  As argued in our previous papers, it would be more natural to modify the
standard approach to quantum theory by requiring that i) one unitary
irreducible representation (UIR) of the symmetry algebra should describe a
particle and its antiparticle simultaneously. This would automatically explain
the existence of antiparticles and show that a particle and its antiparticle
are different states of the same object. If i) is adopted then among the
Poincare, so(2,3) and so(1,4) algebras only the latter is a candidate for
constructing elementary particle theory. We extend our analysis in
hep-th/0210144 and prove that: 1) UIRs of the so(1,4) algebra can indeed be
interpreted in the framework of i) and cannot be interpreted in the framework
of the standard approach; 2) as a consequence of a new symmetry (called AB one)
between particles and antiparticles for UIRs satisfying i), elementary
particles described by UIRs of the so(1,4) algebra can be only fermions; 3) as
a consequence of the AB symmetry, the vacuum condition can be consistent only
for particles with the half-integer spin (in conventional units) and therefore
only such particles can be elementary. In our approach the well known fact that
fermions have imaginary parity is a consequence of the AB symmetry.
