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I want to ask about unitary group $\rm SU(2)$ and $\rm SU(4).$

From my reading, the matrix of $\rm SU(2)$ and $\rm SU(4)$ is unitary matrix.

I make the product of $\rm SU(4)\times SU(2)$

It is possible when I make an evolution, the matrix of $\rm SU(4)\times SU(2)$ is not unitary?

Qmechanic
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munirah
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    That's not the tensor product. It's just the Cartesian product. – user1504 Oct 13 '16 at 18:20
  • For an explanation of user1504's comment, see e.g. this Phys.SE post. – Qmechanic Oct 13 '16 at 18:55
  • @user1504, how to make the cartesian product between SU(4)×SU(2) since it cannot be multiply because not same order. please help me – munirah Oct 15 '16 at 00:40
  • The cartesian product is the set of pairs $(g,h)$, with $g\in SU(4)$ and $h \in SU(2)$. Multiplication on the Cartesian product is element-wise; $(g,h) (g',h') = (gg', hh')$. – user1504 Oct 15 '16 at 09:32
  • it will have a 8x8 matrix? but how?. can u give me example for SU(2)xSU(2). and what it mean by g' and h' – munirah Oct 15 '16 at 09:35
  • $g$ and $g'$ are elements of $SU(4)$. They're 4x4 unitary matrices of determinant 1. $h$ and $h'$ are 2x2 unitary matrices of determinant 1. – user1504 Oct 15 '16 at 09:38

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It is possible [..] the matrix of $\rm SU(4)\times SU(2)$ is not unitary?

No, it will be unitary. The tensor product of two unitary matrices is always unitary.

Proving that for yourself makes a good basic exercise, so I won't spoil it.

Craig Gidney
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  • thank you very much. but I want to ask within the matrix in SU(4), the multiplication between the matrix is a normal matrix multiplication or not same? . If not, how the way to multiply them. – munirah Oct 15 '16 at 00:48