  We study covariant open bosonic string field theories on multiple $Dp$-branes
by using the deformed cubic string field theory which is equivalent to the
string field theory in the proper-time gauge. Constructing the Fock space
representations of the three-string vertex and the four-string vertex on
multiple $Dp$-branes, we obtain the field theoretical effective action in the
zero-slope limit. On the multiple $D0$-branes, the effective action reduces to
the Banks-Fishler-Shenker-Susskind (BFSS) matrix model. We also discuss the
relation between the open string field theory on multiple $D$-instantons in the
zero-slope limit and the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model.
The covariant open string field theory on multiple $Dp$-branes would be useful
to study the non-perturbative properties of quantum field theories in
$(p+1)$-dimensions in the framework of the string theory. The non-zero-slope
corrections may be evaluated systematically by using the covariant string field
theory.
