Quantum Field Theory in a Nutshell

Second Edition

Anthony Zee

• Publisher: Princeton University Press
• Year: 2010
• ISBN:
  9781400835324
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Quantum Theory

A fully updated edition of the classic text by acclaimed physicist A. Zee Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available.This expanded edition features several additional chapters, as well as an entirely new section describing recent developments in quantum field theory such as gravitational waves, the helicity spinor formalism, on-shell gluon scattering, recursion relations for amplitudes with complex momenta, and the hidden connection between Yang-Mills theory and Einstein gravity. Zee also provides added exercises, explanations, and examples, as well as detailed appendices, solutions to selected exercises, and suggestions for further reading.

• The most accessible and comprehensive introductory textbook available
• Features a fully revised, updated, and expanded text
• Covers the latest exciting advances in the field
• Includes new exercises
• Offers a one-of-a-kind resource for students and researchers

Leading universities that have adopted this book include:

• Arizona State University
• Boston University
• Brandeis University
• Brown University
• California Institute of Technology
• Carnegie Mellon
• College of William & Mary
• Cornell
• Harvard University
• Massachusetts Institute of Technology
• Northwestern University
• Ohio State University
• Princeton University
• Purdue University - Main Campus
• Rensselaer Polytechnic Institute
• Rutgers University - New Brunswick
• Stanford University
• University of California - Berkeley
• University of Central Florida
• University of Chicago
• University of Michigan
• University of Montreal
• University of Notre Dame
• Vanderbilt University
• Virginia Tech University

Part I Motivation and Foundation

I.1 Who Needs It?

Who needs quantum field theory?

Quantum field theory arose out of our need to describe the ephemeral nature of life.

No, seriously, quantum field theory is needed when we confront simultaneously the two great physics innovations of the last century of the previous millennium: special relativity and quantum mechanics. Consider a fast moving rocket ship close to light speed. You need special relativity but not quantum mechanics to study its motion. On the other hand, to study a slow moving electron scattering on a proton, you must invoke quantum mechanics, but you don’t have to know a thing about special relativity.

It is in the peculiar confluence of special relativity and quantum mechanics that a new set of phenomena arises: Particles can be born and particles can die. It is this matter of birth, life, and death that requires the development of a new subject in physics, that of quantum field theory.

Let me give a heuristic discussion. In quantum mechanics the uncertainty principle tells us that the energy can fluctuate wildly over a small interval of time. According to special relativity, energy can be converted into mass and vice versa. With quantum mechanics and special relativity, the wildly fluctuating energy can metamorphose into mass, that is, into new particles not previously present.

Write down the Schrödinger equation for an electron scattering off a proton. The equation describes the wave function of one electron, and no matter how you shake and bake the mathematics of the partial differential equation, the electron you follow will remain one electron. But special relativity tells us that energy can be converted to matter: If the electron is energetic enough, an electron and a positron (“the antielectron”) can be produced. The Schrödinger equation is simply incapable of describing such a phenomenon. Nonrelativistic quantum mechanics must break down.

You saw the need for quantum field theory at another point in your education. Toward the end of a good course on nonrelativistic quantum mechanics the interaction between radiation and atoms is often discussed. You would recall that the electromagnetic field is treated as a field; well, it is a field. Its Fourier components are quantized as a collection of harmonic oscillators, leading to creation and annihilation operators for photons. So there, the electromagnetic field is a quantum field. Meanwhile, the electron is treated as a poor cousin, with a wave function

(x) governed by the good old Schrödinger equation. Photons can be created or annihilated, but not electrons. Quite aside from the experimental fact that electrons and positrons could be created in pairs, it would be intellectually more satisfying to treat electrons and photons, as they are both elementary particles, on the same footing.

Figure I.1.1

So, I was more or less right: Quantum field theory is a response...