Introduction: Inheriting Defects Yet Achieving Success?

Quantum Field Theory (QFT) is built on the spectral method of quantum mechanics—expressing physical quantities as operator eigenvalues and projecting evolution onto the frequency or momentum space. However, this "spectral representation" inherently carries three major structural limitations:

• Globalization erases local geometry: Local structures such as trajectories, circular currents, and magnetic flux are averaged out, and spin/magnetic moment are easily misinterpreted as abstract labels;

• Representation and measure dependence: The dimension of the wave function changes with the representation, and ignoring the Jacobian factor can easily mislead the gauge problem as "dimension variability";

• Point coupling and divergence: Point particles + point vertices lead to infinite self-energy and "mysterious vacuum", requiring additional handling to connect with experiments.

Given that QFT inherits these intrinsic defects, why has it achieved remarkable success in describing a wide range of phenomena from atomic spectra to high-energy collisions?

The answer lies in this: the true power of QFT does not come from its initial spectral form, but from the decades-long continuous "reification" project—replacing abstract shorthand with observable responses, anchoring free parameters to experimental data, and embedding conservation laws and causality into the theoretical framework.

More profoundly, the success of QFT is essentially the "re-localization and re-dynamization of the spectral method":

• It puts global eigenmodes back into local fields and conserved currents in spacetime (re-localization);

• It allows the spectral structure to evolve dynamically with energy scales, environments, and interactions (re-dynamization).

On this path, Lattice Quantum Chromodynamics (Lattice QCD) embodies this most fully.

I. Three Pillars of QFT Reification

1. Symmetry and Conservation: Binding Measurable Quantities via "Field-Current"

Gauge fields are interpreted as the consistency coordination of local reference directions (phase/magnetic moment orientation), manifested as Ward/Slavnov–Taylor identities and strict charge conservation;

Noether current \(J^\mu\) and energy-momentum tensor \(T^{\mu\nu}\) become the primary objects organizing calculations, and all processes are rewritten as causal chains of "source-field-response".

2. Spectrum and Response: Replacing "Fictional Vocabulary" with Observable Functions

The Källén–Lehmann spectral density, dispersion relations, and optical theorem translate "invisible narratives" into directly observable quantities such as cross-sections and structure functions;

Vacuum polarization → "frequency dispersion of the dielectric function"; running coupling \(\alpha(Q^2)\) → "energy-scale dependence of medium-induced responses";

The LSZ formalism is understood as "far-region mode projection", and detector counting is spectral selection rather than "entity shuttling".

3. Data and Scales: Confining Unknowns in Low-Energy Constants

Renormalization demystified: interpreting "infinity cancellation" as "parameter redefinition under point approximation", with values pinned down by benchmark observations (e.g., \(\alpha\), mass m, g-factors);

Effective Field Theory (EFT): writing the most general action under symmetry and causality constraints, with low-energy constants determined by data for robust extrapolation;

Lattice field theory: performing calculations on finite lattice sites, preserving locality and microcausality, and realizing "physicalization of regularization" with finite resolution.

II. Lattice QCD: Practice of Reification

Lattice QCD not only satisfies the above three pillars but also pushes them to the extreme:

(1) Full Implementation of Three Reification Criteria

Calculable → Measurable: Path integrals are transformed into real field configuration sampling, abandoning shorthand such as "virtual particles" and directly outputting local currents, stresses, and topological charges;

Regularization → Physical resolution: Lattice spacing a and volume V are explicit spacetime resolutions, and divergences are converted into controllable errors;

Parameters → Data anchoring: Scale setting (e.g., \(f_\pi\), Ω baryon mass) realizes a closed loop of "scale calibration by data, followed by prediction".

(2) Overcoming the Three Limitations of Spectral Representation One by One

Spectral Limitations	Lattice Solutions
Globalization erases local geometry	Directly measure \(J^\mu(x)\), \(T^{\mu\nu}(x)\), \(q(x)\); reconstruct magnetization density, pressure distribution, and flux tube imaging
Representation dependence	Work with Euclidean probability measure, unify into representation-independent observables via non-perturbative renormalization (gradient flow, RI/MOM)
Point coupling divergence	Finite lattice spacing provides UV cutoff; vacuum effects = statistical responses of field configurations, not "virtual particle clouds"

(3) Typical Physical Achievements: From "Shorthand" to "Response"

Visualization of confinement mechanism: Flux tube imaging reveals "confined force lines" for the first time;

Proton internal structure: Pressure, shear, and magnetic moment distributions—moving from "abstract spin" to "local angular momentum and magnetization density";

Muon \(g-2\): Hadronic vacuum polarization is input as a "response function version" from lattice calculations, replacing the "virtual particle loop" metaphor;

Advances in real-time physics: Continuous breakthroughs in Minkowski spectral function reconstruction techniques.

III. Spectrum ↔ Field: The Core Translation Mechanism of QFT

The success of QFT lies in establishing a bidirectional mapping between spectral information and local fields:

1. Re-localization: From Global Eigenmodes to Local Propagators

Källén–Lehmann representation:\(D_F(x) = \int_0^\infty d\mu^2 \rho(\mu^2) \Delta_F(x;\mu^2)\)encodes the "global spectral density \(\rho(\mu^2)\)" as a weight distribution of "local two-point functions".

Physical significance of poles and branch cuts:

• Bound states → real-axis poles of the propagator;

• Resonances → complex-plane poles;

• Continuous spectrum → branch cuts (above-threshold multi-particle states).

2. Re-dynamization: Energy-Scale Evolution of the Spectral Structure

Running coupling and renormalization group: \(\mu \frac{dg}{d\mu} = \beta(g)\), constructing a dynamic network of "spectrum-coupling-energy scale";

Dyson–Schwinger equations: Propagator "dressing" leads to redistribution of mass, width, and \(\rho(\mu^2)\);

Effective Field Theory: Compressing high-energy spectra into a tower of low-energy local operators, realizing "controllable cutoff + dynamic feedback".

3. Spectrum-Field Glossary

Quantum Mechanical Spectral Concepts	QFT Local Implementations
Eigenenergy levels	Propagator poles
Continuous spectrum	Branch cuts
Form factors	Vertex function \(F(q^2)\)
Transition strength	Spectral function weight × matrix element squared
Selection rules	Local symmetries / Noether currents

IV. Deep Resonance with Natural Quantum Theory (NQT)

NQT advocates:Confined spectra are ontological, local conservation is constraining, and only differences are measurable.

This is highly consistent with the reification path of QFT:

• Spectra as ontology: Discrete energy levels = natural frequency-locking points; eigenfunctions = circular current modes;

• Local conservation: Electric and energy currents are verifiable in real space, corresponding to \(J^\mu\) and \(T^{\mu\nu}\) in QFT;

• Measurable differences: Phenomena such as the Lamb shift and Casimir force are all spectral differences caused by changes in boundaries/geometry, and QFT is consistent with NQT in observable quantities.

Lattice QCD goes further:

• Finite lattice spacing a provides a "non-point-like" entity scale, echoing NQT’s "finite vortices/phase domains";

• Spin = local angular momentum density, breaking free from abstract labels;

• Vacuum = response background, defined by field configuration statistics + dispersion relations, not a "particle sea".

V. Future Directions: Bridging the Last Mile

Despite significant progress, the following challenges remain to be addressed:

• Ill-posed nature of the inverse spectral problem: Developing more robust spectral reconstruction algorithms (Bayesian, neural operators);

• Real-time dynamics: Reliably extracting Minkowski spectral functions from Euclidean correlation functions;

• Cross-validation: Promoting a four-way closed loop of lattice-EFT-experiment-NQT, e.g.:

• Mapping the 3D distribution of proton magnetization/vorticity to test "frequency-locked circular currents";

• Distinguishing "phase domains" from "charge radius" via low-\(Q^2\) Compton scattering.

Conclusion: Returning Tools to Their Proper Place

The success of QFT lies not in metaphysics but in engineering:

• Taking conservation and causality as guardrails,

• Taking spectrum and response as grammar,

• Taking data and scales as anchors.

It transforms the "limitations of spectral representation" into the "advantages of measurable responses" and replaces "abstract shorthand" with "quantities with physical references".

The great significance of Lattice QCD is that it makes quantum field theory tangible:

• Conservation laws are exact identities on the lattice;

• The vacuum is samplable field configuration statistics;

• The proton is a "microscopic celestial body" with pressure, shear, and magnetization.

This path is naturally in sync with the concept that "quantum = spectralization, and tools return to their proper place":The spectral method is a bridge, not a destination.

The lattice method grounds both ends of the bridge—calculation on one end, physics on the other; what travels between them are observable energy currents, magnetization, and spectral density.

Only continuous reification can achieve both computational accuracy and profound understanding.