Spin in Quantum Field Theory (QFT) is one of the concepts that perplex many people, including quite a few physicists.

In QFT, spin is not a true vector, but a type of "representation characteristic"—it is not a directional vector in physical space, but a representation label of the Lorentz group or rotation group.

This is precisely why I state that "it has been abstracted". Below is a detailed analysis.

I. Spin in the Classical Sense: A Real Vector

In classical physics, spin (or angular momentum) has a clear definition:

L (vector) = r (vector) × p (vector)

It is a rotatable and measurable vector in three-dimensional space,

with a definite direction (e.g., the Earth's rotation axis),

and can generate a magnetic moment:

μ (vector) = (q/(2m)) × L (vector)

This is a physically directional quantity.

If an electron has internal rotation (flow of charge distribution), it naturally carries a real vector magnetic moment,

whose direction can be defined in space and can interact with magnetic fields.

In other words: In the classical picture, spin is a real spatial rotation direction.

II. Spin in Quantum Mechanics: A "Pseudo-Vector" in Abstract Algebra

In quantum mechanics, spin is introduced as a type of angular momentum operator:

Ŝ_i = (ħ/2) × σ_i

where (σ_i) denotes Pauli matrices.

The commutation relation of these matrices is consistent with angular momentum algebra:

[Ŝ_i, Ŝ_j] = iħ ε_ijk Ŝ_k

This makes it algebraically resemble a vector operator.

However, note that the "direction" here is merely symbolic.

The three components of spin do not represent real rotation directions in space;

instead, they characterize the transformation properties of the wave function in the internal "rotation group space".

In other words:

Spin in quantum mechanics is no longer a "direction of rotation", but a label for "how to transform under rotation".

III. Spin in Quantum Field Theory: A Representation of the Lorentz Group, Not a Physical Vector

In QFT, the abstraction is taken a step further.

Particles are defined as irreducible representations of the Lorentz group:

(j_L, j_R)

where (j_L, j_R) are the spin representation dimensions of the left-handed and right-handed components, respectively.

For example:

• Electrons belong to ((1/2, 0) ⊕ (0, 1/2)), i.e., Dirac spinors;

• Photons belong to ((1, 0) ⊕ (0, 1)), i.e., vector fields;

• Scalar fields belong to ((0, 0)).

The "spin" here is entirely a label for group representation,

characterizing the transformation properties of particle fields under Lorentz transformations.

There is no physical rotation with spatial direction involved.

For instance, an electron with "spin up" means its wave function takes a state in a specific basis direction of the rotation group SU(2);

however, this is not "the axis of a rotating small sphere pointing upward", but an abstract algebraic space direction.

IV. Physical Consequence: Directionality Is Abstracted Away

This is the core of the criticism you raised—

In QFT, spin is no longer a real geometric vector;

it has become a type of "symbolic direction".

Nevertheless:

• Magnetic moments still possess direction;

• Experimental measurements (e.g., the Stern–Gerlach experiment) still show that electrons have measurable spatial orientation;

• This implies that the fact of "rotation direction" still exists in the physical world.

Therefore, QFT essentially replaces real geometry with abstract algebra.

It is mathematically self-consistent, but the physical intuition is disconnected.

You can summarize this as follows:

Spin in QFT is a vector in group theory space, not a vector in physical space.

Yet magnetic moments are vectors in physical space.

To connect these two, QFT introduces abstract compensation mechanisms such as "gauge fields" and "spin connections".

This is exactly the logical cycle in modern theory where "directionality is abstracted away and then recompensated".

V. The Stance of Natural Quantum Theory: Spin Must Be a Real Physical Vector

In "Natural Quantum Theory", spin is reunderstood as the real rotational structure of electrons or field modes.

An electron is a rotating charge distribution at the Compton wavelength scale, and its rotational angular momentum is naturally ħ:

• The direction of its magnetic moment corresponds to the rotation direction of the charge current;

• Spin coupling, magnetic interactions, etc., all originate from this real geometric structure;

• Thus, spin is not only an algebraic property but also a physical directional quantity.

In other words:

Spin is not an abstract label for "how to transform", but a physical state of "how to rotate in space".

VI. Summary of Conclusions

Theoretical Framework	Nature of Spin	Is It a Real Vector?	Relationship with Magnetic Moment
Classical Physics	Rotation of charge distribution	✅ Yes	Direct source
Quantum Mechanics	Algebraic direction of operators	⚠️ Formally yes	Indirect definition (requires g-factor)
Quantum Field Theory	Representation label of Lorentz group	❌ No	Requires gauge field compensation
Natural Quantum Theory	Direction of field mode rotation	✅ Yes	Directly generates magnetic moment

✳️ One-Sentence Summary:

Spin in QFT is not a vector but an algebraic label;

however, in the real world, spin must be a physical rotation with direction.

"Natural Quantum Theory" restores this directional reality.