We consider a relativistic field-theoretic framework containing three sectors:

1) Electrically charged matter, represented for simplicity by a Dirac field ψ ( x ) of mass m and electric charge e .

2) Electromagnetic gauge field A μ ( x ) .

3) Magnetic monopole neutrino background, represented by a fermionic field ν ( x ) carrying magnetic charge g . We assume this sector forms a statistically homogeneous and isotropic background characterized by number density n ν , correlation length ℓ c , and correlation time τ c .

We do not commit to a specific ultraviolet completion for the tachyonic magnetic monopole-neutrino sector. Instead, we treat it as an effective field with prescribed two-point correlation structure. The tachyonic character is encoded through an effective mass parameter m ν 2 < 0 , yielding a superluminal dispersion relation at the level of the free-field propagator. Stability issues associated with tachyonic fields are assumed to be regulated by background self-consistency conditions; detailed microphysical realization is deferred.

To treat electric and magnetic charges consistently, we adopt a dual-symmetric formulation of electromagnetism in which both electric and magnetic sources appear.

To incorporate magnetic charge in a local Lagrangian framework, we introduce a dual field-strength tensor. Define

In the presence of magnetic sources, Maxwell’s equations generalize to

where J e ν and J m ν are electric and magnetic four-currents respectively.

A convenient local formulation employs a dual potential description (e.g., Zwanziger-type formulation), introducing an additional gauge potential C μ that couples to magnetic charge. The electromagnetic Lagrangian density becomes

where

Electric charges couple minimally to A μ , and magnetic charges couple minimally to C μ .

The electrically charged particle is described by

where

Here ℏ 0 is introduced as a bookkeeping parameter that will later be replaced by the emergent effective ℏ eff after coarse-graining. At the microscopic level, ℏ 0 may be treated as a scaling parameter for dimensional consistency; the emergent description will identify the physically observable constant.

The monopole-neutrino field ν ( x ) carries magnetic charge g and obeys

with magnetic covariant derivative

We assume m ν 2 < 0 (tachyonic parameter), but the detailed dispersion structure does not enter the nonrelativistic reduction directly. The monopole four-current is

The background state is defined statistically by

where Ξ μ ν encodes density and correlation structure.

The full Lagrangian density is therefore

where the interaction is entirely contained in the minimal couplings to A μ and C μ . No direct Yukawa-type coupling between ψ and ν is introduced; the interaction is mediated through the gauge sector.

Explicitly,

Because the monopole background has vanishing mean current but nonzero fluctuations, the magnetic sector generates a stochastic dual potential C μ whose fluctuations induce correlated fluctuations in the electric sector via dual Maxwell equations.

Upon integrating out the monopole-neutrino field in a statistical ensemble sense, one obtains an effective action for the electric sector:

where Δ S arises from gauge-field fluctuations sourced by monopole current correlations.

To leading order in weak coupling and assuming short correlation time, the induced term produces:

Momentum-space diffusion proportional to g 2 n ν ,Phase accumulation proportional to e g ,A quadratic gradient term in the effective Hamilton-Jacobi equation.

Dimensional analysis then yields the emergent action scale

where ℱ is determined by the normalized current-current correlation function of the monopole background.

In the nonrelativistic limit of the ψ -sector, write

and expand to order v 2 / c 2 . The induced stochastic gauge fluctuations generate a Fokker-Planck-type correction to the classical Hamilton-Jacobi equation. Under the fluctuation constraint

where D p is the momentum diffusion coefficient derived from monopole correlations, the continuity and modified Hamilton-Jacobi equations combine into the linear Schrödinger equation.

Consider a nonrelativistic charged particle (mass m , electric charge e ) moving in a statistically homogeneous and isotropic monopole-neutrino background. After integrating out the fast background degrees of freedom in a weak-coupling and scale-separation limit, the particle experiences:

1) a mean potential V ( x , t ) (can be external and/or slow background mean-field), and

2) a zero-mean stochastic contribution to the gauge-covariant dynamics coming from monopole fluctuations.

A convenient effective form is a stochastic Hamiltonian with a random scalar potential Φ and vector potential A :

With

and two-point correlations determined by the monopole-neutrino current correlator (symbolically)

Markov limit: if the correlation time is short compared to the particle evolution time,

we can approximate the effective forcing as delta-correlated in time (white-noise limit) after coarse-graining.

The Lorentz force from the fluctuating fields is

Under isotropy and weak coupling, the net effect of the rapidly fluctuating field is a momentum diffusion plus (optionally) small friction. The standard coarse-grained form is a Langevin system

where d W t is a Wiener increment and D p is the momentum-diffusion coefficient.

How D p is tied to the monopole background: in the Markov limit, it is fixed by the force autocorrelation (Green-Kubo form)

with F ( t ) = e ( E + x ˙ × B ) evaluated along the coarse-grained trajectory. Since E , B are sourced by the monopole current J m ∝ g , one generically gets

The exact prefactor depends on which correlator dominates ( 〈 E E 〉 , 〈 B B 〉 , etc.).

For the derivation of Schrödinger dynamics, we now move to a configuration-space diffusion picture, which is the natural place where the “quantum potential” term emerges.

Assume the momentum relaxes rapidly on timescales of interest (or equivalently that the effective dynamics of x can be described by a diffusion process after eliminating p ). The resulting coarse-grained motion is modeled by an Itô diffusion:

where:

b ( x , t ) is the drift velocity field (to be determined), ν is the spatial diffusion constant induced by the background.

The probability density ρ ( x , t ) then obeys the Fokker-Planck equation:

Define the current velocity v and osmotic velocity u by

Then the Fokker-Planck equation becomes a pure continuity equation:

To close the system, we need an evolution law for v . The cleanest assumption (and the one that yields Schrödinger exactly) is that the current velocity is potential flow:

where S is an emergent phase/action field.

Now impose a coarse-grained energy balance (equivalently, a least-action principle for the diffusion process). In Nelson-type mechanics, this yields a modified Hamilton-Jacobi equation:

The last term is the “quantum potential” structure, arising from the osmotic contribution of the diffusion (it is not put in by hand; it is the unique scalar built from ρ that appears from this closure under isotropy).

So now we have the coupled system:

Define

Then 2 m ν 2 = ℏ eff 2 / ( 2 m ) , and the modified Hamilton-Jacobi equation becomes

Now define a complex amplitude

A standard (but explicit) computation shows that the pair of real equations above is exactly equivalent to

That is the emergent nonrelativistic Schrödinger equation.

From the derivation, the only place ℏ eff enters is through ν :

So we now need a model-based estimate for ν in terms of monopole-background statistics. A generic (and dimensionally consistent) way to write it is:

1) The background produces a phase increment variance per unit time (phase diffusion) proportional to e 2 A 2 and sourced by monopole charge g and density n ν . In a correlation cell picture, the dimensionless occupancy is N c ~ n ν ℓ c 3 .

2) Because e g already has units of action, the clean emergent scaling is

with ℱ dimensionless. Two common closures are:

Random-walk (phase diffusion) closure: ℱ ( N c ) ∝ N c Coherent cell closure: ℱ ( N c ) ∝ N c

so that

Then

That plugs directly into the Schrödinger equation above.

1) Coarse-grain the monopole background → configuration-space diffusion d x = b   d t + 2 ν   d W .

2) Fokker-Planck + define v = b − ν   ∇ ln ρ → continuity equation.

3) Impose potential flow v = ∇ S m and an energy/least-action closure → modified Hamilton-Jacobi with term − 2 m ν 2 ∇ 2 ρ / ρ .

4) Identify ℏ eff = 2 m ν and define ψ = ρ   e i S / ℏ eff → Schrödinger equation.

5) Tie ν (hence ℏ eff ) to monopole background correlators → ℏ eff ~ e g × dimensionless function of n ν ℓ c 3 .

We assume the following background structure:

1) Neutrinos behave as tachyonic excitations

2) Each neutrino carries an effective magnetic monopole charge g ν , generating a weak but pervasive background field.

3) Ordinary matter interacts indirectly with this background via phase modulation of matter waves rather than direct force exchange.

4) The background field is:

statistically homogeneousisotropic on laboratory scalesslowly varying in time

This allows a mean-field approximation.

Start from a classical action augmented by coupling to the monopole neutrino background:

where:

Φ ν ( x , t ) is an effective scalar potential arising from coherent tachyonic monopole neutrino flux.

This term does not correspond to a classical force; it encodes phase deformation of the particle’s wavefunction.

Assume a matter wave of the form:

The action satisfies a modified Hamilton-Jacobi equation:

This equation is classical in form, but the background field introduces stochastic micro-fluctuations in S .

Because tachyonic monopole neutrinos propagate outside the light cone, the background induces nonlocal phase correlations:

This destroys classical determinism and forces a statistical description.

As a result, physical observables must be promoted to operators acting on ensemble wavefunctions:

Here, ℏ emerges as the variance scale of phase noise induced by the tachyonic neutrino background:

The effective energy relation becomes:

Applying operator substitutions to the wavefunction:

For laboratory conditions:

Φ ν ( x , t ) is:slowly varyingweak compared to m c 2 statistically uniform.

Thus, it can be absorbed into a global or local phase shift:

yielding:

This is exactly the standard Schrödinger equation, first written down by Erwin Schrödinger.

In this model:

Quantum mechanics is emergent, not fundamental;The wavefunction represents coherent phase alignment with a tachyonic monopole neutrino background; ℏ encodes background-induced action noise;Collapse corresponds to local decoherence with respect to the background field.

From Section 4, the wavefunction was written as:

In the TMMN model:

S ( x , t ) contains stochastic contributions from tachyonic monopole neutrino background fluctuations;These fluctuations are:nonlocal,Lorentz-violating at microscopic scales,unresolvable in single-particle trajectories.

Thus, individual outcomes are not deterministic, even though the background dynamics are.

Define the observable density:

Interpretation in this framework:

ρ ( x , t ) = density of phase-coherent alignments with the neutrino background

That is:

A measurement outcome occurs where the particle’s internal phase locks to a locally stable configuration of the TMMN field.

Substitute ψ = A e i S / ℏ into the Schrödinger equation and separate real and imaginary parts.

The imaginary part yields:

This is a continuity equation, ensuring conservation of total probability.

In the TMMN interpretation:

Probability is conserved because background phase flux is conserved;No probability is “created” or “destroyed”—only redistributed by background fluctuations.

Measurement involves strong coupling between:

the particlethe detector,the local neutrino monopole field.

During measurement:

rapid decoherence destroys phase correlations,only the amplitude density survives ensemble averaging.

Thus, the detection frequency satisfies:

This reproduces the Born rule, introduced by Max Born, not as a postulate, but as:

a statistical consequence of tachyonic background phase decoherence.

In this framework:

“Wavefunction collapse” is not fundamental;It corresponds to rapid re-alignment of local phase coherence with the background monopole field.

No superluminal signal is required—tachyonic correlations already exist in the background.

Because TMMN particles carry magnetic monopole charge g ν :

Their background field has topological structure;Phase transport around closed loops acquires quantized circulation.

This forces the wavefunction to become multi-valued, requiring a two-component structure:

These components represent topologically distinct phase windings, not literal rotation.

The minimal algebra preserving probability and topology is:

These are the Pauli matrices, introduced by Wolfgang Pauli, now interpreted as: generators of monopole-induced phase topology.

Including electromagnetic and monopole-induced vector potentials:

After background averaging:

monopole vector potential contributes only through spin coupling;anomalous magnetic moment emerges naturally.

Reducing yields the standard Pauli equation.

At higher energies, the Schrödinger-Pauli framework fails.The TMMN background enforces:

intrinsic nonlocality,CPT asymmetry at microscopic scales,linear time evolution.

Thus, the wave equation must be:

first order in time,first order in space.

Start from:

where Δ ν is a tachyonic background correction.

Factorization demands matrices α i , β :

This yields the Dirac equation, formulated by Paul Dirac:

In this model:

Antiparticles correspond to opposite monopole phase orientation;Negative-energy solutions reflect tachyonic background modes;Spinors encode four distinct topological phase states.

Standard relativistic quantum mechanics is recovered when: