From Quantum Field Theory to Gauge Field Theory: The NQT Perspective

I. Classical Field Theory: Starting from a Physical Image

Maxwellian electrodynamics provides us with a complete and intuitive physical image: fields are real entities filling space, particles are local excitations of these fields, and electromagnetic interactions are completed through the continuous propagation of the field. In this image, the field possesses a clear ontological status—it is not merely a computational tool but physical reality itself. Charged particles have a finite spatial scale, and their electromagnetic fields generate a complete multipole expansion: monopole moment (total charge), dipole moment, quadrupole moment, octupole moment, and so on. Each order of multipole moment corresponds to genuine degrees of freedom within the particle's internal structure.

The core characteristics of this image are: fields are real, particles have scale, and interactions are local.

II. Quantum Field Theory: From Image to Abstraction

The establishment of Quantum Field Theory (QFT) underwent a systematic process of abstraction. The key step in this process was the introduction of the point-particle hypothesis.

When particles are compressed into geometric points, the complete multipole expansion of the classical image undergoes a fundamental collapse. The higher-order multipole moments (quadrupole, octupole, etc.) naturally carried by finite-scale particles lose their spatial basis for existence; only the lowest-order monopole and dipole moments survive. This leads to a series of consequences:

1. Fields Degenerate from Reality to Operators. Classical fields E(r,t)E(r,t) and B(r,t)B(r,t) are replaced by field operators ϕ^(x)ϕ^(x) . Their physical meaning shifts from "real force fields existing in space" to "algebraic tools for creating and annihilating particles." The ontological status of the field is replaced by the image of virtual particle exchange.

2. Divergences and Renormalization. The self-energy of point particles diverges at short distances. To eliminate these divergences, physicists developed the renormalization program—systematically absorbing infinities into the redefinition of parameters. While renormalization is extremely successful computationally, its cost is that physical information about the particle's internal structure is completely erased and no longer appears in the theory's basic framework.

3. Virtualization of the Interaction Image. Feynman diagrams decompose the continuous process of field propagation into discrete vertices of virtual particle exchange. Each vertex corresponds to a local point-like coupling. This formulation is efficient and elegant for perturbative calculations, but the price paid is that combinatorial graphical rules replace the spatiotemporal evolution image of physical fields.

In summary, QFT completes the transition from "having an image" to "pure abstraction": fields become operators, particles become points, interactions become virtual exchanges, and internal structures disappear into renormalization.

III. Gauge Field Theory: The Pinnacle of Abstraction

Built upon the abstract framework of QFT, Gauge Field Theory further elevates the principle of symmetry to a central position.

Yang-Mills theory requires the Lagrangian to be invariant under local gauge transformations. To satisfy this requirement, one must introduce a gauge field, whose dynamics are determined by the structure constants of non-Abelian groups (such as SU(2)SU(2) and SU(3)SU(3) ). While this construction is mathematically profound, it introduces fundamental confusion regarding the physical image:

1. Abstraction of Internal Space. Gauge groups are defined in an "internal space," independent of spacetime coordinates. The weak isospin space of SU(2)SU(2) and the color space of SU(3)SU(3) are treated as abstract mathematical dimensions completely independent of three-dimensional physical space. The physical origin of these "internal degrees of freedom" is no longer questioned.

2. Axiomatization of Gauge Invariance. Local gauge invariance is elevated from a "physical phenomenon needing explanation" to a "fundamental principle requiring no explanation." The theory no longer answers "why nature demands gauge invariance" but instead uses it as a starting point to derive everything.

3. Mystification of the Confinement Problem. Color confinement in QCD—the fact that quarks cannot appear as free particles—lacks a rigorous analytical proof to this day. Popular explanations appeal to semi-intuitive images like "color flux tubes," but these images themselves do not naturally emerge from the theory's basic principles.

Thus, physical theory has moved from the clear field image of the Maxwell era to a highly abstract system of symmetry axioms. The image has vanished, leaving behind group theory structures, fiber bundle geometry, and path integrals.

IV. NQT: Returning from Abstraction to Image

The core stance of Natural Quantum Theory (NQT) is that the aforementioned abstraction process is not an inevitable cost of physical progress, but an unnecessary deviation introduced by the point-particle hypothesis. By restoring the finite scale of particles ( ∼∼ Compton wavelength), one can reverse this abstraction process and reconstruct a complete physical image.

4.1 Restoring the Multipole Structure

When particles possess a spatial scale on the order of λCλC , their electromagnetic fields naturally expand into a complete multipole sequence:

B(r)=Bdipole+Bquadrupole+Boctupole+⋯B(r)=Bdipole+Bquadrupole+Boctupole+⋯

Each order of multipole moment carries independent orientational degrees of freedom. The dipole moment is described by a direction vector m^m^ with two independent components; the quadrupole moment is described by a symmetric traceless tensor QijQij with five independent components; the octupole moment adds seven more components. These orientational degrees of freedom constitute a complete description of the particle's internal structure.

The key insight is this: The transformation groups of these multipole orientational degrees of freedom are precisely the gauge groups of the Standard Model.

4.2 The Physical Origin of Gauge Groups

• U(1)U(1) : Phase Degree of Freedom of the Dipole. The projection angle of a magnetic dipole moment onto a given axis defines a U(1)U(1) phase. Choosing different reference directions is equivalent to performing a U(1)U(1) transformation. This is the physical content of electromagnetic gauge invariance.

• SU(2)SU(2) : Complete Orientational Degree of Freedom of the Dipole. The full orientation of a magnetic dipole moment in three-dimensional space constitutes an S2S2 sphere, whose rotation group is SU(2)SU(2) . Non-commutativity directly reflects the non-commutativity of rotations in real 3D space—rotating first around the xx -axis and then the yy -axis yields a different result than the reverse order. This is not an "abstract internal symmetry" but a physical transformation of magnetic moment orientation in real space.

• SU(3)SU(3) : Joint Orientation of Dipole and Quadrupole. When considering both the dipole moment (3 components, minus normalization, 2 degrees of freedom) and the quadrupole moment (5 independent components, taking their orientational degrees of freedom), the joint orientation space possesses exactly 8 independent generators, corresponding to the 8 Gell-Mann matrices of the SU(3)SU(3) group. The "color degrees of freedom" of SU(3)SU(3) thus acquire an intuitive physical image: they are the joint degrees of freedom of particle multipole orientations.

4.3 The Physical Image of Gauge Fields

In standard theory, the physical meaning of the gauge field Aμa(x)Aμa(x) is abstract—it is a "connection" ensuring consistent parallel transport in internal space.

In the NQT image, the gauge field acquires a clear physical meaning: The gauge field is the physical field that coordinates the consistency of magnetic moment reference directions across different spatial points.

Specifically: Particles at every spatial point possess magnetic dipole moments, and the description of their direction depends on the choice of reference frame. Choosing different reference directions at different spatial points is equivalent to performing a local gauge transformation. The role of the gauge field is to record the relationships between these local choices, ensuring that physical observables (such as the relative orientation of magnetic moments between adjacent particles) do not depend on the choice of reference direction.

This image directly explains the physical necessity of gauge invariance: it is not an abstract axiom imposed on the theory, but a natural requirement when describing physical systems with orientational degrees of freedom—just as general covariance in General Relativity arises from the natural need to describe physical quantities in curved spacetime.

4.4 The Physical Mechanism of Quark Confinement

The mainstream description of color confinement in standard QCD is as follows: The color force lines between quarks, unlike electric field lines, do not spread out into space but are squeezed into a one-dimensional "color flux tube." The energy of this tube grows linearly with the distance between quarks; thus, the energy required to separate quarks tends toward infinity, making free quarks impossible to isolate. While intuitively appealing, this image faces fundamental difficulties: The flux tube hypothesis itself is not a conclusion analytically derived from the QCD Lagrangian but a semi-empirical description based on lattice simulations and phenomenological models. Confinement remains unproven rigorously—it is even one of the Clay Mathematics Institute's Millennium Prize Problems.

NQT offers a fundamentally different physical image.

4.4.1 Natural Decay and Undetectability of Multipole Fields

The field strength of higher-order multipole moments decays rapidly with distance: quadrupole field ∼1/r4∼1/r4 , octupole field ∼1/r5∼1/r5 . The multipole orientational degrees of freedom corresponding to SU(3)SU(3) have significant effects only within the particle scale ( ∼λC∼λC ); at long distances, these higher-order effects rapidly decay to undetectable levels. Therefore, the reason "color" degrees of freedom cannot be observed at long distances is primarily that the corresponding higher-order multipole fields naturally vanish at such distances. This is a direct physical corollary of the mathematics of multipole expansion, requiring no flux tube hypothesis.

However, this only answers "why color is invisible at a distance," not the more fundamental question: Why can quarks not exist as independent particles?

4.4.2 Quarks as Components of a Stable Structure

NQT's answer to this question is: Quarks may not be independent particle entities at all, but rather internal components of a stable pinch structure, much like a vortex cannot detach from a fluid or a knot cannot detach from a rope.

The physical logic is as follows:
A finite-scale particle (such as a proton) possesses a complex internal multipole field structure. These multipole fields—dipole, quadrupole, octupole—form a self-consistent pinch equilibrium state within the particle. Pinch equilibrium refers to a dynamic mechanical balance achieved between field gradient forces, magnetic pressure, and kinetic pressure, maintaining a stable spatial configuration. This equilibrium is holistic: the existence of each multipole component depends on the constraints of the others, just as every archstone of an arch bridge depends on the support of the remaining stones.

In this image, so-called "quarks" correspond to local substructures within this holistic pinch structure—certain topological features or symmetry modes of the internal field distribution. Deep inelastic scattering experiments indeed detect point-like scattering centers (partons) inside nucleons, but a "point-like scattering center" is not equivalent to a "separable independent particle." Just as the mass of a rigid sphere is concentrated at several density peaks which high-energy probes can resolve, these peaks cannot be "pulled out" of the sphere to become independent entities.

More specifically, the three "quarks" inside a proton can be understood as three topological constraint nodes of the magnetic field pinch structure. These three nodes jointly maintain the stability of the overall structure. Their relationship is not "three independent particles bound together by a force" (the potential well model approach) but "three nodes jointly constituting an indivisible field topological configuration." Attempting to separate one node is equivalent to attempting to destroy the overall topological structure. The pinch effect of the field will resist this destruction—the energy invested will not separate the quarks but will instead excite new structures within the field (i.e., producing new mesons). This is entirely consistent with the experimentally observed phenomenon of "jet fragmentation."

4.4.3 Comparison with Traditional Confinement Images

While the traditional QCD confinement image and the NQT image may correspond in phenomenological description, their physical philosophies are diametrically opposed:

• Traditional Image: Quarks are elementary particles →→ bound together by color force →→ confinement is an effect of "force" →→ requires explaining why the force is infinite →→ appeals to flux tubes, area laws, etc.

• NQT Image: Particles are stable pinch field structures →→ "Quarks" are internal nodes of the structure →→ nodes cannot exist independently apart from the structure →→ confinement is a topological-structural necessity →→ no need to explain an "infinite force."

This distinction is analogous to two ways of understanding "why protons in an atomic nucleus cannot be separated infinitely far apart": one calculates the nuclear force potential curve (traditional), while the other recognizes that protons form a holistic collective structural mode within the nucleus, and separating a single proton means reorganizing the entire collective state (NQT spirit).

The advantage of the NQT image is that it transforms confinement from a problem "requiring a special dynamical mechanism to explain" into a common-sense statement that "stable structures cannot be arbitrarily divided." A stable pinch field configuration naturally has inseparable internal substructures—this does not require non-perturbative field theory calculations to prove but is a mechanical necessity of pinch equilibrium.

4.4.4 Testable Implications

This image yields several qualitative implications comparable with experiments:

1. Nature of Quark "Mass": The "mass" of a quark is not the rest mass of an independent particle but a manifestation of localized energy concentration within the field structure. This aligns with the fact in the Standard Model that "current quark masses are far smaller than constituent quark masses"—most of the "mass" comes from the kinetic energy and interaction energy of the field, not the bare mass of the quark itself. The proton mass is about 938 MeV938 MeV , while the sum of the three current quark masses is only about 10 MeV10 MeV ; over 98% of the mass comes from the field structure itself.

2. Jet Phenomenon: The jet phenomenon in high-energy collisions—fragmentation producing numerous mesons rather than free quarks—is precisely the process where, after the pinch structure is locally disrupted, the field reorganizes into new stable configurations. The invested energy does not produce free quarks but excites new modes of the field, which rapidly condense into new pinch steady states (mesons and baryons), fully consistent with experiments.

3. Many-Body Interactions: If quarks are indeed structural nodes rather than independent particles, then the "interaction" between quarks should not be simply described as a two-body potential but must possess an essential many-body character—the properties are determined by the overall configuration of the three quarks, not by the sum of pairwise interactions. This is consistent with the three-body force effects discovered in lattice QCD calculations.

4.5 The Physical Origin of Interaction Strengths

The Standard Model assigns different coupling constants to the electromagnetic, weak, and strong interactions ( α≈1/137α≈1/137 , αW∼1/30αW∼1/30 , αs∼1αs∼1 ) and treats them as independent fundamental parameters. In the NQT image, these vast differences in coupling strength receive a unified physical explanation.

The key insight is: The electromagnetic field itself has no upper limit on intensity. Maxwell's equations are linear; field strength can take any value without an intrinsic cutoff. The sole factor determining field strength is the distribution and geometric configuration of the source.

When the electromagnetic field is confined to an extremely small spatial scale ( r≪λCr≪λC ), maintaining a stable structure requires the pinch effect of the field. Pinching is the centripetal constraining force generated by the current or magnetic field itself—the magnetic field of a current filament exerts a Lorentz force on the current itself, compressing it inward. Inside the particle, the multipole structure of the magnetic field self-consistently generates this pinch constraint: higher-order multipole fields are extremely strong at small scales, and their gradient forces stably bind the field structure within a region on the order of the Compton wavelength.

This leads to a direct corollary: The shorter the distance scale of the multipole structure, the greater the field strength required to maintain stability, and consequently, the stronger the corresponding interaction.

This can be intuitively understood from the scaling relations of energy density. For a dipole field, the field strength at distance rr is B∼μ/r3B∼μ/r3 , corresponding to an energy density u∼B2∼μ2/r6u∼B2∼μ2/r6 ; for a quadrupole field, B∼Q/r4B∼Q/r4 , with energy density u∼Q2/r8u∼Q2/r8 ; and for an octupole field, the decline is even steeper. Therefore, inside the particle ( r∼λCr∼λC and smaller scales), the energy density of higher-order multipole fields far exceeds that of the dipole field, and even more so the monopole field ( u∼q2/r4u∼q2/r4 ). The stronger the field, the more intense the interaction. This is not because "the nature of the forces is different," but because the same electromagnetic field exhibits different intensities at different spatial scales.

Specifically:

• At long distances ( r≫λCr≫λC ), only the monopole field (Coulomb field ∼1/r2∼1/r2 ) is detectable. The field strength is moderate, with a coupling constant α∼1/137α∼1/137 ; this is the usual electromagnetic interaction.

• At the scale r∼λCr∼λC , the dipole field ( ∼1/r3∼1/r3 ) begins to dominate. Maintaining the stable pinch of the dipole orientational structure requires significantly stronger fields, causing the coupling strength to jump to the level of αW∼1/30αW∼1/30 , manifesting as the moderate coupling of the weak interaction.

• At the internal particle scale r≪λCr≪λC , quadrupole and octupole fields ( ∼1/r4,1/r5∼1/r4,1/r5 ) dominate. The pinch constraints of these higher-order multipole structures demand extremely strong fields, driving the coupling constant up to αs∼1αs∼1 or even higher. This is the flip side of "asymptotic freedom" in the strong interaction: strong coupling at short distances.

Therefore, the hierarchy of coupling strengths for the electromagnetic, weak, and strong interactions:

αem≪αW≪αsαem≪αW≪αs

is not a set of independent parameters for three different forces, but a direct reflection of the field strength levels of the same electromagnetic field at three different spatial scales. The pinch effect provides the physical mechanism: maintaining stable multipole structures on smaller scales requires stronger fields, generating stronger interactions.

This image also naturally implies the physical meaning of asymptotic freedom. At extremely short distances ( r→0r→0 ), if particles have a finite scale, there is no true r→0r→0 limit. The field structure inside the particle reaches its maximum strength at the Compton wavelength scale and then tends toward a stable pinch equilibrium. As one probes gradually from afar into the particle's interior, the effective coupling felt first increases (entering the higher-order multipole field region) and then saturates (reaching pinch equilibrium). This is qualitatively consistent with the running coupling behavior of QCD observed in experiments.

4.6 A Unified Image

From the NQT perspective, the electromagnetic, weak, and strong interactions are not three essentially different forces, but rather different manifestations of the multipole electromagnetic structure of the same finite-scale particle at different distance scales:

Distance Scale	Dominant Multipole	Symmetry Group	Field Strength Scaling	Coupling Strength	Corresponding Interaction
r≫λCr≫λC	Monopole (Charge)	U(1)U(1)	∼1/r2∼1/r2	α∼1/137α∼1/137	Electromagnetic
r∼λCr∼λC	Dipole (Magnetic Moment)	SU(2)SU(2)	∼1/r3∼1/r3	αW∼1/30αW∼1/30	Weak
r≪λCr≪λC	Quadrupole + Octupole	SU(3)SU(3)	∼1/r4∼1/r4	αs∼1αs∼1	Strong

The three "forces" are unified into the multipole field structure of a single particle; the three "gauge groups" are unified as the transformation groups of multipole orientational degrees of freedom; and the three "coupling strengths" are unified as the natural scaling of multipole field strengths at different scales. This is a true physical unification—unifying not only symmetries but also the origin of dynamical strengths.

V. Magnetic Quadrupole Moments and the Elliott SU(3) Model

Giant Resonances in atomic nuclei serve as direct evidence for the physical reality of multipole fields. Collective excitation modes such as the Giant Dipole Resonance (GDR) and the Giant Quadrupole Resonance (GQR) demonstrate that multipole degrees of freedom are real, dynamical, and capable of carrying energy—they are not merely mathematical abstractions. Nuclear matter oscillates as a whole in multipole modes, with each mode possessing definite energy, width, and decay channels. This aligns precisely with the core assertion of NQT: multipole structures are physical realities.

The magnetic quadrupole moment is a second-rank traceless symmetric tensor. In three-dimensional space, it possesses 5 independent components, corresponding to the spherical harmonics Y2,mY2,m (where m=−2,−1,0,+1,+2m=−2,−1,0,+1,+2 ). Each component represents an independent orientation-deformation degree of freedom, which can carry distinct energies under symmetry breaking. The splitting of the Giant Quadrupole Resonance into mm -substates in deformed nuclei provides experimental evidence for this energy distribution. In essence: the five degrees of freedom allow for independent energy allocation.

The most critical point is that Elliott’s SU(3) model (1958) has already established a precise connection with the generators of Gell-Mann’s SU(3):

Lx,Ly,Lz⏟3 Angular Momenta+Q2,m (m=−2,…,+2)⏟5 Quadrupole Operators=8 Generators of SU(3)3 Angular MomentaLx,Ly,Lz+5 Quadrupole OperatorsQ2,m(m=−2,…,+2)=8 Generators of SU(3)

In other words, the 3 degrees of freedom of angular momentum plus the 5 degrees of freedom of the quadrupole moment equal the 8 generators of SU(3), exactly constituting the Lie algebra of SU(3). The Elliott model has been highly successful in nuclear physics, describing rotational bands and collective motions in light nuclei.

This implies an exact algebraic isomorphism between Gell-Mann’s SU(3) generators ( λ1,…,λ8λ1,…,λ8 ) and the combined transformations of angular momentum and quadrupole moments in physical space. The linear transformation operations of the Gell-Mann matrices can be fully mapped onto the physical rotation and deformation operations acting on magnetic dipole orientations (3 components) and magnetic quadrupole orientations (5 components).

VI. Conclusion

The historical development of Quantum Field Theory and Gauge Field Theory represents a process where physical images gradually yielded to abstract structures. NQT posits that the root of this abstraction lies in the point-particle hypothesis, which obscures the internal multipole structure of particles. By restoring the finite spatial scale of particles, the complete multipole expansion re-emerges. Consequently:

• Gauge groups acquire intuitive geometric-physical explanations.

• The phenomenon of confinement is naturally understood.

• The three fundamental interactions are unified as manifestations of multipole fields at different distance scales.

The SU(3) gauge group need not be interpreted as a symmetry of an abstract internal space. Instead, it is the geometric group of orientation transformations for magnetic dipoles plus magnetic quadrupoles in real space.

• The triplet of color charge corresponds to dipole orientations.

• The octet of gluons corresponds to the complete set of transformations involving angular momentum plus quadrupole orientations.

The success of the Elliott model in nuclear physics serves as experimental validation of this physical image itself: SU(3) structures can naturally emerge from multipole degrees of freedom in physical space.

From image to abstraction, and then from abstraction back to image—this is not a simple regression, but a pursuit of intuitive comprehensibility in physics at a higher level. The core belief of NQT is that the deep structure of nature can be intuitively understood, rather than being limited to mere algebraic manipulation.