QFT on the non-commutative Moyal space and combinatorics

In this chapter we present the Phi? QFT model on the non-commutative Moyal space and the UV/IR mixing issue, which prevents it from being renormalizable. We then present the Grosse–Wulkenhaar Phi? QFT model on the non-commutative Moyal space, which changes the usual propagator of the Phi? model (based on the heat kernel formula) to a Mehler kernel based propagator. This Grosse–Wulkenhaar model is perturbatively renormalizable but it is not translation-invariant (translation-invariance being a usual property of high-energy physics models). We then show how the Mellin transform technique can be used to express the Feynman integrals of the Grosse-Wulkenhaar model. In the last part of the chapter, we present another Phi? QFT model on the non-commutative Moyal space, which is however both renormalizable and translation-invariant. We show the relation between the parametric representation of this model and the Bollobás–Riordan polynomial.

– OFF-DIAGONAL ESTIMATES FOR THE ORNSTEIN–UHLENBECK SEMIGROUP: SOME POSITIVE AND NEGATIVE RESULTS

We investigate $L^{p}(\unicode[STIX]{x1D6FE})$–$L^{q}(\unicode[STIX]{x1D6FE})$ off-diagonal estimates for the Ornstein–Uhlenbeck semigroup $(e^{tL})_{t>0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$), these estimates hold in an unrestricted sense, while, for sufficiently small $t$, they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup [Formula: see text]. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers [Formula: see text] of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for [Formula: see text]. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.