Sirag Triples

Physical gyroscope and iPad with a digital gyroscope inside.

Nature seems to like to spin. Rotating systems are everywhere, from the Sun, the planets, the moons and Our Galaxy itself, to Ferris wheels, bicycle wheels, hard drives and CDs. The classical physics
that describes rotating systems such as gyroscopes and the Earth is particularly elegant. Classical rotation is certainly admirable but the quantum mechanical description of rotation is one of the most beautiful cultural achievements of mankind comparable to a grand cathedral or symphony.

The first quantum discovery concerning rotation is that spin is quantized. Spin only exists in nature as integral multiples of Planck's constant h. (Systems with integral values of spin are called Bosons; a second class of systems exists called Fermions consisting only of spins with 1/2-integral values.) For the sake of simplicity I will limit this discussion to Bosons.

An object can be spinning with its spin axis pointing in any of three directions. The direction of the axis defines the "direction of spin" Physicists call these 3 axes X, Y, and Z. In accord with the picture above of a spinning iPad, I will also call these three axes Red, Green and Blue (RGB). What it means for a spin to be quantized is that if we measure the spin of an object along the Red direction, the only values we will ever get are 0, 1, 2, 3, 4 ... units of Planck's constant. Spin is digitized!

Any rotating system can be conveniently labeled by its TOTAL SPIN "J" where J is related to the spin values (X), (Y) and (Z) along the X, Y and Z axes by the formula:

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

The sum of the squares of the values of the spins in all three directions is equal to J(J+1). The reason why the sum of the three squares is equal to J(J+1) instead of J^2 is a quantum thing which I take on faith but which I have never been able to understand.

Besides quantization, the next most important quantum discovery is a fundamental linitation on what you can measure and what you cannot -- an extension of the Heisenberg uncertainty principle to rotating systems. For a spin system, what you can measure is the total spin J, plus one of the three Spin components. If you choose to measure Red Spin, then Green and Blue Spin become uncertain.

If you choose to measure Green Spin, then Red and Blue Spin are fuzzy. This Heisenberg restriction is sometimes explained by saying that a Red measurement "uncontrollably disturbs" the Green and Blue Spins. But for me a better way of thinking about this restriction is that quantum systems with one definite spin direction actually exist in nature. But quantum systems with two definite spin directions do not. To ask what is the value of the Green Spin after I have measured the Red Spin is like asking how long is the fourth side of a triangle. Four-sided triangles do not exist; neither do rotating quantum systems that possess two or more well-defined spin directions.

A classical system possesses all of its properties in a well-defined manner, whether we observe these properties or not. For a quantum system only some of its properties are well defined (sometimes called "good quantum numbers"); the remaining properties dwell in a peculiar sort of quantum limbo similar to the fate of Dr Schrödinger's famous cat.

A general spin system possesses two good quantum numbers: its total spin J and the value of one of its spin components, Red Spin, for instance. Thus a typical quantum system may have a (total) spin of 3 Planck units and a Red Spin value of +2. For a spin-3 system, the only possible values that Red Spin is allowed to take are -3, -2, -1, 0, +1, +2, +3. The plus and minus signs indicate whether the spin is in a clockwise or counterclockwise direction.

Summing up this quantum spin lesson. Quantum systems have two good quantum numbers: Total Spin "J" and Red Spin "X" (where "Red" could be any one of the three basic directions.)

Recess: I learned recently that every new iPad has three digital gyroscopes inside it, that measure the gadget's rotation in the X, Y and Z direction. These gyroscope are not made of rotating chunks of metal but are ingenious computer chips called MEMS devices (short for Microelectromechanical Systems). How this digital gyroscope actually works is described here in great detail. Despite its name, the digital gyroscope is not digital in the quantum sense, rather it's a clever implementation of classical mechanics.

Return to class: Quantum mechanics says that deep down every spinning system is a digital gyroscope with certain natural restrictions on which of its properties you can measure.

Although a general quantum gyroscope possesses only two good quantum numbers "J" and "X", there may exist certain very simple systems that (accidentally) possess more than two good observables.

For instance a system with J = 0, possesses no spin at all. Its total spin is zero. And its spin in all three directions is zero. That's 4 good quantum numbers instead of two for this simple system.

Next up is a system with J = 1. If we measure Red Spin and get 0, we know that the Green and Blue spins must be some fuzzy mixture of +1 and/or -1. That is, we know the absolute value of these unmeasurable spins but do not know their sign. This amount of knowledge is equivalent to saying that when we know that Red Spin = 0, we also know that the SQUARE of the Green Spin = 1. And the SQUARE of the Blue Spin =1. So the spin-1 system (because it's so simple) possesses FOUR good quantum numbers, the number J = 1 and the SQUARES of the spins in all three directions.
 

These three square have the value 1, 1, 0 and satisfy the fundamental spin addition formula (EQ 1). Namely, for a spin-1 system (EQ 1) reads: 1 + 1 + 0 = 2.

The spin-2 system or higher spins do not seem simple enough to possess any more good quantum numbers than the standard two-to-a-customer measurement restrictions that apply to all sufficiently complicated rotating systems. So regular physics stops here. And sci-fi physics begins.

Both the spin-0 system and the spin-1 system permit the SQUARES of the spins in all three directions to be good quantum numbers. Suppose there existed a new kind of quantum physics (call it LEGO PHYSICS) for which the squares of all spins were good quantum numbers. How far could LEGO physics be pushed before it ran into trouble?

How do we test LEGO physics? We pick a total spin value, say J = 3, assume the squares of all the spin components are good quantum numbers. For J = 3, these squares can only take the values 0, 1, 4 and 9. The test of spin-3 LEGO physics is whether some combination of these 4 numbers can be found which will satisfy the fundamental spin addition formula (EQ 1). For a Spin-3 system, this is possible: 4+ 4 + 4 = 12 does the trick. 

The question "How far can LEGO physics be pushed?" leads to what I have called the "Sirag Numbers" in honor of Saul-Paul Sirag, a brilliant mathematical physicist whose birthday is August 31. A Sirag Number is defined as any integer J, for which the LEGO conjecture fails. That is, if the sum of three squares that obey the rules of quantum mechanics cannot be made to sum to J(J+1), then that integer J is a Sirag Number. The story of the Sirag Numbers can be picked up here.

Since Saul-Paul's birthday is approaching, I was thinking about giving him another math present.

Let's use up one of the good quantum numbers and set Spin (Z) = 0. Then the fundamental spin equation reduces to:

(X)^2 + (Y)^2 = J(J+1)       (EQ 2)

We apply the LEGO conjecture which allows both (X)^2 and (Y)^2 to be good quantum numbers and ask the question: For what values of X, Y and J, do integral solutions exist? 

Although this question arises (as did the Sirag Numbers) in the context of VERY DUBIOUS PHYSICS, it is a well-posed mathematical question which may lead to interesting results.

The question of the existence of integer solutions to (EQ 2) closely resembles the question of the existence of PYTHAGOREAN TRIPLES (X, Y, Z) which satisfy the simple formula:

X^2 + Y^2 = Z^2      (EQ 3)

This expression (EQ 3) is the Pythagorean expression for the sum of the squares of the lengths of the sides of a right triangle. A Pythogorean triple is a set of three numbers for which the sides of a right triangle are whole numbers, such as the classic 3-4-5 right triangle.

The question of what integers satisfy the LEGO model formula is almost identical to the question of the existence of Pythagorean triples. Indeed the "LEGO formula" (EQ 3), loosely derived from quantum gyroscopes, might be construed as the "quantum version" of the Pythagorean Theorem.

In honor of Saul-Paul's upcoming birthday, I would like to designate any 3-tuple of integers (X, Y, J) that satisfies the "LEGO formula" (EQ 2) a SIRAG TRIPLE. Simple examples of Sirag triples include (0, 0, 0), (1, 1, 1) and (9, 3, 9). Just as there now exists a formula for generating all Sirag Numbers, there must also exist a formula that generates all Sirag Triples, but I don't yet know what it might be.

Happy Birthday, Saul-Paul.

Sauk-Paul Sirag lecturing at Esalen Institute, Big Sur.

Sirag Numbers Enter the Canon

Saul-Paul Sirag lecturing at Esalen Institute

The Sirag Numbers, first arising in the context of "accidental commutation" in the quantum theory of angular momentum (and dedicated to Saul-Paul on his 72nd birthday), were first defined on this blog and partially "tamed" by myself and Saul-Paul Sirag.

Recently these numbers have attracted the attention of the larger mathematical community (thank you, Antti Karttunem) and are now listed in the Online Encyclopedia of Integer Sequences as Integer Sequence A196224. The Sirag Numbers were "completely tamed" by Jack Brennan, Ray Chandler, Charles P. Greathouse and Harvey P. Dale who devised formulas which generate all Sirag numbers. Happy birthday, Saul-Paul! Now that this sequence has been officially recognized, the one thing lacking is to discover an application for Sirag Numbers in the real world.

Also in the news this week, sci-fi author Rudy Rucker paid me a visit at the Quantum Tantric Ashram where we discussed philosophy and literature in my redwood forest retreat. Over tea and stollen, Rudy presented me with copies of his book of paintings Better Worlds and his latest novel, set in Santa Cruz CA, Jim and the Flims which chronicles a throroughly Ruckeresque vision of the afterlife. Hint: the Rucker afterlife is not even close to what you've been taught in school; it more resembles some seriously twisted DMT trip.

Also this weekend Beverly Rubik, a colleague from the Consciousness Theory Group and the Esalen Seminars on the Nature of Reality, invited me to Berkeley to give a talk at her Institute for Frontier Science.

Dr Beverly Rubik, PhD

I took Beverly's invitation as an opportunity to express my latest conceptions about quantum tantra and to read from my work-in-progress Harlot Nature. The talk was well-attended by a responsive crowd which included Dana Ullman, Jim Johnson and Allison Kennedy, the co-founder of the innovative Berkeley-based psychocyberdelic magazine Mondo 2000.

Nick performing at Institute for Frontier Science

After the talk Beverly treated Allison and I to supper at P. F. Chang's Chinese Bistro in Emeryville. Walking to Chang's in the dark our group was surprised by what appeared to be a hidden portal to another reality but when entered turned out to be a time tunnel memorializing the landscape's early inhabitants. Inside the portal, a parade of black stone slabs, each etched with text and pictures, told the story of Emeryville's former human habitation, brief by geological standards. A beautiful and informative piece of public art--the city of Emeryville's Shellmound Memorial.  

Entrance to Emeryville Shellmound Memorial

September Events

Nick contemplating the Sirag Numbers

First of all, I would like to thank all the peeples dat remembers my birthday this year. And thanks for the many sweet birthday gifts including the opportunity to handle an 1895 Winchester 45-70 rifle

On Sunday, Sept 18, Nick will be the Featured Reader at Poet/Speak, a monthly event hosted by Poetry Santa Cruz. Reading will occur at 2 PM at Santa Cruz Main Library Meeting Room, 224 Church St. Santa Cruz, CA. Open Mike signup. A rare chance to experience quantum tantra live.

On Tuesday, Sept 20, MIT professor David Kaiser will describe his new book "How the Hippies Saved Physics" at the University Club in San Francisco from 6 to 9 PM. Some of the "hippies" will be present for interrogation including Jack Sarfatti, Fred Allen Wolf and Russell Targ. $25 including refreshments. For more info contact Michael Sarfatti at sarfatti@alum.mit.edu.

The Sirag numbers have been partially tamed! Mark Buchanan and Dick Shoup produced a list of SN up to 2995 from which Saul-Paul Sirag extracted several quasi-periodic interval pattern of period 32. From Saul-Paul's data, Nick Herbert constructed this morning the basic equations that all Sirag numbers must satisfy. The Herbert equations classify all Sirag numbers as Primary SNs or Secondary SNs. For the Primary SNs the Herbert equations generate true Sirag numbers. For the Secondary SNs, the Herbert algorithm generates true Sirag numbers but also false ones. However the Herbert classification is exhaustive--any true Sirag number will be generated by one of the Herbert algorithms.

The Primary Sirag numbers (SPs) fall in two classes, Even and Odd. Their defining algorithms are:

SE(n) = 12 + 32n where n = 0 -> N

SO(n) = 12 + 32n - 25 where n = 1 -> N

The Even Sirag numbers SE(n) repeat with a period of 32 beginning with the first Sirag number J*(1) =12. The Odd Sirag numbers SO(n) repeat with a period of 32 beginning with J*(3) = 19. Thus the primary Sirag numbers are represented by two superposed periodic sequences separated by the interval "7". These two equations generate true Sirag numbers but fail to generate ALL SIRAG NUMBERS. For instance J*(2) = 15 is not a member of SO(n). J*(2) is a Secondary Sirag number (SS).

The Secondary Sirag numbers (SSs) fall into four classes--SS3, SS4, SS12 and SS13. These SNs are generated by the four equations:

SS3(n) = SO(n) - 3

SS4(n) = SO(n) - 4

SS12(n) = SE(n) -12

SS13(n) = SE(n) - 13

Any Sirag numbers will be found to be described either as a SP or a SS. Thus this classification is exhaustive. The first set of equations (SPs) can be used to generate Sirag numbers; the second set (SSs) are useful only for classification--some of the numbers generated are not true Sirag numbers.

As an example of this classification scheme we can now recognize J*(2) as SS4(1). The famous Sirag number "1939" representing the year of Saul-Paul's birth turns out to be the Primary Sirag number SO(61). Does the largest computed Sirag number "2995" belong to one of these sets?

The skeptical reader may wish to check Herbert's claims by testing to see if any Sirag number refuses to be pressed into one of these six classifications.

Herbert's equations for the Sirag numbers are a generalization of Saul-Paul's discovery that the intervals between consecutive Sirag numbers are dominated by "7"s and "25"s. And that these intervals are further haphazardly divided into sub-intervals "3 + 4 = 7" and "12 + 13 = 25". And these intervals are the ONLY INTERVALS that appear. No pattern has yet been discovered in the distribution of subintervals, Hence the present lack of an algorithm that will generate all Sirag numbers.

Now shout it from the rooftops:

The Sirag Numbers are (partially) tamed!

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