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The trouble with symmetry

One of the greatest advances in theoretical particle physics in the 20th century is Noether’s theorem. If you’ve never heard of it, you’re not alone. It’s an achievement that seldom makes it into popular titles, despite the fact that it’s arguably the greatest single achievement of mathematical physics that’s ever been made. It was conceived of by one of the unsung heroes of the field–Emmy Noether, probably the greatest woman mathematician who ever lived.

What Noether’s theorem tells us is that for every symmetry of a physical system, there is a conserved quantity, and vice-versa. The conservation of energy, for instance, corresponds to symmetry in time. Conservation of momentum corresponds to the symmetry due to translation through space. What Noether’s theorem essentially tells us is that when you’re trying to build a working theory of physics, what really counts are the symmetries. Nail the symmetries and you’ve essentially nailed the problem.

The problem for digital physics is that Noether’s theorem specifically relates to differentiable symmetries. In other words, ones that change smoothly. For symmetries that don’t change smoothly, all bets are off. This means that anyone trying to use a discrete, computational system to model physics is hamstrung right out of the gate.

In order to bridge this gulf, it seems to me that you need some way of describing computation in terms of symmetries, or symmetries in terms of computation. Either way, you need some nice formal way of putting the two notions on the same footing so that a meaningful, discretized version of Noether’s theorem can be derived. In other words, you need some kind of super-math that slides right in there between calculus and the theory of computation.

Though the link may not yet be obvious, this was where I was going with my recent post on Simplicity and Turing machines. But what does simplicity have to do with symmetry? Plenty, I suspect. I propose that we try to bridge the gulf between symmetry and computation with an idea that has elements of both: the idea of a partial symmetry.

But what is a partial symmetry? This terminology doesn’t exist anywhere in math or physics. And what does it even mean? Either something is symmetric or its not. In truth, partial symmetry something I made up, inspired by the reading I was doing on partial orders. And it’s a notion I’m still ironing the bugs out of. It works like this:

Any time you have a system that displays a symmetry, there is informational redundancy in it. Because there is redundancy in it, you can look at that system as the outcome of some sequence of copying operations applied to an initial seed from which redundancy is missing. Consider a clock face. We can treat the clock as a shape that happens to have twelve-fold symmetry, or we can think of it as a segment for describing a single hour that’s been replicated twelve times. This isn’t how we normally think about symmetry, but in spirit it’s not that far from a more familiar idea that mathematicians use called a group action.

A clock made up of twelve segments

However, if your copying operation doesn’t preserve all the information in the initial seed, you don’t have a full symmetry. Consider what happens if, instead of taking those clock segments and lining them up in a circle, you copy and move with each step in such a way that a part of each segment is hidden. You still end up with something that’s got a lot of the properties of a symmetric object, but it’s not fully symmetric. Furthermore, as soon as you do this, the ordering of the sequence of copying operations suddenly matters.

A partially symmetric not-clock with six segments

My proposal is that partial symmetry is equivalent to computation. And that armed with this idea, we can start to look at the symmetries that appear in nature in a new light. That might sound like a bit of a stretch, but in later posts I’m going to try to show you how it works.

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  1. July 12, 2012 at 1:24 am

    Your predictions about the Higgs that you made in a previous post get a bit of support in this article: http://www.technologyreview.com/view/428428/higgs-boson-may-be-an-imposter-say-particle/

    • July 18, 2012 at 6:21 pm

      Thank you for the link! I think it’s great that they’ve found *something* even though its exact nature is still ambiguous. You’re inspiring me to say something about it on this blog.

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