Since Newton's time, deterministic causality has been considered a crucial prerequisite in any fundamental theory in physics. In contrast, the present work investigates stochastic dynamical models for motion in one spatial dimension, in which Newtonian mechanics becomes an emergent property: We present a coherent theory in which a Hamilton-Jacobi equation (HJE) emerges in a description of the evolution of entropy $-ϕ(x,t)=ε\log$(Probability) of a system under observation and in the limit of large information extent $ε^{-1}$ in homogeneous space and time. The variable $ϕ$ represents a non-random high-order statistical concept that is distinct from probability itself as $ε=0$; the HJE embodies an emergent law of deterministic causality in continuous space and time with an Imaginary Scale symmetry $(t,x,ϕ)\leftrightarrow (it,ix,-iϕ)$. $ϕ(x,t)$ exhibits a nonlinear wave phenomenon with a mathematical singularity in finite time, overcoming which we introduce viscosity $ε(\partial^2ϕ/\partial x^2)$ and wave $iε(\partial^2 ϕ/\partial x^2)$ perturbations, articulating dissipation and conservation, which break the Imaginary Scale symmetry: They lead to the Brownian motion and Schrödinger's equation of motion, respectively. Last but not least, Lagrange's action in classical mechanics acquires an entropic interpretation and Hamilton's principle is established.
Since Newton's time, deterministic causality has been considered a crucial prerequisite in any fundamental theory in physics. In contrast, the present work investigates stochastic dynamical models for motion in one spatial dimension, in which Newtonian mechanics becomes an emergent property: We present a coherent theory in which a Hamilton-Jacobi equation (HJE) emerges in a description of the evolution of entropy −ϕ(x,t)=εlog(Probability) of a system under observation and in the limit of large information extent ε−1 in homogeneous space and time. The variable ϕ represents a non-random high-order statistical concept that is distinct from probability itself as ε=0; the HJE embodies an emergent law of deterministic causality in continuous space and time with an Imaginary Scale symmetry (t,x,ϕ)↔(it,ix,−iϕ). ϕ(x,t) exhibits a nonlinear wave phenomenon with a mathematical singularity in finite time, overcoming which we introduce viscosity ε(∂2ϕ/∂x2) and wave iε(∂2ϕ/∂x2) perturbations, articulating dissipation and conservation, which break the Imaginary Scale symmetry: They lead to the Brownian motion and Schrödinger's equation of motion, respectively. Last but not least, Lagrange's action in classical mechanics acquires an entropic interpretation and Hamilton's principle is established.
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.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics
This article explores models of causality of the physical universe which extend beyond the usual quantum and classical descriptions. To facilitate discussion of this topic, a brief review of relevant areas of modern physics is included. Although mathematical logic is not directed in time, the causal description of the physical universe is classically performed in terms of temporal determinism, an initial-value problem in which subsequent states are determined through the action of differential equations, thus avoiding the paradox of effects preceding their causes. Quantum mechanics has reduced this description to the status of a stochastic-causal theory, in which individual future states of a system are not predictable because the probability interpretation of the wave function prevents a complete knowledge of a single reduction of the wave packet. The transactional interpretation of quantum mechanics provides a basis for examining the workings of stochastic causality in terms of time-symmetri c advanced and retarded waves. Cosmic symmetry-breaking provides a contrasting description in which time-directed phenomena derive their explanation. Dual-time supercausality replaces the stochastic model with one in which two complementary causal processes, symmetric- time and directed-time are operating. The mode of interaction of these avoids temporal paradox in the time- directed description, and also explains why quantum mechanics provides an incomplete stochastic model. An important role for supercausal processes in the structural evolution of the universe is proposed, including the emergence of biosystems, biological evolution, and consciousness. 1 : CAUSALITIES AND SUPERCAUSALITIES Since the development of calculus and the Newtonian model of the universe, the evolution of dynamical events has been described through the directional application of time as a parameter. The evolution of a classical system can in principle be causally described in terms of the initial conditions and the differential equations governing its action. In the classical Laplacian universe, the initial conditions and the dynamical equations taken together completely specify the ongoing state at subsequent times. We will refer to this model as temporal determinism. The classical Hamiltonian and Lagrangian dynamical equations are both expressed as differential equations expressing generalized coordinates in terms of increasing time, where = T - V and in the case of a potential, = T + V, giving the total ( T kinetic & V potential ) energy :
Entropic Dynamics (ED) provides a statistical-inferential foundation for physical laws, deriving motion and field equations from principles of entropy maximization rather than quantization postulates. ED reconstructs quantum mechanics by treating the evolution of probability distributions on configuration space as driven by information constraints, yielding the Schrodinger equation as a non-dissipative diffusion process. Building on this foundation, the present work extends the ED framework into a Unified Entropic Dynamics (UED) formulation that encompasses classical, quantum, relativistic, thermodynamic, and gravitational phenomena within a single information geometric principle. By maximizing entropy subject to constraints on diffusion, drift, and gauge covariance over a manifold endowed with a supermetric H_ab, we derive a universal field equation that merges the Fokker-Planck and Hamilton-Jacobi structures into one covariant form. When specialized to different dynamical variables, this equation reproduces the harmonic oscillator, Schrodinger, Maxwell, Klein-Gordon, and gravitational wave equations, thereby revealing a deep equivalence between probabilistic inference and dynamical law. The UED framework demonstrates that spacetime geometry, quantum coherence and thermodynamic diffusion emerge as complementary expressions of the same entropic process, establishing a unified inferential foundation for both microscopic and macroscopic physics. In this formulation, energy, probability, and entropy are intertwined aspects of information geometry, providing a consistent inferential foundation for understanding classical, quantum, and gravitational dynamics as complementary expressions of a single entropic law.
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.