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arXiv:2605.24373

Semiclassical Propagation and the Dynamics of Configuration Space

V. S. Morales-Salgado

quant-phgr-qchep-thmath-phSubmitted May 23, 2026• Announced May 26, 2026• 0 citations
v1 · May 23, 2026v2 · Jun 3, 2026
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Abstract

This work explores the non-relativistic quantum propagator $K(x,t)$ as a solution of the Schrödinger equation. We suppose that the propagator takes the form ${\rm exp}\left(\frac{\mathrm{i}}{\hbar}S+R\right)$, generalizing the usual WKB ansatz by allowing an additive exponent $R$ whose role as a measure of quantumness is investigated. Since $S$ is subject to the assumption that it is a solution of the Hamilton-Jacobi equation, here we are further interested in the role of $R$ and its interpretation as a measure of quantumness. Several systems are studied as concrete examples to illustrate this approach. Furthermore, a proposal to generalize it for fields is put forward. This is tested for some simple systems. Finally, the possibilities to use this approach for the case of systems whose dynamics are controlled by the Hamiltonian constraint are analyzed.

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This work explores the non-relativistic quantum propagator K(x,t)K(x,t)K(x,t) as a solution of the Schrödinger equation. We suppose that the propagator takes the form exp(iℏS+R){\rm exp}\left(\frac{\mathrm{i}}{\hbar}S+R\right)exp(ℏi​S+R), generalizing the usual WKB ansatz by allowing an additive exponent RRR whose role as a measure of quantumness is investigated. Since SSS is subject to the assumption that it is a solution of the Hamilton-Jacobi equation, here we are further interested in the role of RRR and its interpretation as a measure of quantumness. Several systems are studied as concrete examples to illustrate this approach. Furthermore, a proposal to generalize it for fields is put forward. This is tested for some simple systems. Finally, the possibilities to use this approach for the case of systems whose dynamics are controlled by the Hamiltonian constraint are analyzed.
Taking as upstream input the Distinguished Reproducing Kernel fixed-point program, which forces the realized branch (Σ, g*) ≅ CH² together with its exact normalization package, this paper develops the semiclassical limit of exact-normalized closure. The starting point is the exact-normalized evolution law iħ ∂tψ = Hψ with H = −ħ²Δμ*, where Δμ* is the weighted Laplacian determined by the exact-normalized realized measure rather than by the raw metric volume alone. Writing ψ = a exp(iS/ħ), the paper derives the exact coupled phase-amplitude system associated with the selected generator. Its real part gives a Hamilton–Jacobi-type equation with an exact-normalized quantum correction term, while its imaginary part gives the corresponding transport law for the amplitude. In the semiclassical regime, where the exact-normalized quantum correction tends to zero and the phase converges locally, the phase sector reduces to the leading-order Hamilton–Jacobi equation on the realized branch. Thus the classical limit is not introduced externally, but arises as the leading-order phase law of exact-normalized Distinguished Reproducing Kernel evolution. The paper also derives the continuity equation for the transported density ρ = a² and the corresponding characteristic flow field X_S = 2∇S selected by the limiting phase sector. In this way, the realized branch carries a distinguished classical flow that is not imposed independently, but forced by the same exact-normalized structure that governs the quantum evolution law. All theorem statements are purely formal statements about the inherited branch structure. Physical interpretation is introduced only afterward, through explicit postulates identifying the phase with classical action, the semiclassical limit with the classical limit, the transported density with classical density, and the characteristic flow with classical trajectories. The theorem-level outputs of the paper are therefore the exact phase-amplitude system, the exact-normalized quantum correction term, the forced Hamilton–Jacobi law on the realized branch in the semiclassical limit, the continuity equation for the transported density, and the distinguished characteristic flow selected by the limiting phase sector. Gravitational and minisuperspace specializations are deferred to the following paper.
A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [\ensuremath{\partial}(qf{\mathit{q}}_{\mathit{f}}qf​,pf{\mathit{p}}_{\mathit{f}}pf​)/\ensuremath{\partial}(qi{\mathit{q}}_{\mathit{i}}qi​,pi{\mathit{p}}_{\mathit{i}}pi​)], where (qi{\mathit{q}}_{\mathit{i}}qi​,p){\mathit{p}}_{)}p)​ and (qf{q}_{f}qf​,pf{p}_{f}pf​) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature (1,4)(1,4)(1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as Γ=L2/ℓ2Γ= L^2/\ell^2Γ=L2/ℓ2 for states that saturate the uncertainty relations, where LLL and ℓ\ellℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit ℓ→0\ell \to 0ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit L→∞L \to \inftyL→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both ℓ→0\ell \to 0ℓ→0 and L→∞L \to \inftyL→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.
Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the SSS-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.