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.
6 years ago