  It is well-known that the 4-rotation in the 4-dimensional space-time is
equivalent to the CPT-transformation (C is the charge conjugation, P is the
space inversion and T is the time-reversal). The standard definition of the
T-reversal includes the change of the sign of time variable and replacement of
the initial state of the particle (system of particles) by the final state and
vice versa. Since the time-reversal operation changes the state of a particle,
the particle's wave function cannot be the eigenfunction of the corresponding
operator with a certain eigenvalue, as in the case of space parity. Unlike the
CPT-transformation, the separate P, T, or C transformations cannot be reduced
to any 4-rotation. The extended Lorentz group incorporates all the separate C,
P, or T transformations which do not bring the time axis out of the
corresponding light cone. The latter restriction is included in the standard
definition of the time-reversal. In the present communication, we ignore this
restriction.
  This allows to introduce the "time arrow" operator and characterize every
particle by the new quantum number - "time arrow" value. The wave functions of
all particles are eigenfunctions of this operator with eigenvalues equal to
"time arrow" values. The particles with the "time arrow" values opposite to the
"time arrow" value in our universe form another universe (anti-universe). The
existence of anti-universe can be confirmed, in principle, by laboratory
(atomic) experiments. The anti-universe may be also considered as a candidate
to the role of dark matter.
