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Physics LibreTexts

1.1: Poincare Algebra defined

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  • In order to desc¡ibe a physical system which is Lorentz invaria¡t one must construct from the fundamental dynamical va¡iables for the system ten Hermitian operators: \(H\), \(P^{j}\), \(J^i\), K^j}\) where (where \(j = 1,2,3\)) satisfying

    \[ P^j, P^k]=0\]

    \[ P^j, H]=0\]

    \[[J^j,P^k] = i \hbar \epsilon_{ijk}P^l\]

    \[ J^j, H]=0\]

    \[[J^j,J^k] = i \hbar \epsilon_{ijk}J^l\]

    where \(\hbar = h/2\pi\), \(h\) is Planck's constant, \(c\) is the speed of light, \(\delta_{ij}\) is the Kronecker delta symbol and \(\epsilon_ijl\) is the Levi-Civita permutation symbol.