Assessing quantum thermalization in physical and configuration spaces via many-body weak values

We explore the origin of the arrow of time in an isolated quantum system described by the Schrödinger equation. We provide an explanation from weak values in the configuration space, which are understood as operational properties obtained in the laboratory following a well-defined protocol. We show that quantum systems satisfying the eigenstate thermalization hypothesis can simultaneously provide thermalized ensemble expectation values and nonthermalized weak values of the momentum, both from the same operational probability distribution. The reason why weak values of the momentum may escape from the eigenstate thermalization hypothesis is because they are linked only to off-diagonal elements of the density matrix in the energy representation. Therefore, we state that the origin of the arrow of time in isolated quantum systems described by the Schrödinger equation comes from dealing with properties obtained by averaging (tracing out) some degrees of freedom of the configuration space. We then argue that thermalization does not occur in the properties defined in the configuration space, and our argument is compatible with defending that thermalization is a real phenomenon in the properties defined in the physical space. All of these conclusions are testable in the laboratory through many-body weak values.