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Anthony close his essay with a terrific set of questions, which I am just cutting and pasting here:
1. If the speed of light were much slower, so we could really experience the subjectivity of simultaneity implied by Relativity, how to you think it would change our experience of time, space, and the world?
2. If you truly believed that the past, future, and present all exist in just the same way, as in the Unitary Block view, would it change your attitude toward life -- or death, or decisions?
3. If there are aspects of the world that are unpredictable in principle, is there still a sense in which we can say that they are determined? Or do those two ideas go hand-in-hand?
4. I've suggested that part of the controversy over how to think about quantum mechanics is a controversy over whether the classical, macroscopic world emerges from the quantum world, or whether the quantum world is a particular, stripped-down limit of an essentially classical world. How do you see it? What theoretical or experimental findings might lead you to accept one view as more viable than the other?
Please do visit BQO to take part in this discussion and more.
Some of you may have noticed that I enjoy writing about the question of what happens when you fall into a black hole. At the recent FQXi meeting in Vieques, Puerto Rico, "quantum mechanic" Seth Lloyd talked through this problem and discussed a way to potentially escape a black hole, first proposed by Horowitz and Maldacena, which uses quantum teleportation to smuggle information out--but also violates some quantum laws, in return.
Here is Seth, describing it in his inimitable style. He also explaining why, if you are unlucky enough to cross an event horizon, you shouldn't try and fire up the engines of your rocket and accelerate away from the singularity…because you'll only die faster.
For more about the debate over the fate of an unfortunate astronaut heading into a black hole, here's Anil Ananthaswamy's story about Steve Giddings' research, "Black Holes: Paradox Regained." There's also a story I wrote for Nature last year on the firewall debate, which pits general relativity against quantum physics--and tries to answer whether the space traveller would be spaghettified, as traditionally thought, or burnt to a crisp at the event horizon. And a follow-up discussing Stephen Hawking's take on the whole issue.
As promised, this is a follow-up to one of the summary posts from the FQXi conference in Vieques. If you have read those, you may recall that Anthony Aguirre asked the intriguing question: is there any way for us to tell if the universe is infinite or simply really, really big? (George Musser has also blogged about this.) In this blog post I suggest one possible way in which this might be accomplished. I emphasize that much of the content of this post is entirely speculative but it does offer suggestions on how to more rigorously determine the validity of the conjecture. It relies, however, on an assumption that runs entirely counter to my own conclusion in my most recent FQXi essay: that bits (in some fashion) constitute the basic building blocks of the universe.
Consider the following string of binary digits
where, for the moment, we assume that the full string is infinite but that we only have knowledge of the sub-string shown above. Now consider the following maps:
00 —> a
11 —> a
01 —> b
10 —> b
Notice that each digit in the original string ends up getting counted twice.
Thus the original string can be mapped to
where the X and the Y are unknown given that we do not know what precedes the 0 on the left and what follows the 1 on the right. Formally, this second string is doubly-infinite meaning its cardinality is double the cardinality of the first string since every digit in the first string is counted twice (yes, there are actually levels of infinity!).
Now suppose that we have complete knowledge of the doubly-infinite string of a’s and b’s with the exception of one value. It should be relatively clear that, if we know the distribution of 0’s and 1’s in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known unless the original distribution is 50/50 in which case we have no idea (try it!).
So now take any two locations, X and Y, on what we call the “image” string (the strings of a’s and b’s) with the caveat that there are at least two letters between them. Suppose that we have complete knowledge of the doubly-infinite string other than these two locations. From the argument in the preceding paragraph, it should be clear that the instant we know one of these two values (say X), we immediately know the other (Y). A formal proof of this appears in Section 5 of this article by Steve Shea, which is open access.
Do we really need an infinite amount of knowledge?
The way I have described the problem (courtesy of Steve Shea) would seem to imply that we need to know absolutely every value in the doubly-infinite string, i.e. we need an infinite amount of knowledge, if we are to correctly predict the value of Y given the value of X (which also assumes that the original string of 0’s and 1’s is not a 50/50 split). But this gets to the heart of Anthony’s problem: do we really need an infinite amount of knowledge or just a whole lot of knowledge? One could imagine that we could approach a very high probability of correctly predicting Y given our knowledge of X as our knowledge of the doubly-infinite string gets large (though I would like to emphasize that I am not aware of a formal proof of this as yet—Steve’s article has just been published, though he’s been working on it for several years).
Nevertheless, approaching such a high-degree of accuracy with a prediction would also seem to require that the strings really are infinite. One can imagine that if they are not infinite, there could be some sort of "edge" effect caused by the fact that the second string would be ill-defined at the ends and that such an effect could somehow propagate through the string (again, I emphasize that this is speculation at this point and that there is no formal proof of this). For example, we could interpret the mapping (and thus, to some extent, the original string) as being nothing more than a kind of production rule somewhat akin to those in formal language theories: it’s just a rule for generating the string of a’s and b’s (note that in such theories the "start" symbol need not be at the extreme left or right of a string which means it could still be infinite—if you are unfamiliar with the concept of a "start" symbol, a good read is Douglas Hofstadter’s Pulitzer Prize-winning opus Gödel, Escher, Bach). Given that (and assuming that the ratio of 0’s to 1’s in the original string is not unity, i.e. they are not a 50/50 split since, otherwise, there really is no production rule), it should be clear that the a’s and b’s are interdependent which means I could rewrite the production rule simply in terms of the a’s and b’s themselves. Inherent in the original rule, however, there is no way to deal with endpoints. Specifically, the endpoints of the string of a’s and b’s would be ill-defined based on the production rule which would make any neighboring values in the string undefined and so on such that the entire string is ill-defined (e.g. it would be as if there was no "start" symbol). Thus the string must be infinite, at least in this formal system.
I would then conjecture that while the strings themselves must be infinite, we only need a finite (though arguably large) amount of knowledge in order to predict Y given X with a high degree of accuracy. Of course this all hinges on whether or not the original string is completely random. If the original string of 0’s and 1’s is exactly 50/50, the value of Y cannot be predicted with any greater accuracy than 50% (which essentially means it cannot be predicted at all). Likewise, the requirement that the string of a’s and b’s be infinite no longer holds, i.e. it’s really a relic of the production rule.
A universe of qubits
What does any of this have to do with the universe? Let’s just suppose that John Wheeler was correct and that, at its core, the universe is built up from bits—binary digits—or, rather, qubits. Since we don’t actually see the world as 0’s and 1’s (or answers to yes/no questions) it is clear that there is some form of mapping that goes on at some deep level from these binary questions to something slightly less fundamental. Let’s take, as a first approximation, the map introduced at the beginning and suppose that the string of a’s and b’s represents the results of measurements of simple two-level quantum systems. In other words, the a’s and b’s represent our knowledge of the universe at its most fundamental level (notice that it may or may not suggest a deeper reality of 0’s and 1’s or even something else entirely). We might then imagine that the universe (or, rather, our knowledge of it) can be reduced to a sub-string of a very, very long (possibly infinite) string of outcomes of measurements on qubits. This is not so radical an idea, by the way (see Seth Lloyd’s article on this topic: arXiv:1312.4455).
One glitch in this argument, of course, is that we don’t know the order of the measurement outcomes, i.e. the string of a’s and b’s. We have small strings of correlated outcomes, but we don’t know for certain if all of these small strings are part of one larger string, i.e. if we know, for example,
is it necessarily true that
If all of the sub-strings indeed are part of one, larger string it would seem to suggest that all possible qubit measurements in the universe may be correlated in some way since, as long as there exists a production rule, we can connect even the most far-flung elements of the full string of a’s and b’s. This, also, is not such a radical idea if we take the correlations as being equivalent to quantum entanglement (see Buniy and Hsu’s article on this: arXiv:1205.1584v2).
Note that there is one question here that I have not addressed and that is whether knowledge of X necessarily implies that Y is the opposite of X. If the a’s and b’s represent measurement outcomes for entangled qubits then one would seem to expect that the outcomes of X and Y must be opposite one another. There is no requirement in the mathematics that this be the case. In addition the mathematics imply a string of somewhat looser correlations (recall they must be at least two letters apart) in between the entangled pair. It might be possible to address the latter through something like quantum teleportation, but I won’t address that here. Instead I will simply assume that the production rule is such that the sub-strings that are accessible to us appear to behave in such a way that X and Y always behave as if they are entangled.
Is the universe infinite or just really, really large?
Let’s take Buniy and Hsu’s idea as correct in which case all qubit measurements are somehow part of a single, large string. We can access different sub-strings of this larger string via entanglement measurements. Note, however, that X and Y do not necessarily need to be in the same sub-string. So, for instance, an entanglement experiment with qubits might represent the following pair of sub-strings
In other words, we are ignorant of some of the intermediary processes that connect them. Nevertheless, we do know that a determination of X immediately tells us what Y must be since this is measurable in a laboratory. This suggests that the intermediary processes, while perhaps not readily apparent to us, nevertheless exist as long as there is a production rule and as long as the full string is doubly-infinite. In fact, if Buniy and Hsu are correct (and note that their argument is based on cosmological models and thus not dependent on anything I suggest here), the mere presence of entanglement suggests that the string of all qubit measurements in the universe is infinite which would itself suggest that the universe is infinite, but only if the underlying production rule is not completely random. Recall that none of this works if the 0’s and 1’s in the original string are a 50/50 split. If they are, there is no correlation between X and Y.
I caution against rushing to judgement about these ideas. Clearly we see correlations and entanglement in the universe on some level. Does this automatically imply, then, that the universe is infinite and not entirely random at its base level? It clearly does not since we don’t know if Buniy and Hsu are correct. In other words, we have no idea of our measurements of entanglement are merely parts of one large, interconnected string of such measurements. What it does seem to tell us is that if they are correct (and it is a big "if"), then it is highly likely that the universe is both infinite and non-random. (It also would seem to suggest that Max Tegmark might be right after all about the role mathematics plays in the universe.)
Now Anthony was interested in this from the standpoint of the multiverse and one could easily modify these ideas to take that into account. I won’t do it here. I can also imagine someone taking Steve’s results without reference to Buniy and Hsu and coming up with a way to measure whether the universe is infinite or merely very large.
There are a lot of assumptions and conjectures in this post but there are also a lot of concrete starting points for further exploration. Can we definitively prove, mathematically, that these strings must be infinite for this effect to be possible? I offered a heuristic argument above as to why, but a more formal proof would be welcome. Can we really re-interpret Steve’s original mapping in a way that makes it self-referential to the string of a’s and b’s, i.e. can we find a production rule for the string of a’s and b’s such that it would exactly match what we would have obtained using the original mapping? Conversely, if we can’t, can we definitively rule out the possibility that one exists? Are Buniy and Hsu correct in suggesting that all particles in the universe are ultimately entangled and can we actually reduce everything to a set of qubit measurements? What might the results presented here say about the multiverse? Is there some other way to work with these results that might say something useful about the size of the universe?
In my mind, these are all ideas worth pursuing and so I suppose you could interpret this blog post as a challenge: let’s get some answers to these questions!
“There is a difference between whether the universe is infinite or just really really really really really really big,” Anthony Aguirre said at the recent FQXi conference in Puerto Rico. I’m pretty sure I counted six reallys. With that remark, he encapsulated a major debate going on within physics and cosmology right now. Although the conference theme was officially the physics of information, it could just as well have been the physics of infinity, so often did that little sideways ‘8’ put in an appearance. Is the universe finite or infinite? Is nature capable of a finite or infinite number of possible states? Can spacetime be infinitely subdivided or is it made of finite-size cells? The questions seem undecidable. But maybe the finitude or infinitude makes itself felt every time you do a measurement and every time you stir cream into a coffee cup and can’t unstir it out.
The forces of finitude include Max Tegmark, who has been bad-mouthing infinity on Edge.org, in quotes to New Scientist, and in Chapter 11 of his new book. His complaint is what cosmologists call the measure problem: there’s no way of unambiguously counting members of an infinite set. If there’s no way to count, there’s no way to calculate probabilities and therefore no way to relate theory to experiment. The whole empirical framework of science verges on collapse. A finite universe presents no such difficulty. Even Peter Woit, who agrees with Tegmark on little else, finds common ground with him on the measure problem.
The aficionados of infinity include Alan Guth, who argued in Puerto Rico and on Edge.org that a truly infinite universe would neatly explain the arrow of time. When space has no bound, neither does entropy. It keeps on increasing forever, always pointing the way forward for time. The universe need not have begun in a contrived initial state to create the impetus toward increasing disorder.
When two opposing positions can both muster plausible arguments, what you have is less a debate and more a dilemma. If it were up to them, physicists would surely prefer finitude, yet nature seems to have made different plans. The universe is expanding at an accelerating rate and, if it keeps doing so, it is destined to spawn an infinity of baby universes. “It would be cozy if it were finite, but it doesn’t seem to be,” Aguirre told me. “Eternal inflation gives you an infinite universe, and something like eternal inflation is happening now and probably happened in the past. Nature is rubbing infinity in our face.”
Eternal inflation could cease to be eternal if the dark energy that drives it withered away on a timescale of billions of years. But if dark energy were so unstable, Aguirre has argued, we should see signs of its decay somewhere out there. All indications are that time will never end, which means that space probably doesn’t, either.
The brain-melting Boltzmann-brain paradox is one reason that infinite space and infinite time go together. If time were infinite yet space finite, the contents of the universe would cycle through their possible configurations over and over and over again. Molecules would occasionally converge to produce a conscious mind that lasted for a split second, but was under the misimpression it was the product of billions of years of cosmic evolution. Indeed, in the vastness of eternity, such flashes of deluded awareness would vastly outnumber brains that had formed the old-fashioned way, and we’d have to conclude that our observations are implanted memories, like fossils that young-Earth creationists think God planted in rock strata to fool us. It’s a paradox because an empirical science would lead us to the conclusion that empirical science is a sham.
This sort of argument is what Aguirre had in mind as a genuine distinction between a truly infinite universe and a merely ginormous one. “If it’s finite, no matter how big you make it, it still eventually runs into the paradox,” he said. Chance fluctuations that are inevitable in finite space are vanishingly unlikely in infinite space. An infinite universe is ever-changing, never doing the same thing twice, as Sean Carroll eloquently described in his prize-winning essay for the first FQXi essay contest.
The arrow-of-time argument that Guth has been developing also has the potential of distinguishing infinite from finite. The basic idea goes back to a provocative paper a decade ago by Carroll and Jennifer Chen (who has since left physics research to work on energy regulation). Whereas their scenario involved an accelerating universe, Guth gave a supersimple example involving a gas in an infinite void. At some moment you can take as t=0, the gas occupies some minimum volume. From then on, the gas will expand without limit. If the void is finite, the gas will eventually cycle back to its starting point. Time has a clear forward progression only if the void is truly infinite.
The overall history of Guth’s minimalist universe is fully time-symmetric, as the laws of physics demand. Prior to t=0, the gas was also expanding without limit, albeit backwards in time, and again time has a clear forward progression, the reverse of the arrow on the other side. Only around t=0 does the arrow become ambiguous. If any mortal beings are alive for the crossover, they’ll observe curious reversals of fortune such as those that Ken Wharton, who writes science fiction when not doing physics, once imagined in a poignant short story.
Guth’s scenario is classical, but similar intimations of infinity arise in quantum physics. Yasunori Nomura has argued that an infinite range of possible states (that is, an infinite Hilbert space) would make the process of quantum decoherence irreversible, explaining the arrow of time in quantum measurement.
In a funny way, then, the arrow of time we observe in daily life may reflect the infinity of space, and human mortality may hinge on the immortality of the universe. But still. Infinity? Can it be a real thing rather than simply our idealization?
Aguirre, for all his advocacy of infinity, is unconvinced by Guth’s and Carroll and Chen’s arrow-of-time arguments. “They’re brushing certain things under the rug,” he said. For instance, they take for granted that, if the maximum possible entropy is infinite, it doesn’t matter how the universe began. Any possible initial state has finite entropy, so you get the arrow of time for free. But you can’t take anything for granted when it comes to infinity. Guth and the others implicitly rule out initial states with infinite entropy. Is that really justified? Such a state is hard to imagine, but that doesn’t mean it can’t exist, Aguirre said. His musings remind me of one of the strangest concepts in mathematics: the axiom of choice. This is a rule for selecting objects from an infinite collection even when all standard rules fail. The weird thing is that, although mathematicians know that such a rule exists, they don’t know what the rule is. Worse, they know they’ll never know what it is. A state of infinite entropy may likewise exist even if it is impossible to specify.
Other speakers in Puerto Rico proposed ways to evade the paradoxes that imply infinity. Carroll himself argued that Boltzmann brains go away when you take care to distinguish quantum from thermal fluctuations. Andy Albrecht contended that the measure problem evaporates when you think of all probabilities as inherently quantum. Even a tossed coin, he said, ultimately lands on one side or the other because of quantum indeterminism. If so, probabilities aren’t defined in terms of repeated trials, and the inability to count elements of an infinite set is a red herring.
At this rate, physicists may not have to wonder about infinity. Their discussions may go on long enough to prove the point one way or the other.
More audio from the FQXi meeting in Vieques, this time from cosmologist Max Tegmark. As Ian has already blogged, Max has been pondering what qualities conscious matter would have that differentiates it from non-conscious matter.
In his recent paper, available on the arXiv (arxiv.org/abs/1401.1219), Max identified five properties: information, integration, independence, dynamics and utility principles. The first two properties had already been suggested by neuroscientist Giulio Tononi. In Max's talk, which you can download here, he discusses whether we can start with two barebones mathematical objects, the Hamiltonian and density matrix, and use them to understand why we perceive ourselves as living in a 3D space. It's an ambitious hope, but you can listen to see how far he gets with it.
Time From a Timeless World By GEORGE MUSSER
Theoretical physicists commonly say their biggest challenge is to unite general relativity with quantum theory. But at this month’s FQXi conference in Puerto Rico, Carlo Rovelli said they have an even bigger challenge: to unite general relativity...
Interesting Ways to Die (and More) By IAN DURHAM
[picture]The final full day of the conference in Vieques began with a session on quantum gravity. (Some slides and videos from the meeting are now up: here.) Seth Lloyd led off the longer talks by essentially summarizing what we know about in-falling...
The History of Astronomy with Julian Barbour and... By ZEEYA MERALI
This is a quick post to alert you to a new video project by FQXi members Julian Barbour and Flavio Mercati. Backed by FQXi, they are creating a series of films, in English and Italian, on the history of astronomy. Here is part one, in Italian, but...
String theory vs Loop Quantum Gravity: Bousso and... By ZEEYA MERALI
Depending on how familiar you are with FQXi, you may know that we like to cause a little mischief at our conferences. Knowing that we had famed string theorist Raphael Bousso and Carlo Rovelli, one of the originators of loop quantum gravity, in the...
Fluctuations, Schmucuations By GEORGE MUSSER
PUERTO RICO—One of my favorite talks at the recently concluded FQXi conference was Sean Carroll's takedown of the concept of quantum fluctuations in inflationary cosmology. Regardless of the fate of his overall argument—which my fellow FQXI...
Consciousness, Free Will--and Asking Siri Out on a... By IAN DURHAM
As I mentioned in my previous post, they keep us rather busy at this conference so I did not have the opportunity to put additional thoughts down on "paper" until just now as I sit in the Philly airport for the next few hours (it’s January and...
FQXi'ers Debate the Deep Questions of Free Will By GEORGE MUSSER
VIEQUES, PUERTO RICO—Some years ago, while visiting South Africa, I was astounded by the number of languages my friends spoke. It’s not just they knew Xhosa, Zulu, SeSotho, English, and Afrikaans, but that they moved freely among languages to...
Sean Carroll Fluctuates on Boltzmann Brains and... By ZEEYA MERALI
[picture]More audio from the FQXi meeting is now in. Those of you that read Ian Durham's summary of Day 2 will already know that cosmologist Sean Carroll, at Caltech, gave (in his own words) an "inflammatory" talk (with the deliberately boring title...
Cosmologists are from Mars, Quantum Folks are from... By IAN DURHAM
[picture]We’re certainly kept rather busy at these FQXi conferences so I haven’t had a chance to sit down and write another blog post any sooner. At any rate, I’ll try to summarize a bit about yesterday’s talks and try to express any...
Defining Information By ZEEYA MERALI
[picture]Ok, so I'm a little out of synch in terms of blogging, and technically this should have been the first post about FQXi's Physics of Information meeting, but hey, causality's indefinite at the fundamental level, right?