Current Pinch and Particle Stability

1. Introduction: From Electromagnetic Waves to Particle Ontology

In the conventional picture, light is treated as an oscillating electromagnetic field, while particles are treated as pointlike entities characterized by mass, charge, spin, and so on. The connection between the two is usually made via quantization and field operators, but at the ontological level the picture remains fragmented: photons and electrons are said to "derive from fields," yet they are rarely concretely viewed as different vortex structures of one and the same electromagnetic (or unified) field.

Within a one-field ontological framework such as Natural Quantum Theory (NQT), we can propose a more continuous picture:

• Electromagnetic waves (especially circularly polarized light) are understood as propagating vortices of effective displacement charges and currents.

• The higher the effective current density, the stronger the self-generated magnetic field and hence the current pinch; transverse spreading is more strongly suppressed.

• When the field strength and energy density exceed a certain threshold, a circularly polarized electromagnetic vortex can become self-sustained and non-diffusive; this corresponds to the stability of high-energy gamma photons.

• At yet higher energy density and stronger topological closure, the electromagnetic vortex can become fully self-bound, forming a localized stable configuration; this corresponds to the electron and, more generally, to other particles.

• For fermions with intrinsic magnetic moments, there is a further stabilizing mechanism: the topological conservation of magnetic flux.

In this way, the mechanism of current pinch offers a unified field-theoretic account of the stability of "electromagnetic waves – photons – electrons – other particles".

2. Effective Displacement Charge and Current in Electromagnetic Waves

2.1 Alternating Electric Fields as "Pairs of Displacement Charges"

In vacuum, Maxwell’s equations read

$$\nabla \cdot \mathbf{E}=0,\nabla \cdot \mathbf{B}=0,\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t},\nabla \times \mathbf{B}={\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}.$$

In the absence of free charges, $\nabla \cdot \mathbf{E}=0$ is usually read as "no charge". From a field-ontological, fluid-like perspective, however, one may view the electric field as encoding an effective distribution of positive and negative displacement charges:

• In any small volume element, one can construct a pair of effective positive and negative "displacement charges" whose net amount vanishes, but whose local distribution and motion determine the structure of $\mathbf{E}$.

• The time variation of the electric field, $\partial \mathbf{E}/\partial t$, corresponds to the motion of these effective displacement charges, and thus defines an effective displacement current density ${\mathbf{J}}_{\text{disp}}$.

This effective displacement current enters the magnetic field equation via the Maxwell–Ampère law:

$$\nabla \times \mathbf{B}={\mu }_0{\mathbf{J}}_{\text{eff}},{\mathbf{J}}_{\text{eff}}\equiv {ϵ}_0\frac{\partial \mathbf{E}}{\partial t}$$

(in vacuum, where there is no free conduction current).

2.2 Circularly Polarized Light: A Rotating Displacement-Current Vortex

For a linearly polarized plane wave, $\mathbf{E}$ oscillates along a fixed direction; the effective displacement current then has a simple back-and-forth structure, and its transverse pattern is relatively featureless. In contrast, in circularly polarized light, the transverse electric field vector rotates uniformly in the plane orthogonal to the propagation direction:

$${\mathbf{E}}_{\perp }(t,z)=E_0\left(\hat{\mathbf{x}}\cos (kz-\omega t)+\hat{\mathbf{y}}\sin (kz-\omega t)\right),$$

which can naturally be interpreted as:

In the transverse plane, a pair of effective positive and negative displacement charges orbit around the propagation axis, while the whole structure advances at the speed of light $c$.

The corresponding effective displacement current ${\mathbf{J}}_{\text{eff}}$ forms a rotating vortex structure in the cross-section, generating an azimuthal magnetic field $\mathbf{B}$ and an energy flow (Poynting vector) $\mathbf{S}=\frac{1}{{\mu }_0}\mathbf{E}\times \mathbf{B}$ that carries angular momentum. This sets the geometric stage for the current pinch mechanism.

3. Current Density, Field Strength, and Pinch Strength

3.1 The Basic Idea of Current Pinch

In plasma physics, current filaments contract radially under the action of their own magnetic fields: the current pinch effect. At the level of averaged forces, transverse equilibrium can be written schematically as

$${\nabla }_{\perp }\left(\frac{B^2}{2{\mu }_0}\right)+{\nabla }_{\perp }{p}_{\text{matter}}+{\nabla }_{\perp }{p}_E\approx 0,$$

where $p_E\sim {ϵ}_0{E}^2/2$ is the electric pressure.

In the NQT picture, we extend this idea to displacement-current vortices in vacuum:

• The larger the effective displacement current density ${\mathbf{J}}_{\text{eff}}$, the stronger the self-generated magnetic field $\mathbf{B}$.

• The stronger the magnetic field, the larger the magnetic pressure $B^2/2{\mu }_0$, and the stronger the radial squeezing action on the transverse cross-section.

• At the same time, the higher the field strength, the higher the energy density $u={ϵ}_0{E}^2/2+B^2/2{\mu }_0$, and thus the higher the effective mass density ${\rho }_{\text{eff}}=u/c^2$, which gives the electromagnetic vortex greater inertia against deformation.

In short, high current density and strong fields favor radial confinement; low current density and weak fields favor transverse spreading.

3.2 The Intrinsic Link between Wavelength, Energy, and Stability

Consider an electromagnetic vortex (e.g. a circularly polarized beam) over a given transverse area $A$:

• When the wavelength $\lambda$ is long and the frequency $\omega$ is low:

• The field changes slowly in time across the section; the effective displacement current is small.

• The self-generated magnetic field is weak; pinch effects are negligible; the wave packet tends to spread laterally.

• As $\lambda$ decreases (i.e. $\omega$ increases):

• For the same cross-section, the rate of field variation increases dramatically; the effective displacement current grows.

• The self-generated magnetic field and the pinch force become stronger, progressively suppressing transverse spreading.

Thus, from a dynamical perspective, high-frequency, high-field-strength electromagnetic structures are more prone to form self-sustained vortex-like packets: their transverse profile is no longer as free to expand as that of low-frequency plane waves, but tends to maintain a stable cross-sectional shape.

4. Threshold Density and Self-Sustained Modes: From Gamma Photons to Electrons

4.1 Self-Sustained Circular Vortices and Gamma-Photon Stability

Within this picture, it is natural to expect the existence of a threshold in effective current density / energy density, such that a circularly polarized electromagnetic vortex can become self-sustained and non-diffusive:

• When both field strength and frequency are sufficiently large, the displacement-current vortex generates a magnetic field whose pinch action can fully counterbalance transverse spreading.

• In this regime, the electromagnetic vortex behaves as a self-bound propagating mode: its cross-sectional shape and energy distribution are preserved during propagation.

• High-energy gamma photons can then be understood as the quantized manifestation of such self-sustained circular electromagnetic vortices, capable of traveling astronomical distances while preserving directionality and profile.

This offers a field-ontological dynamical explanation for the stability of gamma photons, without treating them as structureless pointlike "energy packets".

4.2 Electrons and Other Particles: Stronger Confinement and Fully Bound Vortices

At still higher energy density and under suitable topological conditions, the electromagnetic vortex can be fully closed and become a localized, self-bound solution—this corresponds to the electron and, more generally, to other particles:

• The electron, as a charged object with a magnetic moment, is modeled as a localized electromagnetic (or unified-field) vortex, with intense internal current loops and closed magnetic flux.

• Current pinch and the balance between electric and magnetic pressures jointly support the radial scale of the vortex.

• In the NQT framework, the ontological spin of the electron is given by the true field angular momentum of this vortex, with magnitude $\hslash$; its magnetic moment and $g$-factor are determined by this same internal structure.

More generally, different particles (proton, neutron, etc.) can be understood as different topological vortex solutions of the unified field equation: their mass, spin, and magnetic moment reflect different current patterns, flux structures, and stability conditions. The underlying dynamical mechanism—self-organized field vortices stabilized by pinch and topology—is common.

5. Fermions with Magnetic Moments: Magnetic Flux Conservation as a Second Stability Mechanism

For fermions with intrinsic magnetic moments, there is an additional, and crucial, topological stability mechanism associated with magnetic flux:

• The internal current loops and vortex flows within the electron produce not only a magnetic field, but a closed magnetic flux tube.

• Topologically, this magnetic flux can be regarded as a quantized "topological charge": as long as the topology of the vortex remains unchanged, the flux cannot be continuously annihilated.

• This is analogous to flux tubes in type-II superconductors, or to cosmic strings and vortex lines in various field theories: a topological invariant underlies the stability of the vortex line.

Thus, for fermionic vortices with magnetic moments, stability is provided by two mechanisms:

1. Dynamical pinch stability: current density and self-generated magnetic field provide pinch and pressure balance, preventing radial expansion.

2. Topological flux stability: the conservation of closed magnetic flux and its quantization furnish an additional topological barrier against the decay of the vortex.

In this light, the fermion’s magnetic moment ceases to be a mere passive "quantum label": it manifests the internal topological structure of the field configuration and plays an active role in stabilizing the particle.

6. Formal Unified-Field Action and Constraint Conditions

To elevate the above picture to a variational and computational framework, we need a formal action functional and corresponding constraints capable of encoding:

• The dynamics of the unified field in vacuum (including Maxwell theory as a limit),

• The pinch and pressure-balance conditions,

• The existence of vortex and self-bound solutions,

• The quantization and conservation of magnetic flux.

6.1 Basic Form of the Unified-Field Lagrangian

Consider a unified field theory whose dominant low-energy sector reduces to electromagnetism. The action is

$$S=\int d^{4}x\mathcal{L},$$

with Lagrangian density

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$$

 + LNL(Fμν) + Ltop(Aμ)Ltop(Aμ)

Interpretation:

1. Linear electromagnetic term

   $${\mathcal{L}}_{\text{EM}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$$

   is the standard Maxwell Lagrangian, describing linear propagation in vacuum.

2. Nonlinear term ${\mathcal{L}}_{\text{NL}}(F_{\mu \nu })$
   This term encodes the nonlinear response of the vacuum at strong fields, crucial for the existence of vortex and self-bound solutions. A generic Lorentz-invariant form is

   $${\mathcal{L}}_{\text{NL}}={\alpha }_1(F_{\mu \nu }{F}^{\mu \nu }{)}^2+{\alpha }_2(F_{\mu \nu }{\tilde{F}}^{\mu \nu }{)}^2+\cdots ,$$

   where ${\tilde{F}}^{\mu \nu }$ is the dual field-strength tensor and ${\alpha }_i$ are constants. Similar structures appear in the Heisenberg–Euler effective Lagrangian of QED, indicating that the vacuum is indeed nonlinear at strong fields.

   In NQT, ${\mathcal{L}}_{\text{NL}}$ can be interpreted as encoding self-interaction associated with current pinch and field energy density, providing the necessary nonlinearity for stable vortices.

3. Topological term ${\mathcal{L}}_{\text{top}}(A_{\mu })$
   To describe quantized magnetic flux and vortex topological charge, one may introduce a topological term, such as a Chern–Simons-type or $\theta$-like term (depending on dimension):

   $${\mathcal{L}}_{\text{top}}=\beta {ϵ}^{\mu \nu \rho \sigma }{A}_{\mu }{F}_{\nu \rho }{F}_{\sigma \lambda }{n}^{\lambda }+\cdots ,$$

   or, in simpler settings, encode flux quantization via constraints rather than explicit Lagrangian terms, e.g. imposing that certain flux integrals are fixed integers.

The unified-field equations follow from the variational principle

$$\delta S=0,$$

and admit, in principle, a richer set of solutions than linear Maxwell waves, including vortex and soliton-like configurations.

6.2 Constraints for Current Pinch and Pressure Balance (Static / Quasi-static Approximation)

When searching for self-bound vortex solutions (electrons) or self-sustained propagating modes (high-energy photons), we are especially interested in stationary or quasi-stationary solutions whose transverse structure does not change significantly in time. For such solutions, we can impose the following constraints in three-dimensional space:

1. Transverse force-balance (pinch–pressure balance)

   In a steady-state approximation, the transverse (radial) force balance can be written schematically as

   $${\nabla }_{\perp }\left(\frac{B^2}{2{\mu }_0}\right)+{\nabla }_{\perp }\left(\frac{{ϵ}_0{E}^2}{2}\right)+{\nabla }_{\perp }{p}_{\text{eff}}=0,$$

   where $p_{\text{eff}}$ is an effective "field pressure" induced by the nonlinear term ${\mathcal{L}}_{\text{NL}}$. This equation expresses the balance between magnetic pressure (pinch), electric pressure, and nonlinear self-pressure.

2. Fixed total energy

   To represent a given particle or photon, the total field energy is fixed:

   $$E_{\text{tot}}=\int d^{3}xu(\mathbf{x})=E_0,$$

   where $u(\mathbf{x})$ is the local energy density and $E_0$ is prescribed. For an electron, $E_0=m_e{c}^2$; for a photon, $E_0=\hslash \omega$ (per mode).

3. Fixed total angular momentum

   For a spinning vortex, the total field angular momentum must have a fixed magnitude and direction:

   $$\mathbf{S}=\int d^{3}x\mathbf{r}\times \left({ϵ}_0\mathbf{E}\times \mathbf{B}\right)={\mathbf{S}}_0.$$

   For the ontological electron, $|{\mathbf{S}}_0|=\hslash$ (spin 1 at the ontological level); for a photon, the spin angular momentum magnitude is $\hslash$ along the propagation direction.

4. Quantized magnetic flux (for fermionic vortices)

   For vortex solutions with intrinsic magnetic moments, we impose quantization of the closed magnetic flux:

   $$\Phi ={\oint }_{\mathcal{C}}\mathbf{A}\cdot d\mathbf{\ell }=n{\Phi }_0,n\in \mathbb{Z},$$

   where $\mathcal{C}$ is a loop encircling the vortex core and ${\Phi }_0$ is a fundamental flux quantum. This expresses the topological charge associated with the magnetic flux and realizes the flux-maintenance mechanism in a formal way.

6.3 The Constrained Variational Problem

Collecting these ingredients, we consider the field configuration $A_{\mu }(x)$ as the solution of a constrained variational problem:

$$\delta \left[S[A_{\mu }]-{\lambda }_E\left(\int d^{3}xu(\mathbf{x})-E_0\right)-{\mathbf{\lambda }}_S\cdot \left(\int d^{3}x\mathbf{r}\times ({ϵ}_0\mathbf{E}\times \mathbf{B})-{\mathbf{S}}_0\right)-{\lambda }_{\Phi }\left({\oint }_{\mathcal{C}}\mathbf{A}\cdot d\mathbf{\ell }-n{\Phi }_0\right)\right]=0,$$

subject, in the static/quasi-static regime, to the transverse force-balance condition

$${\nabla }_{\perp }\left(\frac{B^2}{2{\mu }_0}+\frac{{ϵ}_0{E}^2}{2}+p_{\text{eff}}\right)=0.$$

Here ${\lambda }_E$, ${\mathbf{\lambda }}_S$, and ${\lambda }_{\Phi }$ are Lagrange multipliers enforcing the energy, angular momentum, and flux constraints, respectively.

• For an electron vortex solution, we set $E_0=m_e{c}^2$, $|{\mathbf{S}}_0|=\hslash$, and $\Phi =n{\Phi }_0$.

• For a photon vortex solution, we set $E_0=\hslash \omega$, $|{\mathbf{S}}_0|=\hslash$, and may choose vanishing net flux or a specific mode-dependent flux.

Any solution to this constrained variational problem (if it exists) is, in the NQT sense, an ontological particle or photon configuration: a self-sustained field vortex held together by current pinch, pressure balance, and topological constraints.

7. From Picture to Computation

By introducing current pinch and magnetic-flux topology into a unified-field framework, we obtain a coherent picture in which the stability of "light – photon – electron – fermion" is understood as manifestations of self-organized field vortices:

• Light is no longer merely an abstract "quantum of energy," but a propagating vortex of effective displacement charges and currents.

• High-energy gamma photons are understood as nearly self-sustained circular electromagnetic vortices, whose strong current density and pinch effect maintain transverse stability.

• Electrons and other fermions are more strongly bound, fully closed vortices with quantized magnetic flux; their mass, spin, and magnetic moment emerge from the same vortex–pinch–topology mechanism.

The formal action and constraint framework above is not yet a full solution; it is a starting point for a systematic variational and numerical search for vortex solutions that satisfy given energy, angular momentum, and flux quantization conditions. If such solutions can be found and their parameters match observed particle properties, this would provide strong support for the NQT view:

Particles are not pointlike entities but field vortices stabilized by current pinch and topological flux; spectroscopic quantum numbers then become projections of these ontological structures under specific observational and dynamical conditions.