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# 1.1: Poincare Algebra defined

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.