the same physical variables are indexed by both upper and lower indices. The inverse vielbein is denoted by $f^i_k$, so that $f^i_k e^k_j
=\d^i_j$. The operators with diagonal free two-point functions are \[oops\] \_k = f\^i\_k \_i S\^[-1]{} \_k where we have defined the Hilbert space operator $  S^{-1}$ by the last equality. Our $  S^{-1}$ has exactly the properties of $S^{-1/2}$ in [@sv]. In particular the string basis Hamiltonian is \[sham\] [H]{} = S H S\^[-1]{} which was shown to be exactly the Hamiltonian of the string bit formalism [@v; @vv] whose order $g_2$ splitting/joining interaction agrees with string field theory and order $g_2^2$ contact term also agrees (if a certain truncation of intermediate states is made) [@rsv].

The toy models of Section 3 can now be described in a new notation in which the states $\co_i$ and ${\tilde \co}_j$ span different bases of the Hilbert space, the gauge theory basis and the string basis, respectively. The time evolution problems treated in Section 3 as models for gauge theory can be described as follows. We found (approximately) the state \[state1\] |(t)= \_[i]{} a\^i(t) \_i which evolves with time by the Hamiltonian $H$ and satisfies an initial condition $a^i(0) = 