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Weak gravity conjecture linked to many fields of maths, physics: an essay

Ben Heidenreich, Matthew Reece, and Tom Rudelius (Harvard) have won the 5th place in the 2016 Gravity Research Foundation Essay Contest (I will avoid the general rating of this kind of essay contests):

Axion Experiments to Algebraic Geometry: Testing Quantum Gravity via the Weak Gravity Conjecture
They discuss a refinement of our conjecture that for any type of a "charge" similar to electromagnetism, there must always exist sources for which the non-gravitational force donalds the gravitational one.

The essay shows that the inequality has implications for inflation (naively excluding a long enough inflation and maybe forcing one to talk about specific types of inflation), for AdS/CFT (charged operators with low enough dimensions should exist), and for pure mathematics (because the inequality should hold for compactifications on complicated enough manifolds, and such an inequality therefore sometimes turns into a nontrivial geometric theorem about those).




They start with my #1 favorite motivation for the weak gravity conjecture – the absence of global Lie symmetries in quantum gravity. It's been something I was emphasizing long before our paper but I vaguely remember that it wasn't new for some or several co-authors, either.




You know, the important pre-WGC lore – which I may have known from my adviser Tom Banks since the late 1990s – was that there are no global continuous symmetries in a consistent quantum theory of gravity. In general relativity, even translations are made "local" (diffeomorphism group) and things that are not "local" become unnatural.

However, gauge theories with tiny couplings \(g\to 0\) may seemingly emulate global symmetries as accurately as you want. That should better be impossible as well. If something (global symmetries) is forbidden, physical situations or vacua that are "infinitesimally close" to the forbidden thing should better be banned as well, right? Otherwise the ban would be operationally vacuous. There should exist a finite value of some quantity that tells you how far from the forbidden point you have to be.

And that's what the weak gravity conjecture does (and many types of evidence – from problems with extremal black hole remnants to lots of stringy examples – support the conjecture, at least in "some" form). A light charged particle with \[

m \leq \sqrt{2} eq M_{Planck}

\] must exist. Heidenreich et al. promote their belief in a stronger "detailed" version of the weak gravity conjecture that we have conjectured but we ran into some counter-arguments. They call it the lattice weak gravity conjecture (LWGC): for every allowed vector \(\vec Q\) in the lattice of charges, there must exist an object that is lighter than the 0.0001% of the mass of a black hole with the charge one million times \(\vec Q\).

(I have inserted the factor of one million and the millionth to make sure that you omit the corrections from the smallness of the black hole – you work with the semiclassical estimate of the mass.)

This sounds too strong. I thought that for larger charges, the statement actually isn't true – only several "elementary", low-charge light particles are required by WGC, I thought. If they replaced the word "particle" by a "state" (which may be a collection of many separate particles whose charges, masses add), I think that my doubts would go away.

It's difficult to decide whether the new light states should exist for every \(\vec Q\), every direction in the charge space, almost every direction, every direction in a basis of directions, every direction in a (near) orthogonal basis in some metric, or something else. There may exist some "very natural" specific version of the inequality that would be as provable as e.g. the Heisenberg uncertainty principle but I don't think that the "best, most accurate yet strong one" has been pinpointed yet.

The analogy with the Heisenberg uncertainty principle is meant to be an exaggeration. At least I still believe that the WGC is vastly less fundamental than the Heisenberg uncertainty principle. The uncertainty principle may be connected to many – in some sense "all" – situations in physics. WGC has been "connected" to many things. But I still don't see in what sense it could be considered a principle that "changes the rules of the game" in a way that is at least qualitatively analogous to the change of physics implied by the Heisenberg uncertainty principle.

There may be similarities between the inequalities but there are also differences. One of them is that the Heisenberg uncertainty principle strictly disagreed with the class of theories that had been considered before Heisenberg and pals revolutionized physics. On the other hand, WGC tells you to consider a subset of the theories of gravity-coupled-to-matter that were previously allowed.

The amount of activity dedicated to WGC is greater than what I used to assume a decade ago. (And I surely believe that e.g. matrix string theory is much more fundamentally important than WGC, for example.) On the other hand, I can imagine that this line of research on WGC will turn into something that will be self-evidently fundamental in its implications.

As we said at the beginning, WGC talks about some "minimal difference between two situations" – too decoupled new forces (with too weak couplings and/or too heavy charged particles) are forbidden etc. So this WGC-dictated "minimum distance" could be a consequence of some new kind of "orthogonality" that is indeed analogous to (if not a special case of) the orthogonality of mutually exclusive states in quantum mechanics – which is an assumption that may be used to derive the uncertainty principle.

By saying that the lightest charge particles have to be light enough, the WGC also quantifies the intuition that all the "engines" responsible for a force etc. can never be squeezed into a too small region of space. You need the rather long distances – the Compton wavelength of the light enough particle – for this force to arise. That's a way to say that WGC may be said to be "somewhat similar" to the holographic principle, too. All these things suggest that the information can't hide in too small volumes, or in too inaccessible physical phenomena.

If something seems to be nontrivially correct – it doesn't seem to be quite a coincidence that gravity is the weakest force – the reasons should better be understood well. So people's thinking about it is clearly desirable. On the other hand, no one is guaranteed that it will lead to a full-fledged revolution. If WGC really implied that no model of inflation may exist, I would personally not believe such a conclusion, anyway (except if someone gave me a truly convincing full definition of quantum gravity with all the proofs; or at least some viable alternative to inflation). Maybe it's a mistake of mine but I still happen to think that the "case of inflation" is still much stronger than the "case for any particular strong version of WGC applied to instantons". (Whether the inequality for 1-forms may really be applied to 0-forms seems disputable to me, too. Note that the energy-time "uncertainty principle" must be interpreted differently, if it is possible at all, than the momentum-position uncertainty principle, and one must often be very careful when he generalizes things to "related situations".)

The subtitle "Testing Quantum Gravity via the Weak Gravity Conjecture" must be provocative for assorted Šmoits. Not only the essay dares to talk about the testing of quantum gravity. It's worse than that: quantum gravity and string theory are being tested according to a conjecture co-authored by a guy who insists that Šmoits and their apologists are just stinky piles of feces. ;-)

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