Discussion:
Size of strings compared to size of elementary particles
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Ulrich Thiel
2005-04-20 08:41:57 UTC
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Hi!

I'm a novice so please excuse my question.
So far as I've 'understood' string theory, strings are very small
(they're supposed to be as long as the Planck length) and that certain
vibrational patterns of a string correspond to certain elementary particles.
If this is right I wonder how a small string can be identified as an
electron for example which has a tremendous size compared to the size of a
string. I don't understand this. Is it my classical view of an electron
as a small 'ball' that's wrong here?

It would be nice if somebody could help me here.

Uli
Urs Schreiber
2005-04-20 09:56:04 UTC
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Post by Ulrich Thiel
I'm a novice so please excuse my question.
So far as I've 'understood' string theory, strings are very small
(they're supposed to be as long as the Planck length) and that certain
vibrational patterns of a string correspond to certain elementary particles.
This is often said, but is a little subtle. For superstrings, which are the
strings that can have fermions in the spectrum, the "vibrational patterns"
that leads to different particles in the massless sector are actually not
true vibrations but fermionic vibrations, if you wish. See the discussion
here:

http://golem.ph.utexas.edu/string/archives/000334.html .
Post by Ulrich Thiel
If this is right I wonder how a small string can be identified as an
electron for example which has a tremendous size compared to the size of a
string. I don't understand this. Is it my classical view of an electron
as a small 'ball' that's wrong here?
Yes, your classical view is wrong. Electrons appear in all accelerator
experiments as pointlike. No substructure or extension is being observed, up
to the currently available precision. The reasoning that leads to the
"classical electron radius" is outdated since the conception of quantum
mechanics and quantum field theory.
Ulrich Thiel
2005-04-20 15:45:06 UTC
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On 2005-04-20, Urs Schreiber <***@uni-essen.de> wrote:
Hi!

Thanks for the answer.
Post by Urs Schreiber
Yes, your classical view is wrong. Electrons appear in all accelerator
experiments as pointlike. No substructure or extension is being observed, up
to the currently available precision. The reasoning that leads to the
"classical electron radius" is outdated since the conception of quantum
mechanics and quantum field theory.
They really appear pointlike? I didn't know this although I remember now the
deBroglie wave description of electrons (which is also outdated I
suppose). Is it right that string theory tells us now that electrons are not
pointlike and they only appear to be pointlike because the size of a string
is so small that we cannot see it with our technical equipment?
But what about protons? Do they also appear to be pointlike?

Thanks for the answer again and sorry for the disturbance of the
scientific atmosphere ;)

Uli
Urs Schreiber
2005-04-20 15:58:54 UTC
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Post by Ulrich Thiel
They really appear pointlike?
Yes. You collide them in accelerators with other stuff, like with
positrons, and you don't see any inelastic scattering or anything which
would reveal an extension or composition of either electrons or positrons.
Post by Ulrich Thiel
I didn't know this although I remember now the
deBroglie wave description of electrons (which is also outdated I
suppose)
This is not outdated in the sense that the idea of a "classical electron
radius" is. The deBroglie wavelenth as well as the Compton wavelength of
any particle are, however, not the same as the fundamental extension of
this particle. For practical purposes, like for instance in solid state
physics, it is often helpful to think of electrons as being smeared over a
region of the deBroglie wavelength or something. But the electron itself
is pointlike.

You may be familiar with the textbook discussion of the hydrogen atom.
Note that here both the electron and the nucleus are taken to be
pointlike. The configuration of the system is described by the coordinates
x_n of the nucleus and those x_e of the electron. Their potential energy
is proportional to the inverse of the distance between these two
positions. Clearly all this assumes that both objects are points.

Now, this is an approximation. We know that the nucleus is not really a
point. It may consist of several protons and neutrons, or at least one
proton in the case of the H-atom. These are held togther by lots of
gluons, too.

So in most QM textbooks somewhere in the later chapters on perturbation
theory you will see the effects discussed that appear once we realize the
mistake of having treated the nucleus as pointlike.

The electron, however, is pointlike to such a good approximation at least
that so far no deviation from its pointlike-ness could be measured.
Post by Ulrich Thiel
Is it right that string theory tells us now that electrons are not
pointlike and they only appear to be pointlike because the size of a string
is so small that we cannot see it with our technical equipment?
Yes!
Post by Ulrich Thiel
But what about protons? Do they also appear to be pointlike?
Protons are composed of quarks. QUurks are elementary particles in the
standard model of particle physics, protons and neutrons are not. All
elementary particles appear pointlike to all our current equipment.

And all these apparently pointlike particles of the standard model would
turn out to be really little wiggly strings on very small scales, if
perturbative string theory is the right description of nature.
Post by Ulrich Thiel
Thanks for the answer again and sorry for the disturbance of the
scientific atmosphere ;)
No problem. You should pester everybody around here until they break down
and give you a good scientific answer to a good layman question. :-)
Ulrich Thiel
2005-04-22 08:41:17 UTC
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Post by Urs Schreiber
Protons are composed of quarks. QUurks are elementary particles in the
standard model of particle physics, protons and neutrons are not. All
elementary particles appear pointlike to all our current equipment.
Okay, thanks for clarification!
Post by Urs Schreiber
No problem. You should pester everybody around here until they break
down and give you a good scientific answer to a good layman question. :-)
Good to know :)


Uli
pirillo
2005-05-02 13:15:08 UTC
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I think ultimately (and even in the intermediate steps)]
the size of the string 1) does not matter 2) is Ill defined.
One first has to define what one means by string size.
1) Conceptually
2) Experimentally
To some people string size may just be the value
of a coupling constant.
Urs Schreiber
2005-05-05 11:00:53 UTC
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Post by pirillo
I think ultimately (and even in the intermediate steps)]
the size of the string 1) does not matter 2) is Ill defined.
One first has to define what one means by string size.
1) Conceptually
This is another FAQ. The last time this came up was here:

http://groups.google.de/group/sci.physics.strings/msg/01c88b017e3ccc62?hl=de
The following was my reply at that time. (There is of course much room for
improving on that reply.)
Post by pirillo
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=dec722c5.0303061133.1bf83085%40po...


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
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.


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Post by pirillo
To some people string size may just be the value
of a coupling constant.
That's not quite right. The value of the coupling constant in 10D string
theory is related to the dilaton which again is related to the circumference
of an extra dimension.
pirillo
2005-05-10 08:19:27 UTC
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Post by Urs Schreiber
Post by pirillo
To some people string size may just be the value
of a coupling constant.
That's not quite right. The value of the coupling constant in 10D string
theory is related to the dilaton which again is related to the
circumference
Post by Urs Schreiber
of an extra dimension.
What if there's no extra compactified dimension, what then.
What if I visualize a bosonic string living in 4 spacetime dimensions
See, the word "string size" is nearly meaningless without adding a lot
of qualifiers. If it's the average size [as you prescibed wrt a given
cutoff
procedure] , wrt a string state then I expect this to wary as the state

varies, what state are you talking about?
Urs Schreiber
2005-05-11 19:10:23 UTC
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Post by Ulrich Thiel
Post by Urs Schreiber
Post by pirillo
To some people string size may just be the value
of a coupling constant.
That's not quite right. The value of the coupling constant in 10D
string
Post by Urs Schreiber
theory is related to the dilaton which again is related to the
circumference
Post by Urs Schreiber
of an extra dimension.
What if there's no extra compactified dimension, what then.
Then it's still not true that the string size is the value of a coupling
constant.
Post by Ulrich Thiel
What if I visualize a bosonic string living in 4 spacetime dimensions
The you have a noncritical string and are in pretty deep waters.
Post by Ulrich Thiel
See, the word "string size" is nearly meaningless without adding a lot
of qualifiers.
True. So go ahead and specify precisely which notion of "string size" you
are interested in.
Post by Ulrich Thiel
If it's the average size [as you prescibed wrt a given
cutoff
procedure] , wrt a string state then I expect this to wary as the state
varies, what state are you talking about?
Indeed. That procedure I mentioned gives you an operator and taking the
expectation value of that operator in a given state of the string gives
the rms size of the string in that state.
pirillo
2005-05-10 08:20:30 UTC
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Do you integrate over the wholeworldsheet (X-X_0 )^2 ,
or do you integrate over the sigma coordinate only?
Urs Schreiber
2005-05-11 19:06:10 UTC
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Post by pirillo
Do you integrate over the wholeworldsheet (X-X_0 )^2 ,
or do you integrate over the sigma coordinate only?
You want to average that over space _and_ time to get the rms size of the
string. I seem to recall that this is discussed in the references that I
provided.
pirillo
2005-05-12 07:46:08 UTC
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Post by Urs Schreiber
You want to average that over space _and_ time to get the rms size of
the string. I seem to recall that this is discussed in the references
that I provided.
Oh, so you do sort of a " average size for the whole history"
operator which say first computes the average distance from cm at each
time and then averages this over the whole history.

Hmmm?

I thought in curved backgrounds you did this at one instant
and found that the string became larger as time passed.
Or, were you saying more like there's an external parameter
which labels a family of spacetimes (the string lives in)
and as you vary this parameter, the quantity you described
above varies, although --if one changes the target space,
then one changes the states.
Urs Schreiber
2005-05-12 08:33:28 UTC
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Post by pirillo
Post by Urs Schreiber
You want to average that over space _and_ time to get the rms size of
the string. I seem to recall that this is discussed in the references
that I provided.
Oh, so you do sort of a " average size for the whole history"
operator which say first computes the average distance from cm at each
time and then averages this over the whole history.
Hmmm?
I am sorry, but I have a hard time understanding what you are confused
about.

The idea we are talking about is not particular to string theory at
all but seems to be just a matter of common sense: You have some fluctuating
something and want to get an idea of its rough size. So you average the
distance of all its points from its center of mass and, since its
fluctuating, average these distances over some period of time. If that piece
of something is systematically growing or shrinking on larger time scales
you will want to take the time average over an interval which is large
enough compared to the fluctuations but small enough to be local in time
with respect to the long-term behaviour.

To be frank, I feel that the discussion of this point is getting a little
off-topic for sci.physics.strings.
pirillo
2005-05-17 08:11:50 UTC
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Post by Urs Schreiber
you will want to take the time average over an interval which is
large enough compared to the fluctuations but small enough to be local
in time with respect to the long-term behaviour.
Yeah, but you said integrate over the "whole" world sheet
not over a "short time band" on the worldsheet. So you're averaging
over the "whole" infinite history. I'm just saying what you seemed to
say --- not what you meant. Which I now think is to integrate over a small
time band.
Post by Urs Schreiber
To be frank, I feel that the discussion of this point is getting a
little off-topic for sci.physics.strings.
And questions about Gerbes, Calabi Yau manifolds and all these objects
which are purely mathematical are not! Ha :-)
I think this "is" very stringy!

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