  The main aim of this article is to show some intimate relations among the
following three notions: (1) the metaplectic representation of
$Sp(2n,\mathbb{R})$ and its extension to some semigroups, called the Olshanski
semigroup for $Sp(2n,\mathbb{R})$ or Howe's oscillator semigroup, (2)
antinormally-ordered quantizations on the phase space
$\mathbb{R}^{2m}\cong\mathbb{C}^{m}$, (3) path integral quantizations where the
paths are on the phase space $\mathbb{R}^{2m}\cong\mathbb{C}^{m}$. In the Main
Theorem, the metaplectic representation $\rho(e^{X})$
($X\in\mathfrak{sp}(2n,\mathbb{R})$) is expressed in terms of generalized
Feynman--Kac(--It\^{o}) formulas, but in real-time (not imaginary-time) path
integral form. Olshanski semigroups play the leading role in the proof of it.
