The Complete Definition of the Electron in the Standard Model—and Its Ontological Dilemma

I. Core Definition: The Electron as an Excitation of a Quantum Field

In the Standard Model (SM), the entire definition of the electron is given by the relevant terms in the following Lagrangian density:

Le=ψˉe(iγμDμ−me)ψeLe=ψˉe(iγμDμ−me)ψe

where the covariant derivative is:

Dμ=∂μ+ig′YLBμ+igτa2WμaDμ=∂μ+ig′YLBμ+ig2τaWμa

This is almost everything the Standard Model has to say about the electron. Breaking it down term by term, the "definition" of the electron is assembled from several independent, mutually non-derivable elements.

II. All Attributes Assigned to the Electron by the Standard Model

2.1 Dirac Spinor Field — The Sole Dynamical Structure

The electron is defined as a four-component Dirac spinor field ψe(x)ψe(x) , belonging to the (12,0)⊕(0,12)(21,0)⊕(0,21) representation of the Lorentz group. This implies:

• The electron is a spin-1/2 fermion.

• The field transforms according to spinor rules under Lorentz transformations.

However, this only specifies the electron's transformation behavior under spacetime symmetries. It addresses none of the electron's internal structure, spatial distribution, or the physical mechanism of spin. In essence, it is merely an "abstract representation."

2.2 Mass — An External Parameter (Yukawa Coupling)

The electron's mass is not derived from the dynamics of the spinor field but is manually introduced via Yukawa coupling with the Higgs field:

LYukawa=−yeLˉeΦeR+h.c.LYukawa=−yeLˉeΦeR+h.c.

After spontaneous electroweak symmetry breaking, the Higgs field acquires a vacuum expectation value ⟨Φ⟩=v/2⟨Φ⟩=v/2 , and the electron gains mass:

me=yev2me=2yev

Crucially: The Yukawa coupling constant ye≈2.94×10−6ye≈2.94×10−6 is a free parameter, determined experimentally and manually input into the model. The Standard Model does not explain why yeye takes this value, nor why the electron mass is exactly 0.5110.511 MeV. For the electron, mass is an external label, not an intrinsic attribute.

2.3 Electric Charge — A Quantum Number of Gauge Coupling

The electron's charge arises from its representation under the U(1)Y×SU(2)LU(1)Y×SU(2)L gauge group:

• The left-handed electron eLeL belongs to an SU(2)LSU(2)L doublet Le=(νe,e)LLe=(νe,e)L with hypercharge Y=−1Y=−1 .

• The right-handed electron eReR is an SU(2)LSU(2)L singlet with hypercharge Y=−2Y=−2 .

After electroweak symmetry breaking, the electric charge is given by the Gell-Mann–Nishijima formula:

Q=T3+Y2Q=T3+2Y

For the left-handed electron: Q=−12+−12=−1Q=−21+2−1=−1 .
For the right-handed electron: Q=0+−22=−1Q=0+2−2=−1 .

However, why are the hypercharges assigned as Y=−1Y=−1 and Y=−2Y=−2 ? This is again manually input. The Standard Model does not explain the reason for charge quantization (in the pure SM, U(1)U(1) charges could theoretically take any real value; integer quantization requires additional mechanisms like Grand Unification or magnetic monopoles).

2.4 Chiral Structure — Left-Right Asymmetry

The Standard Model requires the left-handed and right-handed components of the electron to belong to different gauge representations:

eL∈(2,−1),eR∈(1,−2)eL∈(2,−1),eR∈(1,−2)

This is the mathematical encoding of parity violation in weak interactions. But why nature chose this specific chiral structure? The Standard Model offers no explanation; it merely codifies experimental facts.

2.5 Lepton Number — A Global Conserved Quantity

The electron is assigned a lepton number Le=+1Le=+1 (anti-electron Le=−1Le=−1 ), which is a conserved charge of a global U(1)U(1) symmetry. However, in the Standard Model, lepton number conservation is an accidental symmetry, not derived from first principles, and it can even be violated by non-perturbative effects (such as 't Hooft instanton processes).

2.6 Generation (Family) — Pure Repetition

The electron is the first generation of charged leptons. The Standard Model offers no explanation for "why there are three generations." The electron, muon ( μμ ), and tau ( ττ ) have identical structures in the Lagrangian; the only difference lies in the numerical values of their Yukawa coupling constants ( ye≠yμ≠yτye=yμ=yτ ). These three values are all free parameters.

2.7 Anomalous Magnetic Moment — The Only "Derived" Attribute

The only property of the electron that the Standard Model can calculate (rather than manually input) from its basic structure is its anomalous magnetic moment:

ae=g−22=α2π+…ae=2g−2=2πα+…

This is the result of QED radiative corrections, matching experiments with a precision of order 10−1310−13 . However, this is essentially a perturbative calculation based on existing structures (spinor + charge + mass), not a definition of the electron itself.

III. Summary: The Standard Model's "Definition" Checklist for the Electron

表格

Attribute	Source	Derived?
Spin 1/2	Dirac Spinor Representation	Assumed
Mass ( meme )	Yukawa Coupling Constant ( yeye )	Manually Input
Charge ( Q=−1Q=−1 )	Assignment of Hypercharge ( YY )	Manually Input
Chiral Structure	Different Representations for L/R Components	Manually Input
Lepton Number ( Le=1Le=1 )	Global U(1)U(1) Symmetry	Accidental Symmetry
Generation	Triple Repetition in Lagrangian	Manually Input
g−2g−2	QED Radiative Corrections	Only Derived Quantity
Spatial Structure	—	Completely Missing
Physical Mechanism of Spin	—	Completely Missing
Origin of Mass (Value of yeye )	—	Completely Missing
Reason for Charge Quantization	—	Completely Missing

IV. Are There Other Defenses for the Standard Model?

Beyond the content of the Lagrangian above, the Standard Model has no other defense regarding the definition of the electron. Possible "defenses" often cited include the following, but none add substantial content:

1. "The electron is the lowest excitation state of the electron field."
   This is a general statement of Quantum Field Theory applicable to all particles; it does not constitute a special definition of the electron. It merely states "a quantum of the field is a particle," which is a circular definition.

2. "All properties of the electron are determined by symmetry."
   This is an overstatement. While symmetries constrain the electron's possible behaviors (e.g., spinor transformation rules), all specific numerical values (mass, charge, mixing angles) are free parameters outside the symmetry framework. Symmetry provides the "format of the table," but it does not fill in the "content of the table."

3. "The electron is an irreducible representation."
   The Standard Model defines the electron as an irreducible component of the SU(3)C×SU(2)L×U(1)YSU(3)C×SU(2)L×U(1)Y gauge group under a specific representation. But this is merely classification, not explanation. Asking why nature chose this specific set of representations yields no answer from the Standard Model.

4. Defense via Renormalization Group and Effective Field Theory.
   One might argue that the Standard Model is a low-energy effective theory and thus need not explain the origin of these parameters. However, this admission precisely concedes that the Standard Model is not a fundamental theory of the electron, but merely a phenomenological description.

V. The Ontological Void

The Standard Model's definition of the electron is essentially a checklist of attributes (most of which are manually input), not a physical model. It tells us what the electron does (how it transforms, how it couples), but it does not tell us what the electron is.

From the perspective of Natural Quantum Theory (NQT), this void is exactly what needs to be filled. If the electron is a certain topological structure of the electromagnetic field (or a unified field), then:

• Spin becomes the rotational mode of the field.

• Charge becomes a topological quantum number.

• Mass becomes the localized integral of field energy.

• The gg -factor becomes a geometric consequence of the topological structure.

All those "manually input" parameters would transform into calculable topological invariants. The Standard Model's attribute checklist would evolve from an empirical table into a mathematical deduction of a unified physical image.

This is precisely the motivation behind the discussion that "elementary particles are different topological structures of the electromagnetic field": NQT does not seek to overturn the computational achievements of the Standard Model, but rather to provide the missing ontological foundation upon which it rests.