Introduction: A Neglected Fundamental Disaster

Conservation of angular momentum, together with conservation of energy and momentum, constitutes one of the most basic pillars of physics, forming the framework for our understanding of the universe's operation. From the motion of planets to the structure of atoms, the law of conservation of angular momentum pervades all areas, guiding our cognition of nature.

However, within the framework of the point-particle model for the microscopic world, angular momentum—this most fundamental physical quantity—has actually become an abstract number that can be arbitrarily assigned. This is not a technical detail issue, but a principled disaster that strikes at the very foundation of physics. When we can casually assign "spin" values to particles without explaining their physical origin, physics has deviated from the path of pursuing truth.

Part I: What Does the Arbitrariness of Angular Momentum Assignment Reveal?

Electron Spin: A Seemingly Reasonable Start

Let us begin with the electron's spin of 1/2. In 1925, Uhlenbeck and Goudsmit proposed that the electron possesses intrinsic angular momentum with a value of ħ/2. This proposal was initially strongly opposed by Pauli, because if the electron were truly rotating, its surface velocity would far exceed the speed of light.

However, the subsequent discovery of Thomas precession seemed to provide a certain physical basis for electron spin:

• The electron undergoes acceleration during its orbital motion in an atom

• Relativistic effects lead to the precession of the reference frame

• This precession precisely produces a factor of 1/2, explaining the fine structure

At the very least, this provides a traceable physical process, even though it still does not explain the fundamental origin of spin. The factor of 1/2 resulting from relativistic effects suggests that spin has a profound connection with spacetime structure, rather than being a simple classical rotation.

Photon Spin: A Natural Outcome of Vector Fields

The photon's spin of 1 appears to have a more solid physical basis:

• The electromagnetic field is a vector field, inherently possessing directionality

• Circularly polarized light carries angular momentum, an experimentally verifiable fact

• The angular momentum density ε₀(r × E × B) has a clear field-theoretic expression

The photon's spin is inherently related to the vector nature of the electromagnetic field, which seems reasonable. Each photon carries ±ħ of angular momentum, corresponding to left-handed and right-handed circular polarization, which has a classical electromagnetic counterpart. Considering the photon's relationship with unit charge and angular momentum quantization, a spin of 1 is indeed plausible.

Quark Spin: Complete Arbitrariness

But when we turn to quarks, the situation becomes absurd. Let us carefully compare these data:

Electron:

• Charge: e = -e

• Mass: m = 0.511 MeV

• Spin: s = 1/2

Up quark:

• Charge: e = +2e/3

• Mass: m ≈ 2.3 MeV

• Spin: s = 1/2

Down quark:

• Charge: e = -e/3

• Mass: m ≈ 4.8 MeV

• Spin: s = 1/2

Strange quark:

• Charge: e = -e/3

• Mass: m ≈ 95 MeV

• Spin: s = 1/2

Charm quark:

• Charge: e = +2e/3

• Mass: m ≈ 1,275 MeV

• Spin: s = 1/2

See the problem? Masses differ by nearly 1000 times, charges exhibit unprecedented fractional values, yet spins are precisely equal!

This makes no physical sense. If spin truly reflects some form of internal motion or structure, why would particles with such drastically different masses have identical angular momentum? Why can charges be fractional (1/3, 2/3) while spins are exactly 1/2?

Part II: The Deep Crisis of Angular Momentum Conservation

The Clarity of Angular Momentum in Classical Physics

In classical physics, every quantity of angular momentum has a clear physical origin:

Orbital angular momentum: L = r × p

• r is the position vector

• p is the momentum

• Clear physical image: an object moving around a point

Rotational angular momentum: L = Iω

• I is the moment of inertia, reflecting mass distribution

• ω is the angular velocity

• Clear physical image: an object rotating on its axis

Whether it is the Earth orbiting the Sun or a gyroscope spinning, angular momentum always corresponds to a specific form of motion. You cannot create angular momentum out of nothing, nor can you make it disappear arbitrarily.

The Fundamental Dilemma of the Point-Particle Model

But in the point-particle model, this clarity is completely lost:

No spatial scale → No moment of inertia

• Radius of a point particle: r = 0

• Moment of inertia: I = mr² = 0

• Impossible to generate rotational angular momentum

No internal structure → Unable to explain fixed values

• Why ħ/2 instead of ħ/3?

• Why are the spins of all electrons identical?

• How can consistency be guaranteed without structure?

Arbitrary assignment only → Undermining objectivity

• Spin becomes a label attached to particles, "without classical counterpart"

• Loses physical content

• Reduced to a purely classificatory tool

Logical Chaos in Conservation Laws

If angular momentum can be arbitrarily assigned, conservation laws immediately fall into chaos:

Reaction conservation problem: In particle reactions, we require total angular momentum conservation. But if the spin of each particle is arbitrarily assigned, this conservation becomes a numerical game rather than a physical law.

For example, in β decay: n → p + e⁻ + ν̄ₑ

• Neutron (spin 1/2) → Proton (spin 1/2) + Electron (spin 1/2) + Antineutrino (spin 1/2)

To ensure conservation, we must carefully arrange the spin directions of each particle. But what is the physical origin of these spins? Why is the neutrino also exactly spin 1/2?

Reference frame transformation problem: Angular momentum should transform according to definite rules in different reference frames. But how does the "intrinsic" spin of a point particle transform? It is neither orbital angular momentum nor classical rotation, and its behavior under Lorentz transformations is entirely arbitrarily stipulated.

Orbital-spin coupling problem: In atoms, the electron's orbital angular momentum L and spin S couple to form the total angular momentum J. But if S is merely an abstract label, what is the physical mechanism of this coupling? Why can a real orbital motion interact with an abstract quantum number?

Part III: g-Factor—A Failed Remedy

The Introduction of the Landé g-Factor

When physicists attempted to calculate the magnetic moment of particles, they encountered a serious problem. According to classical electromagnetism, the relationship between the magnetic moment μ of a charged particle and its angular momentum L should be:

μ = (q/(2m))L

But experiments revealed that for spin angular momentum, this relationship requires multiplication by a g-factor:

μ = g(q/(2m))S

The introduction of the g-factor itself acknowledges the failure of the spin concept.

The Chaotic Current Status of g-Factors

Let us examine the g-factors of different particles:

Leptons:

• Electron: g = 2.00231930436256...

• Muon: g = 2.00233184110...

• Tau: g ≈ 2.002

Nucleons:

• Proton: g = 5.585694702

• Neutron: g = -3.82608545

Quarks (theoretical values):

• Up quark: g ≈ 2

• Down quark: g ≈ 2

• Strange quark: g ≈ 2

What do these enormous differences in values indicate?

No unified relationship between spin and magnetic moment

• If spin is fundamental, why are such different corrections needed?

• The proton's g ≈ 5.6 and the neutron's g ≈ -3.8 differ by nearly 10 times!

Independent measurement required for each particle

• g-factors cannot be calculated from first principles

• Must be determined experimentally

• This is not theory, but parameter fitting

g-factors of composite particles reveal internal structure

• The anomalous g-factors of protons and neutrons clearly indicate they have internal structures

• Yet we insist that electrons are point particles?

The "Success" of QED: Precise Mistake

Quantum Electrodynamics (QED) can calculate the electron's g-factor with extraordinary precision:

g/2 = 1 + α/(2π) + higher-order corrections...

The agreement between theory and experiment reaches 12 decimal places! This is often hailed as the most precise prediction in physics.

But let us think calmly: If we need to calculate up to the 4th power of α (involving 891 Feynman diagrams!) to obtain the correct result, is this really a success? Or does it indicate that our basic picture is wrong?

This is like fitting planetary orbits with 100 epicycles—precise, but incorrect.

Part IV: Compton Scale—The Forgotten Key

Restoring the Physical Scale of Particles

If we acknowledge that particles have physical extension on the scale of the Compton wavelength λC = ħ/(mc), many difficulties are immediately resolved:

Angular momentum gains a physical origin:

• Characteristic scale: rC ~ ħ/(mc)

• Characteristic velocity: v ~ c

• Angular momentum: L ~ mrCv ~ ħ

This is no coincidence, but an inevitable result of dimensional analysis! On the Compton scale, angular momentum of the order ħ naturally emerges.

Differences between particles become understandable:

For the electron (λC ≈ 2.4×10^-12 m):

• Relatively simple internal structure

• g-factor close to 2

• Reflects a basic electromagnetic configuration

For the proton (λC ≈ 1.3×10^-15 m):

• Complex quark-gluon structure

• g-factor significantly deviating from 2

• Reflects composite nature

For the neutron:

• Although electrically neutral, internal charge distribution generates magnetic moment

• Negative g-factor reflects a unique internal configuration

Implications of Quarks' Fractional Charges

The fractional charges of quarks, combined with integer spin units, strongly suggest a unique internal structure:

Possible picture:

• The "1/2 spin" of quarks may correspond to specific internal motion modes

• Fractional charges imply charge is not concentrated at a point

• But distributed in some geometric structure

Relationship with the gluon field:

• Quarks are confined within hadrons

• Perhaps quarks themselves are topological excitations of the gluon field

• "Spin" reflects the angular momentum of this topological structure

Part V: The Necessity of Theoretical Reconstruction

The Patchwork Nature of Current Theory

Let us honestly examine how the Standard Model is constructed:

Standard Model = Point particles + Arbitrary spin assignment + g-factor correction + Renormalization + Symmetry breaking + ...

Each additional patch covers up yet another failure of the point-particle assumption. This is not an elegant theory, but a collection of barely functional empirical formulas.

A Theoretical Framework Based on Physical Reality

A true physical theory should satisfy:

1. Angular momentum has a clear physical origin

• Derived from motion of internal structures

• Or angular momentum density distribution of fields

• Traceable to specific physical processes

2. Spin values are determined by dynamics

• Not arbitrarily assigned parameters

• Result of structural stability

• Similar to the quantization of atomic orbitals

3. Magnetic moments arise naturally

• g-factor is not an independent parameter

• Reflects internal charge-current distribution

• Calculable from first principles

Part VI: Profound Implications—A Paradigm Crisis in Physics

The Philosophical Significance of the Angular Momentum Crisis

The arbitrary assignment of angular momentum is not only a technical issue, but also touches the essence of science:

Loss of objectivity: When basic physical quantities become arbitrary assignments, is physics still describing objective reality? Or is it constructing convenient mathematical models?

Destruction of causality: If the properties of particles have no physical origin and are just "as they are," we abandon the pursuit of "why"—is this still science?

Distortion of aesthetic standards: We praise the "success" of the Standard Model, ignoring that it requires 19 free parameters. Is this "success" true understanding, or elaborate fitting?

Implications for Other Fields

Particle physics:

• The Standard Model is not the ultimate theory

• Quarks may not be fundamental

• New physics lies at the λC scale, not the Planck scale

Quantum field theory:

• Renormalization is not a technical trick, but a sign of theoretical flaws

• The vacuum is not empty, but full of structure

• The wave-particle duality of fields and particles has a deeper origin

Cosmology:

• Dark matter may be related to the extended structure of particles

• Resolution of the vacuum energy problem requires finite scales

• Physics of the early universe may need to be re-understood

Possibilities for Experimental Verification

This is not pure theoretical speculation, but a testable scientific hypothesis:

High-energy experiments:

• Search for structural effects at energy scales close to λC

• Precision measurement of the energy dependence of g-factors

• Look for deviations in spin-mass correlations

Precision measurements:

• Upper limits on the electron electric dipole moment

• Relationship between neutrino mass and spin

• g-factors of antimatter

Astrophysical observations:

• Particle behavior in extreme environments

• Matter states inside neutron stars

• Quantum effects near black holes

Conclusion: Reconstructing the Foundation of Physics

The Core of the Crisis

We are facing not a minor revision of theory, but a fundamental conceptual crisis. When basic conserved quantities like angular momentum can be arbitrarily assigned, physics has deviated from the path of seeking truth.

The point-particle model has led us into a dead end:

• Spin has no physical origin and can only be arbitrarily assigned

• g-factors become independent parameters requiring individual measurement

• Conservation laws degenerate into mathematical identities, losing physical content

What Is the Way Forward?

The solution is not to continue patching, but to acknowledge that particles have physical scale:

The Compton wavelength is no coincidence, but a key:

• It gives the characteristic scale of particles

• At this scale, quantum effects are comparable to relativistic effects

• Angular momentum is naturally quantized to the order of ħ

A paradigm shift from points to structures:

• Particles are local excitations of fields with definite spatial extension

• Spin originates from internal electromagnetic configurations

• Differences between particles reflect structural variations

The Mission of Science

True science should not be satisfied with "being able to calculate," but strive to "understand the essence." When our theories require more and more arbitrary parameters, when basic concepts become increasingly abstract, when physical images are completely lost—this is not progress, but confusion.

The angular momentum crisis is a warning: We need to re-examine the foundations of quantum mechanics, return to physical reality, and not indulge in mathematical formalism. Only when angular momentum regains clear physical content, when spin is no longer an arbitrary label, and when g-factors can be derived from first principles—can physics stand on a solid foundation again.

This is not only about the elegance of theory, but about our true understanding of nature. In the 21st century, it is time to end the conceptual chaos inherited from the 20th century and establish a microscopic theory truly based on physical reality. The Compton wavelength points the way—there lies the truth of nature, waiting for us to discover.

The future of physics does not lie in more complex mathematics, but in deeper physical insights. When we no longer need g-factors, no longer arbitrarily assign spins, and no longer rely on renormalization—then, we will truly understand nature.