1 - Relativistic Quantum Mechanics

Summary

Uniting the operator and state-space formalism of quantum mechanics with special relativity through a unitary representation of the Poincaré group.

Introduction

Chapters 1 to 5 constitute the first part of this book. They develop the theory of the scalar field from its roots in special relativity and quantum mechanics to its fruits in cross sections and decay rates. The technique of quantization developed here will be applied in the second part — Chapters 6 to 9 — to spinor and vector fields.

Taking quantum mechanics, with its formalism of state space, Hamiltonian, and observables, together with relativity, with its emphasis on invariance under Lorentz transformations, as the two major pillars or principles in our understanding of particle physics, the purpose of this chapter is to introduce a framework in which both principles coexist.

Section 1.1 clarifies the concept of a state space, putting physical states, position eigenstates, and momentum eigenstates into proper relationship. Section 1.2 takes the first step towards a relativistic quantum theory by promoting the energy and momentum observables into a Lorentz vector. Section 1.3 uses this vector to construct a unitary representation of translations on state space, Section 1.4 uses an independent construction to build a unitary representation of the Lorentz group, and Section 1.5 shows that these two representations determine a unitary representation of the Poincaré group.