  The pp-wave/SYM correspondence is an equivalence relation, H_{string}= Delta
-J, between the Hamiltonian H_{string} of string field theory in the pp-wave
background and the dilatation operator Delta in N=4 Super Yang-Mills in the
double scaling limit. We calculate matrix elements of these operators in string
field theory and in gauge theory.In the string theory Hilbert space we use the
natural string basis,and in the gauge theory we use the basis which is
isomorphic to it. States in this basis are specific linear combinations of the
original BMN operators, and were constructed previously for the case of two
scalar impurities. We extend this construction to incorporate BMN operators
with vector and mixed impurities. This enables us to verify from the gauge
theory perspective two key properties of the three-string interaction vertex of
Spradlin and Volovich: (1) the vanishing of the three-string amplitude for
string states with one vector and one scalar impurity; and (2) the relative
minus sign in the string amplitude involving states with two vector impurities
compared to that with two scalar impurities. This implies a spontaneous
breaking of the Z_2 symmetry of the string field theory in the pp-wave
background. Furthermore, we calculate the gauge theory matrix elements of Delta
-J for states with an arbitrary number of scalar impurities. In all cases we
find perfect agreement with the corresponding string amplitudes derived from
the three-string vertex.
