Carl Brannen directs us to a stackexchange thread where he asks the nagging question: can we write a unitary matrix in the form $AMC$, where $A$ and $C$ are phase diagonals, and $M$ is a magic matrix whose rows and columns sum to $1$. For some time now, a few of us have struggled to prove a theorem along these lines.

It appears that a real mathematician may have finally come to our rescue. Stanford postdoc Samuel Lisi says on the thread that the theorem results from Floer homology, which is a sophisticated subject that is certainly associated to our version of M theory. If Samuel is correct (could another good mathematician please look at his post?) then we immediately have the following remarkable physical consequence: the MNS and CKM mixing matrices must be expressible through magic matrices, an ansatz with which we have worked for some time now.

8 years ago

Nice quote from the wikipedia article:

ReplyDeleteThe symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture.Ah, so my 2 short comments there were deleted, it seems. Well, don't expect me to participate then.

ReplyDeleteDid you see Carl's instructions about how to get enough reputation points to leave a comment under an answer? At the moment you are posting discussion comments as whole new answers, but the Stack Exchange convention is that discussion occurs in the comment section of the answer being discussed.

ReplyDeleteYeah, OK, Mitchell, I get it but I just don't care.

ReplyDeleteIt's not really a "convention". Stack Exchange is a privately owned company that quite strictly enforces how its website is used. The stupid feature is that they won't let you post comments until after your reputation reaches a certain level.

ReplyDeleteThe whole thing was originally designed for computer programmers asking how to fix bugs and write code. It probably made sense in that context but they've had their share of difficulty getting the mathematical community to adapt. The physics community seems to get along with it. That could be because the physics sample size is smaller or it could be because physicists are bosons and don't mind all doing the same thing.

OK, well, maybe we can have the discussion you wanted here. The proof doesn't look complicated. The unitary mapping between MUBs defines a Hamiltonian flow, a Clifford torus moved by a Hamiltonian flow always overlaps its initial position somewhere, and the existence of this overlap (for the Clifford torus in the projective space of rays in Hilbert space) guarantees the existence of the unbiased vector.

ReplyDeleteSounds simple, yes, but I am not familiar with the details of the Floer homology, which I suspect is not quite that simple.

ReplyDeleteI did see your first deleted comment (not the second) where you said there might not be a simpler proof, because these matrices have a motivic origin. I think that's pretty unlikely. It's like saying you need schemes to prove Pythagoras's theorem, because 3^2 + 4^2 = 5^2, 3 and 5 are prime, and primes are associated to knots by Spec Z. What's more likely is that such theorems are trivial instances of the more profound relationships. Anyway, here's a proof of the Clifford torus theorem due to Dmitry Tamarkin.

ReplyDeleteCool, Mitchell! Thanks. Yes, you are probably right about the theorem. Anyway, I answered 2 questions on the exchange, and now have commenting rights. But there is no action there now!

ReplyDeleteThey undeleted one of Kea's comments on that proof page. I'm going to see if I can understand the idea behind it. Perhaps there's a simple proof that makes the result easy to obtain.

ReplyDeleteBy the way, one of Kea's answers on the zeta function generalization is quite striking and should be linked here. I'm sure she's discussed it in a blog (which I have trouble keeping up with): http://math.stackexchange.com/questions/30409/

Now as to getting more rep points on SE, it should be noted that every time you make an edit on a post it brings it to the top of the question list and so exposes it to more people who might vote it up. However, after something like 5 or 10 edits, it becomes "community wiki" and no further reputation points accrue.