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 

Quantum Spin Challenge

Diana Warnock spinning in Boulder Creek

The quantum theory of angular momentum is one of the most beautiful creations of the human mind. Furthermore the math of quantum spin is a closed book--all its beauty published in plain sight like the rules of the game of chess, enshrined in textbooks, this wide-open wonder is even taught to undergraduates--no more mystery, quantum spin's intrinsic beauty diluted by familiarity.

As one of its initiates, I believed I knew all the quantum rules. But I was surprised by a startling paper by Russian physicist Alexander Klyashko who, in deriving a new proof of the Kochen-Specher theorem, seemed to invoke a new rule in the quantum theory of angular momentum--a rule I had never heard of.

For physicists only:

"Everyone knows" that the spin components J(x), J(y), J(z) do not commute. Consequently, when one of these spins is measured, the other two must remain uncertain--a generalization of the Heisenberg Uncertainty Principle to spin systems. Heisenberg says: Only COMMUTING OBSERVABLES can be simultaneously observed.

Also "everyone knows" that if you choose one spin component, say J(z), that the SQUARE OF THE TOTAL SPIN J^2, also commutes with J(z), so both J^2 and J(z) can be simultaneously measured. In
the quantum jargon, J^2 and J(z) are said to be "good quantum numbers".

But Klyashko claims (unless I am mistaken) that there is another set of mutually commuting spin variables that is not in any textbook.

Klyachko claims that the SQUARES OF ALL SPIN COMPONENTS mutually commute.

Now it is easy to verify that this assertion (which I am attributing to Klyashko) is true for spin-1/2 and for spin-1 systems by direct calculation. But is (what I am calling) the Klyashko Assertion "Commutation Conjecture" true for all values of spin, or is it only true for a few special cases?

Using the standard spin calculus, I have tried and failed to prove "Klyachko's Assertion" the Commutation Conjecture.

[One of my teachers was Sid Drell who introduced us to the physicist's version of the mathematician's "impossibility proof". Drell's version goes like this: "I'm a really smart person. And I can't prove this. Therefore it must be impossible."]

Now since I'm really smart. And I can't prove the "Commutation Conjecture" for all spins, I conclude (using the Sid Drell criterion), that the Commutation Conjecture is false.

And I hereby issue "the quantum spin challenge" to my colleagues who are more intimate than I with the open beauty of quantum angular momentum mechanics.

1. Either prove the Commutation Conjecture for all spin values.

2. Or come up with a counter-example--a spin system whose spin components squared do not commute.

First person to successful meet the Quantum Spin Challenge and send their proof to me at quantaATcruzioDOTcom will receive a copy of "Physics on All Fours", a chapbook of quantum tantric rants illustrated by my son Khola.

Who will be first to take up this challenge?

Gentlemen (and ladies) start your (quantum spin) engines.

[Within 24 hours, Jeffrey Bub at U of Maryland informed me: 1. that Klyashko makes no such claim  (hence the name change from Klyachko's Assertion to Commutation Conjecture) and 2. demonstrated that the CC holds only for spin-1/2 and spin-1 and that furthermore this fact is well known in the field so that 3. my "Quantum Spin Challenge" arises only out of Nick Herbert's ignorance of the facts and hence 4. my Challenge is (as Jeffrey so kindly puts it) "not needed". Jeffrey wins the prize and I thank him for correcting my foolishness. I apologize to Alexander Klyachko for connecting his name with this whole sordid mess.]

QT Blog: The Origin Story

Mudhead--Hopi Kachina clown and trickster

Nick Herbert never intended to start a blog. As my blog site guru Harry Hutton remarked:

Been doing this nonsense for three years now, and where's it got me? Nowhere. It has simply widened the circle of people who think I'm a dick. That's all it has achieved.

My reluctance to enter the blogosphere was further strengthened when I discovered that the virtual real estate entitled "quantumtantra.blogspot.com" was already occupied by an entity who called itself "Mudhead" after the Hopi Kachina trickster spirit. Mudhead's site specialized in paying homage to the beauty of the naked female form, following Fugs-founder Tuli Kupferberg's mantra "God is a spread shot". Each day Mudhead's site featured lots of holy pictures from the Temple of Tuli, regular dispatches from a Canadian photographer obsessed with hi-res closeups of the vertical smile, as well as hot links to other sites with similar enthusiasms. After reading my favorite physics blogs I often enjoyed dipping into this blog for a taste of Mudhead's version of quantum tantra.

But about three years ago, Mudhead's site was suddenly abandoned without explanation and its domain name tossed up for grabs for any entity who wished to take possession. 

How could I resist? Mudhead had done a fine job in establishing a beach head into the unexplored quantum tantric realm and, faced with its vacancy, I simply could not allow lesser minds to pervert this 

splendid site. That in short is this quantum tantra blog's origin myth. Thank you, Mudhead, whoever and wherever you may be. And I hope you are not disappointed with what I have chosen to do with your site.

Tuli Kupferberg: "God is a spread shot."

Quantum Physics Tells Us

Quantum wave equation -- Erwin Schrödinger (1925)

QUANTUM PHYSICS TELLS US 
THE WORLD IS UNLIKE
ANYTHING OUR BODIES KNOW


The equations harbor what look like small
migratory birds blown in on stiff winds. 
What land could they possibly
have come from?  Just look at them,
pecking randomly at the white air,
feeding on something
we don’t see.

In those moments when you think
some part of this world belongs to you,
consider what we call atoms,
for which we have no adequate metaphor,
and of which we are made.
These fellow gods
will not be ruled by anything,
even our notion of cause.  They
are elemental wildness.
 
—Len Anderson

Len Anderson, Physicist (UC Berkeley PhD)