7.3: The Jacobi Identity
- Page ID
- 29570
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Another important identity satisfied by the Poisson brackets is the Jacobi identity
\begin{equation}
[f,[g, h]]+[g,[h, f]]+[h,[f, g]]=0
\end{equation}
This can be proved by the incredibly tedious method of just working it out. A more thoughtful proof is presented by Landau, but we’re not going through it here. Ironically, the Jacobi identity is a lot easier to prove in its quantum mechanical incarnation (where the bracket just signifies the commutator of two matrix operators, \(\begin{equation}
[a, b]=a b-b a)
\end{equation}\) Jacobi’s identity plays an important role in general relativity.