# Noncommuting Series Expansion

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1. Mar 16, 2016

### fuchini

1. The problem statement, all variables and given/known data
Need to show that $$[a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}$$

2. Relevant equations
$$[a,a^\dagger]=1$$

3. 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.

Last edited: Mar 16, 2016
2. Mar 17, 2016

### 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.

3. Mar 17, 2016

### 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.

4. Mar 17, 2016

### 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.

5. Mar 18, 2016

### fuchini

Thanks, that did the trick!