Physicists fly high on the idea that our three-dimensional world is actually two-dimensional, that we live in a hologram, and that we’re all projections on the boundary of space. Or something like this you’ve probably read somewhere. It’s been all over the pop science news ever since string theorists sang the Maldacena. Two weeks ago Scientific American produced this “Instant Egghead” video which is a condensed mashup of all the articles I’ve endured on the topic:
To begin with, physicists haven’t believed this since Minkowski doomed space and time to “fade away into mere shadows”. Moyer in his video apparently refers only to space when he says “reality.” That’s forgiveable. I am more disturbed by the word “reality” that always creeps up in this context. Last year I was at a workshop that mixed physicists with philosophers. Inevitably, upon mentioning the gauge-gravity duality, some philosopher would ask, well, how many dimensions then do we really live in? Really? I have some explanations for you about what this really means.
Q: Do we really live in a hologram?
A: What is “real” anyway?
Q: Having a bad day, yes?
A: Yes. How am I supposed to answer a question when I don’t know what it means?
Q: Let me be more precise then. Do we live in a hologram as really as, say, we live on planet Earth?
A: Thank you, much better. The holographic principle is a conjecture. It has zero experimental evidence. String theorists believe in it because their theory supports a specific version of holography, and in some interpretations black hole thermodynamics hints at it too. Be that as it may, we don’t know whether it is the correct description of nature.
Q: So if the holographic principle was the correct description of nature, would we live in a hologram as really as we live on planet Earth?
A: The holographic principle is a mathematical statement about the theories that describe nature. There’s a several thousand years long debate about whether or not math is as real as that apple tree in your back yard. This isn’t a question about holography in particular, you could also ask that question also in general relativity: Do we really live in a metric manifold of dimension four and Lorentzian signature?
Q: Well, do we?
A: On most days I think of the math of our theories as machinery that allows us to describe nature but is not itself nature. On the remaining days I’m not sure what reality is and have a lot of sympathy for Platonism. Make your pick.
Q: So if the holographic principle was true, would we live in a hologram as really as we previously thought we live in the space-time of Einstein’s theory of General Relativity?
A: A hologram is an image on a 2-dimensional surface that allows one to reconstruct a 3-dimensional image. One shouldn’t take the nomenclature “holographic principle” too seriously. To begin with actual holograms are never 2-dimensional in the mathematical sense; they have a finite width. After all they’re made of atoms and stuff. They also do not perfectly recreate the 3-dimensional image because they have a resolution limit which comes from the wavelength of the light used to take (and reconstruct) the image. A hologram is basically a Fourier transformation. If that doesn’t tell you anything, suffices to say this isn’t the same mathematics as that behind the holographic principle.
Q: I keep hearing that the holographic principle says the information of a volume can be encoded on the boundary. What’s the big deal with that? If I get a parcel with a customs declaration, information about the volume is also encoded on the boundary.
A: That statement about the encoding of information is sloppy wording. You have to take into account the resolution that you want to achieve. You are right of course in that there’s no problem in writing down the information about some volume and printing it on some surface (or a string for that matter). The point is that the larger the volume the smaller you’ll have to print.
Here’s an example. Take a square made out of N2 smaller squares and think of each of them as one bit. They’re either black or white. There are 2N2 different patterns of black and white. In analogy, the square is a box full of matter in our universe and the colors are information about the particles in the inside.
Now you want to encode the information about the pattern of that square on the boundary using pieces of the same length as the sidelength of the smaller squares. See image below for N=3. On the left is the division of the square and the boundary, on the right is one way these could encode information.
There’s 4N of these boundary pieces and 24N different patterns for them. If N is larger than 4, there are more ways the square can be colored than you have different patterns for the boundary. This means you cannot uniquely encode the information about the volume on the boundary.
The holographic principle says that this isn’t so. It says yes, you can always encode the volume on the boundary. Now this means, basically, that some of the patterns for the squares can’t happen.
Q: That’s pretty disturbing. Does this mean I can’t pack a parcel in as many ways as I want to?
A: In principle, yes. In practice the things we deal with, even the smallest ones we can presently handle in laboratories, are still far above the resolution limit. They are very large chunks compared to the little squares I have drawn above. There is thus no problem encoding all that we can do to them on the boundary.
Q: What then is the typical size of these pieces?
A: They’re thought to be at the Planck scale, that’s about 10-33 cm. You should not however take the example with the box too seriously. That is just an illustration to explain the scaling of the number of different configurations with the system size. The theory on the surface looks entirely different than the theory in the volume.
Q: Can you reach this resolution limit with an actual hologram?
A: No you can’t. If you’d use photons with a sufficiently high energy, you’d just blast away the sample of whatever image you wanted to take. However, if you loosely interpret the result of such a high energy blast as a hologram, albeit one that’s very difficult to reconstruct, you would eventually notice these limitations and be able to test the underlying theory.
Q: Let me come back to my question then, do we live in the volume or on the boundary?
A: Well, the holographic principle is quite a vague idea. It has a concrete realization in the gauge-gravity correspondence that was discovered in string theory. In this case one knows very well how the volume is related to the boundary and has theories that describe each. These both descriptions are identical. They are said to be “dual” and both equally “real” if you wish. They are just different ways of describing the same thing. In fact, depending on what system you describe, we are living on the boundary of a higher-dimensional space rather than in a volume with a lower dimensional surface.
Q: If they’re the same why then do we think we live in 3 dimensions and not in 2? Or 4?
A: Depends on what you mean with dimension. One way to measure the dimensionality is, roughly speaking, to count the number of ways a particle can get lost if it moves randomly away from a point. The result then depends on what particle you use for the measurement. The particles we deal with will move in 3 dimensions, at least on the distance scales that we typically measure. That’s why we think, feel, and move like we live in 3 dimensions, and nothing wrong with that. The type of particles (or fields) you would have in the dual theories do not correspond to the ones we are used to. And if you ask a string theorist, we live in 11 dimensions one way or the other.
Q: I can see then why it is confusing to vaguely ask what dimension “reality” has. But what is the most confusing thing about Moyer’s video?
A: The reflection on his glasses.
Q: Still having a bad day?
A: It’s this time of the month.
Q: Okay, then let me summarize what I think I learned here. The holographic principle is an unproved conjecture supported by string theory and black hole physics. It has a concrete theoretical formalization in the gauge-gravity correspondence. There, it identifies a theory in a volume with a theory on the boundary of that volume in a mathematically rigorous way. These theories are both equally real. How “real” that is depends on how real you believe math to be to begin with. It is only surprising that information can always be encoded on the boundary of a volum if you request to maintain the resolution, but then it is quite a mindboggling idea indeed. If one defines the number of dimensions in a suitable way that matches our intuition, we live in 3 spatial dimensions as we always thought we do, though experimental tests in extreme regimes may one day reveal that fundamentally our theories can be rewritten to spaces with different numbers of dimensions. Did I get that right?
A: You’re so awesomely attentive.
Q: Any plans on getting a dog?
A: No, I have interesting conversations with my plants.
Are you beating about the bush again Sabine? Perhaps I can be a little more direct:
ReplyDeleteNo, we don't live in a hologram. Really.
That's Emperor's New Clothes woo. It is popscience quackery that is right up there with time travel and the multiverse. Do not fall for it for one moment.
"Do we really live in a metric manifold of dimension four and Lorentzian signature?"
ReplyDeleteNote that a manifold is a kind of space that can be cut up into pieces which have a certain kind of one-to-one correspondence with pieces of R^n. "One-to-one correspondence" is by no means the same as "is the same thing". So for example you could define spacetime to be the set of all events, where "events" means "actual happenings", eg "the assassination of JF Kennedy" or "Sabine arrives in Stockholm". From that point of view, it is quite easy for me to believe that we indeed inhabit a 4-dimensional manifold with a geometric structure.
"There’s 4N of these boundary pieces and 24N different patterns for them. If N is larger than 4, there are more ways the square can be colored than you have different patterns for the boundary. This means you cannot uniquely encode the information about the volume on the boundary.
ReplyDeleteThe holographic principle says that this isn’t so. It says yes, you can always encode the volume on the boundary. Now this means, basically, that some of the patterns for the squares can’t happen. "
I have a problem with this counting argument, in that it immediately fails once the number of squares becomes infinite. The cardinality of Aleph-Null is the same as the cardinality of Aleph-Null x Aleph-Null, etc., if I'm not mistaken. So maybe the holographic principle says that infinities actually occur in nature.
Arun,
ReplyDeleteInteresting point, though I suppose you could take the limit of the ratio to evade it. Best,
B.
John,
ReplyDeleteI'm not 'beating about the bush', I am extracting the scientifically accurate origin of the inevitably misleading statement 'we live in a hologram'. This is interesting physics and just saying this wording is nonsense doesn't do it justice. Best,
B.
Rastus,
ReplyDeleteA metric manifold (I should have added differentiable) is more than a set of events. A local one-to-one correspondence is of course not the same as a global one in case that's what you're trying to say. Best,
B.
Is organic chemistry 3-D? Cubane, dodecahedrane, and all sorts of buckeybubbles enclose volume. They can all be stretched flat into 2-D Schlegel diagrams in which no skeletal bonds cross. Of some 96 million molecules in Chemical Abstracts, about a dozen have crossing Schlegel diagrams and are therefore not topologically flat.
ReplyDeleteFalsification of the holographic universe is then construction of an object (math will do) that cannot be encoded by a hologram, that cannot be constructed with Fourier transforms. Let's have a nice, juicy pathological function, folks.
Addendum - challenge the holographic universe with Kuratowski's theorem.
ReplyDeleteThe state of "mathematical reality" as the state of physics can be mapped rather simply whereas the interpretations within a less general view grow exceedingly complex.
ReplyDeleteOne soul's "time is the fourth dimension " as a physical statement is another soul's philosophic statement.
It is as simple as uncertainties in basic language programing as the nodes over paths of matching faces in a game of dominoes and deciding if the "go to " instruction explains gravitation.
Yes, the issue is what is boundary in a boundless universe where point ideal singularities address path linear continuity in terms of adjacent dimensions involving real structures as questions of the continuum hypothesis. But more is going on.
The shortest action or distance from one point or pixel cell linearly to another of n (say 4) steps in n axes, the initial pixel arbritary, suggests an open or closed loop n+1 hierarchy.
This "lost comet " principle can be further complicated that in each point in say 16 such pixels the same path structures may exist at each pixel and so on. The pattern can run parallel or range to a sort of random differences.
In this time of fragmenting information systems in the evolving internet (or what is still hidden in our biological codes including thinking systems, there should be a unified translator between all bases that ground a computer system that we can find a reasonable unified physics and better define our terms keeping the dimensions straight.
It is about time these questions were raised as Sabine has done in this blog post.
Thank you for useful blog.I thought there was a ongoing experiment to show if we are in a hologram, but I cannot find any information on it.
ReplyDeleteActually, the situations when fields on a boundary determine the fields within the bounded regions are quite common. See for example the Cauchy integral formula, or more general, the Cauchy-Pompeiu formula.
ReplyDeleteIn general, think at boundary value problems. This works for Maxwell's equations, but also for other gauge fields and for gravity (when restricted to 3D space). So it seems that this is the rule.
Of course, one may think that this wouldn't work when thinking of black and white squares. But in this case partial differential equations, which impose constraints to the admissible fields, should be replaced by some constraints imposed to the admitted patterns of squares.
Cristi
ReplyDeleteif the number of points on a line segment equal that of an n dimensional volume made into a grid of cubes which are black and which white may not have the same color between dimensions. Is this a restraint as math or physics? 3D tic tac toe has to be played not getting three colors in a row to win.
Or is it constrained by our interpretations?
If as some assert time is an emergent illusion in 3D and not constrained as some motions not constructable or vanishing from that completely described by information in surfaces inside a hologram (alternatively within a black hole)
From eternity in a sea shell to a universe in an atom of iron to Sagan cosmos of a universe inside an electron to Tyson's change of view as to what happens when we fall into a black hole, the last cosmos picture of universes in universes and so on looks like the one here more up to date with new CMB data and speculations.
The real problem is do lines on the boundary describe completely lines Inc a plane where there can be more curves than lines so at least from human evolved perception the boundary may be a fuzzy mix with the area so may not exist at all.
Uncle Al, again good points and that theorem you mentioned which has rare physical effects organically, as rare as protons exchanged at a QM distance in organic structures does not apply to projective planes we do not know enough about and certainly does not shore up GR distance either.(I am not saying you concerns with chirality does not need deeper investigation.)
In all this the interpretation of intervals as time or even pair production we have not clearly defined nor how linear information where it is naturally Euclidean physics does equal information in both boundary and that it contains.
Sabine, I hope my posts here are clear enough. The fluid and dynamic patterns over defects persist so for me the problem seems more to explain such physical irregularity in nature rather than uniform flatness. It seems to me statements as to where Planck scales apply are really talking about different intervals within the same partial models of space, conceptually different where neutral secquences reduce the same.
The true significance of holographic model can be illustrated by its testable predictions and real observable effects. For example the shape of some nebulae resembles the shape of atom orbitals, the shape of dark matter fibers resembles the dark flow at the boundary of observable Universe, etc. These effects are generally weak because the holographic model gets broken with extradimensions and thus remains limited to few low dimensional examples of it.
ReplyDeleteNo, I'm not worried about the global/local distinction. What I am driving at is that when people say the word "manifold" they often have in mind a picture of something very abstract, consisting of tiny little structureless things, unlike the world in which we live, where *events* can be very complicated things like the assassination of JFK. But in fact the definition of a manifold is not like that at all: it's just a special kind of set with certain rules. The elements of that set can be arbitrarily complicated. To take a simple example, consider the bundle of frames F(M) over a manifold. The elements of F(M) are themselves complicated things -- they are bases of the tangent spaces at points in M. And yet F(M) is a manifold, and so is F(F(M)) etc. The elements of F(F(F(M))) are very complicated things indeed, but it's still a manifold.
ReplyDeleteSo the spacetime of our experience could easily *be* a manifold. As for the metric structure, that is natural: of course there is some sense in which the assassination of JFK is farther away in space and time than "Sabine's first arrival in Stockholm", and it is natural that there should exist a quantitative measure of that sort of fact.
All this is relevant to the question: do we live on the boundary or in the bulk? Well, the spacetime of our experience is certainly not asymptotically AdS, so we certainly do not live in the bulk. We might live in a [sufficiently complex] boundary, [leaving aside possible topological complications!] but we don't know for sure. If our universe can be represented as a boundary, then we live on the boundary; the [supposed] fact that there is a mathematical duality with physics in an AdS bulk is interesting and important, but doesn't change the fact that we *don't* live there.
Somehow it is difficult to be properly grateful for this well-written, interesting, and humorous essay while knowing that you will probably never publish a book of such essays; but I think I can do it.
ReplyDeleteRastus,
ReplyDeleteif we think of space as a topology of manifolds the question of distance between two events may be no more distinguishable than what is a manifold. We evidently live in low dimensions as a fact intrinsic to nature. Riemann knew his boundaries in what in math he did not know and left how his ideas would be applied to physics to future researchers.
There are facts on JFK by the way that are solid and not debates of fading holographic emerging conspiracy.
Peter Spit has essays with deep comments from our constellation of characters that compares speculations
on the vortex theory era to string theory speculation including knot theory within the boundaries of what we can know, plausibly pursue, or test that suggests models ruled out beyond their day.
Can we not imagine and compute as distance so many meters in the "time " direction? Can the surface volume of a hypercube not be seen as a torus that is dynamic also if a vortex? Do we live in that boundary volume?
Where is the firewall some interpret as beyond space or are they concrete walls that isolate a society into a vital mystical dimension of partioning say as in the historical interpretation of ancient right to the West Bank. Who remembers the Etrucians?
Solutions if solvable could begin with new models of physics for world problems.
I imagine units or concepts like h or c or distance may be described as central values of scale, not just minimums or maximums
but that too for now is pure speculation.
Some clarifications according to my view in order to avoid misunderstandings:
ReplyDeleteThe holographic principle stems from the Bekenstein-Hawking area formula for entropy and implies quite directly the holographic bound whenever gravity is included. Whenever gravity is included the degrees of freedom are scaled according to the surface area of the boundary.
And that's why *every* theory of QG is struggling to reproduce the Bekenstein-Hawking area formula.
Due to the fact that this contradicts the usual QFT counting of degrees of freedom it has become a benchmark for *all* theories of QG not just String theory. Every theory of QG should provide a realization of the holographic principle if they want to be taken seriously.
String theory did just that; it provided a realization of the holographic principle for spacetimes with an AdS boundary i.e. AdS/CFT.
For the same reason critics of String theory want to decouple AdS/CFT from String theory and advocate that AdS/CFT is quite general and applies to every QG not just String theory i.e. the QG in the bulk could be any theory of QG and not just String theory.
So overall we should not give the impression that this is some peculiarity of String theory, it has become basically a requirement that every theory of QG should comply to.
driod33,
ReplyDeleteThere is the Holometer, in case that's what you mean. Best,
B.
Yes that's it thank you.
DeleteIt appears to have gone quiet.
Giotis,
ReplyDeleteI basically agree with you. The only other option is to discard the strong interpretation of the BH entropy. That brings its own problems and it is arguably not a very popular option. Best,
B.
@Giotis "Every theory of QG should provide a realization of the holographic principle if they want to be taken seriously."
ReplyDeleteTaken seriously by who? String Theorists?!
Experimental verification is the only final word
Also notice how Q is structuring his/her questions to force Sabine to accept holography. And who by the way elevated this conjecture to a principle?!This is what String theorists do. Tgeys seek wider acceptance by the physics community and use threats like " you will not be taken seriously if you do not accept this" .The Soanish inquisition conss to mind with such behavior.Only exoerimental verification gives the verdict!
ReplyDeleteWell said Stuart.
ReplyDeleteAre we still fighting the Carlsberg Pilsner SUSY wars for the landscape of gravitation. Lubos Motl you must be reading this to have such a passionate reply and demeaning personal attack on Sabine and her sensible take on theory. So in my Facebook status I have a personal memo for you. She evolves and adapts while your evolution seems frozen. It is clear to me that you do not understand string theory nor where holography may legitimately.apply. Nor simple topology. And I am just a lay person.Let me school you. Your representation of vibrations of a tetrahedron assuming it has a center of some description or not has 4D 15 of 16 possibilities. Is one doubled in the center as if a super positioning or is it zero so in a sense non existing? And this is the simplest case of 1+1 to the nth of self dual structures embedded in a much wider reference frame. This BTW comes from an IBM survey of microwave molecular vibrations.Like Sheldon said "I don't feel sorry that I understand,but I do feel sorry they do not understand"(more or less phrased). If Sudoku was invented by Czecks how can you not see this no matter which way you rotate Feynman diagrams. Well maybe you are right about global warming so be careful entering the Texan horizon you don't find yourself in a fundamentalist hotter place.
ReplyDeleteHas anyone considered that since a BH is considered a quantum object maybe the reason for that can be understood through dimensional analysis? As an example, perhaps gravitational interaction IN ANY NEUTRALLY CHARGED PARTICLE is due to the suppression of degrees of freedom, that is, a change to storage of information in just 2 dimensions. One could view the holographic principle in a BH as the information being stored strictly in its angular momentum.
ReplyDeleteThis is not an excuse to say we live in a holographic universe. It just means that under special circumstances, neutrally charged combinations of particles cause all degrees of freedom EXCEPT angular momentum to be inhibited.
Sorry Sabine for going back on my word of not making comments here. Your posts make it irresistible to me.:-)
After upgrading Firefox, I was surprised and annoyed to find that RSS bookmarks in tabs no longer worked, though apparently there is a reason for this. I then installed some add-on to give me the old functionality, and more. One feature is that one can hold the mouse over a link to a post or comment and get a preview of the text. This is sometimes useful if one doesn't want to read through all comments, but first check to see if they are interesting. Of course, lack of context can sometimes be confusing. Thus, when reading the previous comment, I was at first confused since BH is a comman abbreviation for Büstenhalter (bra) in German. Substitute "bra" for "BH" or "black hole" in the previous comment for a laugh.
ReplyDeleteDo we live on planet Earth as really as, say, we live in our own Mind? This is a very interesting topic of discussion; thank you kindly for it. Most days I trend towards the Platonist ideal but I’m a big fan of University of Chicago mathematician Louis Kauffman, and certain of his works introduced me to radical constructivism with its Epistemological Corollary:
ReplyDelete“Reality is neither rejected nor confirmed, it must be considered irrelevant.”
How beautiful is that? So even though I occasionally find myself in disagreement with Radical Constructivists, I have found many of their works beneficial to my own thought processes. Some recommended reading perhaps?
From one of the early proponents of RC, Ernst von Glasersfeld, “A Constructivist Approach to Experiential Foundations of Mathematical Concepts Revisited” (http://www.univie.ac.at/constructivism/journal/articles/1/2/061.glasersfeld.pdf).
An interesting paper from Alexander Riegler, “Towards a Radical Constructivist Understanding of Science” (http://www.univie.ac.at/constructivism/pub/fos/pdf/riegler.pdf).
And from Peter Cariani, “Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics” (http://www.univie.ac.at/constructivism/journal/articles/7/2/116.cariani.pdf).
This last paper I don’t necessarily agree with entirely but I like his treatment of Godel’s incompleteness and Turing’s Halting. The author’s main premise is that we need to constrain formal systems to only those which avoid troublesome infinities but I see obvious problems with this, for instance, from the perspective of convergent series. I mean, the natural exponent is essentially a convergent series and I wonder if it could even exist, or, more practically, have been discovered, if one eliminated divergent series from consideration. And even at this stage of the knowledge acquisition process, if one eliminates uncomputables from consideration, how could one know for certain that a few epistemologically essential elements were not also eliminated? To me, it sometimes seems that certain computables rely on their uncomputable counterparts and those computables seem fundamental to our collective epistemological exercise – the Platonist question and “naked” singularities aside.
Anyway, when it comes right down to it, I tend to agree with George Ellis and Max Tegmark . . . although I feel Tegmark’s Mathematical Multiverse should probably consist of only those axiomatic systems which enable coherent (as opposed to consistent) self-reference, i.e. backreaction!
Haha.....Plato and Bacon would be proud of you. The characterizations, as if two parts of one. I mean after all, there is a Q&A going on, and , I see it on my 2 dimensional screen.
ReplyDeleteSort of like watching pool balls on a table, a white ball hitting another with direction and meaning. Then seeing sound generate a "click"(this is a dramatization of a ball hitting another ball) into another space?
Hmm....a conformal field theory of what space? :)
As a 2 dimension presentation of three dimensional space?
ReplyDeleteYou see the post under that comment. Hmmm that was back in 2006.
Amara was on to something back then too with regard to BICEP?:) See?
ReplyDeleteGreat article. Definitely have strong Platonist tendencies myself. We need more write ups like this to reject all of the crackpot notions that come about when people discuss ideas like the holographic principle.
ReplyDeleteI love this blog! Keep it up Sabine!
ReplyDeleteEven on bad days you make kids (of all ages) giggle.
ReplyDeleteGreat dialogue. Great Rant.
Vertical learning curve.
Real or not.
Eliciting over thirty responses on rant raving bad days. Not bad.
Philosophers. Give thanks.
The area theorems of black hole horizons are derived classically, are they not? The laws of thermodynamics apply to the classical world as well. So holography would seem to be a classical phenomemon; but we are saying that we need string theory AdS/CFT to understand it?
ReplyDeleteArun, the entropy of the BH is nothing else that then the number of the microscopic *quantum* states, coarse grained to the large macroscopic object we call Black Hole. Of course identifying these microscopic degrees of freedom is the whole point; it is a task of the QG theory.
ReplyDeleteWhich holographic difference between low generations of dimensions? 2 to 3, 3 to 4, 4 to 5? Virtual particles are quantized too.
ReplyDeleteWes, a long comment was lost in posting. The philosophic part followed Abbagano "there are no necessary realities " which in the QM spirit of things some as necessary are not forbidden.
Arun,
ReplyDeleteAs Giotis says, you need some theory for the microscopic degrees of freedom to construct a map from the bulk to the boundary. The area theorems are derived classically, yes, but without further knowledge they're area theorems, not thermodynamic relations. To make the connection, you need to know how to count microstates. Best,
B.
Actually it is the second time I find something useful in the discussions on this blog. The first thing was Wheeler's bag of gold which turned out to be a good example in something I wrote recently on the holographic principle and some theorems of topology... I could post a link to the arxiv preprint if it's permitted (?).
ReplyDeleteIn fact there are several reasons not to believe in the Holographic principle despite the validity of some conjectures (like AdS/CFT) in some particular theories. In fact I think I have a good proof that the existence of the holographic principle is undecidable in the context of full quantum gravity. :)
Yes, arxiv links are fine.
ReplyDeleteGreat! :) Then here it is :
ReplyDeletehttp://arxiv-web3.library.cornell.edu/pdf/1404.1800v1.pdf
I call it "2" because it is a distillation and revision of this one
http://arxiv.org/pdf/1403.2039v1.pdf
which even I find a bit too hard to understand now (I think I wanted to pack in too many things and it came out pretty impossible to understand)
Apart of the obvious (but unintended) advertisement of my preprint, nice comments would be nice :)
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ReplyDeleteinMatrix.ru:
ReplyDeleteI didn't delete your comments because you are Russian (which I didn't know) but because you didn't read the comment rules. No theory advertising here, only arxiv and journal references.
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ReplyDelete