Quantum Action Quantization

• Quantum-action-based quantization is a framework that formulates quantum dynamics using a restricted action principle over coherent states, yielding explicit ℏ corrections.
• It eliminates operator-ordering ambiguities by identifying classical variables with coherent state expectation values, ensuring a smooth classical limit.
• The approach applies broadly, from single-particle systems to quantum field theory and quantum gravity, and integrates information-theoretic extensions.

Quantum-action-based quantization is a paradigm in which the quantization of classical systems is formulated directly in terms of the action functional, treating classical and quantum mechanics within a unified variational framework. This approach diverges from canonical quantization and path-integral methods by emphasizing coherent-state restrictions, information-theoretic extensions, and explicit appearance of quantum corrections within augmented classical actions. The methodology is applicable to a wide class of systems, from single-particle mechanics and constrained systems to quantum field theory and quantum gravity. It also establishes deep connections between action principles, quantum interference, and information theory.

1. Fundamental Principles

At the core of quantum-action-based quantization lies the quantum action functional, typically of the form

$$S_Q[|\psi\rangle] = \int dt\, \langle \psi(t) | (i\hbar \partial_t - \hat{H}) | \psi(t) \rangle$$

with variations over normalized Hilbert-space vectors $|\psi(t)\rangle$. Unlike traditional quantization, which promotes classical variables $(q,p)$ to operators $(Q,P)$ and postulates commutation relations, the quantum-action-based approach restricts variations to a parametrized manifold of coherent states $|p(t), q(t)\rangle$, corresponding to semiclassical paths in phase space (Klauder, 2012, Geloun et al., 2012, Adorno et al., 2014).

This restriction yields a restricted quantum action

$S_{Q,R}[p,q] = S_Q[|\psi\rangle \to |p,q\rangle]$

which computes to the classical action plus explicit $\mathcal{O}(\hbar)$ corrections, encoding quantum fluctuations through the geometry of the underlying coherent states.

Crucially, this framework establishes that classical mechanics is embedded as a limiting case within quantum theory ($\hbar \rightarrow 0$), while retaining controlled quantum corrections for finite $\hbar$. The identification of classical variables with expectation values in coherent states removes ordering ambiguities and unifies dynamical evolution (Klauder, 2012).

2. Coherent States, Fiducial Vectors, and Geometric Structures

The construction of coherent states is essential. For canonical systems on $\mathbb{R}^n$, coherent states are typically defined via the Heisenberg-Weyl group: $|\psi(t)\rangle$0 where $|\psi(t)\rangle$1 is a fiducial (e.g., ground) vector with vanishing expectation values for $|\psi(t)\rangle$2 and $|\psi(t)\rangle$3. For systems with nontrivial topology (e.g., a particle on the circle $|\psi(t)\rangle$4), the procedure requires self-adjoint extensions of $|\psi(t)\rangle$5, appropriate boundary conditions, and a spectral family indexed by a topological parameter $|\psi(t)\rangle$6 (Geloun et al., 2012).

The overcompleteness and normalization of coherent states ensure a resolution of the identity and bring forth a geometric (Fubini–Study) metric on the parameter space, leading to an explicit identification of the classical phase-space variables with expectation values in the quantum domain (Adorno et al., 2014, Klauder, 2012). For systems with additional symmetries or statistics (e.g., bosons, fermions, anyons), the choice of fiducial vector encodes the statistical sector and determines the structure of $|\psi(t)\rangle$7 corrections (Adorno et al., 2014).

3. Quantum Action, Classical Limit, and $|\psi(t)\rangle$8-Corrections

Restricting the quantum action to the coherent-state manifold yields

$|\psi(t)\rangle$9

where the $(q,p)$0 terms are explicit and computable as averages of higher moments (quantum fluctuations) associated with the chosen fiducial vector. For example, with $(q,p)$1,

$(q,p)$2

This structure ensures that quantum mechanics is not merely a deformation of classical dynamics, but a natural extension that incorporates quantum fluctuations—recovering classical physics as a singular limit (Adorno et al., 2014, Klauder, 2012, Geloun et al., 2012).

On nontrivial configuration manifolds (e.g., $(q,p)$3), topological and group-theoretical effects produce boundary terms and nonvanishing variances that persist in the semiclassical limit, exemplifying the geometric and topological subtleties of the approach (Geloun et al., 2012).

4. Information-Theoretic and Extended-Action Approaches

Several recent frameworks generalize the quantum-action-based method by explicitly incorporating information-theoretic penalties into the action. For instance, spin quantization emerges from an extended least action principle augmented with a Tsallis (or Rényi) relative entropy term quantifying distinguishability of hidden spin orientations (Yang, 2024). The extremization with respect to probability densities on the orientation sphere $(q,p)$4 enforces quantized eigenvalues as a mathematical consequence in a singular limit.

More broadly, one can reconstruct quantum dynamics as an inference problem over action space: assigning exponential weights to action—labeled alternatives according to maximum-entropy principles yields complex amplitudes, unitary evolution, and ultimately the path integral prescription. Here, Planck’s constant $(q,p)$5 defines the fundamental resolution scale beyond which different action contributions are operationally indistinguishable (Luiz et al., 10 Feb 2026).

Analogous information-theoretic logic underpins action-based variational formulations for constrained quantum systems, where the quantum potential (arising from Fisher information or relative entropy) is systematically embedded in the extended action, circumventing operator-ordering ambiguities and supporting simultaneous enforcement of quantization and constraints (Yang, 26 Sep 2025).

5. Generalizations: Constrained Systems, Field Theory, and Quantum Gravity

Quantum-action-based quantization is adaptable to constrained systems, quantum field theories, and quantum gravity.

For constrained quantum systems (e.g., those with vanishing total momentum), the simultaneous variation of the extended action (with information term and Lagrange multipliers for constraints) yields the full quantum dynamics and naturally derives the quantum potential. This method generalizes Dirac quantization, applying even when constraints do not correspond to linear operators (Yang, 26 Sep 2025).

In quantum field theory, promoting time to a quantum degree of freedom ("quantum time") and passing to a second-quantized action operator in Fock space produces a manifestly covariant quantum-action-based formulation. This framework avoids the collapse to standard equal-time QFT that arises in Dirac quantization of classical field actions, and introduces a new notion of "spacetime state" encoding both spacelike and timelike entanglement in a non-Hermitian operator, with correct recovery of ordinary correlation functions and S-matrix elements in suitable limits (Diaz, 27 Feb 2026).

In quantum gravity, action-based techniques underpin symplectic quantization, where the kinetic term is constructed as a functional over the metric and its "proper-time" derivative, introducing an "intrinsic time" parameter. The resulting action-preserving microcanonical ensemble yields the standard gravitational path integral with an emergent cosmological constant upon suitable integration over momentum variables (Gradenigo, 2021). Relatedly, quantization of minisuperspace actions (e.g., for inflation driven by the quantum potential) is naturally formulated in this framework, providing mechanisms for inflation and reheating purely from the quantum fluctuations in action (He et al., 2014, Salih, 2009).

6. Alternative and Specialized Quantum-Action-Based Schemes

Various specialized forms of quantum-action-based quantization have been developed:

• Action-Angle Coherent-State Quantization: For periodic and integrable systems, coherent states constructed from action–angle variables yield quantization schemes tailored to match arbitrarily prescribed spectra, generalizing both Bohr–Sommerfeld and canonical rules. Here, Bayesian selection of probability densities over action space achieves exact reproduction of observed eigenvalues and supports the construction of well-behaved angle operators (Gazeau et al., 2011).
• Quantum Action Variable Quantization: In the quantum Hamilton–Jacobi approach, the quantum reduced action $(q,p)$6 satisfies a third-order differential equation, with action quantization imposed via Milne’s rule $(q,p)$7. The method makes explicit the microstate structure hidden in the Schrödinger representation and provides deterministic, microstate-dependent trajectories, consistent with all standard quantization results (Floyd, 2015).
• Discrete and Computational Settings: Discrete-time quantum automata and quantum walks admit quantization via action functionals on Hilbert-space sequences, ensuring unitary evolution and allowing direct derivation of discrete conservation laws. Manifest covariance can be maintained by including dynamical lattice coordinates (Debbasch, 2018).
• Optimization and Statistical Mechanics: Quantum-action-based methodology provides a foundation for quantization-based optimization, unifying gradient flow, thermodynamic, and Schrödinger equations. Quantization is interpreted as a finite action-resolution effect, and quantum tunneling enables sampling algorithms to escape local minima even in complex combinatorial scenarios (Seok et al., 12 Mar 2026).

7. Conceptual Implications and Outlook

Quantum-action-based quantization provides a unifying and flexible framework, embedding classical dynamics, quantum corrections, and even information-theoretic principles within a single variational structure. Classical mechanics is not merely a limiting regime, but a submanifold within a quantum theory parameterized by finite $(q,p)$8, and quantum corrections are geometrically and physically interpretable. The explicit role of the action, variational procedures, and coherent-state geometry resolves longstanding ambiguities in canonical quantization, supports extensions to constrained and topologically nontrivial systems, and rigorously connects quantization to foundational questions in information theory and inference.

This approach is actively explored in foundational, mathematical, and applied physics contexts, including quantum field theory, constrained systems, quantum gravity, spin systems, and emerging computational paradigms (Klauder, 2012, Geloun et al., 2012, Adorno et al., 2014, Luiz et al., 10 Feb 2026, Yang, 26 Sep 2025, Diaz, 27 Feb 2026, Gradenigo, 2021, Gazeau et al., 2011, Floyd, 2015, Seok et al., 12 Mar 2026).