Introduction

What could the Standard Model of particle physics possibly have in common with biblical accounts of the Israelites’ travels in the Sinai Desert, Kabbalistic doctrines related to the unfolding of spiritual worlds, or the arrangement of the letters in the Name of G‑d? To make connections or parallels between such unrelated concepts may sound farfetched. However, this is exactly what we are going to do in this essay.

Remember that in structural analysis, we do not concern ourselves with the specifics or the nature of the objects at hand—we are interested only in the interrelationships among the objects, the high-level structure, or the storyline. So, let us not worry that particle physics speaks of subatomic particles, whereas the Torah speaks of the arrangement of Jewish tribes around the Tabernacle in the desert—topics as distant as they could be. What interests us is the interrelationships among the particles or between the tribes. We will be looking for parallels in the structure, no matter how dissimilar the objects in this structure may be.

Surprisingly, the encampment of the Jewish people in the desert, as described in detail in the Torah portion Bamidbar (Numbers 1), exhibits remarkable structural resemblance to the arrangement of the elementary particles in the Standard Model. In this essay, we will explore this and other structural parallels and their significance.

Jewish Encampment in the Desert

The encampment of the Israelites during their journeys through the wilderness of the Sinai Desert was highly ordered. At the center of the camp was the Tabernacle (Mishkan), a portable sanctuary where the divine presence dwelled. It housed the Ark of the Covenant (Aron HaKodesh), which contained the tablets with the Ten Commandments (Luchot) as well as the Torah scroll written by Moses. The Tabernacle was a vital part of divine worship, and its central location signified the importance of G‑d in the life of the community.

Surrounding the Tabernacle were the tents of the Levites—the descendants of Levi—the tribe tasked with the care and transportation of the Tabernacle. The Levites were divided into three families (Numbers 3:17): the family of Gershon—the Gershonites, who camped behind the Tabernacle (Numbers 3:21-26); the family of Kohat—the Kohathites, who camped along the south side of the Tabernacle (Numbers 3:27-31); and the family of Merari—the Merarites, camped along the north side of the Tabernacle (Numbers 3:33–37). Each had specific duties related to different parts of the Tabernacle. Aaron and his sons camped on the east side, toward the entrance of the Tabernacle (Numbers 3:38). Aaron was also a descendant of Levi, and Aaronites were a subset of Levites. In this sense, Aaronites may be considered one of the Levite families, their elevated status notwithstanding. Therefore, we can say that the Tabernacle was surrounded by four families of Levites.

Beyond the Levites, the twelve tribes of Israel were encamped, three tribes in each direction (north, south, east, and west). Each tribe had a designated leader and a tribal standard—a flag or banner—with its unique emblem. The tribes were arranged as follows (Numbers 2:2–31):

1. The camp of Judah on the east: Judah (leader), Issachar, and Zebulun;
2. The camp of Reuben in the south: Reuben (leader), Simeon, and Gad;
3. The camp of Ephraim on the west: Ephraim (leader), Manasseh, and Benjamin;
4. The camp of Dan in the north: Dan (leader), Asher, and Naphtali.

Let us arrange these tribes and families in a table as follows:

Now, let us compare this arrangement of the Tribes of Israel to the arrangement of elementary particles in the Standard Model.

The Standard Model

The Standard Model of particle physics is the theoretical framework describing all known elementary particles and their interactions, except for gravity. This model is based on quantum mechanics and quantum field theory and includes both matter particles (fermions) and force-carrying particles (bosons).

Fermions: They are the twelve elementary particles that make up matter (or twenty-four, if you count fermions’ antiparticles, because each fermion has its own “evil twin” – an antiparticle, which has the opposite electric charge). All fermions have spin ½.^[1] Fermions obey the Pauli exclusion principle, according to which two or more identical particles with half-integer spins (i.e., fermions) cannot occupy the same quantum state within a quantum system simultaneously.^[2] Fermions are divided into two main categories—six quarks and six leptons:

• Quarks. These come in six “flavors”: up (u), down (d), charm (c), strange (s), top (t), and bottom (b).^[3] Up and down quarks are light, very common, and stable (they do not easily decay); they make up protons and neutrons (for example, a proton is made of two up quarks and one down quark, whereas a neutron is made of two down quarks and one up quark). The other four types are heavier and unstable (they exist only for a brief moment and quickly decay into other particles). Quarks are unique, because they carry a fractional electric charge and engage in strong interactions—they are kept or “glued” together by the strong nuclear force.
• Leptons. These also come in six types. The electron (e) is the most familiar, along with its two heavier counterparts, the muon (µ) and the tau (τ) particle. Each of these has an associated neutrino (ν) (the electron neutrino ν_e, the muon neutrino ν_µ, and the tau neutrino ν_τ). Neutrinos are elusive particles that interact very weakly with matter.

Bosons: These particles carry (or, as we say in physics, mediate) the fundamental forces and are responsible for particle interactions. They are quanta of the three fundamental fields—the electromagnetic, strong nuclear, and weak nuclear forces (again, the Standard Model does not account for gravity).

• Gluons mediate the strong nuclear force, the force responsible for gluing quarks together (which is why they are called “gluons”) within protons and neutrons and for holding protons and neutrons together within atomic nuclei.
• Photons are particles of light; photons mediate the electromagnetic force, which governs interactions between electrically charged particles.
• W and Z bosons mediate the weak nuclear force responsible for beta decay.
• Higgs bosons are associated with the Higgs field—a field of energy permeating all space that gives particles their mass.*

The particles in the Standard Model are arranged according to their properties, including their masses, charges, and the ways they interact. The arrangement resembles the periodic table of elements—but for elementary particles instead. Developed in the early 1970s, the Standard Model is remarkably successful. Physicists have found all the particles the Standard Model has predicted.

The First Parallel

Although the Torah speaks of the encampment of the Israelites in the Sinai Desert, whereas the Standard Model speaks of subatomic particles—concepts as far removed from one another as could be—as mentioned before, in structural analysis, we do not concern ourselves with the nature of the objects per se, but focus instead on the structure of their interrelationships. Indeed, the Israelites’ encampment in the wilderness exhibits surprising structural parallels with the Standard Model. Here are some of these parallels:

1. There are sixteen elementary (or seventeen, of the two carriers of weak nuclear force—W and Z bosons—are counted separately) particles in the Standard Model. And there are sixteen groups of Israelites.
2. The sixteen elementary particles comprise twelve fermions (the particles of matter) and four bosons (the force-carrier particles). Similarly, the sixteen units of Israelites comprise twelve tribes of Israel and four families of Levites (including Aaronites).
3. According to the Standard Model, all elementary particles are arranged in a four-by-four table:

Let us recall that, as discussed above, the Israelites encamped around the Tabernacle on all four sides, with three tribes on each side:

• the camp of Judah on the east: Judah (the leader), Issachar, and Zebulun;
• the camp of Reuben in the south: Reuben (the leader), Simeon, and Gad;
• the camp of Ephraim on the west: Ephraim (the leader), Manasseh, and Benjamin; and
• the camp of Dan in the north: Dan (the leader), Asher, and Naphtali.

The four families of Levites (including the Aaronites) also immediately surrounded the Tabernacle in four camps—each family on one of the four side of the Tabernacle. Taken together, as in the Standard Model, the Israelites camping in the Sinai Desert are arranged in a four-by-four

formation:

Interestingly, the Standard Model does not explain its own four-by-four arrangement. In fact, this arrangement seems accidental to the Standard Model, which classifies elementary particles into three generations of fermions, each with four particles—two quarks and two leptons—and four bosons (or five, depending on whether we count the W and Z bosons as distinct particles or as two forms of a single quantum of the weak field). The arrangement of these particles in a four-by-four grid is done mostly for clarity and simplicity, but this arrangement doesn’t have any physical significance in terms of the structure of the model. The arrangement of particles in the Standard Model is primarily dictated by their properties and the nature of their interactions. Quarks participate in strong, weak, and electromagnetic interactions, whereas leptons participate in weak and electromagnetic interactions (neutrinos only participate in weak interactions). The three generations are identical in terms of the types of particles they contain and the interactions these particles participate in, but differ in the masses of the particles. In contrast, the four-by-four arrangement of the encamped Israelites, in addition to the arrangement by the tribes and families of Levites, has an additional explanation in Jewish mysticism, where everything is rooted in the four letters of the Tetragrammaton—the proper name of G‑d.

In Kabbalah, it is axiomatic that all of creation—both of spiritual and physical worlds—is rooted in the four letters of the Tetragrammaton—primeval creative energies. This esoteric doctrine leads us to another incredible parallel. As mentioned above, all fermions (particles of nature) are divided into two families—quarks and leptons. The first generation of fermions (shown in the first column of Table 2) has two quarks and two leptons—four particles altogether. In our metaphor, these four particles correspond to the four letters of the Tetragrammaton. But where do we see the division of these four particles into two families—quarks and leptons? Sure enough, the Tetragrammaton is also divided into two units of two letters each—yud and heh (Y‑H),^[4] and waw and heh (W‑H), so the parallel still holds.  The unexpected depth of this parallel suggests that it is not coincidental.

But the connection doesn’t stop here. As Kabbalah explains, the first group, Y‑H, represents the hidden world (alma d’itkasia—the “world of Concealment”). The second group, W‑H, represents the revealed world (alma d’itgalia).^[5] And this is exactly how it plays out in the material world of elementary particles. Quarks are hidden; they have never been observed because they exist only in combinations, such as in protons or neutrons, which are composed of quarks. They cannot be observed directly—they are hidden—like the alma d’itkasia (“hidden world”). Electrons and neutrinos, on the other hand, are ordinary particles observed in nature—they exist in the alma d’itgalia (the “revealed world”).

We also notice that just as among bosons, the Higgs boson, the so-called G‑d particle, stands out as the quantum of the field that endows other particles with their masses, so too among Levites, the Aaronites. stand out as priests.

The Problem with the Standard Model

The search for basic building blocks of matter can be traced back to antiquity. Ancient Greek atomists saw all matter as made of four basic elements—fire, water, air, and earth. Other cultures, including Hellenic Egypt, India, and Judea, produced similar ideas. By the nineteenth century, chemists believed all matter was made of 100 or so indivisible atoms. Now we know better. Now we know better. We know that atoms are made of electrons orbiting around the nuclei, which, in turn, are made of protons and neutrons. But, as it turned out, this was not the last word in this story. According to the Standard Model, even protons and neutrons are not elementary (indivisible) particles but are comprised of quarks. Quarks, along with electrons, are indeed elementary—that is, indivisible—particles. Other elementary particles include neutrinos—very light neutral particles that weakly interact with matter.

When speaking of four elements, Greek atomists got something right—the number of the basic building blocks of our ordinary matter turned out to be four—up quark, down quark, electron, and neutrino—the elementary particles that occupy the first column of Table 2.

But what about the rest of the fermions? The charm quark is almost identical replica of the top quark (that is, it has the same properties) with one difference—it is heavier than its sibling. Likewise, the strange quark replicates the down quark but is heavier; and the muon, which is almost an exact replica of the electron (it has the same spin and charge), only heavier; and the muon neutrino, which is an exact but heavier replica of the neutrino (also called the electron neutrino). These particles occupying the second column in the fermion table are much heavier than their counterparts in the first column (but identical in all other respects), and are unstable—they quickly decay into their stable siblings. These particles are called second-generation fermions.

The third-generation fermions occupy the next column in the table. The strange phenomenon of heavier clones plays out here as well: these particles are identical to their ordinary siblings from the first column, but much heavier and even more unstable. Second and third-generation particles do not form atoms and are not found in the universe except in cosmic rays (and created artificially in particle accelerators). They exist for split seconds, only to quickly decay into ordinary quarks, electrons, and neutrinos—the first-generation particles. We don’t have a clue why nature would produce second- and third-generation clones, which seemingly play little, if any, role in the universe.

The Jewish esoteric doctrine of Kabbalah may shed some light on this mystery.

The Second Parallel in the Seder Hishtalshelut

Jewish philosophers and mystics also spoke of four elements—aish (“fire”), ru’ach (“air”), mayim (“water”), and afar (“dust”).^[6] While the names may have been borrowed from the Ancient Greeks, the meaning was quite different. To call them “elements” is a misnomer. In Hebrew, they are called arba yesodot (“four foundations”). Instead of indivisible elements (atoms) of matter, in these four elements, Jewish mystics saw four spiritual energies corresponding to the four letters of the Tetragrammaton—yud-heh-waw-heh (Y-H-W-H). Fire corresponds to the letter yud (Y), air to the first heh (H), water to waw (W), and dust to the second heh (H).^[7] All matter would ultimately emerge from these four foundational energies or what contemporary physicists would call fields.^[8]

According to Arizal, the universe of Tikkun is made of four worlds—Atzilut (the world of “Emanation”), Beriyah (the world of “Creation”), Yetzirah (the world of “Formation”), and Assiyah (the world of “Action”). This hierarchical structure is called Seder Hishtalshelut (an ontological chainlike order of creation). Each of these four worlds corresponds to one of the four letters of the Tetragrammaton (Y-H-W-H): Atzilut (the world of “Emanation”) to yud (Y), Beriyah (the world of “Creation”) to the first heh (H), Yetzirah (the world of “Formation”) to waw (W), and Assiyah (the world of “Action”) to the second heh (H). Let us summarize these correspondences in a table:

The four foundations (elements) are present in each of the four worlds. We have, therefore, a primeval hierarchy of spiritual energies or fields organized in a four-by-four table:

Structurally, this table is the mirror reflection of Table 2 of the Standard Model—the first column in Table 5 corresponds to the fourth column in Table 2; the second column in Table 5 corresponds to the third column in Table 2; the third column in Table 5 corresponds to the second column in Table 2; and the fourth column in Table 5 corresponds to the first column in Table 2. If we read Table 2 right to left, as we read Hebrew, the two tables are structurally identical. This establishes a clear structural parallel between the arrangement of elementary particles in the Standard Model and the arrangement of four elements in the four worlds of Seder Hishtalshelut.

One does not need to be a person of faith or a spiritual person to see this parallel—however surprising, it is undeniable. One can also understand it in a metaphorical sense. But for a spiritual person, this is more than a parallel or a metaphor. The spiritual structure summarized in Table 5 may be seen as the primordial spiritual source of the Standard Model. The first column represents the spiritual source of the bosons—quanta of fundamental fields. The world of Atzilut precedes the creation—it is the world of pure emanation, the world of spiritual energies, which correspond to fundamental fields: Aish of Atzilut is the source of photons, Ru’ach of Atzilut is the source of gluons, Mayim of Atzilut is the source of W and Z bosons, and Afar of Atzilut is the sours of Higgs bosons.^[9]

The world of Beriyah is the world of creation. It is in this world that the spiritual source of matter is first created. Aish of Beriyah can, therefore, be seen as the source of the top quark, Ru’ach of Beriyah—as the source of the bottom quark, Mayim of Beriyah—as the source of tau, and Afar of Beriyah—as the source of tau neutrinos. What we call in the Standard Model the third generation of particles is actually the first generation of proto-particles. In quantum field theory, mass is measured in electron-volts—units of energy. The world of Beriyah is the first created world—the rarefied world of high energy. This may help us to understand why corresponding third-generation particles are so heavy (that is, high-energy) and unstable—there is no place for coarse matter in Beriyah. The source of the second generation of particles—charm quark, strange quark, muon, and muon neutrino—are proto-particles of the world of Yetzirah. Only the proto-particles of the lowest world—Assiyah—give rise to stable matter: up and down quarks, electrons, and neutrinos—relatively light particles appropriate for the relatively low energy of Assiyah—the only world in which matter can exist.

The structural parallel between the Seder Hishtalshelut (the chainlike ontological order of creation) and the Standard Model sheds light on the mystery of the second and third generation of particles.^[10]

Conclusion

Structural parallels are not just an exercise in curiosity—they often offer rich metaphors and symbols that help advance our understanding or provide intuitive insight into what otherwise appears as a highly abstract mathematical model. An amazing parallel between the Standard Model on the one hand and the encampment of the Israelites in the Sinai Desert, the arrangement of the four elements in the four spiritual worlds, and the structure of the Tetragrammaton on the other is nothing short of amazing. These parallels enrich our understanding of the Standard Model, at least on a qualitative level, and provide powerful intuition into the mathematics of particle physics.

Endnotes

[1] Spin is a quantum-mechanical analog of angular momentum.

[2] This principle was formulated in 1925 by the Jewish-Austrian physicist Wolfgang Pauli.

[3] The top quark (t) sometimes is called “truth,” and the bottom quark (b) sometimes is called “beauty.” Note that the names of quarks are completely arbitrary and do not represent any real properties of quarks. There is no “up” or “down” in the quantum world.

[4] This is a distinct name of G‑d—one of the seven principal names of G-d—pronounces as “Kah.” We frequently find this name in Psalms. For example, “Extol He Who rides upon the highest heavens with His Name, ‘Kah’” (Psalms, 68:5). The book of Psalms ends with the verse, “Let every thing that hath breath praise the Lord (Kah). Hallelujah (praise-Kah)” (Psalms 150:6).

[5] Rabbi Schneur Zalman of Liadi, Tanya, Likutei Amarim, 26.

[6] Midrash Rabba, Numbers, 14:12.

[7] Zohar Vayera 23.

[8] Sha’ar HaGilgulim (Gate of Reincarnations) 18:4. (See also “The Four Elementals,” Chabad.org, https://www.chabad.org/kabbalah/article_cdo/aid/1372760/jewish/The-Four-Elementals-184.htm, retrieved December 5, 2021.)

[9] On an intuitive level, this taxonomy offers an insight into why Higgs bosons endow other particles with mass. As we see in our parallel, Higgs bosons correspond to the element of afar (“dust” or “earth”). This element corresponds to the sefirah of Malchut, also associated with the planet Earth. Intuitively, we associate mass with earth, because mass is proportionate to weight (according to the principle of equivalence, one of the cornerstones of Einstein’s general theory of relativity), and the weight of an object is the measure of the gravitational attraction of this object to Earth. Thus, it is fitting that Higgs bosons, which are parallel to afar, are responsible for the mass of all elementary particles.

[10] On a more speculative basis, we should note a possible parallel between the division of all elementary particles into two groups—bosons and fermions—on the one hand, and the division of the Tetragrammaton into two groups of letters on the other. All elementary particles possess an important characteristic—spin—the quantum analog of classical angular momentum. Thanks to spin, electrons act as little magnets, as if they were spinning around their axes. In reality, electrons do not spin, but they do possess a magnetic field. The nature of the spin is not relevant to the present discussion. Suffice it is to say, all bosons have integer spin (e.g., photons, gluons, and W/Z bosons have spin 1, Higgs bosons have spin 0, and hypothetical gravitons—spin 2), but all fermions have spin ½. Recall that in our model, the spiritual source bosons is in Atzilut—the wholesome world of perfection and unity. There are no halves they—everything is whole. Fermions, on the other hand originate in the three worlds of BiYA (Beriyah, Yetzirah, and Assiyah. These worlds, where the divine light gets progressively diminished, are characterized by fragmentation and “halfness.” This is, for example, where the concept of half-coin (machtzet hashekel) originates. It is, therefore, fitting that the particles originating from these lower worlds would have their spin of one-half.

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* Brout-Englert-Higgs mechanism is responsible for the mass of the elementary particles. While mass is often confused with weight, the two have very different meanings. Weight is the measure of gravitational attraction on an object to Earth and is proportional to the gravitational mass of the object—the gravitational equivalent of an electric charge that describes how the object reacts to the gravitational field. In contrast, the mass (more properly described as the inertial mass) is the measure of resistance to a change in motion—the greater the inertial mass of an object, the greater force is required to cause the object to accelerate (or decelerate, or change its course). Brout-Englert-Higgs mechanism that endows particles with (inertial) mass in mathematically complex. However, to have some intuitive understanding of how it works, we can imagine the Higgs field filling the space with some sticky substance, which creates “friction.” Particles interacting with the Higgs field get caught up in this sticky substance. Moving through this substance causes “friction,” which results in inertia—the resistance to a change in motion. This mechanical analogy is just an oversimplified metaphor, but it helps visualize the Brout-Englert-Higgs mechanism. Indeed, while there is no mechanical friction involved, particles moving in space interact with the Higgs field, which, through the exchange of energy, slows down moving particles creating resistance to a change in motion that we call inertia.