What Are the Commutators of Vector Operators?

[thecommexokid]

I've been trying to work out some expressions involving commutators of vector operators, and I am hoping some of y'all might know some identities that might make my job a little easier.

Specifically, what is $\left[\mathbf{\hat A}\cdot\mathbf{\hat B}, \mathbf{\hat C}\right]$? Are there any useful identities to express this in terms of simpler commutators?

Any help is appreciated.

 

[brydustin]

Are you sure you mean a "vector operator", typically we talk about matrix operators when discussing the commutator relationships (or group elements in a more general setting).
Vector operator: http://en.wikipedia.org/wiki/Vector_operator

However, your question is straightforward,
[S,T] = ST - TS (by definition)

Start with [AB,C] = ABC - CAB (+ ACB - ACB )
= ABC - ACB + ACB - CAB
= A(BC - CB) + (AC - CA)B
= A[B,C] + [A,C]B

Therefore we conclude [AB,C] = A[B,C] + [A,C]B
to be an identity.
Does that answer your question... you could have looked anywhere on the internet to get this... so I'm guessing this isn't what you want.

 

[brydustin]

You can also construct an identity using the so called anti-commutator: http://mathworld.wolfram.com/Anticommutator.html
You should probably do this for practice.

 

[thecommexokid]

Thank you for the [STRIKE]reply[/STRIKE]replies.

[S,T] = ST - TS (by definition)

Start with [AB,C] = ABC - CAB (+ ACB - ACB )
= ABC - ACB + ACB - CAB
= A(BC - CB) + (AC - CA)B
= A[B,C] + [A,C]B

Therefore we conclude [AB,C] = A[B,C] + [A,C]B

It seems to me that you're being pretty cavalier about vector multiplication, what with the way you're just putting vectors in a row next to each other without any dots or parentheses. For instance, what do you mean when you write “ABC”, when A, B and C are vector operators?

I would think that you should define $[{\bf{\hat S}},{\bf{\hat T}}] = {\bf{\hat S}} \cdot {\bf{\hat T}} - {\bf{\hat T}} \cdot {\bf{\hat S}}$, and therefore start your derivation with
$$[{\bf{\hat A}} \cdot {\bf{\hat B}},{\bf{\hat C}}] = ({\bf{\hat A}} \cdot {\bf{\hat B}}){\bf{\hat C}} - {\bf{\hat C}}({\bf{\hat A}} \cdot {\bf{\hat B}}).$$
But from there, I'm not sure how you can safely proceed, if you're being rigorous with your dots and parens. For instance — and correct me if I'm wrong on this — but I don't think $({\bf{\hat A}} \cdot {\bf{\hat C}}){\bf{\hat B}}$ is equal to ${\bf{\hat A}}({\bf{\hat C}} \cdot {\bf{\hat B}})$, so your next step seems iffy.

 

[brydustin]

Yeah, sorry I don't know. I thought you were intending for matrix operators. Good luck.