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I am reading dr. Joshua Qualls lectures on conformal field theory. https://arxiv.org/abs/1511.04074

In section 2.4 Conformal group he defined the generators

$$ \begin{aligned} J_{\mu,\nu}&=L_{\mu,\nu}\\ J_{-1,\mu} &=\frac{1}{2}(P_{\mu}-K_{\mu})\\ J_{0,\mu} &= \frac{1}{2}(P_{\mu}+K_{\mu})\\ J_{-1,0} &=D \end{aligned} \tag{2.28} $$

where $P,K,D$ and $L$ are translational, special conformal transformation, dilatation and angular momentum generator. then he gave this commutation relation

$$ [J_{mn},J_{pq}]=i(\eta_{mq}J_{np}+\eta_{np}J_{mq}-\eta_{mp}J_{nq}-\eta_{nq}J_{mp}).\tag{2.29} $$

I was wondering how to derive this? in Weinberg's QFT 1, he considered infinitesimal transformation for $U(\lambda,a)$ and then work out the final commutation relation. how to do this using the same way that Weinberg did but now for full conformal transformation? or any other derivation you may know, please share then.

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The conformal algebra in d dimensional Minkowski space is isomorphic to so(d+1,n+1) where n is the number of time-like directions.

So in 4d the conformal algebra can be recast as so(4,2)

Just plug in the definitions and confirm the commutation relations hold. It’s an exercise that everyone studying CFTs should do.

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  • $\begingroup$ oh i see. but then what does $J_{0,\mu},J_{-1,\mu}$ refers to? and how they know that this form will work? $\endgroup$ May 11, 2018 at 16:56

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