1. orthodox,
2. an exhaustive treatment of quantum field theory,
3. meant for those already skilled in the field, or
4. a presentation of the latest, most modern approach to quantum field theory (QFT).
1. an experiment in pedagogy,
2. intended for new students of QFT,
3. an attempt to present material in the easiest possible way to assimilate, and
4. an introduction to only the most fundamental and central concepts in QFT.
The primary emphasis of this text is pedagogic. The objective has been simply to make the subject matter as easy as is humanly possible to learn.
If a student finds material easier to grasp, he or she can learn more quickly, understand more rigorously to greater depth, and gain substantially more enjoyment and self-satisfaction from his or her efforts. If we can, in fact, make the learning process more efficient, time savings for both students and teachers alike can be invested with greater profit in additional research, administration, or even further study.
In an effort to achieve this objective, I have consulted with and emulated professional educators (those who, unlike myself and virtually all other technical textbook writers, make learning and education, per se, their central focus in life.) Additionally, I have solicited and continue to solicit guidance and suggestions from graduate students studying the specific topics treated herein. And perhaps most importantly, I have used my own notes, compiled when I was first studying QFT myself, in which I delineated ways in which the subject could be presented in a more student-friendly manner. In this sense, the text incorporates “peer instruction”, a teaching tool of recognized, and significant, merit.
In the course of doing all of this 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 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. It is my hope that this situation will be changing in the future - hopefully the near future - and that this text will be one small stepping stone in that direction.
From my side, in writing this book, I have repeatedly tried to visualize the presentation of material as a new student would. The student's viewpoint is one we quickly lose when we as teachers and researchers gain more and more familiarity with a given subject, and yet it is a perspective we must maintain if we are to be effective educators. It is my sincere hope that the methodologies I have employed herein have allowed me to remain sympathetic to, and in touch with, such a perspective.
However, different students consider different things pedagogic, so there are no hard and fast rules, and I do not insist that the teaching techniques incorporated herein, some of which are described below, are universally applicable. However, I do believe that most students would consider many of the following, which I have employed in the text, to be of pedagogic value.
1) Holistic previews
The entire book, each chapter, and many sections begin with simple holistic overviews of the material to be covered. This approach should allow the student to gain some 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. Hopefully, these overviews will 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.
2) Schematic diagram summaries (Unifying Charts)
Enhancing the “birds-eye road map” approach are subject area summaries in block diagram form called Unifying Charts. Unlike the chapter previews, these are mathematical and contain considerable theoretical depth. Their advantages are severalfold.
First, in learning any given material we are seeking, most importantly, an understanding of the kernel or conceptual essence - the main idea(s), so to speak. The numerous small parts are not nearly as significant as the whole and it is important to be able to visualize that whole readily. This is what the Unifying Chart schematic summaries do. (See the end of almost any chapter.) In one glance, one can see the meaning, development, and fundamental ideas contained within a given subject area. A picture is worth a thousand words and a Unifying Chart is a "snapshot" of those thousand words.
Second, the chart summaries can be used to advantage even when reading through the chapter for the first time. The holistic overview perspective can be more easily maintained by continual reference to the schematic at the end.
Thirdly, comparison with similar diagrams in related areas can reveal parallel underlying threads running through seemingly diverse phenomena. (See, for example, electroweak symmetries and strong symmetries Unifying Charts ? and ?, pages ? & ?.) This not only aids the learning process but also helps to reveal some of the subtle workings and unified structure inherent in Mother Nature as well.
Further, review of material for qualifying exams or any other future purpose is greatly facilitated. Refreshing of memory and even deepening of understanding can be garnered from one or two summary sheets rather than time consuming ventures through dozens of pages of text.
Still further, the charts can be used as quick and easy-to-find references to key relations at future times, even years hence.
3) 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 special “boxes” separate from the main body of the text.
4) Only basic concepts without peripheral subjects
I believe it of primary importance in learning 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. The peripheral material not only consumes precious study time, but serves 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).
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 the standard, more extensive, texts.
5) Optimal "return on investment" exercises
All too often students get tied up for what seems 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 in a fen of mathematical manipulation.
Later on, when you are practicing researchers and time pressure is not so great, you will have ample opportunity to work through more involved problems down to every last minute algebraic detail. If you are firmly in command of the concepts and principles involved, the calculations, though often lengthy, become trivial. If, however, you never got quite grounded in the fundamentals because study time was not efficiently used, then research can go slowly indeed.
6) 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. If you as a student find cases where this is not so, where you end up "head scratching" between two steps for quite some time, please write and so inform me. I will then supply the missing links in subsequent editions of the book. The next generation of students will thank you for doing this, and I will include the names of all those who make such contributions in significant measure in the forward of the next edition.
7) 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 to note that Richard Feynman, in his autobiography, says,
"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 almost no one teaches that way. This book is an attempt to teach in that way.
8) 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 and its central message is highlighted 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.
Note that with the previews before and summaries after the main text, we start with wholeness, end with wholeness, and in between cover the parts. Yet even when doing the parts we still, via the margin notes, keep an eye toward the whole.
Additionally, the extra space in the margins can be used by you the student to make your 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, nine times out of ten, more translucent to me when reviewing later for exams or other purposes than are those of the author.
9) 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 to the new student. (See “A Nontrivial Manifesto” by Matt Landreman, Physics Today, March 2005, 52-53.)
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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 filed 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.
In either case I invite comments and suggestions for improvement from students and professors alike. Your responses will be both the report card for my efforts and the key to more rapid and fulfilling learning experiences for those yet to come.
Good luck to the new students of quantum field theory. I hope your studies are rewarding and fruitful.
Robert D. Klauber