Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions

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 a...

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Bibliographic Details
Published in:Probability theory and related fields Vol. 183; no. 3-4; pp. 909 - 995
Main Authors: Albeverio, Sergio, Borasi, Luigi, De Vecchi, Francesco C., Gubinelli, Massimiliano
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022
Springer Nature B.V
Subjects:
Algebra
Applied mathematics
Banach spaces
Brownian motion
Differential equations
Economics
Euclidean geometry
Euclidean space
Fermions
Finance
Gaussian process
Hilbert space
Homomorphisms
Insurance
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Measurement
Operations Research/Decision Theory
Partial differential equations
Probability
Probability theory
Probability Theory and Stochastic Processes
Quantitative Finance
Quantum field theory
Quantum theory
Random variables
Spacetime
Statistics for Business
Theoretical
Vector spaces
White noise
Grassmann algebras
Euclidean fermion fields
81S20
Stochastic quantization
Non-commutative probability
Constructive quantum field theory
Non-commutative stochastic partial differential equations
Yukawa model
60H15
81T08
ISSN:0178-8051, 1432-2064
Online Access:Get full text
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Summary: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.
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ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-022-01136-x