  The study of thermalization in isolated quantum many-body systems has a long
history, dating back to the time of developing statistical mechanics. Most of
quantum many-body systems in nature are considered to thermalize, while some
systems are known not to reach the thermal equilibrium states. Presence or
absence of thermalization is strongly related to the concepts of chaos and
integrability, both of which have also been intensively studied for a long
time. A central problem in this field is to clarify whether a given system
thermalizes or not, which has been tackled by vast literature but not yet been
resolved. Here we prove that this problem is undecidable. The result of
undecidability even applies to the case that the system is restricted to a
one-dimensional shift-invariant systems with nearest-neighbor interaction and
the initial state is a fixed product state. To this end, we construct a family
of Hamiltonians encoding dynamics of a reversible universal Turing machine,
where the fate of a relaxation process is dramatically changed depending on
whether the Turing machine halts or not. Our result indicates that there is no
general theorem, algorithm or systematic procedure determining the presence or
absence of thermalization in any given Hamiltonian.
