Abstract: Quantum tunneling is conventionally described through semiclassical WKB methods and imaginary-time formulations of the Feynman Path Integral, which successfully reproduce transmission probabilities but leave the local kinematic structure of the barrier transition largely implicit. In this work, we introduce a finite-rate temporal integration framework in which localized quantum states evolve through successive stochastic re-integrations over a characteristic interval . Within this picture, a macroscopic potential barrier acts as an open fluctuating electromagnetic environment rather than a rigid static boundary. As a localized state partially penetrates the barrier, transient geometric interlocking suppresses forward integration and prolongs the effective interaction window. The under-barrier configuration is modeled not as a single coherent leap across the forbidden region, but as a metastable sequence of stochastic local re-integrations continuously driven by environmental fluctuations. Transmission through thick barriers therefore emerges as a rare cumulative process requiring repeated successful geometric alignments across many finite integration intervals, naturally producing exponential suppression with barrier thickness. To formalize this mechanism, the tunneling process is represented as a coupled stochastic escape problem governed by bipartite Langevin dynamics and an associated Fokker–Planck description. The transition probability acquires a bilinear structure arising from simultaneous compatibility conditions between particle-side and environment-side fluctuations. In the self-consistent limit, this naturally produces the quadratic probability weighting characteristic of the Born measure, while remaining compatible with normalized Hilbert-space amplitudes. Within this framework, the Euclidean-time contribution appearing in standard tunneling theory admits a local kinematic interpretation as the extended stochastic integration interval required for compatible transition geometries to accumulate across the barrier. The formalism further predicts enhanced sensitivity of tunneling dynamics to high-frequency external probing, potentially relevant to attoclock tunneling-delay experiments and strong-field manifestations of the Quantum Zeno effect.
Abstract: Quantum tunneling is conventionally described through semiclassical WKB methods and imaginary-time formulations of the Feynman Path Integral, which successfully reproduce transmission probabilities but leave the local kinematic structure of the barrier transition largely implicit. In this work, we introduce a finite-rate temporal integration framework in which localized quantum states evolve through successive stochastic re-integrations over a characteristic interval . Within this picture, a macroscopic potential barrier acts as an open fluctuating electromagnetic environment rather than a rigid static boundary. As a localized state partially penetrates the barrier, transient geometric interlocking suppresses forward integration and prolongs the effective interaction window. The under-barrier configuration is modeled not as a single coherent leap across the forbidden region, but as a metastable sequence of stochastic local re-integrations continuously driven by environmental fluctuations. Transmission through thick barriers therefore emerges as a rare cumulative process requiring repeated successful geometric alignments across many finite integration intervals, naturally producing exponential suppression with barrier thickness. To formalize this mechanism, the tunneling process is represented as a coupled stochastic escape problem governed by bipartite Langevin dynamics and an associated Fokker–Planck description. The transition probability acquires a bilinear structure arising from simultaneous compatibility conditions between particle-side and environment-side fluctuations. In the self-consistent limit, this naturally produces the quadratic probability weighting characteristic of the Born measure, while remaining compatible with normalized Hilbert-space amplitudes. Within this framework, the Euclidean-time contribution appearing in standard tunneling theory admits a local kinematic interpretation as the extended stochastic integration interval required for compatible transition geometries to accumulate across the barrier. The formalism further predicts enhanced sensitivity of tunneling dynamics to high-frequency external probing, potentially relevant to attoclock tunneling-delay experiments and strong-field manifestations of the Quantum Zeno effect.
Quantum tunneling is conventionally described by the Schrödinger wave equation with a prescribed static potential barrier, providing accurate transmission probabilities but offering limited insight into the microscopic dynamics of particle–barrier interactions. In this work, we develop a dynamical second-quantization framework in which the barrier is modeled as an ensemble of quantized internal modes rather than as an externally imposed classical potential. The tunneling particle interacts directly with these microscopic barrier excitations through a coupled particle–barrier Hamiltonian, from which the Heisenberg equations of motion are derived. Collective coherent excitations of the barrier modes give rise to an emergent effective barrier that naturally recovers the conventional rectangular barrier and the WKB transmission limit under appropriate conditions. Unlike standard treatments, the present formulation explicitly incorporates microscopic barrier dynamics and provides a unified description of particle–barrier coupling within a second-quantized formalism. The framework further suggests that repeated tunneling events may experience different microscopic interaction histories, motivating a statistical interpretation of tunneling times. Because both the particle and barrier are treated within the same operator formalism, the theory provides a natural foundation for extension to relativistic quantum transport, photonic barriers, cavity quantum electrodynamics, and other structured quantum media.
Abstract This work examines the role of finite temporal structure in quantum state formation by relaxing the assumption of instantaneous state reduction. Introducing a finite integration interval Δτ as a temporal regulator, we formulate a modified evolution in which standard unitary dynamics and state projection appear as limiting regimes of a single kinematic process. In slowly varying environments, the formulation reproduces adiabatic unitary evolution. However, transitions into macroscopic measurement environments correspond to strongly non-adiabatic regimes, where finite-rate state formation induces a systematic mismatch between the instantaneous Hamiltonian and the realized state. Within a density matrix formulation, this mechanism naturally generates both decoherence-like dynamics and non-adiabatic driving terms. When combined with stochastic environmental fluctuations, the resulting continuous integration process yields statistical outcomes consistent with the Born rule as a macroscopic limit. The framework preserves the empirical predictions of standard quantum mechanics while providing a continuous, finite-rate interpretation of state projection and measurement.
We present a technical validation of Geometric Quantum Tunneling (GQT) within the MMA-DMF framework. In this approach, barrier penetration is not postulated: it emerges as a deterministic, noise-assisted escape process generated by a scalar-field generalized Langevin dynamics whose stochastic sector is thermodynamically closed by a strict fluctuation--dissipation relation [1, 2] and regulated by a finite ultraviolet (UV) cutoff at a fundamental scale M≃1.00×102 TeVM \simeq 1.00 \times 10^{2}\,\mathrm{TeV}M≃1.00×102TeV. We report 1D benchmark results showing (i) Schrödinger-like dispersion calibration at Nelson equilibrium, (ii) WKB recovery via exponential transmission scaling with barrier width (linear lnT\ln TlnT vs width with R2>0.99R^{2} > 0.99R2>0.99 and a consolidated WKB recovery error of 1.2%), (iii) quantitative isomorphism against transfer-matrix quantum mechanics across energy regimes (deep-tunneling RMSE 0.042, resonant RMSE 0.085, classical-transport RMSE 0.011), and (iv) the distinctive “geometric wall” prediction that tunneling collapses as V0/M→1V_{0}/M \to 1V0/M→1 (rapid suppression already near V0≈0.95MV_{0} \approx 0.95MV0≈0.95M and an inferred cutoff near 0.98M0.98M0.98M in the consolidated summary). We include mandatory ablations via the geometric toggle αgeo\alpha_{\mathrm{geo}}αgeo (tunneling vanishes as αgeo→0\alpha_{\mathrm{geo}} \to 0αgeo→0), and stability/convergence constraints requiring Δt≲0.01 M−1\Delta t \lesssim 0.01\,M^{-1}Δt≲0.01M−1 with validated production operation at Δt=0.005 M−1\Delta t = 0.005\,M^{-1}Δt=0.005M−1 and an energy drift slope 3.4×10−63.4 \times 10^{-6}3.4×10−6 over tmax=1000t_{\max}=1000tmax=1000. Finally, we specify a reproducibility and integrity layer (bit-exact seed replay across hardware targets, deterministic checkpointing, and SHA-256 manifesting) intended to make the full pipeline externally auditable. The consolidated evidence supports that MMA-DMF reproduces standard low-energy tunneling phenomenology while making a falsifiable high-energy prediction tied to UV regulation.
Abstract We validate Geometric Quantum Tunneling (GQT) within the MMA–DMF framework, where barrier transmission is not imposed as a postulate but arises as a rare, noise-assisted escape process generated by deterministic scalar-field dynamics coupled to a structured vacuum sector. The stochastic forcing is thermodynamically constrained by a strict fluctuation–dissipation closure and regulated by a finite ultraviolet (UV) cutoff at a fundamental scale 𝑀 ≃ 1.00 × 102 TeV. Using a one-dimensional rectangular barrier as a controlled benchmark, we report: (i) Schrödinger-like wavepacket dispersion at Nelson equilibrium; (ii) recovery of semiclassical WKB scaling through an exponential dependence of transmission on barrier width (linear ln 𝑇 vs. width with 𝑅2 > 0.99 and a consolidated WKB recovery error of 1.2%); (iii) quantitative agreement with transfer-matrix quantum mechanics across deep-tunneling, resonant, and classical-transport regimes (RMSE 0.042, 0.085, and 0.011, respectively); and (iv) a distinctive near-cutoff prediction in which transmission is strongly suppressed as 𝑉0/𝑀 → 1 (rapid suppression already near 𝑉0 ≈ 0.95𝑀, with an inferred cutoff near 0.98𝑀 in the consolidated summary). Control tests include ablations that disable the geometric coupling, for which tunneling vanishes as expected. Numerical stability and convergence are verified with a step-size constraint 𝑑𝑡 ≲ 0.01𝑀−1; validated runs use 𝑑𝑡 = 0.005𝑀−1 with a small measured energy-drift slope (3.4 × 10−6) over 𝑇max = 1000. Data, scripts, and numerical artifacts required to reproduce the reported benchmarks are provided in the accompanying replication package.
Quantum tunneling is conventionally described using the stationary Schrödinger equation with an externally imposed potential barrier. In this work, we present a dynamical formulation of quantum tunneling in which the barrier is modeled as a structured medium possessing quantized internal modes that interact coherently with an incident electron. Using the Heisenberg operator formalism and a second-quantized representation of the barrier medium, we derive coupled dynamical equations governing the electron–barrier interaction. Under a continuum-mode and mean-field approximation, the collective response of the barrier modes generates an effective potential that reproduces the conventional rectangular barrier model and the standard tunneling transmission probability obtained from the Schrödinger equation. Within this framework, a tunneling traversal time is naturally defined from the dynamical evolution of the electron and is shown to depend on the barrier width, barrier height, and incident electron energy. Numerical simulations illustrating the transmission probability and tunneling-time behavior are presented. The results provide a complementary microscopic interpretation of tunneling processes in structured quantum media and may be relevant to nanoscale transport, photonic barriers, and coherent quantum devices.
The question of how long a particle takes to tunnel through a potential barrier has remained unresolved for decades. Attoclock experiments report traversal times consistent with zero, while Larmor clock measurements yield small but nonzero durations---a discrepancy recently traced to the two methods measuring fundamentally different quantities. We propose a resolution based on a discrete field-quantum network model whose continuum limit is the nonlinear Klein--Gordon equation. In this framework, a particle is not a point object localized on one side of a barrier: it is an extended structure whose electromagnetic and gravitational stress fields---carrying a significant fraction of its total energy---penetrate the barrier and extend to the far side. When a detector on the far side couples resonantly to this pre-existing stress tail, the particle's global structural integrity fails instantaneously, delivering its full energy to the detection point without any transit through the barrier. The ``tunneling time'' is near-zero not because the particle moves superluminally, but because it was already present on both sides before detection. This picture naturally explains the attoclock's null result (no transit occurs), the Larmor clock's nonzero reading (spin precession accumulates in the extended field within the barrier), and the established exponential decay of tunneling probability with barrier width (the stress-field amplitude decays exponentially in classically forbidden regions). No new parameters or modifications to quantum mechanics are introduced.