Friday, August 26, 2011

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.]

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