Quantum field theory over \(\mathbb F_q\)
The electronic journal of combinatorics, Tome 18 (2011) no. 1
We consider the number $\bar N(q)$ of points in the projective complement of graph hypersurfaces over $\mathbb{F}_q$ and show that the smallest graphs with non-polynomial $\bar N(q)$ have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class $\bar N(q)$ depends on the number of cube roots of unity in $\mathbb{F}_q$. At graphs with 16 edges we find examples where $\bar N(q)$ is given by a polynomial in $q$ plus $q^2$ times the number of points in the projective complement of a singular K3 in $\mathbb{P}^3$. In the second part of the paper we show that applying momentum space Feynman-rules over $\mathbb{F}_q$ lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.
@article{10_37236_589,
author = {Oliver Schnetz},
title = {Quantum field theory over \(\mathbb {F_q\)}},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/589},
zbl = {1217.05110},
url = {http://geodesic.mathdoc.fr/articles/10.37236/589/}
}
Oliver Schnetz. Quantum field theory over \(\mathbb F_q\). The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/589
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