  The investigation of thermalization in isolated quantum many-body systems has
a long history, dating back to the time of developing statistical mechanics.
Most quantum many-body systems in nature are considered to thermalize, while
some never achieve thermal equilibrium. The central problem is to clarify
whether a given system thermalizes, which has been addressed previously, but
not resolved. Here, we show that this problem is undecidable. The resulting
undecidability even applies when the system is restricted to one-dimensional
shift-invariant systems with nearest-neighbour interaction, and the initial
state is a fixed product state. We construct a family of Hamiltonians encoding
dynamics of a reversible universal Turing machine, where the fate of a
relaxation process changes considerably depending on whether the Turing machine
halts. Our result indicates that there is no general theorem, algorithm, or
systematic procedure determining the presence or absence of thermalization in
any given Hamiltonian.
