  The non energy-weighted Gamow-Teller(GT) sum rule is satisfied in
relativistic models, when all nuclear density-dependent terms, including Pauli
blocking terms from nucleon-antinucleon excitations, are taken into account in
the RPA correlation function. The no-sea approximation is equivalent to this
approximation for the giant GT resonance state and satisfies the sum rule, but
each of the total $\beta_-$ and $\beta_+$ strengths is different in the two
approximations. It is also shown that the energy-weighted sum of the GT
strengths for the $\beta_-$ and $\beta_+$ transitions in RPA is equal to the
expectation value of the double commutator of the nuclear Hamiltonian with the
GT operator, when the expectation value is calculated with the ground state in
the mean field approximation. Since the present RPA neglects renormalization of
the divergence, however, the energy-weighted strengths outside of the giant GT
resonance region become negative. These facts are shown by calculating in an
analytic way the GT strengths of nuclear matter.
