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

Correlation functions involving Dirac fields from homotopy algebras II: the interacting theory