The Explicit Formula and a Propagator

Abstract

I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. But in the end, I argue that the new formulation of the Explicit Formula signals a specific link with Quantum Fields, as opposed to the Hilbert-Polya operator idea (which leads rather to Quantum Mechanics).