Karen Yeats
The Hopf Algebraic Approach to Dyson-Schwinger Equations
Friday August 3 to Tuesday August 7
Dyson-Schwinger equations are very useful. Mathematically, they are
nice in how they mirror the recursive decomposition of Feynman
diagrams into subdiagrams. This simple combinatorial observation is
surprisingly powerful as it gives us hints as to how to unwind the
combinatorial difficulties from the analytic ones. The Hopf algebras
of renormalization developed by Connes and Kreimer, are the main tool
which lets us exploit this. These lectures will begin by describing
the Hopf algebraic approach to Dyson-Schwinger equations.
One key observation of this approach, which is best described in the
work of Walter van Suijlekom, is that the Slavnov-Taylor identities
for the coupling constants correspond to certain Hopf ideals.
Furthermore, this relates to how graph insertion behaves in Hochschild
cohomology. The lectures will continue by explaining these
connections without expecting prior algebraic experience.
However, there is much more that we can do. In my PhD thesis I
developped a reduction process which reduces standard Dyson-Schwinger
equations to simpler systems of ordinary differential equations. The
approach is to exploit the nesting of subgraphs into graphs, and
introduce new combinatorial objects that pull some of the analytic
side of the Dyson-Schwinger equations into the same combinatorial
framework. I will explain these reductions and their consequences.
The biggest difficulty with this approach is that it retains as input
a modified series built from the primitive Feynman diagrams. To
proceed we either need to approximate (see the work of Marc Bellon) or
to understand this function better. Recently some progress has been
made on the latter approach. We will discuss the problems and
progress.
Finally we will discuss applications of this approach to QCD.
References:
Dirk Kreimer and Karen Yeats, An Étude in non-linear Dyson-Schwinger
Equations. Nucl. Phys. B Proc. Suppl., 160, (2006), 116-121.
Dirk Kreimer and Karen Yeats, Recursion and growth estimates in
renormalizable quantum field theory. Commun. Math. Phys. 279, no.2,
(2008), 401-427.
Walter D. van Suijlekom, Renormalization Hopf algebras for gauge
theories and BRST-symmetries, in Motives, Quantum Field Theory, and
Pseudodifferential Operators, Clay Mathematics Proceedings. 12 (2010).
Guillaume van Baalen, Dirk Kreimer, David Uminsky and Karen Yeats, The
QCD beta-function from global solutions to Dyson-Schwinger
equations. Ann. Phys. 325, 2, (2010), 300-324.
Marc Bellon, An Efficient Method for the Solution of Schwinger-Dyson
equations for propagators. Letters in Mathematical Physics 94, 1
(2010) 77-86.
Karen Yeats, Rearranging Dyson-Schwinger equations.
Memoir. Am. Math. Soc. 211, no. 995, (2011).