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When I study the (conventional) quantum field theory, I learned about the short distant cutoff in renormalization in order to regularize the momentum integral and remove the infinities. I'm thinking about the possible 'cutoffs' in Newtonian mechanics. For example, Newton's law of gravity for two particles of mass $m_1$ and $m_2$ is $G\frac{m_1.m_2}{r^2}$, when they are getting infinitely close to each other, the force blows up. As another example, suppose there is a non-stopping ball that falls under gravity, at an infinitely long time its speed reaches infinity. Are there any cutoffs in these two classical cases to remove the infinity? Do they have anything in common with QFT in terms of the idea of cutoffs?

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    $\begingroup$ Newton's law assumes that a distance r between the two masses is meaningful. That's only the case for arbitrarily small r when dealing with point masses. $\endgroup$
    – chepner
    Feb 6, 2023 at 18:32
  • $\begingroup$ See my updated answer. :) $\endgroup$ Feb 6, 2023 at 18:51

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No there is no equivalent cutoff in Newtonian Gravity. In fact in any classical theory you will not come upon the need to regularize and renormalize, because those arise strictly from quantum corrections to the classical theory.

The example you gave is not an infinity that calls for regularization and renormalization. Of course, a ball falling in a uniform gravitational field with no other forces will physically accelerate indefinitely (leaving out special relativity). Therefore this infinity is correct and 'physical'.

Edit: To address the second example you gave in your updated post, about how the gravitational force in Newtonian gravity appears to blow up as two particles get closer together... This is only an issue for point-particles with zero spatial extent, which as far as Newton was concerned did not exist. If a point particle existed, the first issue you would have address is how you could have an infinite mass-density!

Consider two super small particles - say, two spheres of constant mass-density $\rho$ with infinitesimal radius $r$. Their masses are $\frac{4}{3}\rho \pi r^3$. The closest they could possibly be to each other is when they are touching, and the separation is $d=2r$. The gravitational force would therefore be:

$$F_g = G\frac{m_1 m_2}{d^2} =\frac{4\pi^2 G}{9} \rho^2 r^4$$

So if you keep a non-infinite mass-density $\rho$, the maximal gravitational force between these two particles will actually rapidly decrease as you decrease their size $r$.

If point-particles do exist, assuming you can justify the infinite mass-density, then clearly they cannot come infinitely close. The simple way to remedy this is not via a UV/IR cutoff, but by simply saying that at sufficiently small distances there will be a repulsive force between them. This is indeed what happens in the real world - electrical repulsion and the Pauli exclusion principle (i.e. Pauli repulsion). Note that this is not a UV cutoff - it is inappropriate say that anything which alleviates short-distance infinities is a UV cutoff, that would be an abuse of language.

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    $\begingroup$ I actually strongly disagree that the infinity is "physical" -- to me saying something is "physical" implies that you view it as a robust prediction of the best theory we have and expect the prediction to be confirmed by an experiment. I don't think we ever expect infinite force situations to be physical. A cutoff should appear because the point particle approximation breaks down for any two finite size objects that get close enough. Or, if we have black holes, eventually you need to account for general relativity. $\endgroup$
    – Andrew
    Feb 6, 2023 at 15:41
  • $\begingroup$ @Andrew In a theoretical Galilean universe where there is nothing except a massive point particle and a constant gravitational field, wouldn't that particle keep accelerating forever? Of course falling objects in our universe, whether in space or on Earth, do not accelerate indefinitely and therefore we must amend the underlying theory appropriately. But the appropriate amendments in this case are absolutely not UV or IR cutoffs as we encounter in QFT. The appropriate amendments would be a realistic drag force, or special relativity which limits the speed to be less than that of light. $\endgroup$ Feb 6, 2023 at 18:25
  • $\begingroup$ @Andrew I agree that nearly any infinity that arises in a physical calculation signals a failure of that calculation, but not all need to be remedied with regularization and renormalization. For example, a ball rolling on a frictionless surface will roll forever, to infinity, which is obviously not physical. This infinity is handled by simply limiting your domain and/or range of interest, i.e. only looking within a certain range of time and space. $\endgroup$ Feb 6, 2023 at 18:33
  • $\begingroup$ @Andrew Also, the OP modified their post to include the first question about the blow-up of the Newtonian force as objects get close together. I'll edit my post to answer this. $\endgroup$ Feb 6, 2023 at 18:35
  • $\begingroup$ "In a theoretical Galilean universe where there is nothing except a massive point particle and a constant gravitational field, wouldn't that particle keep accelerating forever?" -- my point is that this scenario isn't very physical. $\endgroup$
    – Andrew
    Feb 6, 2023 at 19:01
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At small separations the point-mass equation fails to take into account that bodies are extended, so some fraction of the mass of the other object is closer than r, and some is further away. We may need to sum the forces over the integral of the volume of the other body, and we might need to start caring about distortion of the body due to differences in forces for the near side and the far side.

Particularly notable is that it is unlikely that gravity is the dominating force if the two bodies are colliding -- for spherical bodies, r being smaller than the sum of the radii of the bodies themselves -- instead, one mass is likely to be pushing directly on the other, and possibly distorting it.

Even if they pass through each other like ghosts, as r tends towards zero, the force begins to drop: at r=0, planet A experiences no net force as the force from the left side of planet B exactly balances the force of the right side of planet B. I think that the net force can be modelled as having strength proportional to r (as opposed to 1/r2, such that the force at contact remains the same, but you'd want to do some proper reading on that. "Gravitational Softening" is worth searching for; this answer goes into more detail, but critically this is a fudge rather than an actual model of something physical.

Many modelling approaches which examine the forces at discrete timesteps will fail if the assumption that the forces do not significantly change between those timestamps. Should a body pass close to another planet at a specific timestep, it will be given that force for the entirety of that timestep, which will lead to an unphysical acceleration.

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