- #1

- Homework Statement:
- See below

- Relevant Equations:
- N/A

I messed up somewhere, but don't know why! We consider this sequence of infinitesimal transformations,$$U = e^{i\varepsilon K_{\mu}}e^{i\varepsilon K_{\nu}}e^{-i\varepsilon K_{\mu}}e^{-i\varepsilon K_{\nu}}$$with ##K_{\mu}## and ##K_{\nu}## being two generators. I said, this simplifies to$$\begin{align*}

e^{i\varepsilon K_{\mu}}e^{i\varepsilon K_{\nu}}e^{-i\varepsilon K_{\mu}}e^{-i\varepsilon K_{\nu}} &= (I+i\varepsilon K_{\mu})(I+i\varepsilon K_{\nu})(I-i\varepsilon K_{\mu})(I-i\varepsilon K_{\nu}) + \mathcal{O}(\varepsilon^5) \\

&= \left[I + i\varepsilon(K_{\mu} + K_{\nu}) - \varepsilon^2 K_{\mu} K_{\nu} \right]\left[I - i\varepsilon(K_{\mu} + K_{\nu}) - \varepsilon^2 K_{\mu} K_{\nu} \right] + \mathcal{O}(\varepsilon^5) \\

&= I+ \varepsilon^2 (K_{\mu}^2 + K_{\mu} K_{\nu} + K_{\nu} K_{\mu} + K_{\nu}^2) -2 \varepsilon^2 K_{\mu} K_{\nu} + \mathcal{O}(\varepsilon^3) \\

&= I + \varepsilon^2 (K_{\nu} K_{\mu} - K_{\mu} K_{\nu}) + \varepsilon^2(K_{\mu}^2 + K_{\nu}^2) + \mathcal{O}(\varepsilon^3)

\end{align*}$$but the textbook only quotes$$U = I + \varepsilon^2 (K_{\nu} K_{\mu} - K_{\mu} K_{\nu}) + \mathcal{O}(\varepsilon^3)$$I wondered why I ended up with an extra term? Thank you.

e^{i\varepsilon K_{\mu}}e^{i\varepsilon K_{\nu}}e^{-i\varepsilon K_{\mu}}e^{-i\varepsilon K_{\nu}} &= (I+i\varepsilon K_{\mu})(I+i\varepsilon K_{\nu})(I-i\varepsilon K_{\mu})(I-i\varepsilon K_{\nu}) + \mathcal{O}(\varepsilon^5) \\

&= \left[I + i\varepsilon(K_{\mu} + K_{\nu}) - \varepsilon^2 K_{\mu} K_{\nu} \right]\left[I - i\varepsilon(K_{\mu} + K_{\nu}) - \varepsilon^2 K_{\mu} K_{\nu} \right] + \mathcal{O}(\varepsilon^5) \\

&= I+ \varepsilon^2 (K_{\mu}^2 + K_{\mu} K_{\nu} + K_{\nu} K_{\mu} + K_{\nu}^2) -2 \varepsilon^2 K_{\mu} K_{\nu} + \mathcal{O}(\varepsilon^3) \\

&= I + \varepsilon^2 (K_{\nu} K_{\mu} - K_{\mu} K_{\nu}) + \varepsilon^2(K_{\mu}^2 + K_{\nu}^2) + \mathcal{O}(\varepsilon^3)

\end{align*}$$but the textbook only quotes$$U = I + \varepsilon^2 (K_{\nu} K_{\mu} - K_{\mu} K_{\nu}) + \mathcal{O}(\varepsilon^3)$$I wondered why I ended up with an extra term? Thank you.

Last edited by a moderator: