If “quantization is an art”, it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.If “quantization is an art”, it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.

Jan 1, 20250 citations

Is Time a Cyclic Dimension? How to Second-Quantize Fields by Means of Periodic Boundary Conditions. https://doi.org/10.1016/j.aop.2022.169182

Dolce, Donatello

If "quantization is an art", it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.If "quantization is an art", it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.

Jan 1, 20250 citations

Quantum-Classical Correspondence: Dynamical Quantization and the Classical Limit

A. O. Bolivar

Jan 11, 201316 citations

Quantization of space-time and the corresponding quantum mechanics

Miklós Banai

Foundations of PhysicsDec 1, 198512 citations

Quantization and Mirror Reduction Do Not Commute in Hamiltonian Embeddings of Nonreciprocal Dynamics

Gaurav Sarmah, Ramakrishna Podila

Hamiltonian embeddings can represent dissipative nonreciprocal classical dynamics exactly on invariant manifolds of enlarged reciprocal systems. We show that canonical quantization of such an embedding need not commute with reduction to the target dynamics. For the mirror construction recently introduced for pairwise nonreciprocal interactions, the invariant manifold is Lagrangian. Exact quantum enforcement of the mirror condition therefore removes the dynamical sector rather than producing a quantum analogue of the reduced flow. If the constraint is imposed only semiclassically, contraction of the target dynamics generates inverse-transpose expansion in the conjugate mirror directions. Quantum uncertainty then gives $\ln D\geΞ(t)+n\ln[π/(σε)]$ , where $Ξ=-\ln|\det M|$ is the accumulated contraction and $D$ the torus Hilbert-space dimension. Exact finite-dimensional Weyl evolution of one- and two-degree-of-freedom nonreciprocal models confirms the resulting logarithmic correspondence time, including its predicted change when the initial localization scales with $\hbar$ . Thus the classical embedding is exact, but its direct canonical quantization is a singular semiclassical construction.Hamiltonian embeddings can represent dissipative nonreciprocal classical dynamics exactly on invariant manifolds of enlarged reciprocal systems. We show that canonical quantization of such an embedding need not commute with reduction to the target dynamics. For the mirror construction recently introduced for pairwise nonreciprocal interactions, the invariant manifold is Lagrangian. Exact quantum enforcement of the mirror condition therefore removes the dynamical sector rather than producing a quantum analogue of the reduced flow. If the constraint is imposed only semiclassically, contraction of the target dynamics generates inverse-transpose expansion in the conjugate mirror directions. Quantum uncertainty then gives $\ln D\geΞ(t)+n\ln[π/(σε)]$, where $Ξ=-\ln|\det M|$ is the accumulated contraction and $D$ the torus Hilbert-space dimension. Exact finite-dimensional Weyl evolution of one- and two-degree-of-freedom nonreciprocal models confirms the resulting logarithmic correspondence time, including its predicted change when the initial localization scales with $\hbar$. Thus the classical embedding is exact, but its direct canonical quantization is a singular semiclassical construction.

quant-pharXiv:2609.28494Sep 4, 2026

\ln D\geΞ(t)+n\ln[π/(σε)]

, where

Ξ=-\ln|\det M|

is the accumulated contraction and

D

the torus Hilbert-space dimension. Exact finite-dimensional Weyl evolution of one- and two-degree-of-freedom nonreciprocal models confirms the resulting logarithmic correspondence time, including its predicted change when the initial localization scales with

\hbar

. Thus the classical embedding is exact, but its direct canonical quantization is a singular semiclassical construction.