Physical foundations of quantization by freezing the Grassmannian partners of time in the classical path integral: supertime interpretation, the origin of the identification $\Delta=\hbar$, and the scope over observables

In the path-integral formulation of classical mechanics (the CPI) built on the Koopman-von Neumann operatorial approach, Abrikosov, Gozzi and Mauro showed that quantization can be implemented by freezing to zero two Grassmannian partners $\theta,\bar{\theta}$ of the ordinary time $t$, and that the Feynman quantum path integral is recovered provided a constant $\Delta$ with the dimension of an action, which the freezing limit leaves behind, is identified with $\hbar$. The authors left open the physical status of this construction: what $\theta,\bar{\theta}$ and their relation to $t$ via the universal symmetries of the CPI mean for the role of time in quantization; whether the identification $\Delta=\hbar$ can be derived rather than imposed; and whether the procedure quantizes all observables without the metaplectic structures that Geometric Quantization needs. Later work reformulated the freezing as a choice of supermetric on supertime, extended the procedure to spin and coadjoint-orbit systems, and proposed anomaly-based mechanisms for the emergence of $\hbar$, while the 2025 reframing of the CPI as an AKSZ sigma model (a graded-geometric construction of gauge-invariant action functionals) leaves the connection to geometric quantization open; no derivation of $\Delta=\hbar$, no settled interpretation of $\theta,\bar{\theta}$, and no general no-metaplectic theorem (or counterexample) is known.

Background

Most of this background follows the conventions of Abrikosov, Gozzi and Mauro, 'Time and Geometric Quantization' (Modern Physics Letters A 18 (2003) 2347), and its extended companion 'Geometric dequantization' (Annals of Physics 317 (2005) 24), where the question below was posed.

In the early 1930s Koopman and von Neumann reformulated classical mechanics as a theory of operators acting on a Hilbert space of wave functions $\psi(q,p)$ on phase space, evolving under the Liouvillian $\hat{L}=-i\,\partial_p H\,\partial_q+i\,\partial_q H\,\partial_p$, the operator that generates Liouville evolution of phase-space distributions; this is called the Koopman-von Neumann (KvN) approach. Gozzi, Reuter and Thacker later gave this operatorial classical mechanics a path-integral form, called the Classical Path Integral (CPI): a functional integral over phase-space paths $\varphi^a=(q^1\ldots q^n,p_1\ldots p_n)$ and over auxiliary variables ($\lambda_a$ commuting, $c_a,\bar{c}_a$ Grassmannian, i.e. anticommuting) whose weight gives value one to the classical trajectory and zero to all others. The Grassmann partners $c_a$ of the phase-space variables can be identified with the basis of differential forms $d\varphi^a$, and the CPI Hamiltonian is the Lie derivative of the Hamiltonian flow, so the CPI carries the full geometric content of classical mechanics. It also possesses hidden Becchi-Rouet-Stora-like (BRS-like, i.e. gauge-theory-ghost-type) and supersymmetry-like invariances, which are universal in the sense that they hold for every Hamiltonian $H$.

These symmetries make it natural to assemble the $8n$ variables $(\varphi^a,c_a,\bar{c}_a,\lambda_a)$ into a single superfield $\Phi^a(t,\theta,\bar{\theta})=\varphi^a(t)+\theta c_a(t)+\bar{\theta}\,\omega^{ab}\bar{c}_b(t)+i\bar{\theta}\theta\,\omega^{ab}\lambda_b(t)$, where $\omega^{ab}$ is the symplectic matrix and $(\theta,\bar{\theta})$ are two Grassmannian partners of time: the base space of the theory becomes the three-dimensional 'supertime' $(t,\theta,\bar{\theta})$, an instance of the superspace construction used in supersymmetric field theories. Expanding the classical action $S[\varphi]$ and the CPI action $S_e$ on this superspace shows that both belong to the same supermultiplet, and the generating functionals of classical and quantum mechanics take the same functional form, with $\varphi^a\to\Phi^a$ and $\int dt\to\int i\,dt\,d\theta\,d\bar{\theta}$.

The central observation of the source paper is that quantization - passing from the CPI to the Feynman quantum path integral (QPI) - is achieved by 'freezing' $\theta,\bar{\theta}\to 0$ (technically, inserting $\delta(\bar{\theta})\delta(\theta)/\hbar$ into the weight, together with a Fourier transform that selects a polarization). Because $\bar{\theta}\theta$ carries the dimension of an action, the limit must be taken on the dimensionless ratio $\bar{\theta}\theta/\Delta$, which leaves a constant $\Delta$ of action dimension in the resulting quantum weight; only if one identifies $\Delta$ with $\hbar$ does one obtain exactly the Feynman path integral $\int\mathcal{D}q\,\mathcal{D}p\,\exp\big(\frac{i}{\hbar}\int dt\,L\big)$, in the coordinate or in the momentum polarization. The two steps (Fourier transform plus freezing) both change the kinetic term that generates the commutators - this is how $[\hat{q},\hat{p}]=i\hbar$ arises from the classical $[\hat{q},\hat{p}]=0$ - and simultaneously implement the polarization, a double role reminiscent of Geometric Quantization (GQ). GQ is the mathematical framework that constructs a quantum Hilbert space from a classical phase space in two steps: prequantization of the symplectic form on an enlarged (KvN-like) Hilbert space, followed by a polarization that cuts the number of variables down to a quantum-mechanical set. A known difficulty of GQ is that some observables cannot be quantized on the polarized space without extra structures, notably metaplectic (half-form) corrections, i.e. contributions from a double cover of the symplectic group.

The source paper notes that its freezing procedure resembles GQ but differs intrinsically (in GQ the commutators stay KvN-like and the operators change; here the commutators themselves change in the limit), and conjectures that this difference 'most probably' removes the metaplectic problem. Its conclusion states that the universal CPI symmetries can turn $\theta,\bar{\theta}$ into the ordinary time $t$, that time or its Grassmann partners therefore play a role in quantization, and that this last issue was under investigation at the time. The identification $\Delta=\hbar$, however, is imposed by hand, and the physical meaning of supertime is not given. The companion paper proposes to derive the identification as an anomaly - a symmetry of the classical action that fails to survive because the path-integral measure is not invariant under it - or as a regularization scale of a super-reparametrization-invariant extension of the CPI, in which classical and quantum mechanics appear as different gauge choices or supermetrics.

Problem Statement

Give the quantization-by-supertime-freezing procedure of the classical path integral a physical and mathematical foundation. The source paper and its companion present three linked questions that jointly define one objective - a physically grounded account of quantization as a reduction of supertime $(t,\theta,\bar{\theta})$ to ordinary time:

(1) Supertime interpretation. The CPI carries universal BRS-like and supersymmetry-like invariances that mix $t$ with its Grassmann partners $\theta,\bar{\theta}$ (including a symmetry rescaling the overall action, realized as a $t\leftrightarrow\theta$ transformation, which is lost in quantum mechanics). Determine what $\theta,\bar{\theta}$ are physically - in particular whether they can be interpreted as, or continuously converted into, the ordinary time variable via these universal symmetries - and what the freezing limit $\theta,\bar{\theta}\to 0$ then means for the role of time in quantization.

(2) Origin of the identification $\Delta=\hbar$. In the freezing limit the dimensionless ratio $\bar{\theta}\theta/\Delta$ is sent to zero, leaving a constant $\Delta$ with the dimension of an action in the resulting quantum weight, and the Feynman path integral is recovered only if $\Delta$ is set equal to $\hbar$ by hand. Determine whether this identification can be derived within the CPI framework, or prove that $\Delta$ is necessarily an independent input; equivalently, explain where the quantum of action comes from in this picture.

(3) Scope over observables and systems. The source paper states only as a probability that the freezing procedure, unlike Geometric Quantization, quantizes observables without extra metaplectic (half-form) structures. Establish the exact domain of validity of the two-step procedure (Fourier transform plus $\theta,\bar{\theta}\to 0$): the Hamiltonians, observable classes, polarizations, and systems with coadjoint-orbit degrees of freedom (spin-type systems whose phase space is an orbit of a symmetry group rather than a flat phase space) for which it reproduces the correct quantum theory, and exhibit an explicit counterexample if the no-metaplectic claim fails.

Resolving the three parts within (or in explicit quantitative comparison with) the CPI conventions of the cited papers requires a physically definite interpretation of $\theta,\bar{\theta}$ - one that fixes what they are, how they relate to the ordinary time via the universal symmetries, and what the freezing limit $\theta,\bar{\theta}\to 0$ does to the role of time, definite enough to exclude competing readings - a derivation (or an impossibility proof) of $\Delta=\hbar$, and a characterization of the observable scope.

References

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