Speed of Open Strings: A First Course in String Theory

[Jimmy Snyder]

In Zwiebach's book "A First Course in String Theory", I read that the endpoints of an open string move at light speed. Do any of the interior points of a string also travel at light speed?

 

[selfAdjoint]

No. And the more recent thing about open strings is to fix those ends on branes with Dirichlet boundary conditions (i.e. end points don't move around, but string is free to wiggle anyhow apart from that). That's where D-branes came from.

 

[Jimmy Snyder]

Thanks

Thanks, SelfAdjoint. Actually, I have a bunch of questions I would like to ask. Since they all are motivated by Zwiebach's book, the first question is: Do you have a copy of the book?

Your comments about Dirichlet conditions on the endpoints are helpful. In fact, in my original post I should have written: FREE endpoints of an open string move at light speed.

 

[nrqed]

No. And the more recent thing about open strings is to fix those ends on branes with Dirichlet boundary conditions (i.e. end points don't move around, but string is free to wiggle anyhow apart from that). That's where D-branes came from.

The ends *do* move around, even if they are fixed to a D-brane. Their motion is simply constrained to the space enclosed by the D-brane (like if itès a D3 brane, the string moves in 3 spatial dimensions). Am I not right?

Pat

 

[nrqed]

Thanks, SelfAdjoint. Actually, I have a bunch of questions I would like to ask. Since they all are motivated by Zwiebach's book, the first question is: Do you have a copy of the book?

Your comments about Dirichlet conditions on the endpoints are helpful. In fact, in my original post I should have written: FREE endpoints of an open string move at light speed.

I am also going through Zwiebach's book presently. So go ahead with your questions. I might not be able to help but at least you'll have at least one person to discuss with.

Which chapter are you going through now?

Pat

 

[Jeebus]

The ends *do* move around, even if they are fixed to a D-brane. Their motion is simply constrained to the space enclosed by the D-brane (like if its a D3 brane, the string moves in 3 spatial dimensions). Am I not right?

I don't really know for sure, but my guess is yes. The reason is that selfAdjoint said Dirichlet boundary conditions. That means, Dirichlet boundary conditions have endpoint fixed to move only on some manifold. So, they can fluctuate becaues of the gravity that they interact with. However, I would definitely say yes for a Neumann boundary condition, but Dirichlet I don't know really.

 

[Jimmy Snyder]

Hi nrqed,

I am just starting Chapter 12. I found Chapter 10 exceedingly difficult because my background in QM is rather weak. I need help understanding one point from page 98. The question that started this thread was directed toward clearing the matter up.

On page 98, it says that at each point on the world-sheet there is a timelike vector tangent to the world sheet. I can't understand the proof given in the book and I wonder if someone can explain it to me. I sent an e-mail to Professor Zwiebach pointing out that since free endpoints move at light speed, there would fail to be timelike tangent vectors there. What is more, Quick Calculation 6.3 on page 99 indicates that there may be points where there are no timelike tangent vectors. He agreed with me and on his web page related to the book, he publishes these points. However, I also pointed out to him that these facts may render the proof on page 98 flawed. He has not commented on this, so my current situation is that I don't understand the proof and I don't completely trust it either.

At the same time, I consider the matter to be of utmost importance because the existence of the timelike tangent vector is at the heart of the parameterizations used in the following chapters.

I would be ever so grateful to anyone who could provide me with a proof that I could understand.

 

[nrqed]

Hi nrqed,

I am just starting Chapter 12. I found Chapter 10 exceedingly difficult because my background in QM is rather weak.

Hi there. Have you had QFT before? If not, I can see how chap 10 may be tough going.

I need help understanding one point from page 98. The question that started this thread was directed toward clearing the matter up.

On page 98, it says that at each point on the world-sheet there is a timelike vector tangent to the world sheet. I can't understand the proof given in the book and I wonder if someone can explain it to me.

I have to admit that I find this page confusing. At first, Ithough that he was focusing on classical strings. In that case it seemed natural to me that there would be a timelike tangent vector at every point because there would always be a frame in which each section of th estring would be at rest. And I thought that was the end of the story.

But on a second, more careful reading, I got confused by some oh his statements. For example, he says that "the string is not made of constituents that whose position we can keep track of". I am a bit baffled by this. If the string is classical, we can for sure keep track of th emotion of each piece (we could paint one, say!).

I sent an e-mail to Professor Zwiebach pointing out that since free endpoints move at light speed, there would fail to be timelike tangent vectors there. What is more, Quick Calculation 6.3 on page 99 indicates that there may be points where there are no timelike tangent vectors. He agreed with me and on his web page related to the book, he publishes these points. However, I also pointed out to him that these facts may render the proof on page 98 flawed. He has not commented on this, so my current situation is that I don't understand the proof and I don't completely trust it either.

Could you give me the URL?

At the same time, I consider the matter to be of utmost importance because the existence of the timelike tangent vector is at the heart of the parameterizations used in the following chapters.

I would be ever so grateful to anyone who could provide me with a proof that I could understand.

I also hope somene else will jump in. If not, it would be worth posting on the string google group.


Regards

Pat

 

[Jimmy Snyder]

Hi Pat,

Thanks for your careful reading of my post. No, I have never studied QFT at all. I did read the first few pages of Professor Zee's "QFT in a Nutshell", so I have seen the simple calculation of the 'sum of histories'. I know it's a prerequisite for Professor Zwiebach's book, but I thought I could pick it up as I went.

I forgot to ask you how far you have gotten in the book.

Here is the URL you requested.

http://xserver.lns.mit.edu/~zwiebach/firstcourse.html

I must thank you for an important insight. When you said that you could paint one of the points, you made me realize what he means when he says:

"the string is not made of constituents that whose position we can keep track of"

His statement is not one that you could prove, it is an assumption. It is equivalent to saying that the string has no underlying structure, it is the fundamental 'thing'.

I wonder if the statement that a timelike tangent vector exists at each point on the world-sheet is not itself an assumption, and his 'proof' is irrelevant (to add to its growing list of deficiencies).

 

[nrqed]

Hi Pat,

Thanks for your careful reading of my post. No, I have never studied QFT at all. I did read the first few pages of Professor Zee's "QFT in a Nutshell", so I have seen the simple calculation of the 'sum of histories'. I know it's a prerequisite for Professor Zwiebach's book, but I thought I could pick it up as I went.

I was just asking because Zwiebach wrote his presentation in such a way that one does not have to know QFT beforehand and he introduces what he needs in chap 10, but that part is mych easier for someone who has some background in QFT. I can see that it would be much more tough without a background in QFT.

I forgot to ask you how far you have gotten in the book.

I am going though chap 9. I had worked out most of the basic stuff by following GSW, but I had worked out things in the covariant formalism (that Z covers in chap 21). So I have some catching up to do on the light-cone front (no pun intended ).

One of my first goals is to work out eq 12.162, to see clearly the need for D=26. I have always been disappointed in the past to work out the maths in string books or papers and to reach the point of restricting the number of spacetime dimensions just to be told that "the result is too long to prove" or that the proof was "beyond" the level of the book. It was annoying since it sounds like the first thing one would want to work out!

In any case, I was going through chap 9 when I made the horrible mistake to start leafing through the later chapters. DON'T DO THAT! I haven't been able to put the book down. I have read parts of almost all the chapters up to chap 23 because I was too happy to finally find a reference that was taking the time to explain things!

Here is the URL you requested.

http://xserver.lns.mit.edu/~zwiebach/firstcourse.html

Ok Thanks!

I must thank you for an important insight. When you said that you could paint one of the points, you made me realize what he means when he says:

"the string is not made of constituents that whose position we can keep track of"

His statement is not one that you could prove, it is an assumption. It is equivalent to saying that the string has no underlying structure, it is the fundamental 'thing'.

Indeed. But I guess that even though it's not a classical string (in the sense of being made of atoms), it is still classical (in the sense of not "quantum fuzzy") and that each piece evolves at a speed below (or equal to) the speed of light. Accepting that, then it seems to me that the existence of a timelike tangent vector everywhere (except possibly at the end points of an open string) follows. Does that sound reasonable to you?

I wonder if the statement that a timelike tangent vector exists at each point on the world-sheet is not itself an assumption, and his 'proof' is irrelevant (to add to its growing list of deficiencies).

I also feel there is something not quite convincing. But what do you think of my above argument?


Regards

Pat

 

[Jimmy Snyder]

Hi Pat,

I don't know how to quote your post, so I just repeated it below.

it is still classical (in the sense of not "quantum fuzzy") and that each piece evolves at a speed below (or equal to) the speed of light. Accepting that, then it seems to me that the existence of a timelike tangent vector everywhere

The way I think of it is that although we can visualize the world-sheet, and we can think about the plane tangent to the sheet at any point, we cannot trace a curve on the sheet and say that the curve is the world line of any physical point. For the same reason, I don't think we can identify any piece of the world-sheet and view it as the subsheet of a piece of the string.

If your approach were correct, then why would the endpoints be treated special? Indeed, I asked Professor Zwiebach that same question, but he didn't answer me.

 

[selfAdjoint]

Perhaps he sees the open string as a "thing with endpoints" but not further decomposable? This whole discussion seems odd to me because when they start really working with the worldsheet they do distinguish points on it, for example the "vertices".

 

[nrqed]

Hi Pat,

I don't know how to quote your post, so I just repeated it below.

When replying, click on the "QUOTE" button at the bottom of the post you are replying to.

it is still classical (in the sense of not "quantum fuzzy") and that each piece evolves at a speed below (or equal to) the speed of light. Accepting that, then it seems to me that the existence of a timelike tangent vector everywhere

The way I think of it is that although we can visualize the world-sheet, and we can think about the plane tangent to the sheet at any point, we cannot trace a curve on the sheet and say that the curve is the world line of any physical point. For the same reason, I don't think we can identify any piece of the world-sheet and view it as the subsheet of a piece of the string.

I see what you are saying and it makes sense. We can't identify a small length element "dl" of the string and keep an eye on it, following it as if it was a point particle. However, the way I think about it is that if we look at a piece of the worldsheet, even though I can't pick out a line and say that this is the worldline of a specific piece of the string, I still can find a tangent vector that is timelike. For this to be possible, all I need to know is that no part of the string will move faster than c.

Again, I agree that I can't say that this timelike vector is tangent to the motion of a specific piece of the string. Still, I think that if none of the pieces moves faster than c, it will be possible to find a timelike vector tangent to the worldsheet. Does that make sense?

Notice that it seems to me that all we can say is that no piece moves faster than c. So I don't seen how to rule out that the tangent vectors be null instead of timelike. That restriction is something I don't understand. And it maybe what you are really worried about.

If your approach were correct, then why would the endpoints be treated special? Indeed, I asked Professor Zwiebach that same question, but he didn't answer me.

I don't know if you mean special in the sense that we can follow the endpoints or special in the sense that it moves at c.

In the first case, the endpoints canactually be followed so we can talk about the worldline of that small piece of string.

On the other hand, as I said above, the exclusion of null tangent vectors is something I don't quite understand. Why can't we say that we can always find vectors tangent to the worldsheet which are either null or timelike (instead of saying that there must be at least one timelike tangent vector)? I don't know. Maybe that's your whole point.

The end points follow worldlines with timelike tangent vectors, so why can't all the points in the string follow the same kind of motion, right? Why can't all the string move at c, without vibrating or spinning, for example? That I don't know.

Pat

 

[Jimmy Snyder]

selfAdjoint, thanks for your input. You bring up a number of interesting issues.

Perhaps he sees the open string as a "thing with endpoints" but not further decomposable?

Wow! This is great food for thought. I wonder if the endpoints of a string can be considered constituents of the string. If so, that would mean that the world is made of strings and endpoints. Zwiebach never speaks of an open string without it's endpoints, but I don't see why not. The worldsheet of such a string would have no edge.

This whole discussion seems odd to me

That's my fault. I started this thread off with a bad question. I intend to start a new thread with a better question.

because when they start really working with the worldsheet they do distinguish points on it, for example the "vertices".

The issue is not distinguishing points on the worldsheet, or on the string. The issue is taking two distinct points on the worldsheet and identifying them with a single point on the string. This is what Zweibach claims cannot be done. Since we can't do it, it renders my initial question ridiculous.

 

[Jimmy Snyder]

Hi Pat,

I don't seen how to rule out that the tangent vectors be null instead of timelike. That restriction is something I don't understand. And it maybe what you are really worried about.

Bingo.

I don't know if you mean special in the sense that we can follow the endpoints or special in the sense that it moves at c.

My fault. I meant the latter.

Why can't we say that we can always find vectors tangent to the worldsheet which are either null or timelike (instead of saying that there must be at least one timelike tangent vector)? I don't know. Maybe that's your whole point.

Bingo again.

Why can't all the string move at c, without vibrating or spinning, for example?

Wow! If the string is associated with a photon, isn't that what is happening? Why is there a timelike vector at any point on the world sheet of a photon?

I intend to abandon this thread and start a new one. Thanks to all for your invaluable help. You have helped me to formulate the question better.

 

[nrqed]

selfAdjoint, thanks for your input. You bring up a number of interesting issues.



Wow! This is great food for thought. I wonder if the endpoints of a string can be considered constituents of the string. If so, that would mean that the world is made of strings and endpoints. Zwiebach never speaks of an open string without it's endpoints, but I don't see why not. The worldsheet of such a string would have no edge.

I really don't see what you would mean by this. An open string necessarily has end points! I really think there is no other way to think about it.

What *is* special about the end points is that we can follow them and therefore assign a worldline to them.

Regards, and thanks for the interesting discussion.

Pat