A few very simple notes on Superstring Theory.

Version : 0.5
Date : 14/04/2012
By : Albert van der Sel
For who : For anyone who likes a very simple introduction in Superstring Theory and Branes. Hopefully, it's any good.




1. Just a few notes on the History and Motivation.

1.1 Kaluza-Klein Theory:

As is the case with any theory in physics (or other science), the history of how it came into existence, and
how it evolved, is just absolutely facinating material by itself !
Ofcourse, it's impossible to say anything of relevance in such a short note like this one.

However, I would like to mention just one special historical fact that contributed to some fundamental ideas
of what is called "Superstring Theory" today.
Around the time Einstein worked and completed his General Theory of Relativity (GTR), some other remarkable work surfaced.
It's nowadays called the "Kaluza Klein" (KK) theory. It's up to the historians to determine how much Nordström, Klein,
and Kaluza, each contributed to the theory.

As you might know, GTR essentially is a unified description of gravity as a geometric property of space and time, especially
the curvature of space. In this sense, it's a 4 dimensional description using 3 spatial dimensions and the time dimension.
The GTR Field Equations of Einstein are certainly a cornerstone in Physics and Cosmology, up to this very day.

Remarkably, the Kaluza-Klein theory extends GTR to a five-dimensional spacetime framework, thereby using one additional
spatial dimension. The objective was unification of gravitation as described by Einstein, and electromagnetism as
is decribed by the Maxwell equations. It's very remarkable to see how the KK theory can be "projected" to 4 the dimensional
descriptions of the Einstein relations, and the equations of Maxwell.

One problem though, was the unobservable extra dimension. Especially for those days, it might be argued that a 5th dimension was probably
regarded as a "too wild" idea for the majority of physicists.
However, the KK authors expressed the idea that the additional spatial dimension was compactified, meaning that
it's metric, or size, is so small, that it will essentially go "unnoticed". It is as if a "one-dimensional line", if you could zoom in in an incredable amount, would actually
resemble a "cilinder", where the additional dimension wraps or circles around the "line".
This way, where this extremely small periodic dimension "goes back on itself", means that it goes undetected on a larger scale.

It's really a fact that modern Superstring theory has borrowed the concept of compactification of additional "hidden" dimensions.

Presently, Superstring Theory uses a 10-dimensional "spacetime" which has 9 spatial dimensions and one time dimension.
Of those 9 spatial dimensions, 6 of them are "compactified" (or curled up) and go essentially "unnoticed".
If we consider M-Theory, (a sort of "extension", or 11D Supergravity), we live in a 11-dimensional "spacetime".
Later more on this.

Note: If Kaluza Klein is really new to you, you might search for it using a search engine on the Internet.
Undoubtly, facinating stuff will pop up.



1.2 Fundamental values:

We know that the speed of light (c), the Gravitational Constant (G), and (the reduced) Planck's constant ℎ,
are truly fundamental constants. On basis of those constants, some other very special values can be derived.
If we do our maths correctly, we then get an astounding fundamental "length", "time", and "Energy or Mass"

For example, the socalled "Planck's Length" lp = √( ℎG/c3),
which is an absolutely astounding short "distiance". It's about 1.6 x 10-35 meter. If we consider the fact that this length is about 10-20 smaller than a proton,
we can't deny it's an astonshing small length.

Similarly, the "Planck's time" lt = √ (ℎG/c5),
about 5 x 10-44 second, is mindboggling short as well.
The Planck's time is about the time a photon would need to cross the Planck's length.

Lastly, the Planck's mass corresponds to an Energy of about 1.2x1019 GeV. That's way, way more than
the current particle accelerators can reach (which is in the order of several TeV)
Just think a minute of this small length and time. What could be the relevance of those values?
  • Many physicist believe it’s the scale, or Energy, where Relativity, Gravity and Quantum Mechanics,
    will somehow “meet” or “converge”.
  • Also, many physicist believe that those values are so fundamental, that they have to play a significant role
    in any theory that tries to describe the physical world.
  • Some say that at the Planck scale, the seemingly orderly space-time fabric, seen at larger scales, ceeses to exists.
    Actually, Superstring Theory postulates strings with a "size" of the Planck length, where the different harmonics
    of the string represent different particles.

Although the remarkable values we have seen above, are known for quite some time (say, already in the first halve
of the former century), as we will see, Superstring theory indeed takes them into account.
In a way, Superstring Theory provides for an explanation of for the "small values".


1.3 Other "stimula" for String theories:

Although the "history" of strings might even be older than the following, it's certainly true that certain studies
around the '70ties on the nuclear force (the strong interaction), proposed a model using vibrating one dimensional "strings".
But quantum chromodynamics (QCD) used a more succesfull model (using "gluons", color), for research on the strong interaction,
resulting in the fact that the focus really shifted to QCD.

However, not much later (middle and late '70ties), especially theoretical physicists renewed interest in
string theories, resulting in a reasonably sound "bosonic string theory".
Then, late '70ties and early '80ties, the train really got momentum and string theory went booming.

Also, in the times mentioned above, we must not forget that the search for a "Theory Of Everyting" (TOE),
was pretty "hot" too. Indeed, for example, Relativity, and especially Gravity, were difficult to reconcile
with Quantum Mechanics.
Indeed, many physicists say that Superstring Theory in many ways might be seen as a theory of Quantum Gravity.
For those physicists (while still many loose ends exists), the theory unites Quantum Mechanics and GTR.

As another example, it was strongly suspected that the (4) fundamental forces actually should have a common "ancester".
Somehow, in the very early stages in the Big Bang, phases have changed as the Temperature cooled down,
resulting in broken symmetries and resulting in the forces and particles as we know them today.
Physicists were motivated for finding a "Theory Of Everyting". Indeed, during those times, it was realised
that string theories were really capable of describing all elementary particles as well as the interactions between them.


1.4 Avoidance of singularities:

Although there still exists a few opponents of Superstring Theories, most physicists appreciate the avoidance
of "singularities" which plaque a lot of other theories.
As we will see in the next chapter, even at a "collision", strings do that over a certain region,
so singularities are sort of avoided in a natural way. This is considered a to be a big pro for Superstring Theory.


1.5 No Direct experimental setups or data:

As said before, there are still quite a few opponents of Superstring Theories. Their main complaint is about the lack
of evidence, or lack of means to "test" the theory. Indeed, the "world" of strings is so small (or the Energy so incredable high),
that "probing" those scales is completely out of our reach. So, the theories sofar are entirely "theoretical".

Well maybe there is hope. String theory is a "TOE", so if our present low energy experiments
(viewed from our current capabilities), somehow makes "supersymmetry" likely, that might be a plus for
Superstring Theory too.

Implictily it is often assumed that the hidden (curled up) dimensions are in the Plank Length realm.
However, some physicists have argumented that (some of) the unnoticed dimensions might be considerably larger.
Indeed, some do not even exclude that one or more are even in the mm scale.
If so, maybe that will enable physicists to device experiments to notice very specific events.
Some serious proposals have been forwarded, and various experiments from hadron, p+p- and e+e-
collisions have been analyzed. However, up to now, no convincing results have been returned.


1.6 Some Competing (and nowadays possibly complementing) Theories :

Other paths have been found as well, to describe particles, spacetime, and to try to unite General Relativity and Quantum Mechanics.
For example "Loop Quantum Gravity" (originally) deviated significantly from string theories.
Many theoretical Physicists are active in that field.

As another approach, "spin networks" and "twistors", which essentially tries to explain the "world" at the Planck scale,
is viewed by many other scientists as a possible promising path to arrive at a discription of spacetime, and
the smallest we can think of.
In particular, it's radically different from the traditional description of spacetime, which treats it as a background continuum.
Instead, in twistor theories, spacetime can be "created", or is the "result" of more fundamental events.

Originally conceived by Witten, nowadays many papers go around, and still get published, that ties twistor and string theories together.
Some go even so far as to speak of a new "string revolution".


Ofcourse, there is so much more to say about the history of String Theories. If you search the Net, you will find many great
articles, where you can read all about the early history, as well as the socalled 1st and 2nd String "Revolutions".




2. Just a few simple notes on the essence of Superstring Theory / M-Theory.

Trying to imagine "how it all looks like" is not easy. Indeed, as always, a "certain" abstraction level is unavoidable.
But that's not really new. Picturing a classical "point particle", or QM "wave function" is not simple either.

String theory goes a bit like this:

All particles are 1 dimensional strings, with "sizes" in the realm of the Planck lenght.
That is not to say they are at that "length" perse.

What we can say is that:

⇒ In original String theories, 10-dim space-time is their natural environment,
so to speak, while we (macroscopic "objects") only experience the 3 streched spatial dimensions (and are unaware of
the 6 compactified dimensions).

⇒ At a certain moment, 5 different String Theories were developed. However, not much later it was realized
that they were sort of manifestations of a "deeper" theory. This unifying 11-dimensional theory is called "M-Theory".
Currently, M-Theory is the working theory for most string physicists.

A string vibrates, where the different vibrating modes (harmonics) determines the sort of particle we have.
Mind you: actually this is a very consistent view:

- Classically, we have many sorts of particles, all with their own set of different properties. You can rightfully ask:
Where do all the "forms" come from, and where do there properties like charge, spin etc.. arise from?

- With Superstrings, only the mode of vibration of the string, is sufficient to "know" what particle it is.

Ofcourse, tons of questions exists. Let's try if we can find some answers.


2.1 Why 10 or 11 dimensions, and how come we don't seem to notice them?

As to the question why we need 10 (or 11 in M-Theory) dimensions, there are several answers.

First, the "calculus" in String theories is massive (or very complex). We don't do any calculus here.

The fundamental formulas in the first (original) String Theories, use a certain factor, namely "1 - (D-2)/24",
as to express the number of dimensions (D) needed to cancel "anomalies", that would otherwise render the theories useless
So, in the first theories, the number of dimensions was 26.

Later (more general) revisions used the factor "8k+2" or "10-D" to cancel anomalies, so, when k=1 we get D=10
as the first number of Dimensions that enables physicists to build a consistent theory.

For most people, the additional (compactified) dimensions really appeals to their imagination.
But do they really exist? And, how come that we don't notice them at all?

For the first question:

Most physicists think they do. See also chapter 1. But there are certainly opponents.

For the second question:

If we can understand that light, and other interactions, just follows the common "stretched" dimensions,
then we are practically already done.
For that matter, there could be even 26 dimensions. It doesn't matter. The interactions we are used to, just "play"
around in our common 4-dim space-time.
Only if the Energy involved gets so terribly high, that the resolving power gets terribly high too, we are able
to "see" the compactified dimensions.

As another reasonable interpretation: Although it's a bit more complicated than the following reasoning, it is true
that if additional dimensions "are periodic", or "returns on themselves", they don't "contribute" in
macroscopically sized events.


2.2 Branes and Open/Closed Strings

In the process of working on the String theories (espescially) by theoretical physicist, they explored different
boundary conditions. One result that sticked around was the "Brane" (or p-brane, D-brane).
You can think of it as a sort of "manifold", a multidimensional construct, on which "strings can end".

Now, you must know that we can have "open" and "closed" strings. The open strings then, would end on the brane.
Theoretically, a brane could have any number of dimensions. However, it's extremely tempting to visualize our common world
as the "Brane" (practically speaking: our Universe).

As a simple analogy: I am sure you can visualize a "plane" in 3D space, in which case that plane would be the "brane",
or in other words: our Universe.

Fig 1.


Indeed, an interesting model emerged: stuff like photons, quarks and leptons, exist in our subspace, a 3D-spatial/1-Time-brane.
But there are also the "hidden" dimensions, known as "the bulk".

One special closed string is the "graviton", which is the force-carrier of Gravity. It's not bound on our brane,
and thus is permitted even to "escape" in the bulk. Needless to say that the theories are quite complex, but quite
a few physicist suspect that this is the reason why, on a atomic scale (which is way, way larger than Plank's length),
gravity seems so weak.


True, this very simple note leaves out about 99.999%, but if you were indeed new to the subject,
hopefully you are triggered to do more explorations !