2020/04/20 by Sergio Albeverio, Albeverio, Sergio, Luigi Borasi +5
Mathematics · Physics and Astronomy · #46L89 #81S20 #81T08 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 60H15 #Probability (math.PR) #Quantum Mechanics and Applications #Secondary 46L53
paper · pdf · doi:10.48550/arxiv.2004.09637
openalex publication_date 2020/04/20 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C∗-algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model.