How Does Dimensional Regularization Simplify Integrals in Quantum Field Theory?

[mhill]

how does dimensional regularization work ?

i see , how can you calculate integrals in d-dimensions of the form

$$\int d^{d} k F( \vec k )$$ ??

and for other cases , let us suppose we have the integral

$$\lim_{\varepsilon\rightarrow 0^+}\int \frac{dp}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{\left(p^2+m^2\right)^1}$$

there is no way this integral can be calculated

 

[Avodyne]

It works by analytic continuation. Consider the integral

$$\int_0^\infty dx\,{x^n\over x^2+1}.$$

For $-1<{\rm Re}\,n<1$, the integral converges, and the result is

$${\pi/2\over\cos(n\pi/2)}.$$

We now define this to be the value of the integral for all $n$.

 

[denian]

in the traditional sense as it leads to infinities

Dimensional regularization is a mathematical technique used in quantum field theory to regulate divergent integrals in higher dimensions. It works by extending the number of dimensions in which the integral is calculated, from the usual 3 spatial dimensions and 1 time dimension to a higher number, typically denoted by d. This allows for the cancellation of infinities that arise in traditional calculations.

In the case of integrals in d-dimensions, the integral is calculated using the same techniques as in 4 dimensions, but with d as the number of dimensions. This means that the integral is still dependent on the function F( \vec k ) and the d-dimensional vector k, but now the integral is over d dimensions instead of 4.

For the second integral, the use of dimensional regularization allows for the integration to be performed in d-dimensions, which will lead to a result that is finite. The limit as \varepsilon\rightarrow 0^+ is taken to get the final result in 4 dimensions. This is because the infinities that arise in traditional calculations are dependent on the number of dimensions, and by extending the number of dimensions to d, the infinities can be cancelled out.

Overall, dimensional regularization is a powerful tool in theoretical physics that allows for the calculation of integrals in higher dimensions and helps to regulate infinities that arise in traditional calculations. It is a crucial technique in quantum field theory and has been used to make many important predictions and calculations in the field.