  We extend the formula for correlation functions of scalar field theories in
terms of quantum $A_{\infty}$ algebras presented in arXiv:2203.05366 to
incorporate Dirac fields. We use a description that is analogous to string
field theory, and the formula for correlation functions takes the same form for
both scalar fields and Dirac fields. We prove that correlation functions from
our formula satisfy the Schwinger-Dyson equations in the free theory. The proof
for interacting theories is presented in the companion paper arXiv:2305.13103
by one of the authors.
