1.0: Prelude to Poincare Algebra

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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 in Chapter 3 and the Poincare Algebra is derived in Chapter 4.

The Poincare generators are a set of ten Hermitian operators: the Hamiltonian and the three components of the total momentum, total angular momentum and Lorentz boost for the system. As discussed in Chapter 3 these operators generate time translations, spatial displacements, rotations and Lorentz boosts, The Poincare Algebra is a set of commutation relations satisfied by the Poincare generators; these commutation relations are the basic equations of relativistic quantum mechanics.

We also present and comment on the Galilei generators and Galilei Algebra which are appropriate for describing a Galilei invariant physical system. The Galilei generators are the Hamiltonian and the three components of the total momentum, total angular momentum and Galilei booster for the system. These operators generate time translations, spatial displacements, rotations and Galilei boosts. The Galilei Algebra is a set of commutation relations satisfied by the Galilei generators; these commutation relations are the nonrelativistic approximation of the Poíncare Algebra and are the basic equations of nonrelativistic quantum mechanics. Nonrelativistic quantum mechanics is arrived at as a nonrelativistic approximation to relativistic quantum mechanics.