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 a...open string
. 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 physics development (althought with that math you can learn a lot about other areas of physics
).
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
