Mercenary in the String Wars
My sojourn in Northern California is now at an end; on Sunday I flew to my parents’ place near Philadelphia for Hanukhrismanewyears. But not before going to Stanford to give a talk to their string theory group about “Computational Complexity and the Anthropic Principle.” Here are the notes from that talk; you can think of them as a Quantum Computing Since Democritus Special Bonus Lecture.
(The best part of the talk — the lengthy arguments with Lenny Susskind, Andrei Linde, and the other stringers and cosmologists, in which I repeatedly used humor to mask my utter lack of understanding — is sadly lost to eternity. Fortunately, I’m sure that new such arguments will erupt in the comments section.)
In preparation for meeting Susskind and the other Stanford stringers, I made sure to brush up on both sides of the String Wars. On the anti-string side, I read Peter Woit’s Not Even Wrong and Lee Smolin’s The Trouble With Physics. On the pro-string side, I read Susskind’s The Cosmic Landscape and also spent hours talking with Greg Kuperberg, who tried to convince me that critics of string theory are as “intellectually non-serious” as quantum computing skeptics or Ralph Nader voters. I heartily recommend all three of the books.
So, what did I learn at Stanford? Among other things, that when you talk to string theorists in person, they’re much more open-minded and reasonable than you’d expect! Of course, when your de facto spokesman is the self-parodying Luboš Motl — who often manages to excoriate feminists, climatologists, and loop quantum gravity theorists in the very same sentence — it’s hard not to seem reasonable by comparison. But I’m not even talking about him.
(Conflict-of-interest warning: I’m painfully well aware that, so long as Luboš is around, I can only ever be the second-funniest physics blogger — even if the world champion in this field isn’t trying to be funny.)
In general, I’ve found that tolerance for alternative ideas, willingness to engage with counterarguments, rejection of appeals to authority, and so on are all greater when talking to string theorists in person than when attending their talks or reading their books and articles. Maybe that’s to be expected — to some extent it’s true of every field! But with string theorists, the magnitude of the difference always astonishes me.
Alright, let me get more concrete. One of the few nontrivial points of agreement between string theory and loop quantum gravity seems to be that, in any bounded region of spacetime, the number of bits of information is finite: at most ~1069 bits per square meter of surface area, or (equivalently) at most ~1 bit per Planck area. In loop quantum gravity, this is basically because one bit of information is “stored” in each Planck area. In string theory, it’s much more subtle than that: the bits of information can’t be put into any sort of one-to-one correspondence with the Planck areas on the horizon, but they both add up to the same number. (Ignoring a factor of 4, which being a complexity theorist, I don’t care about.)
Now, much of my conversation with Susskind and fellow string theorist Steve Shenker focused on the following question: isn’t it a bizarre coincidence that the Planck areas and the bits of information should both add up to the same number, if there’s no “dual” description of string theory in which each bit (or rather qubit) is stored in a Planck area? Susskind agreed with me that such a “local” description of string theory (local on the boundary, not in the bulk) would be desirable — and that, if there isn’t such a description, then that by itself is a fundamental fact worthy of more attention. I’d expected Susskind and Shenker to brush aside my question as idle pontificating; instead, they seemed to want to reinvent string theory that very afternoon so that my question would have an answer!
When it became clear that no such reinvention of the theory was forthcoming (at least that afternoon), I suggested the following. We’ve got this one proposal, string theory, which has had some spectacular technical successes (like “explaining” the Bekenstein-Hawking entropy), but which, setting aside its other well-known problems, offers no “local” description of spacetime in terms of qubits and quantum circuits at the Planck scale. Then we’ve got this other proposal, loop quantum gravity, which has had fewer successes, but which does attempt such a local description at the Planck scale. So, if we agree that such a local description is our eventual goal, then shouldn’t an outsider guess that string theory and loop quantum gravity are probably just different footprints of the same beast — much like the different string theories themselves were found to be different limiting cases of an as-yet-unknown M-theory?
Susskind agreed that such a convergence — between the “top-down” picture of string theory, which grew out of conventional high-energy physics, and a “bottom-up” picture in terms of qubits at the Planck scale — was possible or even likely. He stressed that his opposition was not to the idea of describing spacetime in terms of local interactions of qubits, but rather to the specific technical program of loop quantum gravity, and to the exaggerated claims often made on that program’s behalf. When I reminded him that other people complain about exaggerated claims made on string theory’s behalf, he replied that the two cases were not even remotely comparable.
All in all, it was an extremely productive and enjoyable visit — one in which the conversation topics ranged over (among other things) the explanatory role of the Anthropic Principle, the possibility that the entire universe arose as a quantum fluctuation, the prospects for an efficient quantum algorithm for Graph Isomorphism, the relation between thermodynamics and quantum error-correction, and whether or not Gerard ‘t Hooft actually disbelieves quantum mechanics. Susskind told me, half-jokingly, that the Stanford string theory group was the world’s hotbed of anti-Landscape sentiment, and the arguments that I saw and participated in on my visit gave me no reason to doubt him.
So what are we to make of the fact that, on the one hand, the string theorists are such swell folks in person, and on the other hand, even the most cursory glance at their writings will reveal that the charge of triumphalist arrogance is far from undeserved? Well, to the anti-stringers, the obvious interpretation will be that the string theorists don’t really believe their own pablum: that they say one thing in public and a completely different thing in private. To the pro-stringers, the obvious interpretation will be that, beneath the façade we all erect around ourselves, the string theorists are just scientists like anyone else: grasping at the truth, struggling to learn more, convinced that string theory is the best idea we have but ready to ditch it if something better comes along. As usual, it all depends on where you’re coming from.
Alas, as tidy as this resolution sounds, it doesn’t help me pick sides in the String Wars currently raging through the blogosphere. But then again, why do I need to pick sides? I like hanging out with the loop quantum gravity people at Perimeter Institute. I like the fact that Lee Smolin’s publisher sent me a free review copy of The Trouble With Physics. I like the recent paper by Denef and Douglas on computational complexity and the string Landscape. And I like getting an all-expenses-paid trip to Stanford to have a freewheeling, day-long intellectual conversation with the string theorists there.
I have therefore reached a decision. From this day forward, my allegiances in the String Wars will be open for sale to the highest bidder. Like a cynical arms merchant, I will offer my computational-complexity and humor services to both sides, and publicly espouse the views of whichever side seems more interested in buying them at the moment. Fly me to an exotic enough location, put me up in a swank enough hotel, and the number of spacetime dimensions can be anything you want it to be: 4, 10, 11, or even 172.9+3πi. Is it more important for a quantum gravity theory to connect to the Standard Model, or to build in background-independence from the outset? Can one use the Anthropic Principle to make falsifiable predictions? How much is riding on whether or not the LHC finds supersymmetry? I might have opinions on these topics, but they’re nothing that a cushy job offer or a suitcase full of “reimbursements” couldn’t change.
Someday, perhaps, a dramatic new experimental finding or theoretical breakthrough will change the situation vis-à-vis string theory and its competitors. Until then, I shall answer to no quantum-gravity research program, but rather seek to profit from them all.
Update (12/23): The indefatigable Luboš Motl has put up a new jeremiad against me. Taking my ‘For Sale’ announcement completely seriously, Luboš writes:
It is absolutely impossible for me to hide how intensely I despise people like Scott Aaronson … He’s the ultimate example of a complete moral breakdown of a scientist. It is astonishing that the situation became so bad that the people are not only corrupt and dishonest but they proudly announce this fact on their blogs…
In fact, I have learned that the situation is so bad that when I simply state that Aaronson’s attitude is flagrantly incompatible with the ethical standards of a scholar as they have been understood for centuries, there could be some parts of the official establishment that would support him against me. There doesn’t seem to be a single blog article besides mine that denounces Aaronson’s attitude…
The difference between [the] two of us is like the difference between a superman from the action movies who fights for the universal justice on one side and the most dirty corrupt villain on the other side. It’s like the Heaven and the Hell, freedom and feminism, careful evaluation of the climate and the alarmist hysteria, or string theory and loop quantum gravity…
I can’t tell you how proud I am to have become “the most dirty corrupt villain” in Luboš’s cosmology, and no longer just an anonymous bystander. Thanks so much, Luboš, and Merry Christmas to you too!
Update (12/24): Man oh man, I had no idea that people would take my offer so seriously! Because of this, I now feel obligated to provide a financial disclosure statement. The Stanford string theorists did not actually pay my way to California, although they offered to — most of my expenses were covered by Umesh, my adviser at Berkeley. Stanford paid for (1) one night’s hotel stay in Palo Alto, and (2) one lunch, consisting of a small cheese pizza and an iced tea.
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Comment #1 December 21st, 2006 at 1:24 pm
So, do you not recommend conversations with Greg Kuperberg? (-:
Comment #2 December 21st, 2006 at 1:28 pm
Hmmm … Scott’s physics allegiance is open to the highest bidder? Then the University of Washington bids: one tenure track faculty position in quantum system engineering!
Seriously. This is the first tenure track position every advertised in quantum system engineering (to our knowledge), and we are looking for the strongest athlete (theory or experiment, macroscale or nanoscale).
What’s the connection to string theory? The connection is by way of model order reduction, information theory, and projection onto Kahler manifolds.
In our view, the connection between quantum system engineering and string theory conceptually resembles the connection between Green functions and field theory (see, e.g., Julian Schwinger’s biography), and the opportunities for young people are similarly great.
For details, see the University of Washington Mechanical Engineering Department web page, or follow the link here.
Comment #3 December 21st, 2006 at 1:36 pm
So, do you not recommend conversations with Greg Kuperberg? (-:
Not at all! It’s just that conversations with Greg are an acquired taste — a unique experience that you’ll need to prepare yourself for by reading several representation theory textbooks and drinking large amounts of coffee.
Comment #4 December 21st, 2006 at 3:20 pm
I know from experience that arguing with Lenny Susskind can be extraordinarily enlightening. Some time ago, after spending a day with Lenny, I wrote in a message to Paul Ginsparg: “When I listen to Lenny Susskind, I really believe that information can come out of a black hole.” Paul interpreted this as an intentional double entendre, but actually I think it was unintentional.
Comment #5 December 21st, 2006 at 3:34 pm
It seems to me that string theorists (eg. Denef/Douglas) are interested in complexity theory for the dullest possible reason. They study classical algorithms that can simulate string theory, rather than studying the capabilities of a string computer. I predict you will soon get bored with your new wife and divorce her.
Comment #6 December 21st, 2006 at 4:44 pm
rrtucci: If there were any evidence for string computers more powerful than garden-variety quantum computers, I’m not the only one who’d jump at the chance to study it. Alas, to the extent that we understand string theory at all, it seems to be just another unitary quantum theory.
Comment #7 December 21st, 2006 at 4:50 pm
[…] Scott Aaronson has adopted a sensible attitude towards the controversy over string theory, announcing in a new posting entitled Mercenary in the String Wars that his allegiances in this “War” are for sale to the highest bidder. I encourage all my extremely wealthy financial backers to take him up on this. […]
Comment #8 December 21st, 2006 at 5:07 pm
The real question about seminars at Stanford, of course, is how long did it last?
As for a local conception holography, such speculation ins’t unprecedented (for example, this paper by Banks.)
Comment #9 December 21st, 2006 at 5:14 pm
The real question about seminars at Stanford, of course, is how long did it last?
3:30-5. I didn’t want drag it out longer than that.
Comment #10 December 21st, 2006 at 5:36 pm
I bid 36 beers for your allegiance, Scott. Which side am I on? Neither! Let the bidding begin! Should we put you on eBay?
p.s. San Francisco is not “northern” California. Viva la state of Jefferson!
Comment #11 December 21st, 2006 at 5:40 pm
Scott, string theory is your rich (but full of warts) girlfriend that wants to snag you as her trophy husband. But you know deep inside that quantum computing is your Carmen, the only physics woman that turns you crazy, and makes you beg like a puppy.
Comment #12 December 21st, 2006 at 5:45 pm
rrtucci: I got it the first time.
Comment #13 December 21st, 2006 at 5:48 pm
Dave:
Should we put you on eBay?
Sure!
p.s. San Francisco is not “northern” California.
San Francisco = Northern California
LA = Southern California
Yreka = Who the hell goes there?
Comment #14 December 21st, 2006 at 6:17 pm
In coexistence with the exact state counting as in D1-D5-brane systems, there has always been the more heuristic but more versatile principle known as “string black hole correspondence” principle.
Roughly, the idea there is to consider long semiclassical strings. Their extension gives them a mass, as usual, and their wigglyness gives them an entropy. Their mean square size (when regarded as random walks in space) gives them a diameter and surface area.
If you plug in the numbers for a semiclassical string of such length that the Schwarzschild radius derived from its mass agrees with its root mean square diameter, you can interpret its entropy as a random walk in space with the corresponding black hole entropy.
You don’t get the coefficient of 4 this way, but all the rest. And for a wide variety of black holes.
And people did suggest to interpret this random walk as taking place on the “exteded horizon” of the respective blak hole.
In as far as this works, it does provide something like a picture of bit stored in Planck areas.
For more details and more links to literature, see this:
http://golem.ph.utexas.edu/string/archives/000379.html
Comment #15 December 21st, 2006 at 6:24 pm
Thanks, urs!
Comment #16 December 21st, 2006 at 8:04 pm
[…] Scott Aaronson, well-known around these parts for thinking that a priori constraints on conversations with super-intelligent aliens are more important insights into the fundamantal workings of the universe than dark energy and the holographic principle, is suffering from a bit of Stockholm syndrome. He has visited the Stanford high-energy theory group (intellectual hotbed of agressive Landscapism), given an interesting talk on Computational Complexity and the Anthropic Principle, and discovered to his bemusement that string theorists are quite open-minded and reasonable people! When faced with an interesting new idea, they are even willing to consider it! And their objections to Loop Quantum Gravity seem to be based on physics, rather than just prejudice! Who would have thought? (Also linked from Not Even Wrong.) […]
Comment #17 December 21st, 2006 at 8:47 pm
Very funny indeed!
Let me pick up one physics point I disagree with, a central one about the holographic principle you mentioned. In one approach to LQG spacetime comes from spin chains, and has naturally finitely many states (or “one unit of information”) per Planck *volume*, not per Planck surface *area*. In fact I’d say this is one of the main point of disagreement between the two approaches…
Comment #18 December 21st, 2006 at 8:51 pm
But will you be able to get published if you don’t work on M-theory, but on a rival?
Comment #19 December 21st, 2006 at 8:53 pm
BTW, let me know your charges, please. I need some good clear, well organized review articles on my site, but am too busy with programming work to do much myself.
Comment #20 December 21st, 2006 at 9:00 pm
Nigel: Writing a “well-organized review article” for your website? $2,000,000.
Comment #21 December 21st, 2006 at 9:18 pm
Thanks!
Comment #22 December 21st, 2006 at 9:21 pm
I don’t know about exagerated claims… I think the biggest problem is language. LQG says things in a way that’s not using the vocabulary of QFT one to one. (The reason being that we think we can’t since it’s conceptionally insufficient). Then somebody from particle physics reads it and thinks it’s all bull since it sounds terribly grandiose, when in fact it isn’t…. For example the finite quantization of Yang Mills coupled to Gravity, sounds great, right? However it’s not claimed to be a theory which has a limit corresponding to Yang Mills on background metrics, that’s not something we know. The claim is that we can construct quantum mechanical representations of the gauge symmetry groups.
That’s the overall theme in LQG to me: Let’s find new ways to construct quantum theories. We’ll have to find new ways to check if they correspond to reality (work in progress, too).
Comment #23 December 21st, 2006 at 9:54 pm
fh, read popular stringy books and journals, and you’ll find the exaggerated claims.
‘String theory has the remarkable property of predicting gravity.’ – Dr Edward Witten, M-theory originator, Physics Today, April 1996.
‘The critics feel passionately that they are right, and that their viewpoints have been unfairly neglected by the establishment. … They bring into the public arena technical claims that few can properly evaluate. … Responding to this kind of criticism can be very difficult. It is hard to answer unfair charges of élitism without sounding élitist to non-experts. A direct response may just add fuel to controversies.’ – Dr Edward Witten, M-theory originator, Nature, Vol 444, 16 November 2006.
Comment #24 December 21st, 2006 at 10:03 pm
Let me elaborate, this point confuses me. In my very rudimentary understanding of spin chains, there is some graph associated with any spacetime, and the volume of the spacetime scales like the number of vertices in that graph. Furthermore there is some finite Hilbert space associated with any vertex. I would think then that the dimension of the resulting Hilbert space scales like the exponential of the volume, not the area. Am I missing something?
Comment #25 December 22nd, 2006 at 3:52 am
Moshe,
It is crucial, isn’t it, how many effective degrees of freedom one attributes to this spin chain? The volume may scale like the number of vertices in the graph, but if one is concerned with the entropy associated with a volume, and ultimately with the vertices, then correlations among the states of the vertices (spins) will reduce the entropy.
It seems reasonable to assume at least one essential reason for such correlations in the context of quantum gravity; the absolute spin states are unobservable. In fact, I would think that one should require a symmetry under a global relabeling of spin states, and then consider promoting this to a local symmetry—ie, a sort of gauge symmetry, such that local relabelings of spin states are absorbed into the dynamics. Another way of putting it is that we want to the dynamics to preserve causal connections among state transitions (spin flips) or events—in effect (and more formally) to generate a causal set—and nothing more. Again, we’re talking about quantum gravity, so we presumably want our model to generate a simplicial representation of the causal structure of spacetime. The notion of a “spin flip dynamics” sketched above is consistent with this expectation.
I’m even more ignorant of spin chains than you are, but I’m supposing that, given the origins of these models in condensed matter physics (?) that this way of looking at them is not altogether familiar, and the corresponding ramifications haven’t been explored in their formal development.
[A sort of pseudo-engineering perspective on this notion is the following. Imagine space as filled by an array of discrete memory elements, plus a matrix of connections and combinatorial logic that enable information transfer into and out of this array across a bounding surface. In this viewpoint, storage density is traded off against data accessibility. If one simply maximizes storage density within the volume, one renders its information content inaccessible, since there is no space left for interconnect paths to (and across) the bounding surface. The “data transfer matrix” can be considered analogous to an assumed dynamics for elements in a spin chain. (NOTE: I found a paper in the last couple of years that develops this idea in the context of entropy bounds, but I can’t seem to locate the shortcut I saved.)]
Comment #26 December 22nd, 2006 at 4:52 am
Didn’t it look odd standing in front of people of Susskind’s caliber and writing liner algebra formulas and claiming another complexity theorem?
Comment #27 December 22nd, 2006 at 5:32 am
“Didn’t it look odd standing in front of people of Susskind’s caliber and writing liner algebra formulas and claiming another complexity theorem?”
It would have looked odd, but LS has been known to rise to such levels in moments of lucidity.
Comment #28 December 22nd, 2006 at 5:33 am
“Suppose there are two competing cosmological models. One model leads to a finite universe, the other leads to an infinite universe. Cosmologists are about to launch a space probe that will test which model is correct. But then philosophers come along and say, “Wait — you don’t have to bother! Obviously the infinite model must be the correct one — since in an infinite universe, we’d be infinitely more likely to exist in the first place!” The question I’d put to anthropicists is this: should the philosophers win the Nobel Prize that would have otherwise gone to the cosmologists? And if not, why not?”
What was the response [if any] to these remarkably sensible questions?
Comment #29 December 22nd, 2006 at 5:41 am
Didn’t it look odd standing in front of people of Susskind’s caliber and writing liner algebra formulas and claiming another complexity theorem?
I know that most string theorists don’t have a complexity theory background, but I worked hard to make the talk accessible to them and was extremely pleased with the level of understanding.
Comment #30 December 22nd, 2006 at 7:08 am
Chris, maybe that is the explanation, maybe not. I simply encountered a surprising and interesting statement (LQG assigns one bit of information for any Planck area), which if true would certainly increase my interest in the subject. Alas, I don’t see any way this can be true, based on my very small knowledge of the subject. My only recourse is to ask the person making the statement for clarifications, so far not very successfully…must be those social inadequacies we string theorists are famous for…
Comment #31 December 22nd, 2006 at 10:50 am
You are obviously taking the free market in ideas seriously Scott.
Comment #32 December 22nd, 2006 at 12:40 pm
Sorry, Moshe! I was simply hoping that someone who knew more than I did would take up your question, and at least initially, my hopes seemed to be realized…
I was just reporting the (possibly-mistaken) impression that I got from reading Baez, Penrose, Smolin, and others. For example, John Baez’s week112 talks about counting spin networks that puncture the event horizon, and that would seem to me a manifestly local description of where the entropy comes from. I don’t care if exactly one bit is assigned to each Planck area — all that matters for me is that the description should be (1) local on the event horizon and (2) discrete, with the granularity given by the Planck scale. I don’t have a strong opinion on whether LQG succeeds in giving such a description, but my understanding was that that was its goal. If Baez’s week112 doesn’t answer your question, maybe we could get him or Lee Smolin to comment directly…
Comment #33 December 22nd, 2006 at 2:50 pm
Yes, in the LQG picture, the area of a surface is induced from the spin network edges piercing that surface. That’s ultimately due to the fact that the “spin network” is really a Wilson line network for the Levi-Civita connection.
But, anyway. In those LQG descriptions of black hole entropy people consider a spacestime with boundary, where the boundary is supposed to be the black hole horizon. The black hole (bulk) itself is not considered.
Some boundary condtion is imposed on something that ensures that this boundary really does behave like a horizon.
Next one would like to count “physical states” of the theory, which induce a metric on spacetime such that that boundary has a certain surface area.
Unfortunately, that’s not possible, directly, because nobody knows how to solve Hamiltonian constraint (or how to even define it).
So the idea is to pull the following trick: one assumes/postulates, that for every possible way to draw points on the boundary and label them with irreps of the group that our connection takes values in, there is exactly one physical state of the theory (solving the Hamiltonian constraint) that induces the corresponding area on the boundary.
(There is no reason why the physical states should look like spin network states. They are bound to be continuous linear combinations of these. I am not sure how or if this is dealt with in this approach.)
Anyway, in the end in these kinds of computations what one does compute is the number of ways to draw points on the boundary and label them with, say, su(2)-irreps (i.e. half-integers), such that all these labels add up, under a certain formula, to a fixed area.
That formula contains one free parameter, the “Immirzi parameter”, so the result of this “state counting” depends on that one parameter.
Comment #34 December 22nd, 2006 at 3:11 pm
What I wrote above was purely from memory. Hopefully somebody who is actually an expert on this will correct whatever needs to be corrected about that account.
Comment #35 December 22nd, 2006 at 3:56 pm
Thanks again, urs!
Comment #36 December 22nd, 2006 at 5:33 pm
Scott, have you ever taken Putnam? I think I remember you from there.
Comment #37 December 22nd, 2006 at 5:43 pm
Thanks very much Scott (and Urs). The point that confused me was the part about “any bounded region of spacetime” which is a more general notion of holography than the one referring to black hole horizons…if I have some granular structure on the Planck scale, seems to me the entropy would scale like the volume (yeah, let’s forget factors of order one), for me and others this is one of the issues telling you it is misguided to quantize directly a local (in the bulk) description of gravity. Of course, you are entitled to count only the different ways the spin networks cross the boundary of a region, but that strikes me as answer analysis…
Comment #38 December 22nd, 2006 at 5:55 pm
Scott, have you ever taken Putnam?
Yeah, way back when I was 17. I only got three of the questions.
Comment #39 December 22nd, 2006 at 7:13 pm
also spent hours talking with Greg Kuperberg, who tried to convince me that critics of string theory are as “intellectually non-serious” as quantum computing skeptics or Ralph Nader voters. I heartily recommend all three of the books.
Although I am happy to be mentioned in this blog, including in this context, this comment is an unfortunate exaggeration of what I really think. I certainly don’t think that all critics of string theory are intellectually non-serious all the time. I also don’t think that all criticisms of string theory are inherently unserious. String theory is a profoundly unfinished business; there is undoubtedly plenty to criticize; and who knows pro- or anti-string theory arguments may arise as the whole theory of quantum gravity is filled in. And, I should emphasize, all of my impressions of string theory, or anything else connected with quantum gravity, are outside impressions. I don’t really know any of these things first hand.
What I did say — again, as an outside impression — is that much of loop quantum gravity specifically strikes me as a Naderite quest to compete with string theory. Again, Nader voters aren’t necessarily unserious either, either in their other thinking or even in their motivations to vote for Nader. My point is that there was a fundamental structural flaw in Nader’s quest to compete with the Democratic Party; there seems to be a similar structural problem in the loop quantum gravity program.
In another conversation I talked about Gerard t’Hooft’s views. Now, t’Hooft is a smart guy who has contributed to string theory, contributed massively to quantum field theory, and so forth. He cannot be called unserious. Nonetheless, as I understand it, t’Hooft has been saying that he would like humanity to replace not only string theory, but even quantum probability by something better. My comment is that his suggestion is not very useful without some plausible ideas for the supposed replacements for these theories.
Comment #40 December 22nd, 2006 at 7:25 pm
Dear Moshe,
Let me give a try at giving a clear answer to your question. To start with it depends on which state space you work in.
0) for kinematical states which have a non-separable basis there are an infinite number of states for fixed volume so no hint of holography.
1) for solutions to the gauss law and spatial diffeo constraints but not the Hamiltonian constraint-which live in a separable Hilbert space- what you say is correct so far as combinatorial information: the number of states given by the combinatorial information in the graph grows at least like the exponential of the volume (I can make a hand-waving argument for exponential in the volume but I don’t know a rigorous result on this.)
2) When you take into account that the same graph can be embedded in an infinite number of ways in a three manifold you get infinitely more diffeo classes of states in each fixed volume. Markopoulou (hep-th/0604120) shows that many of these are associated with new conserved quantum numbers, at least under some simple choices for dynamics, and she suggests the interpretation that these are states with emergent matter degrees of freedom. This is just beginning to be explored, (see hep-th/0603022) but without dynamics imposed I see no reason anything like a Bekenstein bound should be satisfied.
3) In the presence of boundaries with certain boundary conditions imposed the Hilbert spaces at either kinematical or spatial diffeo invariant is a sum over products of bulk and boundary states, where the sum is over the eigenvalues of the operator that measures the area of the boundary. In each eigen-sector one gets a fixed relationship between the log of the dimension of the boundary Hilbert space and the area. These boundary conditions include all known black hole and cosmological horizons.
4) One can conjecture that when the Hamiltonian constraint is also imposed in the case where there are these fixed boundary conditions. one gets a reduction of the dimension of the physical Hilbert space to finite dimensions proportional to the exponential of the area. Given a certain ordering of the Hamiltonian constraint there is a sector of solutions where this is true: see gr-qc/9505028. It is not known whether this is a complete set of solutions.
5) In the case of positive cosmological constant one gets a purely quantum mechanical derivation of the N bound conjectured by Fishler and Banks, see hep-th/0209079.
6) It is still open what holography should mean in a non-perturbative background independent context. Since the volume of the bulk is not expected to be sharply defined, because it doesn’t commute with the Hamiltonian or Hamiltonian constraint, there is no clear notion of number of degrees of freedom “in the bulk”. In its place, some of us suggest a weaker notion of holography connected with information observable on a surface. For some explorations in this direction see gr-qc/9505028, hep-th/0009018, hep-th/0003056, astro-ph/0611695.
Let me know if this helps,
Thanks,
Lee
Comment #41 December 22nd, 2006 at 8:55 pm
Thanks Lee, this is really helpful, and very illuminating. I take it that my first impression is correct: there is description which is local on the boundary only in the same sense that the description is also local in the bulk, no more and no less (beyond the conjecture you mention in 4)
In any event, thanks for taking the time to sit in front of a computer in the holiday season, I’ll take a look at some of the references you mention.
Comment #42 December 22nd, 2006 at 9:49 pm
I’ve been thinking more about this issue of local on the boundary versus local on the bulk. For simplicity, let’s only consider black holes, and not other regions. Then the obvious question is, what’s so special about the event horizon, that the qubits should be stored there and not somewhere else?
It would seem like whatever the answer is, it has to have something to do with the expansion and contraction of light rays. Baez mentions this non-expansion of light rays as the criterion physicists use even to define the event horizon in a purely local way. (Are there any other local criteria, or is this the only one?) In a related context, I remember that Bousso used the expansion and contraction of light rays to define “inside” and “outside” in a local way, for the purpose of stating his covariant entropy bound.
So let me put this question to the experts: does either string theory or LQG suggest what the connection should be between entropy and the non-expansion of light rays? In other words, why does nature like to stick her qubits where the light don’t expand?
I apologize if my question is either obvious or misguided. (Then again, if my goal were to avoid making a fool of myself, it’s a bit late to get started… 🙂 )
Comment #43 December 22nd, 2006 at 10:22 pm
Scott, that is an interesting question, and I think there is probably not a unique answer, just different hunches that people may have. For myself (within string theory that I know more about), I think the only time holographic “screens” were shown to make sense is if they are located in regions where gravity is effectively turned off (collectively known as infinity, or asymptotia). Only in such cases one is entitled to have an ordinary quantum mechanical description of the boundary theory (with well-defined Hamiltonian, say) and one can even define what they mean by entropy, locality, etc. In any other case, including black hole horizons, the boundary metric still fluctuates and the boundary theory itself ought to be gravitational (if this notion even makes sense) and thus equally mysterious as the bulk theory. If it exists at all it is probably not an ordinary quantum mechanical system we know how to talk about.
Not really answering your question, just trying to compete with your stated goal…
Comment #44 December 22nd, 2006 at 11:37 pm
Another problem I have with the mental picture of locality on the horizon is the following. In conventional physics when calculating entropy (as function of temperature, in the canonical ensemble), one has a fixed theory, with fixed degrees of freedom and Hamiltonian, and one just put that theory in different thermal density matrices, corresponding to different temperatures. In the picture of bits living on the horizon of the black hole, I don’t see a direct sense in which the entropy of different black holes is just a different way of averaging over the eigenstates of the same underlying Hamiltonian, especially if there is some compelling reason to throw away the bulk contributions, so the theory is truly a boundary theory- different boundary for each black hole.
Alright, mission accomplished I believe…
Comment #45 December 22nd, 2006 at 11:53 pm
Thanks so much, Moshe!
Your unease seems related to the well-known tension between two different concepts of entropy: entropy as a physical parameter related to temperature and pressure, and entropy as the number of bits. I, of course, always think in terms of the second concept, but I agree with you that a fundamental theory ought to connect somehow to the first one.
Comment #46 December 23rd, 2006 at 1:14 am
Ah, we are now getting to my unfamiliar territory (or maybe I am just forgetting): when I hear stories about bits at the horizon I imagine a bunch of Q-bits (say spins) interacting via some local Hamiltonian. I am not sure in what sense one can define an entropy before specifying that Hamiltonian (in the canonical ensemble it appears in the definition of the density matrix). Is there any kind of entropy one can define without having to specify the Hamiltonian?
Comment #47 December 23rd, 2006 at 2:05 am
Alright, disregard that, I think I get it.
Comment #48 December 23rd, 2006 at 3:23 am
Yeah, you can just take the log of the Hilbert space dimension. It’s an interesting question how you know what the Hilbert space dimension is — but at least to my mind, the state space is certainly conceptually prior to any specific choice of Hamiltonian.
Comment #49 December 23rd, 2006 at 3:55 am
Although string theory is mostly over my head, I have studied general relativity. A fundamental point about event horizons is that they are observer-dependent. By definition, an event horizon for an observer A is any “threshold of no return” for an object B, i.e., any surface in the spacetime manifold beyond which B cannot return to A. The conventional event horizon of a black hole is relative to an observer at infinity. To give another example, if the cosmological constant is negative, then behind all of the stars and galaxies in the sky, there is a cosmological event horizon.
Even without any quantum mechanics, one funny observer-dependent features of event horizons is that the observer A can never see the object B fall through the horizon. Instead, the image of B freezes into the surface of the horizon and fades into red.
So as I understand it, Hawking radiation, and even more Bekenstein-Hawking radiation, is a statement about information rather than about observer-indendepent reality. The statement is that the universe available to any given observer is in principle in a pure state. A object that wants to fall past an event horizon is typically deeply entangled with the universe outside. the model is that the event horizon is a barrier that eventually pushes the entanglement back to the observer. (If not directly back at the observer, back to accessible space.)
Or rather, when I say “the statement”, I am referring to one interpretation of Hawking radiation. Apart from in string theory, there is a debate as to whether Hawking radiation is an encoding of the quantum entanglement that tried to fall past the event horizon, or if event horizons actually create true mixed states. I believe, although I don’t really know, that string theory settles the debate in favor of the former answer in some cases.
Comment #50 December 23rd, 2006 at 5:16 am
Mercenary in the String Wars…
So, what did I learn at Stanford? Among other things, that when you talk to string theorists in person, they’re much more open-minded and reasonable than you’d expect! Of course, when your de facto spokesman is the self-parodying Luboš Motl — wh…
Comment #51 December 23rd, 2006 at 2:18 pm
I don’t understand why people are so sure that QG cannot be local. 25 out of 26 consistent quantum gravities in 2d are local rather than holographic. Extrapolating from 4% to 100% seems to be a rather large step.
I am of course thinking about free string theory in D dimensions, which can be regarded as QG in 2d coupled to D scalar fields. The theory is consistent (unitary) for all D less than or equal to 26, but holographic in the sense of no correlators depending on separation only for D=26.
Comment #52 December 23rd, 2006 at 4:53 pm
The question, of course, is what villian are you? Lex Luther? Dr. Evil? Dick Cheney? And what superhero is Lubos?
Comment #53 December 23rd, 2006 at 5:25 pm
A few points: There is no local description of the event horizon; it is a global object. As I remember it (and I’m on vacation now, so I have no reference books to check), there is something called the apparent horizon which has to do with the nonexpansion of nearby light rays, and (possibly assuming an energy condition) one can prove that any apparent horizon is inside of an event horizon. This distinction is important in dynamical situations where, for example, as an object falls into a black hole, the event horizon reaches out anticipating that the object will fall in. (Checking wikipedia, it appears there might be some terminological tension here — what I’m referring to is the Hawking and Ellis definition which Wikipedia calls the absolute horizon. This is an observer-independent notion.)
In addition to the different notions of entropy out there (entanglement vs. coarse graining), there’s also the different ensembles (microcanonical, canonical, grand canonical). I don’t think it’s pointed out enough that the entropies in the various ensembles are different quantities. There are certainly 1/#particles corrections, and one may also have to assume the existence of the thermodynamic limit — I don’t remember. If true, the last point is particularly important as the thermodynamic limit does not exist in the presence of gravity.
Comment #54 December 23rd, 2006 at 10:06 pm
Dear Moshe,
I am not sure if I agree with your “I take it that my first impression is correct: there is description which is local on the boundary only in the same sense that the description is also local in the bulk…”
In both senses-bulk and boundary- in LQG we are dealing with states in a manifold mod its diffeos. There is still a notion of locality but it is different than in ordinary QFT where there is a background metric and no diffeo invariance so that points are meaningful. When the states live in a manifold mod diffeos there are two distinct notions of locality-in each graph eigenstate there is a notion of microlocality. But in a generic superposition there is no single notion as a state superposes basis elements with different notions of microlocality. If the state is semiclassical there is a macroscopic notion of locality defined by the metric it is semiclassical to, but this may differ from microlocality in any of the basis states in the superposition that makes the semiclassical state. See the paper I mentioned by Markopoulou for more on this and its implications.
Dear Scott,
Indeed, expansion and contraction of light rays are key. On a spacelike two surface in a spacetime there area local quantities called the expansion which measure the expansion and contraction of light rays moving off the two surface. (They are defined by the Raychaudhuri equations.) So the vanishing of the expansion of outgoing null rays is a local condition. Asking that it vanish is part of what are called the “isolated horizon boundary conditions”. These are part of the boundary conditions that we fix in the LQG notion of an horizon boundary. It is fascinating that these boundary conditions induce a Chern-simons theory on the boundary-this is because of the tight connection between topological field theory and GR that is essential for understanding these issues.
The entropy we define in LQG is related to the log of the dimension of the boundary state space, as I described above.
Dear Aaron,
These isolated horizon boundary conditions just mentioned are loca;, you can test if they are satisfied or not on any two surface in a spacetime. They then are distinct from the event horizon which is a global notion as you and Greg say.
For more details on allthis search the literature under “isolated horizon”, in papers by Ashtekar, Baez, Corichi, Krasnov and other authors.
Dear Greg,
I’m not sure what you mean by Hawking radiation being “is a statement about information rather than about observer-indendepent reality”. Semiclassical GR, which is the context where Hawking radiation was discovered and is well understood, is well defined mathematically. The Hawking radiation is measured in terms of a flux of energy and momentum from the black hole horizon to infinity, measured by the expectation value of the operator that measures the energy-momentum tensor, in the matter QFT, in the incoming vacuum state. One can also compute the outgoing state at outgoing null infinity and it is a thermal state. These are invariant, observer independent, physical observables in the theory. There are some interesting issues about what local observers near the horizon see when they measure components of the energy-momentum tensor, this was worked out a long time ago and is completely consistent with the hawking radiation being real, physical and observer independent. This is by now textbook material, I suggest you consult the textbooks (by Birrill and Davies or Wald) for more information.
Thanks,
Lee
Comment #55 December 23rd, 2006 at 10:14 pm
Lee,
Indeed, it seems to me that there is simply no notion of locality for a generic state, either bulk or boundary locality.
Lee, Scott: one can define the log of the dimension of the Hilbert space to be “entropy”, but is that the kind of entropy that appears in the first law of thermodynamics? presumably that is the fact about black holes we are trying to explain.
Comment #56 December 23rd, 2006 at 10:32 pm
Thanks so much, Lee!
Alright, I have another embarrassingly simple question for the experts. Suppose you throw a computer into a black hole; then (from the perspective of someone inside the black hole) the computer performs a long computation; then you wait for the black hole to evaporate and collect all the bits encoded in the Hawking radiation. Question: Will you merely be able to recover the original state of the computer, or will you also be able to recover the output of the computation?
(I realize that if you know the initial state, then you can presumably compute the output yourself — but maybe you don’t want to do so! The issue really boils down to one of computational complexity.)
Let me put the question another way. As Greg reminded us, someone standing outside a black hole will never actually see anything fall in. So there are (at least) two strange facts about black holes: (1) that the bits eventually get out, and (2) that you never see them go in in the first place. An obvious question is whether these two facts are related — e.g., whether there’s a “complementary” description in which objects never fall into the black hole in the first place, but just hover around the event horizon before coming out as Hawking radiation. But if there is such a description, then we would only expect the Hawking radiation to encode the information that was thrown in — not whatever information was computed inside the event horizon.
Comment #57 December 23rd, 2006 at 10:35 pm
Man Lubos must not be taking his meds. Guy sounds like he could use a good punch in the head.
I guess string theorists get touchy when someone points out that what they’re doing isn’t physics.
Scott: It might be entertaining if you could keep baiting him; we could have a pool on the exact date when his head explodes.
Comment #58 December 23rd, 2006 at 10:59 pm
These are invariant, observer independent, physical observables in the theory.
Hawking radiation cannot truly be observer-independent, because event horizons themselves are observer-dependent. It is true that the event horizon and Hawking radiation of a black hole, specifically, is what you and Aaron call “global objects”, and therefore do not appear to be observer-dependent. But this is a subtly misleading version of the story. In fact, a black hole is just one example of an event horizon. Every event horizon emits Hawking radiation. The event horizon of a black hole is really defined relative to a reference observer at infinity. But other kinds of event horizons have no reference infinity and are instead only defined relative to a non-canonical observer.
The best example is the backdrop event horizon of a section of de Sitter space, as you see in a universe with a negative cosmological constant. Current thinking is that the true universe does have a negative cosmological constant, and that the CMB will eventually fade into the Hawking radiation of the backdrop event horizon. This event horizon is fundamentally observer-dependent. Any two observers that drift apart (say they are parked in different galaxies) will see each other recede into each other’s backdrop event horizons. They will afterwards live in disjoint spheres of reality.
On the other hand, it is true that Hawking radiation is “real”, even though it is observer-dependent. That is, it is as real as anything else. We are used to the fact that attributes of particles, such as non-rest-mass, are observer-dependent. But even in quantum field theory in non-linear coordinates, particle number is observer-dependent, because the coordinate change alters the dividing line between virtual and non-virtual particles. (This is indeed related to the existence of Hawking radiation!) If particle number is observer-dependent, then everything is.
Comment #59 December 23rd, 2006 at 11:11 pm
Suppose you throw a computer into a black hole; then (from the perspective of someone inside the black hole) the computer performs a long computation; then you wait for the black hole to evaporate and collect all the bits encoded in the Hawking radiation. Question: Will you merely be able to recover the original state of the computer, or will you also be able to recover the output of the computation?
First of all, it isn’t established that the Hawking radiation is the same qubits that fell in, or entirely new, mixed qubits. The former is a likely consequence of string theory — maybe even confirmed for some kinds of black holes, although I am not sure — while the latter always seemed inelegant to me. As Lee said, the model in which Hawking radiation is rigorously established is semiclassical GR+QFT. This model doesn’t identify the entanglement of the qubits that come out.
But if you do believe that the laws of physics never create a mixed state, then there is still no reason to believe that you would enjoy the results of a computation performed by a computer that fell into the black hole. In order to make the question fair, let’s assume that the computer is fast enough that it finishes its work before, in its own frame, it is crunched by the singularity of the black hole. Even so, you should remember that the observer at infinity sees the computer freeze as it approaches the black hole. The computer’s displayed time at this frozen limit is its own elapsed time when it crosses the threshold of no return to the outside.
There is a feeling that quantum gravity in general and string theory in particular will impose limits on physical computation, in constract to the likely computational boost from quantum probability.
Comment #60 December 24th, 2006 at 1:52 am
Greg: Yes, I was of course assuming that the information does come out — even Hawking agrees about that now!
I realize that an observer at infinity would see the computer freeze as it approaches the black hole — that was exactly the genesis of my question. When the computer finally evaporates as Hawking radiation, will it look like it was frozen all along, or will it “fast forward” to what an observer who was comoving with the computer would have seen?
Comment #61 December 24th, 2006 at 2:27 am
Yes, I was of course assuming that the information does come out — even Hawking agrees about that now!
But you should understand that this is just an opinion, and not a theorem or an established precept.
When the computer finally evaporates as Hawking radiation, will it look like it was frozen all along, or will it “fast forward” to what an observer who was comoving with the computer would have seen?
It won’t look like either — Hawking radiation looks like thermal radiation.
I had taken your question to be whether the Hawking radiation somehow has the same computational value as the computer’s future. It is hard to make sense of this possibility, for several reasons. First, since the entanglement in Hawking radiation is hopelessly scrambled by the unitary operator of the black hole, it is hard to say how it has any computational value. Second, if the event horizon is not that of a black hole, there doesn’t have to be a singularity behind it. If the computer crosses a cosmological event horizon and goes along its merry way for billions of years in a disjoint region of the universe, which computational state is supposed to be reported back to you?
Third, according to a paper that I heard about, there is a sense in which the frozen images of all objects that fall past an event horizon segue into the Hawking radiation. Maybe this point is not an intuitive objection, but rather a direct refutation? It would be nice to see the paper or papers in question that address this point.
Comment #62 December 24th, 2006 at 2:56 am
Greg, deSitter space has a positive cosmological constant, as observations suggest our universe does also. And that information comes out of a black hole is generally accepted to be quite well-established now. It is not rigorously proven, but it’s understood fairly well at a “physics level of rigor.” Also, what does “If particle number is observer-dependent, then everything is” even mean?
Comment #63 December 24th, 2006 at 3:26 am
Dear Moshe,
Microlocality is well defined in LQG, it is just dependent on the state. It is meaningful and significant as the moves which define the evolution of spin networks are microlocal.
Moreover we are finding that there are interesting consequences of Markopoulou’s observation that macro and micro locality are likely mismatched in generic semiclassicial states, see some recent and forthcoming papers about this.
Dear Scott,
There is an alternative possibility for where the information goes after it crosses into an horizon that seems conservative and, to me, sensible. Suppose that, as there is increasing theoretical evidence for from calculations, the spacelike singularity of the black hole is eliminated. Then the evolution continues to the future of where the singularity would have been. Let us call this region the “new region”. There is so far no difficulty because one has a pure state overall with the state in the new region entangled with the state external to the horizon.
What happens then? It depends on whether the apparent horizon evaporates and disappears before the end of the universe or not. Ie is the new region causally connected to future null infinity (scri+) or not. If it is then it means that the information must leak out after the black hole evaporates to the point that there is no longer an apparent horizon. (The definition of the apparent horizon is the boundary of the region of trapped surfaces; these are surfaces whose outgoing and ingoing null rays are all contracting.) In this case there never was a proper event horizon. The other possibility is that the black hole does not evaporate within the lifetime of the universe, so that there is a true horizon. In this case the new region is causally to the future of every event in the original spacetime (as was the original spacelike singularity). It is a new region of spacetime aka a baby universe. In this case there is still a pure state overall, but no observer to the future of the formation of the apparent horizon is in a position to observe all the state and reconstruct the pure state.
I don’t see either case as a crisis for the principles of quantum mechanics because in both cases there can be a global pure state. The question is only whether far future observers can always reconstruct all the pure state or not. I know of no principle that shows they should. Indeed, quantum mechanics can be adequately formulated as a quantum causal history in terms of CP maps in either case.
Thus, the key issue is not what happens at the event horizon, as Greg emphasizes, every null surface is the horizon at least temporarily for some observer. The issues are 1) whether quantum gravity eliminates spacelike singularities or not and 2) whether there are far future null observers who can reconstruct all the info in initial pure states or not. 1) seems to be a genuine quantum gravity problem, while 2) seems to depend on whether you can give a precise meaning to the causal structure of a given quantum spacetime.
Thanks,
Lee
Comment #64 December 24th, 2006 at 3:35 am
Greg, deSitter space has a positive cosmological constant, as observations suggest our universe does also.
I stand corrected; I just got the sign backwards. My comments refer to a positive cosmological constant. (I got the sign backwards because I thought of cosmological acceleration as similar to hyperbolic geometry, which it is in some ways. However, hyperbolic space is negatively curved.)
Comment #65 December 24th, 2006 at 3:49 am
[Scott’s question] depends on whether the apparent horizon evaporates and disappears before the end of the universe or not. …[If it does], then it means that the information must leak out after the black hole evaporates to the point that there is no longer an apparent horizon. … In this case there never was a proper event horizon.
This is not the way that I understood the explanation from some other physicists. What I understood is that when an observer A falls through the event horizon of an observer B, then reality is twinned, sort-of in the style of “many-worlds” theories. (Or rather, it could be so, in a suitable model of quantum gravity.) A sees the entropy/entanglement of B chewed up and spit back out as Hawking radiation. B plunges into another universe, such as a baby universe of a black hole or another branch of de Sitter space in the cosmological example. This can be so even if the black hole evaporates in a finite amount of it.
This is the only way that I can make sense of these stories. If you really believe that Hawking radiation is no more than fallen-in entanglement spit back out, then how can B’s future experiences possibly be compatible with what A sees? The conundrum is particularly acute if B falls through A’s cosmological event horizon, and vice versa. In this case they are both guaranteed to continue to live in their own galaxies, and yet they also both see Hawking radiation.
Comment #66 December 24th, 2006 at 4:48 am
Greg: Yes, I was also assuming the “black hole complementarity” picture that you refer to. But as I’m sure you realize, you’ve just been beating around my question! So again, let’s adopt the perspective of your observer A. Then after A hypothetically manages to unscramble the information that’s encoded in the Hawking radiation, does that information
(1) only encode the bits as they were before they fell into the black hole, or
(2) encode the bits after their subsequent evolution inside the black hole?
Now, I readily admit that there’s a deep ambiguity in my question, having to do with exactly what it means to “unscramble” the bits. If the unscrambling procedure can be an arbitrary polynomial-time reduction, then presumably it can just simulate whatever the evolution was inside the event horizon! Despite this, I’d be astonished if physicists had to think in this abstract, computer-sciencey way to explain what happens when a black hole evaporates. I mean, I might not be able to define what it means for the Hawking radiation to encode the inside-the-hole experiences of observer B, but I’d expect that physicists would know it when they see it! 🙂
There might or might not be a singularity behind the event horizon — one can ask the question in either case.
Now, if I’m mistaken, and the question of whether the Hawking radiation does or doesn’t encode observer B’s experiences is a meaningless one, I’d be delighted to learn that. But nothing you’ve said has convinced me that the question is indeed meaningless.
Comment #67 December 24th, 2006 at 7:30 am
Then after A hypothetically manages to unscramble the information that’s encoded in the Hawking radiation, does that information
(1) only encode the bits as they were before they fell into the black hole, or
(2) encode the bits after their subsequent evolution inside the black hole?
They’re the same thing for the purposes of this problem. Unitary evolution is time reversal invariant, so there’s no information gained (or lost) by evolving the system.
Comment #68 December 24th, 2006 at 7:37 am
As for the issue of “more likely” or “less likely” in some Bayesian sense, I tend to think it’s all nonsensical. Bostrom runs through most of the usual objections (variations on the doomsday argument, the presumptuous philosopher, etc.) and still seems to come out thinking that it was possible to do something sensible. Of course, he’s a thirder in the sleeping beauty problem, so perhaps this is to be expected.
A lot of people try to use quantum mechanics to respond to anti-Bayesians like myself, but the better example is the spectrum of CMBR fluctuations. In particular, do we believe that the low power in the quadropole is significant or not? Does inflation predict the spectrum of fluctuations? It’s a mess.
I’m tempted to believe that quantum mechanics is deterministic with physical state reduction after I think about this stuff too much.
Comment #69 December 24th, 2006 at 8:46 am
They’re the same thing for the purposes of this problem. Unitary evolution is time reversal invariant, so there’s no information gained (or lost) by evolving the system.
Aaron: I know very well that unitary evolution is information-theoretically reversible. But evolving a system backwards or forwards takes computational effort — if it didn’t, we wouldn’t talk about it as “evolution” in the first place. So if you like, my question is the following: is it more natural, more parsimonious, to think of the Hawking radiation as a scrambled version of the qubits as they fell in, or as a scrambled version of the qubits as they evolved inside the black hole before it evaporated?
Comment #70 December 24th, 2006 at 9:38 am
Sorry, Jack — I only realized just now that you were addressing this question to me!
No problem!
“3. The philosophers’ argument only works because, in my thought experiment, I assumed that an infinite universe has prior probability greater than zero. But whether an infinite universe is even possible is something we need a physical theory to tell us.”
Well, that was an astonishingly sensible response, especially at Stanford. So whoever said that evidently believes that, since an infinite universe leads to nonsensical conclusions, a good fundamental theory must *not* allow such a thing.
It never ceases to surprise me that people still don’t see the catastrophic consequences of arguments of the form “Yes I know this situation is ridiculously unlikely, but in an infinite universe it is still bound to happen.” I mean, isn’t it obvious that this kind of argument leads nowhere?
Comment #71 December 24th, 2006 at 7:05 am
“Suppose there are two competing cosmological models. One model leads to a finite universe, the other leads to an infinite universe. Cosmologists are about to launch a space probe that will test which model is correct. But then philosophers come along and say, “Wait — you don’t have to bother! Obviously the infinite model must be the correct one — since in an infinite universe, we’d be infinitely more likely to exist in the first place!” The question I’d put to anthropicists is this: should the philosophers win the Nobel Prize that would have otherwise gone to the cosmologists? And if not, why not?”
What was the response [if any] to these remarkably sensible questions?
Sorry, Jack — I only realized just now that you were addressing this question to me!
If I remember correctly, there were three basic responses:
1. Philosophers might be able to make a confident prediction, but just like physicists in a similar situation, they shouldn’t get a Nobel Prize unless their prediction is confirmed by experiment.
2. In any case, who should win the Nobel Prize is a sociological question, not a scientific one.
3. The philosophers’ argument only works because, in my thought experiment, I assumed that an infinite universe has prior probability greater than zero. But whether an infinite universe is even possible is something we need a physical theory to tell us.
I’ll leave it to you to weigh the merits of these objections.
Interestingly, I don’t recall anyone raising what to me would be the most obvious objection — namely, that it’s unclear whether we really are k times as likely to exist in a world with k times as many observers. One might also imagine that the number of observers has no bearing on how likely we are to exist (provided it’s greater than zero) — and that, in a world with k times as many observers, we’re equally likely to exist, but k times less likely to be any particular observer. See Bostrom for much, much more on these themes.
Comment #72 December 24th, 2006 at 2:03 pm
Dear Scott,
I like to read your blog, you have something to say and you belong to those rare bloggers who have a talent for writing. As to your above post however, I didn’t find your offer that your ‘allegiances in the String Wars will be open for sale to the highest bidder’ particularly funny. Look, what you are promoting here is essentially the message: be smart and nod for those who pay best. I hardly ever comment on other people’s opinions, but in this case I wrote a short piece on my blog regarding this part of your post. I just thought you might like to know, here is the link.
Have a merry Christmas,
Sabine
Comment #73 December 24th, 2006 at 3:37 pm
Yes, I was also assuming the “black hole complementarity” picture that you refer to. But as I’m sure you realize, you’ve just been beating around my question!
No, I’m trying to defeat your question. Let me repeat a point that seems to have been glossed over in much of this discussion: Black holes are not the only source of event horizons and Hawking radiation! If the universe has a positive cosmological constant, then every galaxy that is not in the Milky Way’s local cluster is destined to fall into a backdrop event horizon — i.e., an event horizon behind everything else in the sky. Extraterrestrial astronomers in those galaxies will likewise see the Milky Way fall into their cosmological event horizon. Moreover, all of these event horizons will emit Hawking radiation.
Now if you propose that the computational results of the extraterrestrials’ computers will be encoded in the Hawking radiation that we see, then my question is, how much such computation do you propose will be encoded there? After all, the computers in these other galaxies could continue their work for a long time.
Comment #74 December 24th, 2006 at 4:44 pm
Scott — the question is probably not well defined, but if it were, I’d guess that it woud depend on your choice of a time slicing. As Greg has pointed out, Hawking radiation depends on the choice of an observer.
Comment #75 December 24th, 2006 at 7:00 pm
Dear Greg,
It is interesting to speculate about ideas, and both the holographic principle and the black hole complementary principle are ideas-that is they are not yet fully formulated conjectures within a precisely defined mathematical framework. Semiclassical GR is a well defined framework and it is the setting where Hawking radiation is so far understood. So one should be a bit cautious. Nonetheless, if we jump in and try to take black hole complementary seriously then I agree there are puzzles, one of which is the issue you have expressed that there are many horizons such as in deSitter spacetime and Minkowski spacetime (for accelerating observers) that are dependent on the specification of the timelike worldline of an observer. If there is a new physical principle which applies to every such horizon how in the world is it to be formulated?
Moreover there are a number of puzzles when one tries to extend these ideas to cosmology. One is, does the information contained “within” a surface apply to both sides when that surface is an S^2 in an S^3 spatial slice? In addition, in quantum gravity the geometry and perhaps even the topology to the interior of a two surface can be undetermined. What could the “number of degrees of freedom inside a surface” mean gthen? These lead to the suggestion that there is a weaker form of the holographic conjecture that applies only to the information measurable on the surface. (For a review of the different forms of the holographic principle which were proposed see hep-th/0003056.)
At the semiclassical level I think the right answer is the Bousso bound. This applies to a large class of null surfaces and it has the advantage of being true in a well definede context. So far so good, the only problem is that Bousso’s bound assumes a fixed classical spacetime geometry. How are we to go beyond that, especially given the fact that the quantum geometry and topology to the interior of a two dimensional section through a null surface may be fluctuating?
Here is one proposal for what the holographic principle could mean extended to quantum gravity (hep-th/9910146): Make the primary physical quantum quantum information and geometry emergent. A null surface is then fundamentally nothing but a channel for quantum information. The area of a section through it is then nothing but another name for the channel capacity. That is the relationship between information and area becomes nothing but a definition of area in the same sense that temperature is nothing but a name for averaged kinetic energy.
Thanks,
Lee
Ps what do you’all mean by Hawking radiation depends on the observer?
transforms as a tensor in semiclassical GR, Hence all observers agree on whether or not there is a flux of energy moving outwards far from the black hole.
Comment #76 December 24th, 2006 at 8:40 pm
Lee, maybe if you spent less time denigrating AdS/CFT you would find there the well-defined, mathematical, background-independent framework you seek for making these issues precise, at least for spacetimes with negative cosmological constant.
Comment #77 December 24th, 2006 at 8:41 pm
Greg:
Now if you propose that the computational results of the extraterrestrials’ computers will be encoded in the Hawking radiation that we see,
No, I don’t propose that! I don’t! I don’t! I don’t! I’m just asking the question. And you seem to be, not “defeating” my question or showing its meaninglessness, but rather arguing for a particular answer to it: namely, that because of the analogy to cosmological horizons, the information about what went on inside a black-hole horizon shouldn’t come out in the Hawking radiation.
Comment #78 December 24th, 2006 at 9:10 pm
> my expenses were covered by Umesh
Does this mean it is time for another Umeshism?
Like, “if your blog post about strings does not end with
a lengthy back and forth between Jacques and Peter or Lee and Lubos only to finally end with the discussion of
wooden rockets, then your post was too moderate.”
Comment #79 December 24th, 2006 at 9:19 pm
Scott, the information that comes out of a black hole will be horribly mangled — you have to measure an exponentially large number of Hawking quanta to have any hope of getting information back at all — so I doubt there’s any meaningful sense in which you get the result of a computation back.
Comment #80 December 24th, 2006 at 9:43 pm
Hi Lee,
Ps what do you’all mean by Hawking radiation depends on the observer?
transforms as a tensor in semiclassical GR, Hence all observers agree on whether or not there is a flux of energy moving outwards far from the black hole.
Not sure what they mean, but here’s what I mean: The thing is an expectation value and it needs a definition of particle states, annihilation/creation operators and/or something like plane waves (including the boundary conditions which are crucial in the presence of a horizon). Hawking flux isn’t observer independent in the sense that it’s defined to vanish in the in-state, then it appears in the out state, but it actually depends on how your observer ‘chooses’ a and a^\dag
I guess that’s what Greg meant with: But even in quantum field theory in non-linear coordinates, particle number is observer-dependent, because the coordinate change alters the dividing line between virtual and non-virtual particles. (This is indeed related to the existence of Hawking radiation!) If particle number is observer-dependent, then everything is.
Best,
Sabine
Comment #81 December 24th, 2006 at 10:02 pm
Lee: Nonetheless, if we jump in and try to take black hole complementary seriously then I agree there are puzzles, one of which is the issue you have expressed that there are many horizons such as in deSitter spacetime and Minkowski spacetime.
As I see it, the issue is more a matter of taking semiclassical GR seriously. Even if it is formally well-defined, you can see that it is either contradictory or incomplete as you follow it to its logical conclusions. You have to either accept some form of gravity complementarity or accept that Hawking radiation is an ex nihilo mixed state. If Hawking radiation is an encoding of the entanglement that fell in, then that is simply not causally compatible with any continued life for an observer that falls past the event horizon.
Moreover, if you believe that the universe will develop a de Sitter event horizon, and if you believe the Copernican principle, then we ourselves are observers falling through event horizons of other galaxies.
One is, does the information contained “within” a surface apply to both sides when that surface is an S^2 in an S^3 spatial slice?
Since event horizons are relative, I don’t see that they have to have two sides in any meaningful sense. An observer on the other side typically sees a non-feature: “Here lies a surface important to someone else.” In a way, a stationary event horizon relative to a stationary observer as you see with a black hole is a misleading geometric accident.
what do you’all mean by Hawking radiation depends on the observer?
I have read that a non-inertial change of coordinates in quantum field theory is a subtle business, called a “Bogoliubov transformation”, in which zero-point energy transforms to a non-ground state. Without such a subtlety, an object that falls into a black hole would witness an infinite blue shift of the outgoing Hawking radiation, and therefore would never fall in at all! I believe that the appropriate Bogoliubov transformation changes the Hawking radiation so that objects really do fall through event horizons. Indeed, it has to be so if the real universe is de Sitter, because we are constantly falling through someone else’s de Sitter horizon.
Scott: And you seem to be, not “defeating” my question or showing its meaninglessness, but rather arguing for a particular answer to it
Okay, I can accept that characterization. 🙂
namely, that because of the analogy to cosmological horizons, the information about what went on inside a black-hole horizon shouldn’t come out in the Hawking radiation.
Yes, except that I consider it more than analogy. As I understand it, all event horizons should follow the same rules.
Comment #82 December 24th, 2006 at 10:30 pm
Greg Kuperberg, who tried to convince me that critics of string theory are as “intellectually non-serious” as quantum computing skeptics or Ralph Nader voters.
I started this discussion by cowardly stepping back from this remark, but I realize now that it is close to a direct quote and I remember what I meant when I said it. Certainly I did not mean it in this ad hominem vein of Scott’s phrase; I was referring to particular arguments. Also I was referring only to existing arguments. But yes, actually I do get the impression that existing criticisms of string theory are intellectually non-serious.
In fact, so are many existing explanations of string theory! Any lay explanation of any part of science is intellectually non-serious. You cannot read a lay account of Fermat’s Last Theorem and really know that Fermat’s Last Theorem has been proven. The question is whether an intellectually non-serious argument has an intellectually serious counterpart. My impression is that this is true of at least some of the hype in favor of string theory. Certainly I know first-hand that some of the hype in favor of quantum computation can be supported seriously. (Note the emphasis on the word some.) I do not know of any serious, technical substantiation of any existing lay argument that begins, “the topic of quantum gravity needs alternatives to string theory because…”
Comment #83 December 24th, 2006 at 10:35 pm
Scott, the information that comes out of a black hole will be horribly mangled — you have to measure an exponentially large number of Hawking quanta to have any hope of getting information back at all
anonymous: No, that’s incorrect. The time that a black hole takes to evaporate grows like the cube of the radius, while the information content grows like the square of the radius. Everything’s polynomial in the number of bits.
Of course, recovering the information might be forever beyond our practical abilities, but that has no bearing on the question. 🙂
Comment #84 December 24th, 2006 at 11:10 pm
Lubos, as usual, has it all wrong. The tradition of eminent scientists selling out to the highest bidder has a long and honored tradition. Gallileo was court scientist to the Duke of Tuscany (Medicis). His successor was Evangelista Torricelli. Kepler was court mathematician to Rudolph II, and so it goes.
Comment #85 December 25th, 2006 at 12:24 am
As a followup to Eli Rabett’s fine historical review, and more broadly in support of Ecclesiastes 1:9-11 (“What has been will be again, what has been done will be done again; there is nothing new under the sun. Is there anything of which one can say, “Look! This is something new”? It was here already, long ago; it was here before our time. There is no remembrance of men of old, and even those who are yet to come will not be remembered by those who follow.”) … here a a few more BibTeX entries:
@book{Hoskin:2003,
author = {M. Hoskin},
title = {The Herschel Partnership, as Viewed by Caroline},
publisher = {Science History Publications},
year = 2003,
pages = {86},
jasquote = { William Hershel writes in 1785: “In a letter which Sir J. Banks laid before his Majesty, I have mentioned that it would require 12 or 15 hundred pounds to construct a 40-ft telescope, and that moreover the annual expenses attending the same instrument would amount to 150 or 200 pounds. As it was impossible to say exactly what some might be sufficient to finish so grand a work, I now find that many of the parts take up so much more time and labour of workmen, and more materials than I apprehended they would have taken, and that consequently my first estimate of the total expence will fall short of the real amount.” Author M. Hoskin comments: “Not for the last time in the history of astronomy, an astronomer seeking support had been modest in his initial demands, knowing that the funding body, confronted later with a choice between writing off all the money spent so far or coughing up more, would cough up.”},}
@book{Diamond:1992,
author = {J. M. Diamond},
title = {The Third Chimpanzee: the Evolution and the Future of the Human Animal},
publisher = {HarperCollins},
year = 1992,
jasquote = { From Epilogue: Nothing Learned and Everything Forgotten, page 366: “[New Guinea explorer Arthur Wichman] grew disillusioned as he realized that successive explorers committed the same stupidities again and again: unwarranted pride in overstated accomplishments, refusal to acknowledge disastrous oversights, ignoring the accomplishments of previous explorers, consequent repitition of previous errors, hence a long history of unnecessary sufferings and deaths. The bitter last sentence that concluded Wichman’s last volume was: “Nothing learned, and everything forgotten!”},}
@book{Campbell:95,
author = {T. Campbell},
title = {The Old Man’s Trail},
publisher = {Naval Institute Press},
year = 1995,
pages = {141},
jasquote = {Duan talked to them for a long time, leading them skillfully toward the answers he needed to form a plan. He was following a longstanding rule of survival. Whenever an individual or organization threatened his life needlessly through their stupidity, he figured out a way to keep himself from ever getting in their grasp again.},}
@inBook{Knuth:95,
author = {M. Petkov\v{s}ek and H. S. Wilf and D. Zeilberger},
title = {$A=B$},
publisher = {A.~K.~Peters, Ltd.},
year = 1996,
note = {Forward to book.},
jasquote = {Science is what we understand well enough to explain to a computer. Art is everything else we do. [\ldots] Science advances whenever an Art becomes a Science. And the state of the Art advances too, because people always leap into new territory once they have understood more about the old.},}
@book{Deutsch:97,
author = {D. Deutsch},
title = {The Fabric of Reality},
publisher = {Penguin Press},
year = 1997,
pages = {315},
jasquote = {Knowledge does not come into existence fully formed. It exists only as the result of creative processes, which are step-by-step, evolutionary processes, always starting with a problem and proceeding with tentative new theories, criticism and the elimination of errors to a new and preferable problem-situation. This is how Shakespeare wrote his plays. It is how Einstein discovered his field equations. It is how all of us succeed in solving any problem, large or small, in our lives, or in creating anything of value.} }
@article{Saletin:04,
author = {S. A. Saletin},
title = {The Peculiar Institution},
journal = {New Your Times Book Review},
pages = {9},
year = 2004,
edition = {September 26},
jasquote = {Every movement that seeks to change society has two great tasks. The first is to discredit the old order. The second is to offer a new one. Without the assurance of a new order, the debate becomes a choice between order and chaos, and order wins.}, }
Comment #86 December 25th, 2006 at 2:07 am
Hi Greg,
I have read that a non-inertial change of coordinates in quantum field theory is a subtle business, called a “Bogoliubov transformation”, in which zero-point energy transforms to a non-ground state. Without such a subtlety, an object that falls into a black hole would witness an infinite blue shift of the outgoing Hawking radiation, and therefore would never fall in at all! I believe that the appropriate Bogoliubov transformation changes the Hawking radiation so that objects really do fall through event horizons.
? Sorry, but that statement doesn’t make any sense to me. Hawking radiation is usually computed for the observer at infinity, as Lee correctly states. It is possible to compute the exp. value of the stress-energy tensor for an infalling observer, and as far as I can recall it’s finite. (See e.g. Birell+Davies). The whole issue of blue shift is seriously misunderstood. The real problem there isn’t that anything is actually blue-shifted, but that if one does the calculation on the horizon one needs to include arbitrarily high energies, i.e. wavelengths below the Planck length, which makes the whole computation doubtful (essentially, the integrals go up to infinity). This was the reason why Unruh suggested a modified dispersion relation which just wouldn’t have these wavelengths below the Planck scale.
Without the subtlety of Bog. trafo, there would be no Hawking effect at all. Besides this, as far as I know, nobody has ever computed the Hawking radiation for a black hole with an infalling object.
Best,
B.
Comment #87 December 25th, 2006 at 4:03 am
There is another interpretation of the black hole entropy, and one which will be more friendly to those of us more familiar with information theory than with isolated horizons or the Hamiltonian constraint. It says that black hole entropy is nothing more than the entanglement entropy between the fields on the inside of the horizon, and the fields outside the horizon.
When you try to compute the entanglement entropy of the vacuum state for a region of flat space in standard QFT, you get the standard QFT answer of infinity. But when you do it for a spin network in LQG, you get a finite answer. It depends only on the number of links crossing the boundary of the region you trace out, and each spin-j edge contributes log 2j+1 to the entropy.
This is close to the area operator, which has sqrt j(j+1) for each edge. However it’s known that there is some ambiguity in the definition of the area operator, so it isn’t clear whether the proportionality between area and entropy should be exact in LQG or if there should be some “quantum corrections”.
Comment #88 December 25th, 2006 at 5:39 am
Dear William,
There are a few problems with the entanglement entropy interpretation of the black hole entropy. One is that the entanglement entropy is proportional to the number of species of elementary particles in the QFT; this is a straightforward consequence of your argument. But the black hole entropy depends only on the properties of the black hole, it knows nothing about the particle content of the QFT.
There are quantum corrections in LQG to the black hole entropy formula, there is a term proportional to Ln(Area). See recent papers by Livine and Terno and by Modesto about this, among others. This does not come from an ambiguity in the definition of the area operator, it is there given the usual definition of it.
Dear Greg,
“As I see it, the issue is more a matter of taking semiclassical GR seriously. Even if it is formally well-defined, you can see that it is either contradictory or incomplete as you follow it to its logical conclusions.”
But if you don’t take semiclassical GR seriously, at least to the point of believing it gives a good approximation for large black holes and/or small Lambda, you shouldn’t believe in any of the calculations that lead to the conclusions that black holes and horizons have entropy and radiate. So the question can only be how to embed it as an approximation to an exact theory of quantum gravity. It seems to be that the fact that is a tensor is a significant fact we should not loose sight of when we seek to understand how the semiclassical theory could emerge as an approximation to an exact theory.
“You have to either accept some form of gravity complementarity or accept that Hawking radiation is an ex nihilo mixed state.”
As I discussed above there is a third and more conservative choice, which requires only that the singularity is eliminated by quantum gravity. Then, whatever happens after that, global pure state evolution is guaranteed. Given that causal structures of spacetimes are dynamically determined in the classical limit, I don’t see how one could ask for more in a quantum theory of gravity. That is, I don’t see why the definition of the theory should require a global causal structure that allows local observers in the far future to reconstruct the whole global pure state. What is required is much less, which is only that observers that see a whole initial Cauchy surface can reconstruct the pure state on it, which we know can be satisfied.
Given that this is a discussion among scientists, we can use a more precise language than “observer dependence”. Our understanding of semiclassical GR is much improved due to rigorous work done by several people over the last twenty years, such as Bob Wald, Stefan Hollands and others. They have shown that at an operator algebra level, given the spacetime, there is a single, well defined entity which defines the QFT on that spacetime. If I am remembering right all issues of renormalization can be dealt with at this level. Once one defines the theory algebraically one has to choose the state. This is where the “observer dependence” comes in. One can choose the state so that it is annihilated by operators which are negative frequency as measured by the worldliness of a family of observers. One can call this “observer dependence, but this is just the choice of a state. This is not new, in ordinary QFT on Minkowksi spacetime one also has to choose the state. So there is no more state dependence than there was before in conventional QFT.
When a spacetime has timelike killing fields there is a natural choice of a vacuum state, which is to choose it so observers on worldliness following those killing fields see no particles. But this is still a choice that must be made, one has the freedom to choose differently. Why is it then that the vacuum state we use in Minkowski spacetime is the one that is Poincare invariant? There is no other reason than that we choose it so. If we had a truly fundamental theory we would not be allowed to make a choice here, shouldn’t it come out automatically that inertial observers in the ground state of the gravity theory see no particles?
This is a condition we would like to ask of a truly fundamental theory. It is not, by the way satisfied in string theory, so this is one (of several) serious answers to Greg’s question of why “the topic of quantum gravity needs alternatives to string theory because…”
Lee
Ps for more on the current understanding of QFT in curved spacetime there is a recent review of Wald, gr-qc/0608018. I quote the abstract:
“Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory, but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description when the quantum effects of gravity itself do not play a dominant role. A major impetus to the theory was provided by Hawking’s calculation of particle creation by black holes, showing that black holes radiate as perfect black bodies. During the past 30 years, considerable progress has been made in giving a mathematically rigorous formulation of quantum field theory in curved spacetime. Major issues of principle with regard to the formulation of the theory arise from the lack of Poincare symmetry and the absence of a preferred vacuum state or preferred notion of “particles”. By the mid-1980’s, it was understood how all of these difficulties could be overcome for free (i.e., non-self-interacting) quantum fields by formulating the theory via the algebraic approach and focusing attention on the local field observables rather than a notion of “particles”. However, these ideas, by themselves, were not adequate for the formulation of interacting quantum field theory, even at a perturbative level, since standard renormalization prescriptions in Minkowski spacetime rely heavily on Poincare invariance and the existence of a Poincare invariant vacuum state. However, during the past decade, great progress has been made, mainly due to the importation into the theory of the methods of “microlocal analysis”. This article will describe the historical development of the subject and describe some of the recent progress.”
Comment #89 December 25th, 2006 at 6:40 am
Eli: “The tradition of eminent scientists selling out to the highest bidder has a long and honored tradition. Gallileo was court scientist to the Duke of Tuscany (Medicis). His successor was Evangelista Torricelli. Kepler was court mathematician to Rudolph II, and so it goes.”
These examples are more similar to eminent scientists going to the highest paying university. It is not really the same as making a scientific call based on the highest bidder. Examples for that are rather infamous.
Comment #90 December 25th, 2006 at 7:29 am
When you try to compute the entanglement entropy of the vacuum state for a region of flat space in standard QFT, you get the standard QFT answer of infinity
It’s an old calculation (first due to Srednicki, maybe), that if you divide spacetime into two regions in ordinary QFT, the entanglement entropy is proportional to the area of the boundary. This is because you have to cut things off, and the entropy is dominated by the short distance contribution around the boundary. It’s not at all clear that this has anything to do with the usual notion of black hole entropy, however.
I don’t have too much time right now, but I wanted to mention something about Lee’s comment. The observer determines the choice of state. An inertial observer respects the Poincare’ symmetry, so we get a Poincare’ invariant state. An accelerating observer, on the other hand, sees a different state and that’s why they detect Unruh radiation. I’m not sure I see what the issue is here.
Comment #91 December 25th, 2006 at 3:23 pm
> It is not really the same as making a scientific call based on the highest bidder.
Sure it is, because the old grandmasters had no problem with astrology and calculating a horoscope if they were properly paid for it.
Comment #92 December 25th, 2006 at 4:01 pm
Just to supply an engineering perspective on string theory, suppose Alice is running a functional emulation of a Hilbert space containing five spin-1/2 particles, one spin-1 particle, and one spin-3/2 particle.
The total dimensionality of this Hilbert space is 384, a number that Alice finds inconveniently large for numerical simulations (because she is planning eventually to simulate quantum systems whose native Hilbert space has exponentially many dimensions).
To make progress, Alice projects the dynamical equations of her quantum emulation onto the Kähler submanifold that is the direct product of single spin states: this Kähler submanifold has 11 complex dimensions—a considerable savings over 384 dimensions, and also very good practice for Alice’s future simulation manifolds of exponentially higher Hilbert dimensionality.
What is the scalar Ricci curvature of this 11-dimensional Kähler manifold? It is not hard for Alice to calculate that the Ricci curvature is constant and negative (up to a normalization):
R = -164/⟨Ψ|Ψ⟩
The negative sign of the curvature is important … tells Alice that her Kähler manifold fills the embedding space like the curled petals of a flower. It turns out to be not too hard for Alice to compute the curvature of every product manifold, which turns out to grow quadratically with the number of particles.
The next step is for Alice to increase the Beylkin-Mohlenkamp rank of her representation, with a view to more accurately simulating high-order quantum correlations … it is at this step that Alice begins to read seriously the literature on complex manifolds and string theory.
Paradoxically, Alice finds that the mathematics literature is mainly written in an coordinate-free idiom that is not easily translated into her coordinate-based simulation codes, and yet this same literature is grossly incomplete with regard to guiding the design of those codes. Which is not to complain … Alice finds this situation to be highly stimulating and full of creative opportunity.
Alice’s engineering interest in mathematical projections onto complex manifolds is a striking 21st Century analog to Julian Schwinger’s engineering research in Green functions (Schwinger worked on radar at the MIT Rad Lab during the war). In both cases, mathematical tools that were originally pursued for engineering reasons turn out to be equally central to the development of fundamental physics.
The point being, we’re all climbing the same mountain: mathematicians, physicists, and yes, quantum chemists and quantum system engineers too.
Welcome to the century of Kähler manifolds.
Comment #93 December 25th, 2006 at 5:25 pm
Aaron,
It is not that an accelerating observer sees a different state than an inertial observer. An accelerated detector in the Minkowski vacuum becomes excited as if it were in a bath of thermal radiation with a temperature proportional to the acceleration. This has nothing to do with “observer dependence”, it is a straightforwards consequence of ordinary QFT (it is a kind of quantum radiation reaction). It can be computed in the frames of both inertial and accelerating observers and when the calculations are done right they give the same answer.
You are perhaps confusing this with the fact that the “vacuum state” which is annihilated by operators which are negative frequency with respect to clocks carried by uniformly accelerating observers is a distinct state from the Poincare invariant vacuum. But there is no issue of observer dependence. All observers agree on two facts: The Rindler vacuum and Poincare invariant vacuum are different. In each state uniformly accelerating detectors are excited differently than inertial detectors.
These are intriguing questions and I don’t mean to suggest that they are understood-quite the opposite. (I once spent a lot of time on them: On the nature of quantum fluctuations and their relation to gravitation and the principle of inertia} Classical and Quantum Gravity, 3 (1986) 347-359.) But back when they were discussed a lot we found that simple ideas of observer dependence or a “quantum equivalence principle” were too naive to characterize the results (for example there are puzzles when one studies rotating observers). It may be that the holographic principle or principle of complementarity adds something essential, I am only cautioning that they have to first be formulated so that they are consistent with the results from the semiclassical theory.
Thanks,
Lee
Comment #94 December 25th, 2006 at 5:53 pm
Perhaps we have a terminology conflict here. I’m referring, as you say, to the fact various observers see different vacua. As I’m sure you know, a free falling observer sees no Hawking radiation. Now, obviously, anything physical can be computed however you want and you should get the right answer, but Scott is talking about things involving information, so these considerations might be relevant.
Comment #95 December 25th, 2006 at 6:39 pm
More on Alice’s Adventures in Quantum Gravity
As a shakedown test of her quantum model order reduction codes, Alice Michanikos (her family name) is running a simulation of the IBM single-spin MRFM experiment.
‘”How very interesting!” thinks Alice. “The mass of the IBM MRFM cantilever is 92 picograms, the decorrelation time scale is 100 msec, and the separation of its trajectory for spin-up versus spin-down measurements on this time scale is 280 picometers. Since the IBM team experimentally observed a mean-square spin polarization of unity (versus classical mean-square polarization of 1/3), it must be the case that quantum theories of gravity—once we finish constructing them—will support macroscopic polarizations for Schroedinger cats of this mass, this timescale, and this spatial separation.”
“Hmmm … ” thinks Alice. “If only the gravitational theorists would be so good as provide me with a well-posed mathematical recipe, it would be quite easy for me to include the cantilever’s gravitational metric in my functional emulations. It’s a bit frustrating that Bob Wald’s recent preprint tells me in great detail how quantum fields accommodate to metrics, when as a quantum system engineer, what I mainly need to know is how metrics accommodate to quantum fields.”
“I wonder if it would be possible, even in principle, to monitor the quantum state of the IBM cantilever with an external gravitometer? A mass of 92 picograms, having a quantum spatial superposition of 280 picometers, measured on a timescale of 100 msec, is not much of a signal … but perhaps it is enough.”
“From a fundamental physics point of view, it’s unfortunate that the IBM MRFM cantilever had a thin-sheet geometry and that the quantum spatial separation was along an axis perpendicular to that sheet … this is precisely the test mass geometry that minimizes the Penrose potential energy associated with the quantum superposition. As usual, the theorists are pointing out that a spherical cantilever would have yielded a much more stringent test of quantum gravity … which is perfectly true, but it completely ignores the practical realities of nanotechnological engineering!”
“For engineers, the most fun thing about these considerations is that there is such a close link between the practical imaging applications of MRFM, the fundamental math and physics issues of quantum gravity, and the emerging engineering discipline of quantum system engineering.”
——-
The above being a fictionalized summary of the lively and extraordinarily enjoyable discussions that took place at the CalTech Kavli Institute QEM-2 Workshop on Quantum Electromechanical Systems that was held in Morro Bay earlier this month.
Comment #96 December 25th, 2006 at 7:45 pm
Lee: The discussion at this point may just be a matter of semantics, but your position still seems strange. It reminds me of the play on words, where one person says “I think that it’s raining”, and his friend says, “I agree, you think that it’s raining”. Of course, both observers can agree that the Minkowski vacuum does not equal the Rindler vacuum. Nonetheless, an accelerating observer sees a Rindler horizon and its Hawking radiation in the same vacuum where an inertial observer sees neither. In that sense, Hawking radiation is observer-dependent.
Comment #97 December 26th, 2006 at 9:08 am
Scott: “I shall answer to no quantum-gravity research program, but rather seek to profit from them all.”
When it comes to mathematics (and intellectual profit, of course,), this is precisely the approach. Both string
theory and the older (/newer) approaches to quantum gravity are related to fascinating mathematics and
mathematicians are seeking to “make profit from them all” and they do make a lot of profit some of which is invested back into physics.
Here are three connections between quantum gravity and mathematics.
1. Enumerating random graphs.
The first (which Scott can find also be geographically pleasant) is that the Waterloo based eminent graph theorists William Tutte
developed in the 60s a remarkable enumeration theory for planar graph. His ultimate
motivation was to prove the four color conjecture. The four color conjecture was later proved
by entirely different means (related to even older and more direct approaches than Tutte’s approach) by Appel and Haken in 1976. However, Tutte’s
rich and beautiful theory and methods are important in mathematics and quite fundamental in quantum gravity.
There was recently interesting research in mathematics aiming at proving various quantum-gravity based
conjectures on statistical models on random planar graphs. In these works Tutte’s methods are of importance.
2. SLE and KPZ.
Lattice models based on random (planar) graphs rather than “lattices” are important in quantum gravity. There is a mysterious formula – the KPZ
formula of Knizhnik, Polyakov and Zamolodchikov, relating quantum gravity – statistical physics on random planar models, with
statistical physics on grid models. This have led to some predictions which were proved by Lawler, Schramm and Werner as part of their
work on the SLE models. The body of works of Lawler, Schramm and Werner is regarded as one of the crowning achievements of
mathematics of our time. Wendelin Werner received the 2006 Fields Medal.
Schramm’s SLE model, that was originated in the study of critical planar lattice models like loop-erased-random walks and percolation,
is closely related also to conformal field theory which is quite prominent in string theory. It has applications to other physics areas such as turbulence.
3. A basic question for D=3.
One basic (Greg-tailored??) mathematical question related to moving from 2- to 3- dimensional
models is the following: Is the number of (unlabeled) triangulations of the 3-dimensional sphere with N tetrahedra (only)
exponential in N? A positive answer to this problem (asked by Gromov and by others)
is required even to start extending some stuff from D=2 to D=3 and is quite interesting on its own.
Comment #98 December 26th, 2006 at 9:15 am
Scott: “I shall answer to no quantum-gravity research program, but rather seek to profit from them all.”
When it comes to mathematics (and intellectual profit, of course,), this is precisely the approach. Both string theory and the older (/newer) approaches to quantum gravity are related to fascinating mathematics and mathematicians are seeking to “make profit from them all” and they do make a lot of profit some of which is invested back into physics.
Here are three connections between quantum gravity and mathematics.
1. Enumerating planar graphs and random planar graphs.
The first (which Scott can find also be geographically pleasant) is that the Waterloo based eminent graph theorists William Tutte developed in the 60s a remarkable enumeration theory for planar graph. His ultimate motivation was to prove the four color conjecture. The four color conjecture was later proved by entirely different means (related to even older and more direct approaches than Tutte’s approach) by Appel and Haken in 1976. However, Tutte’s rich and beautiful theory and methods are important in mathematics and quite fundamental in quantum gravity. There was recently interesting research in mathematics aiming at proving various quantum-gravity based conjectures on statistical models on random planar graphs. In these works Tutte’s methods are of importance.
2. SLE and KPZ.
Lattice models based on random (planar) graphs rather than “lattices” are important in quantum gravity. There is a mysterious formula – the KPZ formula of Knizhnik, Polyakov and Zamolodchikov, relating quantum gravity – statistical physics on random planar models, with statistical physics on grid models. This have led to some predictions which were proved by Lawler, Schramm and Werner as part of their work on the SLE models. The body of works of Lawler, Schramm and Werner is regarded as one of the crowning achievements of mathematics of our time. Wendelin Werner received the 2006 Fields Medal.
Schramm’s SLE model, that was originated in the study of critical planar lattice models like loop-erased-random walks and percolation, is closely related also to conformal field theory which is quite prominent in string theory. It has applications to other physics areas such as turbulence.
3. A basic question for D=3.
One basic (Greg-tailored??) mathematical question related to moving from 2- to 3- dimensional models is the following: Is the number of (unlabeled) triangulations of the 3-dimensional sphere with N tetrahedra (only) exponential in N? A positive answer to this problem (asked by Gromov and by others) is required even to start extending some stuff from D=2 to D=3 and is quite interesting on its own.
Comment #99 December 26th, 2006 at 4:29 pm
Quantum gravity is not directly relevant for anything we observe, and the string landscape shows that quantum gravity does not need to be indirectly predictive.
In these conditions it is very hard to do intellectually serious physics.
Comment #100 December 26th, 2006 at 6:25 pm
Gil: Counting the number of triangulations of a 3-manifold is a great question in combinatorial geometry, one that you might even call “Greg-flavored”, although I have not thought much about that specific question. I do have a result with David Eppstein and Gunter Ziegler concerning the fatness of cellulations of 3-spheres. That may have some bearing on this counting question.
In fact it does! In arXiv:math.MG/0212004, Pfeifle and Ziegler construct 2^Omega(n^5/4) triangulations of S^3 with n vertices. Of course, that is not as good as n simplices, but it is a related result.
However, this is not necessarily the same as 3D quantum gravity. There are several known models for 3D quantum gravity, and they are not thought to be the same as a simple unweighted count of triangulations. That definition of gravity is special to 2D.
The larger point is that theories of gravity in different dimensions are not necessarily the same subject, only perhaps different subjects with the same name. 2D gravity and 3D gravity are quantum field theories, but string theorists (and some non-string-theorists) believe that gravity in higher dimensions cannot be a quantum field theory. In particular, if quantum gravity exists, then it presumably has a perturbative version. Many string theorists believe that string theory is the only possible renormalizable, perturbative theory of quantum gravity. (Or I should say string theories, since as they say, there are five sensible ones, in addition to the problematic purely bosonic string theory.)
Comment #101 December 28th, 2006 at 8:29 pm
You’re undermining the market!
You should have held out for dinner (mexican or asian-fusion) and a friendly poker game.
Comment #102 December 29th, 2006 at 1:35 am
As an interested layperson, I would like to put forward a thought that has been nagging at me for some time.
A year ago, condensed-matter physicist George Chapline proposed a black hole alternative he termed a “dark energy star“. It was greeted with some derision–it certainly did not help that on its face it invited comparison to another black-hole alternative, a rather contrived and inelegant notion termed a “gravastar“. The choice of name, carrying as it did the largely unsupported implication that this proposal could account for the existence of cosmological dark energy was also too presumptuous by half.
Unlike the the gravastar however, the dark-energy star idea was motivated by analogy with an observed physical system. Chapline felt that, as a superfluid undergoes a phase transition as the speed of sound approaches zero, one which causes sound waves to dissipate as heat, spacetime should undergo a similar phase transition in the vicinity of an event horizon. He proposed that the effect of this transition would be that matter crossing the horizon would decay into some form of exotic “stuff” similar to dark energy which would exert sufficient negative pressure to prevent further gravitational collapse.
The dark energy star proposal clearly frought with problems, not the least of which being the obvious one that it would seem to turn the event horizon into a mesurable physical boundary, with all attendant havoc for the internal consistency of general relativity.
However, the analogy with superfluids still does look somewhat interesting to me, and I wonder if any physicist might at least be able to give some insight into whether and why this analogy falls down.
I’ll now go far out on a limb and presume for the purposes of two followng thoughts that somehow this analogy truely does hold.
Does this neccessarily make the horizon “special”, and detectable as a physical effect from the outside ? Chapline apparantly believes so, but if the thickness of the transition region is, as one might expect, one the order of the Plank length, it would seem far from obvious that this should be the case.Is it concievable that dark-energy stars, rather than presenting an alternative to black holes, are equivalent to them; could a dark-energy star and a “conventional” black hole somehow be different yet equally valid descriptions of the future of the hidden region?
Comment #103 December 29th, 2006 at 8:33 am
Hi Greg: Thanks for your explanations on quantum gravity.
The connection between the physics and mathematics need not be on a 1-1 basis and sometimes it is more via associations. The crux of matters (or the “basic problems”) for a mathematician may be different than for a physicist. (And of course, different for different mathematicians. (However, note that superexponential growths of the numbers of triangulations in question may be an obstacle to various weighted expansions as well.)
Gromov words (Visions in Mathematics towards 2000, Noga Alon et als, eds, part I p. 156) are:
“Here is another basic problem linked to ‘non-locality of topology’. How many triangulations may a given space X (e.g., a smooth manifold, say the sphere S^n) have? Namely let t(X,N) denote the number of mutually combinatorially non-isomorphic triangulations of X into N simplices. Does this t grows at most exponential in N? Notice that the number for all X built on N n-dimensional simplices grows superexponentially roughly N^N and the major difficulty from a given X comes from pi_1(X) and, possibly (but less likely) from H_1(X), where the issue is to count the number of triangulated manifolds X with a fixed pi_1(X) or H_1(X)
These questions (coming from physicists working on quantization of gravity) have an (essentially equivalent) combinatorial counterpart (we stumbled upon with Alex Nabutovski): evaluate the number t_L(N) of connected 3 valent (i.e. degrees
Comment #104 December 29th, 2006 at 8:37 am
The comment and quoting from Gromov was interuppted in the middle,
“… (i.e. degrees
Comment #105 December 29th, 2006 at 8:41 am
It looks the less equal sign causes troubles
“… (i.e. degrees less or equal to 3) graphs X with N edges such that cycles of length less or equal to L normally generates pi_1(X) (or, at least, generates H_1(X))? Is t_L(X) at most exponential in N for a fixed (say 10^10) L? The questions look just great and no idea how to answer them.
Here is somewhat similar but easier question: What does a random group (rather than a space) look like? As we shall see the answer is most satisfactory (at least to me): ‘nothing like we have ever seen before’ (No big surprise though: typical objects are usually atypical.)”
Gromov, thus considers this question (rooted in quantum gravity) to be a basic problem regarding “non-locality” of topology. The “somewhat similar but easier question” that he mentioned and studied have led to a lot of exciting stuff.
As I said, the problem was raised by others independently, more precisely, by me (just for spheres), and my motivation was purely combinatorial. It seems like a good problem but in mathematics and especially in combinatorics it is hard to predict the future of a problem. Weak-expansion properties of dual graphs for triangulated manifolds are clearly related to this problem, and some older works (rooted in theoretical computer science) of Gary Miller (like the paper by Miller, Teng, Thurston and Vavasis J. ACM 44, 1-29) seem relevant.
Comment #106 December 29th, 2006 at 2:40 pm
Gil Kalai: Your position is completely reasonable. Gromov’s questions are important and certainly worth considering. My only remark is that these questions are rooted in 2D quantum gravity much more than in quantum gravity in general. Since this post about string theory, the message is that high-dimensional quantum gravity is a very different problem.
Comment #107 December 29th, 2006 at 4:05 pm
the message is that high-dimensional quantum gravity is a very different problem.
Greg: Yeah, I agree with you that this seems like a key point. I won’t feel like I have any handle on this quantum-gravity business until I understand why the 2+1-dimensional case is so different from the case of 3+1 and higher dimensions. Yes, I know it’s because in 2+1 dimensions you can make all the degrees of freedom topological and in 3+1 dimensions you can’t — but that just pushes the lump under a different corner of the rug.
Comment #108 December 29th, 2006 at 5:18 pm
Just to help equation-posters, here are two useful tables of HTML character codes that (AFAICT) work well with pretty much all browsers, and in particular, work well with Scott’s WordPress host: Symbols and Special Characters.
Comment #109 December 29th, 2006 at 9:40 pm
[…] I was informed by Chad Orzel’s blog Uncertain Principles that I could rent or buy a theoretical physicist of my very own! I always wanted one of those. The product is Scott Aaronson, who advertised in his blog Shtetl-Organized that he’s a Mercenary in the String Wars. Just think about how much fun it would be to hire Scott to make some pompous physics geek’s head explode at the next water cooler lecture! I’m still waiting to see what the high bid is after a week so I can beat it by a penny just like on E-Bay… […]
Comment #110 December 30th, 2006 at 7:55 pm
Greg Scott, It looks that Gromov particular question is related to verious possible extensions of ideas/calculations from 2-D to 3-D QG (or perhaps QFT).
Comment #111 December 31st, 2006 at 4:06 am
First, to get the terminology straight, N-dimensional gravity conventionally means N total dimensions. In fact, both of the cases N+0 and (N-1)+1 are interesting when N is small, not only for gravity but also for quantum field theory in general. The short answer to Scott’s question is (as you might expect) that quantum gravity is renormalizable in 3+0 and 2+1 dimensions, but non-renormalizable in higher dimensions. Renormalizability plays a fundamental role in perturbative quantum field theory. If a QFT is not renormalizable, then it is basically ill-posed. It means that you haven’t really defined a model of what you want, although of course with new ideas you might come up with one.
2D quantum gravity has a trivial Lagrangian, and amounts to defining or computing the volume of the space of metrics on a surface. (I.e., the theory posits that you integrate a constant function on its domain, which is the same as finding the volume of the domain.) I suppose that this counts as “superrenormalizable”, although I am not sure if that term is meant to apply to this case.
3D quantum gravity is, according to celebrated old paper by Witten, equivalent to Chern-Simons field theory, which is a renormalizable quantum field theory. This fact makes it much easier to hope for a combinatorial model of quantum gravity in this dimension, as you might get by counting triangulations, or by a weighted sum over triangulations.
String theory is renormalizable in 25+1 ordinary dimensions, or in 9+1 supersymmetric dimensions. (I do not know, although I may have been told, whether string theory is renormalizable in N+0 dimensions for any N. Maybe a knowledgeable string theorist can come forward with the answer.) Renormalizability is the fundamental reason to focus on these choices for the dimension (*). What the string theorists are saying is that they would far prefer a mathematically well-defined model to a Rorschach test, even at the expense of working in the “wrong” number of dimensions. It may not turn out to be wrong after all, if some of the dimensions are too small to easily perceive.
(*) I have the feeling that non-renormalizability is not the only way to describe the diseases of string theory in dimensions other than 10 or 26. I think that it is one valid description.
Comment #112 December 31st, 2006 at 12:13 pm
String theory is renormalizable in 25+1 ordinary dimensions, or in 9+1 supersymmetric dimensions.
I’ve always wanted to understand where the 26 and 10 come from at an intuitive, number-theoretic level, and have never gotten a good answer. Here’s a really stupid question: is the fact that 25=5^2 and 9=3^2 at all relevant here?
Comment #113 December 31st, 2006 at 4:40 pm
Renormalizable isn’t the word I would use for string theory. The associated gravitational theories aren’t renormalizable, for example. The issue is the consistency of the worldsheet description.
For Scott’s question, there’s a lot of stuff on where the famed 26 comes in. If you go to light-cone gauge, it come about because it’s 24 + 2 (the two being the two dimensionality of the worldsheet) and 24 has a long history is mathematics. It’s related to the -1/12 value of the zeta function, for example.
I wrote a usenet post about this a long time ago when I was learning string theory.
On the other hand, you can also compute the conformal anomaly geometrically. In that case, the 26 arises because it is 13*2.
I don’t know the exposition for the number 10 (and 8), but I’m sure someone has worked it out. 24 is the dimension of the Leech lattice, for example, and 8 is the dimension of the E_8 root lattice, and both are even unimodular lattices.
Comment #114 December 31st, 2006 at 4:41 pm
That smiley face was an 8 followed by a parenthesis.
Comment #115 December 31st, 2006 at 9:46 pm
Sorry, Aaron is of course correct. It is true that string theory is, among other things, a cure for the non-renormalizability of gravity as a quantum field theory. However, as best I can regurgitate, it achieves this because it miraculously needs no renormalization, at least not at the level at which worldsheets are interpreted as Feynman diagrams. Instead, if you try to define string theory in the wrong dimension, the resulting disease is called “the conformal anomaly”. Again, as I understand it, the conformal anomaly means that the amplitude of a string trajectory has an ambiguous phase.
Comment #116 January 2nd, 2007 at 6:22 am
Happy New Year, Scott! But speaking as a P-vs-NP person, it feels like you just dumped 10^500 – 1 universes on my front lawn! (One universe was already there.) And we haven’t even finished picking up all the fallen tree stuff from the Great October 2006 Buffalo Ice Storm yet…
Seriously, I’d like to ask you and people in general about 3 things:
(1) Are there any people who believe in Shor but not in Grover? How have arguments progressed since your thesis. both on the Wolfram/Fredkin front and interpreting your “quantum robot” D&C result? Would that extend to say (in any sense) that observing the winner in a lottery with N tickets can be done with O(root-N) units of “effort”?
(2) Can anyone help bugfix my description of “Quantum Basketball” here? It’s motivated by an analogy I’m trying to make semi-privately about digital simulations with few degrees of freedom, and the Dec. 20 item in Susan Polgar’s Chess Blog was a convenient place to park it. My ability to describe EPR with the same analogy in my latter comment makes me more confident about my former comment (please excuse some unwanted carriage returns).
(3) My former comment in Polgar’s blog ranges speculatively into a 2006 paper Downarowicz and Lacroix (linked there), whose Web discussion (also linked there) gets into Wolfgang Pauli’s interest in Kammerer’s “Law of Series” seemingly with analogy to QM. But I think complexity theory might mute the basic point of Downarowicz-Lacroix. The only level I can get my mind up to now is a very simplistic one: their basic point is “you can’t be more random than random”, so any deviation must attract—but if strong pseudorandom generators exist then you CAN be “more random than random” so maybe that balances out. (??) It seems at least to raise discussion points I’d be curious to hear more on.
Anyway, Jonathan Buss and I have been working on a paper so I’m planning to drive up to Waterloo in about a week—when would be a good day/time to try to catch you?
Cheers, —Ken Regan Site, also with some new chess stuff.
Comment #117 January 3rd, 2007 at 1:50 am
I bid a signed copy of Milton Friedman’s “Free to Choose”. Signed by me, not him.
Comment #118 January 3rd, 2007 at 2:11 am
Isn’t this the old QG observable problem? Namely the only known good invariant at this time is say the SMatrix, or in other words physics at infinity (usually with nice asymptotic far fields).
From what I gather, and in so far as I understand it, entropy in stringy physics outputs a global quantity, and is far more subtle than a naive counting of individual physical states (which seem to be hopelessly observer dependant, no one can agree on what it means to partition a BH).
Comment #119 January 3rd, 2007 at 4:01 am
One of the things I hate about “critical thinking” textbooks is that they give a list of ways we supposedly shouldn’t reason — the genetic fallacy, the fallacy of composition, and so forth — with no real taxonomy. Students forget that the use of (almost) every one of those fallacies emerges because of the great successes that the fallacy gives the reasoner in certain situations.
So many people want to uncritically accept anthromorphic reasoning, or reject anthropomorphic reasoning, that the attempt Scott makes to get at the justification of reasoning is a real breath of fresh air. Maybe like C.S. Peirce, they need him over at Popular Science magazine (which I believe was still predicting the flying cars, even in Peirce’s day).
Another note: I disagree with the poster that argues above that it is impossible to know the proof of a theorem from a popularization. If the popularization tells me what is presumed, and that only the rules of some accepted logical system are used, then I want to say I know that theorem — in fact, I might know the theorem better than someone who thought, by her direct examination of the actual steps in the reasoning, that it depended upon the Axiom of Choice, when a more restricted set of axioms would do.
We depend in the modern world on computing machines epistemologically, just as we depend on telescopic machines epistemologically. I have seen craters on the moon, and I have come to know the truth of theorems I could have never proved.
Comment #120 January 3rd, 2007 at 6:55 pm
We depend in the modern world on computing machines epistemologically, just as we depend on telescopic machines epistemologically. I have seen craters on the moon, and I have come to know the truth of theorems I could have never proved.
You might be a philosopher, Drew, but you’re all right. 🙂 To say I agree with you about computers being “Platonic telescopes” is an understatement. As you might know, Dijkstra reached for the same analogy in his famous remark about computer science, that it’s “no more about computers than astronomy is about telescopes.”
Comment #121 January 3rd, 2007 at 7:19 pm
Hi Ken,
A few responses to your first set of questions (I have nothing to say about the other two):
Are there any people who believe in Shor but not in Grover?
There are certainly people who think that Grover’s algorithm can’t be applied to “physical” databases.
Also, it’s possible to define a model of quantum computing — in which the “answer register” starts out in a maximally mixed state — that can implement Shor’s algorithm but not Grover’s algorithm. But I don’t know of anyone who actually believes that model.
How have arguments progressed since your thesis. both on the Wolfram/Fredkin front and interpreting your “quantum robot” D&C result?
The Wolfram front has been pretty quiet. Nothing really new about the quantum robot result either.
I talked to Fredkin a few months ago at Perimeter. He said he believes that P=BQP, and that the universe is a classical, nonrelativistic cellular automaton made of pixels many orders of magnitude larger than the Planck scale. I wasn’t sure how to continue the conversation after that.
Would that extend to say (in any sense) that observing the winner in a lottery with N tickets can be done with O(root-N) units of “effort”?
That’s exactly the content of Grover’s algorithm, and of Andris’s and my spatial search implementation of it.
Comment #122 January 3rd, 2007 at 7:55 pm
Golly, Scott has created the Never-Ending Thread!
(1) With reference to the advertised tenure-track position in quantum system engineering (see the second comment on this thread) our QSE Group’s newly-posted essay What is quantum system engineering? (click here) is posted in the hope of stimulating strong applicants, mainly by assuring folks that we still get to worry plenty about P vs NP when working as engineers.
(2) Ken Regan’s enjoyable chess-related post encourages me to confess that I too occasionally post P vs NP-related material on the computer chess blogs (click here). My own interest arises from geometric descriptions of interval arithmetic, which is a point of view not too dissimilar from yesterday’s Dowling and Nielsen preprint The geometry of quantum computing (click here).
Evidently, we’re all climbing the same mountain. So, Happy New Year to all, and may we all meet at the summit in fine weather!
Comment #123 January 3rd, 2007 at 8:33 pm
As a followup, folks who (correctly but pedantically) object to statements like “chess is NP-hard” are directed to David Eppstein’s excellent website Computational Complexity of Games and Puzzles.
Comment #124 January 4th, 2007 at 12:47 am
As you might know, Dijkstra reached for the same analogy in his famous remark about computer science, that it’s “no more about computers than astronomy is about telescopes.”
Or rather, computer science is no more about computers than human anatomy is about stethoscopes. Which is to say that Dijstra is only half right.
There are certainly people who think that Grover’s algorithm can’t be applied to “physical” databases.
I think that that is a strange question. It depends on the meaning of “can’t”. Yes, you could, in principle, but for what purpose? A classical search in an indexed database can be done in logarithmic (or polylogarithmic) time. If there is a fundamental reason that it cannot be indexed, then I am not sure why it should be called a database. I am sure that Grover’s algorithm should never have been described in terms of “databases”, because it is really an algorithm for combinatorial searches.
The Wolfram front has been pretty quiet.
That’s a relief. You know, the best review of Wolfram’s book is a single extra punctuation mark: A New Kind-Of Science.
I wasn’t sure how to continue the conversation after that.
If nothing else, you could have credited him with great candor. 🙂
Comment #125 January 4th, 2007 at 4:26 am
I talked to Fredkin a few months ago at Perimeter. He said he believes that P=BQP, and that the universe is a classical, nonrelativistic cellular automaton made of pixels many orders of magnitude larger than the Planck scale. I wasn’t sure how to continue the conversation after that.
Wow, Fredkin is a “Shor Shorter” (do you have a standard name already?)—he believes factoring and discrete-log etc. are in P! I guess the conversation didn’t extend to whether physically realizing Grover’s O(\sqrt(N)) time is compatible with his “linear speed-up theorem” (has anyone seen a “proof” of that?), which is kind-of what I was asking.
I meant to clarify that “observing the lottery winner” included doling out the N tickets to begin with. I.e. is “preparing” a Grover-type problem symmetric with the effort of solving one? Call the winning ticket “black”, the rest “white”. Suppose first that you have N different spatial locations, in a square in 2D space or etc. If you have to deal a white or black ticket to each, I guess that means at least N units of effort, counted serially. But if the locations default to “white”—or if the representation of the problem is more implicit—then since the N possible problem setups max at logN bits of information each, it is not a violation for each to require O(sqrt(N)) or less effort to prepare. What good papers address this kind of information-representation / state-preparation issue, where “linear-vs.-quadratic” not “poly-vs.-superpoly” (like in state-preparation papers I know) is the issue?
I guess that’s enough “talking through my hat” for your blog, thanks! Comments on my quantum-basketball skills (as general science-writing) welcomed… And indeed, I’ve referenced Eppstein’s computer-chess notes myself—they seem to be the only ones that clearly say the nearly-50,000 bits used for Zobrist keys (== subset-XOR hashing) should be random. Many chess engines use PRGs of tiny initial entropy to generate the keys (I have one example on my site; there are more), and I speculate whether that plus their chaotic flogging of open-address hash tables might cause funny effects. They are concrete exponential-time algorithms. Note in my CRC Handbook chapter (27) notes with Eric Allender and Michael Loui on my site, right away we cite the concrete form of Stockmeyer’s “cosmological” lower bound (which we are revising from the recent journal version—it’s under 458 bytes of ASCII text), which is the right way of viewing statements like “chess is NP-hard”.
Comment #126 January 4th, 2007 at 5:20 pm
People, Even Physicists, Respond to Incentives…
Scott Aaronson is my kind of physicist. In all the debates about string theory, he finally decided,
From this day forward, my allegiances in the String Wars will be open for sale to t……
Comment #127 January 5th, 2007 at 5:01 pm
Dijkstra reached for the same analogy in his famous remark about computer science, that it’s “no more about computers than astronomy is about telescopes.”
Perhaps the comment is famous because many find it dumb. I think computer science is inextricably tied to computers the same way physics theory is inextricably tied to experiments.
Comment #128 January 7th, 2007 at 7:17 am
Hi Lee,
Sorry for the high-latency reply. I’m aware of the species problem; in fact Ted Jacobson has a good paper on why it isn’t really a problem: http://arxiv.org/abs/gr-qc/9404039.
Basically you avoid the species problem if you believe that gravity is “induced” as in Sakharov’s approach, or by Jacobson’s approach of deriving gravity from thermodynamics of spacetime.
Another way to avoid the species problem is if gravity and matter are somehow unified, so that the number of species cannot vary. This would presumably be the case in something like string theory, or in a model like the one you developed with Sundance and Fotini where matter content is derived from spin networks.
It’s an old calculation (first due to Srednicki, maybe), that if you divide spacetime into two regions in ordinary QFT, the entanglement entropy is proportional to the area of the boundary. This is because you have to cut things off, and the entropy is dominated by the short distance contribution around the boundary. It’s not at all clear that this has anything to do with the usual notion of black hole entropy, however.
I think Srednicki was the first to do this, although there is an earlier paper by Bombelli et. al. where they derive a similar result for a half-plane instead of a sphere: http://prola.aps.org/abstract/PRD/v34/i2/p373_1.
There is also a paper by Rafael Sorkin, where he proves that the entanglement entropy satisfies a generalized second law. http://arxiv.org/abs/gr-qc/9705006. So the entanglement entropy
– Is a well-defined entropy coming from a coarse graining of the spacetime into two regions defined by the event horizon
– scales like the area (assuming a short distance cutoff)
– satisfies the generalized second law of thermodynamics
I would say that the evidence is pretty good that it has something to do with the macroscopic definition of black hole entropy.