Introduction

In our discussion of quantum field theory, we have so far focused on a nonrelativistic treatment of quantum mechanics. Quantum fields, however, are inherently relativistic objects, and we need a consistent framework to describe spacetime. This is where special relativity comes in.

This will assume that the reader at least has a basic understanding of special relativity. As a review, we will begin with Galilean relativity, which is a historical precursor to special relativity. Then, through a series of thought experiments, we will arrive at special relativity. It is described by the Lorentz transformations, which consists of three boosts and three rotations. One important aspect of Lorentz transformations in the context of quantum field theory is that all physical laws must be invariant under these transformations. We also discuss Minkowski geometry, including its metric, hyperbolic geometry, and the light cone. Finally, we attempt to make quantum mechanics relativistic by introducing the Klein-Gordon equation (which will ultimately fail, and we see why we need a field theoretic approach).

With the review of special relativity complete, we will then discuss more advanced topics. We will begin with some group theory, including the Poincaré group and the Lorentz group. Both of these are Lie groups, which are groups that are also smooth manifolds. The Poincaré group is the group of all isometries of Minkowski spacetime, while the Lorentz group is the group of all Lorentz transformations. Related to this are Weyl and Dirac spinors, which can be thought of as representations of the Lorentz group.

Lastly, we will touch on general relativity, which is a relativistic, classical field theory of gravity. This will involve a discussion of differential geometry, including the metric tensor, geodesics, and the covariant derivative. The first part will be a derivation of the Einstein field equations. We choose to use an approach involving the Bianchi identities instead of the Einstein-Hilbert action (which will be discussed in the next chapter). Then, we discuss a few solutions to the equations, such as the Schwarzschild metric, the Kerr metric, and the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The Schwarzschild metric describes the spacetime around a spherically symmetric, non-rotating mass, while the Kerr metric describes the spacetime around a rotating mass. The FLRW metric describes a homogeneous and isotropic universe, which is the basis of modern cosmology.

It might initially seem unusual to discuss general relativity in a quantum field theory course. However, it actually proves very useful; concepts in differential geometry are extensively used in quantum field theory. Furthermore, learning both theories essentially constitutes a full understanding of modern theoretical physics. This includes understanding why they are currently incompatible, and why we need a theory of quantum gravity.