Relativistic Quantum Mechanics

The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis is given to those elements of the formalism which can be carried on  to Relativistic Quantum Fields , which underpins the theoretical framework of high energy particle physics.

We begin with a brief summary of special relativity, concentrating on 4-vectors and spinors. One-particle states and their Lorentz transforma- tions follow, leading to the Klein–Gordon and the Dirac equations for probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3.

Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field.

Special Relativity

Einstein’s  special relativity is a necessary and fundamental part of any formalism of particle physics.  We  begin with its brief summary.  For  a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4). Here we are only going to present conclusions without derivations avoiding group theory language and aiming at a presentation of key concepts at a qualitative level. A chapter about spinors in (5) is recommended. 

The basic elements of special relativity are 4-vectors (or contravariant

4-vectors) like a 4-displacement² x^µ = (t, x) = (x⁰, x¹, x², x³) = (x⁰, xⁱ) or a 4-momentum p^µ = (E, p) = (p⁰, p¹, p², p³) = (p⁰, pⁱ). 4-vectors have real components and form a vector space. There is a metric tensor g_µν = g^µν which is used to form a dual space to the space of 4-vectors.

This dual space is a vector space of linear functionals, known as 1-forms (or  covariant 4-vectors), which  act on  4-vectors.  For  every 4-vector x^µ,