  We present expressions for correlation functions of scalar field theories in
perturbation theory using quantum $A_\infty$ algebras. Our expressions are
highly explicit and can be used for theories both in Euclidean space and in
Minkowski space including quantum mechanics. Correlation functions at a given
order of perturbation theory can be calculated algebraically without using
canonical quantization or the path integral, and we demonstrate it explicitly
for $\varphi^3$ theory. We show that the Schwinger-Dyson equations are
satisfied as an immediate consequence of the form of the expressions based on
quantum $A_\infty$ algebras.
