) Gaussian measure. The functional measure defined by the solution contains in general the logarithm of a spinor determinant in addition to a sum of the local action and the topological charge. We discuss the solutions of stochastic equations and the functional measure for the Wess-Zumino, ${\mathrm{CP}}^{\mathrm{n}}$, and Higgs models.