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.

Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions / S. Albeverio, L. Borasi, F.C. De Vecchi, M. Gubinelli. - In: PROBABILITY THEORY AND RELATED FIELDS. - ISSN 0178-8051. - 183:3-4(2022 Aug), pp. 909-995. [10.1007/s00440-022-01136-x]

Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions

L. Borasi
Secondo
;
F.C. De Vecchi
Penultimo
;
2022

Abstract

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.
Grassmann algebras; Euclidean fermion fields; Stochastic quantization; Non-commutative probability; Non-commutative stochastic partial differential equations; Constructive quantum field theory; Yukawa model;
Settore MAT/06 - Probabilita' e Statistica Matematica
ago-2022
21-mag-2022
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1019283
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