This chapter introduces the formal “second quantization” method for bosons in quantum field theory. It is shown that phonons (sound particles) and photons (light particles) are simple extensions of the physics of a spring-like oscillator. The connection of boson states to classical waves is shown in a discussion of “coherent states.”

Wigner’s idea, that to each elementary particle is associated an irreducible representation of the Poincare group gives fundamental importance to these representations. They are non-trivial mathematical objects. We strive to give a mathematically sound and complete description of the physically relevant representations, and the multiple ways they can be presented, while avoiding the pitfall of relying on advanced representation theory. The representations corresponding to massive particles depend, besides the mass, on a single non-negative integer which corresponds to the spin of the particle. The representations which correspond to massless particles depend on an integer, the helicity, which is a property somewhat similar to the spin. We investigate that action of parity and the operation of taking a “mirror image” of a particle. Finally we provide a brief account of Dirac’s equation.

Having described motion, we can now explain it. We introduce the conserved 4-momentum, and with it the ideas of energy-momentum, conserved mass, and scattering.

The laws of Newtonian mechanics have to be changed to be consistent with the principles of special relativity introduced in the previous chapter. This chapter describes special relativistic mechanics from a four-dimensional, spacetime point of view. Newtonian mechanics is an approximation to this mechanics of special relativity that is appropriate when motion is at speeds much less than the velocity of light in a particular inertial frame. We begin with the central idea of four-vectors, defined as a directed line segment in four-dimensional flat spacetime, and how to manipulate them. Special relativistic kinematics shows how four-vectors are used for describing the motion of a particle in spacetime terms. Concepts such as four-velocity and four-momentum are introduced. We will posit the principle of extremal proper time for a free particle in curved spacetime, and use it to derive the free particle equation of motion.

This chapter covers the early years of quantum theory, a time of guesswork, inspired by problems presented by the properties of atoms and radiation and their interaction. Later, in the 1920s, this struggle led to the systematic theory known as quantum mechanics, the subject of Chapter 5. Quantum mechanics started with the problem of understanding radiation in thermal equilibrium at a non-zero temperature. It was not possible to make progress in applying quantum ideas to atoms without some understanding of what atoms are. The growth of this understanding began with the discovery of radioactivity.

There are no restrictions on how many bosons can occupy a single particle state, which has important consequences for their thermodynamic behaviour.Photons, quanta of the electromagnetic field, can be viewed as bosons with zero chemical potential, which allows the derivation of the thermodynamic properties of blackbody radiation, including the Stefan--Boltzmann Law.Non-interacting bosons with non-zero chemical potential can exhibit Bose--Einstein condensation at low temperatures, and interacting bosons may form a superfluid state.Low energy excitations in materials -- lattice vibrations (phonons) and spin waves (magnons) -- also behave as bosons, and are important for understanding the specific heat of materials at low temperatures.Of particular note is the Debye model which gives a simple account of the contributions of phonons to specific heat.