The deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, galaxies, super clusters), with quantum mechanics, which describes the other three fundamental forces acting on the atomic scale.
The development of a quantum field theory of a force invariably results in infinite (and therefore useless) probabilities. Physicists have developed mathematical techniques (renormalization) to eliminate these infinities which work for three of the four fundamental forces – electromagnetic, strong nuclear and weak nuclear forces - but not for gravity. The development of a quantum theory of gravity must therefore come about by different means than those used for the other forces.
The basic idea is that the fundamental constituents of reality are strings of the Planck length (about 10^{−33} cm) which vibrate at resonant frequencies. Every string in theory has a unique resonance, or harmonic. Different harmonics determine different fundamental forces. The tension in a string is on the order of the Planck force (10^{44} newtons). The graviton (the proposed messenger particle of the gravitational force), for example, is predicted by the theory to be a string with wave amplitude zero. Another key insight provided by the theory is that no measurable differences can be detected between strings that wrap around dimensions smaller than themselves and those that move along larger dimensions (i.e., effects in a dimension of size R equal those whose size is 1/R). Singularities are avoided because the observed consequences of "Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of a string, at which point it would actually begin expanding.
Our minds have difficulty visualizing higher dimensions because we can only move in three spatial dimensions. One way of dealing with this limitation is not to try to visualize higher dimensions at all, but just to think of them as extra numbers in the equations that describe the way the world works. This opens the question of whether these 'extra numbers' can be investigated directly in any experiment (which must show different results in 1, 2, or 2+1 dimensions to a human scientist). This, in turn, raises the question of whether models that rely on such abstract modelling (and potentially impossibly huge experimental apparatus) can be considered scientific. Six-dimensional Calabi-Yau shapes can account for the additional dimensions required by superstring theory. The theory states that every point in space (or whatever we had previously considered a point) is in fact a very small manifold where each extra dimension has a size on the order of the Planck length.
Superstring theory is not the first theory to propose extra spatial dimensions; the Kaluza-Klein theory had done so previously. Modern string theory relies on the mathematics of folds, knots, and topology, which were largely developed after Kaluza and Klein, and has made physical theories relying on extra dimensions much more credible.
Theoretical physicists were troubled by the existence of five separate string theories. This has been solved by the second superstring revolution in the 1990s during which the five string theories were discovered to be different limits of a single underlying theory: M-theory.
String Theories | ||
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Type | Spacetime dimensions | Details |
Bosonic | 26 | Only bosons, no fermions means only forces, no matter, with both open and closed strings; major flaw: a particle with imaginary mass, called the tachyon |
I | 10 | Supersymmetry between forces and matter, with both open and closed strings, no tachyon, group symmetry is SO(32) |
IIA | 10 | Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions spin both ways (nonchiral) |
IIB | 10 | Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions only spin one way (chiral) |
HO | 10 | Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is SO(32) |
HE | 10 | Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is E_{8}×E_{8} |
The five consistent superstring theories are:
Chiral gauge theories can be inconsistent due to anomalies. This happens when certain one-loop Feynman diagrams cause a quantum mechanical breakdown of the gauge symmetry. The anomalies were canceled out via the Green-Schwarz mechanism.
The major problem with their congruence is that, at sub-Planck (an extremely small unit of length) lengths, general relativity predicts a smooth, flowing surface, while quantum mechanics predicts a random, warped surface, neither of which are anywhere near compatible. Superstring theory resolves this issue, replacing the classical idea of point particles with loops. These loops have an average diameter of the Planck length, with extremely small variances, which completely ignores the quantum mechanical predictions of sub-Planck length dimensional warping, there being no matter that is of sub-Planck length.
There are five ways open and closed strings can interact. An interaction in superstring theory is a topology changing event. Since superstring theory has to be a local theory to obey causality the topology change must only occur at a single point. If C represents a closed string and O an open string, then the five interactions are, symbollically:
OOO + CCC + OOOO + CO + COO
All open superstring theories also contain closed superstrings since closed superstrings can be seen from the fifth interaction, they are unavoidable. Although all these interactions are possible, in practice the most used superstring model is the closed heterotic E8xE8 superstring which only has closed strings and so only the second interaction (CCC) is needed.
$A\_N\; =\; int\{Dmu\; int\{D[X]\; exp\; left(-frac\{1\}\{4pialpha\}\; int\{\; partial\_z\; X\_\{mu\}(z,overline\{z\})\; partial\_\{overline\{z\}\}\; X^\{mu\}(z,overline\{z\})\}dz^2\; +\; i\; sum\_\{i=1\}^\{N\}\{k\_\{i\; mu\}\; X^\{mu\}(z\_i,overline\{z\}\_i)\; \}\; right)\; \}\}$
The functional integral can be done because it is a Gaussian to become:
$A\_N\; =\; int\{Dmu\; prod\_\{0+1\}\{\; |z\_i-z\_j|^\{2alpha\; k\_i.k\_j\}\; \}\; \}\; math>$
This is integrated over the various points $z\_i$. 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 $theta$. 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.
$partial\_z\; rightarrow\; partial\_z\; +iA\_z(z,overline\{z\})$
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.
Superstring Model | Invariant |
---|---|
Heterotic | $partial\_zX^mu-ioverline\{theta\_\{L\}\}Gamma^mupartial\_ztheta\_\{L\}$ |
IIA | $partial\_zX^mu-ioverline\{theta\_\{L\}\}Gamma^mupartial\_ztheta\_\{L\}-ioverline\{theta\_\{R\}\}Gamma^mupartial\_ztheta\_\{R\}$ |
IIB | $partial\_zX^mu-ioverline\{theta^1\_\{L\}\}Gamma^mupartial\_ztheta^1\_\{L\}-ioverline\{theta^2\_\{L\}\}Gamma^mupartial\_ztheta^2\_\{L\}$ |
$int\_\{-infty\}^\{infty\}\{exp(\{a\; x^4+b\; x^3+c\; x^2+d\; x+f\})dx\}$
In the case of membranes the series would correspond to sums of various membrane interactions that are not seen in string theory.