FIELDS OF COLOR
The theory that escaped Einstein
Rodney A. Brooks
2075 Whippet Way
Sedona, AZ 86336
Rodney A. Brooks
2075 Whippet Way
Sedona, AZ 86336
In this final chapter we present a summary and overview of Quantum Field Theory. We will describe its structure and its achievements and out of fairness, we will also describe the gaps in the theory. We will start with the three pillars on which this amazing edifice rests.
The field principle. The first pillar is the assumption that nature is made of fields. However, the concept of a field as a property of space does not come easily. It eluded the great Newton, who could not accept action-at-a-distance. It wasn’t until 1845 that Faraday, inspired by patterns of iron filings, conceived the idea of fields, and it took another 50 years before the concept was accepted without invoking an imaginary ether. The use of colors is my attempt to make the field picture more palatable. Once you are comfortable with the field picture of nature, I think you will find it more satisfying than the particle picture. While particles are more natural to our thinking, they raise the question: what’s inside them? What is an electron made of? The concept of a point particle–a singularity in space–-is even more disturbing. I hope you will agree with me that space with properties makes more sense than particles placed in space.
The quantum principle (discreteness). The quantum principle was introduced in 1900 by Max Planck, who showed that EM radiation is emitted from hot objects in discrete units called quanta (Chapter 3). A further step in discretization was made in 1922 by Otto Stern and Walther Gerlach. Their classic experiment (Fig. 10-1) showed that the angular momentum of an electron can have only two values—nothing in between.
This led to the use of Hilbert space to describe angular momentum and other physical properties that have discrete values. The concept of discreteness was later extended to field strength by Julian Schwinger, who called it measurement algebra. Even though field strength is a continuous property, it is treated mathematically as the limit of increasingly finer discrete values and is described by vectors in an infinite-dimensional Hilbert space. Because of this, field intensity is not a simple number; we can only talk about its expectation value.
The dynamics of the fields are then described by Hilbert operators that act on the vectors. It is these operators that appear in the field equations, not the field intensities themselves. However, you do not need to understand Hilbert algebra, or even know the field equations, to grasp the basic concepts of QFT. Just remember that Hilbert space is not real; it is a mathematical tool and Hilbert operators are not to be confused with the physical fields that exist in real three-dimensional space.
The relativity principle. The third pillar is the assumption that the field equations are the same for all uniformly-moving observers. This is Einstein’s Principle of Relativity, as described in Chapter 7. QFT is the only theory that successfully combines the relativity and quantum principles.
Occam’s Razor. I’m tempted to add another pillar, but it’s really more of a wish than a rule. I’m referring to Occam’s razor: “No more things should be presumed to exist than are absolutely necessary”, attributed to William of Ockham (1285-1349). In other words, all other things being equal, the simplest explanation is the 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 Julian Schwinger said, “nature does not always select what we, in our ignorance, would judge to be the most symmetrical and harmonious possibility”. 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 QFT 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 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)
On these pillars rests the most successful theory ever constructed—a theory that explains almost everything, from the tiniest atomic nucleus to the most remote star. We will now describe its essential features.
Fields. In this book we have described seven fields that are included in QFT: four force fields, two matter fields, and the recently discovered Higgs field. The force fields include gravity, electromagnetic forces, and the strong and weak nuclear forces; the matter fields include the lepton and baryon fields. However, as we will see below, the strong and baryon fields are actually effective fields that are made of more basic but “invisible” fields called quarks and gluons. Each of these fields is actually a group of interrelated “sub-fields” that share the same equations. For example, the EM field consists of electric and magnetic fields that are described by the QFT equivalent of Maxwell’s equations.
Quanta. The fields of QFT exist in units called quanta. Each quantum is a holistic piece of field that lives a life and dies a death of its own. The photon is a quantum of the EM field and protons and neutrons are quanta of the baryon field. Quanta may be free and travel through space, or they may be bound, like an electron in an atom. If a quantum is absorbed or changes its spin state, it does so as a unit. This all-or-nothing behavior is reminiscent of particles, making it difficult for many physicists to give up their belief in particles. Quanta are sometimes called excitations in a field, but that term doesn’t do them justice. Excitations can have any magnitude and can diminish as they travel and slowly die away like water waves. Quanta, on the other hand, have a fixed energy that doesn’t change as they travel, except when energy is transferred to another entity.
Self-fields. In addition to quanta, there are self-fields. Self-fields do not have a life of their own; they are created by a source and remain permanently attached to that source. Examples of self-fields are the gravitational field of the earth (Fig. 2-5), the electric field around an electron (Fig. 6-7), and the strong field around a nucleon (Fig. 4-2).
The vacuum field. Finally, there is the vacuum field. Even in regions where there are no quanta or self-fields, there is a background field called the vacuum field. The vacuum field is especially important in the case of the Higgs mechanism (see below).
Spin. In QFT there are no particles, so there is nothing to spin on its axis. Thus the concept of spin is more abstract and not easy to picture. Because of this, the term helicity is sometimes used instead. Helicity is related to the number of field components and how they change when viewed from different angles, but despite this abstract definition, helicity gives rise to angular momentum, just as does a spinning particle. The force fields have integer spin or helicity: 0 for the strong field, 1 for the EM and weak fields (which are sometimes combined into the electroweak field), and 2 for the gravitational field. The matter fields (lepton and baryon) have a spin of ½, and it is this half-integer spin that leads to the Exclusion Principle, which explains why quanta of the matter fields act more like particles than do quanta of the force fields.
Quantum collapse. If fields evolved only as described by the field equations, nothing of significance would happen, because these equations don’t describe the transfer of energy or momentum. For example, they don’t describe how a photon transfers its energy to a photoreceptor in your eye. I call this process quantum collapse because when a quantum transfers its energy it disappears from all space; it can’t continue to exist without energy. As Art Hobson wrote, referring to the two-slit experiment:
The entire spread-out field… must deposit its quantum of energy all at once, in a single instant, because the field cannot carry some fraction of one quantum—it must always contain either exactly one or exactly zero quanta of energy. When the field deposits its quantum of energy on the viewing screen, the entire spread-out field must instantaneously lose this much energy. — A. Hobson (H2007, p. 303)
Quantum collapse is similar to collapse of the wave function in QM, but the concepts are very different. The QM wave function describes the probability that a particle is in a given place or that the system is in a given state, and when a measurement is made the probabilities “collapse” into a certainty. In QFT the collapse is a physical process and it happens whether or not a measurement is being made.
There are two aspects of quantum collapse that bother many physicists. One of them is that, so far as we know, it is random; QFT does not explain when or how it happens. All we know is that the probability of collapse is related to the field intensity at a given point (Born’s rule). The idea of randomness was troubling to Einstein:
In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist. – A. Einstein (letter to Max Born, 1924)
However, 25 years later Einstein softened his stand. In a 1950 speech to the International Congress of Surgeons, after describing the “overwhelming evidence” for giving up causality, he concluded by saying:
Will this credo survive forever? It seems to me a smile is the best answer. – A. Einstein (Physics Today, June 2005, p. 47-48)
While the problem of randomness is not solved by QFT, at least it has been pinned down to a specific event. It is no longer a vague phenomenon related to the role of the observer; in QFT quantum collapse happens with or without an observer.
The other problem is non-locality. Many physicists are bothered by the fact that the collapse occurs at the same time at widely separated points. Einstein called it “spooky action at a distance”. If an emitted photon spreads out over light-years and then is absorbed by an atom, how, Einstein asked, could a part of the field that is many light-years away “know” this instantly? Non-locality is especially bothersome in the case of two entangled photons. Now the equations of QFT do not permit non-local effects, but quantum collapse is not described by these equations, so there is no theoretical reason that it can’t occur. Besides, non-locality has been experimentally demonstrated, and it does not lead to any paradoxes or inconsistencies.
Quantum collapse is necessary if quanta are to act as indivisible holistic entities, and it offers a simple solution to both the measurement problem and the entanglement problem. It may not be what we expected, but neither did we expect that the earth is round or that matter is made of atoms. Just as we learned to live with those concepts, we can learn to live with quantum collapse. To me, the real paradox is why physicists can’t accept it.
Quarks and gluons. Quarks and gluons are the basic fields that constitute the strong and baryon fields. However, because of the Principle of Confinement (see below), they do not exist in free form. Indeed, we would not have known about them if not for the large number of hadrons (quanta of the strong and baryon fields) that were discovered in the latter half of the 20th century—the so-called “subatomic zoo”. In 1961 Yuval Ne’eman (an Israeli physicist) and Murray Gell-Mann saw a pattern in the zoo that led Gell-Mann to predict a new particle (the omega-minus). Three years later Gell-Mann and George Zweig showed that this pattern would result if hadrons were made of more basic fields that Gell-Mann called quarks, a word taken from the novel Ulysses by James Joyce. Later, gluons (again Gell-Mann’s term) were introduced to hold the quarks together. The field theory that describes quarks and gluons was given the name (by guess whom) quantum chromodynamics (QCD), because arbitrary colors are used to describe different states of quarks. While QCD has its own name, it is part of Quantum Field Theory.
Mass. In classical physics mass is a measure of inertia, but in QFT it is a number that appears in the field equations. The effect of the mass term is to slow down the speed at which a field evolves and propagates, so mass plays the same inertial role in QFT that it does in classical physics. But this is not all that mass does. This same term also causes the fields to oscillate, and the greater the mass, the higher the frequency.
Energy. In classical physics, energy means the ability to do work, and work is defined as force exerted over a distance. This definition, however, doesn’t provide much of a picture, so in classical physics energy is a rather abstract concept. In QFT, on the other hand, the energy of a quantum is represented by oscillations in its field intensity as described by Planck’s famous equation (see Chapter 3): the more energy, the faster the oscillations. Using our color analogy, we might say that the oscillations cause the color to “shimmer”, and the faster the shimmer, the greater the energy of the field.
We will now look at some of the achievements of QFT—achievements that emerge from the theory as easily as raindrops falling from the clouds, or better yet, like presents appearing under the Christmas tree. Some of these have already been mentioned and some will be pleasant surprises. We will start with the most famous equation in physics history.
E = mc2. I know, I promised there would be no equations and, except for a few footnotes, I’ve kept my promise. But I think you will forgive me for making an exception for the world’s most famous equation, the only equation to have its biography written (B2000). And the thing is this: E = mc2 pops right out of QFT. Einstein had to work hard to derive it, and he published it in a separate paper written after his break-through Relativity paper. Yet, in QFT this equation follows as a simple consequence of the field equations. Since the mass term in the equations causes fields to oscillate and since oscillations are related to energy by Planck’s Law, it doesn’t take an Einstein to see that there is a relationship between mass and energy. In fact, any schoolboy can combine the two equations and find (big drum roll, please) E = mc2. Not only does this equation tumble right out of QFT, its physical meaning can be seen in the field oscillations, or shimmer, caused by mass. This is another example of how the bottom-up approach to Relativity, based on how fields behave, provides explanations for effects that otherwise seem paradoxical.
The subatomic zoo. In a tour de force of computer calculation, it was recently shown that the field equations for quarks and gluons account for the mass and spin of all the hadrons in the sub-atomic zoo. The calculation used three measured masses to fix the parameters of the theory and then, using these parameters, excitations of the fields were calculated. Fourteen such excitations were found, and their mass and spin were in close agreement with the fourteen known hadrons, ranging from the proton and neutron to the exotic charmonium. Equally important is that there were no excitations corresponding to the quarks and gluons themselves, thereby providing a theoretical basis for the Principle of Confinement. As Frank Wilczek wrote:
Through difficult calculations of merciless precision that call upon the full power of modern computer technology, they’ve shown that unbendable equations of high symmetry account convincingly and in quantitative detail for the existence of protons and neutrons, and for their properties. They’ve demonstrated the origin of the proton’s mass, and therefore the lion’s share of our mass. I believe this is one of the greatest scientific achievements of all time. — F. Wilczek (W2008, p. 127)
Exclusion principle. When the exclusion principle was introduced by Wolfgang Pauli in 1925, it was only a guess—an empirical postulate. It is only in QFT that this important principle is given a theoretical basis.
In my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from general assumptions… If we search for a theoretical explanation of this law, we must pass to the discussion of relativistic wave mechanics. – W. Pauli (Nobel lecture, 1945)
The theoretical explanation follows from the spin-statistics theorem of QFT, which states that quanta with half-integer spin (called fermions after Enrico Fermi) cannot be in the same quantum state. Not only does this theorem explain why electrons in an atom must be in different states of energy, angular momentum and spin (as per Pauli’s original conjecture), it also explains why matter quanta do not superpose, as force fields do, but are always seen as separate entities. And, as we saw in Chapter 4, it explains why there is a limit to the number of neutrons in a nucleus.
Classical limit. Because the exclusion principle does not apply to force fields with integer spin, quanta of force fields can overlap and build up. Even though each quantum acts individually, the total effect is like that of a classical field.
The Higgs field. The Higgs field interacts with all other fields and, because it has a large vacuum expectation value, generates the effective mass of these fields. As we saw in Chapter 5, the Higgs field was the final ingredient that made electroweak unification possible. It is the newest field in QFT, the Higgs boson having been detected in 2012.
Dark Matter. As early as 1933, astronomical observations showed that there is more matter in the universe than was previously thought. Astronomers now believe that ordinary matter constitutes only 5% of the total mass of the universe, while something called dark matter makes up 27%. (The rest is made of a related substance called dark energy, which we won’t go into.) This conclusion is based on its apparent gravitational influence on other objects. Dark matter is found primarily around galaxies and is believed to be a million times less dense than normal matter. It is not “seen” because it doesn’t emit or absorb light—that is, it doesn’t interact, or interacts feebly, with the EM field.
QFT offers two possible field explanations for dark matter. One possibility, suggested by Steven Weinberg and Frank Wilczek, is a new field with quanta called axions.
I’m very fond of axions, in part because I got to name them. I used that opportunity to fulfill a dream of my youth. I’d noticed that there was a brand of laundry detergent called “Axion”, which sounded to me like the name of a particle. So when theory produced a hypothetical particle that cleaned up a problem with an axial current, I sensed a cosmic convergence. The problem was to get it past the notoriously conservative editors of Physical Review Letters. I told them about the axial current, but not the detergent. It worked. – F. Wilczek (W2008, p. 203)
Attempts have been made to detect axions with an apparatus called DAMA (for DArk MAtter.) that is buried under the Gan Sasso mountain in Italy. According to the DAMA scientific team, a signal was detected recently that is consistent with dark matter, but other scientists have treated this finding with skepticism and new experiments are underway.
The other possible explanation for dark matter is a field suggested by an approach to QFT called supersymmetry. Either of these hypothesized fields would be sufficiently stable and interact feebly enough with normal matter to qualify.
Now we will take a look at what is not covered by QFT—the gaps in the theory.
Renormalization. The most obvious gap is that QFT does not describe the interaction of a quantum with its self-field—an interaction that Richard Feynman called a “silly” idea (see Chapter 6). The problem is that, according to QFT the result of this interaction turns out to be infinite, which is obviously not correct. The problem was solved for EM fields by the simple expedient of replacing the infinite quantities by their experimentally-determined values. However, the fact that the infinities occur in the first place shows that something is going on at the single quantum level that is not described by the QFT equations.
Quantum collapse. We know that quantum collapse occurs and we know the probabilities, but we have no theory to describe it. Maybe someday we will have such a theory, but in the meantime we must accept that it happens. It may not be what we expected but, like Einstein, we can always smile.
A speculation. Quantum collapse and renormalization both involve something that happens at the single quantum level—something not described by QFT. Renormalization is needed because QFT doesn’t explain how a quantum interacts with its self-field. Quantum collapse is a mystery because QFT doesn’t describe when or how a quantum transfers energy. It is possible that these two problems are related, and that if one gap is filled, the other will also be filled.
The next two gaps are not unique to QFT, but are present in every theory.
Whys and Wherefores. QFT does not explain why the numbers that appear in the equations have the values they do. The most famous example is the fine structure constant that describes the strength of the interaction between matter fields and the EM field. This constant was once thought to have a value of 1/137, and this, as you might imagine, led to some numerological attempts to explain why nature had chosen an integer — and such an unusual one at that. Sir Arthur Eddington even claimed that the number could be obtained by pure deduction. These attempts were abandoned when more precise measurements showed that the actual value is 1/137.04.
Many physicists still wonder why the constants in the equations are what they are, and there are attempts to find explanations that are more sophisticated than the 1/137 saga, including something called superstring theory. There are also less sophisticated attempts, such as the so-called anthropic principle, which states that if the equations were different from what they are, the human race could not exist. QFT does not supply answers, nor does any other theory. QFT does an amazing job of explaining the world we live in, but why the world is what it is, in my opinion, is a teleological—if not theological—question.
Consciousness. Finally, we come to the grand-daddy of mysteries. How dare physicists talk about “theories of everything” when they can’t explain what goes on behind their very noses!
As humans, we can identify galaxies light years away, we can study particles smaller than an atom. But we still haven’t unlocked the mystery of the three pounds of matter that sits between our ears. – B. Obama (remarks by the president on the BRAIN Initiative, April 2, 2013)
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 and feelings that our 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:
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”, 1952)
I see consciousness as a more pressing problem than the question of why the field constants have the values they do. My concern began when I was thinking about pain. Pain surely cannot be explained by fields or particles or relativity, or even quantum field theory, nor can pleasurable sensations, like the enjoyment of music or the intense pleasure of an orgasm. Then one day we had a visitor – a young computer hot shot. As we were sitting down for dinner, I asked him “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 0’s and 1’s into the right memory banks. Of course this was nonsense, so I told him he had flunked the test and couldn’t have any dinner. I didn’t really; we fed him, but it gave me the idea to write a book.
I was going to give it the title “Do Robots Have Orgasms?” (the illustration shows the cover designed by my nephew Roger L. Brooks), but I eventually abandoned the project because I figured that a book that boiled down to one word (“no”) wouldn’t sell. However, I had already written or sketched chapters with proposed explanations, and while working on the Quantum Mechanics chapter, I realized that all the quantum mechanical explanations ignored QFT. That’s when I realized that QFT is ignored almost everywhere, as if it never existed. Or if it is mentioned, it is not the “fields only” version created by Julius Schwinger. And that’s why I wrote the book that I wrote.
Is there any prospect that the consciousness mystery will ever be solved? Ambrose Bierce didn’t think so. To quote from his Devil’s Dictionary:
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. – Ambrose Bierce (B1958, p. 87)
Physicists have lived with and accepted the enigmas of QM for almost a hundred years as “shut up and calculate” became the rule. The leading journal in physics even invoked a policy that “papers on the foundations of quantum mechanics were to be rejected out of hand” (C2019).
It is truly surprising how little difference all this makes. Most physicists use quantum mechanics every day in their working lives without needing to worry about the fundamental problem of its interpretation. Being sensible people with very little time to follow up all the ideas and data in their own specialties and not having to worry about this fundamental problem, they do not worry about it. — S. Weinberg, (W1992, p. 84)
However, this is beginning to change. The world, if not the physics community, is finally waking up to the fact that “the emperor has no clothes”. In 2018 alone four books appeared, complaining about the enigmas of quantum mechanics:
Beyond Weird, winner of the 2018 physics Book of the Year award (B2018)
What Is Real? The unfinished quest for the meaning of quantum physics (B2018a)
Through Two Doors at Once: The elegant experiment that captures the enigma of our quantum reality (A2018)
Third Thoughts, Chapter 14 “The trouble with Quantum Mechanics” (W2018).
There have also been articles—and even a TV episode—complaining about these problems:
“Even physicists don’t understand Quantum Mechanics: Worse, they don’t seem to want to”, NY Times, Sep. 7, 2019 (C2019)
“Quantum physics may be even spookier than you think”, Scientific American, May 2018 (B2018b)
“Reimagining of Schrodinger’s cat breaks Quantum Mechanics”, Nature, Sep. 18, 2018 (C2018)
“Einstein’s Quantum Riddle”, NOVA TV, 1/2/2019.
The time is clearly ripe for the QFT answer presented in this book to be rediscovered and accepted as an answer to these problems, and if the physics community won’t do it, maybe the public will. That’s why I have written this book for a lay audience. After all, you don’t ignore the “mature work” of a Nobel laureate who was called the “heir-apparent to Einstein’s mantle”. You don’t ignore solutions to major problems offered by outstanding physicists, even if only a few. You don’t ignore “one of the greatest scientific achievements of all time”. And you don’t ignore a quantum theory of gravity developed by a Nobel laureate. It is time to spread the word.
Quantum Field Theory, as formulated by Julian Schwinger, is built on three pillars:
From these pillars there emerges a picture of a world made entirely of fields that exist in the form of quanta. Each quantum is a holistic unit that lives a life of its own and evolves according to deterministic field equations, except that when it transfers energy it collapses. The fields also have internal properties, such as helicity, or spin. Matter fields (electron, proton, etc.) have a helicity of ½ units and force fields have integer helicity (0, 1, or 2). There are also self-fields that are generated by, and remain attached to, matter quanta, and there are vacuum fields that exist even if no quanta are present.
There are also gaps in the theory:
QFT not only answers Einstein’s enigmas, but also provides many other solutions and explanations. Some examples described in this book are:
Yet this marvelous theory has been forgotten and ignored, while the physics world ignores the problems and sails blithely on, following the “shut up and calculate” rule. However, recently there has been a new awakening, and books and articles have been published pointing out that QM doesn’t make sense. So, dear reader, you have a choice:
Wave or particle? 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. — R. Oerter (O2006, Chap. 6: “Feynman‘s Particles, Schwinger‘s Fields”, p. 128)
And when you make your pick, I hope you will choose QFT, the only theory that makes sense. And maybe the day will finally come when the QM ship made of particles floating on a sea of paradox is abandoned for smoother sailing on the seas of quantum fields.
Click here to learn more about Fields of Color.
 The reader should be warned that many physicists do not accept quantum collapse. They believe that a superposition of states continues until a later time. For example, Roger Penrose suggested that collapse occurs when the gravitational energy exceeds a certain amount (P1994, p. 339).
 When I asked Dirk Bouwmeester (Physics Dept., UCSB) a similar question, his reply was “In for a penny, in for a pound.” In other words, it doesn’t matter if the distance is a nanometer or a light-year, the principle is the same. If you can accept one, you should be able to accept the other.
 Fuwa M, Takeda S, Schrödinger Zwierz M, Wiseman HW, Furusawa A. “Experimental proof of nonlocal wavefunction collapse for a single particle using homodyne measurements.” Nat Commun 2015 Mar 24; 6: #6665
 Zweig used the word ace to describe these new quanta, but Gell-Mann’s “quark” prevailed.
 These colors are not to be confused with the equally arbitrary colors I have used to help visualize fields.
 It is straightforward math to show that the frequency of oscillation is given by f = mc2/h, where h is Planck’s constant.
 Planck’s Law says that the energy of a quantum is given by E = hf, where f is frequency and h is Planck’s constant. Combining this with the equation in footnote 6 gives E = mc2.
 S. Aoki et al., “Quenched Light Hadron Spectrum”, Phys Rev Lett 84(2): pp. 238-41. (2000)
An achievement that few people are aware of. E.g., “We have a perfectly fine theory that describes the elementary particles called quarks and gluons, but no one can calculate how they come together to make a proton.” (H2018)
 The planets Neptune and Pluto were discovered for the same reason.
 As pointed out in this book, this fact is explained by the holistic nature of quantal they act as units.