MMOnline – Информационный портал о мехмате МГУ

27 мая 2004 года (четверг) в 15:40 в конференц-зале НМУ, Б. Власьевский, 11 состоится очередная лекция семинара Глобус «Non-Commutative Worlds». Лектор – Louis Kauffman (Univ. of Illinois, Chicago).

This talk is about non-commutativity in physics, calculus and topology. We begin by pointing out how the difference quotient in classical discrete calculus can be readjusted to satisfy the Leibniz rule( (fg)' = f'g + fg') at the expense of embedding it in a non-commutative framework, and reexpressing the derivative as a commutator, Df = [f,J], for an appropriate operator J with the property that f(x)J = Jf(x + \delta) where \delta is the increment in the discrete calculus. This suggests reformulating multivariable calculus entirely in terms of commutators. Inside this algebra, $\cal G$, we set up a flat reference world (where all the derivatives commute with one another). The commutators for this flat world have the formal appearance of basic quantum mechanics. The non-commutative world as a whole is full of curvature in the sense of non-commuting derivations The dynamical law is dX_{i}/dt = \cal A_{i} where \cal A is some time-varying element in {\cal G}^n. There is a pivot between commutators and Poisson brackets. Everything said with commutators can be said with Poisson brackets, but the interpretations shift. We did not start with Poisson brackets. We started with commutators and found our way to Poisson brackets. We then show how Hamiltonian mechanics, gauge theory formalism and certain aspects of geometry (e.g. the Levi-Civita connection corresponding to a given metric) fit naturally into this non-commutative world with the Jacobi identity as the key. Here we see a new approach to the formalism of differential geometry, based not on the concept of parallel translation, but rather on the concept of an abstract trajectory in an initially algebraic world. Such algebraic worlds are not yet spaces. The mystery is that, via the use of Poisson brackets, spaces are associated with such worlds, and classical and classical quantum mechanics can arise. We shall discuss the meaning of these transitions. We will compare this foray into non-commutativity with patterns from quantum groups, knot invariants and the use of the Jacobi identity in the study of Vassiliev invariants of knots and links and in the coloring of graphs.

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