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Do Random Walks Help Avoid Fireworks?

July 4, 2014

Musings on gravity and quantum on the 4th of July

Cropped from Crumbel source

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 as 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.

Today her book has led me to think further about the relationship between gravity and quantum mechanics.

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 ultimately 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 {A} and {B} be electrically neutral objects of mass {m} separated by some distance {d} 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:

\displaystyle  \frac{\pi}{2\sqrt{2}}d^{3/2}/\sqrt{2mG}.

See this for the detailed calculation. In any event, in a finite time they will collide: the exact bound is not so critical.

In an Einsteinian world they would also move toward each other, and would collide after some time that also depends on the given parameters, {m} and {d}. 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 {A} and {B} 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 {1}. In three dimensions the probability of returning to the start of the random walk is strictly less than {1}. George Pólya proved this back in 1921. Does this make a difference here?

Pólya’s result was originally for {d}-dimensional square latices, but as noted in this 1998 PhD thesis by Peter Doyle, it extends to other regular lattices and to {d}-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 {d=2} 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

\displaystyle  \int_{a}^{\infty}\frac{dr}{r^{d-1}},

while the other expresses the probability of non-repeating divergence in terms of the integral

\displaystyle  \int_a^{\infty}\frac{dr}{(\sqrt{r})^d}.

Even though one exponent is {d-1} and the other is {d/2}, both have {d=2} as the critical value between divergence and convergence. Doyle hints that the latter resolves a “chicken-or-egg” type question of whether the “reason” for {d=3} being the first whole number giving convergence is anchored in mathematics or physical reality, though he minimizes its significance. We, however, still wonder about the full physical significance of this fact, beyond references that stay on quantum walks such as this paper by Martin Štefaňák, Tamás Kiss, and Igor Jex.

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—beyond known issues about quantum field theory predictions for the vacuum?

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

11 Comments leave one →
  1. July 5, 2014 11:54 pm

    Something to think about …

  2. July 6, 2014 8:24 am

    One thing I’ve noticed over fifty years of observing the practice and philosophy of science is that reductionism acts as a kind of gravitational force.

    Are there counter pressures that preserve us from Ockham’s Overly Simplistic Singularity?

  3. Foster Boondoggle permalink
    July 8, 2014 1:14 pm

    If what you’re asking is: “what is the physics of two gravitating neutral particles, accounting for quantum effects”, isn’t this quite straightforward? There are gravitational bound-state solutions to the Schrodinger equation (with sizes vastly larger than the corresponding ones for atoms). The ground state would be a hydrogen-like spherically symmetric configuration of the two particles. This is a semiclassical picture, and doesn’t appear to involve any QFT-like virtual particle concepts. At low energies as you’ve stipulated, QFT effects play no role.

    In asking about “fireworks” — presumably the consequence of some sort of inelastic scattering (such as pair annihilation) — this scenario *does* predict eventual fireworks in 3-d (and higher), as the amplitude of the ground state wave function at zero separation is non-zero, so the probability for an eventual scattering / annihilation event is also non-zero.

    • tcmJOE permalink
      July 11, 2014 9:28 am

      I’d also expect that by graviton emission, the system would eventually settle down to a S_0 ground state eventually, in a similar fashion to how other electronic states will eventually decay down to their ground state.

      Unless there’s something special about graviton spin (L = 2) that really distinguishes it from photons (L = 1)?

  4. July 9, 2014 8:41 am

    Tangentially related is the discussion of neutrons falling in a gravitational field that is given by Kobakhidze (2011): “Once more: gravity is not an entropic force“.

  5. Paul permalink
    July 9, 2014 11:00 am

    Curious paper about one-way functions

    Do you think so?

  6. abel permalink
    July 9, 2014 2:44 pm

    In the last paragraph, what is r?

    • July 10, 2014 1:54 pm

      It is distance, similar to ‘r’ in the “Random Walks” section.


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