Dirac, flat and curved

Time as a fourth generator forces 4 × 4 (1928). In 1929 Weyl and Fock–Ivanenko bent it: a tetrad e writes the metric as g = eᵀηe and the matrices become Γμ = eaμγa. Space, time and gravity in one line:

(Γ(p) − m)(Γ(p) + m) = (g(p, p) − m²) · 1

Dirac: the sixteen anticommutators

Entry μν is {γμ, γν} computed from the 4 × 4 matrices: 2ημν·1. Click one.


    


    

      
Bend the frame (1929)

      
Move the tetrad. The metric g = eᵀηe changes; {Γμ, Γν} − 2gμν stays zero.

      
      
      
      
      
      
gμν =

      
max |{Γμ, Γν} − 2gμν| = 
        det e = 

      
A pure boost leaves g = η: it changes the frame, not the geometry.

    


    

      
The square root of the metric

      
Drag p = (p⁰, p¹). On the shell g(p, p) = m² (blue) Γ(p) − m is singular:
        the Dirac equation has solutions there and only there. Grey: the flat shell.

      
      
      
g(p, p) = det(Γ(p) − m) = 
        (g(p, p) − m²)² = 

      
Γ(p)² − g(p, p)·1: max entry 

    


    

      
Gravity does not move the 4

      
Rank of the 16 ordered monomials Γμ₁⋯Γμₖ, computed from the current tetrad.
        Sixteen independent matrices need n² ≥ 16, so n ≥ 4. Push the z-scale to 0 (a singular tetrad) and it drops.

      
    

  


  
Lean (nrs3-pauli-dirac):
    1928, no_dirac_two_by_two and isLeast_dim (the least dimension is 4);
    1929, IsClifford.transform, IsClifford.symbol_mul_self (Γ(p)² = g(p, p)·1),
    IsClifford.dirac_factorization, tetrad_isClifford and Dirac1929.isLeast_dim
    (for every invertible tetrad the least dimension is still 4). The metric is a fixed background at one point:
    nothing here says how matter curves spacetime.