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A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.

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Torsionfree condition of the covariant derivative operator

In R. Wald's book, this condition is written as: $$\nabla_a \nabla_b f=\nabla_b \nabla_a f.$$ But aren't $\nabla_a$ and $\nabla_b$, like, the exact same derivative operator with a re-naming of index? …
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