A Primer on Quantum Field Theory

By Eric M. Jones

“Everything you’ve learned in school as obvious becomes less and less obvious as you begin to study the universe. For example, there are no solids in the universe. There’s not even a suggestion of a solid. There are no absolute continuums. There are no surfaces. There are no straight lines.” ―R. Buckminster Fuller

WHAT’S GOING ON HERE? This is what a black hole does to space-time in polar coordinates. Sort of. This is similar to the “Rubber Sheet Experiment” so beloved by science presenters and so very wrong. It shows the distortion of space time, as a two-and-a-half dimensional representation of a four-dimensional phenomenon. The picture gives one  the impression that one could get close—maybe at a different angle—and somehow get a different set of forces. But you can’t ... it’s four dimensions.

[Left, two-and-a-half-dimensional representation of the gravitational force surrounding a black hole. The well is infinitely deep. Gravity is the first Quantum Field.]

A general model of the first-discovered Quantum Field—that of Gravity—is poorly represented by a stretched rubber sheet.

Is this what the gravitational field of a black hole looks like? No. The conical nature of the distortion “funnel” would indicate that the black hole has only some finite gravity. But the gravitational pull of a black hole is infinite at its center, so the distortion funnel is infinitely deep. Thus the tunnel walls would be perfectly parallel ... and continue forever.

But the reason we want to look at this is that there is no “matter” anywhere here. This is purely a Gravitational Quantum Field in space. If any matter enters the black hole, the “mouth” of the field increases its diameter. Matter gets converted into field intensity. What we call the center of the black hole has zero diameter—not “almost zero diameter.” Really, actually, perfectly, zero diameter.

Some time ago I published an article here (“Journey to the Bottom of Nothing,” 12-DEC-2013) that described the universe’s solidity as The Grand Illusion, or at least the part about it being “made out of something.” The universe is fundamentally not made of any kind of solid matter. Because if some solid matter constituted the Lego blocks of reality, we would smash them with a big hammer (or shoot at them with extremely high-velocity Lego bullets) to peek inside. But surprisingly, even many of the world’s greatest physicists have long refused to let go of the idea that there are “particles” of some sort, even if they are “particles” of charge like the electron. Younger physicists have grown up with the notion of QFT, Quantum Field Theory, and it appears to be correct—perfectly correct, although not quite complete. And all of quantum mechanics, and even relativity, comes out of QFT as a natural consequence. Reality is made of nothing but energy fields that react with each other in different ways. “Particles” are merely the quanta for each field. A photon is the quanta of the electromagnetic field.

A recent lecture by Physicist Sean Carroll states this remarkable fact quite well: “Every particle is actually a field. The universe is full of fields, and what we think of as particles are just excitations of those fields, like waves in an ocean. An electron, for example, is just the smallest excitation of the electromagnetic field. That is, it is just a quanta of the EM field.” (Actually photons are quanta of the EM field. Electrons are quanta of the Dirac Field—part of the EM field, but close enough. Carroll might have been misquoted.)

It has always been taught “classically” that there are only four fundamental forces—gravitational, electromagnetic, strong nuclear, and weak nuclear. These cover from the very smallest to the very largest forces in our universe. But if particles don’t exist except as quantum fields, then this old definition is due for a rewrite. There are lots of fields. Even the Higgs field is a quantum field. There are almost certainly others not yet detected. Dark energy and dark matter might both be quantum fields. Each field can be thought of as a diffuse, pervasive and permanent characteristic of three-dimensional space. In fact, the best way to view a “field” is not the rubber sheet model, nor any two-dimensional model at all. Rodney A. Brooks in his quite readable book “Fields of Color: the Theory That Escaped Einstein,” represents quantum fields as colors in 3D space. Not bad, but I would have preferred a non-visual analogue, perhaps odors, tastes, temperatures, tones, whatever. Maybe even a mix of them.

Let’s go back a bit. Early physics (and the physics still taught to students today, unfortunately) is represented as the science of one thing made of matter acting on another, or being acted upon. But long ago this was discovered to be an illusion too. Nothing touches anything. Only the repulsion of (usually) electrostatic fields make us feel that things touch. Scientists had a hard time accepting that invisible gravity from the Earth reached out and touched something like the Moon. Or that the Sun’s gravity affected the Earth. Isaac Newton and his contemporaries made it possible to calculate Gravity’s gross effects. Even after Newton, the idea that Gravity and Magnetism and Electrostatics were forces floating around separately seemed impossible.

Physics evolved from:
One thing touching another, to:
Something’s field touching something, to:
Fields interacting with fields.

So here we are: nothing solid exists. Reality is just Quantum Fields, and we’re not yet sure how many.

Thomas Young’s 1801 Double-Slit experiment was an attempt to disprove Newton’s “light corpuscules,” but it left behind the “Duality,” the idea that light could be a particle or a wave simultaneously. For almost a century this was viewed as a paradox. So the Wave-Particle Duality of light (further developed by Albert Einstein), is only a way to simplify the math; it actually blocks the progress to understanding.

Nobody was happy with it, but some thought there being some intermediate form of light transmission between particles and waves just showed the limitations of human minds. Duality now looks completely dead. The mystery of the double slit does not depend on “particles” in the least. It’s all fields all the time.

Richard Feynman was once asked who the world’s greatest physicist was. His answer surprised many because Einstein was all the rage:

“From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as James Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.”

Now don’t be scared. Those famous tee-shirt equations, written in a mathematical language that Maxwell partly had to invent, defined the properties of the second Quantum Field: electromagnetism. They read like a foreign language, but learning about them is not all that difficult once you know what the symbols mean. Still, they are clearly meant for people who fill blackboards full of this stuff.

There are many ways of writing these equations; however they are written, they say something pretty remarkable. The second equation, for example, says there can’t be magnetic monopoles, even though the equation above it defines electric monopoles, or the force caused by static charges. Remarkably, the fourth equation can be solved for the speed of electromagnetic phenomena in a vacuum, “c” which works out to be 2.99792458 × 108 meters per second (present day, agreed-upon number).

[Right, Maxwell's Equations. As Richard Feynman noted: This is probably the most significant development of the last 10,000 years. But it was forty years before it was really used for anything.]

Maxwell published his equations way back in 1861–1862. Experimenters were excited! It was immediately apparent that light had to be an electromagnetic phenomena, because other experimenters who were at the time independently trying to measure the speed of light were coming up with velocities not one part in a thousand different.

(Note: Maxwell’s calculation of “c” was actually a function of two other constants, the permittivity and permeability of a vacuum, values that are not hard to measure. So Maxwell’s “c” is perfect and absolute depending on the accuracy of the other constants, which describe only the nature of the medium ... nothing else! To believe anything can go faster than light is to believe that there is something that has a permittivity and permeability less than vacuum. So in a very fundamental way, you can’t get a more accurate measurement for “c” for any electromagnetic phenomena in a vacuum. Remember, Maxwell was talking only about the characteristic of the electromagnetic field. All the relativity stuff came later with Einstein.)

It is hard to overestimate the significance of Maxwell’s accomplishment. He put together the basic ideas of Faraday, Gauss, Ampere and other 19th century scientists who were then experimenting with the newly discovered principles of electricity and magnetism. They seemed to be related, but nobody was quite sure how. The telegraph had just been invented. The American Civil War had begun. It was the era of wood-fired steam engines and horses and muskets. Velocipedes (Boneshakers) rode boardwalks on wooden wheels. Canada did not yet exist. People worked under whale-oil lamps and candles. By the 1830s and 1840s, telegraph had revolutionized long-distance communication. But the ideas of Maxwell in the early 1860s were miraculous and interest among scientists in his theories was phenomenal.

During the next few decades, many inventors laid claim to the invention of wireless telegraph and radio. Most were wrong. Finally, it was Heinrich Hertz who, between 1886 and 1889, conducted a series of experiments proving the effects he was observing were the results of Maxwell’s electromagnetic waves.

But Hertz was underwhelmed and could see no purpose for what were soon to be called “Hertzian Waves.” When asked what use might be made of his discovery, he stated that, “It’s of no use whatsoever. This is just an experiment that proves Maxwell was right—we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there.”

Asked about the ramifications of his discoveries, Hertz replied, “Nothing, I guess.” So the electromagnetic field didn’t really seem to amount to much. It was forty years later that Guglielmo Marconi and Reginald Fessenden finally developed “radio,” followed quickly by an avalanche of other entrepreneurs. Then we began to fill the EM spectrum with human activity, and continue to do so today.

For much of the time since Maxwell, scientists imagined an ether that was the medium in which electromagnetism was conducted, much like air conducted sound, or water carried ripples in a pond. To prove the existence of the ether, Albert A. Michelson and Edward W. Morley carried out an experiment where an optical interferometer was mounted atop a large stone block floating in a pool of mercury. The interferometer could be turned as required to measure the optical fringes in different directions. It was believed that “etheric drag” must necessarily influence the speed of light and all electromagnetic phenomena, or even the size of the stone block.

Nobody knew what the ether was, but it had to be a fabric of space, and if so, it had to be fixed relative to something. Perhaps it was centered at the Sun, perhaps at the center of the galaxy (which at the time was believed to be the entire universe). But it had to have some locus where the ether was centered. Maxwell’s electromagnetic waves clearly travelled through the ether.

The results from the Michelson-Morley experiment were negative, that is: no etheric drag could be found. And it was a well-crafted experiment. One could hardly do better today.

Recent arguments among physicists follow along the line of “Well, but there is an ether. It was just defined incorrectly. It is simply the quantum field of electromagnetism. And relativity says it works fine as long as it isn’t thought of as an inertial (fixed) field.” By the way, Einstein, said he, like Newton, stood on the shoulders of giants ... and the tallest giant was Maxwell.

We need to step aside here and ask the question: “Is it possible that the universe is comprehensible to humans at all?” Can you teach TV repair to an octopus? Maybe not. But the universe appears to be comprehensible to humans because of their invented symbolic language called math. Human brains cannot contemplate infinity, but we write and work with it. The Arabs introduced the mathematical zero after the fall of Rome. This was not a minor advance in Western civilization. Humans can’t envision “nothing.” Earlier civilizations certainly understood the concept of “nothing.” But using it in math seemed pointless.

The invention of imaginary numbers, for example, made seemingly impossible things possible. Why is ii=0.20787 ...? Why for god’s sake is eiπ even an integer? Logarithms are merely integers with floating decimal points. Weird. But I can use the concepts without comprehending them in the least.

Likewise, in the 18th and early 19th centuries, physics was taught by simple analogy, electricity became like water in a pipe, temperature was like little balls in a can being shaken hard. Quantum Mechanics removed the underpinnings of that method of teaching. Still people could work with totally incomprehensible physics by using rules, math, and experiments.

Niels Bohr wrote: "Anyone who is not shocked by quantum theory has not understood it.” Being shocked does not mean we can’t understand enough of it to operate with it. If we have blackboards, equations, computers, some very brilliant scientists and big experiments, that’s enough to move forward. So humans can and do advance knowledge they cannot—in any sense of the word—comprehend.

Both the strong and weak forces (now often called “interactions”) were discovered when trying to understand what was happening inside the atom’s nucleus after we broke it apart. After all, electrons are repelled by the strongly positive charge of protons; what keeps everything from simply exploding away?

Before the 1930s, only two force fields were known: gravity, which was much too weak to play any part inside the atom, and the EM field, which was trying to bust apart the nucleus. Thus the strong force was born of necessity. It kept everything together and is often known as the “atomic force.”

The weak force is critically involved in Beta radioactive decay, and has recently become much more interesting because of its interaction with the Higgs mechanism.

All the parts of the atom are just fields, too.

The basic fields are very well (but not yet perfectly) understood. Gravity, electromagnetism, the strong and the weak fields. We have looked briefly at the first two and touched on the third and fourth.

There is now a way of describing how the universe is constructed that gives a satisfying and ultimate description: Quantum Field Theory. This describes the universe as being “merely” interactions between quantum fields, of which there are over half a dozen, and possibly a few more to be discovered. Particles have gone the way of the Dodo bird. There is probably no benefit in trying to visualize these fields, analogies fail us. Better to just use the known properties and data.

Recently a new (there have been many) “Fifth Fundamental Force” has been proposed. Normal electric force acts on electrons and protons, but this new boson interacts with electrons and neutrons. It has an extremely limited range (as do the Strong and Weak forces)—according to a paper published in the journal “Physical Review Letters” by theoretical physicists at the University of California, Irvine.

“The particle is not very heavy, and laboratories have had the energies to discover it since the ’50s and ’60s,” said Timothy Tait, professor of physics and astronomy. “But the reason it’s been hard to find is that its interactions are very feeble.”

***

So what will we discover about all these fields? How useful are they? It took forty years from Maxwell’s equations until the beginning of its routine application. One wonders what will happen decades from now with applications of the other Quantum Fields we are discovering. 

Further Reading

A great short description of Maxwell’s equations.

Heinrich Hertz.

Why Did Einstein Postulate That Speed of Light is Constant?

John Wheeler and Richard Feynman once proposed that perhaps—just perhaps— there was only one electron in the universe—and it sort of time-shared. This idea didn’t go far when the electron was thought of as a “particle.” But there is just one EM field.

Particles as the Quanta of Fields.

Eric M. Jones is the Associate Editor and co-founder of “Perihelion.” He is a design engineer, consultant, entrepreneur, and pilot, working in the experimental aircraft community, NASA, space transportation companies, and the ISS.