Abstract
We introduce the $p$-adic particle-in-a-box as a free particle with periodic boundary conditions in the $p$-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the $p$-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of theories connected by a kind of renormalization group flow. We propose that Berkovich spaces can be used to relate $p$-adic and Archimedean quantities generally.
