  In material science, it was established that as the number of particles $ N $
in a material gets more and more, especially in the thermodynamic limit,
various macroscopic quantum phenomena such as superconductivity, superfluidity,
quantum magnetism, Fractional quantum Hall effects and various quantum or
topological phase transitions (QPT) emerge in such non-relativistic quantum
many-body systems. There is always a reservoir which exchanges energy and
particles with the material. This is the essence of P. W. Anderson's great
insight `` More is different ''. However, there is still a fundamental
component missing in this general picture: How the `` More is different ''
becomes different in a moving inertial frame or a moving sample? Here we
address this outstanding problem.We demonstrate our claims by studying one of
the simplest QPTs: Superfluid (SF)-Mott transitions of interacting bosons in a
square lattice in a sample moving with a constant velocity $ \vec{v} $. We
first elaborate the crucial difference between a moving sample and a moving
inertial frame, stressing the crucial roles played by a reservoir in a grand
canonical ensemble which is needed to study the SF-Mott transition in the first
place. In this work, we mainly present the moving sample case and only discuss
very briefly the moving inertial frame case. It is the moving which mixes the
space and time. We also stress the important roles played by the underlying
lattice. Doing various light or neutron scattering measurements in a moving
sample may become an effective way not only measure various intrinsic
properties of the materials, tune various quantum and topological phases
through phase transitions, but also probe the new emergent space-time structure
near any QPT.
