Thursday, 1 September 2011

Beyond Superstring Theory

 

 

It is commonly believed that the 5 superstring theories are approximated to a theory in higher dimensions possibly involving membranes. Unfortunately because the action for this involves quartic terms and higher so is not Gaussian the functional integrals are very difficult to solve and so this has confounded the top theoretical physicists. Edward Witten has popularised the concept of a theory in 11 dimensions M-Theory involving membranes interpolating from the known symmetries of superstring theory.

It may turn out that there exist membrane models or other non-membrane models in higher dimensions which may become acceptable when new unknown symmetries of nature are found, such as noncommutative geometry for example. It is thought, however, that 16 is probably the maximum since O(16) is a maximal subgroup of E8 the largest exceptional lie group and also is more than large enough to contain the Standard Model. Quartic integrals of the non-functional kind are easier to solve so there is hope for the future. This is the series solution which is always convergent when a is non-zero and negative:

\int_{-\infty}^{\infty}{\exp({a x^4+b x^3+c x^2+d x+f})dx} = e^f\sum_{n,m,p=0}^{\infty}{ \frac{ b^{4n}}{(4n)!}\frac{c^{2m}}{(2m)!}\frac{d^{4p}}{(4p)!} \frac{ \Gamma(3n+m+p+\frac14) }{a^{3n+m+p+\frac14} } }

In the case of membranes the series would correspond to sums of various membrane interactions that are not seen in string theory.


Compactification

 

 

Investigating theories of higher dimensions often involves looking at the 10 dimensional superstring theory and interpreting some of the more obscure results in terms of compactified dimensions. For example D-branes are seen as compactified membranes from 11D M-Theory.

Theories of higher dimensions such as 12D F-theory and beyond will produce other effects such as gauge terms higher than U(1). The components of the extra vector fields (A) in the D-brane actions can be thought of as extra coordinates (X) in disguise. However, the known symmetries including supersymmetry currently restrict the spinors to have 32-components which limits the number of dimensions to 11 (or 12 if you include two time dimensions.) Some commentators (e.g. John Baez et al) have speculated that the exceptional lie groups E6, E7 and E8 having maximum orthogonal subgroups O(10), O(12) and O(16) may be related to theories in 10, 12 and 16 dimensions; 10 dimensions corresponding to string theory and the 12 and 16 dimensional theories being yet undiscovered but would be theories bases on 3-branes and 7-branes respectively. However this is a minority view within the string community.

Since E7 is some sense F4 quaternified and E8 is F4 octonified, then the 12 and 16 dimensional theories, if they did exist, may involve the noncommutative geometry based on the quaternions and octonions respectively. From the above discussion it can be seen that physicists have many ideas for to extend superstring theory beyond the current 10 dimensional theory but so far none have been successful.


Kac-Moody algebras

 

Since strings can have an infinite number of modes, the symmetry used to describe string theory is based on infinite dimensional Lie algebras. Some Kac-Moody algebras that have been considered as symmetries for M-Theory have been E10 and E11 and their supersymmetric extensions.




Levels of magnification:

1. Macroscopic level - Matter
2. Molecular level
3. Atomic level - Protons, neutrons, and electrons
4. Subatomic level - Electron
5. Subatomic level - Quarks
6. String level

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Saturday, 27 August 2011

Astrobiology and Space Exploration

"Mungkin saja kehidupan lain di luar bumi sana sedang menunggu kita untuk dikunjungi"
~Arip~

3. From Astrochemistry to Astrobiology

Stanford University Course

In the Stanford Astrobiology Course, our lectures follow a more or less linear path from the Big Bang all the way to the development of complex life and, finally, space exploration. It is truly amazing how evolutionary principles have operated at the macro, and micro, level ever since the birth of the universe we reside in today.

Physics, research, experimentation, astronomy, extraterrestrial life, planets, asteroids, cosmology, measurements, data, innovation, development, history, science, telescopes, observations, theories, predictions, telescopes, instruments, light, expansion.


 A syllabus for the Winter, 2010 Astrobiology Course can be downloaded here.


The Big Bang created the physical universe. Of course life is part of this physical universe, but the immediate building blocks of life are chemicals. Before the Big Bang, words such as “time” had no meaning, but even in the first few minutes there could be no chemistry since there were no atoms. The nuclei of some of the lighter elements formed within minutes, atoms some time later, and elements heavier than lithium were forged in the supernovae of stars. Thus, we are primarily star dust, although the hydrogen atom you drink tonight may be nearly as old as the Big Bang.

But living organisms are more than a collection of atoms. They are a cauldron of molecules in a solvent. For life on earth, that solvent is water. The building blocks of chemical compounds had to form other molecules as well, especially ones based on carbon. Where could these compounds have been formed? Were they formed on earth or transported from elsewhere?

Most stunning are the recent discoveries in astrochemistry showing that the organic compounds that make up life on earth may possibly be THE language of the universe. In September 2010, the NASA Ames IR Spectroscopic Database was released, along with tools to access it. See the press release.

Recommended Reading

“Chemical Evolution across Space and Time: From The Big Bang to Prebiotic Chemistry”  Eds Lori Zaikowski and Jon Friedrich; Published by American Chemical Society, Wash. DC ISBN 978-0-8412-0

Amallondalla’s is chapter 5: ” Chemical Evolution in the Interstellar Medium:  Feedstock of Solar Systems.
The next volume is entitled:

“Chemical Evolution across Space and Time: From Origins of Life to Modern Society”  Eds Lori Zaikowski and Jon Friedrich; Published by American Chemical Society, Wash.

This Scientific American article  is geared to the general scientifically literate audience:  ”Life’s Far-Flung Raw Materials”,

Bernstein, Sandford, Allamandola, Scientific American, July 1999


"The space effort is very simply a continuation of the expansion of ecological range, which has been occurring at an accelerating rate throughout the evolutionary history of Man..."
~Ward J. Haas, "Biological Significance of the Space Effort," in Annals of the New York Academy of Science, 1966 ~

Below you will find a list of recommended resources. If you would like to learn more about the Big Bang, check out these books, videos, and articles.



Sumber:
1. Stanford University
2. NASA

Ucapan Terima Kasih:

1. Bapak. Prof. Dr. Ing. H. B. J. Habibie.

2. Departemen Pendidikan Nasional

3. Kementrian Riset dan Teknologi

4. Lembaga Penerbangan dan Antariksa Nasional


Disusun Ulang Oleh:

Arip Nurahman

Department of Physics, Indonesia University of Education

&

Follower Open Course Ware at MIT-Harvard University, Cambridge.USA.

Semoga Bermanfaat dan Terima Kasih

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Thursday, 25 August 2011

The Mathematics of Super String Theory

 

 

The single most important equation in (first quantisized bosonic) string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are tachyons, with momentum ki which connect to a string world surface at the surface points zi. It is given by the following functional integral which integrates (sums) over all possible embeddings of this 2D surface in 26 dimensions.


A_N = \int{D\mu \int{D[X] exp \left( -\frac{1}{4\pi\alpha} \int{ \partial_z X_{\mu}(z,\overline{z}) \partial_{\overline{z}} X^{\mu}(z,\overline{z})}dz^2 + i \sum_{i=1}^{N}{k_{i \mu} X^{\mu}(z_i,\overline{z}_i) } \right) }}

The functional integral can be done because it is a Gaussian to become:


This is integrated over the various points zi. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different paramaterisations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):


A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2) } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) }


Which is a beta function. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the grassman coordinates θ. Since there are various ways this can be done this leads to different string theories.

When integrating over surfaces such as the torus, we end up with equations in terms of theta functions and elliptic functions such as the Dedekind eta function. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface.

D-Branes

 

 

D-Branes are membrane-like objects in 10D string theory. They can be thought of as occurring as a result of a Kaluza-Klein compactification of 11D M-Theory which contains membranes. Because compactification of a geometric theory produces extra vector fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action.



\partial_z \rightarrow \partial_z +iA_z(z,\overline{z})


In type I open string theory, the ends of open strings are always attached to D-brane surfaces. A string theory with more gauge fields such as SU(2) gauge fields would then correspond to the compactification of some higher dimensional theory above 11 dimensions which is not thought to be possible to date.


Why Five Superstring Theories?

 

 

For a 10 dimensional supersymmetric theory we are allowed a 32-component Majorana spinor. This can be decomposed into a pair of 16-component Majorana-Weyl (chiral) spinors. There are then various ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities:



Superstring Model Invariant
Heterotic \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}
IIA \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}-i\overline{\theta_{R}}\Gamma^\mu\partial_z\theta_{R}
IIB \partial_zX^\mu-i\overline{\theta^1_{L}}\Gamma^\mu\partial_z\theta^1_{L}-i\overline{\theta^2_{L}}\Gamma^\mu\partial_z\theta^2_{L}


The heterotic superstrings come in two types SO(32) and E8xE8 as indicated above and the type I superstrings include open strings.

Sources:

Wikipedia

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