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Causal Set Theory

An artist impression of a causal set in 2+1 dimensions. All matter we know consists of elementary particles. These particles are excitations in quantum fields, but we can also think about them as discrete entities. If all matter is discrete it begs the question could or should space also be discrete? The answer is we do not know.
Discretizing space does lead to one big problem, since it usually breaks Lorentz invariance. Lorentz invariance is the symmetry introduced in Einstein's theory of special relativity and it has been tested to extremely high precision. Causal Set theory discretizes space-time without breaking Lorentz symmetry.
This works because Causal Set theory discretizes the causal structure of space-time. The idea is based on works by Stephen Hawking and David Malament that together prove that two space-times that have the same causal structure are isomorphic to each other up to conformal transformations. From this Rafael Sorkin et. al. concluded that if one discretized the causal structure and assigned a fundamental volume to each element one could reconstruct the causal structure and the conformal transformations. The resulting structure is a partially ordered set. The partial order is induced by the causal order of events so for two elements x,y we can read x ≺ y as x is to the past of y.

Hesse diagram of a small causal set. Since this sounds rather arcane let us examine this in a simple example. On the left hand site is a little picture, a so called Hesse diagram, of a six element partial order. If we want to interpret this as a space-time then the first even, the big bang if you want, is element 0. It is to the past of all other elements in our little causal set. Immediately after the big bang at 0 are the events 1,2, and 3. These elements are space-like to each other. One might be tempted to say they are happening simultaneously, but since the notion of simultaneity can not be defined in Lorentzian theories this would be wrong. These three elements then have different futures. Element 3 has no future at all, so any signal send to this point or any observer at this event would afterwards just cease to exist. Element 1 has the elements 4 and 5 to its future and element 2 shares the element 5. So two observers could travel to element 5 coming along different ways and then exchange opinions on their different experiences. Once the observers reach elements 4 or 5 their path ends. This is of course due to our extremely small example causal set. If we assume that each causal set element is of Planckian volume, then our entire universe from the big bang to the current day would consist of ∼ 10²⁴⁰ elements.

Discretizing space-time in this way is manifestly Lorentz invariant. The problem is reconstructing space-time from the partial order. Much of the physics we do does depend on the differentiable structure of smooth manifolds and this has to be reconstructed or replaced by other means.

A causal set generated by sprinkling into 2 dimensional Minkowski space. Space-time in Causal Set theory is fundamentally discrete, thus the continuum description is an approximation. One way to define when a continuum description is a good approximation of a Causal Set is that the Causal Set would arise with high probability when the space-time is discretized according to the sprinkling process. In the sprinkling process elements of the space-time are chosen at random, according to a Poisson distribution PN(V,ρ)= (ρ V)N/N! e-ρ V with ρ=1/Vₚₗ if we assume that each element of the causal set has a planck volume Vₚₗ. The partial order between these elements is then induced by the causal order in the manifold.

Some of the questions in Causal Set theory that I have worked on and am thinking about are

Differential structure
Most equations in physics that tell us about how things behave are differential equations. They tell us how infinitesimal changes affect a system. In a discrete context we can not make infinitesimal changes, instead all changes become finite and we need to find finite difference equations that approximate the differential equations. In most discrete approaches one can use differences between nearest neighbours as a good approximation, but in a causal set each element has very many neighbours. In the limit of an infinite size causal set each element has infinitely many neighbours. It has been possible to construct a d'Alembertian operator calculating differences including not only the nearest neighbours but also higher order neighbours. How many orders of neighbours are needes is dependent on the dimension.
In the infinite density limit this d'Alembertian operator tends towards the continuum d'Alembertian operator, however with large fluctuations. These fluctuations can be tamed by introducing an intermediate length-scale. And this intermediate length scale can lead to interesting phenomenological consequences.
The d'Alembertian operator can also be used to define an approximation to the curvature scalar of a causal set and thus defines the so called Benincasa-Dowker action on a causal set. Relevant articles on this are
Recognising structure
As mentioned above it is easy to generate a causal set from a manifold that approximates it using the sprinkling process. However the converse process, recovering some of the continuum structure from the causal set, is very hard. Some properties of the continuum are relatively easy to reconstruct, like dimension and geodesic distance. Others however, like the question whether a causal set approximates a small flat region are difficult.
Computer simulations
Discrete systems are ideally suited to being studied on a computer. Computer simulations are already very useful when the system in question has to be discretized first, but in a discrete system this step can be omitted. There are different ideas how one can calculate the path integral of a causal set using Monte Carlo simulations. One can either use a growth dynamics that generates causal sets according to certain probablity distributions or use a causal set action. Using the Benincasa Dowker action to simulate a special class of 2 dimensional causal sets has led to especially interesting results.