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- https://phys.libretexts.org/Under_Construction/Map%3A_Quantum_Leaps_and_Bounds_(McMillan)/IV%3A_Relativistic_Quantum_Mechanics/1%3A_Introduction_to_the_Poincare_Algebra/1.0%3A_Prelude_to_Poincare_AlgebraTo describe a Lorentz invariant physical system using quantum mechanics it is necessary to determine the Poincare generators of the system in terms of the fundamental dynamical variables of the system...To describe a Lorentz invariant physical system using quantum mechanics it is necessary to determine the Poincare generators of the system in terms of the fundamental dynamical variables of the system. In this chapter we present and comment on the the Poincare generators and the Poincare Algebra.
- https://phys.libretexts.org/Under_Construction/Map%3A_Quantum_Leaps_and_Bounds_(McMillan)/IV%3A_Relativistic_Quantum_Mechanics/1%3A_Introduction_to_the_Poincare_AlgebraTo describe a Lorentz invariant physical system using quantum mechanics it is necessary to determine the Poincare generators of the system in terms of the fundamental dynamical variables of the system...To describe a Lorentz invariant physical system using quantum mechanics it is necessary to determine the Poincare generators of the system in terms of the fundamental dynamical variables of the system. In this chapter we present and comment on the the Poincare generators and the Poincare Algebra. Derivations and some definitions are given later: a Lorentz invariant physical system and Poincare transformations are defined and discussed in detail elsewhere.
- https://phys.libretexts.org/Under_Construction/Map%3A_Quantum_Leaps_and_Bounds_(McMillan)/IV%3A_Relativistic_Quantum_Mechanics/1%3A_Introduction_to_the_Poincare_Algebra/1.1%3A_Poincare_Algebra_definedIn 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, Pj, Ji, K^j}\) ...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, Pj, Ji, K^j}\) where (where j=1,2,3) satisfying [Jj,Pk]=iℏϵijkPl where ℏ=h/2π, h is Planck's constant, c is the speed of light, δij is the Kronecker delta symbol and ϵijl is the Levi-Civita permutation symbol.