$$\require{cancel}$$

# Introductory Remarks to Lorentz Invariant Systems

In order to describe a Lorentz invariant physical system using quantum mechanics one must:

1. specify a set of fundamental dynamical variables for the system;
2. specify the fundamental algebra of the set of fundamental dynamical variables
3. select a complete set of compatible observables for the system;
4. specify the Hilbert space of the system through spectral resolution of the complete set of compatible observables;
5. determine the Poincare generators $$H$$, $$\vec{P}$$, $$\vec{J}$$, $$\vec{K}$$ for the system in terms of the fundamental dynamical variables.

For convenience we give some elements of relativistic quantum mechanics in Section 1.1. We give a number of examples of Lorentz invariant systems in this volume of QI,B. We follow the above steps in each case. This procedure differs from the historical one for the Dirac particle discussed in Chapter 4 but it yields all the usual results. We consider a single spinless particle in Chapter 2, a particle with spin in Chapter 3, a Dirac particle in Chapter 4, a system of particle with spin in Chapter 5 and a simple system involving particle creation and annihilation in Chapter 6. Lists of selected reference books, journal articles and theses follow Chapter 6.