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1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter:

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I aim to consider the product $L^0{}_0(\Lambda_1\Lambda_2).$ Consider the following notation $L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu$. How then, does $$L^0{}_0(\Lambda_1\Lambda_2) = L_1{}^0{}_\mu L_2{}^\mu{}_0?$$

2) To write the J and K generators of the Lorentz group in a compact way, one can write $(M^{lm})^j{}_k=i (g^ {lj}g^m{}_k - g^{mj}g^l{}_k)$ where on the left hand side, it is helpful to think of l and m as indices/labels, and the j and k as rows/columns for the whole matrix. (Apparently) One can write this in an operator representation as $M^{\mu\nu} = i(x^\mu \partial^\nu-x^\nu\partial^\mu).$

a) where does this come from?

b) why and how is it used? What is it operating on, $x^\mu?$

c) how does $\partial^\nu x^\sigma= \frac{\partial}{\partial x_\nu} x^\sigma$ equal $g^{\nu\sigma}$?

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  • $\begingroup$ Regarding c): Work out the 16 values of $\frac{\partial}{\partial x_\nu}x^\sigma$. $\endgroup$ – G. Smith Oct 8 at 15:43

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