Quantum Field Theory

This is a book on quantum field theory. The quantum mechanics solution for the simple harmonic oscillator is reviewed at the outset. In the context of quantum field theory this will be referred to as the first quantization solution because we are going to re-use this solution in a new way (an infinite number of times) to arrive at "second quantization" that will describe a quantized field. Second quantization provides a very convenient and simple pathway to having a quantized field. The proof of the validity of second quantization following from a more rigorous canonical quantization formulation is then given. Many situations of interest lead to calculations that become impractical in the canonical quantization formalism, however, so a transition is then made to the Path Integral formalism, where Path Integral quantization, renormalization, and renormalization group methods are described.

A large number of problems and solutions are provided for the various descriptions throughout, especially those that are fundamental to the construction of quantum electrodynamics in later sections. Thus, the book provides complete material for introductory and advanced graduate courses in quantum field theory, as well as special applications to the Standard Model and possible extensions (circa 2024), and to quantum field theory on curved space-time.

The second part of the book describes thermal quantum field theory and quantum field theory on curved space-time. For thermal quantum field theory, upon shifting (Wick rotating) from real time to imaginary time the statistical mechanical partition function for the system is obtained. Thus, a connection between 'thermality' and complex structure, in the time dimension, is indicated. For quantum field theory on curved space-time, we arrive at an early analysis of black hole thermodynamics, where space-time curvature gives rise to thermality (a thermal vacuum) and particle production effects. Black hole thermality was revealed in Hawking radiation, due to the causal boundary at the horizon. Such thermality is even seen in flat space-time if causal boundaries are induced, such as in the case of an accelerated observer. These topics, with many worked examples, will be described in detail.