Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Applicandae Mathematicae

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

S. Twareque Ali, Eduard Prugovečki

Published May 1, 1986• 26 citations
Publisher
DOI
10.1007/bf00046932

Related papers

INFINITE-DIMENSIONAL STOCHASTIC ANALYSIS AND FOUNDATIONS OF QUANTUM MECHANICS

Andrei Yu. Khrennikov

WORLD SCIENTIFIC eBooks•Apr 1, 2007•0 citations

Proceedings of the Norbert Wiener Centenary Congress, 1994 : Michigan State University, November 27-December 3, 1994

Norbert Wiener Centenary Congress, Vidyadhar S. Mandrekar, P. R. Masani

Wiener-Kolmogorov conception of the stochastic organization of nature by S. Molchanov The Wiener programme in statistical physics. Is it feasible in the light of recent developments? by O. Penrose The mathematical ramifications of Wiener's program in statistical physics by L. Gross Potential capacity before and after Wiener by D. R. Adams Generalized harmonic analysis and Gabor and wavelet systems by J. J. Benedetto Wiener-Hopf equations and linear systems by J. Bart, I. Gohberg, and M. A. Kaashoek Complex harmonic analysis in the aftermath of Paley-Wiener by P. Malliavin Amalgam spaces and generalized harmonic analysis by H. G. Feichtinger Wiener space and nonlinear quantum field theory by I. Segal Wiener and Feynman--path integrals and their applications by S. Albeverio Optical coherence before and after Wiener by J. R. Klauder Wiener and the problem of Hidden Parameters in quantum mechanics by E. A. Carlen Finite path integrals by E. F. Thomas Joint probabilities of noncommuting observables and the Einstein-Podolsky-Rosen question in Wiener-Siegel quantum theory by R. L. Warnock The role of Leibniz and Haldane in Wiener's cybernetics by G. Gale Quantum mechanical coherence, resonance, and mind by H. P. Stapp What is mind that the brain may order it? by K. H. Pribram Shannon-Wiener information and stochastic complexity by J. Rissanen Sharp norm comparison of Martingale maximal functions and stochastic integrals by D. L. Burkholder Stochastic analysis on Wiener space by S. Watanabe Stochastic filtering: A part of stochastic nonlinear analysis by G. Kallianpur The theory of learning before and after Wiener by N. K. Bose Sensory and motor prostheses in the aftermath of Wiener by R. W. Mann On Wiener's thought in the computer as an aid in visualizing higher-dimensioanl forms and its modern ramifications by D. Roseman Norbert Wiener and the future of cybernetics by P. R. Masani Letter to Norbert Wiener from John von Neumann Scientific impact of the work of C. E. Shannon by B. McMillan Citation and award of the Norbert Wiener Centenary Medal to Professor C. E. Shannon (in absentia) by J. Ramaekers Wiener and the art of communication by D. Gabor Academic vita of Norbert Wiener Doctoral students of Norbert Wiener Bibliography of Norbert Wiener Defense Department documents List of participants (Norbert Wiener centenary congress).

•Jan 1, 1997•1 citation

Stochastic quantization and geometry of quantum phase space

Blagovest A. Nikolov, D. A. Trifonov

Bulgarian Journal of Physics•Jan 1, 1988•1 citation

The Schrödinger–Robinson inequality from stochastic analysis on a complex Hilbert space

Andrei Yu. Khrennikov

We explored the stochastic analysis on a complex Hilbert space to show that one of the cornerstones of quantum mechanics (QM), namely Heisenberg's uncertainty relation, can be derived in the classical probabilistic framework. We created a new mathematical representation of quantum averages: as averages with respect to classical random fields. The existence of a classical stochastic model matching with Heisenberg's uncertainty relation makes the connection between classical and quantum probabilistic models essentially closer. In real physical situations, random fields are valued in the L2-space. Hence, although we model QM and not QFT, the classical systems under consideration have an infinite number of degrees of freedom. And in our modeling, infinite-dimensional stochastic analysis is the basic mathematical tool.

Physica Scripta•Feb 11, 2013•0 citations

Chapter 4. Stochastic analysis and quantum mechanics

A. A. Dorogovtsev

De Gruyter eBooks•Dec 31, 1994•0 citations

Quantum Harmonic Analysis

Maurice A. de Gosson

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.

De Gruyter eBooks•Jun 21, 2021•17 citations
Wiener-Kolmogorov conception of the stochastic organization of nature by S. Molchanov The Wiener programme in statistical physics. Is it feasible in the light of recent developments? by O. Penrose The mathematical ramifications of Wiener's program in statistical physics by L. Gross Potential capacity before and after Wiener by D. R. Adams Generalized harmonic analysis and Gabor and wavelet systems by J. J. Benedetto Wiener-Hopf equations and linear systems by J. Bart, I. Gohberg, and M. A. Kaashoek Complex harmonic analysis in the aftermath of Paley-Wiener by P. Malliavin Amalgam spaces and generalized harmonic analysis by H. G. Feichtinger Wiener space and nonlinear quantum field theory by I. Segal Wiener and Feynman--path integrals and their applications by S. Albeverio Optical coherence before and after Wiener by J. R. Klauder Wiener and the problem of Hidden Parameters in quantum mechanics by E. A. Carlen Finite path integrals by E. F. Thomas Joint probabilities of noncommuting observables and the Einstein-Podolsky-Rosen question in Wiener-Siegel quantum theory by R. L. Warnock The role of Leibniz and Haldane in Wiener's cybernetics by G. Gale Quantum mechanical coherence, resonance, and mind by H. P. Stapp What is mind that the brain may order it? by K. H. Pribram Shannon-Wiener information and stochastic complexity by J. Rissanen Sharp norm comparison of Martingale maximal functions and stochastic integrals by D. L. Burkholder Stochastic analysis on Wiener space by S. Watanabe Stochastic filtering: A part of stochastic nonlinear analysis by G. Kallianpur The theory of learning before and after Wiener by N. K. Bose Sensory and motor prostheses in the aftermath of Wiener by R. W. Mann On Wiener's thought in the computer as an aid in visualizing higher-dimensioanl forms and its modern ramifications by D. Roseman Norbert Wiener and the future of cybernetics by P. R. Masani Letter to Norbert Wiener from John von Neumann Scientific impact of the work of C. E. Shannon by B. McMillan Citation and award of the Norbert Wiener Centenary Medal to Professor C. E. Shannon (in absentia) by J. Ramaekers Wiener and the art of communication by D. Gabor Academic vita of Norbert Wiener Doctoral students of Norbert Wiener Bibliography of Norbert Wiener Defense Department documents List of participants (Norbert Wiener centenary congress).
We explored the stochastic analysis on a complex Hilbert space to show that one of the cornerstones of quantum mechanics (QM), namely Heisenberg's uncertainty relation, can be derived in the classical probabilistic framework. We created a new mathematical representation of quantum averages: as averages with respect to classical random fields. The existence of a classical stochastic model matching with Heisenberg's uncertainty relation makes the connection between classical and quantum probabilistic models essentially closer. In real physical situations, random fields are valued in the L2-space. Hence, although we model QM and not QFT, the classical systems under consideration have an infinite number of degrees of freedom. And in our modeling, infinite-dimensional stochastic analysis is the basic mathematical tool.
Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.