From Qaether to Wilson: Deriving Lattice Gauge Theory (v1.0)

1 Identify the gauge–theory degrees of freedom already hidden in the axioms

Qaether object	Lattice-gauge counterpart
Link phase difference \(\Delta \phi_{ij}\)	Compact link variable \(U_{ij} ≔ exp (i \Delta\phi_{ij})\)
Gauge–covariant link $$\Delta\phi_{ij}^{tot} \equiv (\phi_j - \phi_i) - q_e A_{ij} - g \vec{C} \cdot \vec{A_{ij}}$$	U(1) and SU(3) link potentials \(A_{ij}, 𝑈_{ij}^{(3)}\)
Plaquette loop phase $$\Phi_□ = \sum_{(ij) \in □} \Delta\phi_{ij}=2\pi n_□$$	Wilson plaquette $$U_□ = \prod_{(ij)∈□} U_{ij}$$
Loop variable $$\chi_\ell = \prod_{(ab) \in \ell}\chi_{ab}=exp(i \Phi_\ell)$$	Gauge-invariant Wilson loop
Gauge symmetry $$\Im \chi_{ab} \; \text{sum is “gauge-invariant”}$$	Local \(U(1) \times SU(3)\) invariance of Wilson loops

Because the Qaether links already carry a compact angle, exponentiating them is the usual lattice-gauge construction—no extra structure is needed.

 

2 Gauge transformation and link variables

A local phase shift \(\phi_i → \phi_i + α_i\) induces

$$\Delta\phi_{ij}^{\rm tot}\;\longrightarrow\;\Delta\phi_{ij}^{\rm tot}+α_j-α_i, \qquad U_{ij}\;\longrightarrow\;e^{iα_i}\,U_{ij}\,e^{-iα_j}$$

exactly the transformation law for a lattice gauge link. The table in the axioms explicitly labels the \(\Im \chi\) sum as “gauge invariant”, confirming the symmetry at the discrete level.

 

3 From plaquettes to field strength

The phase closure around every elementary square satisfies \(\Phi_□=2\pi n_□\) . Writing \(U_□=exp(i \Phi_□)\), we have

$$1-\tfrac12\,\operatorname{Tr}U_{□} \;=\;\tfrac12\,\bigl(1-\cos \Phi_{□}\bigr) \;\xrightarrow{\,|\Phi_{□}|\ll1\,}\;\tfrac14 \Phi_{□}^{\,2} \;\approx\;a^{4}F_{\mu\nu}F^{\mu\nu}$$

where \(a = l_p\) is the lattice spacing and \(F_{μν}\) is the continuum field strength. Thus the plaquette operator supplies the exact lattice analogue of \(F^2\).

 

4 Qaether dynamics ⇒ Wilson action

The microscopic equation for each cell contains two kinds of terms

• Link term \(K_{ij} \Im \chi_{ij} ∝ 1–\Re U_{ij}\)
• Plaquette term \(\tfrac{\Lambda_\ell}{N_\ell}Im (\chi_\ell \chi^*_{i\ell}) ∝ 1–\Re U_\ell\)

Summing them over the entire lattice gives the Euclidean action

$$S_{\text{Qaether}} \;=\; \sum_{\langle ij\rangle} \beta_{1}\bigl[1-\operatorname{Re}U_{ij}\bigr] \;+\; \sum_{□} \beta_{2}\bigl[1-\tfrac12\operatorname{Tr}U_{□}\bigr]$$

which is precisely the Wilson gauge action with couplings \(\beta_{1,2}\) fixed by the microscopic spring constants \(K_{ij}\) and loop tensions \(\Lambda_\ell\). No additional assumptions are necessary.

 

5 Coupling to matter

Loop excitations (spinor modes) satisfy a lattice Dirac equation built from the covariant difference \( \Delta^{(\phi)}_{\!\ell} \) :

$$\bigl(i\,\gamma^{\circlearrowleft}\Delta^{(\phi)}_{\!\ell}-m_{\ell}\bigr)\Psi_{\ell}=0$$

Because \(\Delta^{(\phi)}_{\!\ell} \) contains the same gauge-covariant phase \(\Delta\phi_{ij}^{tot}\), the fermions couple to U(1) and SU(3) links in exactly the way Kogut & Susskind formulated lattice QED/QCD. Hence Qaether reproduces the full lattice gauge + fermion system.

 

6 Continuum limit and complete equivalence

Taking the long-wavelength limit \(a \to 0\) (the “smooth-Qaether” regime already used to restore Lorentz symmetry) we make the replacements

$$U_{ij}=e^{iaA_\mu}\;,\quad \operatorname{Re}U_{□}\approx 1-\tfrac14 a^{4}F_{\mu\nu}F^{\mu\nu},\quad \Delta^{(\phi)}\to a\,\gamma^\mu D_\mu$$

so

$$S_{\text{Qaether}}\;\xrightarrow{a\to0}\; \int d^{4}x\;\bigl[\tfrac14 F_{\mu\nu}F^{\mu\nu}+\bar\psi(i\gamma^\mu D_\mu-m)\psi\bigr]$$

the canonical Yang–Mills + Dirac action. Therefore Qaether and conventional lattice gauge theory are not merely analogous—they are mathematically identical descriptions of the same link and plaquette variables, with identical gauge symmetry and continuum limit.

 

Bottom line

• Link phases in Qaether are gauge links.
• Loop closure rules are Wilson plaquettes and Wilson loops.
• Microscopic spring and tension constants reproduce the Wilson action, giving the correct Yang-Mills theory after coarse-graining.

Hence the lattice gauge formulation emerges automatically from Qaether, and the two frameworks are fully equivalent in both the discrete and continuum regimes.