This paper formulates an entropic Hamilton–Jacobi wave map within the L(6+7) framework, in which the quantum wave function is not treated as a primary isolated object but as the observable projection of an ensemble of extremal histories in an extended state space. The starting point is the Lohmiller–Slotine/MIT bridge: a quantum wave function can be reconstructed from a multi-valued classical action and the density of the associated classical flow. The proposed extension embeds this bridge into the canonical L(6+7) corpus: the unified variational root, the internal seven-sector, entropic time, stable-history selection, and meso–micro reductions. The central result is a wave-map formula in which the phase of a branch is determined by the full action, the amplitude is determined by a density-entropic weight, and the observable wave arises by projection over the internal seven-sector. In the frozen-sector limit the construction returns to the standard Hamilton–Jacobi action-density map; when the internal sector is active, phase and contrast corrections arise and can be tested in a minimal double-slit model.
This paper formulates an entropic Hamilton–Jacobi wave map within the L(6+7) framework, in which the quantum wave function is not treated as a primary isolated object but as the observable projection of an ensemble of extremal histories in an extended state space. The starting point is the Lohmiller–Slotine/MIT bridge: a quantum wave function can be reconstructed from a multi-valued classical action and the density of the associated classical flow. The proposed extension embeds this bridge into the canonical L(6+7) corpus: the unified variational root, the internal seven-sector, entropic time, stable-history selection, and meso–micro reductions. The central result is a wave-map formula in which the phase of a branch is determined by the full action, the amplitude is determined by a density-entropic weight, and the observable wave arises by projection over the internal seven-sector. In the frozen-sector limit the construction returns to the standard Hamilton–Jacobi action-density map; when the internal sector is active, phase and contrast corrections arise and can be tested in a minimal double-slit model.
Abstract Lohmiller and Slotine (2026) derive, within their formalism, that the Schrödinger wavefunction can be computed exactly from a discrete set of classical extremal action paths — the J-valued action census. SIP-PHY-05 mapped the structural correspondence between this construction and the CDR/GTRS regulatory grammar, and proposed a thermodynamic extension conjecture: that L&S's kinematic density is the zero-dissipation limit of a thermodynamic density incorporating entropy production, with a predicted deviation scaling ε ~ σ/ω. This paper provides independent computational verification of the L&S construction and a systematic parameter-regime test of the thermodynamic extension conjecture. Three principal findings are reported. First, the L&S classical-path reconstruction is verified to fidelity > 0.99999 for wavepacket evolution in a particle-in-a-box (convergence study: infidelity decreases exponentially with image count) and produces exact quantum interference from J = 2 classical paths in the double-slit geometry. Second, the naive global linear scaling conjecture (ε ~ σ/ω) is falsified at 95% confidence: the full-range exponent b = 0.250 ± 0.048 (95% CI [0.195, 0.469]) excludes b = 1. Parameter-regime mapping across temperatures (T/ω ∈ [0.01, 5.0]), initial state amplitudes (|α₀| ∈ [0.5, 8.0]), and evaluation times (t ∈ [0.5, 50.0]) reveals near-linear but measurably sub-linear scaling in the weak-dissipation regime: b_weak = 0.890 ± 0.013 (95% CI [0.872, 0.980], R² = 0.9996), approaching but not reaching b = 1. Third, the strong-dissipation limit is governed by a derivable thermal equilibrium floor: f∞ = 1/(1 + 2n_th), where n_th is the Bose-Einstein thermal occupation number. This floor is exact to machine precision — a theorem, not an empirical observation. The overall picture is domain-of-validity discovery: the thermodynamic correction crosses over from a near-linear perturbative regime to a saturating thermal equilibrium, with the crossover controlled by the dimensionless thermalisation parameter γt. Keywords: Lohmiller-Slotine, exact quantum reconstruction, thermodynamic extension, Lindblad, Wigner function, entropy production, thermal floor, Bose-Einstein occupation, CDR cycle, GTRS, parameter regime map, computational falsification
Recent work shows that the Schrödinger equation can be solved exactly based only on classical least action. The computation is based on first computing the multi-valued action solutions of the Hamilton-Jacobi equation, computing the classical propagated density along all stationary action multipaths, and finally constructing the exact kernel and wave function based on these classical multi-valued quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic, as is the case for basic examples such as e.g., the double-slit, quantum tunneling, or the Einstein-Podolsky-Rosen experiment. For more complex cases, residual Bohm-like quantum potentials along each stationary action path can be eliminated through a time-rescaling.
This paper proposes an extended version of the above construction which avoids the need for time-rescaling even in the most general cases. This is achieved by introducing in the Hamilton-Jacobi equation a real viscosity term, which classically simply corresponds to replacing a lumped parameter model with a continuum model including a volumetric stress tensor. This keeps the earlier construction unchanged, except that the real stationary action multipaths are now real stationary viscous action multipaths defined with respect to an associated modified Lagrangian. As in the earlier version, the construction extends straightforwardly to the special and general relativistic contexts, thus describing in a common language the quantum, classical, and relativistic cases.
Note that the global Bohm potential term in the lumped Hamilton-Jacobi equation associated with a Madelung transform does not have a classical equivalent, while the volumetric stress tensor term in our viscous Hamilton-Jacobi equation directly represents the real paths of a classical deformed particle field.
We develop a dissipative extension of classical mechanics based on a complex, and more generally quaternionic, action principle that endows every classical system with an intrinsic environment. Decomposing the action into conservative and divergence-induced components yields two coupled Hamilton Jacobi equations describing a dynamically intertwined system environment pair. This motivates a Dual Sector or Dual Environmental Interpretation (DSI/DEI), in which the additional degrees of freedom behave as an image sector exchanging energy, information, and phase with the system. Applying a generalized Madelung transform produces a nonlinear dissipative wave equation whose symmetric equilibrium limit reduces to the Schroedinger equation, with the quantum potential and linearity emerging from balanced intersector coupling. In this framework, the wavefunction is not fundamental but encodes the interaction geometry between system and environment, providing a classical origin for interference, amplitude phase coupling, and probabilistic structure. Extending the imaginary structure to multiple independent directions yields a multienvironment generalization capable of representing measurement-like processes, nonMarkovian memory, and entanglement type correlations. The formulation unifies aspects of dual-system models, hydrodynamic approaches, and non-Hermitian dynamics within a single action-based framework, and suggests that quantum mechanics corresponds to a stable symmetric phase of a broader dissipative classical theory.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density ρ=R2 coupled to Hamilton's principal function S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair (R,S) into a single complex field. Starting from a completely general complex ansatz ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose ∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.
In this paper, we develop quantum mechanics of quasi-one-dimensional systems upon the framework of the quantum-mechanical Hamilton–Jacobi theory. We will show that the Schrödinger point of view and the Hamilton–Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result of this, a wavefunction in Hamilton–Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. Using the quasi-linearization technique, we derive quantum correction functions in every order of . The quantum correction functions will remove the turning-point singularity that plagues the WKB-series expansion already in zeroth order and thus provide an extremely good approximation to the full solution of the Schrödinger equation. In the language of quantum action it is also possible to elegantly solve the connection problem without asymptotic approximations. The use of quantum action further allows us to derive an equation by which the Maslov index is directly calculable without any approximations. Stationary quantum trajectories will also be considered and thoroughly discussed.