*Leonard got it right – and he got the well-deserved reward, too.*

The truth is that everything in the AdS/CFT has to be stringy for both sides of the duality to work and for their equivalence to hold. After all, one of the sides in AdS/CFT is a quantum theory of gravity in an Anti de Sitter space and string/M-theory is the only known consistent quantum theory of gravity – and most likely, the only mathematically possible theory of quantum gravity.

In this blog post, let me mention some of the aspects of the unavoidable "stringiness" of the theories participating in the AdS/CFT pairing. First, the AdS/CFT duality began with the 1997 paper by Maldacena that is approaching 10,000 citations now.

Enriched by some experience with black hole entropy calculations in string theory that had seemed to work "unexpectedly well", better than expected, Juan almost immediately declared that the gravitational theory is equivalent to a non-gravitational, lower-dimensional one. To prove that, he needed to take the branes in the full string/M-theory and to construct the decoupling limit.

The most famous example deals with a stack of \(N\) D3-branes in type IIB string theory. They have 3+1 dimensions in their world volume and 6 transverse spatial dimensions. Yes, the total number of dimensions in type IIB string theory is and has to be 10. The correct, stringy total spacetime dimension may be extracted from the CFTs in the most famous examples of the AdS/CFT pairs – one hint that the CFTs really know about the whole string/M-theory.

The key observation that allowed Maldacena to complete his first, somewhat heuristic proof of the equivalence is the fact that since the mid 1990s, string theory has offered you (at least) two ways to describe the dynamics (degrees of freedom and their time evolution) of this stack. One of them is the perturbative D-brane calculus. The objects embedded in the surrounding flat space are viewed as loci where fundamental strings of string theory are allowed to end. These open strings attached to these D-branes may be excited and their interactions may be computed via the usual rules of perturbative string theory (the world sheets are allowed to have boundaries because we deal with open strings and some of the boundary conditions are Dirichlet boundary conditions because there are D-branes).

The other description of the system boils down to the observation that a large enough stack of D-branes has a pretty large mass and behaves like a gravitational source – really a higher-dimensional counterpart of a black hole, a black \(p\)-brane – that is curving the space around it.

You could think that all the open strings

*as well as*all the wiggles on the curved gravitational field have to be incorporated to describe what may happen to these objects, the stack of branes. However, Maldacena was sensible, original, and ingenious enough to realize that this would be a double counting. In fact, at weak coupling, the open strings are "the whole story". Similarly, at strong coupling, the objects really look like black holes without any visible strings attached, so one only needs the gravitational degrees of freedom.

Andy Strominger would be telling me about his lunch conversations with Joe Polchinski in Santa Barbara sometime in 1994 or so. Strominger would talk about black \(p\)-branes for half of the lunch; Polchinski would discuss something else, the emergent D-branes, for the other half of the lunch. Only when the year 1995 arrived, they have realized that they were talking about the same physical object! ;-)

*Not everyone may be hired as a drunk homeless man in Prague! The same holds for cops in Prague. The main railway station became a concert hall, too. Those who can play Ježek's Bugatti Step impress me a lot.*

A funny thing is that these two descriptions (open strings attached do D-branes; closed strings with a curved gravitational field) may be used for any value of the coupling, so they could be equivalent. Maldacena has presented a "decoupling" argument:

Consider all processes around this object (stack of branes) that occur very slowly in the Schwarzschild time (that's equivalent to low energy).Great. It worked, tests could have been done for the supergravity part of it (correlators of the conformal field theory, or CFT for short, corresponded to scattering in the anti de Sitter i.e. AdS space: Gubser, Klebanov, Polyakov and Witten), and so on. But the derivation used all of string theory and people could continue with tests of other features of string theory beyond the supergravity.

The open-string description of the D-branes at low energies reduces to the lowest-energy excitations of the open strings. In the case of D3-branes, the relevant description is Yang-Mills theory, the dynamics describing the interactions of the massless states of the open strings.

In the gravitational, closed-string description, low-energy or low-frequency processes are those that occur close to the event horizon of the black \(p\)-brane. You know that everything that happens to a guy right before he falls through the black hole event horizon looks increasingly slowed doowwwnnnn... to the observer at infinity.

The equivalence between the two, open-string-based and closed-string-based, pictures survives in the low-energy limit as well. Consequently, the \(SU(N)\) Yang-Mills theory in \(p+1\) dimensions is equivalent to the gravity in the near-horizon geometry of the black \(p\)-brane geometry – and this near-horizon geometry happens to be the \(AdS_{p+2}\) space in the case of extremal \(p\)-branes.

All of the tests have worked so far; the seemingly "non-gravitational" CFT on the boundary was shown to contain all the stringy objects and phenomena that were predicted by string theory. It quickly became clear that by analyzing the emergent gravitational description of the non-gravitational CFTs, one could have discovered all of string theory "from a different angle".

Let me mention several characteristic features of string theory that were found in the seemingly non-gravitational CFTs.

First, Witten has found the wrapped branes of string theory in the CFT: they were given by nothing else than the baryonic operators. Recall that in a proton, the three quarks have different colors so their fields are really contracted against the antisymmetric \(\varepsilon_{abc}\) symbol with three color indices. Operators similar to this "creation operator for the proton" exist in \(SU(N)\) gauge theories (there are \(N\) indices in the antisymmetric tensor) and one may ask what they correspond to in the gravitational language. They are branes wrapped on various cycles of the compact space in the \(AdS_{p+2}\times{\mathcal M}\) manifold. The paper has nearly 500 citations so you may see lots of extra extensions of Witten's observations.

So the CFT contains wrapped branes. But does it also contain strings? You bet.

The clearest demonstration of the presence of excited, vibrating type IIB strings in the gauge theory was the 2002 paper by Berenstein-Maldacena-Nastase (BMN) who found these strings in a limit of the AdS space, the Penrose \(pp\)-wave limit of the geometry. The gauge theory has three complex scalars \(A,B,Z\) and one may write operators of the type\[

{\rm Tr}(ZZZZZZ {\bf A} ZZZZ {\bf B} ZZZZZ)

\] and so on. It is a trace of a long product so the operator looks like a closed string. The general "string bits" in this closed string are given by the complex scalar \(Z\) but there may be impurities such as \(A,B\) – I wrote them in the boldface purely for the sake of clarity. BMN calculated the matrix elements of the Hamiltonian between the states "created" by these trace-like operators and they verified that the Hamiltonian coincides with the Hamiltonian for vibrating type IIB strings in the \(pp\)-wave curved limit of the AdS space (which admits a description using free but massive fields in the light-cone gauge).

*A string of beads resembles the way how closed strings appear in the AdS/CFT correspondence. The strings between the beads are pretty much the QCD flux tubes. Each bead carries two "hands" which correspond to the two indices of the adjoint fields.*

Various people have verified that the BMN picture of the closed type IIB string (the BMN paper is approaching 1,500 citations) also contain the right stringy interactions, see e.g. this paper of ours. All of this works perfectly in the \(pp\)-wave limit but it's clear that the strings continue to exist outside the limit and their behavior and interactions gets deformed to exactly what you would expect in the general AdS space.

(BMN also presented a massive BFSS-like DLCQ model for the \(pp\)-wave background of string theory and I am convinced it is right, too. Lots of interesting maths follow from that.)

It's not just excited vibrating strings and wrapped branes you may find in the CFT. You also find the characteristically stringy shapes of the compactification space. For example, in 1999, Klebanov and Witten started with the analysis of massive deformations of the CFT. They found out that the dual gravitational side involves various orbifolds and conifolds and they may be deformed (and they actually do deform) much like the conifold – a special case of a Calabi-Yau manifold – in string theory. So not only the supersymmetric AdS/CFT pairs have the right total spacetime dimension on the gravitational side, \(d=10\) or \(d=11\) – the latter in the known M-theory examples of AdS/CFT that are known as well. The allowed shapes of the compactified "extra" dimensions agree with what string theorists have known to be possible for quite some time, too.

I could continue for quite some time. You really find everything you have a reason to look for. Whenever some stringy objects or processes can be seen to leave the traces in the strong (gravitational) coupling limit of the CFT, you will find them. This diverse agreement makes us almost certain that even the aspects we can't fully calculate yet almost certainly work properly, too. But I also want to mention several "disclaimers" concerning the claim that the stringiness is absolutely paramount in AdS/CFT.

First, I want to say that the AdS/CFT correspondence has been used to calculate the entropy of various black holes. The "essence" of these calculations only requires one aspect of string theory that you could "separate" from the rest of string theory and treat as a string-theory-independent aspect of AdS/CFT. To count the microstates, one needs the Cardy's formula for the number of highly excited states which is a universal trick in CFTs, regardless of their stringy or gravitational interpretation; and one needs to realize that via AdS/CFT, the master 2D CFT example of that always gets mapped to a three-dimensional BTZ black hole that may be found in the "heart" of almost any black hole whose entropy has been accurately calculated in AdS/CFT.

So the previous paragraph implies that "not all of string theory is needed for one particular calculation". However, if you want to be able to perform "all meaningful calculations", you will need a whole consistent theory – and you will need all insights and objects from string theory, too.

Second, there exist examples of AdS/CFT pairs that "marginally" look non-stringy. My favorite example is the pure 3D AdS gravity linked to a 2D CFT with the monster group symmetry by Witten. This is very interesting but I think that one may still interpret this vacuum e.g. as an "exceptional" compactification of some 27-dimensional bosonic M-theory on something like the 24-dimensional Leech lattice. Because this vacuum is non-supersymmetric, it isn't and cannot be easily linked to the usual \(d=10\) or \(d=11\) vacua of string/M-theory. But otherwise, the theory still smells "stringy". You should surely expect no confirmation of the fantasies of the people who imagine that quantum gravity is something completely different than what string theory dictates it to be.

My second example of a "potentially non-stringy AdS/CFT pair" will have a clearer resolution. There is something called "the Vasiliev theory" or "higher-spin theory" with infinitely many fields of arbitrarily high spins (like in string theory). However, the number of fields isn't growing exponentially with the mass (in string theory, it is). Nevertheless, in 2002, Klebanov and Polyakov proposed that this bizarre quantum-gravity-like theory is equivalent, via the AdS/CFT correspondence, to the critical \(O(N)\) vector model. It is a theory whose main fields don't transform in the adjoint (matrix) representation of the group like in the most notorious AdS/CFT example; instead, they transform in the fundamental (column) representation.

For years, I've been puzzled by the question whether these folks would claim that the Vasiliev theory is a consistent theory of quantum gravity unrelated to string theory. It had to be either inconsistent, or non-gravitational, or it had to be a vacuum of string theory, I thought. Needless to say, I was right. In 2012, Xi Yin, Shiraz Minwalla, and Chi-Ming Chang and Tarun Sharma who co-wrote their paper showed that the Vasiliev theory and/or the vector model may be obtained as a limit of type IIA string theory compactified on \(AdS_4\times{\mathbb C \mathbb P}^3\) which allows them to prove the equivalence with the help of some usual stringy arguments. The understanding of the origin is good enough that the authors even guess where all the terms in the Vasiliev equations come from; the right hand sides probably arise from the world sheet instantons. It just happens that this AdS/CFT pair was also linked to the ABJ theory found in another recent development, the membrane minirevolution.

For the single monstrous CFT, such a stringy construction may be not known equally explicitly but because it has worked in all other examples, one should say that it is extremely likely that all consistent examples of the AdS/CFT correspondence may be shown to be vacua of string/M-theory and all the objects and processes predicted by string/M-theory may be found in these holographic definitions of quantum gravity, too. Quantum gravity and string/M-theory are ultimately synonyma and the AdS/CFT correspondence has provided us with thousands of pieces of new circumstantial evidence that this is the case.

Incidentally, the Physics Stack Exchange question mentioned the "Rehren duality". Karl-Henning Rehren's "holography" mentioned in another answer over there is a vacuous pseudoscience that has nothing to do with the depth of the actual holography in quantum gravity. What is done in that paper is just rewriting fields \(\phi(x^0,x^1,x^2,x^3,x^4)\) as \(\phi_{x^4}(x^0,x^1,x^2,x^3)\) i.e. rewriting one of the coordinates as an "index" of the fields, and claiming that coordinates may be reduced by one (or any number, for that matter). However, nothing changes about the physical "number of dimensions" by this sleight-of-hand. See some TRF blog posts about Rehren, especially one on the temptation of rigor.

Previous articles on a similar topic:

Unity of strings (2004)So please, if you hear a Sh(mo)ithead talking about the AdS/CFT correspondence's not being dependent on string/M-theory again, reach for your firearm and press the trigger. Let me emphasize that this is no manslaughter, it is self-defense.

Unity of strings (2006)

Two roads from \(\NNN=8\) SUGRA to string theory (2008)

Are AdS/CFT and AdS/CMT relevant for unification? (2009)

Why unification sits at the core of string theory (2010)

Unification as a source of certainty (2011)

Unity and uniqueness of string theory became a heresy (2011)

Just to make sure... check that gun barrel is not pointing ones own head while squeezing the trigger :-) Otherwise it might sting a bit...

ReplyDeleteHahaha, so Andy Strominger and Joe Polchinski were NOT talking past each other during lunch after all, very funny story LOL :-D !

ReplyDeleteAnd I just love the nice "course grained" explanation of how Maldacena proofed the AdS/CFT business... :-)

As one can see from one of my earlier Physics SE questions, I once got confused to by people saying or writing that on the AdS side you just need "a" quantum gravity which can be supergravity too, so the nice examples of what stringy effects are there on the CFT side are good for me to read too ;-)

Going to the corresponding question now to vote ...

Cheers

Having been peripherally involved in the ABJ triality paper, I have a question. As far as I understand it, Prof. Shiraz tells me that one still has to 'complete' the construction of the higher spin fields (say as a perturbation theory in M/N where M & N are the ranks of the two unitary gauge groups in ABJ and as M approaches N, we get ABJM, sorry for being unclear) as another corner of the string/M-theory space. I mean, it hasn't yet been shown that Vasilev theory (in the bulk) indeed arises as what was conjectured in the paper, isn't it?

ReplyDeleteDear Umesh, I agree, the proof isn't complete but they have presented diverse circumstantial evidence that the type IIA strings decay into string bits that are created by the Vasiliev fields.

ReplyDeleteNice article Lubos. Just for my understanding, I would like to clarify one point: definition of strong and weak coupling. Is the definition in ST same as QFT where roughly g~1 would be strong and g<<1 would be weak?

ReplyDeleteYes, it's the same thing as long as "g" represents the (dimensionless) string coupling.

ReplyDeleteA theory is weakly coupled if the higher-order corrections in the perturbation theory are (much) smaller than the lower-order (or leading) terms.

If there are many colors, the actual coupling that decides about the being strongly coupled isn't just g^2, it is g^2*N, because the number of colors N appears as an extra factor with each loop. So if g^2*N is (much) greater than one in a gauge theory with adjoints, it is strongly coupled.

Similarly, there may be more complicated operators in BMN for which some expansion parameter isn't 1/N for the planar expansion but J^2/N or something like that, and so on.

I look forward to reading this extensive post. I ran into a couple of recent articles yesterday that are stated as strong new quantitative evidence supporting Maldacena.

ReplyDeletehttp://www.huffingtonpost.com/2013/12/11/universe-hologram-physicists_n_4428359.html

I hope to be able to read the there linked articles more fully too but am pretty sure that the "possible" in the headline should be removed.

Thanks, I think your translation is superb. Do you actually translate literature? If so, I think it’s harder than doing mathematics.

ReplyDeleteAs for Reich-Ranicki’s politics, I am now sure you are right. It’s been a long time since I read “The Author of Himself” and I have forgotten most of it. I looked at it again and now I see that I was somewhat affected by a few comments and episodes that remained more strongly in my memory that the rest of the book, probably because they struck some particularly sensitive chord. One was the downplaying of Stalinism. One was the remark: “But I venture to point out, in all modesty, but with a hint of satisfaction that I never devoted a single article to a famous, frequently admired and highly praised German prose writer. I am referring to Ernst Junger. His work is alien to me. I feel entitled to keep silent.”

Well, he is entitled, of course. But there are many other authors he is happy to discuss whose work is equally alien to me. One was Theodor Adorno, whom Ranicki apparently somewhat admired. To me, on the other hand, his work was as alien as possible, and I am not sure whether I despised him more for being the teacher and spiritual father of Herbert Marcuse or for his views of music, which belongs together with those of Comrade Zhdanov, though of course fortunately without the ability to cause any real damage. The third thing was, I guess, what seemed to me at that time, Ranicki’s one sided account of the break down of his relations with Joachim Fest.

But now having quickly re-read these passages, my impressions are different. In fact, Ranicki says clearly that he owned more to Fest than to any any other man in that stage of his life, and that the break was a great pain for him: “ The man whom I owe the most gratitude also inflicted the most pain on me. I cannot stop questioning it, I cannot forget it, neither the one nor the other.”

In the end it all comes back to the old Historikerstreit (http://en.wikipedia.org/wiki/Historikerstreit ). In it Ranicki found himself siding with a lot of left wing historians and the marxist philosopher Habermas, and probably this left in my memory the impression that belonged to a leftist “intellectual circle”. Actually, re-reading the memoir again this no longer seems true of his literary criticism, so now I withdraw that statement.

Re-reading Ranicki’s story made me remember something. Here is a link to a page where one can buy a DVD with the film about my uncle, which I wrote about once on the occasion of my trip to the Ukraine.

http://www.artfilm.ch/aus-galizien-in-den-aargau-dvd&lang=en

The DVD costs 35 franks so I only would recommend it to people for whom this is a negligible amount and who have some special reason to be interested in these kind of Holocaust related stories. This is by no means great cinema and one can certainly buy better films for this amount. However, the story itself is not less interesting than Ranicki’s.

Yes, my thoughts exactly! I wanted to ask you what you'd think of a 'proof', which I wanted to outline as the M/N perturbation theory - which was also suggested by Prof. Shiraz. I wanted to know whether you think this would only be of some technical interest, because I have attempted to look into making the details concrete, but it takes some doing.

ReplyDeleteThanks for these nice explanations Lumo, and for supporting my Quora question :-)

ReplyDeleteThis is very nice of you

Sorry for the drag, but the Klebanov-Polyakov duality is still valid, and by itself can't have a stringy embedding, right? I would imagine that the correct supersymmetric Vasiliev theory was what the ABJ triality paper discusses. Should one construe the K-P example as a symptom that AdS/CFT is more 'general' in some sense? I want to know this because many (knowledgable) folks I know use this example (to argue that somehow AdS/CFT is 'more general', which surely contradicts my opinion), and I would love to have your opinion about it. Thank you very much.

ReplyDeleteI will put the DVD on my things-to-buy list. Agree entirely with your remarks on Adorno, both the politics and the abuse of music. Ernst Jünger was an oddity, I don't know what to think of him. Thanks for your kind words on my translation :)

ReplyDeleteVery interesting comments about Polish poetry. I had a feeling that it is indeed truly remarkable. I remember reading an obituary of a clearly outstanding Russian poet who thought that his major purpose in life is to translate one (rather modern) Polish pet. So he spent most of his life translating all the poetry of that Polish poet; apparently, he produced very fine translations, but they remained unread by the general public. (I forgot both names.) Another example is Philip Rosenau (http://en.wikipedia.org/wiki/Philip_Rosenau), an Israeli mathematician. I know him a little bit, and he is probably much more a poet than a mathematician. I think he was born in Poland after the WWII, but left it very young. He avoided coming back to Poland (as Ranicki and for the same reasons) and he has written a lot of poetry in Hebrew, but eventually gave up and now (I believe) he writes poetry mostly in Polish, he visits Poland for literary events... I think his poetry is known in Poland, but hardly known in Israel.

ReplyDeleteI used to have a big tome of Mickiewicz (translated into Russian), but hasn't read... I can imagine that he is untranslatable, I don't think Pushkin is translatable either. By the way, I would not call the latter a Romantic poet. He started as such, but the majority of his work does not seem Romantic. OK, perhaps it was a push to consider Pushkin as an earlier realist, ideologically this seemed more ``advanced''. By the way, Pushkin influenced Russian language more than any other Russian writer; this is a common slogan, yes, but it does seem correct. Can you say the same about Mickiewicz?

Dear Umesh, the Klebanov-Polyakov is still valid but it *is* embedded into string theory by Shiraz et al. - both sides are embedded - at least the N=6 case.

ReplyDeleteRight, I get it now. Thanks, I see what you're saying.

ReplyDeleteLOL :-D

ReplyDeleteEugene,

ReplyDeleteThere's a little of the sadist in you. I must say it's not a very endearing trait.

Look, there's no need for cruelty — you can leave that sort of thing to the continentals. If these people are bothering you my advice would be to dispatch them quickly and have done with it.

Now let's hear no more of it, there's a good fellow.

:)

Hi Lubos: Off topic for this blog, but in the line of stuff you are talking about. In connection with divergent series, you pointed out the fascinating result that

ReplyDeleteD= 2-2/(1+2+3+...) gives 26 for bosonic strings if you interpret with very famous mathematicians that the series is actually -1/12. Can you quote a similar equation which gives D=10 for a fermionic or SUSY case if it is not too involved for the blog?

Dear Kashyap, the calculation of D=26 above is based on the light-cone gauge and the squared mass of the first excited level.

ReplyDeleteIn SUSY theories, the first excited level is kind of manifestly massless, due to Bose-Fermi cancellation, so I think that a very analogous calculation of the critical dimension doesn't work there.

In the RNS formalism, there is an analogous calculation. Conformal ghosts carry c=-3k^2+1 and we have k=3 for the bc ghosts, giving c=-26 which is cancelled by 26 bosons.

For the superstring, there are also bc ghosts with c=-26 but there are also c=+11 beta-gamma superconformal ghosts with k=2 and a sign flip because these ghosts are bosons. This gives -26+11=-15 from ghosts which is cancelled by 10 from 10 dimensions and 10/2 from their fermionic RNS superpartners.

There are also other conditions to at least constrain D=10 from spacetime SUSY. It's no coincidence that D=10 obeys

D-2 = 2^((D-4)/2)

Dear Lubos, if in fact our Lopsided Universe is an Open Universe (http://physics.aps.org/articles/v6/98) would space-time really be AdS ? What would be the implications for cosmology of such AdS/CFT scenario? Thanks

ReplyDeleteNice post, Lubos, but I don't think the stringy embedding of the Klebanov-Polyakov conjecture is clear yet. It has been proposed that the 3-dimensional O(N) vector model has to be coupled to O(N) Chern-Simons theory in order to impose the singlet constraint. Then the dual theory should include a topological closed string sector that is dual to the Chern-Simons theory. So, there are speculations that the Vasiliev theory is a kind of open string sector that needs to be coupled to a topological closed string sector, but nobody has yet written down such a theory. Also, it appears that O(N) vector models in dimensions greater than three, in the singlet sector, are also dual to Vasiliev type theories in AdS_5 and higher dimensions (some checks of this were made in a recent paper by Giombi, Klebanov and Safdi). Not much is yet known about these higher dimensional Vasiliev theories and their relations to string theory.You may be right that all these higher spin theories are related to string theory, but it could be a rather different kind of string theory than what we are used to.

ReplyDeleteInteresting! But if the closed string sector is topological, does it affect the local dynamics of the Vasiliev theory at all? If it only affects some global behavior or boundary conditions, then let the sector be added as string theory wishes, right? "The" Vasiliev theory itself doesn't make any preferences about those issues, does it?

ReplyDeleteLubos why you didn't mention the GKP string?

ReplyDeletehttp://arxiv.org/abs/hep-th/0204051

Higher spin theory must be some (tensionless) limit of String theory.

There is a vast literature about the issue

Tom

ReplyDeleteLubos,

ReplyDeleteAs one holding the view that our mathematics is solely the creation of our skull’s neurons, existing without reference to any metaphysical notions (Platonic forms, etc.), the criticism that string theory relies over much on model esthetics seems somewhat valid to me. With the insufficiency of accelerator energies likely true, what are your thoughts on the likelihood of the CMB’s fine structure providing strong empirical evidence for string theories?

I'm more interested why you insist to. Are such a fool of yourself?

ReplyDeleteI don't know what you can possibly mean by the comment about the neurons - something that isn't obvious nonsense.

ReplyDeleteMathematics may have been written down by mathematicians whose neurons worked but its rules are completely rigid and therefore absolutely impersonal and universal. Another civilization that has never communicated with ours will have the same list of primes, and so on, and so on.

We are really discovering mathematics, just like we are discovering other planets and continents. This is a basic philosophical point. But even if you choose some completely different philosophical attitude than the "Platonism" of pre-existing maths, you should be able to see that from these vague and superficial comments, it is extremely far to invent a criticism of string theory. String theory isn't philosophy. String theory is a very particular rock-solid mathematical structure of laws and objects optimally ready to describe a physical Universe.

Any criticism that only builds on some vague speculations on maths' being juust in neurons that suddenly morphs into a criticism of string theory is self-evidently a proof of a brain defect of the critic, isn't it?

Because I forgot about it, Giotis, and it wasn't filling too much of my brain at any earlier time, anyway. Thanks for the addition. ;-)

ReplyDeleteOk then:-)

ReplyDeleteBTW can you decipher Douglas's answer in this question in the video below? (check 1:06:40)

http://workshops.ift.uam-csic.es/WMH126/discussion.html

Lubos,

ReplyDeleteBe sure to check that none of your derivations rely on the axiom of choice or transfinite induction at any point. Some in the constructivist camp, who are far from idiots, raise strong doubts about such blithe passage from empirical to mathematical.

Because 1 + 2 + 3 +... does not equal -1/12.

ReplyDeleteWhen we say 1 + 2 = 3, and 1 + 2 + 3 +... = -1/12, does the "+" symbol in both equations mean the same thing? If so, then the above infinite sum diverges and does not equal -1/12.

If 1+2+3+... = -1/12 is needed to satisfy some physics theory, then perhaps the physics theory (i.e. string theory) is not quite right and/or complete.

When we say 1 + 2 = 3, and 1 + 2 + 3 +... = -1/12, does the "+" symbol in both equations mean the same thing? If so, then the above infinite sum diverges and does not equal -1/12.

ReplyDeleteIf 1+2+3+... = -1/12 is needed to satisfy some physics theory, then perhaps the physics theory (i.e. string theory) is not quite right and/or complete.

fantastic :D is there amethod to prove the zeta regularization for 1+2^{m}+3^{m}+....

ReplyDelete