One of the most important, powerful and beautiful ideas in modern physics is Noether’s theorem. This essentially says that the fundamental laws of physics are a manifestations of symmetry in the universe.
So if the universe has rotational symmetry, then it must also obey the law of conservation of angular momentum, if it has a time symmetry, then energy must be conserved and so on.
It’s hard to understate the profound significance of this approach. It seems to tease apart the very fabric of the universe to reveal a very powerful beauty beneath.
And yet, peer more closely at Noether’s theorem and you soon find its severe limitations. It turns out that this approach can only be applied to certain types of systems that have continuous symmetries.
That specifically excludes discrete systems, which proceed step-by-step. These include systems such as Turing machines, which one or two readers may be familiar with.
Take for example, Conway’s famous game of life, in which life-like forms can be generated using a cellular automaton. This takes place on a square grid which is symmetrical under quarter-turn rotations, but not under continuous rotations. And in this world, time advances in discrete steps rather than continuous ones.
Clearly Noether’s theorem cannot apply. So what happens to the conservation laws? In the game of life do we have to abandon conservation of energy, angular momentum and the like?
Today, Tommasso Tofoli at Boston University and Silvio Capobianco at Tallinn University of Technology in Estonia tackle exactly these questions. Their answer is a relief, of sorts–they find a family of discrete systems that obey a Noether-like theorem and show why.
The system they study is called an Ising spin model. It is a 2D array of elementary magnets that can each point either up or down. Each magnet is coupled to its four nearest neighbours by the mathematical equivalent of a rubber band. The band is stretched if the neighbour spins in the opposite direction and loose if it spins in the same direction.
The question that Toffoli and Capobianco study is how this system behaves, how the spins flip from one state to another, but first they impose an important limit on the kind of interactions that can occur.
This condition is that a spin will flip only if doing so leaves the sum of the potential energies of the four surrounding bonds unchanged. This can happen if two of the neighbours have parallel spins while the other two have antiparallel spins. This type of system is called a microcanonical Ising model.
This condition has important consequences. It means that potential energy is always conserved.
But think about this in more detail and it becomes a little difficult to pin down exactly what we mean by energy. The number of spin up and down magnets can of course change dramatically so this is not what it is conserved. However, the boundary between them must always be the same length. So if we define the length of this line as energy, then this is what is naturally conserved.
(Of course, the magnets, rubber bands and potential energies are not real but merely useful ways to think about this system.)
That may seem an arbitrary definition for energy but Toffoli and Capobianco go on to show that it has the same mathematical properties of energy in the real universe (defining energy in our world is itself mighty hard to do).
Of course, there is another aspect of this system that is easy to forget but crucial for conservation. This is the structure of the discrete space-time in which all the action takes place, in other words, the 2D grid and the time steps over which change takes place.
The climax of Toffoli and Capobianco’s paper is their demonstration that energy can only be conserved if the space-time is invariant, that all directions and times in this Ising Universe are essentially equivalent.
In this way, they show how a Noether-like theorem can apply in a discrete universe.
That’s hugely significant. It means that the same rules of symmetry that have been powerfully applied to modern physics can also apply to the many new disciplines that are beginning to exploit discrete models. These include many social sciences, complexity science, economics, web science and of course, the biggie: computer science.
In effect, these guys have used symmetry to derive conservation laws in a virtual world for the first time.
But the significance goes deeper still. The thing that links all these disciplines is information. They are all part of of a new thrust in modern science which ignores the superficial properties of physical reality and instead focuses on a deeper bedrock: the information on which the universe is built.
Although they don’t say this explicitly, what Toffoli and Capobianco are studying is the role that Noether-like theorems can play in this new world of information-based science.
Of course, it raises many questions too. Toffoli and Capobianco give just a single example of a discrete system in which a Noether-like theorem applies. What many people will want to know is how this can be generalised. For example, can it be made to apply to Conway’s game of life?
Either way, Toffoli and Capobianco have made a promising start. As they say themselves: “This is just the beginning of what promises to be a productive line of research.”
Ref: arxiv.org/abs/1103.4785: Can Anything From Noether’s Theorem Be Salvaged For Discrete Dynamical Systems?
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