05 December 2012

Causal Dynamical Triangulation

Causal Dynamical Triangulation

I've been reading up on Causal Dynamical Triangulation (CDT) (by Loll, Ambjoern, and Jurkiewicz). It's an attempted unified field theory related to Loop Quantum Gravity (LQG), which you may have read the Scientific American article on a few years back.
What it (like LQG) has to recommend it is that the structure of space emerges from the theory itself. Basically, it proposes a topological substrate (spin-foam) made of simplexes (lines, triangles, tetrahedrons, etc). Spatial curvature emerges from how those simplexes can join together.

Degeneration and the arrow of time

The big problem for CDT in its early form was that the space that emerged was not our space. What emerged was one of two degenerate forms. It either has infinite dimensions or just one. The topology went to one of two extremes of connectedness.
The key insight for CDT was that space emerges correctly if edges of simplexes can only be joined when their arrows of time are pointing in the same direction.

So time doesn't emerge?

But some like to see the "arrow of time" as emergent. The view is that it's not so much that states only mix (unmix) along the arrow of time. It's the other way around: "time" has an arrow of time because it has an unmixed state at one end (or point) and a mixed state at the other.
To say the say thing in a different way, the rule isn't that the arrow of time makes entropy increase, it's that when you have an entropy gradient along a time-like curve, you have an arrow of time.
The appeal is that we don't have to say that the time dimension has special rules such as making entropy increase in one direction. Also, both QM and relativity show us a time-symmetrical picture of fundamental interactions and emergent arrow-of-time doesn't mess that picture up.

Observables and CDT

So I immediately had to wonder, could the "only join edges if arrows of time are the same" behavior be emergent?
In quantum mechanics, you can only observe certain aspects of a wavefunction, called Observables. Given a superposition of a arrow-matched and arrow-mismatched CDT states, is it the case that only the arrow-matched state is observable? Ie that any self-adjoint operator must be only a function of arrow-matched states?
I frankly don't know CDT remotely well enough to say, but it doesn't sound promising and I have to suspect that Loll et al already looked at that.

A weaker variant

So I'm pessimistic of a theory where mismatched arrows are simply always cosmically censored.
But as far as my limited understanding CDT goes, with all due humility, there's room for them to be mostly censored. Like, arrow-mismatched components are strongly suppressed in all observables in cases where there's a strong arrow of time.

Degeneration: A feature, not a bug?

It occured to me that the degeneration I described earlier might be a feature and not a bug.
Suppose for a moment that CDT is true but that the "only join edges if arrows of time are the same" behavior is just emergent, not fundamental. What happens in the far future, the heat death of the universe, when entropy has basically maxxed out?
Space degenerates. It doesn't even resemble our space. It's either an infinite-dimensioned complete graph or a 1-dimensioned line.

The Boltzmann Brain paradox

What's good about that is that it may solve the Boltzmann Brain paradox. Which is this:
What's the likelihood that a brain (and mind) just like yours would arise from random quantum fluctuations in empty space? Say, in a section of interstellar space a million cubic miles in volume which we observe for one minute?
Very small. Very, very small. But it's not zero. Nor does it even approach zero as the universe ages and gets less dense, at least not if the cosmological constant is non-zero. The probability has a lower limit.
Well, multiplying an infinite span of time times that gives an infinite number of expected cases of Boltzmann Brains exactly like our own. The situation should be utterly dominated by those cases. But that's the opposite of what we see.

Degeneracy to the rescue

But if CDT and emergent time are true, the universe would have degenerated long before that time. Waving my hands a bit, I doubt that a Boltzmann Brain could exist even momentarily in that sort of space. Paradox solved.

Is that the Big Rip?

(The foregoing was speculative and hand-waving, but this will be far more so)
Having described that degeneration, I can't help noticing its resemblance to the Big Rip, the hypothesized future event when cosmological expansion dominates the universe and tears everything apart.
That makes me wonder if the accelerating expansion of space that we see could be explained along similar lines. Like, the emergent arrow-of-time-matching isn't quite 100% perfect, and when it "misses", space expands a little.
This would fit with the weaker variant proposed above.


For one thing, it's not clear how it could explain the missing 72.8% of the universe's mass as dark energy was hypothesized to.


Now my hands are tired from all the hand-waving I'm doing, so I'll stop.

Edit: dynamic -> dynamical

Meaning 2

Meaning 2


I relayed the definition of "meaning" that I consider best, which is generally accepted in semiotics:
X means Y just if X is a reliable indication of Y
Lameen Souag asked a good question
how would [meaning as reliable indication] account for the fact that lies have a meaning?


"Reliable" doesn't mean foolproof. Good liars do abuse reliable indicators.
Second, when we have seen through a lie, we do use the term "meaning" in that way. When you know that someone is a liar, you might say "what she says doesn't mean anything" (doesn't reliably indicate anything). Or you might speak of a meaning that has little to do with the lie's literal words, but accords with what it reliably indicates: "When he says `trust me', that means you should keep your wallet closed."

Language interpretation

Perhaps you were speaking of a more surface sense of the lie's meaning? Like, you could say "Sabrina listed this item on Ebay as a 'new computer', but it's actually a used mop." Even people who considered her a liar and her utterances unreliable could understand what her promise meant; that's how they know she told a lie. They extract a meaning from an utterance even though they know it doesn't reliably indicate anything. Is that a fair summation of your point?
To understand utterances divorced from who actually says them, we use a consensus of how to transform from words and constructions to indicators; a language.
Don't throw away the context, though. We divorced the utterance from its circumstances and viewed it thru other people's consensus. We can't turn around and treat what we get thru that process as things we directly obtained from the situation; they weren't.
If Sabrina was reliable in her speech (wouldn't lie etc), we could take a shortcut here, because viewing her utterance thru others' consensus wouldn't change what it means. But she isn't, so we have to remember that the reliable-in-the-consensus indicators are not reliable in the real circumstances (Sabrina's Ebay postings).
So when interpreting a lie, we get a modified sense of meaning. "Consensus meaning", if you will. It's still a meaning (reliable indication), but we mustn't forget how we obtained it: not from the physical situation itself but via a consensus.

The consensus / language

NB, that only works because the (consensus of) language transforms words and constructions in reliable ways. If a lot of people used language very unreliably, it wouldn't. What if (say) half the speakers substituted antonyms on odd-numbered days, or when they secretly flipped a coin and it came up tails. How could you extract much meaning from what they said?

Not all interpretations are created equal

This may sound like All Interpretations Are Created Equal, and therefore you can't say objectively that Sabrina commited fraud; that's just your interpetation of what she said; there could be others. But that's not what I mean at all.
For instance, we can deduce that she committed fraud (taking the report as true).
At the start of our reasoning process, we only know her locutionary act - the physical expression of it, posting 'new computer for sale'. We don't assume anything about her perlocutionary act - convincing you (or someone) that she offers a new computer for sale.
  1. She knows the language (Assumption, so we can skip some boring parts)
  2. You might believe what she tells you (Assumption)
  3. Since the iterm is actually an old mop, making you believe that she offers a new computer is fraud. (Assumption)
  4. Under the language consensus, 'new computer' reliably indicates new computer (common vocabulary)
  5. Since she knows the language, she knew 'new computer' would be transformed reliably-in-the-consensus to indicate new computer (by 1&4)
  6. Reliably indicating 'new computer' to you implies meaning new computer to you. (by definition) (So now we begin to see her perlocutionary act)
  7. So by her uttering 'new computer', she has conveyed to you that she is offering a new computer (by 5&6)
  8. She thereby attempts the perlocutionary act of persuading you that she offers a new computer (by 2&7)
  9. She thereby commits fraud (by 3&8)
I made some assumptions for brevity, but the point is that with no more than this definition of meaning and language-as-mere-consensus, we can make interesting, reasonable deductions.

(Late edits for clarity)