First Applications of Noether ’ s Theorem

and the last two (actually Lorentz transformations already include both) can be written x → x = ∑ ν Λ μ νx ν = Λνx ν , (using the Einstein summation convention), where the matrix Λ is a real matrix satisfying the pseudoorthogonality condition ΛμνgμρΛ ρ τ = gντ , which is required so that the length of a four-vector is preserved, x′ := xxμ = x . All together, this symmetry group is called the inhomogeneous Lorentz group, or Poincaré group.