# Why no change of variable to polar coordinates inside multi-loop integral ?

1. Jun 12, 2010

### zetafunction

why no change of variable to polar coordinates inside multi-loop integral ??

given a mul,ti-loop integral

$$\int d^{4}k_{1} \int d^{4}k_{2}.................\int d^{4}k_{n}f(k_{1} , k_{2},.....,k_{n})$$

which can be considered a 4n integral for integer n , my question is why can just this be evaluated by using a change of variable to 4n- polar coordinates ?

one we have made a change of variable and calculated the Jacobian, and integrated over ALL the angular variables we just have to make an integral

$$\int_{0}^{\infty}drg(r)r^{4n-1}$$ which is just easier to handle

2. Jun 12, 2010

### mathman

Re: why no change of variable to polar coordinates inside multi-loop integral ??

I don't what specific integral you have in mind, but it depends very much on the form of f as it depends on the k's. You seem to imply that it can be represented as a function g of one variable. This may be true for some particular f, but it certainly is not true in general.

3. Jun 13, 2010

### zetafunction

Re: why no change of variable to polar coordinates inside multi-loop integral ??

for example

$$\iint dx dy \frac{x^{3}}{1+xy}$$ its divergent if taking the limits (0,oo)

making a change of variable to polar coordinates one gets

$$\int du \int_{0}^{\infty}dr\frac{r^{4}cos^{3}(u)}{1+(1/2)r^{2}sin(2u)}$$

integrating over the angular variable 'u' you have now a simple one dimensional integral

4. Jun 13, 2010

### mathman

Re: why no change of variable to polar coordinates inside multi-loop integral ??

In general if you have an m dimensional integral and integrate over m-1 dimensions, you will have a one dimensional integral. In your general case (4n) I am not sure what you mean by polar coordinates.

This question belongs in the mathematics forum. There isn't apparent connection with Beyond the Standard Model (physics).

Last edited: Jun 13, 2010