- #1
LagrangeEuler
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What is the difference between groups ##SU(n)## and ##SO(n,\mathbb{C})##? They look completely the same.
Unitary matrices satisfy ##A\cdot \bar A^\tau=1## whereas orthogonal matrices satisfy ##A\cdot A^\tau =1##. There is no complex conjugation in the definition of orthogonal groups.LagrangeEuler said:What is the difference between groups ##SU(n)## and ##SO(n,\mathbb{C})##? They look completely the same.
Your personal opinion doesn't change the definition.LagrangeEuler said:I disagree.
A definition can't be incorrect, and as ##\mathbb{R}\subseteq\mathbb{C}## we get ##\operatorname{SO}(n,\mathbb{C}) \neq \{1\}.## Whether this is reasonable is a different question and the reason why we introduced unitary matrices in the complex case. However, some theorems in linear algebra require an algebraically closed field, so it makes sense to allow any field, not just the real numbers.If you have complex entries in some matrix then condition ##A \cdot A^T=1## is incorrect.
LagrangeEuler said:[tex]
\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix} [/tex]
if you diagonalise this matrix you will get
[tex]\begin{bmatrix}
e^{i \theta} & 0\\
0 & e^{-i\theta}
\end{bmatrix}[/tex]
end then try to emply condition ##A \cdot A^T=1##. You will see that is not valid.
It looks fine to me.PeterDonis said:It seems like you have made a mistake in your diagonalization.
The property is not invariant under replacing a matrix with its complex diagonalization.PeterDonis said:The matrix you start with does satisfy ##A \cdot A^T=1##, and that property should be invariant under diagonalization, so the matrix you end with should satisfy it too.
Infrared said:The property is not invariant under replacing a matrix with its complex diagonalization.
$$\frac{1}{\sqrt{3}}\left(LagrangeEuler said:I disagree. If you have complex entries in some matrix then condition ##A \cdot A^T=1## is incorrect.
Yes. But you still need to have a group. Show me that group. An infinite group of matrices like this.mfb said:
Please see this text.fresh_42 said:Your personal opinion doesn't change the definition.
A definition can't be incorrect, and as ##\mathbb{R}\subseteq\mathbb{C}## we get ##\operatorname{SO}(n,\mathbb{C}) \neq \{1\}.## Whether this is reasonable is a different question and the reason why we introduced unitary matrices in the complex case. However, some theorems in linear algebra require an algebraically closed field, so it makes sense to allow any field, not just the real numbers.
Have a look at https://www.amazon.com/s?k=Gantmacher+Matrix+theory&ref=nb_sb_noss and argue with Gantmacher whether this is "incorrect". Minor difficulty: he passed away in 1964.
Not a problem. I just ask why some people make a difference between ##SU(n)## and ##SO(n,\mathbb{C})##? Could you please show me the two dimensional representation of group ##SO(n,\mathbb{C})##?Infrared said:It looks fine to me.The property is not invariant under replacing a matrix with its complex diagonalization.
@LagrangeEuler Why is it a problem that the diagonalized form is no longer in ##SO(n,\mathbb{C})##?
Because they're different groups. The matrix @mfb gave is an element of ##SO(2,\mathbb{C})## but not of ##SU(2)##.LagrangeEuler said:Not a problem. I just ask why some people make a difference between ##SU(n)## and ##SO(n,\mathbb{C})##?
I don't know what you're asking for here. You've been given the definition and a non-real example. You can take ##\begin{pmatrix}z & -w\\ w & z\end{pmatrix}## where ##z^2+w^2=1## to get the whole group.LagrangeEuler said:Could you please show me the two dimensional representation of group ##SO(n,\mathbb{C})##?
##\operatorname{SO}(n,\mathbb{C})## is a group.LagrangeEuler said:Yes. But you still need to have a group. Show me that group. An infinite group of matrices like this.
So? Myself has listed some isomorphisms here:LagrangeEuler said:Please see this text.
##\varphi\, : \,\operatorname{SO}(2,\mathbb{C})\longrightarrow \operatorname{GL}(2,\mathbb{C})## with ##\varphi(A)(\mathbf{v}):=A\cdot \mathbf{v}.##LagrangeEuler said:Not a problem. I just ask why some people make a difference between ##SU(n)## and ##SO(n,\mathbb{C})##? Could you please show me the two dimensional representation of group ##SO(n,\mathbb{C})##?
SU(n) and SO(n,C) are both mathematical groups used in theoretical physics and other fields of science. The main difference between them is that SU(n) is a subgroup of the special unitary group, while SO(n,C) is a subgroup of the special orthogonal group over the complex numbers.
SU(n) and SO(n,C) are related through the concept of Lie groups, which are continuous groups of symmetries in mathematics. SU(n) is a subgroup of SO(n,C), meaning that all elements in SU(n) are also elements in SO(n,C), but not vice versa.
In physics, SU(n) and SO(n,C) are used to describe symmetries in physical systems. SU(n) is often used to describe the symmetries of quantum systems, while SO(n,C) is used to describe the symmetries of relativistic systems.
One example of a physical system that exhibits SU(n) symmetry is the behavior of subatomic particles, such as quarks and leptons. These particles have spin, which is a quantum property that can be described using SU(n) symmetry.
The representations of SU(n) and SO(n,C) differ in their dimensionality. SU(n) has a finite-dimensional representation, while SO(n,C) has an infinite-dimensional representation. This means that the elements in SU(n) can be described using a finite number of parameters, while the elements in SO(n,C) require an infinite number of parameters.