Dear Mauro,
I appreciate the excellent analysis. I will have to break my response into a couple of segments, so I will post them as new posts rather than replies. To the points on which we seem to agree, I have little more to add, though I am interested in the "Planck-scale experiment" you referenced. One point is that the generalization of covariance I have in mind is much more general than semigroup representations. For the points on which we may disagree, I will itemize.
1. Regarding transitivity, I must insist on distinguishing between the "causal order" (of a classical universe) and the "binary relation generating the causal order." On large scales, the intransitivity I am talking about is as simple as the fact that the statement "Jane talked to Bill, then Bill talked to Susan," is not the same as the statement "Jane talked to Bill, then Jane and Bill talked to Susan." In either case, Susan received information from Jane, so the two statements are indistinguishable in their causal orders. However, in the first instance, the information is transmitted only through Bill, whereas in the second case it is transmitted both through Bill and directly. Thus, there are two different binary relations that generate the same causal order: the intransitive one in which information passes only through Bill, and the transitive one in which information also reaches Susan directly from Jane. These two are a priori different. At an ordinary scale, this is obvious to everyone.
For fundamental physics, the reasoning is as follows. Many scientists (by no means all!) agree that "causality," however you define it, is one of the most fundamental concepts in physics. The question then becomes: how do you define/describe causality? Well, a cause and effect certainly seem to define a direction; you can imagine an arrow pointing from the cause to the effect. This is completely local. Include lots of causes and effects (vertices), and arrows (directed edges) without yet imposing any other conditions, and you get a directed graph, which is equivalent to a binary relation on the set of vertices. At this stage, there is nothing to rule out cycles, and certainly nothing to impose transitivity, which are both generally nonlocal phenomena. There is a "causal order" generated by this directed graph, which is the relation defined by closing the graph relation under transitivity. I put "order" in quotes because this is still more general at this stage than the usual definition of a partial order; it may still have cycles, for instance.
This is all purely classical. To obtain a quantum theory, you need the superposition principle. The appropriate version of this in this case is a path sum over a configuration space of classical causal universes; i.e., directed graphs. I will explain why this is the appropriate version below. The question then becomes, "which graphs should be included in the configuration space?" This is the first real choice in the entire procedure, and involves a judgment about what types of graphs correspond to physical reality. My personal guess would be "acyclic locally finite directed graphs," but I want to make it clear that these are second-level assumptions that come further along in the development. I prefer acyclicity because we don't seem to observe causal cycles, and I choose local finiteness because I suspect that volume has something to do with counting (not necessarily as simple as Sorkin's "order plus number equals geometry", but in the same spirit).
I particular, it makes an a priori difference if you include only transitive graphs (graphs in which there is an edge between two vertices whenever there is a path between them). It's conceivable that this difference would fall out of the path sum, but I see no justification for assuming this at the outset.
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