Stochastic quantum mechanics and quantum spacetime. A consistent unification of relativity and quantum theory based on stochastic spaces
type: Item
instance of: scholarly article
title: Stochastic quantum mechanics and quantum spacetime. A consistent unification of relativity and quantum theory based on stochastic spaces
zbMATH Open document ID: 0536.60100
author: Eduard Prugovečki
published in: Fundamental Theories of Physics
publication date: 1984-01-01T00:00:00Z
review text: This book contains two parts respectively entitled Stochastic Quantum mechanics (three chapters) and Quantum spacetime (two chapters), plus two appendices (elements of measure theory and elements of operator theory on Hilbert spaces). In chapter I and chapter II are successively defined, in the nonrelativistic case and the relativistic case, respectively: confidence measures \(\mu_{\alpha}\) peaked at the reading \(\alpha(=\) ''smoothed'' confidence intervals), stochastic values \((\alpha,\mu_{\alpha})\), stochastic spaces (\({\mathcal X},{\mathcal A},\underset \tilde{} {\mathcal X})\) where \({\mathcal A}\) is a Boolean \(\sigma\)-algebra of subsets of \({\mathcal X}\), and \(\underset \tilde{} {\mathcal X}\) the set of all the stochastic values \((\alpha,\mu_{\alpha})\), stochastic configuration space \({\underset \tilde{} {\mathbb{R}}}^ N_{conf}=\{(q,\chi_ q);q\in {\mathbb{R}}^ N\}\) with \(\mu_ q(dq')=\chi_ q(q')dq'\), stochastic momentum space \({\underset \tilde{} {\mathbb{R}}}^ N_{mom}=\{(p,\chi_ p);p\in {\mathbb{R}}^ N\}\) with, respectively, \(\chi_ q(q')=\chi_ o(q'-q)\) and \(\chi_ p(p')=\chi_ o(p'-p)\), their Cartesian product called stochastic phase space with \(\chi_{q,p}(q',p')=\chi_ q(q')\chi_ p(p')\), stochastic localizability and systems of covariance (respectively generalizations of sharp localizability and systems of imprimitivity), the phase-space representations of the Galilei group and of the Poincaré group, their generators, the stochastic probability currents and their (classical) sharp point limits when the confidence measures become Dirac measures, the corresponding smoothed propagators and their sharp-point limits. In chapter 3, statistical mechanics is extended on these stochastic phase spaces, unifying classical and quantum formulations: informational completeness, master Liouville spaces, Boltzmann equations. In chapter 4, Born's reciprocity theory (which seems to be the kernel of this book where a smoothing theory of some conditional e…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 60A99
zbMATH DE Number: 3852186
zbMATH Keywords: confidence intervals
zbMATH Keywords: nonrelativistic
zbMATH Keywords: relativistic
zbMATH Keywords: imprimitivity
zbMATH Keywords: propagators
zbMATH Keywords: statistical mechanics
zbMATH Keywords: Born's reciprocity
zbMATH Keywords: relativistic canonical commutation relations
zbMATH Keywords: field theory
zbMATH Keywords: Feynman
zbMATH Keywords: Klein-Gordon
zbMATH Keywords: Dirac
zbMATH Keywords: Fermions
zbMATH Keywords: Stochastic Quantum mechanics
zbMATH Keywords: Quantum spacetime
zbMATH Keywords: stochastic localizability
zbMATH Keywords: quantum field theory
zbMATH Keywords: weak and the strong laws
zbMATH Keywords: Hermite polynomial
zbMATH Keywords: Schrödinger equation
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description: scientific article; zbMATH DE number 3852186