Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.     

Entropy Numbers of Trudinger-Strichartz Embeddings of Radial Besov Spaces and Applications

The asymptotic behaviour of entropy numbers of Trudinger–Strichartzembeddings of radial Besov spaces on Rⁿ into exponential Orliczspaces is calculated. Estimates of the entropy numbers as wellas estimates of entropy numbers of Sobolev embeddings of radialBesov spaces are applied to spectral theory of certain pseudo-differentialoperators.     

Solubility of Lanthanide Chlorides ( Ln Cl3) in Alkali Metal Chlorides ( M Cl): Thermodynamics and Electrical Conductivity of the M 3 Ln Cl6 Compounds

Summary.  Several compounds may exist in LnCl₃–MCl mixtures. Those corresponding to the M ₂ LnCl₅ and MLn ₂Cl₇ stoichiometries are formed in a few systems only, with diverse stability strongly dependent on both the corresponding lanthanide and alkali metal. On the other hand, M ₃ LnCl₆ that occur in most systems have a far larger stability range and melt congruently. These latter compounds were investigated in the present work by differential scanning calorimetry and electrical conductivity measurements. The thermodynamic and transport properties were correlated to structural features and related to the mechanism of compound formation. Corresponding author. E-mail: Marcelle.Gaune-Escard@polytech.univ-mrs.fr Received October 2, 2002; accepted November 6, 2002 Published online April 24, 2003 RID="a" ID="a" This paper is dedicated to Professor H. Gamsj?ger on the occasion of his birthday     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Analysis of the normal modes of molecules with D3h symmetry: Infrared spectra of monomeric s-triazine and cyanuric acid

Normal modes of s-triazine and cyanuric acid were calculated at the DFT(B3LYP)/6-311++G(d,p) level. These modes were analyzed in terms of potential energy distribution (PED), computed using a specially designed set of symmetry coordinates. The normal-mode analysis was described in detail and the PED matrix definition used in the calculations was provided. Particular attention has been devoted to the degenerate vibrations. The experimental infrared spectra of s-triazine and cyanuric acid isolated in low-temperature Ar matrices have been recorded and interpreted by comparison with the theoretically predicted normal modes. In the spectrum of matrix-isolated s-triazine, the IR bands originating from ¹³C and ¹⁵N isotopologues with one of the ring atoms substituted by a rare isotope were detected. These bands were identified thanks to the excellent agreement between the experimentally observed and theoretically predicted isotope shifts.     

A convenient synthesis of deuterium labeled tertiary aliphatic nitro ketones and nitriles - starting materials for preparation of deuterated cyclic nitrones, isomeric hydroxylamines, and corresponding C-nitroso compounds

Tertiary aliphatic β- and γ-nitro nitriles and ketones deuterated in (several) selected positions had been synthesized. The deuterated nitro compounds served as a starting material for the corresponding deuterium labeled nitrones or hydroxylamines (reducing with aluminum amalgam). Further oxidation of the last two groups of compounds with sodium periodate or m-CPBA afforded the relevant deuterated tertiary C-nitroso compounds.     

2-Substituted nitrones and isomeric hydroxylamines - obtained via aluminium amalgam reduction of nitro nitriles and ketones—a new access to convenient intermediates for nitroso carbonyl compounds preparation

Substituted five-membered cyclic nitrones (pyrroline N-oxides) have been obtained in good to high yields from tertiary γ-nitro ketones and nitriles employing aluminium amalgam as a reducing agent in moist diethyl ether or THF. Attempts to obtain cyclic amino nitrones from α- or β-nitro nitriles failed and only the corresponding hydroxylamines have been isolated. Both nitrones and hydroxylamines have been used for synthesis of tertiary C-nitroso nitriles or ketones.     

Fundamental Problems in the Unification of Physics

We discuss the following problems, plaguing the present search for the “final theory”: (1) How to find a mathematical structure rich enough to be suitably approximated by the mathematical structures of general relativity and quantum mechanics? (2) How to reconcile nonlocal phenomena of quantum mechanics with time honored causality and reality postulates? (3) Does the collapse of the wave function contain some hints concerning the future quantum gravity theory? (4) It seems that the final theory cannot avoid the problem of dynamics, and consequently the problem of time. What kind of time, if this theory is supposed to be background free? (5) Will the dynamics of the “final theory” be probabilistic? Quantum probability exhibits some essential differences as compared with classical probability; are they but variations of some more general probabilistic measure theory? (6) Do we need a radically new interpretation of quantum mechanics, or rather an entirely new theory of which the present quantum mechanics is an approximation? (7) If the final theory is to be background free, it should provide a mechanism of space-time generation. Should we try to explain not only the generation of space-time, but also the generation of its material content? (8) As far as the existence of the initial singularity is concerned, one usually expects either “yes” or “not” answers from the final theory. However, if the mathematical structure of the future theory is supposed to be truly more general that the mathematical structures of the present general relativity and quantum mechanics, is a “third answer“ possible? Could this third answer be related to the probabilistic character of the final theory? We discuss these questions in the framework of a working model unifying gravity and quanta. The analysis reveals unexpected aspects of these rather wildly discussed issues.     

Laser frequency stabilization by magnetically assisted rotation spectroscopy

We present a method of Doppler-free laser frequency stabilization based on magnetically assisted rotation spectroscopy (MARS) which combines the Doppler-free velocity-selective optical pumping (VSOP) and magnetic rotation spectroscopy. The stabilization is demonstrated for the atomic rubidium transitions at 780 nm. The proposed method is largely independent of stray magnetic fields and does not require any modulation of the laser frequency. Moreover, the discussed method allows one to choose between locking the laser exactly to the line center, or with a magnetically-controlled shift to an arbitrary frequency detuned by up to several natural linewidths. This feature is useful in many situations, e.g. for laser cooling experiments. In addition to presenting the principle of the method, its theoretical background and peculiarities inherent to the repopulation VSOP are discussed.