Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Ordering problem in quantum mechanics: Prime quantization and a physical interpretation

S. Ali, H. D. Doebner

Published Feb 1, 1990• 23 citations
Publisher

Abstract

A method for quantizing a classical system, based upon the notion of localization on phase space, is developed. This method, to which is given the name prime quantization, is based upon the mathematical theory of positive-operator-valued measures and their relationship to reproducing kernel Hilbert spaces, and to a notion of generalized kinematical variables related to phase space. For a system with ${\mathrm{openR}}^{n}$ as its configuration space, it has the advantage of being able to connect the well-known problem of ordering in quantum mechanics to the choice of a measuring apparatus in a joint measurement of position and momentum (within the limits of the uncertainty principle). Moreover, for such a system, a choice of ordering in the present context also turns out to be a choice of polarization in the method of geometric quantization.

DOI
10.1103/physreva.41.1199

Related papers

From classical to quantum mechanics: “How to translate physical ideas into mathematical language”

Hervé Bergeron

Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].

arXiv:quant-ph/0102084•Journal of Mathematical Physics•Sep 1, 2001•12 citations

An Introduction To Geometric Prequantization

Geraci, Joseph

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 $M$ and the non-degenerate 2-form $ω$, which comes with $M$. 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 ($\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_i$) 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.

arXiv:0911.0964•Nov 5, 2009•4 citations

Phase space quantization and the operator moment problem

Jukka Kiukas, PEKKA J. LAHTI, Kari Ylinen

We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Journal of Mathematical Physics•Jul 1, 2006•23 citations

Mirror Geometry as the Physical Origin of Quantum Uncertainty

Andrzej Statowski

This work introduces a geometric reinterpretation of Heisenberg’s uncertainty principle using a mirror as a physical operator that separates conjugate variables into two independent information spaces. Position is encoded in the real space, while momentum is transformed into a reflection angle in the mirror space. This separation preserves quantum mechanics while allowing full state information without violating Heisenberg’s inequality. The model naturally reproduces entanglement as a correlation between the original and reflected states. ─── Conceptual Introduction (for general readers) Quantum mechanics states that position and momentum cannot be measured simultaneously with arbitrary precision. This is usually interpreted as a fundamental limit of nature. In this work, we propose a different view: A mirror creates a second information space. 1. In this space: the position of a particle remains in the real world, the momentum of the particle becomes a reflection angle. Thus: position lives in one space, momentum lives in another, and the conflict between them disappears. The uncertainty principle is not broken — its geometry is changed. This simple idea leads to a powerful reinterpretation of quantum information. ─── Formal Model (for scientific readers) 2.1 Two Hilbert Spaces We define two independent Hilbert spaces: [ \mathcal{H}R \quad \text{(real space)} ] [ \mathcal{H}{R'} \quad \text{(mirror space)} ] The full state becomes: [ |\Psi\rangle = |\psiR\rangle \otimes |\psi{R'} 2. \rangle ] Position and momentum no longer coexist in the same space. ─── 2.2 Mirror Operator The mirror is defined as a linear operator: [ \hat{M}: \mathcal{H}R \rightarrow \mathcal{H} {R'} ] acting as: [ \hat{M}|\psiR(x)\rangle = |\psi{R'}(x')\rangle ] with: [ x' = f(x) ] where (f) is the geometric reflection. ─── 2.3 Momentum Becomes Reflection Angle In mirror space, momentum is replaced by a geometric observable: [ \hat{\theta} = g(\hat{p}) ] Example mapping: [ \theta = \arctan\left(\frac{p}{p_0}\right) ] Thus: position → real space momentum → angle in mirror space And therefore: [ [\hat{x}, \hat{\theta}] = 0 ] This is the mathematical heart of the model. ─── 2.4 Modified Uncertainty Relation Standard quantum mechanics: [ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} ] In the mirror model: [ \Delta x \in \mathcal{H}R ] [ \Delta \theta \in \mathcal{H}{R'} ] Thus: [ \Delta x \cdot \Delta \theta \quad \text{is not constrained by Heisenberg} ] because (x) and (\theta) are not conjugate variables in the same space. ─── 2.5 Entanglement as Mirror Correlation The original and reflected states form a natural entangled pair: [ |\Psi\rangle = \sumi ci |xi\rangle \otimes | \thetai\rangle ] A change in position immediately changes the reflection angle: [ x \rightarrow x + \delta x ] [ \theta \rightarrow \theta + \delta\theta ] This reproduces quantum entanglement as a geometric correlation. ─── Main Conclusion A mirror acts as a geometric operator that: separates conjugate variables into two spaces, 3. 1. transforms momentum into reflection angle, allows full state information without violating Heisenberg, provides a physical model of entanglement, extends quantum mechanics without contradicting it. This is the third pillar of the broader theoretical framework developed in earlier works.

•Sep 10, 2026•0 citations

Ordering problem in geometrical quantization

Dehai Bao, Zhongyuan Zhu

The operator ordering problem is investigated in the framework of geometrical quantization in the Schrodinger coordinate representation. It is shown that, for quantization of polarization-preserving classical observables pf(q), correct Hermitian quantum operators are derived; for quantization of polarization-unpreserving classical observables p 2 f(q), the BKS kernel method actually defines an ordering rule which is not Weyl's rule, but is the GCT-invariant rule proposed by DeWitt (1957).

Journal of Physics A Mathematical and General•Apr 21, 1992•3 citations

Localization of relativistic particles and uncertainty relations

Daniel R. Terno

Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.

arXiv:1308.0479•Physical Review A•Apr 21, 2014•24 citations
A method for quantizing a classical system, based upon the notion of localization on phase space, is developed. This method, to which is given the name prime quantization, is based upon the mathematical theory of positive-operator-valued measures and their relationship to reproducing kernel Hilbert spaces, and to a notion of generalized kinematical variables related to phase space. For a system with openRn{\mathrm{openR}}^{n}openRn as its configuration space, it has the advantage of being able to connect the well-known problem of ordering in quantum mechanics to the choice of a measuring apparatus in a joint measurement of position and momentum (within the limits of the uncertainty principle). Moreover, for such a system, a choice of ordering in the present context also turns out to be a choice of polarization in the method of geometric quantization.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
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.
We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
This work introduces a geometric reinterpretation of Heisenberg’s uncertainty principle using a mirror as a physical operator that separates conjugate variables into two independent information spaces. Position is encoded in the real space, while momentum is transformed into a reflection angle in the mirror space. This separation preserves quantum mechanics while allowing full state information without violating Heisenberg’s inequality. The model naturally reproduces entanglement as a correlation between the original and reflected states. ─── Conceptual Introduction (for general readers) Quantum mechanics states that position and momentum cannot be measured simultaneously with arbitrary precision. This is usually interpreted as a fundamental limit of nature. In this work, we propose a different view: A mirror creates a second information space. 1. In this space: the position of a particle remains in the real world, the momentum of the particle becomes a reflection angle. Thus: position lives in one space, momentum lives in another, and the conflict between them disappears. The uncertainty principle is not broken — its geometry is changed. This simple idea leads to a powerful reinterpretation of quantum information. ─── Formal Model (for scientific readers) 2.1 Two Hilbert Spaces We define two independent Hilbert spaces: [ \mathcal{H}R \quad \text{(real space)} ] [ \mathcal{H}{R'} \quad \text{(mirror space)} ] The full state becomes: [ |\Psi\rangle = |\psiR\rangle \otimes |\psi{R'} 2. \rangle ] Position and momentum no longer coexist in the same space. ─── 2.2 Mirror Operator The mirror is defined as a linear operator: [ \hat{M}: \mathcal{H}R \rightarrow \mathcal{H} {R'} ] acting as: [ \hat{M}|\psiR(x)\rangle = |\psi{R'}(x')\rangle ] with: [ x' = f(x) ] where (f) is the geometric reflection. ─── 2.3 Momentum Becomes Reflection Angle In mirror space, momentum is replaced by a geometric observable: [ \hat{\theta} = g(\hat{p}) ] Example mapping: [ \theta = \arctan\left(\frac{p}{p_0}\right) ] Thus: position → real space momentum → angle in mirror space And therefore: [ [\hat{x}, \hat{\theta}] = 0 ] This is the mathematical heart of the model. ─── 2.4 Modified Uncertainty Relation Standard quantum mechanics: [ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} ] In the mirror model: [ \Delta x \in \mathcal{H}R ] [ \Delta \theta \in \mathcal{H}{R'} ] Thus: [ \Delta x \cdot \Delta \theta \quad \text{is not constrained by Heisenberg} ] because (x) and (\theta) are not conjugate variables in the same space. ─── 2.5 Entanglement as Mirror Correlation The original and reflected states form a natural entangled pair: [ |\Psi\rangle = \sumi ci |xi\rangle \otimes | \thetai\rangle ] A change in position immediately changes the reflection angle: [ x \rightarrow x + \delta x ] [ \theta \rightarrow \theta + \delta\theta ] This reproduces quantum entanglement as a geometric correlation. ─── Main Conclusion A mirror acts as a geometric operator that: separates conjugate variables into two spaces, 3. 1. transforms momentum into reflection angle, allows full state information without violating Heisenberg, provides a physical model of entanglement, extends quantum mechanics without contradicting it. This is the third pillar of the broader theoretical framework developed in earlier works.
Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.