## minimum size of a string

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>How would string theory be changed if the minimum size of a string wasn\'t\n10^ -34 metres?\nThe current radius of the universe is about 4 x 10^26 metres.\nDoes string theory predict that a string can be this size at the most?\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>How would string theory be changed if the minimum size of a string wasn't
$10^ -34$ metres?
The current radius of the universe is about 4 $x 10^26$ metres.
Does string theory predict that a string can be this size at the most?

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On Mon, 28 Jun 2004 10:40:07 $-0400,$ alistair wrote: > How would string theory be changed if the minimum size of a string wasn't > $10^ -34$ metres? > The current radius of the universe is about 4 $x 10^26$ metres. > Does string theory predict that a string can be this size at the most? What happens with the string depends on the target space time you embedd it into. If the spacetime is something like a closed universe, then I think the string could wrap around it. But actually, the energy of a string increases with it's length, and you'd get very very massive objects. So one normally thinks of strings to be very short, and even then the energies of the massive excitations are huge. Rene. -- René Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China



First of all, what do you mean by "the size of a string" 1) Do you mean minimum radius sized ball which contains the string? 2)Do you mean the length of the string? what do you mean?

## minimum size of a string

MANDRO:

First of all, what do you mean by "the size
of a string" 1) Do you mean minimum radius
sized ball which contains the string? 2)Do
you mean the length of the string? what do
you mean?

ALISTAIR writes:

I mean the length of an open string.


kurious wrote in message news:... > MANDRO: > > First of all, what do you mean by "the size > of a string" 1) Do you mean minimum radius > sized ball which contains the string? 2)Do > you mean the length of the string? what do > you mean? > > > ALISTAIR writes: > > I mean the length of an open string. > It would have to be the planck length. David



Well, I just made my statement so you'd think about this. You know, as Lubos informed me, the length of the average string is infinity. So what's your proposed resolution now?



"Ulmo" wrote in message news:53ca460a.0407011218.7b191562-10....google.com... > kurious wrote in message news:... > > MANDRO: > > > > I mean the length of an open string. > > > > It would have to be the planck length. If the mininum length of a string is the Planck length, does that mean the wavelength of the string's vibrations are sub-Planck length? Can anything of a sub-Planck length have any meaning? Cheers, Simon.



mandro wrote in message news:... > First of all, what do you mean by "the size > of a string" 1) Do you mean minimum radius > sized ball which contains the string? 2)Do > you mean the length of the string? what do > you mean? Both questions seem interesting to me, so please go ahead with both.



Well, I already said, that I'd been informed that the average length of a string is infinity. So, one may be able to try the other way of looking at it i.e., to catch it in a ball of a certain constrainig radius. $I was/am$ confused about this for a long time and several people told me stuff I didn't understad. The main reason for this is that I was looking at it as wavefunctions on the space of smooth loops (I think that's) what most amateurs have in mind when they thik of a string. However, over time, I have come to believe that what these guys are actually doing is that they as looking at it as a QFT in 2d space time (which I've known for years)but what I didnt know well was what the configuration space was for that theory and "how" to apply it to decide questions like this for string theory. I now believe that what these guys do, is to take the space of field configs on a one-dimenstional space say [0, $\pi] (2-d$ space time) to be a space that, at least, contains C[0, $\pi]$. Now, there's some spiel about most functions in $C[0,\pi]$ having infinite length, where the length is defined via some suitable limit of smooth functions. So, evidently the length of most objects in the string configuration space is infinite. Thus it's foolish to say that a string state has finite length (unless you allow $\delta$ functions in your space of wavefunctions). Now, the argument for the conataining in a ball part I don't know, but I'm sure how the experts view this is to ask the question "Can I have states where the value of the fields in configuration space are all contained in a certain ball" e.g., analize the case of a scalar field and ask this question.



"mandro" schrieb im Newsbeitrag news:dec722c5.0407131057.2602b41b-10....google.com... > Well, I already said, that I'd been > informed that the average length of > a string is infinity. Yes, but by regularizing (normal ordering) the observable which measures the size of the string, one obtains a finite value which is physically very interesting, since it can be related to black hole entropy considerations. I recall that you, mandro, have asked these questions before, and I think I had answered most of them, for instance in the thread http://groups.google.de/groups?selm=...ing.google.com But maybe I wasn't pointing you to enough literature. Anybody interested in these questions should have a look at the very nice paper Thibault Damour, Gabriele Veneziano: Self-gravitating fundamental strings and black-holes http://www.arxiv.org/abs/hep-th/9907030 and references given there, where the observable measuring the rms size of a string is given in equations (2.$9)-(2$.11). The idea is quite simple: The mean squared diameter of the string is the average of $(X-X_0)^2,$ taken over the worldsheet, where $X_0$ is the center of mass coordinate. Now expand X in terms of worldsheet Fourier modes as usual and then integrate over the worldsheet coordinates in order to average. The result is (2.11), which says that the rms size is proportional to $\sum_{n=1}^\infty \frac{1}{n^2} (\alpha_{-n} \cdot \alpha_n + \alpha_n\cdot \alpha_{-n})$. Clearly, when you take the expectation value of this guy in any string state you'll get an infinite contribution from pulling the annihilators $\alpha_n$ through the creators $\alpha_{-n}$. This is a common quantum effect and is removed by normal ordering$. It$ has been argued that this infinite contribution to the string's length has a proper physical meaning - but the point is that the remaining finite part has, too. In particular, the finite part is related to string/black hole correspondence, which I have tried to review here: http://golem.ph.utexas.edu/string/archives/000379.html . In Paris I had a chance to look at Barton Zwiebach's new textbook on string theory (my own copy has not arribed yet) and I saw that there, too, a very nice summary of the string/black hole correspondence along the lines summarized at the above link is given. So maybe mandro and others will benefit from having a look at that book.



Urs Schreiber wrote in message news:<2ln50rFeihpsU1-100000@uni-berlin.de>... > > Mandro wrote: Well, I already said, that I'd been > > informed that the average length of > > a string is infinity. > > Yes, but by regularizing (normal ordering) > the observable which measures the > size of the string, one obtains a finite > value which is physically very > interesting, since it can be related to > black hole entropy considerations. > > I recall that you, mandro, have asked these questions > before, and I think I had answered most of them, > for instance in the thread I wouldn't go as far as saying you answered them. Maybe more like tried to answer them. Firstly I don't think you were totally articulate in answering them (probably because people don't usually talk about this stuff)--although I must admit you know your stuff and are a fairly good expositor. I guess it's just that things like this don't have just one answer, i.e., it's more like a family of answers that depends on 1) What aspects you want to concentrate on, 2) How you decide to define the "string length" observable , and the string length measuring procedure etc. Naturally, all these aspects are usually not obtained in just one reply, unless the person replying is a supersmart alien, who sees all the possible aspects of this and gives them in his reply. Thus I don't think you can just give one aspect and say "case closed". > The idea is quite simple: The mean squared diameter > of the string is the > average of $(X-X_0)^2,$ taken over the worldsheet, [Moderator's note: I guess that Urs agrees that in any low-energy state of a vibrating string, the squared diameter has a logarithmically divergent (infinite) expectation value. As soon as we impose a cutoff, the quantity becomes finite and comparable to $l_{string}$ (squared), and imposing a finite cutoff is something we can afford for any particular physical question. The pieces of the string that are "infinitely far" are associated with infinite frequencies that are averaged to zero rapidly, and their effect on physics is very small. Nevertheless, it is not necessarily true that the squared diameter or the length of the string are necessarily the most interesting physical observables. Interesting observables are those that can be measured in experiments. The length or the diameter of a string cannot be measured so easily - you would need a point-like probe that is able to resolve substringy distances and quickly fly along the string - but string theory does not allow for the existence of very small very fast probes. LM] What do you mean by mean square diameter? [Moderator's note: I thought that Urs immediately defined it for you. Is not it more constructive to learn how to read first? LM] By the way, the objects that you describe here, are they operators, or simply expectation values? [Moderator's note: I am afraid that Urs will have a hard time to answer such a question. The expectation values in quantum mechanics are always expectation values of some operators, so whenever we talk about "simply" expectation values in quantum mechanics, we also talk about operators. One of your unsolvable problems, mandro, one that I have wasted 200 kilobytes of e-mails with, is that you are not able to understand that X - the position - or $X(\sigma) - a$ position of a point along the string - are operators, too, and they are totally analogous to P, J, or other operators. I discourage Urs from trying to answer these questions of yours because it seems as a clear waste of time. You are focusing in the position representation of the states in the Hilbert space, and you do not want to see that various operators are equally good operators, regardless whether they commute with position or not. LM] As Always Thanks



Urs Schreiber wrote in message news:<2ln50rFeihpsU1-100000@uni-berlin.de>... > "mandro" schrieb im Newsbeitrag > news:dec722c5.0407131057.2602b41b-10....google.com... > > > Well, I already said, that I'd been > > informed that the average length of > > a string is infinity. > > Yes, but by regularizing (normal ordering) the observable which measures the > size of the string, one obtains a finite value which is physically very > interesting, since it can be related to black hole entropy considerations. What about the old hadronic string? It was described, in the 1970's, as a straight string having a pair quark/antiquark in the extremes, and rotating about relativistic speeds. To get a constant force $-for$ quark confinement-, it was postulated (or proofed, perhaps?) that string energy is proportional to its length. So in that time the length has a valid meaning. Alejandro



I'll let the more erudite brainiacs respond to this, but I will venture the following, I don't think the old hadronic theory is exactly the same as the standard string theories. Come to think of it, even though I've heard countless times that string theory as it is known today evolved from a theory of looking at hadrons as a string with a quark at each end, I really dont understand this, or "how" it relates to the modern string. Just to say well, dahh they both mention the word "string" will no satisfy me either, that's like saying that classical mechanics already hints at string theory because it can describe strings. I.e., strings appear in both cases, but in totally different contexts. And, from this point of view, they have no relation to each other.



mandro wrote in message news:... > I'll let the more erudite brainiacs > respond to this, but I will venture > the following, I don't think the > old hadronic theory is exactly the > same as the standard string theories. On the other side, if we were to consider "string theory" to be only the five theories of strings and the results depending mathematically of these theories, most of the string papers should be assigned to other branchs of theoretical physics. Some authors recognize this by speaking of "string inspired" results.