The geometric dequantization framework of Abrikosov, Gozzi and Mauro turns the path integral of quantum mechanics into the path-integral form of the Koopman-von Neumann operatorial formulation of classical mechanics by extending time tt to a supertime (t,θ,θˉ)(t,\theta,\bar\theta) with two Grassmann partners, replacing phase-space variables with superfields, and substituting ∫dt→iℏ∫dt dθ dθˉ\int dt \to i\hbar\int dt\,d\theta\,d\bar\theta; quantization is the reverse dimensional reduction θ,θˉ→0\theta,\bar\theta\to 0, equivalent to geometric quantization. The authors left the physical foundation of these rules open: whether classical and quantum mechanics are different gauge fixings of a superdiffeomorphism-invariant theory on supertime, whether the transformation between them is obstructed by an anomaly, whether ℏ\hbar can be produced as the regularized Jacobian of that transformation, and what physical meaning attaches to the Grassmann partners of time. Later literature realized only background, non-dynamical versions of this picture - supermetric selections reproducing both path integrals (2010), a general-dreinbein formulation (a frame field on supertime) with a proven classical-versus-quantum curvature distinction (2019), and the identification of the classical path integral as a gauge slice of an AKSZ (Alexandrov-Kontsevich-Schwarz-Zaboronsky) sigma model (2025) - explicitly falling short of a genuinely gauge-invariant formulation with ℏ\hbar emerging from a Jacobian or anomaly. The problem remains open.

Background

Koopman-von Neumann (KvN) mechanics is the operatorial formulation of classical mechanics proposed in the early 1930s: the Liouville equation for a probability density on the 2n2n-dimensional phase space with coordinates φa=(q1,…,qn;p1,…,pn)\varphi^a=(q^1,\dots,q^n;p_1,\dots,p_n) is recast as a Schroedinger-like equation for wave functions ψ(φ)\psi(\varphi), with mutually commuting operators φ^a\hat\varphi^a and commuting auxiliary operators λa\lambda_a that shift the eigenvalues of their conjugates; classical mechanics is thereby given a Hilbert-space implementation distinct from the quantum one. Gozzi and collaborators later gave this formulation a functional form, the classical path integral (CPI), to be distinguished from the quantum path integral (QPI). The CPI becomes compact when time tt is extended to a three-dimensional supertime (t,θ,θˉ)(t,\theta,\bar\theta), where θ,θˉ\theta,\bar\theta are Grassmann variables (anticommuting numbers with θ2=θˉ2=0\theta^2=\bar\theta^2=0), and each phase-space coordinate is embedded in a superfield

Φa=φa+θca+θˉ ωabcˉb+iθˉθ ωabλb,\Phi^a=\varphi^a+\theta c^a+\bar\theta\,\omega^{ab}\bar c_b+i\bar\theta\theta\,\omega^{ab}\lambda_b,

where ωab\omega^{ab} is the matrix defining the Poisson bracket, ca,cˉac^a,\bar c_a are Grassmann partners interpretable as the differential forms dφad\varphi^a or as Jacobi fields (deviations between neighboring classical trajectories), and λb\lambda_b is an auxiliary commuting variable. The CPI is invariant under universal symmetries - Becchi-Rouet-Stora- (BRS-) like and supersymmetry-like transformations, plus ghost-number and ghost-conjugation (conserved charges that grade the Grassmann variables) - the classical analogues of the rigid symmetries of gauge theories.

Abrikosov, Gozzi and Mauro proved that the QPI is turned into the CPI ('geometric dequantization') by three substitutions, valid only inside path-integral expressions: φa(t)→Φa(t,θ,θˉ)\varphi^a(t)\to\Phi^a(t,\theta,\bar\theta), ∫dt→iℏ∫dt dθ dθˉ\int dt\to i\hbar\int dt\,d\theta\,d\bar\theta, and iℏ δ/δJ(t)→δ/δJ(t,θ,θˉ)i\hbar\,\delta/\delta J(t)\to\delta/\delta J(t,\theta,\bar\theta) for differentiation with respect to an external source JJ. Conversely, quantization is the dimensional reduction θ,θˉ→0\theta,\bar\theta\to 0, which the same group proved equivalent to geometric quantization (the standard framework that constructs quantum Hilbert spaces from a classical phase space with a choice of polarization). The rules are geometric but postulated, and the authors themselves list what is not understood. (i) The physical meaning of the Grassmann partners of time is unknown: dimensional analysis suggests Δt∼θθˉ\Delta t\sim\theta\bar\theta rather than θ∼dt\theta\sim dt, and the proposed pictures - a Grassmann 'temperature' or frequency associated with Grassmann time that would become infinite in quantum mechanics, echoing the identification of time with inverse temperature - are explicitly qualitative. (ii) The integration measure is extended to supertime but the time derivative d/dtd/dt is not, and this asymmetry 'needs for sure further study'. (iii) Most ambitiously, the authors propose to derive the construction from a gauge principle: a Lagrangian L~inv\widetilde{\mathcal{L}}_{\mathrm{inv}} invariant under local supertime reparametrizations t′=t′(t,θ,θˉ)t'=t'(t,\theta,\bar\theta), or even field-dependent ones t′=t′(t,θ,θˉ;Φ)t'=t'(t,\theta,\bar\theta;\Phi) - superdiffeomorphisms, i.e. coordinate changes of the supermanifold - with a supermetric (a metric on the supermanifold) so that both classical and quantum mechanics appear as different gauge fixings of a single invariant theory (gauge fixing here in the path-integral sense, implemented by a Faddeev-Popov determinant, the Jacobian factor associated with fixing a gauge redundancy). Since gauge-equivalent descriptions must describe the same theory, the impossibility of rotating classical into quantum mechanics would have to be attributed to an anomaly - the failure of a classical symmetry to survive in the functional integral, detected as a non-trivial Jacobian of the integration measure under the transformation, typically requiring regularization - plausibly affecting the large or field-dependent reparametrizations, which can be anomalous even in systems with finitely many degrees of freedom. The authors conjecture that ℏ\hbar (or a cutoff on the action volume) is precisely the regularizing parameter of that Jacobian, which would derive the factor iℏi\hbar in ∫dt→iℏ∫dt dθ dθˉ\int dt\to i\hbar\int dt\,d\theta\,d\bar\theta from first principles; they announced this computation as work in progress. Relatedly, they note that quantum mechanics is anomalous with respect to the universal symmetry that rescales the overall action, since the presence of the quantum of action ℏ\hbar forbids such a rescaling.

Problem Statement

For mechanical systems with finitely many degrees of freedom, replace the postulated dequantization/quantization rules of the geometric-dequantization framework with a dynamical gauge principle - in the authors' words, understand the physics behind them. Concretely:

(a) Construct - or prove the impossibility of - a supertime reparametrization-invariant action on the supertime supermanifold (t,θ,θˉ)(t,\theta,\bar\theta), with a dynamical supermetric (or dreinbein, the supersymmetric analogue on the (t,θ,θˉ)(t,\theta,\bar\theta) supermanifold of the vierbein frame field of general relativity) and the CPI superfields as matter content, whose gauge fixings reproduce both the classical path integral and the quantum path integral, including the Faddeev-Popov treatment of the gauge fixing. The authors note as further context that the CPI likely possesses universal symmetries beyond the known ones and that field-dependent supertime reparametrizations have not been studied much in the literature.

(b) Within such a construction, compute the Jacobian of the superdiffeomorphism connecting the two gauge choices, exhibit its regularization, and determine whether it produces ℏ\hbar - thereby deriving from first principles the factor iℏi\hbar in ∫dt→iℏ∫dt dθ dθˉ\int dt\to i\hbar\int dt\,d\theta\,d\bar\theta, and explaining why no gauge transformation turns classical into quantum mechanics. (If (a) is resolved negatively there is no invariant-action transformation between the two gauge choices, hence no mechanism producing ℏ\hbar.)

(c) Resolve, within such a framework (or, if (a) is resolved negatively, independently of any invariant action), the physical status of the Grassmann partners of time: whether the identification Δt∼θθˉ\Delta t\sim\theta\bar\theta is physically meaningful, whether a Grassmann 'temperature' or frequency can be rigorously defined so that the quantization limit θ,θˉ→0\theta,\bar\theta\to 0 admits the proposed physical reading (analogous to time as inverse temperature), and why the integration measure is extended to supertime while the time derivative d/dtd/dt is not.

(a)-(c) jointly serve one overarching objective - a gauge-principle foundation of the classical-quantum map - with (b) the mechanism that would realize (a) and (c) the physical interpretation the authors require of any such construction. A construction of (a) is realized through (b) and (c); an impossibility proof of (a) leaves no superdiffeomorphism connecting the gauge choices and hence no Jacobian, with (c) then standing independently of any invariant action. The completion boundary is exact: the background-supermetric versions already available in the literature (a fixed, non-dynamical dreinbein without Faddeev-Popov sector and without any Jacobian computation) do not close the problem on either branch.

References

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