  We present a theoretical framework called Lorentz quantum mechanics, where
the dynamics of a system is a complex Lorentz transformation in complex
Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the
unitary transformation in Hilbert space. In our Lorentz quantum mechanics,
there exist three types of states, space-like, light-like, and time-like.
Fundamental aspects are explored in parallel to the usual quantum mechanics,
such as matrix form of a Lorentz transformation, construction of Pauli-like
matrices for spinors. We also investigate the adiabatic evolution in this
mechanics, as well as the associated Berry curvature and Chern number. Three
typical physical systems, where this Lorentz quantum dynamics can arise, are
presented. They are one dimensional fermion gas, Bose-Einstein condensate (or
superfluid), and one dimensional antiferromagnet.
