In an earlier post, I asked a question about unitary transformation between two given matrices. Some really nice solutions were provided. However, what I'm really looking for is a unitary matrix U between two sets of matrices (that is, the same U is a unitary transform for all 5 corresponding pairs simultaneously). I verified that the ones provided do not achieve the goal for the following two sets,
G1={{{0, 0, 0, I}, {0, 0, I, 0}, {0, -I, 0, 0}, {-I, 0, 0, 0}}, {{0, 0, 0, -1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {-1, 0, 0, 0}}, {{0, 0, I, 0}, {0, 0, 0, -I}, {-I, 0, 0, 0}, {0, I, 0, 0}}, {{0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}}, {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}}};
G2={{{0, 0, 0, -I}, {0, 0, -I, 0}, {0, I, 0, 0}, {I, 0, 0, 0}}, {{0, 0, 0, -1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {-1, 0, 0, 0}}, {{0, 0, -I, 0}, {0, 0, 0, I}, {I, 0, 0, 0}, {0, -I, 0, 0}}, {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}}, {{0, 0, -1, 0}, {0, 0, 0, -1}, {-1, 0, 0, 0}, {0, -1, 0, 0}}};
This problem arises from the need to switch between two gamma matrix conventions in physics. I know a solution exists up to a constant phase factor.