I. Introduction: When the Tool Usurps the Master's Role

The original intention of physics is to learn from nature. Theories are the language of experience, not the judges of nature. However, since the 20th century, the evolution of quantum mechanics has witnessed an astonishing reversal—mathematical formalism has become authoritative, while physical reality has been forced to give way.

This inverted scientific relationship has spawned the so-called "Instrumental Quantum Theory (IQT)": a theoretical system that replaces physical truth with formal consistency.

In its framework, equations take precedence over experiments, and operators are more credible than physical quantities. Faced with contradictions, people do not revise the model but modify nature. This is the ultimate form of "mutilating the foot to fit the shoe"—when the shoe (mathematical tool) does not fit, one cuts off the toes (physical reality).

II. Electron Spin: From Physical Phenomenon to Algebraic Dogma

The history of electron spin marks the beginning of "cutting the toe to fit the shoe."

Before the advent of the Dirac equation, atomic spectra had revealed the half-integer splitting of electron angular momentum. At that time, the physics community already knew that this splitting originated from Thomas precession—a relativistic correction caused by the electron’s motion in the orbital magnetic field. This effect halves the observed angular momentum projection but does not mean the electron’s intrinsic angular momentum is truly "reduced."

In other words:

The true angular momentum of the electron is an objective, unalterable physical quantity. It is what it is. The "½" displayed in spectra is merely a result of the precession effect, not the definition of the intrinsic spin itself.

However, the mathematical formalism of the Dirac equation requires a wave function with an anticommutative structure, which naturally yields a ½ spin solution. Instead of verifying the physical rationality of this assumption, the physics community forced nature to submit to algebra.

The electron is no longer regarded as an entity with spatial structure and magnetic current distribution, but defined as a point particle obeying anticommutative algebra. To maintain formal consistency, all fundamental particles with magnetic moments are uniformly classified into the "spin ½" category.

This is a classic example of mutilating the foot to fit the shoe:

The electron’s magnetic moment could have been understood as the spatial distribution of a circular current;

yet to preserve the symmetry of the equation, its physical structure was abandoned;

and then the g-factor was introduced as a correction to remedy computational deviations.

As a result, Instrumental Quantum Theory achieved formal neatness but lost physical truth.

III. Angular Momentum: Nature’s Non-Negotiable Order

Angular momentum is one of the most basic and strictly conserved quantities in the physical world. It is independent of measurement methods and cannot be "redefined" by mathematical formalism. The angular momentum of an electron is what it is—a fact of nature, not a degree of freedom for equations.

However, to accommodate algebraic structures, Instrumental Quantum Theory treats angular momentum as an "adjustable parameter": when the equation demands ½, it is defined as ½; when experiments show slight deviations, "anomalous magnetic moment" is introduced for compensation.

Natural Quantum Theory (NQT) argues that this approach essentially modifies physics for the sake of formalism. Thomas precession has long revealed that the halving of angular momentum in spectra is a kinematic effect, not an inherent property of the physical entity. The Dirac equation mistakenly "ontologizes" this observed projection, thus establishing a precedent for "algebra governing physics."

IV. Wave Function Dimension: The Physicalization of a Mathematical Oversight

The wave function is inherently a spectral tool used to represent the amplitude distribution of a system. However, if integral variables are not made dimensionless during integration, the dimension of the wave function varies with spatial dimensions. This is a purely mathematical flaw, yet Instrumental Quantum Theory interprets it as a "deep quantum principle": the dimension of the wave function can change with representation.

A simple unit error is sanctified as a "singularity of nature." This "mathematical worship" is another manifestation of mutilating the foot to fit the shoe.

V. Uncertainty Principle: The Physical Misinterpretation of Mathematical Reciprocity

In Fourier transforms, there exists a reciprocal limitation between time and frequency. This is a mathematical fact, not a physical prohibition. But in Instrumental Quantum Theory, this mathematical reciprocity is physicalized as the "Uncertainty Principle."

Consequently, momentum and position—two clearly defined physical quantities—are forcibly "blurred." The limitations of a mathematical method are mistaken for inherent characteristics of nature. This is not physical progress but intellectual regression.

VI. Quantum Entanglement and the Surrender of Causality

Quantum entanglement experiments show that measurement results of different systems exhibit statistical correlations. From the perspective of Natural Quantum Theory, this is a form of spectral correlation—global correlation of statistical distributions.

Yet Instrumental Quantum Theory insists on interpreting it as nature’s "nonlocality" or even "violation of causality." When equations contradict logic, it chooses to let logic yield. This is the highest realm of mutilating the foot to fit the shoe—making the world submit to mathematics.

VII. Anticommutative Algebra and the Root of the Spin ½ Dogma

Anticommutative algebra was originally a technical device Dirac introduced to eliminate negative energy solutions. However, it has been misinterpreted as a physical law—all particles with magnetic moments must have spin ½.

This artificial unification obscures the true origin of magnetic moments: magnetic moments arise from local currents, not abstract symmetry. In Natural Quantum Theory, magnetic moments are local and have finite spatial scales; but in Instrumental Quantum Theory, they are compressed into "point operators," losing their physical content.

This not only leads to a misunderstanding of magnetic correlations but also plunges the "coupling between angular momentum and magnetic moment" into mathematical illusion. Thus, a physical quantity derived from spatial currents is transformed into a symbol of algebraic structure.

VIII. The Complete Logic of Mutilating the Foot to Fit the Shoe

Formal Requirements of IQT	Sacrificed Physical Reality
Dirac’s anticommutative algebra	Forcing particles with magnetic moments to have spin = ½
Wave function dimension drift	Ignoring the error of non-dimensionalized integration variables
Uncertainty Principle	Misinterpreting mathematical reciprocity as natural fuzziness
g-factor correction	Replacing theoretical consistency with empirical fitting
Field-theoretic point operators	Concealing the spatial structure of spin/magnetic moment
Misinterpretation of Thomas precession	Ontologizing observational effects as particle properties

In these examples, Instrumental Quantum Theory exhibits striking consistency: when mathematical formalism conflicts with nature, it always forces nature to submit.

IX. The Return of Natural Quantum Theory

Natural Quantum Theory (NQT) aims to restore the realistic foundation of quantum mechanics:

"Quantization" equals spectralization, and the wave function is the spectral representation of classical mechanics;

Spin and magnetic moment are local spatial current phenomena;

Stable particles arise from the natural locking of magnetic flux, which requires external force to change;

Dirac algebra and anticommutative structures are merely computational tools, not physical truths.

NQT does not oppose mathematical precision but rejects the use of mathematics to distort nature. Its mission is not to destroy quantum mechanics but to correct its physical direction.

X. Conclusion: The Self-Redemption of Science

Instrumental Quantum Theory has achieved formal victory but failed in understanding. It has turned theory into religion and formalism into reality.

Natural Quantum Theory reminds us:

Genuine science does not force nature to adapt to equations, but lets equations reveal nature.

If the shoe does not fit, one should modify the shoe, not mutilate the foot. Only in this way can physics firmly step back onto the ground of nature.