  This paper presents an algebraic formulation of the renormalization group
flow in quantum mechanics on flat target spaces. For any interacting quantum
mechanical theory, the fixed point of this flow is a theory of classical
probability, not a different effective quantum mechanics. Each energy
eigenstate of the UV Hamiltonian flows to a probability distribution whose
entropy is a natural diagnostic of quantum ergodicity of the original state.
These conclusions are supported by various examples worked out in some detail.
