Hello again, cubic symmetry, and simulations
Hello all. It’s been a while since I’ve posted anything on this blog. My life has been in flux of late, as I’ve been moving to Princeton, NJ, changing jobs, and having a baby all at the same time. Now that things are starting to settle, it should be a lot easier for me to find time to write.
With that in mind, here’s my take on a recent article that people forwarded to me a while back during my break–the result from Silas Beane at the university of Bonn that claims to have something to say on the subject of the simulated universe. The arxiv blog, as usual, as a good write up.
The gist of the research is this: if the universe is running in a simulation on a cubic lattice, in much the way that current quantum chromodynamics simulations are calculated, then there should be experimentally observable consequences. Beane and his team identify two: the anisotropic distribution of cosmic rays (different amounts of rays in different directions), and a cut-off in the energy of cosmic ray particles. This article generated some excitement because the cut-off matches a phenomenon that’s already been observed.
A great moment for digital physics, right? I’m not convinced. I have a few concerns about this work. For starters, as I have discussed on this blog, there are a huge number of ways of building discrete universe models, of which a 3D lattice is only one. That simulation style has significant limitations, which, while not insurmountable, certainly make it a tough fit for a huge number of observed physical effects, such as relativity and spatial expansion.
Furthermore, in order to make their predictions, Beane and his associates simulated at a tiny scale. This is convenient because you only have to consider a single reference frame, and can treat space as a static backdrop for events. In other words, it’s pretty clear that the main problems with regular lattice simulations are things that their research didn’t touch.
I would find it astonishing, therefore, if we discovered the predicted cosmic ray anisotropy. And this brings me on to my second major concern. People, upon finding no irregularity in the cosmic ray distribution, are then likely to think, “gosh, well the universe was isotropic after all, I guess we’re not in a simulation.”
Except, let’s recall, experiments have already seen the expected energetic cut-off. In other words, the cosmic ray observations we see are perfectly consistent with a universe that’s discrete, but also isotropic. In other words, irregular, like a network. This, perhaps, shouldn’t come as a surprise.
Then, there is my third concern, and this reflects the interpretation imposed on this result. Namely, that a universe that turns out to run on an algorithm must somehow be a simulation running on a computer elsewhere. This, as I’ve also mentioned in previous posts, is just plain wrong.
Algorithms, like equations, are tools we use to build models. One does not have primacy over the other. One is not more natural than the other. A universe that turns out to be algorithmic no more requires a computer to run on than a universe based on differential equations needs a system of valves. The one main difference between algorithms and equations is that you can describe a vastly larger set of systems with algorithms. Equations are nice because, once you’ve figured them out, you can do lots of nifty reasoning. However, the number of possible systems that are amenable to this treatment is vanishingly small, compared to the systems that are not.
Most physicists want the universe to turn out to be completely describable with equations, because it would make life a lot easier for everyone. It’s a nice thing to hope for. It’s just that given the set of options available, it’s not terribly likely.
How to Build A Universe 
Aren’t algorithms just iterative equations?
Great question!
And I think the answer is rather harder to pin down than it might first appear. I would propose that a classical computer scientist would say ‘no’. That’s because algorithms have features like machine state and operate on structures other than numeric variables. However, I suspect that someone like Gregory Chaitin might argue otherwise. After all, you can take classes of iterative systems and make statements about their machine equivalence.
However, in any case, iteration is important. Furthermore, we’re not just talking about any old equations here. Mainstream physics is doing its best to demonstrate that natural systems equate to differentiable functions, and those functions are a tiny subset of the total function space.
They have good reason to try, because we do see a lot of differentiable behavior at the scales we can probe. The problem is that when you’re doing quantum gravity research, you’re talking about describing nature at its smallest scale. It’s relatively easy to create non-differentiable systems that yield differentiable behavior in bulk most of the time. Given this, it seems mighty optimistic to presume differentiability all the way down, particularly because of the horrific implications for the machine-equivalence of the universe.
Hi Alex, it’s great to see you back and blogging and congratulations on the new addition to your family! Terrific post and your explanation of the significance of the cosmic ray test is enlightening. It will be great if there are more How To Build a Universe posts to come.
I’ve had things keep me away also.
One thing that caught my attention is your comment that there are vastly more sets of systems with algorithms. It certainly got me thinking and I’m not quite sure what it means. I’ve always presumed the number of algorithms is countably infinite since there are countably infinite algorithm programs and they can in fact be ordered. Systems of equations I also took to be infinite if for no other reason that an equation can have parameters and parameters can be tweaked so there are infinite equations. Technically there are actually *uncountably* infinite systems if you allow parameters to be any real number, a thing you can’t do with algorithms if you assume them to be computable.
I’d think there are even countably infinite *structures* of equations.
But maybe my thoughts are confused, to me the space of definable systems is a confusing notion.
Hey Ray,
You raise a good point. There are two problems with the statements I made in this post. First, my language wasn’t clear, and secondly, I didn’t back up my statement with a source or reference. To address the first, I should say that I was referring to systems of differentiable equations specifically. To address the second point, there is some work by Leon Chua at Berkeley that I was using as the basis of the remark. He’s done some work that suggests that differentiable functions are rare cases in the set of possible outputs from some general class of representational systems. I confess that while I’ve looked at some of his work, I’m not familiar with the details of his proof.
On thinking about it though, you’re right that there are countably many algorithms. I’ve also heard that there is some equivalence between completely differentiable systems and TM-equivalent ones. (Another piece of math I’d have to track down to be able to justify.) And it seems highly unlikely that the set of completely differentiable systems is finite, and thus it must be the same size as the set of algorithmic ones, at the very least.
There is of course, still the difference in the efficiency of representation in the two systems, and the implications for the kind of universal machine you need to run them.
I guess I don’t have my clear-thinking blog-mojo quite reactivated yet. A sudden case of fatherhood will do that, I’m told. I’ll try to pay a little closer attention in my next post. 🙂
Alex
Sorry for the long delay since my last comment.
Firstly, your blog-mojo is fully up and running as far as I’m concerned. Secondly, my intent was never to call you out – it was a case of piqued interest and I wanted to learn more. (I also enjoyed the irony of a finitist trying to convince someone not only of infinite sets but of different orders of infinite sets.) Your answer gave me a little better understanding of the concept of the set of differentiable systems.
I checked out Chua’s site and was interested to see he’s done work on cellular automata.
Ray
Perhaps you would be interested to read this
http://www.technologyreview.com/view/415054/how-entanglement-could-be-deterministic/
Hooft said : “One could argue that symmetry arguments should not enter into the discussion of the interpretation of quantum mechanics.”
thought i didn’t understand it maybe that would interest you
I believe that Gerard ‘tHooft has met with, and been influenced by Ed Fredkin. I find ‘tHooft’s work fascinating, but I’m also ill-equipped to be able to asses a lot of it. The fact that he’s pursuing still discrete physics options gives me hope. However, there are deterministic ways to model quantum entanglement that are a lot simpler than what he describes.
If you haven’t looked at my Making Waves post, that’s an easy place to start.