Shemini,Phase Space,Hamiltonian Mechanics: Kabbalah and Physics of Seven and Eight

By Alexander Poltorak

Abstract

The Torah portion of Shemini records the climactic eighth day of the Tabernacle’s dedication, the moment when divine fire finally descends. Classical commentators see in the contrast between the first seven days and the eighth the difference between the natural and the supra‑natural. This essay revisits that symbolism from the standpoint of contemporary physics. First, it reassesses the traditional link between seven and the “natural order.” Second, it proposes that the six spatial–momentum coordinates of classical phase space, together with time, provide a more precise physical analogue for the seven midot than the usual six directions of space plus a temporal axis. Finally, it explores an unexpected resonance between the Hamiltonian action integral and the Tetragrammaton, suggesting that the principle of stationary action mirrors the scriptural affirmation, “For I Y‑H‑W‑H do not change.” In so doing, it offers a fresh perspective on how the eighth day of Shemini hints at teleology—the purposeful influx that cannot arise endogenously from nature but must be bestowed ab extra.

1.   Introduction: Shemini and the Mystery of the Eighth Day

The book of Leviticus culminates the inauguration of the Tabernacle with a dramatic narrative (Leviticus 9:23‑24). For seven days, Moses alone erected, dismantled, and instructed. On the eighth (shemini), Aaron officiated, heavenly fire consumed the offerings, and the divine presence became manifest. Keli Yakar^[1] and R. Bahya ben Asher^[2] read this delay as emblematic: seven signifies the enclosed cycle of nature, while eight signals an order that transcends it. Yet the usual supporting examples—seven days of the week, seven years of the Sabbatical cycle (shemitah), and seven shemitot of Jubilee—belong to covenantal time‑keeping, not to physics. Does the created world itself whisper the secret of seven and eight?

2.   Why Seven? An Initial Challenge

Outside a few charming coincidences—seven colours in the visible spectrum, seven notes in the diatonic scale—hard physical laws seldom privilege the integer 7. Nor does Scripture suggest that rainbow optics or musical acoustics compelled the divine choice of a week. The received explanation, therefore, risks circularity: religious observance proves seven’s “naturalness,” which in turn validates religious observance. A more robust account should locate the number in nature’s own architecture.

3.   Space, Time, and Sefirot

As I explained in detail in my earlier essay, “The Mystery of the Eighth Day,” the number seven is related to the six directions in space, plus time; the number eight adds purpose. To briefly recap, we live in a three-dimensional space. The number 6 plays a significant role in a 3-dimensional space. In such a space, every cube has six surfaces. Another way to look at it is that, in a three-dimensional space, we encounter six directions.

Kabbalah long ago correlated the six midot—Ḥesed, Gevurah, Tiferet, Netzaḥ, Hod, and Yesod—with the six “depths” enumerated in Sefer Yetzirah 1:5: up, down, east, west, north, and south. Adding Malchut (often identified with temporality) yields seven.

In Kabbalah, these six directions correspond to six midot (“measures,” lower sefirot): Ḥesed, Gevurah, Tiferet, Netzaḥ, Hod, Yesod—that are viewed in the Lurianic Kabbalah as six extremities of the partzuf Ze’ir Anpin (Z”A). Indeed, Kabbalah views space is being constructed from these six midot. Time comes from the last midah (“measure”)—Malḥut—the Nukvah (female partner)of the Z”A.

The six directions in space (or six midot of the Z”A) plus time (or Malḥut) make up the number seven. This is one way to justify the number seven as symbolic of the natural order.

The number eight, on the other hand, adds a purpose. A purpose cannot be an emergent phenomenon and must be imposed on the system from the outside—either by humans, if the system is human-made, or by G‑d, if the system is made or ordained by G‑d—in either case, it must come ab extra.

4.   Phase Space

In addition to the parallel described above, I would like to propose an alternative approach to arriving at the number seven that offers a better fit. For this, we need to consider what is known as a “phase space.”

Simply speaking, phase space is a way to completely describe the state of a physical system by plotting all its key properties simultaneously.^[3] To specify the micro‑state of a point particle, one records both its position (q_x,q_y,q_z) and its momentum (p_x,p_y,p_z). The result is a six‑dimensional manifold Γ. (See, my essay, “The Symbolism of the Menorah.”

The phase space’s dimensionality is directly related to a system’s degrees of freedom. Degrees of freedom refer to the number of independent parameters needed to completely specify the state of a physical system. For instance, a single particle moving in three-dimensional space has three degrees of freedom for its position (x, y, z coordinates) and three more for its momentum or velocity components (p_x, p_y, p_z). These two triplets of coordinates in the phase space are called “generalized coordinates” and traditionally denoted as q₁,q₂,q₃,p₁,p₂,p₃. Crucially, these six generalized coordinates separate naturally into two analogous triads—q-space and p-space—mirroring the twin triads that form the body of Ze’ir Anpin in the Etz Ḥayim diagram.

The dimensionality of the phase space is then twice the number of degrees of freedom. This happens because for each degree of freedom (each independent way the system can move or change configuration), we need to track both its position and momentum. So, for a system with N degrees of freedom, the phase space will have 2N dimensions.

As we see, the number 6 appears naturally in the phase space as its dimension for a single particle moving in a 3D space. We can also conjecture that the six extremities of the partzuf Ze’ir Anpin represent six dimensions in the phase space.

If I may be so bold, I would suggest that phase space aligns more closely with the sefirotic model of Kabbalah than our ordinary three-dimensional space. Although the origin of the notion that the six directions of space are parallel to the six extremities of Ze’ir Anpin goes back to at least Sefe Yetzirah, one of the oldest and most authoritative books of Kabbalah, this parallel omits some details. The six extremities of Zeir Anpin—the six sefirot (or midot) from Ḥesed to Yesod—are arranged in two triads: Ḥesed–Gevurah–Tiferet and Netzaḥ–Hod–Yesod. In the graphical depiction of the sefirot as the Etz Ḥayim (the Tree of Life), these two triads are represented by two identical triangles, one positioned beneath the other.

In both descriptions—as two triads or two triangles—the 3+3 symmetry is inescapable. This structure is absent in the 3D physical space. Furthermore, the second triad of the sefirot, Netzaḥ–Hod–Yesod, is very similar to the first triad, Ḥesed–Gevurah–Tiferet. Both Ḥesed and Netzaḥ are located in the right column of the sefirotic tree, representing centripetal forces, extravert tendencies, and the desire to give. On the other hand, both Gevurah and Hod are located in the left column of the sefirotic tree, representing centrifugal forces, introvert tendencies, and the desire to receive. Tiferet and Yesod are located in the central column of the sefirotic tree, representing the balance of opposing tendencies and the resulting harmony.  So, what makes the second triad “lower” than the first triad? The sefirot of the second triad are derivative of the sefirot of the first triad in a sense that they mean essentially the same thing with one difference—the sefirot of the first triad give (Ḥesed) or withhold (Gevurah) without concern for the receiver (Malḥut); the sefirot of the second triad give (Netzaḥ) or withhold (Hod) commensurate with the abilities and needs of the receiver (Malḥut).

Incredibly, this relationship is precisely mirrored by the phase space. The second triplet of generalized coordinates, p₁,p₂,p₃, indeed comprises derivatives of the first triplet, q1,q2,q3. The first triad in Ze’ir Anpin merely stakes their positions or natural tendencies to give, withhold, or harmonize. Similarly, q-coordinates merely establish the position of the particle in space. The second thread modulates the actions towards the receiver. Similarly, momentum (p) coordinates measure the particle’s projected influence upon its surroundings—the impulse it will impart.

The fit is so precise that it makes me think that, perhaps, we can mathematically model sefirot as functions, where Malḥut (the receiver) serves as the independent variable parallel to time, and all other sefirot are modeled as functions of Malḥut (whereas in phase space p and q are functions of time). This fits very well with the kabbalistic notion that time originates in Malḥut. In this case, the second triplet of the extremities of Ze’ir Anpin—sefirot Netzaḥ–Hod–Yesod—can be thought of as the first derivatives of sefirot of the first triad of sefirot Ḥesed–Gevurah–Tiferet with respect to Malḥut.^[4] This model helps understand the spiritual dynamic of sefirot with mathematical precision.

In any event, the 6 generalized coordinates of the phase space together with time make up the number 7, which indeed appears to represent the natural process (for a single particle).

5.   Hamiltonian Mechanics

Hamiltonian mechanics describes dynamics in terms of the Hamiltonian function, which represents the total energy of the system.^[5] The action S functional is its integral along a path.^[6] Nature selects the trajectory for which δS = 0. In other words, the action is stationary across the true path. The principle of stationary action is the guiding principle of Hamiltonian mechanics. (See my essays, “Principle of Least Action I,” “Principle of Least Action II — Introduction,” “Principle of Least Action III — History,” and “Principle of Least Action — IV Lagrangian Mechanics.”)

It seems that the Tetragrammaton, the four-letter proper name of G‑d (Y-H-W-H), is parallel, as it were (kaviyahu), to the action functional. Kabbalah attributes an all‑inclusive character to the four‑letter Name. Yod embodies Ḥokhmah; the first Heh, Binah; Waw, whose gematria is six, gathers the six midot; the final Heh stands for Malḥut. The Tetragrammaton thus sums the entire sefirotic process.

The prophecy, “For I the Lord (Y‑H‑W‑H) do not change,” (Malachi 3:6) proclaims a constancy of G‑d. If one reads as the metaphoric analogue of the Name—an integral that enfolds every energetic manifestation—then the principle of stationary action becomes a physical echo of that scriptural principle of immutability. The created world unfolds along trajectories that leave the eternal divine signature invariant.

6.   The Eighth Coordinate: Purpose as an External Parameter

Phase space plus time suffice to encode every lawful evolution. What they do not encode is telos—an overarching aim. Teleology, by definition, is imposed from beyond the closed system. In the narrative of Shemini, that purpose manifests in the heavenly fire on the eighth day. The Mishkan’s architectural and ritual perfection still required an external gift from above to burst into life. Likewise, physics knows no equation for goals; it can trace how energy disperses, but not why it should realize a given value. The number eight, therefore, points to the extrinsic variable that tips a completed mechanism into meaningful function.

Conclusion

Classical commentators intuited that the number seven denotes the cycle of nature and eight its transcendence. Modern mechanics reframes that intuition with sharper contours. The six canonical coordinates of phase space, marching with time, yield a mathematically rigorous sevenfold structure that aligns with the kabbalistic midot more closely than the older six‑direction model. The Hamiltonian action integral, moreover, invites a daring identification with the Tetragrammaton, rendering the principle of stationary action a scientific midrash on Malachi 3:6. When the Torah announces on the eighth day that “fire came forth from before the Lord,” it signals the arrival of purpose—an incursion that physics cannot predict, yet whose possibility its very formalism seems prepared to host.

Bibliography

Bahya ben Asher. Commentary on the Torah. Leviticus 9.

Keli Yakar. Commentary on Leviticus 9:23‑24.

Poltorak, Alexander. “The Mystery of the Eighth Day,” QuantumTorah.com, April 14, 2024, (https://quantumtorah.com/the-mystery-of-the-eighth-day/).

Sefer Yetzirah. Critical edition and English translation by Aryeh Kaplan. New York: Weiser, 1990.

Malachi. Tanakh: A New Translation of the Holy Scriptures according to the Masoretic Text. Philadelphia: Jewish Publication Society, 1985.

Poltorak, Alexander. “The Symbolism of the Menorah,” QuantumTorah.com, June 23rd, 2024 (https://quantumtorah.com/the-symbolism-of-the-menorah/).

Goldstein, Herbert. Classical Mechanics. 3rd ed. Reading, MA: Addison‑Wesley, 2002.

Landau, L. D., and E. M. Lifshitz. Mechanics. Vol. 1 of Course of Theoretical Physics. 3rd ed. Oxford: Butterworth‑Heinemann, 1976.

Liboff, Richard. Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. New York: Springer, 2003.

Poltorak, Alexander. “Principle of Least Action I,” QuantumTorah.com, September 4th, 2023 (https://quantumtorah.com/principle-of-least-action-i/); see also “Principle of Least Action II — Introduction,” September 6, 2023 (https://quantumtorah.com/principle-of-least-action-ii-introduction/), “Principle of Least Action III — History,” eptember 10th, 2023 (https://quantumtorah.com/principle-of-least-action-iii-history/), and “Principle of Least Action — IV Lagrangian Mechanics,” October 2nd, 2023 (https://quantumtorah.com/principle-of-least-action-iv-lagrangian-mechanics/).

Zohar. Sefer ha‑Zohar, ed. Daniel Matt. Stanford: Stanford University Press, 2004–2017.

Endnotes

[1] Rabbi Shlomo Ephraim ben Aaron Luntschitz (“Keli Yakar”) (c. 1550 – c. 1625) was a prominent kabbalist and biblical commentator who lived in Safed and is best known for his mystical commentary on the Torah bearing the name “Keli Yakar.”

[2] Rebbeinu Baḥye ben Asher (c. 1255 – c. 1340) was a prominent Spanish rabbi, kabbalist, and ethical philosopher best known for his influential ethical work, Chovot HaLevavot (Duties of the Heart) and his biblical commentary.

[3] Imagine you’re tracking a bouncing ball. To fully describe its state at any moment, you need to know both, where it is (its position) and how fast it is moving (its velocity). Phase space is a mathematical “map” that shows both these properties simultaneously. Every possible state of your ball becomes a single point in this phase space. As the ball bounces, it traces a path through this space. For the bouncing ball, we would have a 2D graph where one axis shows position and the other axis shows velocity. In this simple example, three dimensions of space (x,y,z) were collapsed into a single dimension (x), and the three components of velocity (v_x,v_y,v_z) were collapsed into a single component (v_x). If we were to plot all of these components, we would have to depict a six-dimensional space, which is not possible to fit on a plane. But generally, a single particle has six-dimensional phase space where the first three dimensions are ordinary spatial dimensions and the other three dimensions represents three components of the momentum (the product of mass m and velocity v) of the particle.The power of phase space is that it reveals patterns that aren’t obvious when looking at position or velocity separately. For instance, a pendulum swinging back and forth creates a closed loop in phase space, showing its repeating behavior.

[4] If the generalized coordinates are denoted as q₁,q₂,q₃,their first derivates over time are traditionally denoted as dot on top of the letter: , , . Thus, ,  ,  . Similarly, if we denote the sefirot as follows: Ḥesed = x, Gevurah = g, Tiferet = t, Netzaḥ = n, Hod = h, and Yesod = y, then we can write: , , , where the dot above the letter means the first derivative over Malḥut.

[5] In classical Hamiltonian mechanics, the Hamiltonian for a single particle is a function that represents the total energy of the particle. It is typically expressed in terms of the particle’s generalized coordinates (position) and their corresponding conjugate momenta. For a single particle of mass (m) moving in three-dimensional space under the influence of a potential energy field (V(q), where (q = (q₁, q₂, q₃) = (x, y, z)) are the Cartesian coordinates, the Hamiltonian (H) is given by: H(q,p,t)=T(p)+V(q,t), where:

• T(p) is the kinetic energy of the particle, expressed as a function of its conjugate momentum p = (p₁, p₂, p₃) = (p_x, p_y, p_z). In Cartesian coordinates, the kinetic energy is:

• V(q, t) is the potential energy of the particle, which can be a function of its position (q) and possibly time (t) if the forces are time-dependent.

Therefore, the Hamiltonian for a single particle in Cartesian coordinates is:

In more general curvilinear coordinates (q_i), the form of the kinetic energy in terms of the conjugate momenta (p_i) will be different, depending on the metric tensor of the coordinate system. However, the fundamental definition of the Hamiltonian as the sum of kinetic and potential energy remains the same. The conjugate momentum (p_i) is generally defined as , where L is the Lagrangian of the system.

[6] The action functional is its integral along a path:

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