Infinitesimal Lorentz Transformations

Question

In Weinberg's Gravitation and Cosmology, the author mentions that an infinitesimal Lorentz transformation (in the four-vector representation of the Lorentz group) has the form $$Lambda^{alpha}_{phantom{alpha}beta}=delta^{alpha}_{phantom{alpha}beta}+omega^{alpha}_{phantom{alpha}beta}.tag{$dagger$}$$ It is then straightforward to verify that the $omega$-matrix must satisfy $$omega_{gammadelta}=-omega_{deltagamma}.tag{$*$}$$ I'm okay with that. Then, Weinberg says that the matrix representation $D(Lambda)$ of such a transformation (now a general, say $ntimes n$ representation) must satisfy $$D(1+omega)=1+frac{1}{2}omega^{alphabeta}sigma_{alphabeta}tag{$**$}$$ where $sigma_{alphabeta}$ are a set of matrices which may be chosen to be antisymmetric, by virtue of (*).
I have no idea where the exact form ( ** ) comes from. What's a little confusing to me is that in (†), I believe that $omega$ may be any any linear combination of the generators of the four-vector representation of the Lorentz group and $omega^alpha_{phantom{alpha}beta}$ are its matrix elements, while in (**) $omega$ seems to be a set of infinitesimal parameters, whilst $sigma_{alphabeta}$ are now the generators (I think I know the $1/2$ prevents counting each generator twice).

Rob Tan · Answer

This is what's happening. You have $vartheta^{gammadelta}$ parameters for the $text{SO}(1,3)$; they constitute an antisimmetric matrix in their $gamma,delta$ indexes, so that the actual free parameters are the usual $6$ for the Lorentz transformation. That said consider that an infinitesimal Lorentz transformation will be
$$
{Lambda^alpha}_beta
approx
{mathbb{I}^alpha}_beta
+
frac{1}{2}
vartheta^{gammadelta}
{mathbb{J}^{gammadeltaalpha}}_beta
$$
where you got ${left(mathbb{J}^{gammadelta}right)^alpha}_beta=-{left(mathbb{J}^{deltagamma}right)^alpha}_beta,,forall,alpha,beta$, so that you can think it as an antisymmetric (but just on $gamma,delta$ indexes) matrix of matrices
begin{gather*}
	begin{pmatrix}
	{left(mathbb{J}^{gammadelta}right)^alpha}_beta
	end{pmatrix}
=
	begin{pmatrix}
	mathbb{O}&mathbb{J}^{01}&mathbb{J}^{02}&mathbb{J}^{03}
	
	-mathbb{J}^{01}&mathbb{O}&mathbb{J}^{12}&mathbb{J}^{13}
	
	-mathbb{J}^{02}&-mathbb{J}^{12}&mathbb{O}&mathbb{J}^{23}
	
	-mathbb{J}^{03}&-mathbb{J}^{13}&-mathbb{J}^{23}&mathbb{O}
	end{pmatrix}

mathbb{J}^{01}
=
	begin{pmatrix}
	0&1&0&0
	
	1&0&0&0
	
	0&0&0&0
	
	0&0&0&0
	end{pmatrix},
,
mathbb{J}^{02}
=
	begin{pmatrix}
	0&0&1&0
	
	0&0&0&0
	
	1&0&0&0
	
	0&0&0&0
	end{pmatrix},
,
mathbb{J}^{03}
=
	begin{pmatrix}
	0&0&0&1
	
	0&0&0&0
	
	0&0&0&0
	
	1&0&0&0
	end{pmatrix}

mathbb{J}^{12}
=
	begin{pmatrix}
	0&0&0&0
	
	0&0&-1&0
	
	0&1&0&0
	
	0&0&0&0
	end{pmatrix},
,
mathbb{J}^{13}
=
	begin{pmatrix}
	0&0&0&0
	
	0&0&0&-1
	
	0&0&0&0
	
	0&1&0&0
	end{pmatrix},
,
mathbb{J}^{23}
=
	begin{pmatrix}
	0&0&0&0
	
	0&0&0&0
	
	0&0&0&-1
	
	0&0&1&0
	end{pmatrix}
end{gather*}
P.S.
$$
mathbb{G}^{alphagamma}
{mathbb{I}^delta}_beta
-
mathbb{G}^{alphadelta}
{mathbb{I}^gamma}_beta
doteq
{mathbb{J}^{gammadeltaalpha}}_beta
$$

Infinitesimal Lorentz Transformations

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