Quantum Field Theory for the Gifted Amateur

Book Preface

Quantum field theory is arguably the most far-reaching and beautiful physical theory ever constructed. It describes not only the quantum vacuum, but also the stuff that nature replaces it with. Aspects of quantum field theory are also more stringently tested, as well as verified to greater precision, than any other theory in physics. The subject nevertheless has a reputation for difficulty which is perhaps well-deserved; its practitioners not only manipulate formidable equations but also depict physical processes using a strange diagrammatic language consisting of bubbles, wiggly lines, vertices, and other geometrical structures, each of which has a well defined quantitative significance. Learning this mathematical and geometrical language is an important initiation rite for any aspiring theoretical physicist, and a quantum field theory graduate course is found in most universities, aided by a large number of weighty quantum field theory textbooks. These books are written by professional quantum field theorists and are designed for those who aspire to join them in that profession. Consequently they are frequently thorough, serious minded and demand a high level of mathematical sophistication.

The motivation for our book is the idea that quantum field theory is too important, too beautiful and too engaging to be restricted to the professionals. Experimental physicists, or theoretical physicists in other fields, would benefit greatly from knowing some quantum field theory, both to understand research papers that use these ideas and also to comprehend and appreciate the important insights that quantum field theory has to offer. Quantum field theory has given us such a radically different and revolutionary view of the physical world that we think that more physicists should have the opportunity to engage with it.

The problem is that the existing texts require far too much in the way of advanced mathematical facility and provide too little in the way of physical motivation to assist those who want to learn quantum field theory but not to be professional quantum field theorists. The gap between an undergraduate course on quantum mechanics and a graduate level quantum field theory textbook is a wide and deep chasm, and one of the aims of this book is to provide a bridge to cross it. That being said, we are not assuming the readers of this are simple-minded folk who can be fobbed off with a trite analogy as a substitute for mathematical argument. We aim to introduce all the maths but, by using numerous worked examples and carefully worded motivations, to smooth the path for understanding in a manner we have not found in the existing books.

We have chosen this book’s title with 1After all, with the number of chapters great care.1 Our imagined reader we ended up including, we could have called it ‘Fifty shades of quantum field theory’. is an amateur, wanting to learn quantum field theory without (at least initially) joining the ranks of professional quantum field theorists; but (s)he is gifted, possessing a curious and adaptable mind and willing to embark on a significant intellectual challenge; (s)he has abundant curiosity about the physical world, a basic grounding in undergraduate physics, and a desire to be told an entertaining and intellectually stimulating story, but will not feel patronized if a few mathematical niceties are spelled out in detail. In fact, we suspect and hope that our book will find wide readership amongst the graduate trainee quantum field theorists who will want to use it in conjunction with one of the traditional texts (for learning most hard subjects, one usually needs at least two books in order to get a more rounded picture).

One feature of our book is the large number of worked examples, which are set in slightly smaller type. They are integral to the story, and flesh out the details of calculations, but for the more casual reader the guts of the argument of each chapter is played out in the main text. To really get to grips with the subject, the many examples should provide transparent demonstrations of the key ideas and understanding can be confirmed by tackling the exercises at the end of each chapter. The chapters are reasonably short, so that the development of ideas is kept at a steady pace and each chapter ends with a summary of the key ideas introduced.

Though the vacuum plays a big part in the story of quantum field theory, we have not been writing in one. In many ways the present volume represents a compilation of some of the best ideas from the literature and, as a result, we are indebted to these other books for providing the raw material for many of our arguments. There is an extensive list of further reading in Appendix A where we acknowledge our sources, but we note here, in particular, the books by Zee and by Peskin and Schroeder and their legendary antecedent: the lectures in quantum field theory by Sidney Coleman. The latter are currently available online as streamed videos and come highly recommended. Also deserving of special mention is the text by Weinberg which is ‘a book to which we are 2T.S. Eliot on Ulysses. all indebted, and from which none of us can escape.’2 It is a pleasure to acknowledge the help we have received from various sources in writing this book. Particular mention is due to S¨onke Adlung at Oxford University Press who has helped steer this project to completion. No authors could wish for a more supportive editor and we thank him, Jessica White and the OUP team, particularly Mike Nugent, our eagle-eyed copy editor. We are very grateful for the comments and corrections we received from a number of friends and colleagues who kindly gave up their time to read drafts of various chapters: Peter Byrne, Claudio Castelnovo, John Chalker, Martin Galpin, Chris Maxwell, Tom