A short overview.

Edit: By the way, have a look at this: http://arxiv.org/find/all/1/all:+AND+kane+AND+kumar+acharya/0/1/0/all/0/1 a list of papers relevant to this topic.

This post is intended to dismiss some of the claims that “String Theory isn’t testable“;.  So, let’s first list out some of the claims we hear about the experimental testability of  String Theory, in random discussions,  on comments by trolls on tRF , and by the well-known crackpots, (dubbed “Smoit”, but I suggest “Woilin”):

• String Theory disagrees with well-known Physics!
• Ya, whatever, fine, but it agrees with everything, so that basically means that String Theory isn’t testable! It’s like, pseudo-science!
• Ok, ok, but it isn’t testable at today’s energy scales, alright? Ha!
• Well, fine, whatever, but it has been experimentally disproven!
• Ok, fine, WHATEVER, but it hasn’t been experimentally proven, at least, ok? Ha! How will you counter that?!

Ok, so let’s counter each of them.

————————————————————————————————————————————————————————————————————-  String Theory does agree with well-known Physics.

It is a trivial excercise to show that String Theory agrees with General Relativity.

One starts with the beta functionals, which describe the breaking of conformal symmetry due to the presence of the Dilaton. To keep Conformal Symmetry, these functionals must be set to be 0.

The coupling of the string to the Dilaton Field is described by the following action integral:

${q_\Phi&space;}\ell&space;_P^2\int&space;{\Phi&space;R\sqrt&space;{&space;-&space;\det&space;{h_{\alpha&space;\beta&space;}}}&space;{\text{&space;}}{{\text{d}}^2}\xi&space;}$

To derive the beta functionals, one may do this in Riemann Normal Coordinates (see at Wikipedia or at the n(Cat)Lab).

The breakdown of conformal invariance would then be: \\\

$$\left\langle&space;{T_\alpha&space;^\alpha&space;}&space;\right\rangle&space;=&space;-&space;\frac{{\pi&space;T}}{{{c_0}}}{\left(&space;{{\beta&space;_{\mu&space;\nu&space;}}\left(&space;g&space;\right){h^{\alpha&space;\beta&space;}}{\partial&space;_\alpha&space;}{X^\mu&space;}{\partial&space;_\beta&space;}{X^\nu&space;}&space;+&space;i{\beta&space;_{\mu&space;\nu&space;}}\left(&space;F&space;\right){h^{\alpha&space;\beta&space;}}{\partial&space;_\alpha&space;}{X^\mu&space;}{\partial&space;_\beta&space;}{X^\nu&space;}&space;+&space;\frac{1}{2}\beta&space;\left(&space;\Phi&space;\right)R}&space;\right)_{\operatorname{elec}&space;\operatorname{tromagnetic}&space;{\text{&space;worldsheet}}}}$$

With the beta functionals given by:

$\begin{gathered}&space;{\beta&space;_{\mu&space;\nu&space;}}\left(&space;F&space;\right)&space;=&space;\frac{{\ell&space;_P^2}}{2}{\nabla&space;^\lambda&space;}{H_{\lambda&space;\mu&space;\nu&space;}}&space;\hfill&space;\\&space;\beta&space;\left(&space;\Phi&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{&space;-&space;\frac{1}{2}{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;+&space;{\nabla&space;_\mu&space;}\Phi&space;{\nabla&space;^\mu&space;}\Phi&space;-&space;\frac{1}{{24}}{H_{\mu&space;\nu&space;\lambda&space;}}{H^{\mu&space;\nu&space;\lambda&space;}}}&space;\right)&space;\hfill&space;\\&space;{\beta&space;_{\mu&space;\nu&space;}}\left(&space;g&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{{R_{\mu&space;\nu&space;}}&space;+&space;\ell&space;_P^2\frac{{\delta&space;\left(&space;{{R_{\mu&space;\nu&space;\rho&space;\sigma&space;}}{R^{\mu&space;\nu&space;\rho&space;\sigma&space;}}}&space;\right)}}{{\delta&space;{g_{\mu&space;\nu&space;}}}}&space;+&space;2{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;-&space;{H_{\mu&space;\nu&space;\lambda&space;\kappa&space;}}H_\nu&space;^{\lambda&space;\kappa&space;}}&space;\right)&space;\hfill&space;\\&space;\end{gathered}$

To impose conformal invariance, these beta functionals must vanish, as follows:

$\begin{gathered}&space;{\beta&space;_{\mu&space;\nu&space;}}\left(&space;g&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{{R_{\mu&space;\nu&space;}}&space;+&space;2{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;-&space;{H_{\mu&space;\nu&space;\lambda&space;\kappa&space;}}H_\nu&space;^{\lambda&space;\kappa&space;}}&space;\right)&space;=&space;0&space;\hfill&space;\\&space;{\beta&space;_{\mu&space;\nu&space;}}\left(&space;F&space;\right)&space;=&space;\frac{{\ell&space;_P^2}}{2}{\nabla&space;^\lambda&space;}{H_{\lambda&space;\mu&space;\nu&space;}}&space;=&space;0&space;\hfill&space;\\&space;\beta&space;\left(&space;\Phi&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{&space;-&space;\frac{1}{2}{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;+&space;{\nabla&space;_\mu&space;}\Phi&space;{\nabla&space;^\mu&space;}\Phi&space;-&space;\frac{1}{{24}}{H_{\mu&space;\nu&space;\lambda&space;}}{H^{\mu&space;\nu&space;\lambda&space;}}}&space;\right)&space;=&space;0&space;\hfill&space;\\&space;\end{gathered}$

These are the field equations for the graviton, dilaton, and photon fields respectively. Notice that they have a rather fundamental basis, conformal invariance. We need to focus on this one:

${\beta&space;_{\mu&space;\nu&space;}}\left(&space;g&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{{R_{\mu&space;\nu&space;}}&space;+&space;2{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;-&space;{H_{\mu&space;\nu&space;\lambda&space;\kappa&space;}}H_\nu&space;^{\lambda&space;\kappa&space;}}&space;\right)&space;=&space;0$

This is obviously the field equation for gravity. Notice that I have removed one term on the way. This term is ${&space;+&space;\ell&space;_P^2\frac{{\delta&space;\left(&space;{{R_{\mu&space;\nu&space;\rho&space;\sigma&space;}}{R^{\mu&space;\nu&space;\rho&space;\sigma&space;}}}&space;\right)}}{{\delta&space;{g_{\mu&space;\nu&space;}}}}}$. This is because I have assumed that the Riemann Curvature Tensor is negligibly small.

I don;’t need to.

${\beta&space;_{\mu&space;\nu&space;}}\left(&space;g&space;\right)&space;=&space;\ell&space;_P^2\left(&space;{{R_{\mu&space;\nu&space;}}&space;+&space;\ell&space;_P^2\frac{{\delta&space;\left(&space;{{R_{\mu&space;\nu&space;\rho&space;\sigma&space;}}{R^{\mu&space;\nu&space;\rho&space;\sigma&space;}}}&space;\right)}}{{\delta&space;{g_{\mu&space;\nu&space;}}}}&space;+&space;2{\nabla&space;_\mu&space;}{\nabla&space;_\nu&space;}\Phi&space;-&space;{H_{\mu&space;\nu&space;\lambda&space;\kappa&space;}}H_\nu&space;^{\lambda&space;\kappa&space;}}&space;\right)&space;=&space;0$

However, in the limit of little gravity, and no dilaton, this becomes the ordinary vacuum Einstein Field Equation.

String Theory also agrees with the Minimal Supersymmetric Standard Model (MSSM) as shown by [1] (pdf  here).    Upon Supersymmetry breaking, this means that it also agrees with the Standard Model j.

————————————————————————————————————————————————————————————————————-  String Theory is testable.

What does String Theory predict?

It predicts scattering amplitudes, caisimir energy, superpartners, gravitons, an infinitude of particles in a mass spectrum, gravitons, extra dimensions, AdS/CFT, and what not?  Talking about AdS/CFT, see this recent paper  by Raju and Papadogmias [2] (pdf here)   and this one by Papawdogmias and Raju [3] (pdf here).         This means the prediction of certain operators in the conformal boundary.

————————————————————————————————————————————————————————————————————-

String Theory is testable at today’s energy scales.

Firstly, that isn’t a valid deleteion argument, as it  is still testable.

Secondly, the Supersymmetry-related predictions of String Theory just depend on a certain number of parameters, called the Supersymmetry breaking parameters. For an $\mathcal{N}=1$ supersymmetric string theory (like a $G\left(2&space;\right&space;)$ manifold compactification of M-Theory), it is in fact possible to test the effects of supersymmetry, because the supersymmetry breaking  energy parameter is low enough!.

————————————————————————————————————————————————————————————————————-

String Theory has withstood experimental tests.

Huh, no. The only experimental result in contrary to the predictions of String Theory is probably [4] (PDF here). Other than this,  String Theory has in fact been supported by experimental predictions. Also, the experiment does not rule out compactification lengths smallernthan half a milimetre.

————————————————————————————————————————————————————————————————————-

String Theory has had experimental verification repeatedly.

See this article at the Mathematics and Physics Wikia (Introduction to String Theory)  for the entire list.

Note that the 125 GeV Higgs actually serves as an experimental support for String Theory.

————————————————————————————————————————————————————————————————————-

So basically, these criticisms of String Theory are just some ingeniously crafted Markov Chains, cooked up by a computer repairman at the Mathematics Department of the University of Columbia, aka the “Troll King“, and popularised by the popular media, such as ”Scientific American”, a magasine devoted to making people unscientific and dog-ma-believing.

\oint_C\left(\mbox{Random}^{\mathrm{rubbish}}\right)\cdot\mbox{d}\vec r = 0 …

I have used MathJax in this post, only to find that MathJax isn’t supported here. For the correct version, view my post at Psi Epsilon (MathJax).

Many of us may have heard of the AdS/CFT correspondence.

A $D$ – dimensional string theory in Anti-de-Sitter (AdS) space is exactly equivalent to $D-1$ dimensional Conformal field theory (CFT), such as Quantum Yang-Mills theory, etc.

That just sounds a bit crazy right? How can a string theory be equivalent to a mere CFT, of all things?

But in reality, the confusion only arises from the way it is phrased. It should be phrased in terms of the Holographic principle. Then you ask, “What is this Holographic principle?”.

The information stored inside a reigon is completely described by the information on its boundary.

Ugh…… The information inside a water bottle (which is the information in the water) is equivalent to the information on the bottle’s surface itself, which is the information in plastic? Is this alchemy, or something?

But holography is a law of nature and there’s nothing wrong about it. Let us start with some obvious examples.

1. Stokes’s theorem

Ok, consider a field originating from a certain point. To make things simple, let us say its a vector field, and it is actually the field of forces (field of the electromagnetic force, as opposed to an electromagnetic field, but the latter would work too). Now, let us say there is some 2-dimensional surface $S$ , with a boundary curve $C$. The work done by the force field along this curve, is given by:

$$\oint \vec f\cdot\mbox{d}\vec r$$

This really just follows from $\mbox{d} W=\vec F\cdot\vec r$. You may already start to see where this is going! What is the flux of the curl of the force field through the surface? We know that it is, equal to:

$$\iint_S\left(\nabla\times\vec f\right)\cdot\hat n \mbox{ d}S$$

(Admittedly, it is foolish to say the integral of the “curl of the force field”, because it has a very limited physical meaning, unless you use stokes’ theorem).

Now what does Stokes’ theorem, more specifically, the Kevin-Stokes’ theorem, say?

$$\oint \vec f\cdot\mbox{d}\vec r = \iint_S\left(\nabla\times\vec f\right)\cdot\hat n \mbox{ d}S$$

In other words, the flux of the curl through the surface, is exactly equivalent to the work done on the boundary, which is, the curve! .

2. Gauss’s theorem

Consider the sum of the divergences within a volume $V$. Then, Gauss’s theorem tells us that that is equivalent to :

$$\iint_S\vec f \cdot \hat n \mbox{ d}S = \iiint_V \nabla\cdot\vec f\mbox{ d}V$$

I.e. a property of the reigon is equivalent to a property of the surface.

3. Black holes

Consider two observers, observer A, and observer B, . Observer B is falling into a black hole, whereas observer A is outside. Then, for simplicity, say, the black hole, is Schwarzschild, so that the time dilation is then:

$$\frac{\mbox{d}t}{\mbox{d\tau} = \frac1{\sqrt{1-\frac{r_s}r}}$$

Which is an obvious result from the Schwarzschild metric.

Then, this means that Observer A is going to observe that Observer B’s time scales get shrunk, so that Observer B will appear to move towards the black hole slower, and slower, and finally stop at the event horizon. However, for Observer B himself, everything will appear normal, from his reference frame. I.e. what is going on inside the black hole (as observer B observes it) seems to go on on the surface of the black hole (the event horizon, of the black hole, now you know why it’s called an “event horizon”.) for Observer A. This is also a resolution to the Hawking information Paradox. The information is encoded on the event horizon, which is why it doesn’t disappear.

So, this just means that the information inside a reigon is completely encoded on to its boundary. So, this means, that,… ?

It is the Holographic principle.

$D$ – dimensional Anti-de-Sitter space has a $D-1$-dimensional boundary, which is governed by a Conformal field theory, and the Anti-de-Sitter space itself, is governed by a string theory.
Proof that $\left(\mathbb R^3,\times\right)$ is a Semi-Group