Common · Knowledge
Jacobi identity
Property of some binary operations, such as the cross product and any ring's commutator
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed).
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The identity is named after the German mathematician Carl Gustav Jacob Jacobi. He derived the Jacobi identity for Poisson brackets in his 1862 paper on differential equations. The cross product a × b {\displaystyle a\times b} and the Lie bracket operation [ a , b ] {\displaystyle [a,b]} both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Let + {\displaystyle +} be a binary operation and × {\displaystyle \times } be an anti-commuting binary operation, and let 0 {\displaystyle 0} be the identity element for + {\displaystyle +} . The Jacobi identity is Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form a × ( b × c ) {\displaystyle a\times (b\times c)} , the variables x {\displaystyle x} , y {\displaystyle y} and z {\displaystyle z} are permuted according to the cycle x ↦ y ↦ z ↦ x {\displaystyle x\mapsto y\mapsto z\mapsto x} .
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