  We extend the formula for correlation functions of free scalar field theories
and Dirac field theories in terms of quantum $A_{\infty}$ algebras presented in
arXiv:2305.11634 to general scalar-Dirac systems. We obtain the result that the
same formula as in the previous paper holds in this case. We show that
correlation functions from our formula satisfy the Schwinger-Dyson equations.
We therefore confirm that correlation functions from our formula express
correlation functions from the ordinary approach of quantum field theory.
