Introduction To String Theory

General Relativity and Quantum Mechanics

The two greatest achievements of 20th century physics have been

  1. General Theory of Relativity
  2. Quantum Mechanics

While the former successfully describes the Universe on a Large Scale, the latter solved mysteries of the world of small. Ever since there has been a constant effort by the physicists to form a single Unified Theory of Everything (TOE) which would describe the entire universe ranging from quarks to black holes.

Obviously,  such a TOE should essentially incorporate within it the features of both General Theory as well as Quantum Theory. But the very basic fact that these two theories disagree with each other has kept the dream of a TOE away. There have been unsuccessful efforts to find a Quantum Theory of Gravity that describes Gravity using the principles of Quantum Mechanics.

One might wonder about the need to unify GR and QT as both deal with the universe on completely different scales. Why not just use GR when dealing with Gravity and Large scale Universe and use QT for the world of the small?
The problem lies with certain parts of space time called Singularities. For ex: Blackhole Singularities. These are the regions of spacetime with dimensions of the range of  an atom and at the same time with  gravitational fields so strong that the effect of General Relativity cannot be ignored. Thus, if we have to describe singularities then, we must unite Quantum Mechanics and General Relativity.

One of the fundamental issues on which General Relativity (GR) and Quantum Theory (QT) disagree is Time. While GR rules out the idea of absolute time, QT does not. However the absolute time of QT is not the same as the classical Newtonian one, infact the absolute time of QT is not measurable at all!

Also, unacceptable to GR is the microscopic universe of QT filled with Quantum Foam. The world of the small is heavily chaotic with the constant creation and annihilation of Quantum particles. The space here is highly irregular and this is something which GR can't agree with. according to GR, space might also be tightly curved,  but it is always smooth and there is absolutely no place for quantum irregularities. While Quantum Theory talks about the occurrences of events in terms of probabilities, General Relativity completely rules out probabilities, as Einstein put in his famous quote "God never plays dice".

Thus the Quantum Theory and General Relativity are completely inconsistent with each other.

String Theory Enters the Scenario:

String Theory has emerged as the most promising candidate for the post of a Theory of Everything.

Originally String Theory was born to explain the interactions between hadrons. In more simple terms, String Theory tried to explain Strong Nuclear Force between particles like protons and neutrons. But the success of Quantum Chromodynamics in explaining these interactions made physicists forget the String Theory for quite sometime.

Once again String Theory entered into the picture, but now with a completely different look when it was found that this could successfully prove the existence of gravitons. Before going into the details of this...

What actually the String Theory all about?

We know that the quantum theory looks at the nature as particles and the General relativity looks at the nature geometrically as a 4 dimensional spacetime.
But String Theory insists that the basic entities of matter at the fundamental level of the order of Planck's length are Strings (see Fig 1) , similar to the Strings of a musical instrument like Veena or Guitar, with different modes of vibration. 
Fig 1. A String

The beauty of String Theory is the basic idea of explaining the different fundamental particles of nature as the various modes of vibration of the same string! While a string vibrating with a given mode becomes a top quark, the same String when it starts to vibrates in some other mode might be a photon!

Things turn out more interesting when we look at the heavy mathematics of this theory. As we look at the behavior of particles formed when a string vibrates in different modes, there comes a particular mode of vibration at which the resulting equations become exactly same as those of General Relativity! Thus the particle formed at this particular mode of vibration is none other than our old friend whom we have been searching for years together, The Graviton.

One of the fundamental problems which the String Theory solved was the infinity regarding the Inverse Square Laws. When the distance between two particles are taken to be zero, both the Inverse Square Laws predict that the respective Forces between the particles become infinite. For ex, if the distance between two electrons is zero, then the Gravitational Force of attraction as well as the Electromagnetic Force of repulsion becomes infinite and infinities are not always welcome for us.

This problem for Electromagnetism was solved by Quantum Theory whose uncertainty principle smears out the value r=0. But the problem remained unsolved in the case of gravity. Remember that even  Newton's Formula of Gravitation is not incorrect, but is a good approximation of General Relativity. String Theory solved this infinity problem immediately as there is no question of Zero distance between two points here. Here interactions between Strings always take place at a small but certainly finite distance. 

The various String Theories we have today are based on the ideas like whether a String is an open one or a closed one and even whether we want to include fermions in the theory or not. 

If we consider only bosons, i.e, the Force carrying particles and both open as well as closed Strings, we get what is called the Bosonic String Theory. This String Theory makes sense only in 26 dimensions! and remember there is no room for fermions and hence for matter here. But then if you look  carefully, you find that this is not the Theory we are looking for, due to the obvious reason that it doesn't deal with Fermions. To be a Theory of Everything, it must explain everything! Also one more drawback with the bosonic String Theory is that it explains a particle called Tachyon which has imaginary mass! This is something which we cannot agree with.

So we will concentrate only on those String Theories which explain not only bosons, but also fermions. But to do so, these theories demand a special type of symmetry called supersymmetry to exist in nature. According to this, each fermion has its bosonic partner. For ex: electron which is a fermion has its bosonic particle called selectron! But then one has never found a selectron. But that does not also mean that there are no selectrons!! perhaps they are hidden in the extra dimensions! Yes even when we consider fermions the Universe of String Theory still has more dimensions than our usual four. But unlike bosonic String Theory which talks about 26 dimensions, here the numbers come down to 10 i.e, 9 space dimensions and one time dimension (The science fiction writers might be disappointed as the number of time dimensions continues to be one!!). Nevertheless the other 6 space dimensions also are not the ones which we can explore into as they are tightly compacted at subatomic distances.. So it might be that only a few fundamental particles are able to wander around in these extra dimensions!! or are they?? 

The String Theories which talk about SuperSymmetry in nature and thus explain bosons as well as fermions are called SuperString Theories. These theories do not need Tachyons (The particles with imaginary mass which bosonic String Theory explains) to exist. Currently physicists have filtered out five different types of SuperString Theories called I, II A, II B, HO and HE.

While SuperString Theory I differs from the other in the respect that it has both open as well as closed Strings, the other Super String Theories talk only about closed strings. Unlike II A and II B, HO and HE are heterotic String theories. In heterotic string theories the strings moving to right are different from those moving left. heterotic string theories have emerged to be the most promising candidates for a Theory of Everything. But still we have two of them and still a lot needs to be done before we finally have a TOE.

Nowadays physicists love to call a TOE as M-Theory (Mother of all theories!). Let us hope that we will find one very soon.

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This is a Mathematical Branch which deals with Shapes without going into the Geometric details. To be more precise topology deals with Connectedness of Shapes. Two shapes are to said to have the same connectedness if we can deform one into another without tearing. For ex: a cup and a doughnut have the same connectedness.

Region of spacetime so tightly curved by gravity that even light cannot escape.

Planck's Length:
is given by the formula (hG/c3)1/2
, where h is the Planck's constant, G is the Gravitational constant and c is the velocity of light. It's magnitude is about 10-35 m. This is the length at which the strength of the Gravitational Force is equal to that of the other fundamental forces of nature.

is a quantum particle zero mass of spin 2, which is believed to be the carrier of the Gravitational Force.

are the particles that obey Pauli's Exclusion principle. These are the fundamental particles that make up matter. Ex: electron

are the particles that do not obey
Pauli's Exclusion principle. These are the Force carrying particles of nature. Ex: graviton