Dirac Equation in (-,++++) Notation

[pleasehelpmeno]

Just to clarify in the dirac equation $(i\gamma^{\mu}\partial_{\mu} -m)\psi=0$

Is it equal to $(-i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i} -m)\psi=0$ in (-,++++) notation?

 

[Bill_K]

No, for a repeated index with one index up and one index down, just sum:

γ^μ∂_μ = γ⁰∂₀ + γ¹∂₁ + γ²∂₂ + γ³∂₃

You only need to use the Minkowski metric if both indices are up, or both indices down:

η^μνγ_μ∂_ν = - γ₀∂₀ + γ₁∂₁ + γ₂∂₂ + γ₃∂₃

or η_μνγ^μ∂^ν = - γ⁰∂⁰ + γ¹∂¹ + γ²∂² + γ³∂³