story

This is going to be a quick post. Trying to look up common commutation relationships is a pain. I’m going to compile a list of these relationships that I’ve found useful. This post should be treated as a reference. I will likely update this post as time goes on.

quick introduction

Unfortunately, a lot of useful algebras are not commutative. The commutator describes how close two elements are to commuting and often encapsulates some interesting properties.

$[A,B] = AB - BA$

For the rest of this post, capital letters $$A$$, $$B$$, $$C$$ etc. will be used to denote elements of a non-commutative algebra and lowercase letters will be used to denote elements of the field.

basic commutator algebra

Addition within the commutator, $[A+B,C]=[A,C]+[B,C]$

Multiplication within the commutator, $[A,BC]=B[A,C]+[A,B]C$

This one shows up sometimes when composing unitary transformations, $\frac{\partial}{\partial t}(e^{tA}Be^{-tA})=e^{tA}[A,B]e^{-tA}$

Baker-Campbell-Hausdorf (-lite)

The full BCH relationship is very general and unwieldy. Luckly, in a lot of situations the commutator commutes with the original elements truncating the BCH formula after the first commutator term. Small note: this might actually be known as the Zassenhaus formula, with the real BCH being its dual, but I think most people would identify this as the BCH relationship.

If $[A,[A,B]]=[B,[A,B]]=0$ then, $e^{t(A+B)} = e^{tA}e^{tB}e^{-\frac{t^2}{2}[A,B]}$

The two relationships below are also closely related to the BCH expansion,

$e^BAe^{-B}=A+[B,A]$

$Ae^B = e^BA+[A,B]e^B$

the first one is also known as the Hadamard lemma.

Jacobi identity

Another one that shows up frequently is the Jacobi identity.

$[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$

Rewritten, the Jacobi identity can be seen as a generalization of associativity,

$[[A,B],C] = [A,[B,C]] - [B,[A,C]]$

The standard cross product also satisfies the Jacobi identity (the lowercase letters below are elements of the vector space $$\mathbb{R}^3$$),

$(a \times b) \times c = a \times (b \times c) - b \times (a \times c)$

revision history

May 9 – started post