On Tuesday, September 3, 2002, at 02:21 PM, scerir wrote:

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> Tim May: > I don't have a comprehensive theory of time, > but I am very fond of "causal time." > > Sometimes we read papers saying there is now > experimental evidence that quantum phenomena > are "a-causal" or "non-causal" or "out-of-time". > > See, in example, these recent papers > http://arxiv.org/abs/quant-ph/0110124 > http://arxiv.org/abs/quant-ph/0201036 > > Now, can lattices capture also those important > features? I haven't read the papers, just the abstracts. I could wait to comment for a few days or weeks until I've had a chance to absorb the papers, if ever, or comment now. First, it looks like these events are the usual "entangled states," which can be spacelike (the usual example of particles separated by light years). Second, for such spacelike intervals, they are outside each others' light cones in the extreme cases, so it would not be expected for any partial ordering to exist. Third, my own idiosyncratic view is to look at entangled particles as a single system, regardless of separation. Fourth, as to the mechanics of lattices: the essence of a partially-ordered set (poset) is that it does not require trichotomy, where either a is less than b, a is greater than b, or a is equal to b. In a chain, a linear form of a lattice, trichotomy holds. So, the integers obey trichotomy, as one integer is either less than, greater than, or equal to any other integer. Orders which obey trichotomy are said to be well-ordered. But not all sets are well-ordered. If the ordering relation is set inclusion, then a series of sets need not obey trichotomy. Some sets may be disjoint, with one neither including the other, being included by the other, or equal. In terms of causality, not even getting involved with speed of light issues and light cones, it is quite possible to say "event A neither caused event B nor was caused by event B nor is the same as event B." That is, the events A and B are incommensurate, or disjoint...they fail trichotomy. Clearly, most events all around us are such examples of incommensurate. They form posets. What a lattice does is to formalize the notions of order and to say there is only one edge between two events, and nothing in between (no other nodes in between). If two events are separated by many instants of time, many other events, then the lattice is made up of the smallest identifiable events. The events look like a lattice. (As I said, the Web has many nice pictures. No point in my spending 20 minutes drawing an ASCII lattice here, having it reproduced poorly, when entering "lattice poset" into Google will turn up nice pictures.) So, I would say from reading the abstracts that the Bell example just fits the ecample of a poset, where two events, which may or may not be entangled, are spacelike to each other. (This is the essence of the usual "instantaneous action" of EPR/delayed choice experiments.) --Tim