How to construct a general 4 by 4 unitary matrix?
- From: Barrow <GRseminar@xxxxxxxxx>
- Date: Tue, 4 Mar 2008 06:37:50 -0800 (PST)
Dear all,
I wanna construct a general 4 by 4 unitary matrix in terms of the 2 by
2 sub-matrices.
That is, I wanna divide the 4 by 4 unitary matrix(U) into four pieces,
such that the matrix U can be
written as a_ij where i,j = 1,2 and each a_ij is a 2 by 2 matrix.
I have tried to to use UU^\dagger = U^\dagger U = 1 to list the
constraints satisfied by a_ij
But I don't know how to solve these a_ij.
And, I knew the general element of a group can be written as the
linear combination of the generators of the group. But I don't know
how to construct and generators of the U(4) group. Moreover, I
searched google but in vain.
Actually, my purpose is to find out the unitary transformation matrix
which serves the change of representations of the gamma matrices.
(gamma matrices can have different representations, e.g. Dirac
representation, chiral representation, Majorana representation...etc.
Those are connected by a unitary transformation U\gamma^\mu U^\dagger)
Any ideas will be appreciated so much!
Sincerely Barrow
.
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