Arthemy Kiselev (RuG)
Deformation quantisation of field theory models: iterated variations in Kontsevich's star-products
This talk will consist of two parts. First, we recall the concept of associativity-preserving deformation quantisation on finite-dimensional smooth real Poisson manifolds. By definition, the usual, commutative multiplication in the ring of functions is then deformed to the star-product --- which contains a given Poisson bracket in the leading deformation term and which, although in general no longer commutative, does stay associative. Kontsevich proved (1997) that the entire deformation series is completely determined by the Poisson structure in its linear part; for effective construction of higher-degree terms in the power series of star-products, he developed the technique of summation over weighted graphs.
In the second part of the talk we shall extend the finite-dimensional Poisson set-up and graph summation technique to the infinite-dimensional Poisson geometry of field theory models. This extension became possible by using methods from the geometry of iterated variations (2013); nontrivial yet straightforward, the new approach allows for deformation quantisation of local functionals such as the models' action or observables. For instance, we shall derive the variational analogue of associative Moyal's star-product, then addressing the mechanism of associativity preservation for less elementary variational Poisson geometries.
(The associativity of scattering, or the triangle equation, is an empiric fact which must be respected by the various quantisation schemes. Satisfying this requirement by construction, the variational deformation quantisation of field theory models is also regular, that is, contains no built-in divergences.)