diff --git a/Physlib/Relativity/PauliMatrices/Relations.lean b/Physlib/Relativity/PauliMatrices/Relations.lean index 806a46097..d272c11bb 100644 --- a/Physlib/Relativity/PauliMatrices/Relations.lean +++ b/Physlib/Relativity/PauliMatrices/Relations.lean @@ -18,6 +18,12 @@ for the Pauli four-vectors. The current way this result is proved is by using tensor tree manipulations. There is likely a more direct path to this result. +## References + +* Dreiner, Haber and Martin, *Two-component spinor techniques and Feynman rules for quantum + field theory and supersymmetry*, arXiv:0812.1594, equations (2.54) and (2.55), in the + conventions `g = diag(+1, -1, -1, -1)` and `ε⁰¹²³ = +1`. [ref: Dreiner:2008tw] + -/ @[expose] public section @@ -164,9 +170,9 @@ lemma leviCivita_mul_pauliDual : · decide · rfl -/-- Equation (2.26), the three-Pauli identity -`σ^μ barσ^ν σ^ρ = g^{μν} σ^ρ - g^{μρ} σ^ν + g^{νρ} σ^μ - + i ε^{μνρκ} σ_κ`, with barred and lowered forms expressed through index dualization `τ`. -/ +/-- The three-Pauli identity +`σ^μ barσ^ν σ^ρ = g^{μν} σ^ρ - g^{μρ} σ^ν + g^{νρ} σ^μ + i ε^{μνρκ} σ_κ`, +with barred and lowered forms expressed through index dualization `τ`. -/ lemma pauliContr_mul_pauliContrDown_mul_pauliContr : ({ σ^^^ | μ α β ⊗ σ^^^ | ν τ(α') τ(β) ⊗ σ^^^ | ρ α' β' = ((((η | μ ν ⊗ σ^^^ | ρ α β') + (-((η | μ ρ ⊗ σ^^^ | ν α β')))) @@ -191,9 +197,9 @@ lemma pauliContr_mul_pauliContrDown_mul_pauliContr : ({ apply Physlib.RatComplexNum.toComplexNum_eq_add_neg_add_add_iff.mpr decide +revert +kernel -/-- Equation (2.27), the conjugate three-Pauli identity -`barσ^μ σ^ν barσ^ρ = g^{μν} barσ^ρ - g^{μρ} barσ^ν + g^{νρ} barσ^μ - - i ε^{μνρκ} barσ_κ`, with barred and lowered forms expressed through index dualization `τ`. -/ +/-- The conjugate three-Pauli identity +`barσ^μ σ^ν barσ^ρ = g^{μν} barσ^ρ - g^{μρ} barσ^ν + g^{νρ} barσ^μ - i ε^{μνρκ} barσ_κ`, +with barred and lowered forms expressed through index dualization `τ`. -/ lemma pauliContrDown_mul_pauliContr_mul_pauliContrDown : ({ σ^^^ | μ τ(α) τ(β) ⊗ σ^^^ | ν α β' ⊗ σ^^^ | ρ τ(α') τ(β') = ((((η | μ ν ⊗ σ^^^ | ρ τ(α') τ(β)) diff --git a/docs/references.bib b/docs/references.bib index dee4808c2..ded357f7d 100644 --- a/docs/references.bib +++ b/docs/references.bib @@ -33,8 +33,7 @@ @article{Dreiner:2008tw doi = {10.1016/j.physrep.2010.05.002}, journal = {Phys. Rept.}, pages = {1-196}, - year = {2010}, - note = {not currently cited by any module; kept for future use} + year = {2010} } @article{Lohitsiri:2019fuu,