arXiv:2601.23259

Geometric Quantization by Paths, Part III: The Metaplectic Anomaly

Patrick Iglesias-Zemmour

Published Jan 30, 2026• 0 citations

Abstract

In the previous parts of this work, we established the Prequantum Groupoid $\mathbf{T}_ω$ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, $E_0 = n\hbar/2$. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

DOI: 10.48550/arxiv.2601.23259

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arXiv:2601.23259

Geometric Quantization by Paths, Part III: The Metaplectic Anomaly

Patrick Iglesias-Zemmour

Published Jan 30, 2026• 0 citations

arXiv HTML PDF

Abstract

In the previous parts of this work, we established the Prequantum Groupoid

\mathbf{T}_ω

as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator,

E_0 = n\hbar/2

. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

DOI: 10.48550/arxiv.2601.23259

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Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $\tau$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in $1+1$ dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation, however, suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued from $\mathbb{R}$ to $\mathbb{C}$ and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories already for the free theory and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time $1$ -dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $\tau$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in $1+1$ dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation, however, suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued from $\mathbb{R}$ to $\mathbb{C}$ and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories already for the free theory and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time $1$-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.

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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

A Derivation of Geometric Quantization via Feynman's Path Integral on Phase Space

Joshua Lackman

arXiv:2405.17273May 27, 20240 citations

In the previous parts of this work, we established the Prequantum Groupoid

\mathbf{T}_ω

E_0 = n\hbar/2

. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

\tau

which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in

1+1

dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation, however, suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued from

\mathbb{R}

\mathbb{C}

and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories already for the free theory and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time

1

-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.

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.