ON THE RECONSTRUCTION OF A UNITARY MATRIX FROM ITS MODULI

We study the problem of reconstructing a unitary matrix from the knowledge of the moduli of its matrix elements, first in the case of a symmetric matrix, which could be the S matrix for n coupled channels, second in the case of a non-symmetric matrix like the Cabibbo-Kobayashi-Maskawa matrix for n generations of quarks and leptons. In the symmetric case we find conditions under which the problem has 2(n2-3n)/2 solutions differing in a non-trivial way, but also situations where one has continuous ambiguities. In the non-symmetric case we show that for n > 3 there may be continuous ambiguities, of which we give an exhaustic list for n = 4. We give indications that there is also a set of moduli for which one has 2(n2-3n)/2 discrete solutions, but no rigorous proof.