How can strings have multiple windings?

[ConcreteF]

Hello all.

I am currently reading 'The Elegant Universe' and I'm puzzled by one key thing that is really starting to 'niggle' me!

How can a one-dimensional string have multiple windings along a two-dimensional circular spatial dimension ("Garden Hose"). For this to be possible the string has to cross itself at one point (for two windings) or more, and surely you would need a third dimension to allow the string to pass over or under itself. With only two dimensions the string would have to pass through itself wouldn't it?

Sorry to ask such a 'lay' question but I'd really appreciate any clarification on this matter.

Many thanks,

CF

 

[tom.stoer]

I agree that the string must cross itself, but that does not mean that there exists a kind of interaction at the contact point. The string itself does neither feel nor see this contact point, it is a purely mathematical artifact.

 

[Fra]

How can a one-dimensional string have multiple windings along a two-dimensional circular spatial dimension ("Garden Hose"). For this to be possible the string has to cross itself at one point (for two windings) or more, and surely you would need a third dimension to allow the string to pass over or under itself. With only two dimensions the string would have to pass through itself wouldn't it?

You seem to raise the intuitive question of what is the physical significance of several windings if there is no additional dimension to distinguish the revolutions? I mean, how do you physically distinguish this from just a more "massive" string with less revolutions?

I think it's a good question loosely related to T-duality in string theory, which suggest that the winding number, radius of circular dimensions is indistinguishable from a different more massive string with lower winding number.

This duality, including some others in string theory, suggests IMO that the litteral picture of a "string" does not quite make sense. It's more to be seen as a mathematical abstraction whose real "meaning" (if any) is yet missing.

/Fredrik

 

[arivero]

What you say is that, equal to a point particle moves with x=f(t), a string moves with x=f(t,u), with f(t,u)=f(t,u+2pi). So the no-problem in the particle case, where x can be the same at different cases, becomes a problem in the string case, where x can be the same not only for different t but also for different u and equal t. While you have noticed the problem in the winding, note that even an string with trivial winding number can have this problem.

 

[ConcreteF]

Thank you all for taking time to answer my question.

It seems that I need to allow myself to let go of my tangible understanding of a string and just accept it as a mathematical 'thing'!

If I understand correctly then, a string can 'pass through' itself an unlimited number of times without any consequence. And, that there is no difference between a string with multiple windings or one, larger one (of the same length) without any windings, they are effectively the same.

Thanks again.

CF

 

[arivero]

Thank you all for taking time to answer my question.

It seems that I need to allow myself to let go of my tangible understanding of a string and just accept it as a mathematical 'thing'!

If I understand correctly then, a string can 'pass through' itself an unlimited number of times without any consequence.

Yes. If it helps, think about bosons in particle quantum theory.

And, that there is no difference between a string with multiple windings or one, larger one (of the same length) without any windings, they are effectively the same.

No, it is a topological problem, you can not deform one string to another of different winding around a circle. For instance for a two-windings string, which should be "bivalued", you can try to move it back and forth but all you get is a zone "fourvalued" and the rest of the circle still "bivalued", and so on.

 

[Fra]

I'm no string theorist and not even a string advocate/fan - I have my own quite biased understanding of the string logic, but for some more info check

http://en.wikipedia.org/wiki/T-duality
http://www.slimy.com/~steuard/research/MITClub2004/slide29.html - this has a nice/simple picture.

I personally prefer an information angle, and here a string is merely one possible microstructure. Two different microstructures can be "dual" to each other in the sense that they can encode the same information, but the difference can still be one of fitness and persistive when you consdier the (time)EVOLUTION of the microstructures. One microstructure can even deform into another one with different topology. This one can appreciate without string theory at all.

/Fredrik