Power Series For Function of Operators

1. Oct 19, 2005

ghotra

Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute.

My first guess was:
$$F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j$$

However, I don't think this is correct as it is possible to have operators between x and p.
So then I thought that this might be the correct expansion:
$$F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i b_{ij} p^j + c_{ij} p^i d_{ij} x^j$$

Obviously, I'm just guessing here and could use some help. Why do I care about this? I am trying to derive a formula for

[F(x_1,....,x_n), G]

Suppose I know how G commutes with each of the operators x_i. I want to get an expansion for G commuting with some function F of the x_i operators. I'll work on the details of that, but first I need some help with the power series of F.

Last edited: Oct 19, 2005
2. Oct 19, 2005

dextercioby

There exists a simple rule, if you'd be applying your discussion to quantum mechanics.

Daniel.

3. Oct 19, 2005

ghotra

So here is what I am actually trying to do. I have:

$$[P^j,\phi_r(x)] = -i \hbar \frac{\partial\phi_r(x)}{\partial x_j}$$

and

$$[P^j,\pi_r(x)] = -i \hbar \frac{\partial\pi_r(x)}{\partial x_j}$$

For a function $F\left(\phi_r(x),\pi_r(x)\right)$, I need to show the following:

$$[P^j,F\left(\phi_r(x),\pi_r(x)\right)] = -i \hbar \frac{\partial}{\partial x_j}F\left(\phi_r(x),\pi_r(x)\right)$$

I was thinking of considering the various commutator relations:

$$[P^j,\phi^n_r(x)]$$

but since the operators don't commute, there are (too) many possible combinations to consider. I would be interested in knowing this trick you speak of.

Thanks.

4. Oct 21, 2005

dextercioby

Use the Poisson bracket and Dirac's rule giving the canonical quantization.

Daniel.

5. Oct 25, 2005

ghotra

Could someone spell this out for me? I have convinced myself that there is no pretty way to write a power series for a function of operators (that do not necessarily commute). It seems like you'd have a sum of an infinite product...each term in the product with their own index...so you are summing over an infinite number of indices.