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Wheeler once said that spacetime would be highly curved at very small scales because of the uncertainty principle for energy-momentum. In which case the spacetime becomes very bumpy and not smooth anymore, which Wheeler called spacetime foam. It seems that such a picture doesn't bother us because in most cases we are dealing with physics of larger scales and the spacetime becomes smooth again at an averaged level over the large scale.

But when we extend the picture to cosmology, problems appear even at a semiclassical level. Now let's consider the $\phi^4$ scalar theory with $$V(\phi)=\frac{\lambda}{4}\phi^4.$$ For the vacuum, because of the uncertainty principle, $\phi$ cannot stay at 0 every where and all the time. If that the field would have definite configuration and definite velocity (field momentum) which violates the uncertainty principle. At the Planck scale $l_p$, the energy should have an uncertainty of $M_p$. Thus for every Plack volume, $\phi$ may take values between $-M_p/(\lambda)^{1/4}$ and $M_p/(\lambda)^{1/4}$ so that $V(\phi)=\frac{\lambda}{4}\phi^{4}\sim M_p^4$.

However in such case, this small patch of space will be driven to inflation. $$a(t)=a_0\exp(Ht),$$ where $$H=\left(\frac{8\pi}{3}V(\phi)/M^{2}_p\right)^{1/2}.$$ Note that the analysis here applies not only to the early universe but also the current universe. But such inflation at small scales will surely cause problems such as the inhomogeneity in our universe which is of course not the case of our observed universe.

So what's the problem with the analysis given above?

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  • $\begingroup$ Things that are smooth, like the bubbles comprising a foam, don't have angles, so I don't quite follow the relation you're trying to make between Wheeler's visualization and a bumpy spacetime. $\endgroup$
    – Edouard
    Mar 13 at 20:11

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The picture in this post is basically wrong. The uncertainty is used for the length scale of the whole system. For example, if we consider a particle moving in a box of volume $L^3$, we would say that the particle has momentum uncertainty ~$\hbar/L$ and therefore energy uncertainty $\hbar^2/(2m L^2)$. Hence for a large box, the energy uncertainty is nearly zero. But we can not say in this case that, we can look at a smaller region, and in that region, there is a large energy uncertainty which may lead to a large total energy uncertainty if we add them up. That is, we should distinguish the scale of the whole system and the distance from point to point which can be arbitrary small. Likewise, the uncertainty principle does not force us away from the spacetime concept between two arbitrary close points. But it tell us that, a quantum system with a very small scale is meaningless since a black hole will be formed from the large energy fluctuations.

I hope this answer is not misleading.

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  • $\begingroup$ I understand that the Planck length is the smallest meaningful distance, but I've tended to assume that smaller scales are bound to exist, even if only as an evolved characteristic of a reality that cannot be destroyed by the machinations of the more malicious of its inhabitants. (When did they evolve? Maybe an eternity ago, as I tend to think that the universe --or multiverse--may be as old as it may be large.) $\endgroup$
    – Edouard
    Mar 13 at 20:18
  • $\begingroup$ Whether it wears a bathrobe, has a beard, has good days and bad, etc., I dunno.... $\endgroup$
    – Edouard
    Mar 13 at 20:20

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