The Entangled Twins

Entanglement is often called the most baffling and the most quintessential aspect of quantum mechanics. What is entanglement, in a nutshell? Two particles born out of one reaction (or two particles that interacted through a collision) remain connected, no matter how distant from each other. A change in the status of one particle instantaneously causes a change in the status of the other particle. Einstein called it “spooky action at a distance.”

Entanglement is often associated with a certain symmetry and corresponding conservation laws. For example, the law of conservation of angular momentum requires that the spin (the quantum-mechanical analog of the angular momentum) of two entangled particles always point in the opposite directions. This means that, if two entangled particles have their spin in a state of superposition of Up (↑) and Down (↓), and we collapse the wavefunction of one of these particles thereby fixing its spin, say, in the Down (↓) direction, the wavefunction of the other twin-particle is immediately collapsed as well, fixing the spin in the opposite, Up (↑), direction. The entangled particles act as twins that instantaneously react to changes in each other’s mood. One difference between entangled particles and twins is that entangled particles do not look like each other, as do identical twins. In fact, they are often a mirror (i.e., opposite) image of each other—the consequence of the underlying symmetry.

The Torah portion Toldot is all about one set of biological twins who, like our entangled particles, were complete opposites. Speaking of Rebecca’s difficult pregnancy, the Torah says,

And the children struggled within her… And the Lord said to her, “Two nations are in your womb, and two kingdoms will separate from your innards, and one kingdom will become mightier than the other kingdom, and the elder will serve the younger… and behold, there were twins in her womb. (Gen. XXV, 22-24)

Two twin brothers are born, Esau and Jacob. The twins share many characteristics of entangled quantum-mechanical objects. The entanglement of the twins is hinted at by the Torah in the later verse:

…[H]is brother [i.e., Jacob] emerged, and his hand was grasping Esau’s heel. (Gen. XXV, 22-26)

Twins, as the entangled particles, remain connected!

Their oppositeness is further emphasized as they grow up:

And the youths grew up, and Esau was a man skilled in hunting, a man of the field, whereas Jacob was an innocent man, dwelling in tents. (Gen. XXV, 27)

There is an apparent symmetry — as twins, they can be swapped with no change to the sum — there would still be two sons born to Rebecca, one Jacob and another Esau, one righteous and one evil. It is this symmetry that allows Jacob later to actually play out this swapping by pretending he was Esau to obtain his father’s blessing.

There are conservation laws at play here as well. For example, Esau sells his birthright to Jacob. He can’t just gift it to Jacob, he must receive a compensation for his birthright to obey the conservation law. As a result of the transaction, Esau loses his birthright and Jacob gains it—the symmetry is preserved!

                 Selling His Birthright

As another example, after Isaac blesses Jacob, who was pretending to be Esau, and the real Esau belatedly demands his blessing, Isaac tells him:

Your brother came with cunning and took your blessing. (Gen. XXVII, 35)

And here comes an astonishing manifestation of true entanglement. Just as with the entangled particles, if one has the spin Up (↑) the other must assume the spin Down (↓), or vice versa, so here too, Isaac tells his son, Esau:

Behold, I made him a master over you… and you shall serve your brother, and it will be, when you grieve, that you will break his yoke off your neck. (Gen. XXVII, 37-40)

If the fortune of Jacob is up (↑), he is a master over Esau, and the fortune of his brother, Esau, is down (↓). But if Jacob misbehaves and loses his merit to have the blessing, the fortune of Esau can turn up (↑), thereby making Jacob subservient to him and bringing his fortune down (↓). We have seen this played out in Jewish history time and again.

This story ends with a puzzling statement of Rebecca who tells Jacob:

Behold, your brother Esau regrets [his relationship] to you [and wishes] to kill you…Why should I be bereft of both of you on one day? (Gen. XXVII, 42-45)

This is hard to understand. If Esau were to kill Jacob, Rebecca would lose only one son, as she would still have Esau. Why would she be “bereft of both”? We can easily understand it from the point of view of quantum mechanics. One particle can destroy another particle, with which it is entangled only if it is its anti-particle, say, electron and positron. The collision of a particle and anti-particle causes the annihilation of both particles. Indeed, evil Esau was an “anti-particle” of righteous Jacob. Thinking about Esau’s murderous plans to kill Jacob, was Rebecca worried about “the thrust of two-widows” with both twins simultaneously killing each other?

Symmetry plays a vitally important role in physics in general and in quantum mechanics in particular. However, symmetry breaking is even more important in quantum field theory and the Standard Model. In a particular type of symmetry breaking, called spontaneous symmetry breaking, the underlying laws remain invariant (i.e., unchanged) under symmetry transformation, but the resulting system is asymmetrical. It is spontaneous symmetry breaking that is responsible for the Higgs mechanism and Higgs Boson, the so-called “G‑d Particle,” that endows particles with mass. In the Torah portion Toldot, we have an example of such spontaneous symmetry breaking.

The Haftorah (weekly reading of the Prophets) connected to this Torah portion begins with the following paragraph:

I loved you, said the Lord, and you said, “How have You loved us?” Was not Esau a brother to Jacob? says the Lord. And I loved Jacob. And I hated Esau… (Malachi I, 2-3)

This is a beautiful example of spontaneous symmetry breaking. The underlying laws of family relations are symmetrical — “Was not Esau a brother to Jacob?” Yet, at the end, G‑d “spontaneously” chooses Jacob, breaking the symmetry.

This concept plays out explicitly on Purim. Haman, who wanted to destroy the Jewish people, dared not attack them directly — he knew full well that the G‑d of the Hebrews, Almighty G‑d, would protect them. He devised an ingenious plan of reaching the spiritual level where Jacob and Esau were brothers and there was no difference for G‑d between Jews and Gentiles, or so he thought. “Was not Esau a brother to Jacob?” He devised a kind of symmetrical dice (in Hebrew, “pur” — hence Purim) that he used to cast to determine the month and day of the attack. He relied on symmetry to reach a level where he would be unopposed by G‑d.

Apparently, Haman did not study quantum field theory. He knew nothing about symmetry breaking — “And I loved Jacob. And I hated Esau” — that was his mistake that ultimately caused his downfall. The moral of this story: it pays to study physics!