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I recently read The Elegant Universe by Brian Greene to understand more about relativity, quantum mechanics, and the conflict between them. What it says is that "the notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales". I still don't understand why this is. Why can't small scales be random and with quantum foam, but when you zoom out, space curves smoothly like relativity predicts?

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There are many very closely related questions on this topic, but my reason for attempting an answer to this particular post is to try and explain why we just can't ignore (if they really exist), the problems of small scale spacetime discontinuities, as the OP suggests in the last line of the post.

Theory suggests, without any experimental evidence, as far as I know to back it up, is that there is an uncertainty in how much energy there is in one place at any one time. If this energy is treated as mass, it will cause distortions in spacetime that result in a breakdown of GR equations, which are based on continous functions.

See the box on the right for related questions.

Why can't small scales be random and with quantum foam, but when you zoom out, space curves smoothly like relativity predicts?

The random motions (if they are really there) won't go away just because GR works fine if we don't look too closely. We have to look closely to try and explain the big bang and black holes.

My answer is unsophisticated (and lots of people will say wrong), but that's the quick answer, as I understand it. Look at Vacuum Fluctuations by Matt Strassler, who gives a far better explanation than I could.

You could also Google "quantum gravity" for more reasons and details as to why we are trying to reconcile GR and QM, and a list of possible solutions to the problem of having one theory for large scales and another for the microworld.

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The most important thing to realize is that the important quantum fluctuations are those that couple to a real field. The vacuum has a collection of harmonic oscillator fields that define the so called zero point energy (ZPE). The ZPE modes that are purely vacuum can be removed from the physics. Normal ordering raising and lowering operators removes the "dangling commutator" corresponding to these ZPE modes.

This is fine except when it comes to gravity. Gravitation interacts with mass-energy of any sort, and this includes the energy density of ZPE modes. Normal ordering does not easily get us out of the problem with the vacuum. The sum over vacuum modes define this vacuum energy $$ E_{vac}~=~\frac{\hbar}{2}\sum_{\omega=0}^{2\pi/T_p}\omega~\rightarrow~\frac{\hbar}{2}\int_0^{2\pi/T_p}d\omega~=~\frac{\pi\hbar}{T_p}. $$ This is the Planck energy $\simeq~2\times 10^{9}j$ or about $10^{19}GeV$. The density of vacuum energy is then $\rho~\simeq~E_{vac}^4~=~10^{76}GeV^4$. Gravitation couples to the vacuum energy density and gives the energy equation for the FLRW cosmology $$ \left(\frac{\dot a}{a}\right)^2~=~\frac{8\pi G\rho}{3c^2}~=~\frac{\Lambda}{3c^2} $$ The cosmological constant is $\Lambda~\simeq~10^{-52}m^{-2}$ or in Planck units of length this is $10^{-87}\ell_p^2$. This is then $124$ orders of magnitude smaller than what we would predict from how gravitation would interact with the ZPE.

Something is terribly wrong! The M-theory approach to this problem involves a lot of work with wrapped D-branes or string and flux of Yang-Mills fields through these objects. Some progress has been made at reducing this number, but physics still predicts something many orders of magnitude smaller than what we expect from the small cosmological constant of the observable universe.

It also needs to be pointed out that the attempts to measure quantum foam have been null. The highest energy ZPE modes are thought to be quantum gravity fluctuations. This is the quantum foam. A photon travelling a great distance will interact with this foam in a weak way. The shorter the wavelength the photon is the more probable it is it will couple to the foam. As a result there is an expected dispersion of photons from distant sources. The Fermi spacecraft measured photons across the EM spectrum from distant ( billion light years etc) burstars and found no dispersion. In fact the data suggests that spacetime is smooth down to a couple of orders magnitude smaller than the Planck scale.

The idea of quantum foam is then in trouble. One might say that the above measurements are across a large baseline. This means $\Delta x~\rightarrow~\infty$ and so the uncertainty in momentum is $\Delta p~=~0$ This is not quite the same as a Heisenberg microscope measuring a tiny region of space. This may also be one reason why the vacuum energy plays such a tiny role in the universe at large. Maybe the vacuum energy contribution of physics has some strange scaling principle.

The AdS/CFT correspondence is bearing out a curious relationships between nonlocal and local physics. The AdS/CFT correspondence indicates that the bulk gravity in the anti-de Sitter spacetime $AdS_n$ of dimension $n$ is equivalent to the conformal field theory $CFT_{n-1}$ on the conformal boundary of dimension $n-1$. It also turns out that if all the fields in the $AdS_n$ are nonlocal, and we tend think of quantum gravity as likely being nonlocal, then the $CFT_{n-1}$ is local. The converse also holds. This suggests that the physically real fields in the universe are nonlocal and this could massively reduce the number of physically real degrees of freedom. This would mean a new method for removing this massive vacuum energy we expect and thus finding how the universe has a small cosmological constant.

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It does not conflict. Just like Maxwell's electromagnetic field theory is not inconsistent with the later discovered quantum theory of electrodynamics, with photons the excitation so of the field. One, the quantum theory covers the high energy small distances (down to the Planck length, or really before that down to the Weinberg-Salam electroweak unification), and classical electromagnetism still emerges as a low energy larger distance accurate approximation.

The bigger issue and problem is that we still don't understand, or have an appropriate nor verified by any measurements theory, of what quantum gravity is. The main technical reasons are stated sometimes as the standard canonical quantization being non-renormalizable, or various other reasons. Partly it is the difficulty of how it is that one can think of spacetime, e.g., points in space, as not really being well defined. So various theories have been proposed, the most well known being string theory where the basic entity is a string and not a point (but still not background independent), and loop quantum gravity, background independent. As is known, neither one, not any other, is problem free, nor backed up by any new evidence.

The statements made by Brian Greene and others reflect the estimation most gravitational physicists have that quantizing gravity, if that is what leads to a correct theory of gravity at very high energies towards the order of the Planck length, is going to involve having to give up, at that microscopic level, the idea that spacetime is a set of points that we can treat as a sort of manifold with a continuous set of points at which a semi-continuous metric can be defined. That we will need to somehow, and we really don't know how, deal with the violent, high energy, nature of spacetime at those scales. Some call it quantum foam, maybe quantum string or branes, maybe loops. But none of them believe that whatever theory emerges as true (no easy feat), that it does not get reconciled with General Relativity at the larger scales and lower energies. We just don't know how, the concept nor the detailed theory for it.

Ther is no conflict. We just don't know what quantum gravity, or gravity in the very small scales, really is.

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