If you take $n$ independent quantum fields $\hat{\varphi}_r(x)$ with $r = 1,\dots,n$, each of which satisfy the Heisenberg equations $\partial_t \hat{\varphi}_r(x) = i [\hat{H},\hat{\varphi}_r(x)]$, $\partial_t \hat{\pi}_r(x) = i [\hat{H},\hat{\pi}_r(x)]$, and commutation relations between $\hat{\pi}_r$ and $\hat{\varphi}_s$, the matrix elements of these $\hat{\varphi}_r(x)$ operators have to satisfy
$$\langle \Psi_p'|\hat{\varphi}_r(x')|\Psi_q'\rangle = S_{rs}(a) \langle \Psi_p|\hat{\varphi}_s(x)|\Psi_q\rangle, x' = ax + b, a \in O(1,3)$$