The derivation of the irreducible representations of the Lorentz group

Question

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following:

In the real Lie-algebra of the Lorentz group the following the commutation relations are valid: $[ M_\mu, M_\nu] = \epsilon_{\mu\nu\lambda} M_\lambda$, $[ N_\mu, N_\nu] =-\epsilon_{\mu\nu\lambda} M_\lambda$ and $[ N_\mu, N_\nu] = \epsilon_{\mu\nu\lambda} N_\lambda$.

An infinitesimal Lorentz group element is given by $Id +\vec{\alpha}\bf{M}+\vec{v}\bf{N}$, $\vec{\alpha}$ and $\vec{v}$ (rotation vector and velocity vector, velocity of light c=1) are real parameters. If a different basis is chosen $M^{\pm}_\mu = \frac{1}{2}(M_\mu\pm iN_\mu)$ the commutation relations go over in the following relations:

$[ M^\pm_\mu, M^\pm_\nu] = \epsilon_{\mu\nu\lambda} M^\pm_\lambda$, and $[M^+_\mu, M^-_\nu]=0$.

This is the Lie-algebra of the direct sum $so(3)\oplus so(3)$ (or $su(2) \oplus su(2)$ if you like) or and therefore all irreducible representations of the Lorentz group group are given by the irreducible representations of $SO(3) \times SO(3)$. The problem I have is that in order to make this argument it is necessary to pass through complex representations of the Lorentz group since the decomposition of the Lie-algebra of the Lorentz group only works with complex numbers. An infinitesimal Lorentz group element is now given by $Id + (\vec{\alpha} - i\vec{v})\bf{M}^+ (\vec{\alpha} +i\vec{v})\bf{M}^-$ but now the parameters are complex. Doing it this way it would not change the reducibility of representations according to Sexl, Urbantke.

Actually I have a counter example: Look at the following Lorentz group example. $$\left(\begin{array}{c} E'_1 \\E'_2 \\E'_3 \\B'_1 \\B'_2 \\B'_3 \end{array}\right)=\left(\begin{array}{cccccc} 1 & 0 & 0& 0 & 0&0\\ 0 & \gamma & 0 & 0 & 0& -\gamma v\\ 0 & 0& \gamma & 0 &\gamma v & 0\\ \gamma & 0 & 0& 1& 0 & 0\\0 & 0& \gamma v& 0 & \gamma & 0 \\ 0 & -\gamma v &0 & 0& 0& \gamma \end{array}\right)=\left(\begin{array}{c} E_1 \\E_2 \\E_3 \\B_1 \\B_2 \\B_3 \end{array}\right)$$

This is a real irreducible representation of the Lorentz group. However, if now complex representations are considered, the basis can be changed to a complex basis and within this basis the representation is reducible and decomposes into 2 irreducible complex representations:

$$\left(\begin{array}{c} E'_1 + iB'_1\\E'_2+iB'_2 \\E'_3+iB'_3 \\E'_1-iB'_1 \\E'_2 -iB'_2\\E'_3 -iB'_3\end{array}\right)=\left(\begin{array}{cccccc} 1 & 0 & 0 & 0&0 & 0\\ 0 &\gamma & i\gamma v& 0 & 0& 0\\ 0 &-i\gamma v & \gamma & 0 &0& 0\\ 0& 0& 0& 1 & 0 & 0\\ 0 & 0& 0& 0& \gamma & -i\gamma v \\ 0 &0& 0& 0 & i\gamma v& \gamma \end{array}\right)\left(\begin{array}{c} E_1 + iB_1\\E_2+iB_2 \\E_3+iB_3 \\E_1-iB_1 \\E_2 -iB_2\\E_3 -iB_3\end{array}\right)$$

With real parameters the representation is irreducible whereas using complex numbers it is reducible. That means, using real or complex numbers makes a difference in representation theory.

Therefore I cannot understand why Lorentz group representations can be (so easily) classified according to the given argument.

I learnt in the meantime using complex numbers in Lie group theory is rather convenient, but in physics almost all Lie groups are $\bf{real}$ groups and the $\bf{real}$ representations have to be classified and understood. I hope somebody here has a deeper understanding than I have and can explain it to me.

Answer 1

OK, you seem to have stumbled on the Weyl unitarian trick which validates complexification maneuvers so I won't dwell on it here. The non-unitary reducible 6dim rep you started with is this group's adjoint. But then, complexifying, you end up with the 3dim self-dual forms, together adding up to the equivalent curvature form rep.

Suppose you complex-conjugate your 2nd expression. That interchanges the 1,2,3 block with the 4,5,6 block, in both vectors and matrices. The reduced 6x6 matrix can be transformed to the equivalent complex conjugated one by the similarity symmetric matrix that has 3d identity matrices in the 2-off diagonal 3x3 sub-blocks and 0s in the 2 3x3 diagonal blocks. (it clearly squares to the identity).

Comparing with the WP article, you see that you are combining two 3dim irreps to a 6dim one...

Edit to comport with @ACuriousMind's valid point below. Indeed. You may see that complexification hardly matters if you consider the complex similarity transformation converting your 1st expression to the 2nd one, vectors and matrices, $$S=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1\!\!1 & i 1\!\!1 \\ 1\!\!1 & -i1\!\!1 \end{array}\right)= (S^T) ^{-1},$$ where $1\!\!1$ is the 3x3 identity matrix.

Answer 2

You seem to be mixing representations. The real tensor six-dimensional adjoint representation is irreducible(the antisymmetric tensor in Minkowski space is clearly irreducible) . You seem to be comparing it with the direct sum of 2 irreducible complex three-dimensional representations (1,0) and (0,1) that are indeed built using a complexified algebra. But they are not the same object, the latter is not the adjoint representation of the Lorentz group. It is not a decomposition into representations of smaller real dimension.