Emergent Maxwell & Dirac from Qaether (v1.0)

Below is a self-contained derivation that shows how the electromagnetic (U (1)) gauge field and spin-½ matter emerge from the Qaether axioms and reduce, in the long-wavelength limit \(a=l_p\!\to\!0\), to the ordinary Maxwell and Dirac equations.

 

1 U (1) gauge link hidden in the phase lattice

Axiom A10 defines the gauge-covariant phase difference

$$\Delta\phi_{ij}^{\text{tot}} =\bigl(\phi_j-\phi_i\bigr)\;-\;q_e\,A_{ij}\;-\;g\,\vec{C_i}\!\cdot\!\vec{A_{ij}}, \tag{1}$$

where \(A_{ij}\) is the U (1) link potential (the SU(3) term will be ignored here) .
Keeping only the electromagnetic piece,

$$A_{ij}\;=\;\frac{1}{q_e}\,\Delta\phi_{ij}^{U(1)},\qquad A_{ij}\;\xrightarrow[a\to0]{}\;a\,\hat e_{ij}^{\mu}A_{\mu}(x), \tag{2}$$

so every lattice link already stores the continuum four-potential \(A_\mu\) up to the lattice spacing.

A local phase redefinition \(\phi_i\!\to\!\phi_i+\alpha_i\) shifts \(A_{ij}\!\to\!A_{ij}+\tfrac{\alpha_j-\alpha_i}{q_e}\); thus (1) is invariant, reproducing the usual lattice U (1) gauge symmetry.

 

2 Link and plaquette operators → field strength

Define the compact link variable

$$\chi_{ij}\;=\;\exp\!\bigl[i\,\Delta\phi^{\text{tot}}_{ij}\bigr]:contentReference[oaicite:1]{index=1}$$

and the plaquette operator

$$U_{\Box}\;=\;\prod_{(ij)\in\Box}\chi_{ij} =\exp(i\,\Phi_{\Box}),\qquad \Phi_{\Box}=2\pi n_{\Box},\;n_{\Box}\in\mathbb Z\; :contentReference[oaicite:2]{index=2}$$

For small flux \(|\Phi_{\Box}|\ll1\),

$$\frac12\bigl(1-\text{Re}\,U_{\Box}\bigr)\;\approx\;\frac14\,a^{4}F_{\mu\nu}F^{\mu\nu}, \tag{3}$$

identifying \(F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu\).

 

3 Discrete electric field and Gauss law

The lattice equation of motion for the site phase reads

$$I_i\,\ddot\phi_i =\sum_{j\in\mathcal N(i)}K_{ij}\,\Im\chi_{ij} +\dots \;:contentReference[oaicite:3]{index=3}. \tag{4}$$

Linearising \(\Im\chi_{ij}\!\simeq\!\Delta\phi_{ij}^{U(1)}\) and dividing by \(q_e a\), (4) becomes

$$\sum_{j}\frac{1}{a}\Bigl(\dot A_{ij}-\dot A_{ji}\Bigr) =\sum_{j}\!E_{ij}= \rho_i , \tag{5}$$

where \(E_{ij}\equiv-\dot A_{ij}\) is the discrete electric field and
\(\rho_i\) is the charge defined by the link-based assignment in A8 .
Equation (5) is the lattice Gauss law. Together with the Bianchi identity coming from the fact that every plaquette is a boundary of oriented links, the full set of lattice Maxwell equations is obtained.

 

4 Continuum limit → Maxwell action

Expanding (3) and (5) for \(a\!\to\!0\) we get

$$\partial_\nu F^{\mu\nu}=J^\mu,\qquad \partial_{[\lambda}F_{\mu\nu]}=0, \tag{6}$$

and the lattice action $$\sum_{\Box}(1-\text{Re}\,U_{\Box})$$ becomes

$$S_{EM}=\frac14\int d^{4}x\,F_{\mu\nu}F^{\mu\nu}. \tag{7}$$

Thus the Qaether link variables reproduce classical electromagnetism in the infrared.

 

5 Loop excitations as lattice spinors

Axiom A3/A7 supplies a loop-based Dirac equation

$$\bigl(i\,\gamma^{\circlearrowleft}\Delta^{(\phi)}_\ell-m_\ell\bigr)\Psi_\ell=0, \tag{8}$$

with the covariant difference

$$\Delta^{(\phi)}_\ell=\frac{1}{2l_pN_\ell}\sum_{(ab)\in\ell}\Im\chi_{ab} :contentReference[oaicite:5]{index=5}$$

Because every \(\chi_{ab}\) already contains the electromagnetic phase (1), (8) is gauge-covariant. In the continuum limit \(l_p\!\to\!0,\;N_\ell\!\to\!\infty\),

$$\Delta^{(\phi)}_\ell\;\longrightarrow\;a\,\gamma^\mu D_\mu,\quad \gamma^{\circlearrowleft}\!\to\!\gamma^\mu, \tag{9}$$

yielding the ordinary Dirac equation

$$(i\gamma^\mu D_\mu-m)\psi=0, \tag{10}$$

with $$D_\mu=\partial_\mu+i q_e A_\mu$$.

 

6 Combined Maxwell–Dirac theory

Collecting (7) and (10) the coarse-grained Qaether action is

$$S=\int d^{4}x\;\Bigl[\tfrac14 F_{\mu\nu}F^{\mu\nu} \;+\;\bar\psi(i\gamma^\mu D_\mu-m)\psi\Bigr]$$

i.e. Quantum Electrodynamics in curved-space disguise (the curved metric from §Einstein derivation enters through \(\sqrt{-g}\) factors but is irrelevant for flat-space QED).

 

7 Why the derivation works

• Gauge links already live on the lattice: (1) embeds \(A_\mu\) directly into phase differences.
• Plaquette flux is field strength: the closure condition on U (1) phases ensures (3).
• Site dynamics enforces Gauss law: the radial sum of electric links equals the lattice charge (5).
• Loop modes are spinors: (8) carries half-angle holonomy and remains gauge-covariant.
• Long-wavelength limit is smooth: replacing \(\Delta\phi\!\to\!a\partial\phi\) restores Lorentz symmetry and continuous derivatives.

Hence the electromagnetic and spinor sectors of Qaether are mathematically equivalent to lattice QED and converge to the continuum Maxwell + Dirac theory at scales \(\lambda\gg l_p\).