256×256 Gamma-Matrix QED Calculations: Compton, Muon-Pair Production, Møller, Bhabha

We provide Mathematica notebooks and PDFs for four QED processes computed with a two index Gamma basis realized in a 256×256 representation. Version 6 revises the metric weighting so that  $g^{\mu \nu} g_{\nu \sigma} = \delta^{\mu}_{\ \sigma}$  holds in code. 

In the flat limit the 256×256 results coincide with standard 4×4 outcomes. In curved spacetime, each scattering process can be evaluated using explicit local metric values,allowing numerical examination of deviations from the flat-space limit.

This version enables direct construction of the effective Gamma operators with given local metric inputs,making the physical impact of curvature visible in computed observables.

Addendum for Ver.6 (new attachment)

• We include the verification notebook “05_gamma256_anticommutation_check Ver6”.
  It bulk-tests all 16x16 combinations of the two-index 256x256 basis (via determinant and related checks), confirming both the flat-space anticommutation of the 16 basis blocks and the metric-weighted mixed anticommutation:

$$
\begin{aligned}
\Gamma_{\mu\nu}(x) &:= \Gamma_{\mu}{}^{\rho}\,g_{\rho\nu}(x),\\[2pt]
\{\widehat{\Gamma}_{\mu}(x),\,\widehat{\Gamma}_{\nu}(x)\} &= 2\,g_{\mu\nu}(x)\,I_{256},\\[2pt]
g^{\mu\nu}(x)\,g_{\nu\sigma}(x) &= \delta^{\mu}{}_{\sigma}.
\end{aligned}
$$

  The script prints: “Gamma matrices … anticommutation relation confirmed OK” and “Mixed anticommutation relation OK”.

• The tests use an explicit constant metric with off-diagonal entries (e.g.,$g_{02}=(1/10)^2$,$g_{23}=1/20)^2$, and scaled diagonals $g_{11}=g_{22}=g_{33}=(100/99)^2$), demonstrating that the matrix formalism remains consistent beyond diagonal metrics. Representative pairs reduce to  “scalar × identity” for identical index pairs and to the zero matrix for different pairs, as expected.

• Practically, Ver.6 constructs $\widehat{\Gamma}_{\nu}(x)$ directly from local metric inputs (no spin connection) and documents numeric checks that prevent spurious $g^2$ double counting in the algebra.