References for causal sets                       HOME

We can, of course, invent causal sets or generate them artificially for the purpose of studying them. But in nature, causal sets come from observation. We learn causal sets by observing nature. Interactions are their natural source. In Physics, and interaction takes place when two physical objects, that are different and independent, approach each other and affect each-other's state. For example, a beam of light impinging on a photosensitive element that generates a signal is an interaction and gives rise to one causal pair where the beam is the cause and the signal is the effect. But not the only cause: the geometry is also a cause. The interaction happens if and only if the sensor is located in the trajectory of the beam.

Causal sets are very valuable for us, and we treasure them to the point that we are constantly seeking them. A child who asks "why" is asking for a causality. A scientist who studies a disease, a detective who investigates a crime, a viewer who is watching the news, are all seeking causality. Causality is what allows us to predict, and predicting gives us more options and helps us to live better and survive.

Because causal sets are so valuable for us, we save them and preserve them in many different ways: literature, manuals, theories, computer programs, pictures, and many others. The original causal sets can be recovered fro there, at least in some case. One of the most useful sources for recovery are computer programs. A causal set is an algorithm, and so also is a computer program. A computer program is a giant causal set, encoded in some human-readable programming language. Small programs can be converted manually, but specialized language-specific software is needed to make conversion more agile and less error prone. To my knowledge, no such software exists.

But causal sets by themselves have been vastly ignored. There is very little specific literature about them and their properties. Here is what I have found so far.

Most of what has been written so far about causal sets is specific for quantum gravity. This includes the "A non-technical introduction to causal sets" by Rafael Sorkin, Geometry from order: causal sets also by Rafael Sorkin, the Wikipedia page on causal sets, the Wikipedia page on causal structure, the Wikipedia page on Rafael Sorkin, a Non-technical introduction to causal sets by Rafael Sorkin, and Causal sets and quantum gravity. Causal set theory and the origin of mass ratio is an article by Carey Carlson where he uses pure causal reasoning to derive causal models for several fundamental particles and uses the models to calculate the proton/electron mass ratio. The result is 1836, while the measured value is currently accepted to be 1836.15. This is a fantastic success for the causal theory.

In the same category, there is also The scalar curvature of a causal set, by D.M. T. Benincasa and F. Dowker, arXiv: 1001.2725v4 [gr-qc] 1 Nov 2011. This paper is also about quantum gravity, but it has a unique feature: it proposes an approximately local action functional for causal sets.

That is because causal sets are considered as a particular case of partially ordered sets. Even partially ordered sets came to attention only recently, and several books have been published about them. But nearly all the books are concerned mostly with infinite partially ordered sets. As far as I know, there is only one book  about finite partially ordered sets: Finite Ordered Sets by N. Caspard, B. Leclerc, and B. Monjardet. It has some theorems for partially ordered sets, but nothing about causal sets. The term does not even appear in the index. There is also a Library of Lisp functions for partially ordered sets, but I am not familiar with this material.

But causal sets are in fact a specialization of partially ordered sets. They have such interesting and powerful properties that they deserve a place for themselves. Unfortunately, very few people have realized this and Wikipedia has not allowed me to write a page on causal sets proper. There is preciously little literature on causal sets proper. This is what I have:

Ten simple rules for dynamic causal modeling, by K. E. Stephan, W. D. Penny, R. J. Moran, H. E. M. den Ouden, J. Daunizeau, and K. J. Friston. Neuroimage, 49 (2010) 3099-3109. Many references are included.

Causal sets from simple models of computation, by Tommaso Bolognesi, arXiv:1004.3128, April 19, 2010.

A glossary for causal sets

However, none of these authors appears to be aware of my work, specifically of the following three properties that I have proposed and apply to any causal set model of a dynamical system:
(1) the causal set is the simplest mathematical object that has a metric;
(2) the simplest metric of causal sets represents action in the dynamical model.
(3) as a physical model, causal sets satisfy Lee Smolin's third rule for competing hypothesis, also known as Occam's razor.
I also propose that, as a result of the third property, and of the fact that all physical systems this side of the black holes are causal, the third property should be reversed as follows: Lee Smolin's third rule and Occam's razor follow as a consequence of the first and second properties of causal set dynamical models.

My papers in WASET, Complexity, and JAGI should also be included here. Note that, being unaware of causal sets, I referred to them as canonical matrices, or as partially ordered sets. Now I know I meant causal sets in all cases.