O(N) and O(N) and O(N)

Understand

Three related analyses of $\phi^4$ theory with $O(N)$ symmetry are presented.

• In the first, we review the $O(N)$ model over the $p$-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context.
• We demonstrate the existence of a Wilson-Fisher fixed point using an $\epsilon$ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point.
• Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms.