推 BBSealion :THX! 04/21 23:00
※ 編輯: JohnMash 來自: 112.104.114.174 (04/22 06:49)
※ 引述《BBSealion (海獅)》之銘言：
: 我好像應該先弄懂這個問題才對XD
: 對一個N*N維的矩陣A
: 限制其為Unitary會減少幾個自由度呢?
Let A=e^(iR)
A.A^H=I=e^(iR) e^(-iR^H) where M^H is the hermitian conjugate of M
hence, R=R^H
There are 2*[1+2+...+(N-1)]+N=N^2 independent real variables.
: 還有限制 det(A)=1 會減少幾個自由度(這個是1吧?)
A=e^(iR)
because R=R^H is hermitian,
R=U^H.D.U
where U is unitary and D is real diagonal.
D=diag[d_1,d_2,...,d_N]
det(A)=det(e^(iU^H.D.U))=det(U^H.e^(iD).U)=det(e^(iD))
=e^(i(d_1+d_2+...+d_N))=1
d_1+d_2+...+d_N=0
Trace(D)=0
Trace(U^H.D.U)=Trace(R)=0
R is traceless.
Hence, A=e^(iR) is SU(N) if and only if R is traceless hermitian matrix
There are N^2-1 independent real variables.
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