Exact Results in Quantum Theory

The Kontsevich star-product admits a well-defined restriction to the class of affine – in particular, linear – Poisson brackets; its graph expansion consists only of Kontsevich’s graphs with in-degree ≤ 1 for aerial vertices. We obtain the formula *aff mod ō(ℏ7) with harmonic propagators for the graph weights (over n ≤ 7 aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet–Felder–Willwacher, that they match the computations using the kontsevint software by Panzer, and the resulting affine star-product is associative modulo ō(ℏ7). We discover that the Riemann zeta value ζ(3)2/π6, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of *aff mod ō(ℏ7) because all the Q-linear combinations of Kontsevich graphs near ζ(3)2/π6 represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula *red mod ō(ℏ7) with only rational coefficients. We also discover that the mechanism of associativity for the star-product up to ō(ℏ6) is different from the mechanism at order 7. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod ō(ℏ6), whereas at order (ℏ7) and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step. (Joint work with R. Buring, see arXiv:2209.14438, arXiv:2309.16664, and arXiv:1702.00681 in Experimental Mathematics.)