Quantum Physics Web Assignment
------String Theory
The 20th Century might be one of the most great harvest centuries for theoretical physics. There were two great revolutionary moves in the searching of the fundamental theory of the universe, the quantum theory, which were founded by Werner Heisenberg, Erwin Schrodinger, Wolfgang Pauli and many other talented scientists; the relativity, special and general, both were developed by Albert Einstein nearly single handed.
Among all of them, the combination of the quantum theory and the special relativity can be considered as the most promising theory so far, the relativistic quantum field theory. For example, the prediction of the theory says that the magnetic moment of an electron is 1.001159652193 Bohr magneton, and the data from the experiment records 1.001159652188 Bohr magneton, as can be seen, they are very close.
However, even the relativistic quantum field theory explains the properties of the fundamental particles very well, the fact is all of the results are obtained under a non-realistic condition, that the gravity is weak which can be considered negligible. General relativity also has a similar problem, even though it gives a great view of the universe, but it assumes that the universe is classical and quantum theory does not exist.
Then is there a fundamental theory to combine the two of them? The string theory might be the solution.
“String” in string theory can be roughly understood through the study of the strings on the string instruments.
Here are the strings on a guitar. When it is played by a musician, the strings on the guitar vibrate under the tension which is give by the fingers. Music notes are created, and the music notes are called the excitation modes of the strings.
The wavelength and the frequency of the excitation modes have the relationship as
In string theory, the classical view that consider fundamental particles as point mass breaks down, instead they “look” like “strings”. A typical string is a 1-demesional object, and it is considered to have no thickness but do have a typical length of 10^-33 cm.
And this little string, behaves like a guitar string that can be excited to excitation modes under tension. Unlike a guitar string which produces standing waves, the string floats around in spacetime and sweeps a world sheet.
A classical string equation can be written as
where and are coordinates on the world sheet.
From the solutions, two types of strings can be obtained, open-string and closed-string. The ends of open string are floppy and travel freely in spacetime. There are also two types of closed strings, right-movers and left-movers, which move in opposite directions around the string.
There are six kinds of string theories currently which play the main role, and they are charactered into two categories, the Bosonic string theory, and the superstring theory. These string theories differ in the following characters. First, if the theory involves open string or closed string, second, does the theory apply to bosons or fermions.
For the second condition, that is the difference between the Bosonic string theory and the superstring theory.
The Bosonic string theory, which was developed in the late 1960s, only apply for the bosons, which means that only forces are considered in this theory but not matter, and it takes account both open and closed strings, it also predicts the appearance of a special class of fundamental particles, the tachyons. For mathematically reasons, the Bosonic string theory appears to have 26-dimensions, 25 spatial dimensions and 1 time dimension. However, the Bosonic string theory is not very realistic, which the fermions are part of the real world.
The superstring theories are based on the idea of supersymmetry, which means all of these five theories consider supersymmetry as part of their assumption, but they differ in details.
Only I theory involves both open string and closed string whereas the rest only has closed string in their idea; IIA theory states that the massless fermions spin both ways, but IIB theory states that they only spin in one way; HO and HE differs the right moving string and the left moving string, and they have different group symmetry. And the tachyons do not exist in the superstring theory.
There are the ideas of compact dimensions in the superstring theory, which that there are other dimensions that are very small, in Planck length, which are even smaller than the smallest know particles, that means the compact dimensions may not be measurable in nowadays technology. All superstring theories apply under 10 dimensions for mathematically reasons, 4-dimesion spacetime as the ordinary universe, and 6 compact dimensions, which in the compact dimensions, the numbers are anticommute, i.e. X*Y=-Y*X.
In fact, the five superstring theories are not isolated, they are related by duality. And this “Theory of Everything” (at least, for everything about string theory) is called the M-theory.
Instead of having these open or closed string, M-theory provides something called the p-brane. P-brane does not only just include the 1-dimension string but also other dimensions objects as well. The “p” in p-brane, states the dimensionality, e.g. a dimensionless point is a 0-brane and a 1-dimension string is a 1-brane.
P-brane comes from the solutions of Einstein’s equation, it is a spacetime object. And in order to solve the equation, the condition requires the low energy limit of superstring theory. D-brane is a specially kind of p-brane, it can be considered as the open ends strings “attach” their ends on the brane.
The fact is that the string theory yet does not have any experimental observations to support the pure mathematical model, so it can be considered as a completely philosophical idea, it is still not clear that if the string theory is the solution of the fundamental theory of physics.