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Symmetry

From Quantum to Classical Within the Framework of Integral Quantization

Ligia M. C. S. Rodrigues, Evaldo M. F. Curado, Diego Noguera, A. C. Maioli

Published Feb 25, 2026• 0 citations
Publisher

Abstract

Integral quantization is a powerful framework for mapping classical phase-space functions—defined on a symplectic manifold—onto quantum operators in a Hilbert space. It encompasses several quantization methods, such as coherent-state quantization, and inherently incorporates operator symmetrization. The formalism relies on a choice of weight function, whose flexibility allows for a family of possible quantizations. In this work, we address the inverse problem: given a quantum operator, how can one determine a classical phase-space function whose integral quantization reproduces exactly that operator? We propose a systematic method, within the integral quantization framework, to construct such a classical function, which depends on the chosen weight. We demonstrate that quantizing the resulting function recovers the original operator, thereby establishing a consistent two-way mapping between classical and quantum descriptions. The method is applied to several physically relevant operators: the projector, a mixed-state density operator, the annihilation operator, and an entangled state. We also analyze how quantum entanglement manifests in the structure of the corresponding classical phase-space function.

DOI
10.3390/sym18030403

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Integral quantization is a powerful framework for mapping classical phase-space functions—defined on a symplectic manifold—onto quantum operators in a Hilbert space. It encompasses several quantization methods, such as coherent-state quantization, and inherently incorporates operator symmetrization. The formalism relies on a choice of weight function, whose flexibility allows for a family of possible quantizations. In this work, we address the inverse problem: given a quantum operator, how can one determine a classical phase-space function whose integral quantization reproduces exactly that operator? We propose a systematic method, within the integral quantization framework, to construct such a classical function, which depends on the chosen weight. We demonstrate that quantizing the resulting function recovers the original operator, thereby establishing a consistent two-way mapping between classical and quantum descriptions. The method is applied to several physically relevant operators: the projector, a mixed-state density operator, the annihilation operator, and an entangled state. We also analyze how quantum entanglement manifests in the structure of the corresponding classical phase-space function.
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Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some symplectic manifold MMM and the non-degenerate 2-form ωωω, which comes with MMM. Geometric prequantization is a process by which one does this in a mathematically "rigorous" manner and we shall attempt to just introduce the methods here. We do this by exploring this contruction for (R2n,∑i=1ndpi∧dqi\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_iR2n,∑i=1n​dpi​∧dqi​) which avoids using complex line bundles in any non-trivial way. One should note however that the Hilbert space one obtains is in fact too "big", in the sense that it has too many functions in order to correspond with actual physically significant Hilbert spaces. Geometric quantization remedies this situation but it should be noted that not all manifolds are prequantizable. We shall not discuss either of these issues however.
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We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.