The Special Theory of Relativity, published by Albert Einstein in 1905, was a game changer. It ushered the era of new physics dubbed by some as the “Jewish physics.” And Jewish physics it was.

At the core of Special Relativity is the notion of the relativity of motion – all motion is not absolute, as Newton thought, but is relative to something else, to a particular point of view. In physics, we define a point of view as a frame of reference. If you are standing in a moving train, leaving the railroad platform, in your frame of reference (which is always at rest) the train is at rest relative to your frame of reference. The railroad platform, on the other hand, is moving with respect to your frame of reference. For someone standing on the platform waving goodbye, the railroad platform is, of course, at rest, and it is the train and you that are speeding away – all is relative.

Mathematically, the Special Theory of Relativity describes reference frames as coordinate systems in a four-dimensional spacetime called Minkowski space. Besides three familiar spatial dimensions x, y, z, (or up-down, forward-backward, and right-left):

it adds the forth temporal dimension t.

The laws of nature should be invariant with respect to coordinate transformation. This means that no matter what coordinate system you chose, it should not affect the laws of nature – they are the same at every point in space and every moment in time. In mathematics, this is called symmetry.

One type of symmetry in space is rotational symmetry. If we rotate our coordinate system around an axis, say the z-axis, this should leave something unchanged. That something is a vector in the Minkowski space. Herman Minkowski was Einstein’s math professor in college, who first proposed to unify space and time into a 4-dimensional spacetime continuum, the World, as he called it.

Rotations around an axis of the coordinate system leave the magnitude and the direction of a vector unchanged, or, in the language of mathematics, invariant. Any coordinate transformations in Minkowski space are called Lorentz transformations after Hendrich Antoon Lorentz, Dutch physicist who first mathematically described time dilation and length contraction at high velocities. Because there are three spatial axes around which we can rotate the coordinates, there are three such Lorent rotations.

Hendrik_Antoon_Lorentz

Hendrik Antoon Lorentz

Another type of symmetry is a Lorentz boost – the transition from one frame of reference to another frame of reference moving with a constant velocity along a straight line. Since such a frame can move with respect to our chosen frame of reference in three different directions, there are three such Lorentz boosts.

All such transformations that leave a vector of the Minkowski space invariant make up the so-called Lorentz group. Because there six types of such transformations – three rotations and three boosts – the Lorentz group has six parameters or six dimensions.

Henri_Poincare

Henri Poincaré

Besides rotations and boosts, there are other coordinate transformations – translations – when we move the whole coordinate system along one of the four axes (in three directions in space and in time). These four translations that leave the magnitude of a vector invariant (although they move the vector to another place) when added to the Lorentz group expand it to what is called Poincaré group. Henri Poincaré was a French mathematician who contributed to the development of special relativity.

Because there are four axes – three spatial and one temporal – there are four additional parameters in the Poincaré group. Together with six parameters of the Lorentz group, the Poincaré group has a total of ten parameters or ten dimensions. Let us summarize these parameters in the following table

Poincaré Group

Three rotations Three rotations around the X, Y, Z axes
Three boosts Tree Lorentzian boosts
Three translations Three translations along spatial axes X, Y, Z
One translation One translation along the time axis T
Ten Ten parameters or dimensions

 

Alternatively, we can group these parameters as

Three rotations Three rotations around the X, Y, Z axes
Three boosts Tree Lorentzian boosts
Four translations Four translations along X, Y, Z, T axes
Ten Ten parameters or dimensions
Emmy_Noether

Emmy Noether (1882-1935)

The Poincaré group describes all symmetries and, therefore, the geometry of the Minkowski space. Thanks to the famous Noether Theorem formulated by Jewish-German mathematician Amalie Emmy Noether, we now know that every symmetry corresponds to a physical conservation law. For example, the translational symmetry is responsible for the momentum conservation, rotational symmetry is responsible for the conservation of angular momentum, and time-symmetry is responsible for energy conservation. Thus, the Poincaré group not only defines the geometry of relativity but the physics of it as well. It is the foundation of the Theory of Relativity and Quantum Field Theory – the Standard Model. It is indeed the foundation of modern physics.

According to Kabbalah, the physical world we live in is a reflection of higher spiritual worlds that have a similar structure. It is logical to assume, therefore, that the spiritual space and time in the spiritual worlds above have the same geometry as the physical world below. If the physical world is just a reflection of the spiritual realm, we should expect to find the Poincaré group in the spiritual space.

The Kabbalah teaches us that the structure of the spiritual worlds is built around ten sephirot. (Few things irk me more than when people compare ten sephirot to ten dimensions in the string theory. It’s a tortured analogy – besides the number ten, they have nothing in common!)

Just the fact that there are ten sephirot and ten parameters of the Poincaré group would not be enough to draw an analogy. But let us look at the structure of the ten sephirot:

Ten Sephirot

Three Categories of ChaBaD Three intellects – Chochmah, Bina, Daat
Three Primary Emotions – HaGaT Chesed, Gevurah, Tiferet
Three Secondary Emotions Netzach, Hod, Yesod
One Female Principle Malchut
Ten Ten Sephirot

 

Or, alternatively,

Three Categories of ChaBaD Three intellects – Chochmah, Bina, Daat
Three Primary Emotions – HaGaT Chesed, Gevurah, Tiferet
Four lower sephirot – NeHYM Netzach, Hod, Yesod, Malchut
Ten Ten Sephirot

 

Sometimes, the four lower emanations, Netzach, Hod, Yesod, Malchut, are grouped together. For example, when discussing the differences between Shabbat, Sabbatical year, Shemita, and Jubilee, Yovel, the Arizal says that during Shabbat, four lower sephirot – Netzach, Hod, Yesod, Malchut (NHYM – pronounced Nehim)are elevated together (see Sefer Likutim, Behar). In the language of the Zohar, these for lower sephirot make up the “lower body.”

If you have not yet noticed the similarity, let me now bring ten parameters of the Poincaré group and ten sephirot together:

Number Parameters of Poincaré Group Sephirot
Three Three rotations around the X, Y, Z axes Three intellects – Chochmah, Bina, Daat (HaBaD)
Three Tree Lorentzian boosts Chesed, Gevurah, Tiferet (HaGaT)
Three Three translations along spatial axes X, Y, Z Netzach, Hod, Yesod (NeHY)
One One translation along the time axis T Malchut
Ten Ten parameters or dimensions Ten Sephirot

Alternatively,

Number Parameters of Poincaré Group Sephirot
Three Three rotations around the X, Y, Z axes Three intellects – Chochmah, Bina, Daat
Three Tree Lorentzian boosts Chesed, Gevurah, Tiferet
Four Four translations along spatial axes X, Y, Z, T Netzach, Hod, Yesod, Malchut
Ten Ten parameters or dimensions Ten Sephirot

We see that both the Poincaré Group and the Ten Sephirot can be presented as three triads plus the tenth element, which in the Poincaré Group is a translation in time and in sephirot is Malchut. The Kabbalah teaches that time comes from Malchut (of the world of Emanation – Atzilut). Alternatively, both constructs may be presented as two triads and a quartet – four translations in the Poincaré Group corresponding to four lower sephirot that move (translate) together in tandem on Shabbat. The first three sephirot, Chochmah, Bina, and Daat (ChaBaD) are parallel three rotations around axes X, Y, and Z. The second triad of sephirot, Chesed, Gevurah, and Tiferet (ChaGaT) are parallel three Lorentz boosts. And the four lower sephirot, Netzach, Hod, Yesod, and Malchut (NeHYM) are parallel four translations in Minkowski space. Thus, ten sephirot can be viewed as ten parameters (or dimensions) of Poincaré Group. It all now fits together.

These uncanny structural similarities suggest that indeed, just as in the physical space Poincaré group define the geometry, in the spiritual space, ten sephirot define ten parameters (or dimensions) and the structure of the Poincaré group. The Jewish physics indeed!

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