# Commutator and Anti-Commutator

## Identities

### Lie algebra identities

- $[A + B, C] = [A, C] + [B, C]$
- $[A, A] = 0$
- $[A, B] = -[B, A]$
- $[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0$ is called Jacobi identity

### Additional identities

- $[A, BC] = [A, B]C + B[A, C]$
- $[A, BCD] = [A, B]CD + B[A, C]D + BC[A, D]$
- $[A, BCDE] = [A, B]CDE + B[A, C]DE + BC[A, D]E + BCD[A, E]$
- $[AB, C] = A[B, C] + [A, C]B$
- $[ABC, D] = AB[C, D] + A[B, D]C + [A, D]BC$
- $[ABCD, E] = ABC[D, E] + AB[C, E]D + A[B, E]CD + [A, E]BCD$
- $[A, B + C] = [A, B] + [A, C]$
- $[A + B, C + D] = [A, C] + [A, D] + [B, C] + [B, D]$
- $[AB, CD] = A[B, C]D + [A, C]BD + CA[B, D] + C[A, D]B$
- $[[A, C], [B, D]] = [[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C]$

### Exponential identities

- $e^{A} = \exp(A) = 1 + A + \frac{1}{2!}A^2 + \cdots$
- The solution of $e^{x}e^{y} = e^{z}$ if $X$ and $Y$ are non-commutative to each other is $Z = X + Y + \frac{1}{2} [X, Y] + \frac{1}{12} [X, [X, Y]] - \frac{1}{12} [Y, [X, Y]] + \cdots$. It’s called Baker-Campbell-Hausdorff formula.
- $e^{A + B} = \lim_{n \to \infty} \left(e^{A/n} e^{B/n} \right)^{n}$ is known as Trotter product formula. It says for sufficiently large $n$, we can ignore the noncommutativity between $A$ and $B$.
*It can be used to formulate the Feynman path integral, see here.* - $e^{A} B e^{-A} = B + [A, B] + \frac{1}{2!}[A, [A, B]] + \frac{1}{3!}[A, [A, [A, B]]] + \cdots$

Permalink athttps://www.physicslog.com/math-notes/commutator

Published onMay 3, 2021

Last revised onJul 2, 2023

References