  We investigate the ordering dynamics of the voter model with time-delayed
interactions. The dynamical process in the $d$-dimensional lattice is shown to
be equivalent to the first passage problem of a random walker in the
$(d+1)$-dimensional strip of a finite width determined by the delay time. The
equivalence reveals that the time delay leads to the dimensional crossover from
the $(d+1)$-dimensional scaling behavior at short time to the $d$-dimensional
scaling behavior in the long time. The scaling property in both regimes and the
crossover time scale are obtained analytically, which are confirmed with the
numerical simulation results.
