Stochastic quantum mechanics and quantum spacetime. A consistent unification of relativity and quantum theory based on stochastic spaces
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 expectation theory is described, see for instance \textit{K. R. Parthasarathy}, Probability measures on metric spaces. (1967; Zbl 0153.191), chapter 5, section 8: ''Regular conditional probability''; the reviewer conjectures here a way to define Feynman propagators with the martingale theorem) and the relativistic harmonic oscillator quark model are written down according to this measurement formalism which leads to a mass formula (external and internal relativistic canonical commutation relations). In chapter 5, field theory on quantum spacetime is proposed for smoothing the difficulties of local quantum field theory, with corresponding Lagrangian field theory, Dirac bispinors representations, treatment of photon localizability, renormalization. Well, this stochastic quantum mechanics is not the one of I. Fenyes, E. Nelson, F. Guerra, etc. But these two stochastic theories are not incompatible, the author's theory is an image theory through ''smoothed conditional expectation'', Fenyes' theory is a source theory, and they could be fused if necessary. I would emphasize a new fact about Fenyes' theory, which in reviewer's mind is nothing but a Brownian-Broglian theory where mechanics is built upon the strong large numbers law through Huygens' principle: A paper of \textit{D. S. Shucker}, J. Funct. Anal. 38, 146-155 (1980; Zbl 0447.60042) opens a new era, almost sure convergence is at last introduced in the stochastic theory at its right place through a proof which in fact mimics the so-called equivalence theorem connecting the weak and the strong laws [cf. \textit{T. Hida}, Brownian motion. (1980; Zbl 0432.60002) page 30], but the right proof of the strong law is based on martingale property (which gives both convergence and structure). Here Doob's inequality becomes \[ -const\quad T_ i^{-1}-T_ f^{-1}\leq E(N)(a-b)+E[M_{T_ i}-b)^+] \] where \(T_ i\) is some initial time, because the process \(M_ t\) is not a martingale but tends to be a martingale (in Shucker's paper \(M_ t=X_ t/t)\), and can be the generating exponential (asymptotical) martingale which we can develop into Hermite polynomial (asymptotical) martingales. So, we can construct a great fugue, of which the subject \(S_ n/n\) and the counter-subject \((S_ n/n-p)^ 2-pq/n\) are the germs of the Poincaré-Cartan form pdq-Edt, of which the variations, developments and reversements run through Wiener process, Schrödinger equation, Klein- Gordon equation, spinorial equations: the author's theory is nothing but a part of the stretta \((=the\) end) of this great fugue which might include, mainly, Wheeler's geometrodynamics or something like that.
relativistic canonical commutation relations
scientific article; zbMATH DE number 3852186
scientific article