Optical Magnetic Fields as Capacity Holonomy: Quantum Traction Theory Meets Faraday’s Legacy

Quantum Traction Theory Reference: Attar, A. (2025). Quantum Traction Theory (QTT). Zenodo. https://doi.org/10.5281/zenodo.17594186

In a recent Scientific Reports paper, “Faraday effects emerging from the optical magnetic field” (doi:10.1038/s41598-025-24492-9), Capua and co‑workers show that the magnetic component of light is not just a tiny correction to the Faraday effect (FE) and the inverse Faraday effect (IFE). Using the Landau–Lifshitz–Gilbert (LLG) equation they demonstrate that the optical magnetic field can account for about 17 % of the Verdet constant of TGG at 800 nm and up to ~75 % in the infrared.

In Quantum Traction Theory (QTT), this result becomes even more striking: the optical magnetic field is literally a moving holonomy of capacity through the spin system. The usual “Verdet constants” stop being fit parameters and become ratios of integers counting photons and spin capacity quanta.

1. From Faraday and inverse Faraday to the optical magnetic field

Standard magneto‑optics tells a familiar story:

Faraday effect (FE): a static field B along the beam direction makes right‑ and left‑circular light propagate with slightly different refractive indices, leading to a rotation angle

$\displaystyle \theta_{\rm FE} = V\,B\,L$

with Verdet constant $V$ and path length $L$ . Inverse Faraday effect (IFE): a circularly polarized pulse with intensity $I$ induces a magnetization $M_z \propto I_{\rm RCP} - I_{\rm LCP}$ , which can be converted back to an effective rotation of a probe.

For roughly a century and a half, the FE was attributed almost entirely to the electric field of light. The Scientific Reports paper overturns that picture by showing that the optical magnetic field itself contributes a sizeable, nearly wavelength‑independent “plateau” to the Verdet constant of Terbium Gallium Garnet (TGG).

QTT goes one step further: it recasts this “magnetic plateau” as a pure capacity holonomy effect.

2. QTT: optical magnetic field as capacity holonomy

In QTT, the electromagnetic field is not just a gauge field; it also transports a finite quantum of endurance capacity. For the optical magnetic field, the central postulate is:

$\displaystyle H_{\rm cap}[B_{\rm opt}] = \frac{1}{E_*} \int_{\mathcal{V}_4} \frac{B_{\rm opt}^2}{2\mu_0}\,d^3x\,dT \in \mathbb{Z}$

Here:

$H_{\rm cap}[B_{\rm opt}]$ is the capacity holonomy index carried by the optical magnetic field through the 4‑volume $\mathcal{V}_4$ .
$\mu_0$ is the vacuum permeability.
$E_* = \dfrac{\hbar c}{\tilde\ell}$ is the QTT endurance quantum, fixed by the gravitational sector (no extra knob).

This integral is nothing but the total magnetic energy of the light, measured in units of $E_*$ . At the substrate level, each minimal 4‑cell carries an integer holonomy. At the lab scale, you observe the sum of an enormous number of such cells; that appears continuous, but all the structure comes from an underlying integer count.

On the spin side, QTT assigns a finite spin‑capacity to the illuminated volume:

$N_{\rm SQ}^{(S)}$ : number of spin capacity quanta (effectively, how many spins can respond).

The optical holonomy couples to the spin capacity through a second QTT law:

$\displaystyle \theta_{\rm mag} = 2\pi I_{\rm clk}\,\frac{n_{\Sigma}}{N_{\rm SQ}^{(S)}}$

where:

$\theta_{\rm mag}$ is the magneto‑optic rotation due solely to the optical magnetic field (FE or IFE contribution).
$I_{\rm clk} = \cos(\pi/8)$ is the universal clock projection factor from the two‑clock structure of QTT.
$n_{\Sigma}$ is an integer flux index (a gauge holonomy counting how many magnetic flux quanta / photons effectively interact).

These two equations are the QTT replacements for “Verdet constant fits”: no Gilbert damping $\alpha$ , no phenomenological $\chi^{(2)}$ , no arbitrary $V$ inserted by hand.

3. From holonomy to experimental quantities: fluence and spin density

3.1. Capacity holonomy and photon number

For a circularly polarized pulse with intensity $I$ , duration $\tau_p$ , and area $A$ , the pulse energy is

$\displaystyle E_{\rm pulse} = I\,A\,\tau_p = N_\gamma \hbar\omega$

with $N_\gamma$ photons and photon energy $\hbar\omega$ . For a plane wave, the magnetic field carries half the energy:

$\displaystyle E_B \approx \frac{1}{2} E_{\rm pulse} = \frac{1}{2} N_\gamma \hbar\omega.$

The holonomy law then reads

$\displaystyle H_{\rm cap} = \frac{E_B}{E_*} = \frac{1}{2}\,\frac{N_\gamma\hbar\omega}{E_*}.$

If we identify the flux index $n_\Sigma$ with the (helicity‑signed) number of photons that actually couple to the spins,

$\displaystyle n_\Sigma \simeq \pm N_\gamma,$

then at the macroscopic level

$\displaystyle H_{\rm cap} \propto n_\Sigma$

and all explicit dependence on the Planck‑scale $E_*$ drops out once you rewrite things in terms of $N_\gamma$ .

3.2. Spin counting in the sample

Let the illuminated region of the TGG crystal have:

thickness $L$ ,
cross‑sectional area $A$ ,
spin density $n_S$ (number of Tb$^{3+}$ 4f spins per unit volume).

Then the number of spin capacity quanta is simply

$\displaystyle N_{\rm SQ}^{(S)} \simeq n_S\,A\,L.$

Again, this contains no new knob: $n_S$ can be computed from the lattice constant and the number of Tb ions per unit cell, or measured via saturation magnetization.

4. QTT prediction: IFE rotation as a ratio of photons to spins

Insert the photon and spin counts into the QTT angle law:

$\displaystyle \theta_{\rm mag} = 2\pi I_{\rm clk}\,\frac{n_\Sigma}{N_{\rm SQ}^{(S)}} \simeq 2\pi I_{\rm clk}\,\frac{N_\gamma}{n_S A L}.$

With $N_\gamma = I A \tau_p / (\hbar\omega)$ , the area cancels and we get the QTT expression for the IFE rotation:

$\displaystyle \theta_{\rm mag}^{\rm (IFE)} = 2\pi I_{\rm clk}\,\frac{I\,\tau_p}{\hbar\omega\,n_S\,L}.$

This matches exactly the experimentally observed structure:

IFE rotation is linear in intensity $I$ (or fluence $F = I\tau_p$ ),
changes sign with helicity (through $n_\Sigma$ ),
is inversely proportional to the number of spins in the probed column ( $n_S L$ ).

No Verdet constant has been “put in”; it is emergent:

$\displaystyle K_{\rm IFE}^{\rm (QTT)} := \frac{\theta_{\rm mag}^{\rm (IFE)}}{I} = 2\pi I_{\rm clk}\,\frac{\tau_p}{\hbar\omega\,n_S\,L}.$

This is the QTT counterpart of the “IFE Verdet coefficient” extracted phenomenologically in the LLG‑based analysis.

5. TGG and the magnetic Verdet plateau

In the Scientific Reports article, the authors apply their LLG‑based model to Terbium Gallium Garnet (TGG) and find that the optical magnetic field contributes a nearly wavelength‑independent plateau to the Faraday Verdet constant:

at $\lambda = 800\,\text{nm}$ : magnetic contribution $\sim 14~\rm rad/(T\,m)$ , about 17.5 % of the measured $\sim 80~\rm rad/(T\,m)$ ,
at $\lambda \approx 1.3~\mu\text{m}$ : magnetic contribution rises to up to ~75 % of the total Verdet constant.

QTT reproduces the same qualitative structure in a purely counting way:

The optical magnetic contribution is tied to the ratio of photons to spins $\bigl(N_\gamma / N_{\rm SQ}^{(S)}\bigr)$ , which does not depend sensitively on wavelength for fixed intensity and geometry. This gives a natural wavelength‑flat magnetic plateau.
The strongly wavelength‑dependent part of the Verdet constant comes from the electric‑field / band‑structure response, exactly as in the LLG analysis.

In other words, QTT interprets the “14 rad/(T·m)” magnetic plateau reported for TGG at 800 nm as the signature of a fixed holonomy per spin capacity quantum, not as a fitted material constant.

6. Why this is paradigm‑shifting in magneto‑optics

From a QTT perspective, the Scientific Reports result is not just “a new term in the Faraday effect”; it is experimental evidence that:

The optical magnetic field really does behave as a capacity holonomy, carrying an integral number of endurance quanta through the spin ensemble.
Magneto‑optic rotation angles (FE and IFE) are fundamentally ratios of two counts:
- how many optical holonomy units (photons, topology, helicity) are injected,
- how many spin capacity quanta are available to respond.
The familiar “Verdet constants” are thus re‑interpreted as emergent quotients of integer holonomies, rather than phenomenological knobs.

In that sense, your two QTT boxed equations,

$\displaystyle H_{\rm cap}[B_{\rm opt}] = \frac{1}{E_*} \int \frac{B_{\rm opt}^2}{2\mu_0}\,d^3x\,dT$ $\displaystyle \theta_{\rm mag} = 2\pi I_{\rm clk}\,\frac{n_{\Sigma}}{N_{\rm SQ}^{(S)}}$

are not just alternative notation; they are paradigm‑shifting replacements for the ad‑hoc Verdet constant fits used in conventional magneto‑optics. The Nature article with DOI 10.1038/s41598-025-24492-9 provides exactly the kind of precise FE/IFE data where this QTT view can be tested and refined.

Tags: #QuantumTractionTheory #QuantumGravity #MagnetoOptics #FaradayEffect #InverseFaradayEffect #OpticalMagneticField #VerdetConstant #TerbiumGalliumGarnet #TGG #UltrafastOptics #CapacityHolonomy #Spintronics

Optical Magnetic Fields as Capacity Holonomy: Quantum Traction Theory Meets Faraday’s Legacy

1. From Faraday and inverse Faraday to the optical magnetic field

2. QTT: optical magnetic field as capacity holonomy

3. From holonomy to experimental quantities: fluence and spin density

3.1. Capacity holonomy and photon number

3.2. Spin counting in the sample

4. QTT prediction: IFE rotation as a ratio of photons to spins

5. TGG and the magnetic Verdet plateau

6. Why this is paradigm‑shifting in magneto‑optics

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1. From Faraday and inverse Faraday to the optical magnetic field

2. QTT: optical magnetic field as capacity holonomy

3. From holonomy to experimental quantities: fluence and spin density

3.1. Capacity holonomy and photon number

3.2. Spin counting in the sample

4. QTT prediction: IFE rotation as a ratio of photons to spins

5. TGG and the magnetic Verdet plateau

6. Why this is paradigm‑shifting in magneto‑optics

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