How to Expand Noncommuting Variables in a Formal Power Series?

[fuchini]

Homework Statement

Need to show that $$[a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}$$

Homework Equations

$$[a,a^\dagger]=1$$

The Attempt at a Solution

Need to expand $$f(a,a^\dagger)$$ in a formal power series. However I don´t know how to do it if the variables don´t commute.

 

[DrDu]

$f(a,a^+)= b_0+ b_1 a+b_2 a^+ b_3 a^2 +b_4 aa^+ +b_5 a^+a+b_6 (a^+)^2+\ldots$
So the general term is some product of a and $a^+$ in arbitrary order.

 

[fuchini]

Thanks for answering, but how would it be in terms of derivatives? Normally It would be:

$$f=\sum_{m,n} \frac{a^m a^{\dagger m}}{n!m!}\frac{\partial^{n+m} f}{\partial a^n \partial a^{\dagger m}}$$

But in this case I guess I have to take into account that they're noncommuting.

 

[strangerep]

Start with a simpler case, i.e., $f = (a^\dagger)^n$ only. Use induction (on $n$) to show that the desired formula holds. Once you understand the induction method for this problem, you'll probably work it out for more general $f$ more easily.

 

[fuchini]

Thanks, that did the trick!