“All of physics is either impossible or trivial.
It is impossible until you understand it, and then it becomes trivial”
This book is
1. an attempt to make learning quantum field theory (QFT) as easy as is humanly possible,
2. intended, first and foremost, for new students of QFT, and
3. an introduction to only the most fundamental and central concepts of the theory.
It is not
2. an exhaustive treatment of QFT,
3. concise (lacking extensive explanation),
4. written for seasoned practitioners in the field, or
5. a presentation of the latest, most modern approach to it.
Students planning a career in field theory will obviously have to move on to more advanced texts, after they digest the more elementary material presented herein. This book is intended to provide a solid foundation in the most essential elements of the theory, nothing more.
In my own teaching experience, and in the course of researching pedagogy, I have come to see that “learning” has at its basis a fundamental three-in-one structure. The wholeness of learning is composed of
i) the knowledge to be learned,
ii) the learner, and
iii) the process of learning itself.
It seems unfortunate that physics and physics textbooks have too often been almost solely concerned with the knowledge of physics and only rarely concerned with those who are learning it or how they could best go about learning. However, there are signs that this situation may be changing somewhat, and I hope that this book will be one stepping stone in that direction.
In writing this book, I have repeatedly tried to visualize the learning process as a new learner would. This viewpoint is one we quickly lose when we, as teachers and researchers, gain familiarity with a given subject, and yet it is a perspective we must maintain if we are to be effective educators. To this end, I have solicited guidance and suggestions from professional educators (those who make learning and education, per se, their central focus in life), and more importantly, from those studying QFT for the first time. In addition, I have used my own notes, compiled when I was first studying the theory myself, in which I carefully delineated ways the subject could be presented in a more student-friendly manner. In this sense, the text incorporates “peer instruction”, a pedagogic tool of recognized, and considerable, merit, wherein students help teach fellow students who are learning the same subject.
It is my sincere hope that the methodologies I have employed herein have helped me to remain sympathetic to, and in touch with, the perspective of a new learner. Of course, different students find different teaching techniques to have varying degrees of transparency, so there are no hard and fast rules. However, I do believe that most students would consider many of the following principles, which I have employed in the text, to be of pedagogic value.
1) Brevity Avoided
Conciseness is typically a horror for new students trying to fathom unfamiliar concepts. While it can be advantageous in some arenas, it is almost never so in education. Unfortunately, being succinct, has, in scientific/technical circles, become a goal unto itself, extending even into pedagogy an area for which it was never suited.
In this book, I have gone to great lengths to avoid conciseness and to present extensive explanations.
2) Holistic previews
The entire book, each chapter, and many sections begin with simple, non-mathematical overviews of the material to be covered. These allow the student to gain a qualitative understanding of the “big picture” before he or she plunges into the rigors of the underlying mathematics.
Doing physics is a lot like doing a jig-saw puzzle. We assemble bits and pieces into small wholes and then gradually merge those small wholes into greater ones, until ultimately we end up with the “big picture.” Seeing the picture on the puzzle box before we start has immense value in helping us put the whole thing together. We know the blue goes here, the green there, and the boundary of the two, somewhere in between. Without that picture preview to guide us, the entire job becomes considerably more difficult, more tedious, and less enjoyable. In this book, the holistic previews are much like the pictures on the puzzle boxes. The detail is not there, but the essence of the final goal is. These overviews should eliminate, or at least minimize, the “lost in a maze of equations” syndrome by providing a “birds-eye road map” of where we have come from, and where we are going. By so doing we not only will keep sight of the forest in spite of the trees, but will also have a feeling, from the beginning, for the relevance of each particular topic to the overriding structure of the wholeness of knowledge in which it is embedded.
3) Schematic diagram summaries (Wholeness Charts)
Enhancing the “birds-eye road map” approach are block diagram summaries, which I call Wholeness Charts, so named because they reveal in chart form the underlying connections that unite various aspects of a given theory into a greater whole. Unlike the chapter previews, these are often mathematical and contain considerable theoretical depth.
Learning a computer program line-by-line is immensely harder than learning it with a block diagram of the program, showing major sections and sub-sections, and how they are all interrelated. There is a structure underlying the program, which is its essence and most important aspect, but which is not obvious by looking directly at the program code itself.
The same is true in physics, where line-by-line delineation of concepts and mathematics corresponds to program code, and in this text, Wholeness Charts play the role of block diagrams. In my own learning experiences, in which I constructed such charts myself from my books and lecture notes, I found them to be invaluable aids. They coalesced a lot of different information into one central, compact, easy-to-see, easy-to-understand, and easy-to-reference framework.
The specific advantages of Wholeness Charts are severalfold.
First, in learning any given material we are seeking, most importantly, an understanding of the kernel or conceptual essence, i.e., the main idea(s) underlying all the text. A picture is worth a thousand words, and a Wholeness Chart is a “snapshot” of those thousand words.
Second, although the charts can summarize in-depth mathematics and concepts, they can be used to advantage even when reading through material for the first time. The holistic overview perspective can be more easily maintained by continual reference to the schematic as one learns the details.
Thirdly, comparison with similar diagrams in related areas can reveal parallel underlying threads running through seemingly diverse phenomena. (See, for example, Summary of Classical Mechanics Wholeness Chart 2-2 and Summary of Quantum Mechanics Wholeness Chart 2-5 in Chap. 2, Sects. 2.4 and 2.7.) This not only aids the learning process but also helps to reveal some of the subtle workings and unified structure inherent in Mother Nature.
Further, review of material for qualifying exams or any other future purpose is greatly facilitated. It is much easier to refresh one’s memory, and even deepen understanding, from one or two summary sheets, rather than time consuming ventures through dozens of pages of text. And by copying all of the Wholeness Charts herein and stapling them together, you will have a pretty good summary of the entire book.
Still further, the charts can be used as quick and easy-to-find references to key relations at future times, even years later.
4) Reviews of background material
In situations where development of a given idea depends on material studied in previous courses (e.g., quantum mechanics) short reviews of the relevant background subject matter are provided, usually in chapter introductory sections or later on, in special boxes separate from the main body of the text.
5) Only basic concepts without peripheral subjects
I believe it is of primary importance in the learning process to focus on the fundamental concepts first, to the exclusion of all else. The time to branch out into related (and usually more complex) areas is after the core knowledge is assimilated, not during the assimilation period.
All too often, students are presented with a great deal of new material, some fundamental, other more peripheral or advanced. The peripheral/advanced material not only consumes precious study time, but tends to confuse the student with regard to what precisely is essential (what he or she must understand), and what is not (what it would be nice if he or she also understood at this point in their development).
As one example, for those familiar with other approaches to QFT, this book does not introduce concepts appropriate to weak interactions, such as 4 theory, before students have first become grounded in the more elementary theory of quantum electrodynamics.
This book, by careful intention, restricts itself to only the most core principles of QFT. Once those principles are well in hand, the student should then be ready to glean maximum value from standard, more extensive, texts.
6) Optimal “return on investment” exercises
All too often students get tied up, for what seem interminable periods, working through problems from which minimum actual learning is reaped. Study time is valuable and spending it engulfed in great quantities of algebra and trigonometry is probably not its best use.
I have tried, as best I could, to design the exercises in this book so that they consume minimum time but yield maximum return. Emphasis has been placed on gleaning an understanding of concepts without getting mired down.
Later on, when students have become practicing researchers and time pressure is not so great, there will be ample opportunity to work through more involved problems down to every last minute algebraic detail. If they are firmly in command of the concepts and principles involved, the calculations, though often lengthy, become trivial. If, however, they never got quite grounded in the fundamentals because study time was not efficiently used, then research can go slowly indeed.
7) Many small steps, rather than fewer large ones
Professional educators have known for some time now that learning progresses faster and more profoundly when new material is presented in small bites. The longer, more moderately sloped trail can get one to the mountaintop much more readily than the agonizing climb up the steep vertical face.
Unfortunately, from my personal experience as a student, it often seemed like my textbooks were trying to take me up the steepest grade. I sincerely hope that those using this book do not have this experience. I have made every effort to include each and every relevant step in all derivations and examples.
8) Liberal use of simple concrete examples
Professional educators have also known for quite some time that abstract concepts are best taught by leading into them with simple, physically visualizable examples. Further, their understanding is deepened, broadened, and solidified with even more such concrete examples.
Some may argue that a more formal mathematical approach is preferable because it is important to have a profound, not superficial, understanding. While I completely agree that a profound understanding is essential, it is my experience that the mathematically rigorous introduction, more often than not, has quite the opposite result. (Ask any student about this.) Further, to know any field profoundly we must know it from all angles. We must know the underlying mathematics in detail plus we must have a grasp on what it all means in the real world, i.e., how the relevant systems behave, how they parallel other types of systems with which we are already familiar, etc. Since we have to cover the whole range of knowledge from abstract to physical anyway, it seems best to start with the end of the spectrum most readily apprehensible (i.e., the visualizable, concrete, and analogous) and move on from there.
This methodology is employed liberally in this book. It is hoped that so doing will ameliorate the “what is going on?” frustration common among students who are introduced to conceptually new ideas almost solely via routes heavily oriented toward abstraction and pure mathematics.
In this context it is relevant that Richard Feynman, in his autobiography, notes,
“I can’t understand anything in general unless I’m carrying along in my mind a specific example and watching it go....(Others think) I’m following the steps mathematically but that’s not what I’m doing. I have the specific, physical example of what (is being analyzed) and I know from instinct and experience the properties of the thing.”
I know from my own experience that I learn in the same way, and I have a suspicion that almost everyone else does as well. Yet few teach that way. This book is an attempt to teach in that way.
9) Margin overview notes
Within a given section of any textbook, one group of paragraphs can refer to one subject, another group to another subject. When reading material for the first time, not knowing exactly where one train of the author’s thought ends and a different one begins can oftentimes prove confusing. In this book, each new idea not set off with its own section heading is highlighted, along with its central message, by notations in the margins. In this way, emphasis is once again placed on the overview, the “big picture” of each topic, even on the subordinate levels within sections and subsections.
Additionally, the extra space in the margins can be used by students, to make their own notes and comments. In my own experience as a student I found this practice to be invaluable. My own remarks written in a book are, almost invariably, more comprehensible to me when reviewing later for exams or other purposes than are those of the author.
10) Definitions and key equations emphasized
As a student, I often found myself encountering a term that had been introduced earlier in the text, but not being clear on its exact meaning, I had to search back through pages clumsily trying to find the first use of the word. In this book, new terminology is underlined when it is introduced or defined, so that it “jumps out” at the reader later when trying to find it again.
In addition, key equations the ones students really need to know have borders around them.
11) Non use of terms like “obvious”, “trivial”, etc.
The text avoids use of emotionally debilitating terms such as “obvious”, “trivial”, “simple”, “easy”, and the like to describe things that may, after years of familiarity, be easy or obvious to the author, but can be anything but that to the new student. (See “A Nontrivial Manifesto” by Matt Landreman, Physics Today, March 2005, 52-53.)
The job I have undertaken here has been a challenging one. I have sought to produce a physics textbook which is relatively lucid and transparent to those studying quantum field theory for the first time.
I suspect many physics professors will consider the book too verbose and too simple. I only ask them to try it and let their students be the judges. The proof will be in the pudding - if comprehension comes more quickly and more deeply, then the approach taken here will be vindicated.
Good luck to the new students of quantum field theory! I hope their studies are personally rewarding and professionally fruitful.
Robert D. Klauber
Quantum field theory takes off where the following subjects end. Those beginning this book should be reasonably well versed in them, at the levels described below.
An absolute minimum of two undergraduate quarters. Far more preferably, an additional two graduate level quarters. Some exposure to relativistic quantum mechanics would be advantageous, but is not necessary. Optimal level of proficiency: Eugen Merzbacher’s Quantum Mechanics (John Wiley) or a similar book.
A semester at the graduate level. Topics covered should include the Lagrangian formulation (for particles, and importantly, also for fields), the Legendre transformation, special relativity, and classical scattering. A familiarity with Poisson brackets would be helpful. Optimal level of proficiency: Herbert Goldstein’s Classical Mechanics (Addison-Wesley) or similar.
Two quarters at the undergraduate level plus two graduate quarters. Areas studied should comprise Maxwell’s equations, conservation laws, e/m wave propagation, relativistic treatment, Maxwell’s equations in terms of the four potential. Optimal level of proficiency: John David Jackson’s Classical Electrodynamics (John Wiley) or similar.
Advantageous but not essential, as it is covered in Chap. 2 appendix: Exposure to covariant and contravariant coordinates, and metric tensors, for orthogonal 4D systems, at the level found in Jackson’s chapters on special relativity.