Physical meaning of the Grassmann partners of time and the universal supertime symmetries lost in quantization

In the Koopman-von Neumann functional formulation of classical mechanics, quantization can be implemented by extending the time base to a superspace $(t,\theta,\bar\theta)$ with two Grassmann partners of time, packaging the extended phase-space variables into a single superfield, and freezing $\theta,\bar\theta\to0$ with an action-scale insertion $\bar\theta\theta/\hbar$. This 'dimensional reduction of supertime' has been shown by its authors to reproduce the Feynman path integral and the standard steps of geometric quantization. Two questions raised when the construction was introduced remain without a complete answer. First, what do $\theta,\bar\theta$ physically encode: only dimensional heuristics exist ($\bar\theta\theta$ carries the dimension of an action, $\Delta t\sim\bar\theta\theta\sim\hbar$, reminiscent of the Wiener relation $(\Delta q)^2\sim\Delta t\,\hbar$), together with qualitative pictures that their own authors describe as vague. Second, which universal symmetries of the extended classical formalism — in particular the symmetry that rescales the overall action and mixes $\theta$ with $t$ — are lost in the $\theta,\bar\theta\to0$ projection, and with what consequences for quantum mechanics, including a mechanism sketched in later work but never completed, in which $\hbar$ appears as the regularizing scale of an anomalous Jacobian of the classical functional measure under supertime reparametrizations. The problem is to give the quantization-by-freezing construction a complete physical account: a rigorous operational characterization of $\theta,\bar\theta$ (or a proof that none exists), together with a complete classification of the symmetries lost in the projection and their physical consequences, valid for generic Hamiltonian systems.

Background

Quantum mechanics is standardly formulated on Hilbert spaces, with space-time entering only indirectly. Geometric quantization (N. M. J. Woodhouse, 'Geometric Quantization', 1992) is the mathematical framework that constructs the quantum Hilbert space and the Schrödinger operator from classical phase-space data, via a prequantization step followed by a polarization — a choice of half of the phase-space variables (for example keeping $q$ and discarding $p$) that reduces the (too large) prequantum Hilbert space to the physical quantum one.

A parallel, less familiar route to quantization starts from the observation that classical mechanics itself admits a Hilbert-space formulation. Koopman (1931) and von Neumann (1932) showed that the Liouville equation $i\partial_t\rho=\hat L\rho$ for phase-space probability densities, with $\hat L$ the Liouville operator built from the Hamiltonian $H(q,p)$, can be lifted to an equation $i\partial_t\psi=\hat L\psi$ for wave functions $\psi(q,p)$; this is the Koopman-von Neumann (KvN) theory. Gozzi, Reuter and Thacker (1989) gave this operatorial classical mechanics a path-integral counterpart, the classical path integral (CPI), which assigns weight one to classical trajectories and zero to all others. Writing it requires, besides the phase-space coordinates $\varphi^a=(q^1,\dots,q^n,p^1,\dots,p^n)$, a conjugate variable $\lambda_a$ and two Grassmann variables $c^a,\bar c_a$ per coordinate. Grassmann (anticommuting) variables, satisfying $c^ac^b=-c^bc^a$, are the standard algebraic device for fermions and for differential forms; in the CPI the $c,\bar c$ are precisely the components of differential forms on phase space ($c^a\sim d\varphi^a$) or of Jacobi fields (deviations between neighboring classical trajectories), and the extended $8n$-variable system carries, for every Hamiltonian, a universal BRST-like (BRST: Becchi-Rouet-Stora-Tyutin, the rigid algebraic signature of a gauge symmetry) and $N=2$ supersymmetry algebra — a graded symmetry algebra mixing the bosonic variables $\varphi,\lambda$ with the fermionic $c,\bar c$ (reviewed by E. Deotto and E. Gozzi, 2001).

The $8n$ variables can be assembled 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 new Grassmann coordinates introduced as partners of time, so that the base space of mechanics becomes the supertime $(t,\theta,\bar\theta)$ — a supermanifold with one even and two odd directions. Expanding the classical action $S[\varphi]=\int(p\dot q-H)\,dt$ on $\Phi^a$ produces, in one supermultiplet, both the quantum action $S[\varphi]$ (the $\theta,\bar\theta$-independent part, or body) and the CPI action (the $\theta\bar\theta$ component, up to a surface term). A. A. Abrikosov Jr. and E. Gozzi (2000) then observed that, after a partial Fourier transform that removes the surface terms, the quantum transition amplitude $\langle q_f;t_f|q_i;t_i\rangle=\int\mathcal D q\,\mathcal D p\,\exp\big(\frac{i}{\hbar}S[\varphi]\big)$ is obtained from the classical one by freezing $\theta,\bar\theta\to0$, provided an insertion with the dimension of an action — implemented as $-\frac{i}{\hbar}\delta(\theta)\delta(\bar\theta)$ in the Grassmann (Berezin) integration — supplies the factor $\hbar$: quantization becomes a dimensional reduction of supertime from $(t,\theta,\bar\theta)$ to $t$. The construction was developed by the same school into precise dequantization rules $\int dt\to i\hbar\int dt\,d\theta\,d\bar\theta$ and $\varphi^a\to\Phi^a$, valid at the level of generating functionals (A. A. Abrikosov Jr., E. Gozzi and D. Mauro, 2003 and 2005), and was shown by its authors to reproduce the steps of geometric quantization — prequantization space, polarization, and the passage from the Lie derivative of the Hamiltonian flow to the Schrödinger operator — in a unified way (2006).

Two aspects were flagged as open at the construction's inception and remain without a complete answer. First, what $\theta,\bar\theta$ are physically: in the target space the Grassmann partners of $\varphi^a$ are forms or Jacobi fields, but at the base-space level only dimensional heuristics exist — $\bar\theta\theta$ carries the dimension of an action, suggesting $\Delta t\sim\bar\theta\theta\sim\hbar$, a relation reminiscent of the Wiener process, i.e. the random-walk structure $(\Delta q)^2\sim\Delta t\,\hbar$ that underlies the Feynman path integral. Second, the extended classical formalism has universal symmetries (BRST-like, $N=2$ supersymmetry, ghost-number conjugation — the symmetry exchanging each ghost $c^a$ with its conjugate $\bar c_a$, which carry opposite ghost number, the integer grading of the Grassmann sector — and in particular a universal symmetry that rescales the overall action, mixing $\theta$ with $t$, identified already in E. Gozzi's 1985 work on transformation theory); which of these symmetries survive the $\theta,\bar\theta\to0$ projection into quantum mechanics and which are lost, and what the loss of those that do not survive entails for quantum mechanics — with the suggestion, made in later work, that $\hbar$ enters as the scale of an anomaly, that is, the failure of a classical symmetry to survive quantification signalled by the non-invariance of the functional measure — was left as an ongoing direction.

Problem Statement

Within the CPI/KvN framework, quantization of a Hamiltonian system is implemented by extending the time base to the supertime $(t,\theta,\bar\theta)$, packaging the $8n$ classical path-integral variables into the superfield $\Phi^a$, and freezing $\theta,\bar\theta\to0$ with an action-scale insertion $\bar\theta\theta/\hbar$. The problem is to give this construction a complete physical account, in two linked parts that jointly constitute one objective — understanding what the quantization-by-supertime-freezing procedure actually does — and that the originating literature left open:

(1) Operational meaning of the Grassmann partners of time. Provide a rigorous measurement-theoretic or dynamical characterization of what $\theta,\bar\theta$ encode — going beyond the dimensional identification $[\bar\theta\theta]=\text{action}$, $\Delta t\sim\bar\theta\theta\sim\hbar$ and the qualitative Wiener-process, Grassmann-temperature and coarse-graining proposals — from which the effect of the freezing procedure on the classical ensemble can be derived and checked; or prove that no such operational content exists within the formalism, i.e. that the Grassmann base coordinates are necessarily pure bookkeeping for the differential-form structure with no independent physical meaning.

(2) Universal symmetries lost in the projection. Determine completely which of the universal symmetries of the extended classical formalism (BRST/anti-BRST, ghost-number conjugation, the $N=2$ supersymmetry, and the action-rescaling symmetry relating $\theta$ and $t$) survive the $\theta,\bar\theta\to0$ projection into quantum mechanics, and what the loss of the action-rescaling symmetry entails: in particular, whether it can be realized as an anomaly of the classical functional measure under (field-dependent) supertime reparametrizations, with $\hbar$ emerging as the regularizing scale of the Jacobian, as sketched but never completed; and what observable or structural consequences the symmetry loss has for quantum mechanics.

An answer must hold for generic Hamiltonian systems, not only special potentials (such as the inverse-square potential, whose scale symmetry is already known to be anomalous), and must remain consistent with the established anchors of the construction: the freezing procedure reproducing the Feynman path integral and the dequantization rules, the equivalence with geometric quantization, and the metric/curvature distinction between classical and quantum supertime structures.

References

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