Feynman path integrals on phase space and the metaplectic representation

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 $ρ(e^{X})$ ( $X\in\mathfrak{sp}(2n,\mathbb{R})$ ) is expressed in terms of generalized Feynman--Kac(--Itô) formulas, but in real-time (not imaginary-time) path integral form. Olshanski semigroups play the leading role in the proof of it.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 $ρ(e^{X})$ ($X\in\mathfrak{sp}(2n,\mathbb{R})$) is expressed in terms of generalized Feynman--Kac(--Itô) formulas, but in real-time (not imaginary-time) path integral form. Olshanski semigroups play the leading role in the proof of it.

arXiv:2209.04139Sep 9, 20220 citations

Dfinition of the Feynman path integral in the

P

-representation

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

ρ(e^{X})

(

X\in\mathfrak{sp}(2n,\mathbb{R})

) is expressed in terms of generalized Feynman--Kac(--Itô) formulas, but in real-time (not imaginary-time) path integral form. Olshanski semigroups play the leading role in the proof of it.