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Commutator

Let $\tilde A$, $\tilde B$, ...be Operators. Then the commutator of $\tilde A$ and $\tilde B$ is defined as

\begin{displaymath}[\tilde A,\tilde B]\equiv \tilde A\tilde B-\tilde B\tilde A. \end{displaymath} (1)

Let $a$, $b$, ...be constants. Identities include
$\displaystyle {[}f(x),x]$ $\textstyle =$ $\displaystyle 0$ (2)
$\displaystyle {[}\tilde A,\tilde A]$ $\textstyle =$ $\displaystyle 0$ (3)
$\displaystyle {[}\tilde A,\tilde B]$ $\textstyle =$ $\displaystyle -[\tilde B,\tilde A]$ (4)
$\displaystyle {[}\tilde A,\tilde B\tilde C]$ $\textstyle =$ $\displaystyle [\tilde A,\tilde B]\tilde C+\tilde B[\tilde A,\tilde C]$ (5)
$\displaystyle {[}\tilde A\tilde B,\tilde C]$ $\textstyle =$ $\displaystyle [\tilde A,\tilde C]\tilde B+\tilde A[\tilde B,\tilde C]$ (6)
$\displaystyle {[}a+\tilde A,b+\tilde B]$ $\textstyle =$ $\displaystyle {[}\tilde A,\tilde B]$ (7)
$\displaystyle {[}\tilde A+\tilde B,\tilde C+\tilde D]$ $\textstyle =$ $\displaystyle [\tilde A,\tilde C]+[\tilde A,\tilde D]+[\tilde B,\tilde C]+[\tilde B,\tilde D].$  
      (8)

The commutator can be interpreted as the ``infinitesimal'' of the commutator of a Lie Group.


Let $A$ and $B$ be Tensors. Then

\begin{displaymath}[A,B]\equiv \nabla_A B-\nabla_B A. \end{displaymath} (9)

See also Anticommutator, Jacobi Identities



© 1996-9 Eric W. Weisstein
1999-05-26