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.