# IR and UV divergences

perhaps is a dumb quetion but,

given a IR divergent integral (diverges whenever x tends to 0)

$$\int_{0}^{\infty} \frac{dx}{x^{3}}$$

then using a simple change of variables x=1/u the IR integral becomes an UV divergent integral

$$\int_{0}^{\infty} udu$$ which is an UV divergent integral (it diverges whenever x tends to infinity)

then why we call IR or UV divergences if they are essentially the same thing ??

Avodyne
They are not the same thing. An IR divergence is one that arises from integrals over low momentum or energy, and a UV divergence is one that arises from integrals over high momentum or energy.

yes of course, but from the mathematical point of view a change of variable would turn an IR divergence into a UV one, for a mathematician both functions or divergences would be the same since from the cut-off we can define a function $$\epsilon = 1/ \Lambda$$

with this epsilon tending to 0

malawi_glenn
Homework Helper
But the physical variable is not the same, that is why we call them different

of course is not the same taking the integral

$$\int_{0}^{\infty} d\lambda f( \lambda )$$

or taking the integral $$\int_{0}^{\infty} dp f(p)$$

in the first integral we integrate over the wavelength (meters) whereas in the second we integrate over moment (kg.m/second) but for a mathematician both singularities would seem the same.