Superknot theory

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  • #1
I have never heard of anyone talk about the knotting of superstrings? Is it reasonable to imagine huge collections tangled superstrings stuck together in knots? Would these be huge clusters of knots withing 10 dimensions (in a static state)?

I do realize that I am arguably sitting on a huge cluster of knotted strings called my chair, but this give me a completely different picture of what superstrings might look like - and behave.

For example, over distances of an atoms diameter and larger, there are really only three dimensions. But strings are 10^-34 meters in length and exist in 10 or 11 dimensions. Would huge knot clusters have 3D properties while still having tiny pathways into some invisible hyperdimensional 10-D space?


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  • #2
I have been thinking about that idea, yoou can see some preliminar thinkings about it in my (now temporally frozen) blog

Since the last entries I have been studiying deepeer string theory (i stil am on it) and also triying to see how the knoting of strings could work. First for noninteracting strings (how to define a diferent hilbert space for knoted states, if it makes sense, if it would be some kind of tensor product (or direct sum) of unknoted strings or what, and later the questiong of how the ktnoted states (and the knotting process) would be described in interacting strings.

Anyway I would advise that knoting is a dimensional depending thing in te following sense. Any knot in 3 dimensions dissaperas in 4 d because you always can unknot anything just displaceing the string in the upper dimension. So you would need to think about knoting in the upper dimension available.

Also I would advise you that you need to think about a knoting mechanism. One posibility is that an open string would become closed when beeing intersecting a closed string. Another posibilitie I am triying to study is the "blowin-up" process of algebraic geometry for tow intersecting closed strings cross among them in the blowed up dimesnions (that would imply an extension of the polyakov prescription for the path integral). Also i am triying to see if the idea of knoting makes sense for open strings with extremes attached in the same d-brane. That of course remember one the need of triying to see how knoted states would behave under dualities.

Another question I would point is that the whoe idea of kntoing makes a very clear sense for the bosonic string, but it is not so obvious what it is in superstring theory. Realize that the common way to see superstring theory is like a collection of two-dimensinal fields in worldsheet. But while that worldsheet idea appears in an obvious manner as the surface sweped by a line element in the bosonic string that idea is no so clear in the superstirng. The thing is that in teh open string you can make an easy one-one identification of target space coordinates with twodimensional bosonic fields. On the superstring instead you have for every target space coordinate some super-multiplete. If you want to have an "easy" geometrical picture of that you would need to choose as target space a superspace. But that rise the question that the observed geometric space is not a superspace, you know ;).

Anyway, I see these idea of knoting, at least till now, mainly as a "laboratory of ideas" to try to clarify what string theory that something else.

For example, suposedly there is the posiblity of "fundamental" cosmic strings, that could, in principle, beeing knoted. What would the knoting imply for cosmic strings.

Hope these half cooked ideas could be of any help to you, but hey, if I would have a fully developing of these I wouldn´t post them in a forum or a blog ;).
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  • #3
Thank you Sauron for your thoughts. Your knowledge of Superstring theory is highly developed so I will pondering some of the terminology and its mathematical meaning for a while.

I agree that strings might be able to knot in 3-space but not in higher or lower dimensions. Could that possibly be how a 3space macro universe (one with planets and people) actually manifest?

Knotting requires that the strings not be able to pass through each other. If they could somehow tunnel through each other (by exceding some energy gap) AND if they could themselves become tangled into a knot, than they might behiave a little bit like nucleons, and the more familiar proton and electron particles.

To become knotted, that have to be able to bend? right? Otherwise, they act like a pile of nails that don't really snag to each other.

I guess I'm using familiar objects to understand the subject because my mathematical exploration paused at differential equations and linear algebra.
  • #4
Knotting requires that the strings not be able to pass through each other. If they could somehow tunnel through each other (by exceding some energy gap) AND if they could themselves become tangled into a knot, than they might behiave a little bit like nucleons, and the more familiar proton and electron particles.

I had already unswered that:

a)One posibility is that an open string would become closed when beeing intersecting a closed string. Using the "familiar objects" perspective you ask you could see a closed string like a rim, or hoop (or a ring ;) )and an open string like string :biggrin: . You could pass an extreme of the open string inside the rim and you could tie it with the other extrem to get two closed strings knotted together. That requires to have a theory which combines open and closed strings and once. Usually in the basic expositions of strings people explains about open bosonic strings and closed bosonic strings, but some superstring theories (type I) contain necessarilly open and closed string sectors. Semengly these mechaism of tieing an open string around a closed string in principle would be available only for these kind of strings. But as far as supsedly there are dualitites betwen all the string theories the knoting mechanism of type I strings would translate into something else inthe other (superstring) ones.

b) The other posibilitie can badly be explained in terms of usual objects. I´ll try to explains some of it anyway. The idea comes form considering the situation when two closed strings touch themselves by a point. If you see them as separate strings there is no problem. But mathemathically nothing forbides to see them as a single string (mathemathically a curve). But it wouldn´t be in that case a regular curve. That pont wouldn´t be a regular point because in that point you don´t have a single tangent to the cruve but two (one along the "individula" strings).

Algebraic geometrists have developed a theory to work with these kind of points, it is the "blowin-up" process. The idea is to embed the courves in a projective space (in the case of plane curves it is engought the projective plane). One way to see the projective plane consists in the identification of vectors i.e. directions, of space as points. So the solution is to consider the tangents of the curves as points in the projective plain, from the viewpoint of the projective plane you can have a regular way to see how the tangent of the curve evolves. Well, the "slight" problem is that the blowin-up around a point requires one further dimension that the one in which the curve is naturally defined, that is, for a plane curve you would need that around the self-intersecting point you would have 3 dimensions and so on. I still need to study a lot of algebraic geometry, so i gues these is explanation couldn´t bee as clear, or exact, as it could, sorry for the inconvenience.

Well, that were my two atacks to the problem of knoting. I have to think about your idea of "quantum tunneling" and how it fits in the context of the polyakov integral.

Your knowledge of Superstring theory is highly developed

Thanks but the apreciation but I would be totally dishonest if woldn´t advise you that for someone who, as you admit, has a knoledge of physics limited to linear algebra and diferential equations could get false ideas about the understanding of other people about string theory, or any modern theretical physic development (althought with that math you can learn a lot about other areas of physics :rolleyes:).

For example in these concrete discusion I must advise you about some fact. In the ortodoxy of string theory people seems not to get too seriously the "string as a mathemathical surve" paradigm. In a previous discusion in these forum demistifyer argumented me that you can see the individual points of a string because to probe an string you need another string and they are of a simiar magnitud size and so the whole idea of points in the string makes no sense. Well, that is not entirelly true even from the very viewpoint of known string theory because you can use D-0 branes to probe distances smaller than the size of a fundamental string.

But anyway I find very bizarre that a physical theory could be beyond the mathemathical language in which it is forumlated so that is why I wonder about these idea of string knotting and it´s possible consequences.

Of course the most obvious one is that two knoted strings would travel together. So they would represent, for practical observational purposes, a particle who would be a composite of individual string states (the thecnichal way to say it is that you could represent the hilbert space of the knoted state as a tensor product of the indivual hilbert spaces of the individual strings). That is, if one of the knotes strings is vibrating as an electron and the other as some quark you would get some mixed quark-electron state. By the way, to describe that linked states in the polyakov integral you would need a vertex operator that would represent them, I still need to check exactllly how to build it and if there is a way to do it that preserved the consistency rules for vertex operators (if it has the correct conformal weighth and all that).

Of course nothing like that has been observed and that means that if knoted string states exist they must be very unprobable. But as far as I see if some of them would be oserved it would be candidate of prove of the validity of string theory. And I see candidate because possibly there would be other posiblities to explain that states.

Said all these once agian to advise you that you woudn´t take these ideas too seriously, the "knoted states" paradign is beeing for me a kind of curiosity while studying string theory to try to see what i study from a viewpoint diferent to the conventional one, something that I always try to do when I study something becauses it leads me to consider questions in which otherwise I wouldn´t repair. That´s why I just comment some of the technicalities necessary to put in a proper way the idea but I haven´t actually tried to do them.

But also I must say that I still don´t see a clear reason to discard the idea of knotting so...well, that´s funny :cool:
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  • #5
Please correct me if I'm wrong, but this is my understanding of string theory. We have no earthly idea if there really are tiny physical strings. We only have our mathematical ideas of strings (and various properties) that seem to bloom Maxwell's equations, Relativity, and a bunch of symmetries. But I realize that superstrings, compared to rubber-bands and garden hoses, may not have 1:1 identical properties. Shoelace strings get knots, superstrings probably break and form like bubbles in a boiling pot.

Sorry, quantum physics is so bizzarre, we're like cavemen, " like rock? like stick?"

Now that I think about it, knots do sort of require some linear friction (a property not attributed to superstrings)...
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  • #6
On more thought. String theory is a development of QM. Quantum Mechanics uses wave amplitudes (things that don't exist physically). Wave amplitudes come from describing QM interactions which fields and particles. Then, the wave amplitudes are used to get the probability of finding some particle in some state or predict all of the possible states available.

I hate to say it, but it's like the physical universe that we know and love (nice solid chair) comes out of some strange Alice in Wonderland dice cup.
  • #7
...Sorry, quantum physics is so bizzarre, we're like cavemen, " like rock? like stick?"...
No it's not, it makes perfect sense.
What I hate about humans, is that when a theory is not elegant enough, or clear enough then we don't like it and most probably wrong(I mean this in general).
I can't understand the human desire for everything to be beautiful and symmetrical.
If it was left for humans, then probably we would have made Pi=3.
  • #8
I'm trying to imagine what QM is really like behind the mathematics. Scientists have made QM measurements for almost a hundred years. It's like there's some wavy interconnectedness (hence a need for tensors) and a game of "Gotcha... you little electron!" ... on some craps table with hyperdimensional everchanging dice. The fact that wave amplitudes are not real things but <x_i> = Integral Psi x_ Psi*d_i only enters the material/physical world as the random filling of available states almost seems metaphysical.

It bugs me a little that the Uncertainty Principle leaves just enough space for the New Age crazy people to get stuck in the doorway, but won't either keep them out of science or invite them in. There are many QM scientists who believe that the wave mechanics are the totality of physical matter. But if I have an experiment which makes the momentum standard deviation very tight, then either delta x cannot be probed very accurately (which is what most QM PhD's might say) or I could have a chorus of astrologers tap dancing within my Delta x, sticking there toungues out at me.

Please forgive my ranting. I am probably one of those crazies who got stuck in the doorway between physics and metaphysics.