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?

 

[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 let's us easily do integrals that "prefer" to be integrated to infinity.

 

[pmacias]

I can understand your concern about integrating a physical quantity to infinity. It is indeed a concept that has been debated and discussed in the scientific community for a long time. However, I would like to clarify that when we talk about integrating to infinity, it does not necessarily mean that we are considering an infinite universe. Rather, it is a mathematical tool that allows us to make approximations and simplify complex calculations.

In electrostatics, for example, we often use the concept of bringing charges from infinity in order to determine the work needed to assemble them. This is because it is much easier to calculate the work done on a charge when it is at an infinite distance from all other charges, rather than trying to take into account the interactions with every other charge in the system. Similarly, in quantum mechanics, integrating over all space is a way to take into account all possible positions and states of a particle, without having to specifically consider each one individually.

It is true that the concept of infinity is not physically realizable, and we must always keep in mind that our calculations and approximations are just models of reality. However, these mathematical tools have proven to be incredibly useful in understanding and predicting physical phenomena.

Regarding your point about the universe not being infinite in size, I would like to mention that even though the current prevailing theory in cosmology suggests that the universe will expand forever, it does not mean that the universe is infinite. It is possible that the universe has a finite size and is expanding within that finite space. Therefore, integrating to infinity in this context is still a useful mathematical tool.

In conclusion, while it is important to keep in mind the limitations and approximations of our mathematical models, integrating to infinity is a valid and necessary tool in many areas of physics. It allows us to simplify complex calculations and gain a better understanding of physical phenomena.