Abstract
A Euclidean topological action is always purely imaginary; its stochastic quantization inevitably involves the complex Langevin equation. We show that the standard results for abelian Chern-Simons theory can be reproduced in the stochastic approach with the Maxwell term as a regularization; but if one ignores the factor of i, the Langevin equation will become pathological. Simplification may occur if one uses the generalized Langevin equation with an appropriate, purely imaginary kernel; we exemplify this by stochastic quantization in Minkowski space-time and again the abelian Chern-Simons theory. The stochastic perturbation theory of non-abelian Chern-Simons theory is also studied and the contributions of usual Faddeev-Popov ghosts are verified to be reproducible without gauge fixing
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