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Smolin denounces dualities, promotes trialities

A reader asked me what I thought about a new paper by Lee Smolin

Extending dualities to trialities deepens the foundations of dynamics.
I was told that the question makes sense because he cites your humble correspondent's paper on matrix string theory. Well, unfortunately, I cannot return the niceties because the paper is crackpottery of the most hardcore sort. It's my belief that no paper by this author has been discussed on TRF for more than 5 years – it's perhaps a good moment for an exception.

The author dismisses dualities – a key concept in string theory and modern theoretical physics in general – because they contradict his religiously held principle about the "background independence". And he brings us the new gospel: when the dualities are replaced by trialities, relationships between three entities, the religion is upheld and things are fine. The paper contains something claimed to be "examples" of this triality.




These claims are just plain idiotic at every conceivable level. But let me begin with a citation complaint. Given the fact that the only catchy part of the paper is the concept "dualities vs trialities", he should have cited our paper Dualities vs Singularities (TRF) – the clever title was invented by Tom Banks – which actually showed that (good) dualities are enough to prove that all the (bad) singularities in the moduli space of M-theory on tori are illusions.




The words "singularity" and "duality" sound analogous – they're like the same words derived from the numbers one and two – but their meaning is obviously very different. We respect the meanings and their differences and our observations about a relationship between singularities and dualities are actually correct and very nontrivial.

On the other hand, everything that Smolin says about the dualities and trialities is just pure nonsense. To mention some lethal problems with the paper, let me start with the following list:
  1. the "background independence" as understood by Smolin, i.e. a hysterical fear of any background, is a totally wrong meme that makes all physics research impossible for anyone who accepts it
  2. dualities don't contradict any form of "background independence", anyway; on the contrary, they prove that the underlying theory, e.g. string theory, doesn't take the backgrounds seriously as it identifies physical situations that used to be called "very different backgrounds"
  3. even if dualities contradicted any principle of this sort, trialities would contradict it exactly as much as dualities, so the claim that trialities represent an "improvement" is plain silly
  4. in Smolin's terminology, most of the things he calls "dualities" should actually be called "infinitialities" as they relate infinitely many points
  5. he conflates "dualities" of the old trivial type (like the duality between a vector space and its dual) with "dualities" in string theory (highly nontrivial equivalences)
  6. all his examples of trialities are completely wrong – they're effectively mixing apples with oranges and elephants
  7. his more technical claims about Chern-Simons theory as the de facto mother of all types of theories in physics including string/M-theory are wrong or at least totally indefensible at this point
This list is in no way complete but I think it's sufficient to understand that Lee Smolin is a spherical crackpot – he is a crackpot no matter which way you look at him. What he keeps on writing and saying is wrong at every conceivable level.

Background independence insanity

All phenomena that are studied by physics – or science – take place in a version of spacetime. Some future or futuristic formulations of the laws of physics may work without a spacetime but the established ones do use a spacetime and even the futuristic formulations will have to show that they agree with the laws of Nature operating within a spacetime that have been used for centuries.

The spacetime may be curved, quantum fluctuating, and it may refuse to be uniquely determined for a given situation but there is a spacetime and objects are excitations of the background spacetime or additions placed on top of the background spacetime. If some candidate laws of Nature are incompatible with the existence of any spacetime, they are excluded – they fail to obey the simplest condition we demand in science.

Loop quantum gravity implies that the nearly smooth, nearly flat spacetime we know doesn't exist, so this theory is lethally sick and instantly excluded. Lee Smolin has invented a whole insane ideology based on his favorite "background independence" slogans that tries to sell this lethal vice as a virtue.

There is a legitimate interpretation of the "background independence" in physics but it has virtually nothing to do with Smolin's fantasy. Physics implies many backgrounds and they may dynamically change and the theory ultimately doesn't say that one background is fundamentally better or more natural than others. It allows all of them. But it doesn't mean that we should be disgusted by any laws that are easily formulated relatively to a background. On the contrary, many backgrounds may be good (and "some background" may be necessary) to understand any physical phenomena. And as the principle of background indifference says, it seems that any background – with any number of dimensions, any shape, any topology etc. – may be in principle OK to describe any physical phenomenon!

String dualities challenge the special status of any backgrounds

String dualities are equivalences between pairs (or larger groups) of theories or between sets of physics-like mathematical equations that superficially look very different because they contain different building blocks, different fields, different particles, different shapes of spacetimes, different allowed boundary conditions or branes or fluxes, and/or different interactions. But if you study them carefully, you will find out that there is a one-to-one correspondence between the objects in both or all theories related by the dualities, and the "operation of translation" commutes with the evolution in time. (And some time or spacetime must be at least approximately meaningful at least in the spatial region at infinity, otherwise no meaningful statements about physics may exist.)

So dual theories describe physically indistinguishable physical situations and phenomena! If you live in a universe that is described by some laws of physics and those laws admit a dual description (or many dual descriptions), you must admit that you may also live in the universe obeying the other dual laws. They're really the same laws from the viewpoint of physics even though the mathematics may "look" very different to us and even though we use totally different words and concepts to talk about the phenomena. But when we map the possible observations in "either world" to the mathematical apparatus of the corresponding theory and when we convert the "words about dynamics" to the adequate calculations and include all the corrections and subtleties, we find out that the truly observable predictions of all the dual theories exactly agree!

Quantum field theory and especially string theory is full of these dualities. S-duality, T-duality, U-duality, mirror symmetry, AdS/CFT holographic duality, and so on, and so on. These equivalences were mostly found in the recent 20 years and they totally transformed our idea about the "set of a priori possible physical theories" and the "set of solutions to [environments allowed by the laws of] string theory".

How did dualities affect our discussions about the "background independence"? Well, they have strengthened our certainty that the underlying theory boasts some "background independent" physics. Background dependence is the concept in old-fashioned physics that allows you to say that one background is right, other backgrounds are wrong, and you may always decide which background is the right one.

But dualities imply that many seemingly different backgrounds must be physically identified, so if you say that one of them is right, so are all of its duals. It means that dualities are enough to break the old naive ideas about "right and wrong backgrounds" – they are a sufficient argument that makes a more general thinking about the backgrounds, a "background independent thinking", inevitable.

If you look through Smolin's paper and try to find out why he thinks that dualities contradict his "background independence", you will only find the complaint that dualities depend on a "hidden structure". That's really what it is about. Unlike all of Smolin's papers, claims about the duality have a structure – they have a content. Of course that without a "structure" (which is only "hidden" if you don't understand things well), you can't say anything meaningful about physics at all, and that's one reason why Lee Smolin hasn't said anything meaningful about physics in his life so far.

Dualities and trialities differ by the number but they're conceptually the same

Smolin preposterously claims that "dualities" may be bad for his religious faith – although he never offers a glimpse of evidence why it should be so – but "trialities" are OK. This is of course totally silly. If dualities conflicted with some metaphysical principle, so would trialities, and vice versa.

Sometimes, the term "duality" literally means a relationship between 2 physically identical and mathematically seemingly inequivalent theories or situations. However, it's much more typical that the dualities may relate a theory (or a point in the moduli space of string/M-theory) with a larger number of equivalent points, usually an infinite number.

Infinitialities

Take the \(SL(2,\ZZ)\) modular group relevant for the description of the shapes of two-tori. Is it a duality, a triality, or something else? Well, it is a \(SL(2,\ZZ)\) group of elements that may be described as transformations\[

\tau\sim\tau' = \frac{a\tau+b}{c\tau+d},\quad a,b,c,d\in\ZZ,\quad ad-bc = 1.

\] For each matrix of integers \(((a,b),(c,d))\), you may compute one \(\tau'\) that is equivalent to the original value of \(\tau\). There are clearly infinitely many values of \(\tau\) that are physically identified. The upper half-plane is covered by infinitely many deformed copies of the fundamental region.



So if you use the word "duality" not just as a concept to describe the nontrivial equivalences but you care about the number of backgrounds that are identified, \(SL(2,\ZZ)\) is an "infinitiality", not a "duality". It only becomes a duality if you demand that both \(\tau\) and \(\tau'\) are pure imaginary. In that case, only the \(\ZZ_2\) subgroup of \(SL(2,\ZZ)\) is preserved.

Incidentally, for some special values of \(\tau\), some transformations in the \(SL(2,\ZZ)\) group may yield the same value of \(\tau'=\tau\) so the duality transformation isn't something that relates "different superselection sectors" but rather a (gauge) symmetry within a single sector. You may see that in \(SL(2,\ZZ)\), there are two inequivalent cases how it may occur.

For \(\tau=i\) (or any other value related by \(SL(2,\ZZ)\) transformations), the transformation \(\tau\to -1/\tau\) sends us to \(\tau=i\) again. So this \(\ZZ_2\) subgroup of \(SL(2,\ZZ)\) becomes a symmetry acting in one superselection sector, in one background. The \(\ZZ_2\) fixed point is nothing else than the \(g=1\) fixed point of the "simple" \(g\to 1/g\) strong-weak duality.

Similarly but less trivially, for \(\tau=\exp(i\pi/3)\), the transformation \(\tau\to-1/\tau + 1\) brings us to the same value of \(\tau\), it's a fixed point. But now only the third power of the transformation yields the identity so this is a \(\ZZ_3\) subgroup of \(SL(2,\ZZ)\) that becomes a gauge symmetry within a superselection sector.

There are many larger groups that may appear, too. (You may use the words duality, triality, quadrality, pentality, hexality, heptality, octality, ninality... but they only remember the order of the group while it may be important or useful to know the whole group.)

Also, if you consider e.g. a heterotic compactification on a Calabi-Yau, you may find various equivalent descriptions by the heterotic-K3 duality, mirror symmetry, and many other dualities. So the number of seemingly different backgrounds that are normally identified with a given background thanks to the dualities is usually higher than two – and often infinite. (When it's infinite, the copies are usually produced from a template with different values of integer parameters so the infinite-order dualities usually don't depend on an infinite number of new clever ideas.)

So the number of backgrounds that are identified isn't really the main point – the fact that we may identify the backgrounds that seem different (and not their number) is the point of dualities. Recently, I embedded a cute talk about string theory for mathematicians by Cumrun Vafa where dualities played the central role. You may also want to read a recent review of dualities by Joe Polchinski.

Trivial and nontrivial dualities

Smolin clearly doesn't understand the meaning of the words he is using at the technical level so if two words are "spelled the same", he just can't tell the difference! So he conflates "dualities" as simple as the relationship between a vector space and the space of the linear forms on that vector space – something that, in the presence of the complex inner product, becomes the bra-ket relationship\[

\bra{\psi}\leftrightarrow \ket{\psi}

\] with the nontrivial dualities in string theory (and field theory). Both kinds of "dualities" are some relationships between two entities but they're very different relationships when it comes to their detailed content, intellectual depth, and implications. The bra-ket "duality" wouldn't be counted as a duality in the string-theoretical sense exactly because it's so trivial and mathematically understood.

The notion of a "duality" in the most general sense is omnipresent in mathematics. For example, Platonic polyhedra are clumped into pairs by dualities, too. (There are also several natural trialities, such as the triality in \(SO(8)\) or affine \(E_6\).) But in modern theoretical physics, we mean something much more concrete and nontrivial (or surprising) if we talk about dualities. Maps that involve huge dictionaries translating every object and every process in a theory to something else.

Silly examples of trialities

Smolin's new crackpot paper is generally tragic but yes, I did explode in laughter when I got to his first two big example of the (good) trialities:
In the case of the classical duality between \(x^\mu\) and \(p_\mu\) there is the time derivative \(d/d\tau\). So there really are three elements in that duality\[

x^\mu, p_\nu, \frac{d}{d\tau}

\] ... Once again, there are really three actors in the game\[

\ket\Psi, \bra\Psi, \dagger

\]
Holy crap, it's just so stupid that even Lee Smolin can't be serious.

You see that the first example talks about the "duality" between coordinates \(x^\mu\) and momenta \(p_\mu\). The position space and the momentum space are related by the Fourier transform – another textbook example of a simple duality. You Fourier transform a function twice and you basically get back to the original one. So the functions like \(f(x)\) and its Fourier transform \(\tilde f(p)\) are related by the Fourier transform, an example of a "simple duality".

But what the hell the time derivative is doing in the triplet? The time derivative may get you from positions to velocities but why is this operation \(d/d\tau\) written on the same line – on par – with the coordinates \(x^\mu\) and momenta \(p_\mu\) themselves? The time derivative clearly plays a completely different role in the relationships. The relationship between \(x^\mu\) and \(p_\mu\) is qualitatively different than the relationship of \(d/d\tau\) with either (or both) of them. There is obviously no relationship (I mean a duality map) that would map the coordinates or the momenta to the time derivative (they have a different number of components, among other problems).

It is very obvious that when Lee Smolin talks about a "triality", he means that he is able to write three objects or expressions – totally random objects, with any asymmetric pairwise relationships between them – on the same line. This is just so stupid! A duality or a triality must obviously be an operation, and the objects that are related by a duality or a triality must be (unequal but) commensurable or analogous. They must be objects of the "same kind". You just can't relate coordinates with some completely different operation that sometimes acts on them as "actors" in a triality.

Smolin's example involving the bra-vector, the ket-vector, and the Hermitian conjugation is exactly equally stupid. A duality relates the ket-vectors \(\ket\Psi\) with the bra-vectors \(\bra\Psi\) but there is no third actor. The dagger \(\dagger\) – the Hermitian conjugation – is in no way a "third actor" that could be exchanged with the bra-vectors by a duality or a triality. In fact, the dagger \(\dagger\) happens to be nothing else than a good symbol for the duality between the bra-vectors and ket-vectors itself.

So what he sells as "trialities" are no trialities at all. They're just random sets of three completely incommensurable mathematical symbols with no uniform relationships. It's a much more stupid way to imagine a "triality" than what the simplest rural Christians have in mind when they talk about the Holy Trinity. The actors in the Holy Trinity could be related by a \(\ZZ_3\) or \(S_3\) map. But Smolin's "actors" obviously can't. He is mixing apples with oranges and with elephants.

"Trialities" involving cubic matrix models

It doesn't get any better when Smolin switches from the alleged "bra-ket-dagger trialities" to more complicated mathematical physics such as Chern-Simons theory. Here, his new "triality" is the following list of three "actors":
particle mechanics, string theory, Chern-Simons theory.
They're claimed to be the same thing, or related by a map except that he never describes how it could work. So it's just another silly list of three concepts that can't even be compared.

You know, Chern-Simons theory is a rather complex but not too complex theory that cares about the topology of the spacetime and the knots inside etc. It is more complex than particle mechanics and it cares about very different things. Claiming that there is a duality between them is a hugely bold claim and you should better be extremely careful and detailed about the claim you want to make unless you want to be instantly recognized as a hopeless crank.

String theory is much more complex than either particle mechanics or Chern-Simons theory. It contains many more solutions, each of them contains many more objects and possible phenomena and other things. If you say that this is "dual" or "trial" to some simpler theories, you immediately face a problem because it just doesn't work.

Now, matrix theory describes a superselection sector of string/M-theory in terms of a particle mechanics model – that's why he referred to matrix string theory, too. But it's a very specific (supersymmetric) matrix model of "particle mechanics" and it describes one superselection sector – or one background, if you wish – of string/M-theory. It was a great breakthrough (I primarily mean the BFSS matrix model for 11D M-theory that started it and remained the "master example" as of today) to find this equivalence and the evidence that it's actually true. Lee Smolin has never found anything that would be remotely analogous. He just randomly spits garbage about random things' being equivalent.

Similar comments apply to Chern-Simons theory, and especially Chern–Simons theory. I remember that once when he invited himself to Harvard by misinterpreting an e-mail I sent him, he would tell me that "He didn't believe that M-theory was difficult. It must be simple and Chern-Simons theory was also simple so they had to be the same thing." This was a stunner for me. I've always known he had been a moron but this kind of a moron? String/M-theory may be "simple" in some way but it's still very complex as we see it now – it contains all the good ideas and calculations in physics, after all – so these facts make all assumptions about its "excessive simplicity" extremely unlikely.

But even if string theory were as simple as Chern-Simons theory, it just wouldn't imply that they're the same thing! A genuine physicist just can't just randomly enumerate two or three theories and claim that they're the same. A genuine physicist has at least some standard, some filters that prevent him from polluting his environment with completely generic stinky garbage and noise. A professional physicist must really reduce his or her "error rate" to a very small value because physical arguments depend on each other and on long chains of others – but even if you disagreed, you may agree that a 99% error rate is too much. ;-)

The cubic Chern-Simons-like models have some relationships to descriptions of string/M-theory or quantum field theory. For example, the \(\NNN=4\) gauge theory may be written in the \(\NNN=1\) language using a cubic (but no quartic) superpotential. And the ABJM theories in the membrane minirevolution and Witten's cubic string field theory are two more examples of systems of equations that suggest that "some actions that are at most cubic" are enough to describe as complex theories as supersymmetric quantum field theories or interesting backgrounds of string/M-theory.

But the details – the field content, the precise form of the action, the underlying representation theory etc. – matter a great deal. A real physics paper in 2015 simply can't make blanket statements such as "there is a triality between Chern-Simons theory, particle physics, and string theory", or something like that. Some refined, much more modest versions of such claims have been established for years but the general big claim seems self-evidently wrong and if it is right for some reasons, it's very clear that these reasons haven't even been sketched in Smolin's paper.

The paper is just breathtaking cr*p and I am amazed that Paul Ginsparg hasn't been able to block this "author" from the professional hep-th archive yet.

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snail feedback (28) :


reader Leo Vuyk said...

Still IMHO (as a humble
architect) topological trialities seem
also able to represent torie with threefold internal rotational freedom forming
topological photons and propeller shaped fermions with dual spin states able to
create a Calabi Yau surface. See http://vixra.org/pdf/1103.0002v4.pdf
Q-FFF models


reader Nikolay said...

There are non-trivial examples of "trialities". See for example this paper

http://arxiv.org/abs/1310.0818



Of course this is not the subject of Smolin's paper...


reader Mike said...

'All phenomena that are studied by physics – or science – take place in a
version of spacetime. Some future or futuristic formulations of the
laws of physics may work without a spacetime but the established ones do
use a spacetime and even the futuristic formulations will have to show
that they agree with the laws of Nature operating within a spacetime
that have been used for centuries. The spacetime may be curved, quantum fluctuating, and it may refuse to
be uniquely determined for a given situation but there is a spacetime
and objects are excitations of the background spacetime or additions
placed on top of the background spacetime. If some candidate laws of
Nature are incompatible with the existence of any spacetime, they are
excluded – they fail to obey the simplest condition we demand in
science.'

Hi,

I've made it through only 1/3rd of the article; before I proceed, I want to make sure I understand, at least simplistically, what I've read so far.

If I understand you, the kind of background independence that Smolin promotes is absurd since it's a formulation of reality in which the laws of Nature cannot rely on a space-time, either a 4- or higher-dimensional, background, leading to the absurd conclusion that our 4-dimensional space-time doesn't exist. Is this partially right or completely off mark?

Sincerely,
Mike


reader Tony said...

OT: for whatever reason, I see guest voting available again. I couldn't upvote anything for a few days: tooltip was telling me I have to log in first. Not too late for a possible crackpottery bonanza associated with this post ;)


reader Giotis said...

But we know that Chern-Simons is not the full string theory.

In 3d for example we have the old CS/WZW correspondence and in a *certain limit* of AdS/CFT we have a similar correspondence. See for example Witten's paper

"AdS/CFT Correspondence And Topological Field Theory"


reader Smoking Frog said...

SOMEWHAT OFF-TOPIC
Lubos, is this any good, or is it BS?

Traveling without moving: Quantum communication scheme transfers quantum states without transmitting physical particles


reader Sergio Montañez Naz said...

Lubos, this paper
http://arxiv.org/abs/0705.2980
shows that the duality between the closed topological string description and the open string one has the same structure as the equivalence between a coherent state representation and the position representation in QM. The closed string description is the coherent state representation, where the frequency of the corresponding harmonic oscillator plays the role of the background moduli. Since coherent state representation is overcomplete, that means that the correspondence is not one-to-one.


reader Luboš Motl said...

It's subpar Chinese pseudoscience. One cannot create entanglement between two physical objects without these objects' sharing some information or events in the common past.


Because all the known phenomena we have observed in Nature may be described in terms of quantum fields or particles they create, the event that created the entanglement means that some particle or excitation of these fields had to move sometime in the past through every plane that divides the space into 2 half-spaces with 1 object in each.


reader Luboš Motl said...

Dear Sergio, the problem you mention is seen perturbatively - in this case and others - but when one correctly includes and interprets all the corrections, the problem goes away and the two sides match perfectly.


One must recognize that some parameters we encounter in one of the descriptions are actually collective degrees of freedom, and they have to be quantized, and when it's done correctly, they span the very same Hilbert space on both sides.


reader Sergio Montañez Naz said...

"they span the very same Hilbert space on both sides" That is just what the paper says. But the closed string side is a representation based on a basis that is overcomplete. A wavefunction in a coherent state representation can be easily mapped to the position representation because coherent states span the very same Hilbert space. But the map is not one-to-one because the coherent state basis is overcomplete. This is therefore an example of a duality which is not one-to-one.


reader RAF III said...

Not only that, but it seems that guests can now downvote as well. Yay!


reader Luboš Motl said...

If you manipulate with all the things correctly and make the right conclusions from the right calculations, there is nothing wrong about using overcomplete "bases" (a basis isn't overcomplete, by definition) for any calculation.


Your interpretation that the duality is "not one-to-one" because a natural description on one side uses an overcomplete "basis" is simply invalid.


The duality is a one-to-one map between objects in two descriptions. On both sides of the duality, one may use complete bases or overcomplete bases. If a natural description on one side uses a different (or overcomplete) bases, one must first translate one of the sides for the two sides to agree about the basis and its (over)completeness.


By your own admission, you have failed to do so, but it is your defect, not a defect of the duality.


reader btw said...

If you Fourier transform twice you don't get back the same function btw...


reader Sergio Montañez Naz said...

I think we agree
-There is a one-to-one (in fact unitary) map between both Hilbert spaces that is, states psiclosed and psiopen map one-to-one
-But the representation that arise from closed strings is an overcomplete basis representation, so the map from wavefunctions in this representation and states psiclosed is not one-to-one.
-So the map from wavefunctions in the closed string side to states psiopen in the open string side is not one-to-one
-I do not see anything wrong with that. It is not a defect of the duality, but something we should not forget if we are looking for new dualities.
-May be because my explanation was so short, you understood that I was saying a different thing. In any case thank you for your answer


reader Luboš Motl said...

"The wave functions" and "the Hilbert space" mean the very same thing, so if the map between the latter is one-to-one, and you agreed it was, then the map between the former is one-to-one, too. Your comments make no sense.


reader Luboš Motl said...

I wrote "basically". With normal symmetric conventions, the double Fourier transform brings f(x) to f(-x).


reader John Harley said...

Lee's strange three-way with Time is no surprise, because in his Cracker Jack book "Time Reborn" he regards time as being "real." I don't know what he means but it is orthogonal to how modern physics regards time.


reader Sergio Montañez Naz said...

A "wavefunction" in coherent state representation is the scalar product of coherent states with the state. Since coherent states are centered on (p,q), those wavefunctions are also called "phase space wavefunctions". Torres-Vega and co-workers claim that phase space wavefunctions cannot be obtained from the position representation, one can only go the other way around, that is, the position wave functions can be obtained from the phase-space wave function by a projection
and that is why I said that the map between coherent states
"wavefunctions" and states was not one-to-one. But I have seem from

http://orbit.dtu.dk/fedora/objects/orbit:17760/datastreams/file_4063988/content

that this is certainly not true. The transformation has inverse and, in fact, I wrote it some years ago, but I did not remember it. In any case, notice that the map between
closed string "wavefunctions" of

http://arxiv.org/abs/0705.2980

and open string "wavefunctions" is not explicit because of the overcompleteness of the closed string representation, making the
duality more dificult to see. For instance, because of the fact that the set of coherent states we have used as "basis" have non zero overlap, the sum of the squares of "wavefunctions" is not the square of the norm.


reader Luboš Motl said...

A wave function is an element of the Hilbert space - it's always the same thing.


A wave function = an element of the Hilbert space - may be described in various ways, in various representations i.e. relatively to various bases (or overcomplete "bases").


If two wave functions = elements of the Hilbert space are equal in one representation, they are equal in another representation.


reader Smoking Frog said...

Thanks, Lubos. I was suspicious because the paper as described seemed to me to equivocate or even to contradict itself, what with "counterfactual," "50% error rate," and other things which I forget now.


reader Dilaton said...

Thanks for this very interesting and enjoyable post Lumo ;-)

The dual description of a spherical crackpot is a scalar crackpot:

Whatever you do, whatever crazy transformations you try, you cant get rid of the fact that a crackpot is a crackpot ;-P ;-)


reader LM said...

sure but it depends how the result is represented, which depends on the procedure. You can find a 1:1 using an overcomplete basis, say as an exploratory component. But you represent a 1:1 as carrying fundamental significance if it was matched using an overcomplete basis. And that is obvious.


reader LM said...

sorry, meant to say "But you can't represent a 1:1 as carrying fundamental significance if it was matched using an overcomplete basis. And that is obvious."


reader LM said...

"One must recognize that some parameters we encounter in one of the descriptions are actually collective degrees of freedom, and they have to be quantized, and when it's done correctly, they span the very same Hilbert space on both sides."
- Yes but you have to know in advance what the other side is, so there's no fundamental significance. Which often isn't a problem because the work being done is not fundamental


reader Luboš Motl said...

What you write is totally wrong.


In *every* duality, there exists an independent definition of the theory on *both* sides, either a complete definition, or an approximate or otherwise incomplete one.


But even if it is incomplete, *every* duality implies infinitely many predictions that may still be verified to hold even though they are not guaranteed trivially a priori.


reader LM said...

Hi Lubos, OK sorry for that. I still don't get it but I'm content to take your word for it (for now!). That aside, the process of finding out more led to some intriguing revelations about String Theory. I think I'm starting to see why you (and the others) feel so strongly about it. The dualities are startling. I wish you would be more forthcoming about the problems and the issue with predictions. It would go much further to establishing confidence in the theory by this route. I don't think people mind waiting for a promising idea. But the talk about predictions being 'old' way of doing things...that's seriously disturbing dude! I know you are just being loyal as you see it..


reader Rehbock said...

Catching up. Excellent post. Smolin seems to be creating the background he denies exists by packaging it as if it provides third element in a duality. I don't see that this adds anything other than saying one can always relate two things to a third. But the word thing is intentionally non-scientific and trivial. For Smolin the third is triality but I don't think he says more.


reader Gordon said...

Smolin is just generating a paper out of nothing by promoting triality...reminds me of Unitarianism vs Trinitarianism in Newton's time. Smolin defends his heresies using the Squid Defense---disappear in a cloud of ink...in his case in a popular magazine like Sci-Am etc so that to the general public, he is the go-to expert. Narcissistic personality disorder.