# Integrating a physical quantity to infinity

#### Visceral

This is something that has bothered me for some time, and I can't seem to find any threads on here about it. In a lot of my undergraduate courses in physics, we talk about integrating something physical to infinity. For example, in electrostatics, we talk about the work needed to assemble a collection of charges that we "brought in from infinity." Or in quantum, we integrate to infinity all of the time to satisfy probability (e.g. the normalization condition). As my quantum professor always says, "we integrate over all space," which is usually a sphere with infinite radius. I know we have to make approximations all of the time in physics, and I am fine with that, but this is one that to me doesn't seem valid with all that we know about the universe.

As far as I know, physicists don't think the universe is infinite in size. I have read, though, that the prevailing theory in cosmology is that the universe will probably expand forever. If that is the case, then I can see some validity in integrating space or time out to infinity. What do you guys think? I know this will probably make some of your eyes roll, because for all practical purposes, we can just do this math in order to get a very good approximation of something we are interested in.

Another thing that just occurred to me, is that concepts such as infinite mass or density (e.g. with black holes) is "not physical," yet considering interactions between matter and energy at infinite separation is?

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#### dipole

It depends on the scale of the problem. If you're normalizing the wave function of an electron in a hydrogren atom, then what's the difference whether you integrate $\int^{R_{\textrm{observable universe}}}_0$ or $\int^{\infty}_0$? It won't make any possible detectable difference in your answer, and the fact is that infinity often simplifies the calculation greatly at almost no cost of accuracy.

It's a very useful mathematical tool, and often only a very slight idealization. There's no problem with it.

#### Okefenokee

How small would the universe have to be to make a 1% difference in the solution to your integral? That would be an interesting problem to solve.

#### Visceral

It depends on the scale of the problem. If you're normalizing the wave function of an electron in a hydrogren atom, then what's the difference whether you integrate $\int^{R_{\textrm{observable universe}}}_0$ or $\int^{\infty}_0$? It won't make any possible detectable difference in your answer, and the fact is that infinity often simplifies the calculation greatly at almost no cost of accuracy.

It's a very useful mathematical tool, and often only a very slight idealization. There's no problem with it.
Yes, I understand scenarios where it makes little differences such as the one you have described. But what about cosmological scales? I assume it's the same routine. You are right, there would never be any detectable difference, so in all practical purposes it's the most logical thing to do as it lets us easily do integrals that "prefer" to be integrated to infinity.