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) 

Alternative to Marriage

"I've come to New York to introduce an alternative to marriage." --Jon Keats

With five millennia of history, and a plethora of religious and civil ceremonies, marriage is a popular means of producing families. Yet matrimony isn't the only method of uniting people, nor even is it the most effective technique. Modern science suggests a far more profound alternative, one that does not operate by religious tradition or civil mandate, but rather bonds couples by a law of nature: quantum entanglement. More here.

With Usura

Mug Shot: Ezra Pound (1885-1972)

For an exceptional back-in-time experience -- to hear Ezra Pound reading his own work --- visit the University of Pennsylvania's PennSound Ezra Pound Page. Grander than the Beatles and performing without a back-up track --- EZRA POUND LIVE! --- the poet himself chants With Usura in this mp3 recording. An extraordinary man raising his solitary voice against the enslavement of men and women by economic institutions, Pound's old-fashioned words ring as soundly today as a quark explosion in the Large Hadron Collider.

Ezra Pound
Canto XLV
WITH USURA

With usura hath no man a house of good stone
each block cut smooth and well fitting
that delight might cover their face.

WITH USURA
hath no man a painted paradise on his church wall
harpes et luz
or where virgin receiveth message
and halo projects from incision.

WITH USURA
seeth no man Gonzaga his heirs and his concubines
no picture is made to endure nor to live with
but it is made to sell and sell quickly.

WITH USURA
sin against nature
is thy bread ever more stale rags
is thy bread dry as paper,
with no mountain wheat, no strong flour.

With usura the line grows thick
with usura is no clear demarcation
and no man can find site for his dwelling.
Stonecutter is kept from his stone
weaver is kept from his loom.

WITH USURA
wool comes not to market
sheep bringeth no gain with usura
Usura is a murrain, usura
blunteth the needle in the maid’s hand
and stoppeth the spinner’s cunning.
Pietro Lombardo came not by usura
Duccio came not by usura
nor Pier della Francesca; Zuan Bellin’ not by usura
nor was La Calunnia painted.
Came not by usura Angelico; came not Ambrogio Praedis,
No church of cut stone signed: Adamo me fecit.
Not by usura Saint Trophime
Not by usura Saint Hilaire,
Usura rusteth the chisel
It rusteth the craft and the craftsman
It gnaweth the thread in the loom
None learneth to weave gold in her pattern;
Azure hath a canker by usura; cramoisi is unbroidered
Emerald findeth no Memling.

Usura slayeth the child in the womb
It stayeth the young man’s courting
It hath brought palsy to bed, lyeth
between the young bride and her bridegroom.

CONTRA NATURAM
They have brought whores for Eleusis
Corpses are set to banquet
at behest of usura.

[And so that you don't continually misunderstand. Usury and interest are not the same thing. Usury is a charge made for the use of money regardless of production and often without regard even of the possibilities of production. I now repeat the theme in Canto LI.]

Quantum Magic Confirmed

Alexander Klyachko, the "K" in the KCBS inequality

Quantum mechanics, the most powerful theory of the material world that humans have ever possessed, comes with a heavy price--the renunciation of knowing what is really going on in the world. Use this theory, give up reality. And perhaps the most peculiar fact about the quantum reality mystery is that it's hidden in plain sight. The quantum math is clear, complete and unambiguous; we can calculate with perfect precision exactly what will happen in any experiment that we can imagine. So even though we cannot visualize what's happening "behind the scenes" of the phenomena that we observe, we can use the theory itself to constrain any model that we might wish to construct about the "reality that exists beneath the quantum facts." The results of such indirect reasoning about deep reality have been posted. The conclusion: quantum reality is "magic".

Physics is all about making measurements. But quantum theory says that for every measurement you might like to make (say position) there is a COMPLEMENTARY measurement (in this case momentum) whose accuracy is disturbed by the first measurement. The existence of complementary pairs of observables is an entirely quantum feature not present in classical physics. But for most quantum systems, one can often pick out a set of "compatible observables",  that can be measured without mutual interference. Such measurements form a kind of "classical sector" in the full quantum realm, and one might suppose that the reality underlying this classical sector could be explained using classical concepts.

The Kocken-Specker theorem, published in 1967, proves this supposition to be false. Even the classical sector of quantum physics cannot be explained by an underlying classical reality. It can only be explained by "magic". Simon Kocken tried to evoke the flavor of this "quantum magic" in his "Parable of the Over-protective Seer".

In ancient Nineveh there lived an ingenious wizard-astrologer whose beautiful daughter was besieged by many suitors. To prevent his daughter from marrying an unworthy man, the wizard devised a "suitable suitor" test. On a table in the royal interview room, the wizard placed three jewel boxes. The suitor's task was to state "FULL" or "EMPTY" and then select the two boxes that confirmed his prediction. If, for instance, the suitor said "FULL" and both his chosen boxes contained jewels, then he could claim his princess bride. Dozens of men attempted the test--with precisely the same result. Whatever pair of boxes a suitor would choose, one always contained jewels, the other turned up empty. 

Since there is no way that anyone can fairly prepare three boxes, without two of them being either full or empty, pure logic suggests that the seer used "magic"--some sort of trickery--as part of the "reality" behind his test.

Quantum theory is strange but not strange enough to build such boxes. However Simon Kochen (an American mathematician) and Ernst Specker (a Swiss mathematician)--using the same sort of reasoning that proves the fictional seer must use magic--proved that the quantum world must use a similar "magic", even in its "tame" classical sector, to produce its quantum facts.

[In technical terms, the K-S theorem proves that, even in the classical sector, where all measurements "mutually commute", no NON-CONTEXTUAL REALITY can underlie the quantum facts. In a non-contextual world what box I choose to open (what measurement I choose to make) does not affect the contents of the other boxes (does not affect the outcome of other compatible measurements.) This may sound subtle--and indeed it is. But this is REALITY that K&S are talking about--not mere phenomena. And any knowledge we might gain about reality is certainly worth paying attention to--knowledge about reality being infinitely more precious than jewels in some mad wizard's box.]

K&S's original proof was exceedingly complicated. It involved a diagram with 117 "rays". Their proof was simplified by Asher Peres to 36 rays, then to 18 rays by Adan Cabello, then to 8 rays by Rob Clifton.

Recently Vladivostok-born Alexander Klyachko entered the frey. Klyachko (a visiting math professor at Bilkent University in Ankara, Turkey) devised a proof of the K-S theorem using only 5 rays. Moreover he expressed his proof in the form of a testable inequality (the KCBS inequality).

If a physics experiment obeys the KCBS inequality, then you don't need "magic" to explain it.

However if a physics experiment violates the KCBS inequality, then (by reasoning of the kind that exposed the seer's trickery) the reality behind that experiment must be "magic".

Klyachko's 5-ray proof works for the smallest quantum system (a spin-1 particle) that satisfies the K-S postulates, so his demonstration will no doubt enter the record books as the most concise proof of the existence of "quantum magic". Nature is full of spin-1 particles, but to do complex experiments on such particles is devilishly difficult. Recognizing the practical difficulties of testing his KCBS inequality on actual spin-1 systems, Klyachko cleverly devised an equivalent physical system composed of two photons in the same spatial mode (the so-called "biphoton") that possess exactly the same external quantum properties as an actual spin-1 system.

Inspired by Klyachko's work, Radek Lapkiewicz and colleagues in Anton Zeilinger's lab at the University of Vienna recently prepared a similar optical analog to the spin-1 particle using the considerable resources of modern quantum optics. The LLSLRWZ collaboration tested the KCBS inequality and demonstrated that the inequality was violated. Hence reality has to be contextual. Reality cannot be modeled by ordinary classical concepts. Reality, in short, must be "magic".

In light of this blog's tantric theme and the parable of the magic jewel boxes, one is tempted to suppose that, like Simon Kochen's over-protective seer from Nineveh,  Mother Nature might be using "quantum magic" to prevent adolescent apes with plutonium from tinkering with reality's deep primordial innocence.

Some secrets may be only suitable for grownups.

The Lapkiewicz et al test of KCBS inequality in Vienna

Kubla Khan

Dr. Jabir's KUBLA KHAN: a modern version of Samuel Coleridge's celebrated Pleasure Dome poem conceived during an opium vision--Coleridge's setting it down in writing cut off abruptly by a visit from some anonymous "person from Porlock". Could Jabir in his dervish trance have recaptured in this short video a few of Coleridge's lost stanzas?

Today's the 3rd birthday of this Quantum Tantra blog. This year I added "tags"--a kind of crude table of contents--for more convenient exploration of older posts.

Hippies in the Sunday Times

David Kaiser, MIT science historian, author of How the Hippies Saved Physics

Continuing praise for MIT historian David Kaiser's How the Hippies Saved Physics.

Here's the book's webpage with pictures and reviews.


The buzz for David's book began in the blogs--in my own blog, in Peter Woit's and Chad Orzel's. (Everyone is eagerly awaiting an endorsement from Lubos Motl, controversial physics blogoczar and kingmaker to the lucrative Czech science book market).


That initial buzz around David's book has now expanded to a ROAR with a review by George Johnson in this Sunday's New York Times. A book review in the Sunday Times is the literary equivalent of a private interview with the Pope.

Thank you, George.


George Johnson lives in Santa Fe, New Mexico and is the prize-winning author of The Ten Most Beautiful Experiments, an acclaimed biography of Nobel physicist Murray Gell-Mann as well as many other well-crafted pieces of science writing.

George Johnson concluded his mostly sympathetic review of Kaiser's book with the judgment that much of the work of "the hippies who saved physics" was mere "physics porn"--"titillating but with no follow through".

How could Johnson have known in his witty putdown that one of the goals of my own quantum tantra research is to discover a more intimate way of connecting with nature than merely reading the results of classical instruments?

In writing his review, George Johnson knew nothing of Lifting the Veil of a New Sensual Science, of Tantric Jihad, of Future Science, No More Safe Science, of Physicist, Heal Thyself or of Opening Night.

Unaware of Harlot Nature, Johnson was able to accurately intuit my work as a species of "physics porn".

Will future generations recognize science journalist George Johnson not only for his substantial body of science writing but also as an unconscious herald of the freshest and most unlikely revolution in modern physics?

George Johnson, author of The Ten Most Beautiful Experiments

When the Impossible Happens

August O'Connor encountering a Groffian impossibility

How would history have changed had Sigmund Freud's favorite drug been LSD instead of cocaine?

Using lysergic acid instead of analysis to access the unconscious mind, Freud's career might have resembled Stan Grof's as recounted in When the Impossible Happens. Freud might have escaped his dreary office in Vienna, lived 14 years at Esalen Institute, taken peyote with Indians in Kansas, MDA in Palenque and ketamine in Machu Picchu, guided hundreds of patients thru the tangled jungles of their own personalities using LSD sessions, Holotropic breath work and other radical mind-opening techniques. Like Freud, Grof brings a sophisticated European mind set to bear to evaluate a lifetime of extraordinary experiences. Unlike Tim Leary and Terence McKenna, who were of Irish blood, Stanislav Grof was born in Czechoslovakia and possesses a "Slavic soul" which binds him solidly to the soil even as his yearnings for the infinite bring him face to face with experiences that would shake the sanity of many less solidly grounded men.

I'm sure that if Stan were to encounter Satan (which he does in one of the episodes in this book) he would do so calmly, all senses alert, pushing his fear into the background while striving rationally to assess the true dimensions of the experience. Stan is a compassionate scientist and brave explorer of the heights and depths of human consciousness. Of all of his many books, When the Impossible Happens is the most biographical and offers personal stories of the most astonishing high points of his long career as an passionate courtier of non-ordinary realities.

His book consists of more than 60 short episodes, from Stan Grof's first LSD experience as a fledgling doctor in Prague, to his years with his wife Christina as scholar-in-residence at Esalen, to psychic adventures in Iceland, Brazil, Mexico and Australia. This is a book of experiences not theory. And, as the title suggests, many of his experiences would be considered "impossible" according to our present model of the human mind which holds that consciousness is solely the product of some (as yet unknown) process in the brain--a position called "reductive materialism" by its supporters and "promissory materialism" by its critics. Just as the goal of modern physics is to explain not only the ordinary world but every sort of outlandish physical phenomena made accessible by Hubbles, LHCs and even newer probes on the horizon, so any serious psychologist (mind scientist) should welcome this account of dozens of exotic psychological "quarks" and "gluons" observed during a lifetime of exploring the extreme edges of human experience.

Stan is an splendid story teller and one finds oneself experiencing the emotions that he must have felt as the events were actually happening. I was pleased to discover his account of an extraordinary adventure in which I played a small role--Stan's marriage to Joan Halifax which took place in a volcanic caldera in Iceland, in a Viking wedding ritual designed by Icelandic mythologist Einar Pallson and Joseph Campbell, who gave away the bride. My wife Betsy and I brought the traditional child to the wedding--our son, Khola, who had just celebrated his second birthday a day before with the children of the kitchen staff. We wedding guests arranged ourselves in a ring around the edge of the caldera and after the ceremony, Stan and Joan went around the circle to be congratulated. At the moment when Betsy, Khola and I embraced them, the midnight sun which had temporary dipped below the horizon, suddenly rose and flooded the five of us with its brilliant light.

I know Stan slightly but over the years our paths have crossed several times since that first meeting in Iceland. As a fellow Slav and a member of his wedding I feel a personal bond. I admire him as an expecially bold scientist and always feel a bit high when coming into his presence.

Nick's not the only one who admires Stan Grof. In 2007, Vlaclav Havel, the president of Czechoslovakia, presented the VISION 97 award to Stan for his pioneering research into non-ordinary states of consciousness.

When the Impossible Happens is one of the best tour guides I can recommend into the details of some of those non-ordinary states.

Next month Dr. Stanislav Grof will experience his eightieth trip around the Sun.

Happy Birthday, Stan!

Stan Grof at Esalen SuperSymposium IV