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We introduce the $p$-adic particle-in-a-box as a free particle with periodicboundary conditions in the $p$-adic spatial domain. We compute its energyspectrum, and show that the spectrum of the Archimedean particle-in-a-box canbe recovered from the $p$-adic spectrum via an Euler product formula. Thisproduct formula arises from a flow equation in Berkovich space, which weinterpret as a space of theories connected by a kind of renormalization groupflow. We propose that Berkovich spaces can be used to relate $p$-adic andArchimedean quantities generally.

From $p$-adic to Archimedean Physics: Renormalization Group Flow and Berkovich Spaces