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I have a problem with the Baker-Campbell-Hausdorf formula in third order approximation. I hope there is anyone who can help me through this calculation!

1. The Problem Statement

All Lie group elements can be written as

[itex]U(\alpha_I)=\exp(i \alpha_I T^I).[/itex]

Proof the Baker-Campbell-Hausdorff formula

[itex]U(\alpha_I) U(\beta_I) = U(\gamma_I),[/itex]

up to the third order.

First Order

[itex]U(\alpha_I) U(\beta_I) = (\mathbb{1} + i \alpha_I T^I)(\mathbb{1} + i \beta_I T^I) =

\mathbb{1} + i(\alpha_I + \beta_I)T^I

\Rightarrow \gamma^{(1)}_I = (\alpha_I + \beta_I)[/itex]

Second Order

[itex]U(\alpha_I) U(\beta_I) = (\mathbb{1} + i \alpha_I T^I - \frac{1}{2} \alpha_I \alpha_J T^I T^J)(\mathbb{1} + i \beta_I T^I - \frac{1}{2} \beta_I \beta_J T^I T^J) =

\mathbb{1} + i(\alpha_I + \beta_I)T^I - \frac{1}{2} \left( \alpha_I \alpha_J + \beta_I \beta_J + 2 \alpha_I \beta_J \right)T^I T^J

= \mathbb{1} + i(\alpha_I + \beta_I)T^I - \frac{1}{2} \left[ (\alpha_I + \beta_I)(\alpha_J + \beta_J)T^I T^J + \alpha_I \beta_J[T^I,T^J] \right]

\Rightarrow \gamma^{(1)}_I = \alpha_I + \beta_I[/itex]

[itex]\Rightarrow \gamma^{(2)}_I = \alpha_I + \beta_I - \frac{1}{2} \alpha_I \beta_J [T^I, T^J][/itex]

Third Order

HOW TO CALCULATE THIS?

THX for every help!!

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# Baker-Campbell-Hausdorff formula

Can you offer guidance or do you also need help?

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