Order Underpins Everything

I discovered the work of Kevin H. Knuth, and took a dive into his papers and this recent talk given at the Perimeter Institute. The theme of his research is that a simple ordering relation among elements is more fundamental than, and can be used to derive, more familiar theories. The talk is entitled “The Role of Order in Natural Law”, and was part of a workshop on the topic of laws of nature.

In older papers, such as “Deriving Laws from Ordering Relations”, and “The Origin of Probability and Entropy”, Knuth shows how a partially ordered set, or lattice, can give rise to probability theory and information theory. There are broader applications, too, which follow from his demonstration of how order leads to algebra, and , through assigning numbers to elements, can further be used to derive measure theory; this is the link to order being revealed as foundational to other areas of science and mathematics. The derivation of probability theory, which is the first half of the talk, is easy to follow and interesting.

In a new paper, co-authored by colleague Newshaw Bahreyni, Knuth constructs a causal set (a bit different partially ordered set compared to the lattice) and undertakes to derive special relativity. The steps here are also pretty straightforward; a causal set is one where events are ordered according to a relation where some events have the potential to be informed of or influenced by, other events, but the relationship is not reciprocal. Completely ordered chains of events are selected to be “observers”, and events on chains are quantified by simply assigning integers. Then the relation of off-chain events to the earliest potentially informed events on 2 observer-chains gives rise to time-like and space-like coordinates. The Minkowski metric follows from this analysis, and Knuth and Bahreyni also derive Lorentz transformations.

Knuth also thinks that ordered pairs of numbers also could play a role in the foundations of quantum mechanics. He is a co-author of this paper by Philip Goyal (who also gave a talk at the same workshop), which looks to derive the complex quantum amplitudes from ordered pairs of real numbers. This looks very interesting as well.

From the conclusion of the causal set paper: “Given the belief that physical law represents underlying order, one would expect that given this underlying order, one ought to be able to derive the most fundamental aspects of physical law.”

Knuth’s program , while not yet tackling gravity, shares the spirit of many of the quantum gravity programs I’ve reviewed which explore the idea that the space-time of general relativity itself emerges from a more fundamental network of causally-linked elements. Causal sets themselves have played a featured role in the research program spearheaded by Rafael Sorkin (see old post here). Progress has been made in this program over a number of fronts, although incorporating quantum mechanics explicitly has evidently been a challenge. The arxiv has recent update papers on this causal set program by Sorkin and Joe Henson.

UPDATE 18 June 2010: I finished listening to Goyal's talk, and I am excited by his program. If even the foundations of QM end up being derived from 1) the ordering of events, 2) the logic that derives operationally from this order, and 3)an assumption about the limits on the accessibility of degrees of freedom (the essence of complementarity), this would be a great achievement. Events (which Goyal interestingly thinks about as Aristotelian actualizations) would be fundamental: "The quantum formalism is not explicitly about space and matter" -- these would be secondary notions; and time enters only in the minimal (but crucial) sense of ordering.