We propose the _Quantum Optimal Transport Amplitude Method_ (QOTAM), a new framework in which scattering amplitudes arise from a complex optimal-transport problem on on-shell kinematic space. Incoming and outgoing wavepackets define probability distributions, and we introduce a complex kernel \(K(q|p)\) whose modulus squared transports \(\rho_{\rm in}\) to \(\rho_{\rm out}\) while its phase encodes the classical action. The physical amplitude is obtained from the variational principle \(\delta\mathcal{S}[K]=0\), where \(\mathcal{S}[K]\) includes a transport cost, an \(iS_{\rm cl}/\hbar\) phase term, and regularizers enforcing locality and factorization. Tree amplitudes emerge from linear response of the optimal kernel, and the formulation admits efficient numerical realization via entropically regularized optimal transport. QOTAM offers a geometric–variational approach to scattering with potential applications to high-multiplicity and non-perturbative regimes.
We propose the Quantum Optimal Transport Amplitude Method (QOTAM), a new framework in which scattering amplitudes arise from a complex optimal-transport problem on on-shell kinematic space. Incoming and outgoing wavepackets define probability distributions, and we introduce a complex kernel (K(q|p)) whose modulus squared transports (\rho_{\rm in}) to (\rho_{\rm out}) while its phase encodes the classical action. The physical amplitude is obtained from the variational principle (\delta\mathcal{S}[K]=0), where (\mathcal{S}[K]) includes a transport cost, an (iS_{\rm cl}/\hbar) phase term, and regularizers enforcing locality and factorization. Tree amplitudes emerge from linear response of the optimal kernel, and the formulation admits efficient numerical realization via entropically regularized optimal transport. QOTAM offers a geometric–variational approach to scattering with potential applications to high-multiplicity and non-perturbative regimes.
Quantum scattering connects microscopic dynamics to amplitudes, phase shifts, cross sections, transport coefficients, and resonances. This selective Review organizes the Lippmann–Schwinger equation, resolvents, transfer and scattering matrices, nonequilibrium Green’s functions, and R-matrix theory as related representations of a common scattering problem. Born, Wentzel–Kramers–Brillouin, and eikonal error tests accompany comparisons of numerical scaling, stability, and method selection. Inverse scattering is examined through the effects of phase loss, incomplete aperture, Jacobian null spaces, singular-value decay, model discrepancy, and noise, together with regularized, Bayesian, and learned approaches for stabilizing reconstruction. The Review then discusses probability-level optimal transport and its extension to emerging amplitude-based formulations, using the Quantum Optimal Transport Amplitude Method (QOTAM) as a concrete example. Its Euclidean and real-time variational constructions are summarized and tested across resonant barriers, multichannel and angular scattering, three-dimensional targets, and disordered waveguides. Independent reference calculations verify complex-amplitude closure, unitarity, flux conservation, and optical-theorem relations to benchmark-dependent numerical precision. In inverse problems, QOTAM is found to be competitive with correctly specified phase-sensitive likelihood methods while showing useful robustness in selected model-mismatch tests. These examples illustrate the promise of optimal-transport-based scattering methods. More broadly, the Review emphasizes how physical constraints, uncertainty quantification, identifiability, and computational scalability jointly shape reliable forward and inverse-scattering analyses.
Quantum optimal transport (QOT) is a rapidly developing field. Among the many formulations of this adaptation of classical optimal transport (OT) to spaces of density matrices, we numerically study a family of distances based on a dynamical formulation inspired by the Benamou-Brenier OT formulation. We introduce an interior-point regularized method to compute geodesics between positive semidefinite matrices and visualize the results in terms of integral kernels and densities, inspired by quantum chemistry applications. We show that dynamical QOT may provide a good approximation to certain problems in quantum chemistry with appropriate parameter tuning. We also study the numerical properties of the distances at hand, and the convergence of the objects when the size of the matrices increases.
We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.
Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.