String Theory: Minimal Distance Implications

[nomadreid]

In String Theory, the existence of a minimal size of a dimension, probably the Planck distance, is proposed based on T-duality. At first sight, the argument seems to also imply the minimal size for anything, assuming that string theory or M-theory is correct. (The fact that the distance proposed for the minimal size of a dimension coincides with the minimal size which has been proposed from dimensional analysis is curious in itself.) However, not being a physicist, I am not sure if this extension is correct. Could someone clarify this for me? Thanks.

 

[suprised]

This issue more general than T-duality which requires a compact dimension. It also applies to questions such as to how "thick" a string is (a source of permanent confusion, see other threads).

The point is, how would one measure sub-planckian distances, such as the thickness of a string? In a complete theory such as strings, there are no other probes than strings. So one needs to scatter strings against strings. And what happens if you tune up energy is that more and more modes of the strings get excited, so they turn into some kind of fuzzy mess that actually grows in size with energy. There are other, related pictures which essentially tell that one forms black holes that become larger with larger energy (and thus may even become classical at some point).

In other words, there is simply no way to probe smaller distances than the Planck Length, and questions how strings look "under a microscope" and whether they are infinitesimally thin or have a tiny diameter, are ill-posed.

 

[nomadreid]

belatedly thanks, surprised. (Trouble with Internet.) Am I to gather that you are saying that not only the question about the thickness of strings, but also the question as to the size of a dimension, is ill-posed? I have not, in my readings on string theory, found anything even mentioning the idea of the thickness of a string (except for some mentions of being one-dimensional in space or tubular in spacetime), but there are mentions about the size of the hidden dimensions.

[I presumed that the missing verb in your first sentence was "is" between "issue" and "more".]

From your statement about a "fuzzy mess that actually grows in size with energy" , which would seem to apply also to particles as well as to dimensions, it was not clear to me whether you meant this as a result of the argument based on T-duality or on other arguments. I suspect the latter. If so, is the argument based on T-duality (assuming that we are applying it to compact dimensions) superfluous?

 

[friend]

In other words, there is simply no way to probe smaller distances than the Planck Length, and questions how strings look "under a microscope" and whether they are infinitesimally thin or have a tiny diameter, are ill-posed.

If nothing is smaller than the Planck Length, then what is the smallest wave-length of the wave of a string?

 

[nomadreid]

My question continues to be misunderstood. I am asking about the logic of a particular argument, not about the conclusion. I am not asking whether there is a smaller distance than the Planck distance. What I am asking is how far the validity of the argument which uses T-duality reaches. To make an analogy, of the proofs that work for Euclidean geometry, some work for non-Euclidean geometry, and some don't. Or another analogy: if a school child proves that in a right triangle with legs 3 and 4 has the hypotenuse 5, I can accept the conclusion, but if the reasoning is that 5 is 1 more than the biggest leg (hey, it works for 5,12, 13!), then I don’t accept it.
Or, to put it schematically,
T-duality |- minimal distance for dimensions
T-duality |-? minimal distance for particles

 

[suprised]

If nothing is smaller than the Planck Length, then what is the smallest wave-length of the wave of a string?

In brief, the notion of "length" becomes ill-definded below the Planck scale.

 

[suprised]

Am I to gather that you are saying that not only the question about the thickness of strings, but also the question as to the size of a dimension, is ill-posed?

Yes, the latter is the content of T-duality.

Note that the Planck scale, or "string size" P, and size of a compactification space R are different, independent entities. Each of them implies some "ill-definedness" of some question. The Planck scale P is a cutoff below which no measurement can be made, independently from any compactification and strings. On the other hand, T duality is stringy, because it involves wrapped strings. The size R of a compact space is ill-defined in the sense, roughly, in that one cannot distinguish R from P^2/R.

From your statement about a "fuzzy mess that actually grows in size with energy" , which would seem to apply also to particles as well as to dimensions,
it was not clear to me whether you meant this as a result of the argument based on T-duality or on other arguments. I suspect the latter.

Yes, T-Duality is something else, and arguments based on it apply only to compactified dimensions. There are other, general arguments related as to why the Planck scale is a firm cutoff (formation of black holes eg).

If so, is the argument based on T-duality (assuming that we are applying it to compact dimensions) superfluous?

What I am asking is how far the validity of the argument which uses T-duality reaches. ...Or, to put it schematically,
T-duality |- minimal distance for dimensions
T-duality |-? minimal distance for particles

Depends on what you are asking for. T-duality seems to be a good symmetry of any string compactification, and is strictly valid and has definite consequences, so it is never superfluous. It is however not necessary for arguing that the Planck scale is a minimal length scale below which measurements cannot be done.

 

[arivero]

I was also confused by the T-duality argument. In bosonic strings, T-duality is a self duality, so they can be interpreted in the way of a minimal possible compactification. In superstrings, this is not true anymore, it maps a theory to a diffrent one. Which is a pity because the exact self-duality point usually carries by itself an enhancement of symmetries.

 

[suprised]

I was also confused by the T-duality argument. In bosonic strings, T-duality is a self duality, so they can be interpreted in the way of a minimal possible compactification. In superstrings, this is not true anymore, it maps a theory to a diffrent one. Which is a pity because the exact self-duality point usually carries by itself an enhancement of symmetries.

Well with a combined duality operation acting on two circles simultaneously one maps back to the same superstring.

What is perhaps confusing and not so much known since most often T-duality is explained only for a circle or a torus, is that for more complicated compactification manifolds like Calabi-Yau spaces, the T-dualities form in general a large discrete group, and only one generator corresponds to the inversion of the "size". In fact there can be hundreds of size parameters (governing the sizes of various submanifolds like 2-cycles), and the duality grous acts on all of them in an in general non-transparent way. In general the duality group G is some subgroup of Sp(2h_11+2, Z) where the Hodge number h_11 can be in the hundreds.

So the physical parameter space is identified by the action of G, much like the parameter space of a circle compactification is identified by the operation of Z_2 which acts like R->1/R, ie, the parameter space is given by (positive real numbers)/Z_2. So a theory with a given set of internal sizes is equivalent to many other theories with different internal sizes and theta angles. There can also be non-geometric regimes where eg the whole CY space has zero volume but a bunch of sub-manifolds have non-zero volumes, etc. In order to describe this properly, one needs to give up standard notions of geometry and replace them by some generalized stringy quantum geometry, where notions like dimension, distance, volume, and even of topology are revised.

And no, philosophers won't be able to say any useful word about this (pun intended ;-).

 

[arivero]

And no, philosophers won't be able to say any useful word about this (pun intended ;-).

I missed the pun :-( but I am going to quote a philosopher:

Suppose yn a number of Mathematicall points were indued wth such a power as yt they could not touch nor be in one place (for if they touch they will touch all over, & bee in one place) Then ad thees as close in a line as they can stand together every point added must make some extension to ye length because it cannot sinke into ye formers place or touch it so here will be a line wch hath partes extra partes; {illeg} another of these points cannot bee added into ye midst of this line, for yt implys yt ye former points did not lie so close but yt they might lye closer. The distan\ce/ yn twixt each point is ye least yt can be & so little may an attome be & no lesse: now yt this distance is {illeg} indivisible (& therefore ye matter conteined in it) is thus made plaine:

 

[suprised]

I missed the pun :-( but I am going to quote a philosopher:

Well I meant that philosophizing eg. about how spacetime "should" look at small distances, ie., what the properly generalized notions of geometry are, whether it should be discrete etc, doesn't make much sense without actual computational insights.

 

[crackjack]

@surprised

for more complicated compactification manifolds like Calabi-Yau spaces, the T-dualities form in general a large discrete group

Is this larger symmetry same as what is referred to as mirror symmetry?

In order to describe this properly, one needs to give up standard notions of geometry and replace them by some generalized stringy quantum geometry, where notions like dimension, distance, volume, and even of topology are revised.

Is this requirement for Generalized Geometry a result of the need for tools to study strings in the presence of background fields (whereas, usual geometry is suited only with fluxes turned off)? In this sense, is Generalized Geometry a worldsheet equivalent tool of Pure-Spinors for spacetime?

 

[suprised]

Is this larger symmetry same as what is referred to as mirror symmetry?

Not quite, mirror symmetry can be understood as part of T-duality, but there are many more dualities beyond that. I also was talking only about perturbative dualities which are true order by order in the string coupling. But on top of that are non-perturbative dualities that act on the string coupling as well (S-, U-Dualities).

Is this requirement for Generalized Geometry a result of the need for tools to study strings in the presence of background fields (whereas, usual geometry is suited only with fluxes turned off)? ?

No that's different; there is indeed the notion of "Generalized Geometry" which deals with ordinary geometry on top of which are flux background fields. That's still a very classical un-stringy geometry which appears in supergravity already. What I mean are stringy geometries that are mostly relevant at small distances, where ordinary notions of geometry (dimension, topology) become fuzzy or ambiguous or break down. Also equivalences due to dualities should be included in this generalized notion (ie, many different classical geometries are indistinguishable in string theory, so need to be identified).