Four and N­Generation Mixing Matrix
Bjorken & Dunietz (1987) provides us with a general representation of a 4 × 4 unitary matrix given in Eq. (1). We have calculated the possible combinations and have found that there are eight distinct parametrizations.

The unitary nature of the matrix imposes eight conditions on the connections between adjacent rows and columns, analagous to the six unitarity triangles for the three family case, but for four generations the unitarity condition forms a quadri­lateral in the imaginary plane. We have found only one set of invariants that are independent of their positions in the matrix, i.e. for which one can choose any element to be the “starting point” element Vj,a in the definitions of K, L and M given below (where the invariants are the sums or differences of the imaginary parts of four plaquettes).



where k = j or j + 1 and b = a or a + 1, but if
k = j + 1, then b ≠ a + 1 and similarly, if b =a + 1 then k ≠ j + 1 (Maiani 1976).

Acknowledgments
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Appendix
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