{\huge Do Random Walks Avoid Fireworks?}

Musings on gravity and quantum on the 4th of July \vspace{.5in}

Amanda Gefter is the author of the book titled Trespassing on Einstein's Lawn: A Father, a Daughter, the Meaning of Nothing, and the Beginning of Everything. As the title suggests and Peter Woit pointed out in his review, there are parallels with Jim Holt's book Why Does the World Exist?, which we covered a year ago. Gefter's essay for Edge last year lays out more briefly how Einstein regarded relativity and early quantum as structurally divergent theories.

I am currently reading her book, and it has led me to think further about the relationship between gravity and quantum effects.

The book is terrific. She simply has the ability to explain deep physics to a non-expert, me, in a way that makes it seem clear. Perhaps it is all a shell game, and what she says is misleading. But I think that she really does open the door to physics in a way that few popular authors have been able to do. Read the book and decide for yourself.

What I am thinking about is whether quantum and gravity ``meet'' in ways that maybe haven't been fully appreciated, even though the context is simple: collisions of freely moving bodies. The bodies have to be tiny yet acting mainly under gravity. The question is whether quantum effects would make them avoid collision and fireworks with high probability. Of course today is Independence Day in the US where we have fireworks, though Ken and I feel the World Cup soccer games have already given us plenty.

A Thought Experiment

Let $latex {A}&fg=000000$ and $latex {B}&fg=000000$ be electrically neutral objects of mass $latex {m}&fg=000000$ separated by some distance $latex {d}&fg=000000$ in a universe with nothing else. Okay maybe that is impossible, but this is a thought experiment. In a Newtonian world they would move toward each other according to the inverse law of gravity and would collide in a time that depends on their mass and distance. The exact bound is, I believe:

$latex \displaystyle \frac{\pi}{2\sqrt{2}}d^{3/2}/\sqrt{2mG}. &fg=000000$

In an Einsteinian world they would also move toward each other, and would collide after some time that also depends on the given parameters, $latex {m}&fg=000000$ and $latex {d}&fg=000000$. The time would, of course, be close to the Newtonian time, if the mass and distance are small. In any event, again, in a finite time they would collide.

But the above arguments miss a point. Suppose that the objects $latex {A}&fg=000000$ and $latex {B}&fg=000000$ were neutrons. Then the above conditions seem fine, but there is a problem. The ``nothing else'' clause is false if we allow---as we must---quantum effects. As the particles are drawn together due to gravity, I believe they will encounter virtual particles from the vacuum quantum flux. Suppose that virtual particles appear and disappear: would they not create some gravitational force, even for a short time? Then it seems they would disturb the path of the neutrons?

If this is the case, then a curious possibility seems to arise. The objects may never collide at any time in the future. The reason is based on how random walks work in three dimensions. If we view the particles as moving toward each other but subject to random small tugs, then depending on the strength of the virtual gravity effect, the particles could avoid each other forever. Or they could take a very different time than predicted by either theory of gravity. What happens?

Random Walks

When there is lots of matter, quantum effects are known to forestall overwhelming gravitational force. The engine of electron degeneracy pressure, which prevents certain stars from collapsing beyond the white-dwarf stage, is Wolfgang Pauli's exclusion principle, which forbids two fermions from having the same quantum state. There is more general quantum degeneracy, and in models it is affected by the dimension of the ambient space.

For very sparse matter, however, the fact that I allude to is this: For a random walk in Euclidean space, the nature of the walk depends on the dimension of the space. In one or two dimensions an unbiased walk will return to a initial location with probability $latex {1}&fg=000000$. In three dimensions the probability of returning to the start of the random walk is strictly less than $latex {1}&fg=000000$. George Pólya proved this back in 1921. Does this make a difference here?

Pólya's result was originally for $latex {d}&fg=000000$-dimensional square latices, but as noted in this 1998 PhD thesis by Peter Doyle, it extends to other regular lattices and to $latex {d}&fg=000000$-dimensional Euclidean space in general. There are various different proofs, of which perhaps the shortest is this recent proof by Shrirang Mare drawing on Doyle's thesis.

The ``quick'' reason why $latex {d=2}&fg=000000$ is a critical value is expressed by either of two integrals. One is that the ``resistance'' of the spatial medium to being infinitely penetrated by the walk is expressible by an integral of the form

$latex \displaystyle \int_{a}^{\infty}\frac{dr}{r^{d-1}}, &fg=000000$

$latex \displaystyle \int_a^{\infty}\frac{dr}{(\sqrt{r})^d}. &fg=000000$

Open Problems

The question is: what actually happens? Do the virtual particles create a gravitation effect that moves the neutrons? Or is the effect too small? Or nonexistent? What happens? Is this in some quantum sense a reason why space is quiet?

We are also interested because of some conjectures noted at the end of Doyle's thesis: For any infinite graph $latex {G}&fg=000000$ that is ``highly regular'' in the sense of its automorphism group having only finitely many orbits, the maximum number of vertices within distance $latex {d}&fg=000000$ of a given node grows either as $latex {r^d}&fg=000000$ or as $latex {e^{\Theta(r)}}&fg=000000$, where in the first case of this ``poly/exp gap,'' $latex {d}&fg=000000$ is a positive integer. The further conjecture is that the basic random walk on $latex {G}&fg=000000$ is recurrent if and only if $latex {d \leq 2}&fg=000000$. We noted that a large case of this conjecture was proved by Mikhail Gromov.