Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Deformation quantization of Poisson manifolds, I

Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

On the Behavior of Cross Sections Near Thresholds

Abstract: The energy dependence of the cross section for the formation of a product, near the threshold energy for that formation, is considered. It is shown that the cross section is, apart from a constant, in the neighborhood of the threshold the same function of energy, no matter what the reaction mechanism is, as long as the long-range interaction of the product particles is the same. The same must hold, because of the principle of detailed balance, for the back reaction, i.e., the reaction between particles with very low relative velocities. In this case, the cross section, as function of the energy, depends only on the long-range interaction of the reacting particles. The energy dependence of the cross section is determined for three types of interactions, viz. no interaction, Coulomb repulsion and Coulomb attraction. The rule for a $\frac{1}{{r}^{2}}$ interaction can be obtained from the first case. Reasons are adduced to show that two interactions, the difference of which goes to zero at least as fast as ${r}^{\ensuremath{-}2\ensuremath{-}\ensuremath{\epsilon}}$ with $\ensuremath{\epsilon}g0$, give the same energy dependence of the cross section. Hence, long-range interaction in the above connection should mean an interaction which, at large distances of the particles, does not go to zero faster than ${r}^{\ensuremath{-}2}$. The effect of small perturbations in the long-range interaction is discussed in general.