During the 1940s, Wigner and Racah separately developed a mathematical theory which today is called “The Angular Momentum Algebra” or “The Wigner-Racah Analysis”. The theory greatly simplifies the calculation of the electron structure in atoms, but its profound meaning is potentially a major contribution to the understanding of the structure of nucleons.
Based on historical evidences, one can assume that if elementary particle scientists were familiar with this theory and had reached a profound understanding of its consequences, Standard Model’s QCD would have never been conceived. Racah died in a home accident in 1965 during a family visit in Italy, and Wigner never really dived deep down into particle theory during the late 60s, and stopped publishing in this field. The sad truth is that there are perhaps no elementary particle scientists who really understand the theory’s profound meaning. Comay recalls that a few years back he met a colleague who told him how, when working on a paper describing Racah’s work, he sent it to a major journal, and the editor returned it to him saying that there was no scientist capable of reviewing the article, and that none of the scientists associated to the journal has any idea what it was about… the paper ended up being published in a different journal.
Before getting deep into the details in order to understand the theory and its consequences on the understanding of matter, let me quote a few findings which are easily accounted for when understanding the theory:
– The relatively low energy of the particles Omega- and Delta++, which played an important role in the motivation for the invention of QCD.
– The phenomenon called “the second EMC effect” revealing that the total spin of the proton quarks is close to zero. This effect is called “The proton spin crisis” and was the subject of hundreds of articles.
A book by Amos de Shalit and Igal Talmi explaining Wigner and Racah’s theory was once qualified as “a jabbering book” by a certain elementary particle scientist. I do hope that now, in view of the issues raised here, the reader will realize the possible impact of this theory, and unlike that scientist, will read the article through.
By the way, when Comay told me how that scientist spoke about the book, he got up, picked up a book called “Nuclear Shell Theory” off his bookshelf, and said with insult and sorrow: “How can one talk like that about such a book?”
God plays with many dices
One of the most astonishing things about quantum mechanics is the use of statistics to describe a particle’s state. From the beginning, Quantum Theory claimed that a particle’s location cannot be well defined.
Experiments support the assumption that an elementary quantum particle is a point-like particle. However, the theory’s founders assumed that the particle is present at different probabilities in many different places at the same time. This concept was so bewildering, that Einstein complained that “God does not play dice”. Einstein and his colleagues proposed an experiment, today called the Einstein-Podolsky-Rosen-Bohm experiment, the outcome of which was expected to invalidate Quantum theory. When decades later the experiment was finally carried out, the results fitted Quantum Mechanics’ prediction and not the intuitive reasoning of Einstein and his partners… It turns out that the Laws of Nature do indeed like dice way more than we can grasp.
In the late thirties and early forties of the past century scientists took uncertainty one step further. They looked at a free electron, i.e., out of the atomic shell. In fact, an electron could be composed of 2 electrons and one positron – three particles which when combined together have properties analogous to a single electron. (The positron is the electron’s anti-particle, and from a distance, the couple electron-positron seems like they cancel each other.) The state composed of 2 electrons and one positron is considered as a legitimate state of the electron. That is to say, the electron can be described as a combination of particles in different specific states. Scientists claimed that the electron is simultaneously in a mixture of several states, each having its own probability.
Doesn’t it sound strange? In the case of the electron, the probability of a state composed of 2 electrons and a positron is very small (about one to a million) but experiments showed a tiny deviation from the predictions of the Dirac Equation and confirmed this explanation.
Wigner and Racah’s theory brought out another aspect of the uncertainty of the electron state. The electrons in the atomic shells have different simultaneous states, meaning that they occupy several shells simultaneously. All the combinations allowed by the basic laws do exist, and each has its own probability!
Wigner and Racah developed mathematical tools to calculate the probability of each configuration. The theory they developed for calculating these probabilities is called “Angular Momentum Algebra”. Computer calculations done in the late 1950s and early 1960s showed that for quite a few configurations, probability is not as small as in the case of the free electron. For example, if we take the ground state of the Helium atom, naively considered as simple since it only has one electron shell with 2 electrons in it, 35 (!) distinct configurations were used simultaneously, some of which with a two-digit probability . By the way, the configuration number can be brought down to 10, but even 10 is a large and surprising number for the ground state of a “simple” atom containing only 2 electrons.
When high speed computers were built, Wigner-Racah techniques were used in order to calculate the states of multi-electron atoms.
So far, although this is not one of the hot topics at the front of science, everything is documented. The impact of these physical properties on the understanding of the behavior of quarks in the nucleons is even more farfetched.
Wigner, Racah and the Quarks
Assuming that Comay’s model is correct, and that quarks inside the baryons fill up energy levels inside the nucleons just as electrons do in atoms, i.e., nucleons have an attracting core, the same rules should apply to both. In fact, the configuration approach holds such a fundamental place in physics that they probably exist even without Comay’s model. Using the same principles as for electrons in atoms, quarks, being 3 and not 2 as in the Helium atom, and subject to forces hundred times stronger than the electric forces, are simultaneously found in many possible configurations, and it would actually be impossible to approximate and define one dominant configuration with a significantly higher probability. This yields a mixture of many configurations in which the quarks are found simultaneously.
The situation is actually even more complicated than that: as opposed to electrons in the atom, there are additional pairs of quarks and antiquarks in the nucleon occupying energy levels with a non negligible probability. Measures showed that the number of additional quark-antiquark pairs inside the nucleon is real and concrete- in average, about an extra half pair can be found inside the nucleon.
I suppose that most of the readers, who tried to follow up to here, went out for a cigarette break, even if they are non smokers… but let’s continue anyway.
The proton spin crisis
Another issue is that if we sum up the spins of this large quark configuration mixture inside the proton, the total will be close to zero. Why? Because adding up spins is a directional addition (vector addition, for those who are familiar with this term), which means that spins in opposite directions cancel each other. Since quarks occupy simultaneously a large number of configurations cancelling each other’s spin, the total spin will be close to zero, as the proton spin crisis showed.
If elementary particle scientists were familiar of this demonstrated physical knowledge, the proton spin crisis would not have turned into a crisis at all.
In fact, the result of the proton spin crisis is so obvious, that the question is why Comay didn’t publish his prediction. “I just never thought it would ever be possible to measure something like that, and therefore I never considered suggesting measuring the spin addition,” he answered.
BTW, the proton spin crisis is considered an unresolved problem until today. 
The Omega- and the Delta++ particles
Let me remind here again, that properties of these particles, discovered experimentally, lead the QCD pioneers to look for an out of the box solution and resulted in further developments to their theory. The principle for the Delta++ particle is the same as for Omega-. Delta++ is a baryon, i.e., a particle from the nucleon family, which has 3 u quarks, and its spin is 3/2.
The energy of this Baryon is relatively low. Because of the Pauli Principle, it is impossible to come up with a total spin of 3/2 for the 3 quarks, if only one configuration exists in which all the quarks are in the spatial state of minimal kinetic energy. Scientists were embarrassed by this only because they considered a unique particle configuration, i.e., they thought the particle had only one definite state. Understanding that the particle has several simultaneous configurations is a significant step toward resolving the enigma.
The next part of the explanation about the delta++, published a few years ago , is meant for physicists…
An important point for the understanding of the structure of delta++ is what is called in atomic spectroscopy “exchange integral”. This quantity appears explicitly in a configuration of electrons (or quarks here) in different shells. It turns out that the exchange integral is very fond of states in which spins are symmetric (i.e. parallel) and the spatial function is anti-symmetric. In general, parallel spins lead to an anti-symmetric spatial state which reduces the binding energy, as is formulated in one of Hund’s rules for atomic spectroscopy. This property can be illustrated in the following picturesque manner: Electrons (or quarks here) in an anti-symmetric spatial state are farther apart from each other. Therefore the positive repulsion energy is smaller and the binding energy is larger. This means that the combination of a symmetric spin and anti-symmetric spatial functions is more frequent in low energy states (such as the baryon delta++ or in atoms obeying Hund’s rule).
The following point emphasizes the importance of the previous argument. On the basis of the proton’s dimension and Heisenberg’s Uncertainty Principle, the energy necessary for the spatial excitation of a quark in a proton can be estimated to some 200MeV. On the other hand, excited baryonic states show that the binding energy is larger than 1000MeV. For comparison, in the Hydrogen atom, it is well known that the ratio between the kinetic and potential energy is half.
Therefore, quarks in the proton would find it “really cheap” to “pay” with kinetic energy and save the repulsion energy between the quarks. These considerations provide a qualitative explanation to the relatively low energy level of the Delta++ baryon.
And this is not all
The impact of Wigner and Racah’s arguments goes even further. Regarding the nonvanishing probability of quark-antiquark pairs, any calculation of energy states of quarks in nucleons should take these pairs into account, because unlike the electron calculations – the additional multiplicity of quark configurations does have a more significant impact on the results.
BTW, it seems like most (if not all) elementary particle scientists are unaware, even today, of the necessity of using multiple configurations for describing quark states. Indeed, “The proton spin crisis” has been going on for decades and is not explained even today .
 A. W. Weiss, Phys. Rev. 122, 1826 (1961)
 Wikipedia list of unsolved problems in physics
 E. Comay, in “Has the Last Word Been Said on Classical Electrodynamics?” ed. A. Chubykalo, V. Onoochin, A. Espinoza and R. Smirnov-Rueda (Rinton Press, Paramus, NJ, 2004). (download)