Space-Time Curvature and Relativity

General Relativity is the name given to Einstein’s theory of gravity that described in Chapter 2. As the theory is usually presented, it describes gravity as a curvature in four-dimensional space-time. Now this is a concept far beyond the reach of ordinary folks.. Just the idea of four-dimensional space-time causes most of us to shudder… The answer in Quantum Field Theory is simple: Space is space and time is time, and there is no curvature. In QFT gravity is a quantum field in ordinary three-dimensional space, just like the other three force fields (EM, strong and weak).


This does not mean that four-dimensional notation is not useful. It is a convenient way of handling the mathematical relationship between space and time that is required by special relativity. One might almost say that physicists couldn’t live without it. Nevertheless, spatial and temporal evolution are fundamentally different, and I say shame on those who try to foist and force the four-dimensional concept onto the public as essential to the understanding of relativity theory.

The idea of space-time curvature also had its origin in mathematics. When searching for a mathematical method that could embody his Principle of Equivalence, Einstein was led to the equations of Riemannian geometry. And yes, these equations describe four-dimensional curvature, for those who can visualize it. You see, mathematicians are not limited by physical constraints; equations that have a physical meaning in three dimensions can be generalized algebraically to any number of dimensions. But when you do this, you are really dealing with algebra (equations), not geometry (spatial configurations).

By stretching our minds, some of us can even form a vague mental image of what four-dimensional curvature would be like if it did exist. Nevertheless, saying that the gravitational field equations are equivalent to curvature is not the same as saying that there is curvature. In QFT, the gravitational field is just another force field, like the EM, strong and weak fields, albeit with a greater complexity that is reflected in its higher spin value of 2.

While QFT resolves these paradoxical statements, I don’t want to leave you with the impression that the theory of quantum gravity is problem-free. While computational problems involving the EM field were overcome by the process known as renormalization, similar problems involving the quantum gravitational field have not been overcome. Fortunately they do not interfere with macroscopic calculations, for which the QFT equations become identical to Einstein’s.

Your choice. Once again you the reader have a choice, as you did in regard to the two approaches to special relativity. The choice is not about the equations, it is about their interpretation. Einstein’s equations can be interpreted as indicating a curvature of space-time, unpicturable as it may be, or as describing a quantum field in three-dimensional space, similar to the other quantum force fields. To the physicist, it really doesn’t make much difference. Physicists are more concerned with solving their equations than with interpreting them. If you will allow me one more Weinberg quote:

weinberg“The important thing is to be able to make predictions about images on the astronomers photographic plates, frequencies of spectral lines, and so on, and it simply doesn’t matter whether we ascribe these predictions to the physical effects of gravitational fields on the motion of planets and photons or to a curvature of space and time.” (The reader should be warned that these views are heterodox and would meet with objections from many general relativists.) – Steven Weinberg

So if you want, you can believe that gravitational effects are due to a curvature of space-time (even if you can’t picture it). Or, like Weinberg (and me), you can view gravity as a force field that, like the other force fields in QFT, exists in three-dimensional space and evolves in time according to the field equations.





Most physicists today are Einsteinian “top-downers”. They regard the various relativistic effects as consequences of the Principle of Relativity and that’s the way they present relativity to the public. They believe that deriving these effects as consequences of the way fields behave is somehow illegitimate.

“Look what you’ve done to our beautiful theory,” they say. “You’ve reduced it to mere physical effects. The F-L contraction is not a physical process that occurs because field configurations are affected by motion; it is something that is built into the nature of space. And this time dilation – it’s not that processes happen more slowly, it is a property of time itself.”

While the above is my paraphrasing, note how Einstein’s biographer, Abraham Pais, applied the condescending word “corrected” to the bottom-up explanations given by FitzGerald and Lorentz:

FitzGerald and Lorentz had already seen that the explanation of the Michelson-Morley experiment demanded the introduction of a new postulate, the contraction hypothesis. Their belief that this contraction is a dynamic effect (molecular forces in a rod in uniform motion differ from the forces in a rod at rest) was corrected by Einstein; the contraction of rods is a necessary consequence of his two postulates and is for the first time given its proper observational meaning in the June paper.

The fact is, either approach is correct and one does not preclude the other. Yes, the Principle of Relativity is elegant and the top-down approach is easier to use; physicists love it for that reason. But the field equations are also elegant and they not only contain the Principle of Relativity within them, they also provide a physical explanation for effects that otherwise are paradoxical. We can never know if God started with the Principle of Relativity and derived the field equations or started with the field equations from which follows the Principle. If She started with the principle that the laws of nature should be the same in all moving systems, then She also provided mechanisms to make it happen. And if the mechanisms are there, why not use them? They are real and understandable, and they should not be ignored.


Why Time “Dilates”

Quantum Field TheoryTime dilation is probably the best-known of the relativity effects because of the twin paradox. Here is the scenario: An astronaut leaves on a rocket ship traveling at close to the speed of light. After whizzing around the galaxy she returns to find that her (non-identical) twin brother on Earth is an old man with a long beard while she herself is still young. Now this is certainly mind-boggling. Why should time pass more slowly just because you’re moving? What physical explanation can we find for that?

Intuitive explanation. The explanation is again based on the field nature of matter, described by the field equations. Consider two atoms in a rocket ship (or in its contents). Suppose that one atom creates a field disturbance and when that disturbance reaches the second atom something happens. (It is the interaction among atoms, after all, that causes everything to happen.) Now if the rocket ship is moving, the second atom will have moved farther ahead, so the disturbance must travel a greater distance to get there, even after taking the F-L contraction into account. Since fields travel at a fixed rate, it will therefore take longer for the disturbance to reach the second atom. (Disturbances that propagate in the backward direction have a shorter distance to travel, but this effect turns out to be not as great.) In short, things happen more slowly when you’re moving because the fields have to travel a greater distance.

An analogy. Consider two men on a raft who exchange information by calling back and forth to each other. Suppose further that this exchange of information determines the evolution of events on the raft. That is, when B receives information from A he makes certain things happen, and when B calls back to A, other things happen. The problem is, it takes time for the sound waves to travel from A to B and by the time the sound reaches B, he will have moved to a new position B’. Therefore the sound must travel through a greater distance and the communication will take longer than if the raft were at rest.

LorentzIf the line between the two men is transverse to the motion (upper sketch) the calculation is not hard to do. The result, as it happens, is exactly the same as Lorentz’s result from Maxwell’s equations. The result is the same if the men are aligned in the direction of motion (lower sketch), although the calculation is harder because the time for forward communication is different from the time for backward communication.

NASA routinely observes time dilation in orbiting satellites and corrections are applied to keep atomic clocks on the GPS satellites in sync with clocks on earth. Time dilation has also been seen in particle accelerators. At the CERN accelerator radioactive particles traveling at 99.9% the speed of light are observed to decay 30 times more slowly than they do at rest (S1986, p. 57).

Another analogy. The idea of length contraction and time dilation may be easier to accept when you consider that objects contract and processes slow down when cooled. The only difference between the effect of temperature and the effect of motion is the mechanism: In a cooler chest it is the slowing down of atomic motion that affects rates and interatomic distances, while in moving objects it is the extra distance through which fields must propagate. Would we think it paradoxical if a twin was placed in a cold chamber for 50 years and then emerged to find that her brother was old and she was young? No, we would not; in fact there are firms that offer to preserve people by freezing them. Why then should we not accept that motion can have a similar effect on chemical and physical processes? As Lorentz himself said,

We may, I think, even go so far as to say that… the conclusion is no less legitimate than the inferences concerning the dilatation by heat. – H. Lorentz (L1916, p. 196)


Why Lengths Contract With Motion

George_Francis_FitzGeraldThe idea of contraction was first suggested by a relatively unknown Irish physicist, George Francis FitzGerald. FitzGerald expressed his idea in a short communication to the American journal Science in 1891, ten years after Michelson’s first reported result, and he also suggested a reason.

I would suggest that almost the only hypothesis that can reconcile this [conflict] is that the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocities to that of light. We know that electric forces are affected by the motion of the electrified bodies relative to the ether, and it seems a not improbable supposition that the molecular forces are affected by the motion, and that the size of a body alters consequently.

While FitzGerald referred to the ether, which was believed to be the carrier for light waves at the time, the reasoning holds with or without the ether. Somewhat later his rather timid suggestion that molecular forces are affected by motion was repeated and refined by the most famous physicist of the time.

While FitzGerald was little known outside Ireland, the Dutch scientist Hendrik Lorentz was recognized as the greatest physicist since Maxwell. In 1902 he and Pieter Zeeman received the second Nobel Prize ever awarded for discovering the “Zeeman effect” that led to the discovery of electron spin (Chapter 6). Einstein called Lorentz “the most well-rounded and harmonious personality he had met in his entire life” (P1982, p. 169). Upon Lorentz’s death, Europe’s greatest physicists attended his funeral and three minutes of silence were observed throughout Holland.

Lorentz had not seen FitzGerald’s paper, but he too realized that Michelson’s strange result would make sense if the apparatus contracted along the direction of motion. However he went further than FitzGerald; he did the calculation (not an easy one) using Maxwell’s equations. When he found that the theoretical contraction exactly compensated for the extra travel distance, this was surely one of the great “Eureka” moments in physics, comparable to those of Newton and Einstein.

When Lorentz learned of FitzGerald’s work, he wrote to him to be sure he was not usurping credit,… and thereafter Lorentz was careful to acknowledge FitzGerald’s priority. The contraction is sometimes called the FitzGerald contraction, some¬times the Lorentz contraction, and sometimes the FitzGerald-Lorentz (F-L) contraction. Fig. A-3 shows a modern version of Lorentz’s calcu¬lation done by John Bell with the aid of a computer.

Misconception #1. Some writers claim that the F-L contraction was an ad hoc explanation offered without any theoretical basis. In fact it was based on a deep understanding of how fields behave when in motion and how this behavior affects the molecular configurations.

Surprising as this hypothesis may appear at first sight, yet we shall have to admit that it is by no means far-fetched as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we are able at the present time to make this assertion definitely. If they are so transmitted, the translation will very probably affect this action between two molecules or atoms in a manner resembling the attraction or repulsion between charged particles. Now, since the form and dimensions of a solid body are ultimately conditioned by the intensity of molecular actions, there cannot fail to be a change of dimensions as well. – H. A. Lorentz (E1923, p. 5-6)

Intuitive explanation. While I hope you can accept, as did FitzGerald and Lorentz, that length contraction happens because the field equations require it, it would be nice to have some intuitive insight into the phenomenon. We must recognize that even if the molecular configuration of an object appears to be static, the component fields are always interacting with each other. The EM field interacts with the matter fields and vice versa, the strong field interacts with the nucleon fields, etc. These interactions are what holds the object together. Now if the object is moving very fast, this communication among fields will become more difficult because the fields, on the average, will have to interact through greater distances. Thus the object in motion must somehow adjust itself so that the same interaction among fields can occur. How can it do this? The only way is by reducing the distance the component fields must travel. Since the spacing between atoms and molecules, and hence the dimensions of an object, are deter¬mined by the nature and configuration of the force fields that bind them together, the dimensions of an object must therefore be affected by motion.


Why I Wrote “Fields of Color”

Rodney BrooksThis book started from a chapter in a book I once intended to write, called “Can Robots Have Orgasms?” Here is how that title came about. In my earlier years if I thought about consciousness at all, I probably believed that the mind operates according to the laws of physics and chemistry, like any other organ of the body. But then one day I thought about pain – that strong searing sensation that can make one scream and yell. Pain, I thought, surely cannot be explained by fields or particles or relativity or quantum mechanics or even quantum field theory. It is something that is quite apart, quite different from what happens in a computer or in any other machine. Then I thought about all the other things that computers can’t feel, one of which is the intense pleasure of an orgasm.

Then one day we had a visitor – a bright young computer hot shot. I asked him as we were sitting down for dinner “Do you think a computer can ever experience a sexual orgasm?” Well this young fellow began to tell me how you could create an orgasm by putting the right 0′s and 1′s into the right memory banks. Of course this was ridiculous nonsense, so I told him he had flunked the test and couldn’t have any dinner.

I didn’t really. We fed him, but he did give me the idea for the title: “Can Robots Have Orgasms?” As it happened, I eventually abandoned the book because I figured that a book that boiled down to just one word (“No”) wouldn’t sell. However I had already written or sketched chapters that I called “dead ends” about three explanations that have been proposed for consciousness: Artificial Intelligence, Religion, and Quantum Mechanics. To my way of thinking these all fail to provide an answer, or any hope of an answer. However, as I worked on the chapter entitled “Quantum Mechanics”, I realized that all the quantum mechanical explanations ignored Quantum Field Theory. And then I realized that QFT is ignored everywhere, as if it never existed. And that’s why I wrote the book that I wrote…

Why Quantum Field Theory is Ignored

Given all its successes, you must surely wonder why QFT has remained an unwanted child. For one thing, there is no physical evidence to compel us to believe in fields, or for that matter, to believe in anything. Philosophers tell us that we can’t prove what is real, or even that there is a reality. I cannot prove that my entire life has not been a dream in the mind of some alien being. All that we can do is try to find a theory that explains our observations; and then, if we choose – and only if we choose, – we can believe that the theory represents reality.

So the choice is yours. You can believe that reality consists of particles – tiny spheres or point particles – despite the many inconsistencies and absurdities, not to mention questions like how big the particles are and what are they made of. Or you may choose to believe in wave-particle duality, which is neither fish nor fowl. Or you may want to join those physicists, like Steven Hawking, who don’t worry about reality.

The choice was described this way by Robert Oerter:

Wave or particle? The answer: Both, and neither. You could think of the electron or the photon as a particle, but only if you were willing to let particles behave in the bizarre way described by Feynman: appearing again, interfering with each other and cancelling out. You could also think of it as a field, or wave, but you had to remember that the detector always registers one electron, or none – never half an electron, no matter how much the field has been split up or spread out. In the end, is the field just a calculational tool to tell you where the particle will be, or are the particles just calculational tools to tell you what the field values are? Take your pick.

And when you take your pick, dear reader, I hope you won’t choose the picture of nature that doesn’t make sense – that even its proponents call “bizarre”. I hope that, like Schwinger, Weinberg, Wilczek (and me), you will choose to believe in a reality made of quantum fields – properties of space that are described by the equations of QFT. This is a picture that resolves all three of Einstein’s enigmas (see Appendices), a picture that solves the action-at-a-distance problem that even Newton found unacceptable, a picture based on simple and elegant equations (take my word for that), a picture that explains or is consistent with all the data known to date. And on top of that, QFT provides the most philosophically acceptable picture of nature that I can imagine…. The choice is now up to you.

Choose Quantum Field Theory.

Speed of Light – Why Nothing Can Go Faster

Of course the idea that there is an ultimate speed limit seems absurd. While the speed of light is very high by earthly standards, the magnitude is not the point; any kind of speed limit in nature doesn’t make sense. Suppose, for example, that a spaceship is traveling at almost the speed of light. Why can’t you fire the engine again and make it go faster – or if necessary, build another ship with a more powerful engine? Or if a proton is whirling around in a cyclotron at close to the speed of light, why can’t you give it additional energy boosts and make it go faster?


Intuitive explanation. When we think of the spaceship and the proton as made of fields, not as solid objects, the idea is no longer ridiculous.  Fields can’t move infinitely fast. Changes in a field propagate in a “laborious” manner, with a change in intensity at one point causing a change at nearby points, in accord­ance with the field equations. Consider the wave created when you drop a stone in water: The stone generates a disturbance that moves outward as the water level at one point affects the level at another point, and there is nothing we can do to speed it up. Or consider a sound wave traveling through air: The disturb­ance in air pressure propagates as the pressure at one point affects the pressure at an adjacent point, and we can’t do anything to speed it up. In both cases the speed of travel is determined by properties of he transmitting medium – air and water, and there are mathematical equations that describe those properties.

wilczek_frankFields are also described by mathematical equations, based on the properties of space. It is the constant c in those equations that determines the maximum speed of propagation. If the field has mass, there is also a mass term that slows down the propagation speed further. Since everything is made of fields – including protons and rocketships – it is clear that nothing can go faster than light. As Frank Wilczek wrote,

One of the most basic results of special relativity, that the speed of light is a limiting velocity for the propagation of any physical influence, makes the field concept almost inevitable. – F. Wilczek (“The persistence of Ether”, p. 11, Physics Today, Jan. 1999)

David Bodanis tried to make this point in the following way:

Light will always be a quick leapfrogging of electricity out from magnetism, and then of magnetism leaping out from electricity, all swiftly shooting away from anything trying to catch up to it. That’s why its speed can be an upper limit. – D. Bodanis

However Bodanis only told part of the story. It is only when we recognize that everything, not just light, is made of fields that we can conclude that there is a universal speed limit. 

Now let’s take another look at that proton whirling around in an accelerator, using our colored glasses to visualize the fields. We see the proton as a blob of redness “oozing” (I prefer that term to “leapfrogging”) ahead, as the amount of redness at one point affects the redness at a neighboring point. The process is very fast by our usual norms, but it is not instantaneous. The proton can’t move any faster because the field equations put a limit on how fast the redness can ooze.

The Michelson-Morley Experiment

Albert Michelson

Albert Michelson

The story of relativity did not begin in 1905.  It started in 1881 with an experiment that yielded very surprising results – results that helped lead Einstein to his theory. The experiment was inspired by a proposal made by James Maxwell to determine the earth’s motion through the ether (which was still believed in at the time) by measuring the speed of light in two directions: one parallel to the earth’s motion and the other perpendicular to that motion. By comparing these two measurements, one should be able to calculate the speed of the earth as it passes through the ether. However the measurement accuracy that would be needed (one part in 200 million) was well beyond the capability of the time, so Maxwell concluded that the experiment was impossible. It took a young American physicist to make it possible, and the result that he found caused a revolution in physics unlike any seen before.

Albert Michelson came to the United States at the age of two, the son of Jewish-German parents. After serving in the US Navy (which he rejoined at the age of 62 to serve in World War I) he pursued a career in physics. In 1881, while studying in Europe, he came across Maxwell’s “challenge” and conceived the idea of the interferometer – an instrument that can measure exceedingly small distances by observing optical interference patterns. Using this sensitive instrument, Michelson was able to perform Maxwell’s experiment.

michelson morley interferometer

The central part of the apparatus is a thinly-silvered mirror that splits a light beam into two parts, with one beam traveling through the mirror and the other reflected upward. The two beams are then reflected back to the central mirror, which sends them to a detector. The light paths are equal in length so that if the apparatus is stationary the light beams would take equal times to reach the detector. However if the apparatus is moving, the beam traveling in the direction of motion would have to cover a greater distance because the mirrors and detector move during the time of travel. The transverse beam would also be affected by motion, but not as much. (You can either take my word for this or work it out with some high school algebra.) The resulting difference in travel times would put the beams “out of phase” and would create an interference pattern when they combine at the detector.

Edward Morley

Edward Morley

When the experiment was performed, much to Michelson’s surprise there was no difference between the two directions! The two light beams took the same time to reach the detector despite the extra distance created by the earth’s motion. More accurate experiments were performed later in collaboration with Edward Morley, using more mirrors to extend the path lengths. This improvement in accuracy turned out to be critical, as Michelson had made an error in his first measurement that was pointed out by Hendrik Lorentz. The experiment, now called the Michelson-Morley (M-M) experiment, was repeated many times – at different times of day (as the earth’s surface moves in different directions because of its rotation) and at different seasons of the year (when the earth moves in different directions as it orbits the sun). The answer remained the same: The two light beams took equal times to traverse their paths, regardless of the earth’s motion.

That the speed of light should be independent of motion was most surprising… It makes no sense for a light beam – or anything, for that matter – to travel at the same speed regardless of the motion of the observer. Suppose, for example, that you are observing a very fast train from another train. The apparent speed of the fast train would clearly depend on its direction relative to yours.  If the other train is moving in the opposite direction, it would go whooshing by, but if it is moving in the same direction as you, it would pass very slowly. Yet Michelson, a passenger on a train called earth, found that another train called light always moves at the same speed no matter which way it is moving relative to the earth.

If the M-M experiment had been performed only once, there would have been no problem. We could have simply said this is the frame of reference in which the laws of physics hold, in which Maxwell’s equations apply and light travels with velocity c. But the experiment was repeated with the earth moving in different, and even opposite, directions and the result was always the same.  It is not possible for light to travel with the same velocity in all of these frames of reference unless “something funny” is going on.

The “something funny” turned out to be even more surprising than the M-M result itself.  In a nutshell, objects contract when they move! More specifically, they contract in the direction of motion. Think about it. If the path length of Michelson’s apparatus in the forward direction contracted by the same amount as the extra distance the light beam would have to travel because of motion, the two effects would cancel out. In fact, this is the only way that Michelson’s null result could be explained.

Five Unexplained Mysteries Not Explained by Quantum Field Theory

Despite the many successes of Quantum Field Theory, there are five unexplained mysteries or “gaps” that may someday be filled: 

  • Renormalization is necessary because Quantum Field Theory does not describe how an electron (or other charged quantum) is affected by its self-generated EM field.
  • Field collapse is of two types: spatial collapse, when a spread-out quantum suddenly is absorbed or becomes localized, and internal collapse, when the spin or other internal property of a quantum suddenly changes. Collapse can also occur with two or more entangled quanta. Quantum Field Theory does not describe how and when this occurs, although it can predict probabilities.
  • Whys and wherefores. Quantum Field Theory does not provide an explanation for why the masses and interaction strengths of the various fields are what they are.
  • Dark matter and dark energy are believed to exist in outer space because of astronomical evidence. They also are not explained by the known fields of Quantum Field Theory.
  • Consciousness is something that happens behind our very noses, but is not explained by Quantum Field Theory.

…How dare physicists talk about “theories of everything” when they can’t explain what goes on behind their very noses! But please understand, by consciousness I don’t mean simple information processing, such as can be done by any computer. I mean the sense of awareness, the sensations, the feelings that human and other minds experience every day – from the color red to the beauty of a Mozart sonata or the pain of a toothache. Such sensations are known as qualia. Most physicists don’t want to be bothered with the question, and it is left to philosophers like Charlie Chaplin to worry about it:

Charlie Chaplin

Billions of years it’s taken to evolve human consciousness… The miracle of all existence… More important than anything in the whole universe. What can the stars do? Nothing but sit on their axis! And the sun, shoot­ing flames 280,000 miles high. So what? Wasting all its natural resources. Can the sun think? Is it conscious? – C. Chaplin (film “Limelight”)

I see consciousness as a more urgent problem than the question of why the field constants have the values they do, and I would trade a hundred field collapses for an explanation of why we see colors. Among those physicists who are willing to consider the problem, most believe that consciousness results from the complexity of the brain – that our brains do nothing more than an extremely complex computer or robot could do. (A physicist has been defined as “the atom’s way of thinking about atoms.”) This is known as the Artificial Intelligence (AI) explanation.  However there are a few physicists who believe that the phenomenon of consciousness goes beyond our present knowledge:

steven weinberg

Of all the areas of experience that we try to link to the principles of physics by arrows of explanation, it is consciousness that presents us with the greatest difficulty. We know about our own conscious thoughts directly, without the intervention of the senses, so how can consciousness ever be brought into the ambit of physics and chemistry? The physicist Brian Pippard… has put it thus: “What is surely impossible is that a theoretical physicist, given unlimited computing power, should deduce from the laws of physics that a certain complex structure is aware of its own existence.”  I have to confess that I find this issue terribly difficult. – S. Weinberg

To me it is perfectly obvious that consciousness consists of more than electric or electro-chemical signals, as in a computer or robot. Why do I believe this? For the same reason I believe that it is impossible to make a television set out of wood. If I took the most skilled carpenters in the world, gave them an unlimited supply of wood and said, “Take this wood and make a television set, but don’t use anything except wood”, I know they couldn’t do it. Wood doesn’t have within itself the capability to do the things that a TV set does. Similarly, electrical signals and computer memories don’t have it within themselves the capability to experience the color blue or the sensation of pain. We can’t even define these sensations, much less know how to create them from computer parts.

Some scientists justify their belief in the AI explanation by asking “what else? If it’s not electro-chemical signals (which we understand), then what else is there?” My answer is, I don’t know, but that doesn’t mean there isn’t something else going on. If you the reader have learned nothing else from this book, you have learned that the entire history of physics involved the recognition that there is “something else” going on. Why is this so difficult to believe in regard to consciousness?

Will we ever find an explanation?  Ambrose Bierce didn’t think so:


Mind, n. A mysterious form of matter secreted by the brain. Its chief activity consists in the endeavor to ascertain its own nature, the futility of the attempt being due to the fact that it has nothing but itself to know itself with. – A. Bierce (“The Devil’s Dictionary”)

The Foundations of Quantum Field Theory

occams_razoQuantum Field Theory is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. (To my knowledge, Julian Schwinger is the only person who has presented Quatum Field Theory in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950′s.) 

1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space – the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these properties or field intensities influence each other and change with time. In Quantum Field Theory there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn’t until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable. 

2. The quantum principle (discretization). The quantum principle is the second pillar, following from Planck’s 1900 proposal that EM fields are made up of discrete pieces. In Quantum field Theory, all physical properties are treated as having discrete values. Even field strengths, whose values are continuous, are regarded as the limit of increasingly finer discrete values.

The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment showed that the angular momentum (or spin) of the electron in a given direction can have only two values: + ½ or – ½ Planck units. 

The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values, this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet used by Stern and Gerlach. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet “selects” one of the values with corresponding probabilities, so “measurement” of field intensity at a point will select one of the possible values with corresponding probability (see “Field Collapse” in Chapter 8). It is discretization and superposition that led to Hilbert algebra as the mathematical language of QFT.

3. The relativity principle. There is one more fundamental assumption – that the field equations must be the same for all uniformly-moving observer. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 (see Appendix A). Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.

Occam’s Razor. I’m tempted to add another principle, but it’s really more of a wish than a rule. I’m referring to Occam’s razor, which states in essence, “All things being equal, the simplest explanation is best.” Einstein put it somewhat differently: “A physical theory should be as simple as possible, but no simpler.” The last phrase is important because, as Schwinger said, “nature does not always select what we, in our ignorance, would judge to be the most symmetrical and harmonious possibility” (S1970, p. 393). If the theory were as simple as possible, there would be just one field (or perhaps none!), and the world would be very uninteresting – not to mention uninhabitable. I think it can be said that the equations of Quantum Field theory are indeed about as simple as possible, but no simpler. 

The move from a particle description to a field description will be especially fruitful if the fields obey simple equations, so that we can calculate the future values of fields from the values they have now… Maxwell’s theory of electromagnetism, general relativity, and QCD [quantum chromodynamics] all have this property. Evidently, Nature has taken the opportunity to keep things relatively simple by using fields. – F. Wilczek (W2008, p. 86)

Discovery of Electron Spin

Electron SpinWolfgang Pauli was born in Vienna in the same year that Max Planck introduced quantization. He was five when Einstein published his theory of relativity. At the age of 19, while a student at the University of Munich, he was asked to write an encyclopedia article on Einstein’s theory. The article was so brilliant that when Einstein read it he commented that perhaps Pauli knew more about relativity than he himself did. In 1925, five years before postulating the neutrino, Pauli introduced his Exclusion Principle. He showed that the various atomic spectra would make sense if the electron states in an atom are described by four quantum numbers and if each of these states is occupied by only one electron. In other words, once a given state is occupied, all other electrons are excluded from that state. Pauli realized that one of the numbers must represent energy, which is related to the distance from the nucleus (if we regard the states as orbits), while two other numbers represent angular momentum, which has to do with the shape and orientation of the orbit. However Pauli could find no physical significance for the fourth quantum number, which was needed empirically. The significance was found that same year by two Dutch physics students.

George Uhlenbeck and Samuel Goudsmit were studying certain details of spectral lines known as the anomalous Zeeman effect. This eventually led them to the realization that Pauli’s fourth quantum number must relate to electron spin.  Here is the story in Goudsmit’s delightful and self-effacing words – a story that conveys the groping and uncertainty that exist in the struggle to understand nature, as contrasted with the logic and certainty that are imposed after the battle is won.

The Pauli principle was published early in 1925… if I had been a good physi­cist, then I would have noticed already in May 1925 that this implied that the electron possessed spin. But I was not a good physicist and thus I did not realize this… Then Uhlenbeck appears on the scene… he asked all those questions I had never asked… When the day came that I had to tell Uhlenbeck about the Pauli principle – of course using my own quantum numbers – then he said to me: “But don’t you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates”… I asked him: “What is a degree of freedom?” In any case, when he made his remark, it was luck that I knew all these things about the spectra, and I said: “That fits precisely in our hydrogen scheme which we wrote about four weeks ago. If one now allows the electron to be magnetic with the appropriate magnetic moment, then one can understand all those com­plicated Zeeman-effects. – S. Goudsmit

And so was introduced the idea that the electron spins on its axis (still thinking of the electron as a particle) and that this spin has a value of ½, as contrasted with the photon’s spin of 1.


The next step was to see if this idea is consistent with experiment. During a course at Harvard University, Prof. Wendell Furry gave the following account of how this happened (as best I recall):

After Uhlenbeck and Goudsmit had the idea that electron spin might explain the anomalous spectroscopic results, there remained the crucial task of determining if the effect is in the right direction. This involved the kind of calculation that all physics students have to suffer through in which polarity, direction of spin, direction of magnetic field, the “right-hand rule”, etc., get all confused and make the head spin (no pun intended). In other words, there are many opportunities to go wrong, and many do. Well, the story goes, each man did the calculation and when they compared notes they found they had opposite results. One of them had obviously made a mistake, so they went back to check their calculations. As it happened, each found an error, so they were still in disagreement. At this point they went to their mentor, Paul Ehrenfest, who happened to have a distinguished visitor named Albert Einstein. It was decided that the four of them would do the calculation independently (remember, we are talking about elementary physics here). When they got together the result was 2-2. They finally broke the deadlock by counting Einstein’s vote twice. – W. Furry (reconstructed)

I believe that this story was a joke, making fun of the difficulty physicists have in keeping track of the proper sign. (It is said that the difference between a good and a bad physicist is that the good one makes an even number of errors, so the final sign is correct.) It also pokes fun at the “papal” authority of Ein­stein, although Uhlenbeck did say that Einstein visited Leiden in 1925 and “gave us the essential hint” to complete the calculation.