Abstract: This thesis examines the consequences of non-Archimedean geometry in holography and many-body physics. In this framework real space is replaced by a non-Archimedean field which is almost always a padic field or an algebraic extension of a p-adic field. The corresponding bulk geometry is replaced by a discrete tree. The resulting theories are not only useful as toy models of real theories. They help elucidate which parts of physics do and do not depend on the choice of underlying field.

Chapter 1 sets the stage for the rest of the thesis and gives a brief review of non-Archimedean mathematics and physics.

Chapter 2 is based on work with Steven S. Gubser, Christian Jepsen, and Ziming Ji. We exhibit a sparsely coupled classical statistical mechanical lattice model that interpolates between real and p-adic geometry when varying a spectral exponent. Hölder continuity conditions in both real and p-adic space allow us to quantify how smooth or ragged the two-point Green’s function is as a function of the spectral exponent. This model was motivated by proposed cold atom experiments and serves as our bridge into the nonArchimedean world.

Chapter 3 is based on work with Steven S. Gubser, Christian Jepsen, and Ziming Ji. We study melonic tensor models over real and non-Archimedean fields in tandem. Much attention is paid to the combinatorial structure of potential interaction terms and the perturbative expansion. The SchwingerDyson equation is solved exactly in the p-adic case for a subset of these field theories.

Chapter 4 is based on work with Steven S. Gubser and Christian Jepsen. We examine the bulk dual p-adic field theories whose Green's functions are non-trivial sign characters. These theories constitute the simplest known notion of spin in p-adic physics. The construction is achieved by the introduction of an non-dynamical U(1) gauge field on the discrete bulk geometry and the two point functions for dual operators on the boundary are computed explicitly.