I am trying to follow along the derivation of the commutator relations for the generators of the Galilei Group in Ballentine.

He states that the product of two infinitesimal generators and their inverses is

$e^{i\epsilon K_\mu}e^{i\epsilon K_\nu}e^{-i\epsilon K_\mu}e^{-i\epsilon K_\nu} = I + \epsilon^2(K_\nu K_\mu - K_\mu K_\nu) + O(\epsilon^3)$

However, when I try and derive this myself, I don't get the same answer. My derivation is as follows,

$e^{i\epsilon K_\mu}e^{i\epsilon K_\nu}e^{-i\epsilon K_\mu}e^{-i\epsilon K_\nu} = (I + i\epsilon K_\mu)(I + i\epsilon K_\nu)(I - i\epsilon K_\mu)(I - i\epsilon K_\nu)$

$=(I + i\epsilon K_\mu + i\epsilon K_\nu -\epsilon^2K_\mu K_\nu) (I - i\epsilon K_\mu - i\epsilon K_\nu -\epsilon^2K_\mu K_\nu)$

$=I + \epsilon^2(K_\nu K_\mu - K_\mu K_\nu +K_\mu K_\mu + K_\nu K_\nu) + O(\epsilon^3)$

In other words, I have two extra terms $\epsilon^2K^2_\mu$ and $\epsilon^2K^2_\nu$.

Any insights where I am going wrong here would be appreciated.


As you answer is to be of order $\epsilon^2$, you need to keep all terms of that order when expanding out $$ e^{i\epsilon K_\mu}= 1+ i\epsilon K_\mu- \frac 12 \epsilon^2 K_\mu^2+\ldots $$


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