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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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Commutation rules between itinerant and localized electron operators in s-f Model

{i,\sigma},\,\hat{a}_{i,\sigma}$ and the spin operator of localized electrons $\hat{S}_{i}^{z}, \hat{S}_{i}^{+}, \hat{S}_{i}^{-}$ I am thinking , since these are two different kind of electrons the commutator
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