  The leading term for the average height of the barriers which could separate
pure states in Ising spin glasses is calculated using instanton methods. It is
finite in dimensions $d < 6$. Replica symmetry breaking requires that the
barriers between pure states are infinite in the thermodynamic limit, as finite
barriers allow thermal mixing of pure states. As a consequence, a replica
symmetry broken phase cannot survive when $d < 6$. However, for $d > 6$ no
similar instanton solution exists.
