Fundamental Physics and NKS

Øyvind Tafjord, Wolfram Research

Outline of talk:

NKS insights and fundamental physics

Insights from A New Kind of Science relevant when thinking about fundamental physics:

Traditional fundamental physics is based on PDEs and continuous systems. A New Kind of Science makes real the possibility that a simple, discrete system could do the job.

Intrinsic randomess helps discrete systems act continuous on a larger scale. Might also help in understanding the mysteries of quantum mechanics.

Computational irreducibility puts limits on what we can hope to do, and guides how we should go about doing it. In many cases, we just have to follow the evolution of a system to see what happens.

In a simple enough framework we can do exhaustive searches over all possible rules, without any preconceived notions of what we're looking for (caveat: must chose an appropriate framework!).

Causal Networks and Relativity

We can "slice" a causal network corresponding to the evolution of the underlying system, as on page 516. But not all slicings correspond to an actual evolution.

However, if the underlying rules are "causally invariant", any slice corresponds to a possible evolution state (page 518).

Relative uniform motion is now represented by slicing through the network at different angles, note how all slices correspond to a possible state in the underlying system. We also see there's a maximum possible angle, and correspondingly a maximum speed - which we identify with the speed of light. 

Thus we have the cornerstone of relativity: an absolute upper limit on the speed of causal propagation which is the same for all observers moving at constant velocity.

Network systems as underlying rule

We've been looking at causal networks derived from an underlying string-rewrite rule, but this would probably not be a convenient way of representing the universe. We need a simple, yet flexible structure.

The suggestion in A New Kind of Science is to consider network replacement rules. A network is simply a collection of nodes and connections (where each node has exactly three connections, for simplicity), and the replacement rule takes a part of the network that matches a pattern and rewrites that part into another network (page 512).

What's so great about networks?
	• Simple. One of the simplest non-trivial structures on can imagine
	• Abstract. We don't commit to a particular structure for space, for instance
	• Flexible. Can "represent" space of various dimensionality (page 477), localized structures (particles), etc.
	• Simple rules. Can have simple replacement rules with possibilities for exhaustive searches
	
Of course, having a promising idea doesn't mean that it has to work out perfectly. But starting to correlate this idea with the known features of fundamental physics, one can see that it has the potential for working. Let's explore a couple of those features.

Particles and networks

In it's simplest form, a particle can be thought of a distinguished feature on the network, moving around according to the update rules, as in on page 525.

In reality this is too fragile of a notion of a particle, as any disturbance will destroy it. 

Some higher-level feature, perhaps of topological nature, is more appropriate.

Toy example: Assume the network is for the most part planar. A local non-planar part of the network can then be associated with a particle (page 527).

Part of the challenge of understanding the network physics, will be to identify quantities associated with tangles - quantities that are mostly conserved such that they can be associated with particle properties, such a charge, mass, spin, etc.

Gravity and networks

One of the profound discoveries of the last century was that gravity can be viewed as a consequence of space curvature. Can we see such curvature in the networks?

For starters, how do we even know the dimensionality of the space corresponding to a network? Doesn't that depend on how we draw the network?

If we consider, as an approximation, that each connection in the network corresonds to roughly a unit distance in space (say, the Planck length of (10^(-35)) meter), then there are restrictions for how to draw the network, and we can infer the dimensionality from this. One way to do this is to start at a node and count how many nodes are within a distance r away, call this number N(r). Then check whether N(r) increases as r, (r^2), (r^3), etc - indication of 1, 2, 3, etc dimensions:

Gravity and networks

If N(r) scales exactly like (r^2), the resulting network corresponds to flat 2-dimensional space. A small deviation from this will result in the space being curved.

Let's look at some pictures of 2-dimensional networks. First a flat one, and then some curved ones (page 532). The flat one has all hexagons, while the ones bulging outwards (positive curvature) has a few pentagons strewn about, and the negative curved ones have heptagons. If we view these surfaces from the side instead, they look like this: 


And in fact, in general we have N(r) = a (r^2)(1 - k (r^2) + … ), where k is directly proportional to the scalar curvature (the Ricci scalar) of the surface at that point. Similarly, in other dimensions we have N(r) = a (r^d)(1 - k (r^2) + … ). 

From this starting point of scalar curvature, one can start building the more complicated curvature quantities in the Ricci tensor, by combining the notion of a shortest path between two points and a cylinder along this path. And by then we have all the ingredients to build the gravitational part of the Einstein equations for gravity.

In summary

This whole discussion of curvature on network spaces might cause one to think that these network are somehow fairly static, clean and structured. But those are just gross idealizations of what would really go on!

So let's do a thought experiment and zoom in on my finger and see what we'll find once we've gone past the details of the finger print, the tissue, the skin cells, the molecules, the atoms, the protons, the quarks, and way beyond... Finally we'll start seeing (in an appropriately interpreted way) the nodes and connections of the underlying network. And what a mess it is! 

Viewed in "compressed time", there's seething activity, with connections flipping around constantly, seemingly at random. But zooming out somewhat, we should see some overall features that correspond to our 3-dimensional space, but with all kinds of complicated concoctions which seems like fairly localized units, although some of them melt into eachother in strange ways. That would be our dear elementary particles. And from there on out, things resolve into their well-known nature until we're back to the finger again...

And if this bold hypothesis is right, then everything around us is caused by some simple network replacement rule just doing its work over and over and over again. Recognizing the result of this rule as our world won't be easy, as there are many layers of abstraction between it and the laws of Nature as we know them today.

But that's the challenge of fundamental physics, the NKS way.

init stuff

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