Chapter 1: Introduction
I’m interested in learning things. I hope I can finally understand physics before I leave the earth. – Bill Clinton (MSNBC interview, Feb. 17, 2011)
Welcome to the world of physics — a world you may once have thought you would never enter. Perhaps you were told that present-day physics is too hard and so you didn’t try. Or perhaps you read a popular article or book and found the subject to be incomprehensible. Well, I don’t blame you; you are not the only one. The way modern physics is usually presented, it is incomprehensible. Despite the plethora of popular books on science, there are few people who can do more than mouth the words they have read.
And yet there is a theory that makes perfect sense and that can be understood by anyone. I am referring to Quantum Field Theory (QFT), a theory that has been overlooked or ignored in most books about physics. This theory, in its true sense of “no particles, only fields,” can return us to the good old days when every educated person had a conceptual grasp of science, without needing any mathematical ability.
The good old days. In 1666, when Isaac Newton discovered his law of gravity, people did not need to understand the calculus that Newton invented so that he could calculate the orbit of the moon. It was enough to know that there is a universal force that causes apples to fall to the ground and that this same force holds the moon in its orbit around the earth. There was nothing paradoxical about this concept; indeed it explained so many things that the poet Alexander Pope wrote as Newton’s epitaph:
Nature and Nature’s laws lay hid in night.
God said, “Let Newton be!” And all was light.
Two centuries later, James Clerk Maxwell developed equations that united electric and magnetic forces into a single field called (naturally) the electromagnetic (EM) field, and he showed that light is a travelling oscillation in this field. These concepts also could be grasped and understood by anyone, even those who could not solve Maxwell’s equations.
The discovery of atoms at the beginning of the 20th century posed a more difficult challenge. As Sir Arthur Eddington wrote, after referring to his “ordinary” writing table as Table No. 1:
Table No. 2 is my scientific table. It is a more recent acquaintance and I do not feel so familiar with it. It does not belong to the world previously mentioned – that world which spontaneously appears around me when I open my eyes… It is part of a world which in more devious ways has forced itself on my attention. My scientific table is mostly emptiness. Sparsely scattered in that emptiness are numerous electric charges rushing about with great speed; but their combined bulk amounts to less than a billionth of the bulk of the table itself. – A. Eddington (The Nature of the Physical World, 1927, p. ix)
Yet there was nothing in this new picture that is paradoxical or that violates common sense, and it didn’t take long for people to understand and accept atomic theory. This is an example of how, when evidence demands it, our intuitive picture of things can be replaced by a less intuitive picture that we accept because of common sense.
This pleasant state of affairs ended when Albert Einstein entered the scene. Einstein was without doubt the great light of 20th century physics. His contributions are immense and he is seen as the embodiment of ultimate wisdom. But it is also true that Einstein’s contributions caused much confusion:
…for all of his popular appeal and surface accessibility, Einstein also came to symbolize the perception that modern physics was something that ordinary laymen could not comprehend, “the province of priest-like experts”. — W. Isaacson (I2007, p. 6)
The situation was so bad that poet J.C. Squire added the following lines to Pope’s couplet (A2004, p. 253):
It did not last: the Devil, howling “Ho!
Let Einstein be,” restored the status quo.
The reasons for the confusion – this curtain of chaos cast over physics — are three theories for which Einstein was either wholly or partly responsible:
 It should be noted, however, that Einstein was only human and made his share of errors. A little-known but glaring example, never admitted by Einstein, occurred in 1915 when he threw out valid experimental data in order to obtain agreement with a theory that was later found to be incorrect (P1982, p. 248).
Special Relativity, General Relativity, and Quantum Mechanics (QM). Of course Einstein did not develop the theory of QM — in fact, he eventually repudiated it. However, in his 1905 Nobel-prize paper he introduced its central concept: wave-particle duality. While Einstein may not be the father of QM, he surely is its grandfather.
I refer to these theories as “enigmas” because they involve concepts that just don’t make sense to most people — concepts that are truly paradoxical. To testify to the seriousness of the problem, I call the following witnesses to the stand: Marilyn vos Savant, Chaim Weizmann, and Joseph Heller, with guest appearances by Elsa Einstein, Margot Einstein and Richard Feynman.
Special Relativity. Marilyn vos Savant has the highest IQ score ever recorded, according to the Guinness Book of World Records. The following question and answer appeared in her column “Ask Marilyn”:
- Is there any way for the average person to grasp the theory of relativity? A. In my opinion, no… Without being highly educated in physics, we can only read summaries of the theory, accept the points on faith and then successfully repeat what we’ve learned to others. — M. Vos Savant (Parade magazine, Sept. 9, 2001)
Einstein’s wife Elsa also had trouble with the theory. When asked if she understood relativity, she replied simply, “Understanding relativity is not necessary for my happiness” (I2007, p. 246). If neither Einstein’s wife nor the smartest person in the world could understand relativity, something must surely be wrong. The problem is the many paradoxical statements that arise from the theory. For example, take the statement that nothing can go faster than light. A sensible person can’t help but ask, “Why on earth not?”
General Relativity. Chaim Weizmann, a brilliant chemist and the future president of Israel, once accompanied Einstein on a trans-Atlantic sea voyage. After the voyage was over, Weizmann reported:
Einstein explained his theory to me every day, and on my arrival I was fully convinced that he understood it. – C. Weizmann (S1986, p. xi)
Nor did Einstein have any better luck with his step-daughter, Margot Einstein. While working on a book for the lay reader about relativity,
…he read every page out loud to Elsa’s daughter Margot, pausing frequently to ask whether she indeed got it. “Yes, Albert,” she invariably
 The theory known as special relativity was originally called simply relativity. The word special was added later to distinguish it from Einstein’s theory of gravity, which he called general relativity. In this book the word relativity by itself will refer to the special theory.
replied, even though (as she confided to others) she found the whole thing totally baffling. – W. Isaacson (I2007, p. 232)
The problem with general relativity, as it is usually presented, is that gravity is attributed to curvature in four-dimensional space-time. What is this thing called space-time, one asks, and how can it be curved? Curvature of a two-dimensional surface can be pictured, but curvature in three dimensions, let alone four, is beyond most people’s grasp.
Quantum Mechanics. Joseph Heller was one of America’s great 20th century novelists — author of the classic Catch 22. Having written brilliantly about the psychological world, he felt that as an educated, intelligent human being he should also know something about the physical world. His attempt, however, went sadly wrong.
In recent years I have been… writhing in an exasperating quandary over quantum mechanics, which, to my mind, remains impossible even to define, let alone comprehend. – J. Heller (Now and Then, p. 194)
He was not alone. Richard Feynman, a leading 20th century physicist, said:
I think I can safely say that nobody understands quantum mechanics. — R. Feynman (The Character of Physical Law, 1965)
According to QM, matter is made of particles that do things no self-respecting particle could ever do — things that only waves can do. The idea that particles can behave like waves is a concept that even physicists find paradoxical:
[Wave-particle duality] is a paradox because particles are, by definition, localized entities that follow definite trajectories while waves are not confined to any particular path or region of space. How could the same thing be both confined and not confined, both a particle and a wave? – T. Norsen (“Intelligent design in the classroom”, Forum on Physics and Society, vol. 35, no. 3, 2006)
According to a physicists’ joke, “light is waves on Mondays, Wednesdays and Fridays; it is particles on Tuesdays, Thursdays and Saturdays; and on Sundays we think about it.” The situation is so bad that philosophers invented a philosophy called logical positivism, which states essentially that hope must be abandoned of ever describing a consistent reality.
I am a positivist who believes that physical theories are just mathematical models we construct, and that it is meaningless to ask if they correspond to reality, just whether they predict observations. – S. Hawking (P1997, p.169)
Einstein’s search. Even as the developers of QM proceeded further with particles and wave-particle duality, Einstein came to believe that reality must consist of fields and only fields. He spent the last 25 years of his life searching for a unified field theory — a theory that would not only combine gravity and EM forces (the only two forces known at the time) into a single field, but would also include matter.
What appears certain to me, however, is that in the foundations of any consistent field theory, there shall not be, in addition to the concept of field, any concept concerning particles. The whole theory must be based solely on partial differential equations and their… solutions. – A. Einstein (The theory of relativity, and other essays, p. 34)
Quite naturally, he took as his starting point the gravitational equations that had brought him so much fame and acclaim, trying to expand them in a way that would include the EM field and also matter. He tried first one mathematical approach, then another. At one point his friend Wolfgang Pauli admonished him,
“Within a year, if not before, you will have abandoned that whole distant parallelism, just as earlier you gave up the affine theory…” The following January, [Einstein] admitted to Pauli, “So you were right after all, you rascal.” – W. Isaacson (I2007, p. 344)
Einstein’s search was not successful.
And so it went for another two decades… To the very end, he struggled to find his elusive unified field theory. And the final thing he wrote, before he went to sleep for the last time, was one more line of symbols and numbers that he hoped might get him, and the rest of us, just a little step closer to the spirit manifest in the laws of the universe. – W. Isaacson (I2007, p. 344, 543)
While Einstein was pursuing his lonely and unsuccessful search for a unified field theory, other physicists were taking a different track that led eventually to QFT. This wonderful theory, so little known or appreciated, fulfills Einstein’s desire for a world made only of fields, except that forces and matter are not combined into a single field, as Einstein hoped. In fact, there are at least seven different types of field. Not only is QFT the answer to Einstein’s search, it also resolves his enigmas, and in a way that can be understood by the man (or woman) on the street (see Chapters 7-9).
What is a field? Abandoning the familiar picture of solid particles and replacing it with intangible fields is not easy. It will require a leap of imagination greater than the atomic picture that Eddington struggled with. To put it briefly, a field is a property or a condition of space. The field concept was introduced into physics in 1845 by Michael Faraday as an explanation for electric and magnetic forces. His experiment with iron filings that align themselves in the region around a magnet is done today by every physics student (Fig. 1-1).
Fig. 1-1. Iron filings forming “lines of force” about a magnet led Faraday to believe that there must be a field present in the space around the magnet (www.ndt-ed.org).
However, the idea that fields can exist by themselves as properties of space was too much for physicists of the time to accept. Instead, they invented an invisible substance called ether to carry the EM oscillations. Belief in the ether prevailed for decades, but when no evidence for its existence could be found, despite many attempts, the ether was finally abandoned and physicists accepted that the EM field has an existence in itself. The idea that space can have properties does not come easily, but by the time you finish this book you will be comfortable with the concept of fields.
Field equations. In 1864 James Maxwell developed equations that describe how the strength of the EM field changes with time. These changes are local in the sense that only field intensities at the point of interest affect what happens at that point. However, a change of field strength somewhere else can propagate through space and eventually reach the point of interest, just as a stone dropped in water creates waves that propagate through the water. Light travels through space the same way, with a change in the EM field at one point creating changes at adjacent points, and so on. The way that fields propagate and the speed at which they propagate are determined by the field equations.
What is a quantum? In the centennial year of 1900 Max Planck introduced the idea that the EM field is not a continuous “classical” field, but is made of pieces, or chunks, that he dubbed quanta (from Latin quantum meaning “how much”). While Maxwell’s classical EM field can be arbitrarily small, quantum fields are made of chunks that cannot be further reduced. Quanta may overlap each other, but each one maintains its separate identity; it lives a life and dies a death of its own. In that sense, and in that sense only, field quanta resemble particles.
“Particles” are quanta. In the 1920’s it was found that the particles that make up matter exhibit wave characteristics. This led to the development of QM, with its characteristic wave-particle duality. This nettlesome problem (see Norsen quote above) was solved when QFT came along. In QFT there are no particles; there are only fields. It was Julian Schwinger who, in 1954, completed the formulation of QFT by finally treating force fields and matter fields on an equal basis.
…these two distinct classical concepts [particles and fields] are merged and become transcended in something that has no classical counterpart – the quantized field that is a new conception of its own, a unity that replaces the classical duality. – J. Schwinger (S2001)
QFT vs. QM. It is important to understand that QFT is not just a variation of QM. Conceptually, the two couldn’t be more different. QM describes a world made of particles and the equations give the probability that a particle is at a given point. QFT describes a world made of fields and the equations give the strength of the field at a given point. Another difference is that QM equations deal with numbers, while the fields of QFT are described by vectors in an abstract space known as Hilbert space. Fortunately, it is not necessary to understand the equations or to understand Hilbert space in order to grasp the concepts of QFT.
Fields of color. A special feature of this book is the use of color to depict fields that in themselves are unpicturable. Just as the color blue permeates the sky, so you will be asked to picture space as permeated with color, with different colors representing different fields. Color is an appropriate tool for this job because color is something that does not exist in itself; it exists only as a property of something else. By using colors to represent physical fields, we remind ourselves that fields are a property of space, not a separate substance in space.
The choice of color, of course, is arbitrary. I chose blue to represent the gravitational field, so the gravitational field of the earth is visualized as a blueness in the surrounding space — a blueness that extends not just to the sky, but to the moon and beyond, becoming fainter at greater distances. Similarly, the EM field is pictured as a greenness of space, while the two nuclear force fields — strong and weak — are represented by purple and brown. Finally, I chose yellow and red for the two matter fields, so “particles” like the electron and proton are pictured not as small spheres but as smeared-out blobs of yellow and red.
Limitation of color. There is a limit to how far the use of color can take us. A single color can help us visualize a field that is otherwise invisible, but it cannot inform us about the complexity of the field — and the fields we will encounter are complex. The EM field, for example, contains both electric and magnetic components, each of which has a direction associated with it, yet we will picture it simply as a green field. Also the use of color cannot convey the quantum nature of the fields – the fact that each field is made up of many individual pieces that keep their own identity, overlapping as they may be. The colors can help us picture the fields as “being there”, but they don’t tell the whole story.
When the theories of relativity were experimentally confirmed, the red carpet was rolled out despite (or perhaps because of) the fact that almost no one understood them. Einstein received world-wide acclaim unlike that bestowed on any other scientist, and was named “Man of the Century” by Time Magazine. QM, on the other hand, crept into public awareness without fanfare, but then gathered momentum. Today it, more than relativity, dominates articles about science, epitomizing the incomprehensibility of modern physics (see Candorville comic strip on p. 129). Now QFT represents as big a revolution in our thinking as relativity and QM put together — in fact, it combines and explains them. Yet you will look long and hard to find a book for the public that says quantized fields are the fabric of the cosmos.
A battle of three-rounds. There have been three rounds in the battle between fields and particles, and the battle is still going on. The first round came in 1905 when Einstein reintroduced Newton’s concept of light as corpuscles to explain the photoelectric effect, for which he received the Nobel prize (Chapter 3). The second round took place in the 1920’s when Schrödinger’s hope for a field theory of matter was abandoned because of the particle-like behavior that physicists could not ignore (Chapter 6). The third round occurred in the 1940’s when Feynman’s particle approach to quantum electrodynamics triumphed over Schwinger’s field picture, partly because Feynman’s method was easier to use (Chapter 6).
Einstein rejects QFT. Feynman was not the only physicist to reject QFT. Albert Einstein also rejected it after asking a friend, Valentin Bargmann, to give him a private tutorial on the subject. This is not surprising; at the time QFT was neither complete nor successful, nor was Bargmann its best spokesman. We must also consider Einstein’s great mistrust of the QM theory that preceded QFT — a mistrust based on its probabilistic nature. The idea of randomness in physics was repugnant to Einstein, as expressed in his famous comment, “I cannot believe that God plays dice with the universe.” As his friend and biographer, Abraham Pais, wrote:
I know from experience how difficult it was to discuss quantum field theory with him. He did not believe that… quantum mechanics provided a secure enough basis for relativistic generalizations. Relativistic quantum field theory was repugnant to him… and he did not believe in any of its consequences. – A. Pais (P1982, p. 463)
What if? I can’t help but wonder what might have happened if Einstein had waited a bit longer for his tutorial and chosen a different tutor – perhaps Julian Schwinger.
The year is 1954, the year Schwinger published the final installment of his “Theory of Quantized Fields” papers. The 36-year-old Schwinger comes to Princeton to meet the 75-year-old Einstein, who is to die the next year. “Prof. Einstein,” he begins, in his calm but self-assured manner, “I want to show you a theory that unites all the known forces, as well as matter, into a field theory that incorporates both the principle of relativity as you have formulated it and the quantum principle that you helped introduce. It is a theory that is philosophically satisfying and, more importantly, has produced agreement with experiment to an undreamed of precision.” “Well, go ahead then,” says Einstein, and Schwinger proceeds to expound his theory in Einstein’s Princeton study.
The sessions are intensive, with Einstein asking many probing questions along the way. Finally Einstein says, “This is indeed a beautiful theory, but it seems there are six separate fields – four force fields and two matter fields. [The Higgs field was not known at the time.] As you know, my hope was to find a single field that comprises all forces and matter. This theory of yours does not meet that objective.” “True,” says Schwinger, “but may I remind you what your friend Pauli once said: ‘What God hath put asunder, let no man join together.’ If the God you often refer to chose to have six fields, it is not for us to second-guess Him. Yes, there are six different fields, each with its own properties and equations, but this is the way the universe seems to be made. As Niels Bohr once said to you, “It is not for us to tell God what to do.”
“All right,” says Einstein, “I can accept the six fields, but there is a further problem that is more troubling to me, and that is the probabilistic nature of your theory. I have never been able to believe that God would play dice with the universe.” “I understand,” says Schwinger, “and I admit that QFT has not solved the causality problem. But at least it has localized the probability aspect to a single event called quantum collapse – an event that is not covered by the equations of QFT. That is to say [Schwinger’s favorite expression], QFT itself is causal, but it only takes us so far. At some point a discontinuous event takes place that is not covered by the theory. It is possible that at some time a theory will be developed for quantum collapse and it may turn out to be causal after all. On the other hand, it may be that it is a truly probabilistic process, and perhaps God does play dice with the universe. Either way, it doesn’t detract from what QFT has accomplished.”
Einstein gives in. He is captivated by Schwinger’s elegance and the soundness, both conceptual and mathematical, of his theory. He recognizes that QFT meets his desires for a field theory without “putting together what God hath torn asunder”. He appreciates the explanation that QFT provides for his two theories of relativity. He then announces to the world that QFT is the theory he has been looking for all his life. Of course this turns the tide and QFT becomes accepted throughout the physics and public communities, restoring the popular appreciation and understanding of science that once existed in the time of Newton and Maxwell.
But that didn’t happen. And so the most wonderful theory ever devised — a theory that has produced more precise agreement with experiment than anything before, that encompasses all forces and all matter, that unifies QM and relativity, that resolves Einstein’s Enigmas, and that reintroduces common sense into physics — this theory has largely remained a secret. Today, however, this is beginning to change as more physicists, like Nobel laureate Frank Wilczek, are telling people that nature is made of quantized fields.
The Core theory, which summarizes our best current understanding of fundamental processes, is formulated in terms of quantum fields. Particles appear as secondary consequences; they are localized disturbances in the primary entities – that is, in quantum fields. – F. Wilczek (W2008, p. 236)
The plan. The next five chapters will take you through the evolution of QFT step by step. The five force fields of nature are described in Chapters 2-5, beginning with gravity, the oldest known force, and ending with the two most recently understood ones: the weak field and the Higgs field. The two matter fields are then described in Chapter 6.
Chapters 7-9 describe Einstein’s enigmas and how they are resolved by QFT. In Chapter 7 you will see that the paradoxes of special relativity are natural and understandable consequences of the way fields behave. In Chapter 8 you will learn that gravity is not caused by curvature of space-time; it is a field like the other fields, and space and time are fundamentally different, in accord with our natural perceptions. In Chapter 9 you will see how QFT resolves the wave-particle duality problem, and provides a simple solution to the “Schrödinger’s cat” dilemma, also known as the measurement problem.
Chapter 10 (“The triumph of QFT”) provides an overview of QFT and its many successes, among which are a simple understanding of Einstein’s famous e = mc2 equation (not found elsewhere, to my knowledge) and what Wilczek called “one of the greatest scientific achievements of all time” (that few people know about). It also shows that two of these fields are made out of more basic fields called quarks and gluons. Finally, out of fairness, the gaps in QFT are also described.
What’s missing. This is not a physics textbook. QFT is a complex subject, and this book only skims the surface, sometimes to the point of oversimplification. My aim is to convey the essence of what quantum fields are and how they behave without burdening you with the details and the mathematics. Although to physicists these details are important and the mathematics essential, they would cause anguish to many readers, or at least cause them to lose interest. (I once read that for every equation, you lose a thousand readers.) While the mathematics of QFT is part of its beauty, QFT can be understood, or at least appreciated, at a conceptual level without equations. As Paul Dirac once said,
Mathematics is only a tool and one should learn to hold the physical ideas in one’s mind without reference to the mathematical form. – P. Dirac (D1958, p. viii)
There are also many things happening in physics today that are not included. You will not read here about black holes, supernovas, or the origin or fate of the universe. Cosmology is a fascinating topic, but it is not relevant to an understanding of the laws of nature. Nor will you read about attempts (e.g., superstring theory) to explain why fields have the properties they do. Why and wherefore are also fascinating questions, but QFT does not provide answers. Another noteworthy gap is consciousness, a mysterious process that happens behind our very noses, that neither QFT nor any other theory can even begin to explain.
Common sense. In this book I will frequently invoke common sense, but I must clarify what I mean by the term. I believe that common sense is choosing, wherever possible, the simplest, most intuitively-satisfying explanation that is consistent with our observations. This is not the same thing as accepting intuition blindly. An early example occurred when evidence that the earth is round triumphed over our intuitive belief that it is flat. A more recent example occurred when evidence for atoms dispelled our intuitive notion that matter is solid continuous stuff. If evidence, even very subtle evidence, requires that we change our long-standing intuitive view, we do it because of, not in spite of, common sense. However, the new view must make sense; it must be understandable, even if it wasn’t what we expected. “OK, the earth is a round ball and not a flat pancake; I can live with that”, we said. Or “OK, matter is made of tiny atoms held together by electric forces”, or even, “OK, electrons and protons are not particles, they are little blobs of field. I can live with that.” The job of science is to look beyond our intuition and to give up old ways of thinking when the evidence requires it. That’s what I call common sense.
However, there is one intuitive concept that I will retain at all costs: that there is a reality. Those who don’t believe that, or who are not interested in what it consists of, should not read this book. Those who do read the book will find that QFT offers a picture of reality that may not be what we expected, but one that we accept because of our common sense.
Features. References to the Bibliography are by author’s initial and year of publication, e.g., E1923. There are also many references to speeches by Nobel laureates that can be found at www.nobelprize.org — the best single place I know to learn physics. Footnotes are used for comments that can safely be ignored by the casual reader. I have made abundant and possibly excessive use of quotes, both to add authority to my claims (I am not an expert in QFT) and also because I think it is important to read the words of the scientists who created the theories.
Now if you’re thinking that physics = hard, difficult, impossible, please don’t be discouraged. It is not easy to change our intuitive picture of matter from something that exists in space to an abstract property of space. Yet you will see that there is nothing paradoxical or inconsistent about QFT – it’s just different from what you’re used to. You may even find (like me) that the field picture of nature is more philosophically satisfying than the particle one. On top of this, QFT rests on mathematics that are the most beautiful equations I have ever seen or could imagine. If you will read ahead with, as Einstein once said, “a fair amount of patience and force of will”, you will achieve a basic understanding of the only consistent, non-paradoxical theory of what the world is made of: quantum fields.