The Sirag Numbers

Saul-Paul Sirag, a "hippie that saved physics"

Yesterday, I issued a challenge to my physicist friends to solve a problem that, as Jeffrey Bub informed me, does not even exist. My challenge was based on the fact that for quantum spins 1/2 and 1, the squares of the spin components J(x), J(y), J(z) commute although the spin components themselves do not. I erroneous believed that this result was TRUE FOR ALL SPINS--what I called the "Commutation Conjecture" and asked for either a proof of this conjecture or a refutation. Within a few hours Jeffrey Bub (physicist at University of Maryland) replied that this conjecture was false and that no one in the field had ever believed otherwise. Casey Blood (physicist formerly at Rutgers) also sent a refutation a few minutes later. This contest is now closed.

My so-called "challenge" was a mistake from the start, but it led to an interesting discovery--the "Sirag Numbers"--named after my colleague Saul-Paul Sirag now living in Eugene, Oregon. Conversations with Saul-Paul concerning the quantum spin operators spurred my own refutation of the "Commutation Conjecture" that invokes a brand-new set of numbers.

If we assume that the Commutation Conjecture is true--that for every spin J, the operators J(x)^2, J(y)^2, J(z)^2 commute, then these spin squares are simultaneously observable. And furthermore these three observables must add up to J(J+1).

Now define the Sirag number J* such that the spin squared components of J* no matter how selected cannot be made to sum to J(J+1). If a Sirag number exists then the Commutation Conjecture is refuted because it leads to a contradiction.

Proof that "3" is a Sirag Number (ie, that its spin components squared cannot add up to 3(3+1)

The spin components of a spin-3 system are 3, 2, 1, 0, -1, -2, -3.

The squares of these components are 9, 4, 1 and 0.

It is impossible (by inspection) for 3 of these numbers to sum to 12.

Thus 3 is a Sirag number and the Commutation Conjecture is refuted.

Several questions immediately present themselves:

1. Is the number of Sirag Numbers finite or infinite?
2. Does an algorithm exist for calculating all J*?
3. Have these numbers been discovered before?
4. Is "137" a Sirag Number?

A little research shows that "3/2" is the smallest Sirag Number and that "3", "7/2" and "12" also belong in the class of Sirag Numbers. Today the largest known Sirag Number is "12" but that record is not likely to stand for long.

[Breaking news: the Sirag Numbers have been redefined to exclude 1/2 integer values; "3" has been shown NOT to be a Sirag Number; Mark Buchanan, president of Optical Alchemy, using an ad hoc
iPad app, has declared the lowest THREE Sirag Numbers to be 12, 15 and 19.

The Sirag Number J* is now officially defined by this equation:

X^2 + Y^2 + Z^2 = J*(J* + 1)

A integer J* is a Sirag Number, if no triplet (X, Y, Z) of integers exists that satisfies this constraint.

Saul-Paul's birthday is August 31 and Nick's birthday gift to him is this set of numbers.]

HAPPY BIRTHDAY, SAUL-PAUL!

[[ Sirag Number cognescenti will not fail to appreciate that the Buchanan Program, through some lucky circumstance, has revealed 1. J*(1) = 12, the first EVEN Sirag Number. 2. J*(2) = 15, the first ODD Sirag Number and 3. J*(3) = 19, the first PRIME Sirag Number. The BIG QUESTION now is: What is the value of J*(4), the fourth Sirag Number? ]]

[[[ Late Breaking News!!! On Saul-Paul's birthday, Mark Buchanan calculated ALL SIRAG NUMBERS between 0 and 1000. A partial list: 12, 15, 19, 44, 51, 63, 76, 83, 108, 112, 115, 140, 143, 147. Sad to say, Saul-Paul's favorite number 137 is not on this list, but the numbers 371 and 911 do make an appearance. ]]]

Direct from Mark Buchanan's smoking iPad: the first 92 Sirag Numbers:

12 15 19 44 51 63 76 83
108 112 115 140 143 147 172 179
204 211 236 240 243 255 268 271 275
300 307 332 339 364 368 371 396 399
403 428 435 448 460 467 492 496 499
524 527 531 556 563 575 588 595
620 624 627 652 655 659 684 691
716 723 748 752 755 780 783 787
812 819 844 851 876 880 883
908 911 915 940 947 960 972 979