  The perturbative path-integral gives a morphism of the (quantum) $A_{\infty
}$ structure intrinsic to each quantum field theory, which we show explicitly
on the basis of the homological perturbation. As is known, in the BV formalism,
any effective action also solves the BV master equation, which implies that the
path-integral can be understood as a morphism of the BV differential. Since
each solution of the BV master equation is in one-to-one correspondence with a
(quantum) $A_{\infty }$ structure, the path-integral preserves this intrinsic
$A_{\infty }$ structure of quantum field theory, where $A_{\infty }$ reduces to
$L_{\infty }$ whenever multiplications of space-time fields are graded
commutative. We apply these ideas to string field theory and (re-)derive some
quantities based on the perturbative path-integral, such as effective theories
with finite $\alpha ^{\prime }$, reduction of gauge and unphysical degrees,
$S$-matrix and gauge invariant observables.
