Quantum Force Wave Equation (QFWE)

• QFWE is a broad term for force‐explicit wave formulations that recast quantum mechanics through varied methodological frameworks.
• It encompasses hydrodynamic rewritings, momentum‐field techniques, nonlinear dissipative models, and discrete-causal set approaches.
• Recent proposals extend QFWE to curved spacetime, linking quantum forces to modified geometric and gravitational dynamics.

Searching arXiv for the cited works and closely related QFWE terminology. Searching arXiv for "Quantum Force Wave Equation". Searching arXiv for (Musielak, 2021) and related nonrelativistic wave-equation work. Quantum Force Wave Equation (QFWE) is not a single standardized equation in the arXiv literature. The term is used explicitly in one recent curved-spacetime proposal, but conceptually related constructions appear in several distinct lines of work: hydrodynamic rewritings of Schrödinger dynamics in which the quantum force is explicit, nonlinear dissipative Schrödinger-type equations derived from radiation reaction or friction, momentum-field formulations in which a quantum force is obtained from a momentum-dependent potential, non-localized wave equations written in force-like variables, higher-kinematic chains of wave functions depending on velocity and acceleration, and discrete-causal-set interaction equations for two particles [(Adom, 9 Jul 2025); (Mirza, 2015); (Wu et al., 2010); (Guimarães, 2016); (Shi et al., 2023); (Perepelkin et al., 2018); (Gudder, 2015)]. In that broad sense, QFWE denotes a family of force-explicit wave formulations rather than a canonical PDE.

1. Terminological scope and nonstandard status

The modern literature does not present QFWE as a universally accepted object with a fixed operator content, Hilbert-space structure, or reduction to Schrödinger, Dirac, or Klein–Gordon theory. The phrase is explicit in “Manifestation of Quantum Forces in Spacetime: Towards a General Theory of Quantum Forces” (Adom, 9 Jul 2025), whereas several earlier papers develop closely related constructions without using the label. This suggests that QFWE is best treated as a descriptive umbrella for wave equations in which force-like quantities are made primary.

Within nonrelativistic representation theory, the available abstract of “Nonrelativistic Fundamental Quantum and Classical Wave Equations” reports that irreducible representations of the extended Galilean group yield infinite sets of symmetric and asymmetric second-order differential equations with constant coefficients, that all derived equations are local and their Lagrangians exist, and that the asymmetric equations are Galilean invariant whereas the symmetric ones are not; the same abstract states that Schrödinger, Schrödinger-like, and new asymmetric equations arise after physical constants are fixed (Musielak, 2021). The supplied arXiv stub does not provide the full text, so only abstract-level claims are available for that work.

A persistent misconception is that every QFWE proposal introduces a genuinely new dynamical law. In the cited literature, some constructions are reinterpretations or rewritings of established equations, some are nonlinear phenomenological extensions, and some are discrete or covariant operator identities rather than conventional time-evolution equations.

2. Force-explicit reformulations of quantum dynamics

One major strand writes ordinary quantum mechanics in hydrodynamic or force-balance form. In “On Wave Function Representation of Particles as Shock Wave Discontinuities,” Schrödinger dynamics is recast through the Madelung/Bohm decomposition into a continuity equation and an Euler-type equation,

$$\frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{v}) = 0, \qquad \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\!\cdot\nabla)\mathbf{v} = -\frac{1}{m}\nabla\!\bigl(V + mQ\bigr),$$

with

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$

Here the quantum force per unit mass is $-\nabla Q$. The paper’s central claim is that non-zero quantum force produces nonlinear characteristic speeds $\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$ and can lead to shock-like discontinuities for Gaussian wave packets, whereas localized quantum density waves with vanishing quantum force are shock-free (Mirza, 2015).

A different reformulation appears in “A Quantum Mechanics Picture,” where the wave function is eliminated in favor of a complex, curl-free momentum eigenvalue field $\vec p(\vec r)$. The stationary Schrödinger equation is rewritten as

$$E = \frac{p^2}{2m} + U - \frac{i\hbar}{2m}\,\nabla\!\cdot\vec p,$$

and the corresponding momentum-dependent potential is

$V = U - \frac{i\hbar}{2m}\,\nabla\!\cdot\vec p.$

The associated total force is then

$$\frac{d\vec p}{dt} = - \nabla U + \frac{i\hbar}{2m}\,\nabla(\nabla \cdot \vec p) + \frac{i\hbar}{2m}\,\nabla^2 \vec p.$$

In this formulation, the “quantum force” is the derivative contribution proportional to $\hbar$, and the dynamics is represented by auxiliary “wave particles” whose trajectories collectively reproduce probabilistic structure (Guimarães, 2016).

A third force-explicit route is the zitterbewegung-based construction in “Quantum Wave Mechanics as the Magnetic Interaction of Dirac Particles.” There the modulation $M$ of a Coulomb-like magnetic force between relativistically circulating charges satisfies a Schrödinger-type equation. In the Osiak-relativity version, the modulation obeys

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$0

and, in a time-independent potential,

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$1

In that model, the wavefunction is interpreted as a force-modulation field rather than as a primitive dynamical entity (Lush, 2016).

3. Nonlinear and dissipative QFWEs

A more direct “force wave equation” idea appears in nonlinear Schrödinger-type models derived from radiation reaction and friction. “Nonlinear Quantum Wave Equation of Radiation Electron and Dissipative Systems” starts from an energy balance that includes radiated power or dissipative work, promotes

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$2

and rewrites velocity and acceleration through the phase of $Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$3:

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$4

For a radiating nonrelativistic electron this yields

$Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$5

The extra term is a state-dependent, time-nonlocal radiation-reaction functional. The same paper gives analogous equations for friction laws of the forms $Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$6, $Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$7, and $Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$8 (Wu et al., 2010).

These equations remain first order in time but are nonlinear and integro-differential. Their nonlinearity is non-polynomial, entering through $Q = -\frac{\hbar^{2}}{2m^{2}R}\,\nabla^{2}R.$9 and its derivatives, and their memory term makes them non-Markovian. The paper treats them as phenomenological quantum equations for radiation-corrected or dissipative dynamics rather than as derivations from a Lindblad or master-equation formalism (Wu et al., 2010).

4. Nonlocalized, higher-kinematic, and discrete formulations

Several papers generalize the force-wave idea by enlarging the underlying state space. “A Theory of Complex Adaptive Learning and a Non-Localized Wave Equation in Quantum Mechanics” defines non-localized momentum and force by

$-\nabla Q$0

and introduces a reinforcement coordinate $-\nabla Q$1 with energy

$-\nabla Q$2

The corresponding time-independent non-localized wave equation is

$-\nabla Q$3

with interaction conservation written as

$-\nabla Q$4

In this framework, entanglement is recast as an “interactively coherent state” governed by a conserved interaction invariant rather than by superposition alone (Shi et al., 2023).

“The chain of quantum mechanics equations” moves further by defining a hierarchy of wave functions

$-\nabla Q$5

with

$-\nabla Q$6

Each level carries its own Schrödinger-like equation, generalized quantum potential, and Hamilton–Jacobi structure. The conceptual significance for QFWE is that force and acceleration are explicit state variables rather than derived observables, so one obtains wave equations in jet-like spaces whose coordinates include velocity, acceleration, and higher derivatives (Perepelkin et al., 2018).

A discrete quantum-gravity realization appears in “Wave Equations for Discrete Quantum Gravity.” On a covariant causal set, the left-hand side of the two-particle wave equation is a covariant difference operator

$-\nabla Q$7

while the right-hand side contains force terms. For repulsive electric interaction,

$-\nabla Q$8

for attractive electric interaction,

$-\nabla Q$9

and for the strong force,

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$0

This is one of the clearest discrete realizations of a force-dependent wave equation, with amplitudes summed over noncrossing path pairs and normalized to obtain probabilities (Gudder, 2015).

5. QFWE as a curved-spacetime force law

The most explicit and ambitious use of the term occurs in “Manifestation of Quantum Forces in Spacetime: Towards a General Theory of Quantum Forces” (Adom, 9 Jul 2025). There the generalized wavefunction is

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$1

with field-coupling vector

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$2

The covariant derivative of the path-ordered phase is written

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$3

where $\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$4 is a nonlocal coupling vector built from nested commutators along the path.

The generalized quantum force is defined by

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$5

with time component

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$6

After constructing a third-order covariant operator $\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$7 through repeated covariant differentiation, the paper arrives at its defining QFWE,

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$8

This is not a Schrödinger-type first-order time equation; it is a covariant operator identity relating a force four-vector to a wavefunction acted on by a third-order, gauge-covariant, nonlocal differential operator (Adom, 9 Jul 2025).

The same work couples the force to geometry through a modified Einstein equation. With an LQG-inspired area scale, it derives

$\lambda_{1,2}=u\pm\sqrt{\rho Q_\rho}$9

where $\vec p(\vec r)$0 is built from $\vec p(\vec r)$1. The paper presents this as a semiclassical backreaction law in which spacetime curvature responds directly to the quantum-force operator rather than to an expectation value of the stress-energy tensor. The work also explicitly states that the metric is not quantized and that the framework is not claimed to be a complete quantum-gravity theory (Adom, 9 Jul 2025).

6. Relation to established wave equations, computation, and controversy

QFWE proposals sit within a broader landscape of generalized quantum wave equations. “The unified quantum wave equation” derives a telegraph-type second-order equation,

$\vec p(\vec r)$2

from a quaternionic formulation and argues that Dirac, Klein–Gordon, and Schrödinger equations can be obtained from it (Arbab, 2011). Although this is presented as a unified quantum wave equation rather than a QFWE, it shows that force-oriented generalizations coexist with broader programs of wave-equation unification.

From a numerical perspective, “Quantum Algorithm for Simulating the Wave Equation” develops a Hamiltonian-simulation framework for the standard wave equation, Klein–Gordon equations, and Maxwell’s equations by factoring discretized Laplacians into sparse incidence structures (Costa et al., 2017). The supplied summary explicitly states that the same machinery can be adapted to linear wave-type PDEs with added potential or force terms. This suggests a plausible computational route for linear QFWE variants, although no dedicated QFWE algorithm is introduced there.

The main controversy surrounding QFWE is therefore not a single disputed formula but a problem of heterogeneity. Some formulations are equivalent to hydrodynamic or stationary rewritings of standard quantum mechanics (Mirza, 2015, Guimarães, 2016), some are phenomenological nonlinear extensions for dissipation or radiation loss (Wu et al., 2010), some reinterpret the Schrödinger wavefunction as a classical force-modulation field (Lush, 2016), some operate in nonstandard state spaces such as reinforcement coordinates or higher-kinematic chains (Shi et al., 2023, Perepelkin et al., 2018), and some recast force as a covariant spacetime operator with geometric backreaction (Adom, 9 Jul 2025). A further caution is that the available arXiv record for (Musielak, 2021) does not furnish the full text, so it cannot presently anchor detailed QFWE claims beyond its abstract.

In this state of the literature, QFWE is best understood as a research label for wave equations in which force—classical, quantum, geometric, dissipative, or interaction-theoretic—is elevated from a derived quantity to an explicit structural component of the wave equation itself.