Question on UV and IR divergences

[mhill]

no matter what theory we use , are all UV and IR divergences of the form ??

$$\int_{0}^{\infty} dk. k^{m}$$ where 'm' is an integer

or there are another divergent integral different from a power-law or logarithmic divergence ?? , and another question , can be this true

$$i^{m+n}D^{m}\delta (w) D^{n}\delta (w)= Fourier. transform (\int_{-\infty}^{\infty}dt.t^{n}(x-t)^{m})$$

where i have used the fact that the Fourier transform of a convolution of two functions (f*g) is just the product of the Fourier transform F(w)G(w)

 

[lbrits]

UV divergences are of that form, but it depends on the type of regulator you use. I'm not sure about IR divergences.

I don't know what you mean by $$D^m\delta(\omega)$$. A derivative?

$$D^m \delta(\omega) D^n \delta(\omega) \approx \int\!dt\,t^m e^{i \omega t}\int\!dx\,x^n e^{i \omega x}= \int\!dt\,\int\!dx\, t^m x^n e^{i \omega (x+ t)} = \int\!dx \left[\int\!dt\, t^m (x-t)^n \right] e^{i \omega x}$$
You will have to put in factors of i and pi and such.

 

[denian]

I cannot provide a definitive answer to this question as it depends on the specific theory and calculations being used. However, I can provide some general information about UV and IR divergences.

UV (ultraviolet) and IR (infrared) divergences are mathematical artifacts that arise in quantum field theory calculations when dealing with infinite quantities. These divergences indicate that the theory being used is not complete and needs to be further refined.

In general, UV divergences are associated with high-energy or short-distance behavior, while IR divergences are associated with low-energy or long-distance behavior. These divergences can take various forms, including power-law or logarithmic divergences as mentioned in the question.

There can also be other types of divergences, such as those involving non-local operators or higher-dimensional integrals. The specific form of the divergence depends on the particular theory and calculation being performed.

Regarding the second question, it is possible for the given expression to be true, but again, it depends on the specific theory and calculation being used. The Fourier transform is a powerful mathematical tool that can be used to analyze and manipulate functions, but its applicability to a given problem depends on the assumptions and conditions being considered.

In summary, while there may be some general patterns in the form of UV and IR divergences, it is not accurate to say that all divergences follow a simple power-law or logarithmic form. The exact form of the divergence depends on the specific theory and calculation being used.