A purely geometrical Aharonov-Bohm effect

Aidan Zlotak; Jean-Pierre Gazeau; Romain Murenzi; Tomoi Koide

REVIEW 4 major objections 4 minor 49 references

A purely geometrical Aharonov-Bohm effect

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quantizing a free particle on a punctured plane yields the Aharonov-Bohm gauge field from topology alone, with no external solenoid.

desk verdict A real ACIQ derivation of an AB-like vector potential on the punctured plane, but the flux is set by a free weight parameter and the example violates the paper's own symmetry condition. read the letter →

arxiv 2412.13919 v7 pith:JLMYM7VV submitted 2024-12-18 quant-ph math-phmath.MP

Jean-Pierre Gazeau , Tomoi Koide , Romain Murenzi , Aidan Zlotak This is my paper · ORCID

classification quant-phmath-phmath.MP

keywords Aharonov-BohmeffectaffinecovariantintegralquantizationpuncturedplanesimilitudegroupSIM(2)emergentgaugepotentialtopologicalquantuminterferenceflux

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the Aharonov-Bohm (AB) effect—the shift of electron interference by a magnetic field the electron never touches—can originate from the quantization procedure itself when the plane is punctured by an impenetrable solenoid. The authors quantize a free particle on the punctured plane $\mathbb{R}^2_* = \mathbb{R}^2 \setminus \{0\}$ using affine covariant integral quantization, a method built for phase spaces with translation, rotation, and dilation symmetry rather than plain translation symmetry. They find that the quantized kinetic energy automatically takes the same form as that of a charged particle in the AB vector potential of an infinitely thin solenoid, with a magnetic flux that becomes quantized for suitable choices of the quantization weight. The same calculation produces a repulsive, centrifugal-like scalar potential that keeps the particle away from the singularity and can make the Hamiltonian essentially self-adjoint. If the result stands, the AB effect is a consequence of the topology and symmetry of the punctured plane, not of a classical gauge field generated by an external coil.

What carries the argument

The central machinery is affine covariant integral quantization (ACIQ), applied on the similitude group $\mathrm{SIM}(2)$, whose elements act on the punctured plane by translations, rotations, and dilations. ACIQ starts from a weight function $\varpi(q,p)$, or equivalently a positive operator $M_\varpi$, and forms the transported family $M_\varpi(q,p) = U(q,p) M_\varpi U^\dagger(q,p)$ under the unitary affine representation; the resolution of the identity $\int d^2q\,d^2p\, M_\varpi(q,p)/c_{M_\varpi} = \mathbf{1}$ turns any classical phase-space function $f$ into an operator $\mathrm{Op}^\varpi_f$. The load-bearing identity is the completion of the square in Eq. (V.3), which rewrites the quantized kinetic energy $\mathrm{Op}^\varpi_{p^2}$ as $(P - qA^\varpi(Q))^2 + K^\varpi/Q^2$; this identity exposes the affine vector potential $A^\varpi$ and the scalar potential $K^\varpi/Q^2$. The specific choice of $\varpi$ localized near the identity, Eq. (IV.23), with angular phase $\alpha(\arg q)=e^{i\mu\arg q}$, is what makes $A^\varpi$ nonvanishing and sets the flux $\Phi^\varpi_0 = 2\pi\hbar\mu/q$.

What would settle it

Quantize $p^2/2m$ on $\mathbb{R}^2_*$ with the same ACIQ construction but with the explicitly admissible real weight (IV.23) without the $e^{i\mu\arg q}$ factor. The topology is unchanged, yet Eq. (V.11) gives $\Phi^{\varpi}_0=0$ and the vector potential disappears; if that calculation is correct, the AB gauge field in this framework depends on the freely chosen weight, not on the puncture alone.

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Extended reading notes

Core claim

Working with the similitude group $\mathrm{SIM}(2) = (\mathbb{R}^*_+ \times SO(2)) \ltimes \mathbb{R}^2 \simeq \mathbb{C}^* \ltimes \mathbb{C}$ as the phase space of the punctured plane, the paper quantizes the free-particle kinetic energy $p^2/2m$ through affine covariant integral quantization. For a weight function $\varpi(q,p)$ of the form (IV.23) with phase factor $\alpha(\arg q) = e^{i\mu\arg q}$, the resulting operator is $(P - q A^{\varpi}(Q))^2 + K^{\varpi}/Q^2$, where the vector potential has components $A^{\varpi}_{x_1} = -(\Phi^{\varpi}_0/2\pi)\,Q_2/Q^2$ and $A^{\varpi}_{x_2} = (\Phi^{\varpi}_0/2\pi)\,Q_1/Q^2$, the same functional form as the AB potential of an infinitesimally thin infinite solenoid. The emergent flux is $\Phi^{\varpi}_0 = -i(2\pi\hbar/q)\,\partial_2 \ln\Omega(1) = 2\pi\hbar\mu/q$ for the chosen phase, and taking $\mu \in \mathbb{Z}$ makes the weight $2\pi$-periodic in $\arg q$ and quantizes the flux in units of $h/q$. The scalar coefficient $K^{\varpi} = 2\hbar^2\nu^2$ for the explicit weight is positive and can be made arbitrarily large, giving a repulsive $1/Q^2$ barrier and ensuring essential self-adjointness of the Hamiltonian. On these grounds the paper claims that the AB gauge field emerges from the topological constraint and affine symmetry of the punctured plane rather than from an externally applied classical gauge field.

Load-bearing premise

The load-bearing premise is that the quantization weight $\varpi(q,p)$ takes the admissible localized form with angular phase $e^{i\mu\arg q}$; the value of $\mu$ is a free parameter, so if a different admissible weight is used the emergent vector potential can change or vanish, and the topology of the punctured plane alone does not fix the flux.

Editorial extensions

If this is right

The AB vector potential would become an emergent object: quantizing a free particle on $\mathbb{R}^2_*$ with affine symmetry is enough to produce the same gauge field, so the interference shift should arise without an external solenoid.
The Hamiltonian automatically includes a repulsive $1/Q^2$ scalar potential that keeps wave functions away from the punctured origin and can be tuned through the weight function to make the kinetic operator essentially self-adjoint.
Requiring the quantization weight to be $2\pi$-periodic in the angular variable (choosing $\mu\in\mathbb{Z}$) quantizes the emergent flux in multiples of $h/q$, reproducing Dirac flux quantization through the choice of weight rather than through single-valuedness of the wave function.
The same ACIQ structure previously gave a repulsive scalar potential on the half-line; the new result shows that the rotation and dilation structure of $\mathrm{SIM}(2)$ also generates a vector potential of the AB form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the authors leave implicit: because the flux is proportional to the free parameter $\mu$, ACIQ does not by itself predict the value of the AB flux; an independent physical principle is needed to fix $\mu$.
If the derivation extends to $\mathbb{R}^3 \setminus \{0\}$, the same angular phase mechanism could produce a monopole-type vector potential; the paper lists this as a future direction but does not develop it.
A testable extension would be to compute the interference phase for two admissible weights with the same topology but different $\alpha$; a weight-dependent phase would show that the emergent geometric AB effect is distinguishable from the conventional solenoid AB field.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper does a real computation: applying the ACIQ scheme to the punctured plane, it derives an operator of the form (P - q A)^2 + K/Q^2, with A having the same functional form as the Aharonov-Bohm vector potential. That part is plausible and internally consistent. But the flux carried by A is set by a free parameter mu in the quantization weight, and the specific weight used to get nonzero flux violates the paper's own symmetry condition (IV.5). So the central claim that the AB effect is 'purely geometrical' is not supported as written.

What is genuinely new: this is the first time ACIQ is applied to the punctured plane, and the operator formulas (V.3)-(V.8) are new. The identification of the phase space with SIM(2) is natural, and the derivation is transparent enough to follow. The authors also cite the earlier work of Ohnuki and others on topological generation of gauge potentials; that context is appropriate. To their credit, they are explicit that the derived scalar potential K/Q^2 is repulsive and prevents the electron from reaching the singularity, which means their Hamiltonian is not exactly the standard AB Hamiltonian. They note that the scalar potential would modify the interference, and they flag the need for future work.

The soft spots are real and load-bearing. First, the flux in (V.12) is Phi = 2 pi hbar mu/q, with mu free. No physical or topological principle determines mu; the 'flux quantization' for mu = n is a choice, not a consequence of the punctured-plane topology. Second, the stress-test observation holds up: equation (IV.5), evaluated at p=0, requires alpha(theta)=alpha(-theta). The example alpha(theta)=exp(i mu theta) satisfies that only for mu=0, which would make the flux zero. Either (IV.5) is misprinted (a corrected condition involving complex conjugation would leave mu arbitrary), or the example falls outside the admitted class. In both readings, the AB connection is not uniquely generated by the geometry; the construction yields a weight-dependent family of gauge connections. That is the core problem.

The paper is worth a referee's time because the formalism is nontrivial and the question—whether topological quantization can generate the AB phase—is legitimate. But the revision needs to confront the admissibility of the weight and either fix mu from physical principles or present the result as a family of quantizations, with the 'purely geometrical' language toned down. I would send it to peer review, with a request for major revision.

Referee Report

4 major / 4 minor

Summary. The paper applies affine covariant integral quantization (ACIQ) to quantum mechanics on the punctured plane, whose phase space is identified with the similitude group SIM(2). By quantizing the free-particle momentum and kinetic energy, the authors obtain an effective operator of the form (P - qA)^2 + K/Q^2, where A has the same functional form as the Aharonov-Bohm (AB) vector potential of an infinite solenoid, with flux Phi = 2*pi*hbar*mu/q. They interpret this as evidence that the AB effect emerges from the topological constraint imposed by the impenetrable coil rather than from an external classical gauge field. The paper reviews the ACIQ formalism, derives the operator expressions (IV.27)-(IV.28), introduces the affine vector and scalar potentials, and discusses their semiclassical portraits and implications.

Significance. If fully established, the central claim would be conceptually striking: a quantization procedure on a punctured plane generating an AB-type vector potential purely from the topology and the SIM(2) symmetry, together with an explicitly computed repulsive scalar potential. The algebraic derivation leading to Eq. (V.3) is detailed and appears internally consistent, and the paper makes good use of the previously developed ACIQ framework, providing closed-form formulas for the weight-dependent potentials. These are genuine strengths: the manuscript offers a concrete, checkable operator-level construction. However, the 'purely geometrical' claim in the title and abstract is not supported by the present text, because the magnitude of the emergent flux is fixed by a free parameter mu in the quantization weight, and the specific example that produces nonzero flux appears to violate the paper's own symmetry condition (IV.5). The value of the work lies more in exhibiting a family of AB-type vector potentials within ACIQ than in establishing a unique topological derivation.

major comments (4)

[IV, Eq. (IV.5) and Eqs. (IV.23)-(IV.25)] The nonzero-flux example is not admissible under the stated symmetry condition. Evaluating (IV.5) at p=0 gives alpha(theta) = alpha(-theta); with alpha(theta) = exp(i mu theta), this forces mu = 0. Hence the example used to obtain Phi_0^varpi = 2*pi*hbar*mu/q in (V.12) contradicts the paper's own assumption (IV.24). Either the symmetry condition (IV.5) is misprinted, in which case the corrected condition and the admissibility of the example must be stated, or the main example falls outside the admitted class. As written, the derivation of a nonzero affine flux is internally inconsistent.
[V, Eqs. (V.9)-(V.12)] The 'emergent' flux is determined by the free parameter mu introduced in the weight (IV.25), not by the topology of the punctured plane. The sentence 'if we choose mu = n in Z' in Section V makes this explicit. Without a physical or representation-theoretic principle that fixes mu, the construction yields a weight-dependent family of AB-type potentials, and the claim that the AB gauge field emerges purely from geometry is not supported. This is a load-bearing point for the paper's central thesis.
[V, Eq. (V.3) and condition (V.6)] The reduction to the exact AB form (V.7)-(V.8) relies on the condition partial_1 ln Omega(1) = -2 stated in (V.6). Although this condition happens to be satisfied by the particular weight (IV.23), the general operator (V.2) contains an additional partial_1 ln Omega term in the off-diagonal part of the vector potential, as seen by comparing (V.3)-(V.5) with (V.6)-(V.8). The paper presents (V.6) as an ad hoc condition without discussing how the result depends on it; the claim that ACIQ produces the AB vector potential is therefore only established for a restricted class of weights, not as a general consequence of the method.
[VI] The paper acknowledges in the concluding section that the induced scalar potential can prevent the electron from reaching the singularity and may suppress the standard AB interference. This caveat is central to the physical interpretation: if the scalar barrier is strong, the model may not reproduce the interference pattern of the AB effect. The paper defers a quantitative analysis to future work, leaving the physical relevance of the derived vector potential uncertain.

minor comments (4)

[IV, Eq. (IV.4)] The phrase 'with Tr{U(q,p)M_varpi} denoting the complex conjugation of the expression' is unclear; please clarify the notation and state explicitly whether the inversion formula involves a complex conjugate.
[V, Eq. (V.3)] The 'fictive charge' q introduced in Eq. (V.3) is never physically motivated; since the flux Phi_0^varpi in (V.10)-(V.12) depends on 1/q, the comparison with the physical AB flux requires some discussion of the meaning and possible values of q.
[General] There are numerous typographical and formatting issues, such as missing spaces in the conclusion paragraph and inconsistent superscript/subscript notation in Appendix D; these should be corrected in a final version.
[Figure 1] The caption says the function is normalized by its maximum modulus, but the contour and color representation is not fully described; please specify the plotted quantity, the normalization, and the meaning of the isovalue contour level.

Circularity Check

1 steps flagged · score 8.0 of 10

The 'emergent' AB flux is the free phase derivative α′(0) of the arbitrarily chosen weight; the result is put in by hand, not derived from topology.

self definitional [Section V, Eqs. (V.11)-(V.12), with input from Section IV, Eqs. (IV.23)-(IV.25)]
"With the choice (IV.24), leading to the function Ω in (IV.26), the condition (V.6) is satisfied, and the resulting flux is given by: Φϖ0 = −i 2πℏ/q α′(0) . (V.11) With the more specific choice (IV.25) the flux takes the form Φϖ0 = 2πℏµ/q . (V.12)"

Equation (IV.25) inserts the free constant µ into the quantization weight through α(arg q)=exp(iµ arg q). Equations (V.11)-(V.12) then return the flux as 2πℏµ/q. No equation in the paper determines µ from the punctured-plane topology; the later statement 'if we choose µ = n ∈ Z' is an additional choice, not a consequence of the geometry. Thus the affine vector potential (V.7)-(V.8) is not emergent but is the logarithmic derivative of the arbitrarily chosen ϖ: Aϖ is, by construction, a functional of Ω, and Ω is fixed by the chosen α. Moreover, the stated admissibility condition (IV.5)/(IV.24) requires α(θ)=α(−θ), which exp(iµθ) obeys only for µ=0; so the nonzero-flux example contradicts the paper's own assumptions.

full rationale

The algebraic derivation leading to Eq. (V.3), i.e. the operator identity (P−qAϖ)²+Kϖ/Q², is internally consistent given the ACIQ formalism. The circularity is in the physical interpretation of Aϖ as an emergent, topological Aharonov-Bohm field. The vector potential (V.4) is defined in terms of logarithmic derivatives of Ω(1), and Ω(1) is itself computed from the freely chosen weight function ϖ in (IV.23). The flux formula (V.11) reduces exactly to α′(0), and with the specific ansatz α(θ)=exp(iµθ) the flux is simply 2πℏµ/q (V.12). The parameter µ is not fixed by any property of the punctured plane; it is an input to the quantization weight. Hence the 'derived' AB potential is equivalent, by construction, to the phase winding that was already put into α. The self-citations to [17,18] for the ACIQ framework are not themselves circular, since the framework is developed independently, but they do not rescue the main claim: the emergence claim would require that the flux be forced by the topology, whereas the paper's own equations show it is forced by the choice of ϖ. Additionally, the example used to obtain nonzero flux violates the stated symmetry condition (IV.24) for µ≠0, underscoring that the result is not robust. Score 8 reflects that the central 'purely geometrical' claim reduces by construction to an arbitrary ansatz.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central result depends on a specific weight function with free parameters mu and nu. The flux is set by mu, and the scalar potential strength by nu. The quantization framework itself is a domain assumption. No new physical entities are postulated.

free parameters (4)

mu (phase of weight function alpha) = not fixed; mu in Z for flux quantization
In Eq. (V.12), the magnetic flux is Phi = 2 pi hbar mu / q, so mu directly sets the physical flux. The choice mu in Z is imposed by requiring 2pi-periodicity of the weight, but the integer is not determined by the formalism.
nu (localization parameter) = not fixed; can be made arbitrarily large
Appears in the weight (IV.23) and controls the strength of the scalar potential K = hbar^2 (2 nu + (alpha')^2 - alpha'') (V.13). The paper uses nu to make K large enough to guarantee self-adjointness, so it is a free parameter affecting the predicted dynamics.
sigma (momentum width) = 3.5 in Figure 1
Parameter in the weight (IV.23); enters Omega(1) = pi sigma^2 but cancels in the ratio defining the vector potential. It does not affect the main results but is a free parameter of the construction.
q (fictive charge) = unspecified
Introduced after Eq. (V.3) to write the operator as (P - q A)^2. The combination mu/q sets the flux, but q is not tied to any physical charge.

assumptions (4)

domain assumption The phase space of a particle on the punctured plane R2* is the similitude group SIM(2) = (R+* x SO(2)) semidirect R2.
Section III.1. This identification is the basis for using ACIQ. It is a modeling choice: the punctured plane has dilation, rotation, and translation symmetries, not full translation symmetry.
domain assumption The weight function varpi satisfies Assumption 1 (smoothness, tempered distribution, bounded self-adjoint M_varpi) and the normalization Tr M_varpi = 1.
Section IV. These conditions guarantee the resolution of the identity and the quantization map. They are technical assumptions on the admissible quantizers.
ad hoc to paper The condition partial_1 ln Omega(1) = -2 is imposed on the weight.
Eq. (V.6). This simplifies the vector potential to the AB form. It is not derived from physical requirements, but selected to make the result look like the AB potential.
domain assumption The free Hamiltonian on the punctured plane is H0 = p^2 / 2m.
Sections II and V. The quantization starts from the classical free particle Hamiltonian p^2/2m; the vector potential emerges from the quantization of p and p^2.

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Cite this review

Pith. "Pith review of A purely geometrical Aharonov-Bohm effect." pith.science (2026). https://pith.science/paper/JLMYM7VV

@misc{pith2026241213919,
  author       = {Pith},
  title        = {Pith review of: A purely geometrical Aharonov-Bohm effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLMYM7VV}},
  note         = {Machine review of arXiv:2412.13919}
}

read the original abstract

We present an application of the affine covariant integral quantization (ACIQ) (Adv. Oper. Theory, 5, 2020; Adv. Oper. Theory, 7, 2022) to quantum mechanics on the punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), which comprises translations, rotations, and dilations of the plane. Due to the topology of the punctured plane, our quantization procedure gives rise to an affine vector potential. This potential can be interpreted as the Aharonov-Bohm (AB) gauge field produced by an infinite solenoid. This observation supports a reinterpretation of the AB effect: it emerges from the topological constraint imposed by the impenetrable coil rather than from an externally applied classical gauge field. In addition to this gauge structure, ACIQ also generates a repulsive, centrifugal-like scalar potential, a feature already encountered when applying ACIQ to motion on the half-line, whose phase space is the open half-plane. These results provide a new perspective on the AB effect, highlighting the central roles of topology and symmetry in quantum mechanics.

Figures

Figures reproduced from arXiv: 2412.13919 by the authors.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

49 extracted references · 49 canonical work pages

[1]

Separable functions: 18

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[2]

Position dependent functions 19

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[3]

Covariant

Monomial functions in momentum coordinates 19 C. Quantization with rank-one density operator 21 D. Formulae 23 References 24 I. INTRODUCTION The Aharonov-Bohm (AB) effect is a cornerstone of quantum mechanics, illustrating how topology influences physical observables [1]. Despite experimental verification, its deeper im- plications, especially concerning ...

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[4]

The corresponding Hamiltonian is H0(x, p) = 1 2m p − q Asol 2 = p2 2m − q m Φ0 2πx2 x ∧ p + 1 2m qΦ0 2π 2 1 x2 , x = ∥x∥

We consider a finite magnetic fluxΦ0, which remains dependent on n, J and a, even asa approaches zero. The corresponding Hamiltonian is H0(x, p) = 1 2m p − q Asol 2 = p2 2m − q m Φ0 2πx2 x ∧ p + 1 2m qΦ0 2π 2 1 x2 , x = ∥x∥ . (II.4) Here (x, p) denotes a pair of canonical variables. Due to the presence of the infinitely thin solenoid, the origin of the pl...

work page
[5]

imaginary

(III.6) It results that e1q = q for all q, e2e2 = −e1, and so e2q = e2(q1, q2) = ( −q2, q1). One thus should avoid the confusion between this “imaginary”e2 and the “i” appearing in the canonical quantization p 7→ P = −i∇∇∇, where ∇∇∇ = ( ∂/∂q 1, ∂/∂q2) (reminder that we set ℏ = 1). The Euclidean inner product is equally denoted byRe(q∗q′) or q · q′. Consi...

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[6]

Tr(Mϖ) = 1 2π Z R2∗ d2x bϖp(1, −x) = 1 2π Ω−2(1) = 1

Hence, the condition (IV.7) can be also written as a condition on the functionΩ−2. Tr(Mϖ) = 1 2π Z R2∗ d2x bϖp(1, −x) = 1 2π Ω−2(1) = 1 . (IV.22) Let us give the following example of a functionϖ which complies with conditions (IV.4), (IV.7), (IV.15), and (IV.17). It is a function well-localised about the affine identity(1, 0) in Γ: ϖ(q, p) = e2ν q e−ν(q+1...

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[7]

f (q, p) = p Opϖ p = P + i Q∗ 2 + ∇∇∇Ω(1) Ω(1) . (IV.27)

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[8]

f (q, p) = p 2 Opϖ p 2 = P2 + 2i Q∗ 2 + ∇∇∇Ω(1) Ω(1) · P − 1 Q2 4 + 4e1 · ∇∇∇Ω(1) Ω(1) + △Ω(1) Ω(1) , (IV.28) where ∇∇∇Ω(1) and △Ω(1) are given by ∇∇∇Ω(1) = Z R2∗ d2x x2 ∇∇∇q bϖp (q, x)|q=1 , △Ω(1) = Z R2∗ d2x x2 △q bϖp (q, x)|q=1 . 13 FIG. 1. Phase-space representation of the quantization weight functionϖ(q, p) defined in (IV.23), shown for angle0 rad an...

work page 2021

Show all 49 references

[9]

Separable functions: f (q, p) ≡ u(q) v(p) The formula (IV.19) simplifies to Aϖ u(q)v(p)(x, x′) = 1 cMϖ ˆv(x′ − x) x2 x′2 bϖp x x′ , −. ∗af fu (x) , (B.1) where • the 2D-affine convolution product∗af fon the multiplicative groupR2 ∗ = R+ ∗ ×SO(2) is defined by (f1 ∗af ff2)(x) =...
[10]

Note the particular cases:

Position dependent functions f (q, p) ≡ u(q) The expression (B.1) then yields the multiplication operator Opϖ u(q)ϕ (x) = 1 cMϖ (w ∗af fu)(x)ϕ(x) , (B.3) where w(x) = 2π bϖp(1, −x). Note the particular cases:
[11]

Then we have Opϖ qβ = 2π cMϖ Z R2∗ d2y y2+β bϖp(1, −y) Qβ = Ωβ(1) Ω(1) Qβ

u(q) is a simple power ofq, say u(q) = qβ. Then we have Opϖ qβ = 2π cMϖ Z R2∗ d2y y2+β bϖp(1, −y) Qβ = Ωβ(1) Ω(1) Qβ . (B.4)
[12]

For the special case of Ω(2,0,1)(1) = 0 Ω(2,1,0)(1) = cM ϖ 2π , our matrix is the unit matrix1 2

u(q) = q stands for the classical vector position in the plane, Opϖ q = 2π cMϖ   Ω(2,1,0)(1) Ω (2,0,1)(1) −Ω(2,0,1)(1) Ω (2,1,0)(1)    Q1 Q2   , (B.5) where we have generalized the notation (IV.21) for the following integrals Ωβ,ν1,ν2(u) := Z R2∗ d2y yβ+2 bϖp (u, −y) y...
[13]

∗af fu (x)

Monomial functions in momentum coordinates f (q, p) ≡ u(q) pn i 3 Monomial functions in momentum coordinates 20 Aϖ u(q) pn i (x, x′) = in 2π ccMϖ x2 x′2 δ(x′ 3−i − x3−i) δ(ni) x′ i (x′ i − xi) x2 x′2 bϖp x x′ , −. ∗af fu (x) . (B.8) Applying the corresponding operator onϕ ∈ C ...
[14]

(B.10) Here (∇∇∇ bϖp)(1, −·) := ∂ ∂q1 bϖp)(q, −·), ∂ ∂q2 bϖp)(q, −· q=1

f (q, p) = u(q)p Opϖ u(q)p = 2π cMϖ ( bϖp(1, −·) ∗af fu)(Q) P+ + 4πi cMϖ 1 Q∗ ( bϖp(1, −·) ∗af fu)(Q) + 2πi cMϖ 1 Q∗ ((∇∇∇ bϖp)(1, −·) ∗af fu)(Q) . (B.10) Here (∇∇∇ bϖp)(1, −·) := ∂ ∂q1 bϖp)(q, −·), ∂ ∂q2 bϖp)(q, −· q=1 . More generally and from now on, the gradient vector∇∇∇F...
[15]

f (q, p) = p Then the previous one simplifies to the quantum momentum: Opϖ p = P + i Q∗ 2 + ∇∇∇Ω(1) Ω(1) . (B.11)
[16]

(B.9) yields the quantum kinetic energy (up to the usual factor1/2m): Opϖ p 2 = P2 + 2i Q∗ 2 + ∇∇∇Ω(1) Ω(1) · P − 1 Q2 4 + 4e1 · ∇∇∇Ω(1) Ω(1) + △Ω(1) Ω(1) , (B.12)

f (q, p) = p 2 Eq. (B.9) yields the quantum kinetic energy (up to the usual factor1/2m): Opϖ p 2 = P2 + 2i Q∗ 2 + ∇∇∇Ω(1) Ω(1) · P − 1 Q2 4 + 4e1 · ∇∇∇Ω(1) Ω(1) + △Ω(1) Ω(1) , (B.12)
[17]

f (q, p) = q · p = q1p1 + q2p2 (generator of dilations in the plane) By using(B.6), we get: Opϖ q·p = 2π cMϖ Ω(2,1,0)(1)(Q · P + 2i) − Ω(2,0,1)Q ∧ P +i(e1 · ∇∇∇Ω(2,1,0)(1) − e2 · ∇∇∇Ω(2,0,1)(1)) . (B.13)
[18]

(B.14) 21 Simplified expressions for the above formulas are always possible with a suitable choice of the function ϖ

f (q, p) = q ∧ p = q1p2 − q2p1 (angular momentum) The result is given in terms of the integrals (B.6) as Opϖ q×p = 2π cMϖ Ω(2,1,0)(1)Q ∧ P + Ω(2,0,1)(1)(Q · P + 2i) +i(e1 · ∇∇∇Ω(2,0,1)(1) + e2 · ∇∇∇Ω(2,1,0)(1)) . (B.14) 21 Simplified expressions for the above formulas are alwa...
[19]

Aharonov and D

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Separable functions: 18

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Position dependent functions 19

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Covariant

Monomial functions in momentum coordinates 19 C. Quantization with rank-one density operator 21 D. Formulae 23 References 24 I. INTRODUCTION The Aharonov-Bohm (AB) effect is a cornerstone of quantum mechanics, illustrating how topology influences physical observables [1]. Despite experimental verification, its deeper im- plications, especially concerning ...

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[4] [4]

The corresponding Hamiltonian is H0(x, p) = 1 2m p − q Asol 2 = p2 2m − q m Φ0 2πx2 x ∧ p + 1 2m qΦ0 2π 2 1 x2 , x = ∥x∥

We consider a finite magnetic fluxΦ0, which remains dependent on n, J and a, even asa approaches zero. The corresponding Hamiltonian is H0(x, p) = 1 2m p − q Asol 2 = p2 2m − q m Φ0 2πx2 x ∧ p + 1 2m qΦ0 2π 2 1 x2 , x = ∥x∥ . (II.4) Here (x, p) denotes a pair of canonical variables. Due to the presence of the infinitely thin solenoid, the origin of the pl...

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imaginary

(III.6) It results that e1q = q for all q, e2e2 = −e1, and so e2q = e2(q1, q2) = ( −q2, q1). One thus should avoid the confusion between this “imaginary”e2 and the “i” appearing in the canonical quantization p 7→ P = −i∇∇∇, where ∇∇∇ = ( ∂/∂q 1, ∂/∂q2) (reminder that we set ℏ = 1). The Euclidean inner product is equally denoted byRe(q∗q′) or q · q′. Consi...

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[6] [6]

Tr(Mϖ) = 1 2π Z R2∗ d2x bϖp(1, −x) = 1 2π Ω−2(1) = 1

Hence, the condition (IV.7) can be also written as a condition on the functionΩ−2. Tr(Mϖ) = 1 2π Z R2∗ d2x bϖp(1, −x) = 1 2π Ω−2(1) = 1 . (IV.22) Let us give the following example of a functionϖ which complies with conditions (IV.4), (IV.7), (IV.15), and (IV.17). It is a function well-localised about the affine identity(1, 0) in Γ: ϖ(q, p) = e2ν q e−ν(q+1...

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f (q, p) = p Opϖ p = P + i Q∗ 2 + ∇∇∇Ω(1) Ω(1) . (IV.27)

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[8] [8]

f (q, p) = p 2 Opϖ p 2 = P2 + 2i Q∗ 2 + ∇∇∇Ω(1) Ω(1) · P − 1 Q2 4 + 4e1 · ∇∇∇Ω(1) Ω(1) + △Ω(1) Ω(1) , (IV.28) where ∇∇∇Ω(1) and △Ω(1) are given by ∇∇∇Ω(1) = Z R2∗ d2x x2 ∇∇∇q bϖp (q, x)|q=1 , △Ω(1) = Z R2∗ d2x x2 △q bϖp (q, x)|q=1 . 13 FIG. 1. Phase-space representation of the quantization weight functionϖ(q, p) defined in (IV.23), shown for angle0 rad an...

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