Saturday, August 29, 2015

JJCCTT Device for FTL Signaling

Omer and Nick's JJCCTT Device for FTL Signaling

JJCCTT Device for FTL Signaling

Bell's Theorem proves that quantum reality must be non-local.

Belfast-born physicist John Stewart Bell based his important proof about reality on the EPR Device (named after Einstein, Podolsky and Rosen) which uses TWO ENTANGLED PHOTONS whose wave function ψ can be written:

ψ (EPR) = 1/√2 ( |HH> + |VV> )          (EQ1)

Bell showed (and John Clauser subsequently measured) that this quantum wave function's statistical predictions exceed any result that any merely local reality is able to muster. Thus quantum reality is non-local.

Later Greenberger, Horne and Zeilinger used THREE ENTANGLED PHOTONS described by the GHZ wave function:

ψ (GHZ) = 1/√2 ( |HHH> + |VVV> )        (EQ2)

to prove a "Bell's Theorem without Inequalities". GHZ showed from EQ 2 that a local reality predicts a certain result will never happen, while quantum mechanics says that this same result must always happen. Since quantum mechanics gives the correct prediction, one measurement suffices to prove that quantum reality is non-local.

In recognition of GHZ's concise proof, the quantum state described by EQ 2 is usually called a "GHZ state".

Recently, Omer Dickstein, a physics student at Jerusalem College of Technology, proposed to exploit the non-locality exhibited by FOUR ENTANGLED PHOTONS for faster-than-light signaling. His wave function just adds one more photon to the GHZ state:

ψ (GHZ + 1) = 1/√2 ( |HHHH> + |VVVV> )        (EQ3)

As a place holder I have tentatively called EQ 3 the "GHZ plus one" state. How this state will eventually be designated will depend on what we learn from temporarily construing it as the core ingredient of an FTL signaling machine -- the so-called JJCCTT Device where JJCCTT stands for "Joint Jerusalem-California Collaboration on Transluminal Telecommunication".

The state |HHHH> represents the simultaneous emission of FOUR Horizontally-polarized photons, two of which are measured by Alice, and two of which are measured by Bob, each of whom possess two detectors that register "H" when accepting a Horizontal photon and "V" when accepting a Vertical photon.

The state |VVVV> represents the simultaneous emission of FOUR Vertically-polarized photons, two of which go to Alice, and two of which go to Bob.

One nice thing about the JJCCTT situation is that, unlike the EPR situation where Alice and Bob possess only one detector each (hence can obtain only ONE BIT of information), in the JJCCTT situation both particles possess two detectors (hence each can potentially obtain TWO BITS of digital information. The JJCCTT Device represents, information-wise, a more broad-band channel than its EPR competitor.

One might naively imagine that EQ 3 represents a situation in which the 4-photon source emits EITHER a pulse of 4 H-photons OR a pulse of 4 V-photons but that is not the case at all. In this peculiar process (called "polarization entanglement") illegal for all except quantum systems, the source emits BOTH a quadruple of H-photons and a quadruple of V-photons AT THE SAME TIME.

You might think of EQ 3 as describing a kind of "four-photon Schrödinger Cat state". When unlooked-at, "this cat" is in a superposition of both a 4H-cat and 4V-cat. (4H-cat has four Hazel (yellowish-brown) feet and 4V-cat has four Vermilion (yellowish-red) feet). Each foot simultaneously is both colors. That's when not looked at. But whenever it's looked at, whoever looks will always see this cat with each foot having the same color -- either four Hazel-colored feet or four Vermilion-colored feet -- no matter how far apart the cat's feet are. Bob and Alice might be 100 light-years apart and this feet-coloring process will happen exactly the same way.

According to quantum mechanics, when this ambiguous pulse of 4-light (quantum cat) encounters a detector, the detector "flips a coin" and randomly decides which one of these two possible polarizations it will "make real". Will it be the all-H-state or the all-V-state? It is important to understand that quantum theory tells us that this choice of what polarization will be recorded is made at the detector, not at the source.

Once this decision is made by one detector (we can never really identify which one), the other 3 detectors follow suit, so that each detector, no matter how far it might be separated from all the others, immediately comes to the same conclusion concerning which possible polarization state (H or V) it will also make real.

It is easy to see how this apparent instantaneous conspiracy between far-distant detectors to always record the same polarization, when up until the moment of choice both polarizations were actively possible, might embolden some physicists to attempt to exploit this system to send signals faster than light.

The configuration pictured above won't work as an FTL channel between Alice (the traditional sender) and Bob (the inevitable recipient), because EQ 3 allows only two elemental events to happen, either HHHH or VVVV, none of which are under the control of either Alice or Bob.

But here's how Omer from JCT in Jerusalem plans to change all that. We note that in the START STATE (pictured below), only two things ever happen to Alice and Bob. They either both receive 4 H-photons. Or both receive 4 V-photons. Bob always registers HH or VV in his two polarization detectors. And so does Alice. Never anything else.

START STATE: Only two events can happen, either HHHH or VVVV.

In particular Bob never observes a "cross term" such as HV or VH, where one of his two detectors counts a H-photon and the other counts a V-photon.

If there were something Alice could do with her photons that would produce "cross terms" in Bob's results, then Alice would be able to send a message to Bob at superluminal speed.

The essence of the JJCCTT Project is to examine all the things that Alice can do to her 2 photons, while looking for effects that Alice's actions might have on Bob's 2-photon cross terms.

Here's a hint about what Alice might do to induce cross-terms into Bob's detectors.

Alice might decide, for instance, to "make real" states of Right and Left Circularly polarized light rather than H and V polarized light as in the START STATE. Alice can easily configure her two detectors (using a phase plate and a beam combiner) to register R and L light rather than H and V light.

If Alice's R-and-L-making action causes any amount of R and L light to appear at either one of Bob's detectors, this will have drastic consequences. Because R light incident on an H/V detector (the kind Bob has deployed) will always produce a random mixture of H and V counts -- that will certainly lead to cross terms in Bob's data. Likewise L light incident on Bob's detectors will inevitably produce cross terms via the same procedure.

The 2-photon EPR situation offers some hope that this could happen, via a process that Irwin Schrödinger dubbed "steering". Starting with EQ 1, which represents a perfect correlation of H and V photons between Alice and Bob, Alice can transform her detectors from the Plane-Polarized basis H/V to the Circularly-Polarized Basis R/L, where R and L are Right- and Left-circularly polarized photons. This transformation turns EQ 1 into:

ψ (EPR) = 1/√2 ( |RL> + |LR> )          (EQ4)

One interpretation of EQ 4 is that by Alice's choice to measure R/L polarization rather than H/V polarization, she was able to "steer" Bob's distant photons from a mixture of the H/V eigenstates into a mixture of the R/L eigenstates.

Can the same "steering mechanism" that works for the EPR state work its magic on the GHZ +1 state? If Alice can steer even the tiniest fraction of Bob's H/V photons into a R state and/or a L state, then transluminal signaling will be accomplished via the instant appearance of cross terms {of the form HV or VH) in Bob's two HV detectors.

Any physicist familiar with purported FTL signaling schemes will reflexively credit such a vaguely plausible argument as no more than a hopeful conjecture. And will suspend judgement until seeing some actual calculations. Omer at Jerusalem Center for Technology and Nick at Quantum Tantra Ashram in California are currently calculating the 16-term quantum correlation matrix that encodes the full behavior of the JJCCTT device for whatever detector choices Alice can make to try to signal Bob. These are very elementary calculations. But it is easy to make mistakes.

For the record: It was Omer who suggested that the GHZ + 1 system might be a promising candidate for FTL signaling. And it was Omer who proposed that Alice-controllable cross-terms in Bob's HV detectors might function as an FTL signal. And it was Nick who suggested that Alice might use Schrödinger "steering" to remotely create cross-terms in Bob's HV detectors. And Nick did the graphics.

RESULTS: Omer calculated the correlation matrix for the case where Alice chooses to measure R and L photons rather than H and V photons:

Now eight events can happen, but none produces Bob's HV or VH cross terms.

Next Omer considered rotating both Alice's detectors by 45 degrees so that Alice registers Diagonal (D) and Slant (S) polarized photons instead of H and V.

Again eight events can happen, but none produces Bob's HV or VH cross terms.

Neither of these two efforts on Alice's part succeeds in producing cross terms in Bob's detectors. And indeed a more general calculation that allows Alice to effect any possible combination of rotation and phase change in her detectors gives the same result. Nothing that Alice can do will produce an FTL signal in Bob's detector.

So the JJCCTT proposal fails as an FTL signaling device.

"Science is great, but it’s low-yield. Most experiments fail. That doesn’t mean the challenge isn’t worth it, but we can’t expect every dollar to turn a positive result. Most of the things you try don’t work out — that’s just the nature of the process. Rather than merely avoiding failure, we need to court truth." -- Ferric Fang, microbiologist

Citing the FTL signaling Impossibility proofs of Philippe Eberhard and many others, it would be easy to have anticipated our negative result, These well-known impossibility proofs state, in essence, that 1. YES, quantum Theory is non-local (by inspection); 2. YES, quantum Reality is non-local (proved by John Bell) but; 3. NO, the quantum Facts are as local as can be.

Despite the FTL nature of the Theory that represents the World, despite the FTL nature of the Reality which underlies the World, the World Herself displays not a speck of evidence for any FTL connections.

I wish to thank Omer at JCT for proposing this project and I appreciate the fun we had doing these calculations. But now, as in so many other encounters with quantum reality, we end up where we started, back home again at Physics for Beginners.
Omer and Nick: two collaborators separated by 10 time zones.

Saturday, August 22, 2015

Palm of my Hand


Do you remember the night
I told you
That your underbrush
Was as familiar to me
As the palm of my hand?

But could I really sketch from memory

My hand's Head, Heart and Life Lines?
And how many little creases 
Run across my left hand's Mount of Venus?

Like an alien language

Like the back side of the Moon
Like your underbrush

And quantum reality:
Each as much a mystery
As the palm of my hand.

Tuesday, August 18, 2015

Roaring Twenties Topless Duet

Betsy on stage (circa 1970)
Today is the 13th anniversary of my wife Betsy's death.

I first met Betsy in San Francisco when she was living on Stanyan Street, studying and teaching modern dance at Ann Halprin's Dancer's Workshop on Divisadero St. While searching for something else in my files this morning I ran across some excerpts from Betsy's Journals concerning those days (during the reign of Herb Caen), when Betsy performed "The Topless Duet".


Lying in bed this morning,  I thought about Maude Meehan's suggestion to write about my topless dancing career. Some opening lines drifted into my head.

Eight of us shared the job, four men and four women. We were part of a dance class taught by AA Leath. He was the guy who scored the gig. We performed in pairs and on alternating nights, the Topless Man-Woman Duet at the Roaring Twenties in San Francisco's North Beach. On the marquee on Kearney St we were listed underneath the Topless Girl in the Swing. She was the feature. Every night, at 10, 11, midnight and 1 AM she climbed aboard a pink-cushioned swing hung 2 stories high in a wide stairwell and arched her back for all to enjoy.

"Come on down tonight," AA said, "See what you think of it." Twenty-five bucks a shot, 4 performances a nite. That amounted to about 30 minutes of dancing for $100. Never in a whole decade's career had I ever dreamed of getting paid like that. It was tempting but terrifying. Nevertheless, at 9:30 that night I found myself bound for North Beach in my bouncy little Citroen.

A big place, the Roaring Twenties, with lots of red plush decor. A long bar off to one side, tables all around the swing's staircase and up to the stage. On stage, a young and sprightly rock band. the manager points me to the dressing rooms upstairs. There's two of them, one for the girl-on-the-swing, the stripper and the topless waitresses and the other for the duet contingent. I open the door to find AA stretching and muttering in his usual fashion and with a southern drawl about getting in touch with some psychological phenomenon. Susan, his partner for the evening, was busy wrapping up in the several yards of chiffon which, atop a G-string, comprised her costume.

It was the Winter before the Summer of Love, 1967, and lots of us were driven by the idea that the Beautiful People could make a difference wherever we found ourselves. AA had that in mind for the clientele of the Roaring Twenties. "We'll show them that they can love their bodies". Sounded intriguing, but looking at AA in his G-string there in the dressing room didn't have me convinced.

I went downstairs, found Jim and Nancy sitting at the bar. They were also in AA's class and considering topless employment. the Girl on the Swing was waving goodbye to three tables of men from this week's convention. We ordered 7-Ups and waited, eager but figgety. the topless waitresses moved about, adorned in high heels, tights, miniskirts and pert aprons. Above the waist they were wearing a sheath of numbness.

The band leader announces The Duet and moves his musicians into a lively number. AA and Susan approach the stage through the audience, from opposite sides of the club, each clothed in chiffon. once on stage there is a lot of self-conscious shedding of chiffon. Reduced to G-strings and each other, the dance of seduction begins. I sit entranced, wondering what it would be like to be in Susan's place, all-but-naked, on stage.  It is dark out here in the audience and the stage lighting is soft enough to be kindly. They seem protected by each other and the movement, which shifts from modern-dancey to Fillmoresque and never leaves the realm of self-consciousness. I look at the tables of convention men and a collection of couples watching. They are looking but they don't seem to SEE. AA and Susan skirt each other, do-se-do, retreat and occasionally touch...

More on Roaring Twenties piece. What I remember about what it was like to perform the duet:

i remember that the band, whose name I can't now recall, who eventually was successful enuf to cut a record, played a piece with the lyrics "GI Joe, come back home. You'll have the best old time you've ever known," Real upbeat.

I remember going to buy my G-string downtown at the dance wear store. What a fuss over absolutely nothing! Well, practically nothing.

The commute to North Beach was by cable car.

Children picking up our bones
Will never know that these were once
As quick as foxes on the hill. 
                ---Wallace Stevens

Tuesday, August 11, 2015

Siragian Triangles

The square of the hypotenuse of a Siragian Triangle equals J(J+1)
Pythagoras's Rule for the calculation of the lengths of the sides of a right triangle is familiar to everyone. So well-known is this piece of elemental math that it has been repeatedly suggested that humans might construct giant replicas of the Pythagorean Theorem on Earth's surface as a means of signaling to aliens. As a warning perhaps that humans had progressed to the level of high-school geometry -- and regressed to a high-school level of adolescent and worse behavior toward our fellow beings with whom we share the Earth.

Motivated in part by analogy with the quantum mechanics of angular momentum and partly by mere curiosity, I introduced a modification to Pythagoras's Triangle which I named after my friend Saul-Paul Sirag. The Siragian Triangle differs from the Pythagorean Triangle only by the way in which you describe its hypotenuse. Phythagoras uses the symbol Z. Sirag uses the symbol J where Z and J are related by the expression:

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

The Sirag gambit does not alter the geometry of the right triangle but is merely a conventional change of variables similar to the choice to use Centigrade rather than Fahrenheit degrees to measure temperature.

One of the questions that mathematics have asked about the Pythagorean Triangle is this: How many right triangles have sides whose lengths are integers? The resulting 3 numbers for the integral lengths of two sides and the hypotenuse are called a Pythagorean Triple and can be symbolized PT(X, Y, Z).

The most familiar Pythagorean Triple is the famous 3-4-5 right triangle, but lots of integral-sided Pythogorean Triangles exist.

Similarly if one looks for integral solutions to the Siragian Triangle, one is led to the notion of a Siragian Triple ST(X, Y, J). Shortly after introducing the Sirag naming convention, I calculated a few Sirag Triples on my pocket calculator and committed a little elementary arithmetic in the privacy of my home.

Saul-Paul was immediately inspired to take the question further. "This problem reminds me of my favorite book," he said. His favorite book tells you a lot about Saul-Paul. It's Albert K. Beiler's Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.

(Apparently that branch of math called Number Theory is widely recognized as "the Queen of Mathematics.")

With our meager little Sirag Triple problem in hand, opening the cover of Beiler's book was like walking through a door from a dingy alleyway into the crystal palace of the Queen of Mathematics. Our precise question was not answered in Beiler's book but we did find many closely related math problems that had caught the attention of Western mathematicians as well as Chinese, Arabs, Indians and Greeks. Saul-Paul's favorite book does indeed contain the keys to solving the problem of Sirag Triples, but for the mathematically adventurous, it holds many more treasures besides.

The Square Numbers S (N) = N^2
I learned in Beiler, for instance, about the concept of Square Numbers, which are not numbers that lack hipness, but numbers that can be formed into a square. The Square Number S(8) = 64, for example, represents the number of squares on a chess board.

The Triangle Numbers T(N) = N(N+!)/2
 Also in The Queen of Mathematics Entertains I encountered the concept of Triangular Numbers which, as their name suggests, can be arranged in a triangular array. The most familiar example of a triangular number is T(4) = 10, which pictures the way that pins are set up in a bowling alley.

But what do these cute numerical squares and triangles have to do with the Sirag Triples? Quite a lot, as it turns out. For our work in Her Majesty's service, the Queen of Mathematics Herself bestowed on Saul-Paul and me a most marvelous gift. A gift that shines. A gift that will last forever.

What we expected: Two long strings of numbers.

What we got instead: A mathematical miracle.

To understand the nature of our gift, consider the special case of the isosceles Siragian Triangle, a right triangle that possesses two equal sides, which I will call "X". The Sirag Triple then takes the isosceles form ST(X, X, J). The numerical values in this triple answer this question: What two pairs of integers X and J will satisfy (EQ 2)?

 X^2 + X^2 = J(J+1)           (EQ 2)

By trial and error, and using a pocket calculator, one can compute the first three solutions to the Isosceles Sirag Triple problem. These three solutions are ST(1, 1, 1), ST(6, 6, 8) and ST(35, 35, 49). You can check for yourself that each of these three pairs of numbers satisfies (EQ 2), hence is belonging with full benefits to the illustrious Sirag Triple Society. Here is a larger table of Isosceles Sirag Triples that cost a day and a half of my time using pocket calculator, graph paper and lots of strong coffee.

The first nine Isosceles Sirag Triples
To the casual glance, these two columns of numbers appear completely arbitrary except perhaps for the fact that they both alternate between "even" and "odd" as you progress down each column. However the apparent patternlessness of X and J conceals a remarkably simple relationship, which is the gist of Her Majesty's Mathematical Gift. Her Gift is precisely this: If we let the numbers X in the left column label the Square Numbers S(X), and the numbers J in the right column label the Triangular Numbers T(J), then anytime X and J satisfy the formula (EQ 3), then X, X and J form an Isosceles Sirag Triple.

S(X) = T(J)              (EQ 3)

In plain language, the problem of finding an Isosceles Sirag Triple is exactly the same as finding an integer that is both a Square and a Triangular Number. Dismissing the integer "1", which is a trivial case, the lowest square triangular number is 36, which is the square of 6 but also can form a triangle with base 8. For this case (EQ 3) reduces to S(6) = T(8) = 36. As you can see from the green diagram, numbers that are both Triangular and Square are exceedingly rare. Searching through the first 2 trillion numbers, I was only able to find 9 such magic numbers that satisfy this condition. (Of course, I couldn't actually test all 2 trillion digits to construct the green diagram, but used tricks to get close to the right number. Then I conducted a local search on a few likely suspects to nab the culprit.)

Given the scarcity of square triangular numbers (and their largeness, which rapidly outstrips the capacity of a pocket calculator), how does one actually find and capture more of these rare beasts? The Queen of Mathematics Entertains suggests many complicated strategies for carrying out this Quest. But there is no method shorter and more efficient than that devised recently by Armando Guarnaschelli, an amateur mathematician who lives in Argentina.

The Isosceles Sirag Triple ST(X, X, J) is not strictly speaking a triple since two of its members are alike. These numbers are in effect a Sirag Couple. Saul-Paul, playing around with Siragian Triangles,
has discovered an infinity of such triangles with unequal sides. But all such non-isosceles Siragian Triangles that Saul-Paul has so far derived conform to the type ST(X, J, J); that is, one of the triangle's sides always has the same length as the "half hypotenuse" J. Generalizing from this experience, Saul-Paul conjectures that there are no true Sirag Triples. The only Sirag Triples that exist in nature are in effect Sirag Couples which come in two varieties, namely ST(X, X, J) and ST(X, J, J). Saul-Paul's conjecture may well be true. But it could be suddenly demolished by a single counter example.

I would like to thank Saul-Paul Sirag for inspiring this effort and for putting together most of the pieces. (And thanks also for more chances to learn to spell the word "isosceles"). And deep gratitude to Her High Radiance, the Queen of Mathematics, for deigning to bless our work with such an unexpected and breathtakingly elegant outcome.

Saul-Paul Sirag in front of the Ken Kesey statue in Eugene, Oregon

Saturday, August 8, 2015

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.