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Journal of Physics A Mathematical and General

Metaplectic spinor fields on phase space: A path integral approach

E. Gozzi, M Reuter

Published Nov 21, 1993• 8 citations
Publisher

Abstract

We study spinor fields on the phase space of a generic Hamiltonian system. Under linearized canonical transformations these spinors transform according to the metaplectic representation of Sp(2N). We derive a path integral for their time evolution and discuss their dynamical and geometrical properties. In particular we show that they can be interpreted as semiclassical wavefunctions for the associated Hamiltonian.

DOI
10.1088/0305-4470/26/22/030

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We study spinor fields on the phase space of a generic Hamiltonian system. Under linearized canonical transformations these spinors transform according to the metaplectic representation of Sp(2N). We derive a path integral for their time evolution and discuss their dynamical and geometrical properties. In particular we show that they can be interpreted as semiclassical wavefunctions for the associated Hamiltonian.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables: they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role. (orig.)
The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.
The main aim of this article is to show some intimate relations among the following three notions: (1) the metaplectic representation of Sp(2n,R)Sp(2n,\mathbb{R})Sp(2n,R) and its extension to some semigroups, called the Olshanski semigroup for Sp(2n,R)Sp(2n,\mathbb{R})Sp(2n,R) or Howe's oscillator semigroup, (2) antinormally-ordered quantizations on the phase space R2m≅Cm\mathbb{R}^{2m}\cong\mathbb{C}^{m}R2m≅Cm, (3) path integral quantizations where the paths are on the phase space R2m≅Cm\mathbb{R}^{2m}\cong\mathbb{C}^{m}R2m≅Cm. In the Main Theorem, the metaplectic representation ρ(eX)ρ(e^{X})ρ(eX) (X∈sp(2n,R)X\in\mathfrak{sp}(2n,\mathbb{R})X∈sp(2n,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.
We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a ccc-number function H(p,q)\mathcal{H}(p,q)H(p,q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel T\mathcal{T}T is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$, q=p‾q=\overline{p}q=p​ reflection by studying changes of variables in the path integral. The ordering dependence of T\mathcal{T}T is displayed. The invariance of commutation rules is also discussed.