Ben Niehoff said:
I can't read French well enough to say for sure what this is supposed to be. However:
Google Chrome will translate for me.
Duality
In conclusion, the energy of a particle receives two contributions: one proportional (from the windings), the other inversely proportional (from the Kaluza-Klein compactification) to the compactification radius. The consequences are important:
There is a relation of reciprocity, called duality, between the modes of Kaluza-Klein and the modes of winding. If the compactness radius is large, the Kaluza-Klein mode contributions to the masses of the particles have a spectrum whose levels are close together, while the levels corresponding to the winding modes are spaced apart. If the compactification radius is small, it is the opposite. For a certain intermediate value of the compactification radius, the two spectra are identical: it is the point of duality. If for some values of the compactification radius the spectrum of Kaluza-Klein modes of a first string theory is identical to the winding spectrum of a second string theory and vice versa,
Duality of Kaluza-Klein modes and winding modes.
- Heisenberg's uncertainties are changed. In quantum field theory, they show that in order to explore smaller and smaller distances, the target must be bombarded with particles that can transfer increasingly large pulses to the target, thus with increasingly energetic particles , hence the construction of increasingly powerful particle accelerators. In the theory of strings, by increasing the pulse transferred to the target, we begin by exploring smaller and smaller distances, as shown by the Kaluza-Klein term, inversely proportional to the radius of compactification. But if the pulse transferred to the target is continued to increase, is more likely to cross the minimum distance equal to the radius of duality: the winding modes are then excited, proportional to the compactification radius. The minimal distance is of crucial importance for astrophysics and cosmology, because black holes and the Big Bang, where gravitational fields are extremely intense, can not be described by Einstein's theory or by quantum theory of the fields: if the energy increases indefinitely, the distance would decrease indefinitely to zero! String theory shows that there is a limit to the small distances, so a limit to the magnitude of the energy. If the universe were in contraction, an inverse evolution to the Big Bang, it could not contract forever because, it would contract, its dual part would expand. Physicists are currently very interested in this field.
In 1995, Christophe Hull, Paul Townsend and Edward Witten show that different string theories are linked by a mathematical transformation, called duality S.
In theories, it often happens that it is difficult, if not impossible, to calculate exact processes or physical quantities. We then content ourselves with using the theory of perturbations and making approximate calculations: we express the result as a sum of the successive terms, called the perturbation series. Depending on the degree of precision required, a number of terms greater or less is taken into account in the calculation, the higher it is, the better the approximation. This method only works if, as the number of terms increases, they become smaller and smaller. This is the case of quantum electrodynamics: the magnitude of the successive terms is reduced by the electromagnetic coupling constant (or fine structure constant), equal to 1/137, which makes it possible to obtain excellent approximations even if the sum of the disturbances contains only two or three terms. This is also the case of quantum chromodynamics in the high-energy regime. On the other hand, in the low-energy domain, the coupling constant of the strong interaction becomes supraunitary, the terms of the perturbation series become larger and the result no longer has any meaning: nothing can be calculated . Therefore, in quantum chromodynamics, we can not calculate the mass of the proton even if we know the masses of the quarks. obtain excellent approximations even if the sum of the perturbations contains only two or three terms. This is also the case of quantum chromodynamics in the high-energy regime. On the other hand, in the low-energy domain, the coupling constant of the strong interaction becomes supraunitary, the terms of the perturbation series become larger and the result no longer has any meaning: nothing can be calculated . Therefore, in quantum chromodynamics, we can not calculate the mass of the proton even if we know the masses of the quarks. obtain excellent approximations even if the sum of the perturbations contains only two or three terms. This is also the case of quantum chromodynamics in the high-energy regime. On the other hand, in the low-energy domain, the coupling constant of the strong interaction becomes supraunitary, the terms of the perturbation series become larger and the result no longer has any meaning: nothing can be calculated . Therefore, in quantum chromodynamics, we can not calculate the mass of the proton even if we know the masses of the quarks. in the domain of low energies, the coupling constant of the strong interaction becomes supraunitary, the terms of the series of perturbations become larger and the result no longer has any meaning: nothing can be calculated. Therefore, in quantum chromodynamics, we can not calculate the mass of the proton even if we know the masses of the quarks. in the domain of low energies, the coupling constant of the strong interaction becomes supraunitary, the terms of the series of perturbations become larger and the result no longer has any meaning: nothing can be calculated. Therefore, in quantum chromodynamics, we can not calculate the mass of the proton even if we know the masses of the quarks.
Physicists have therefore channeled their efforts in the search for exactly soluble theories, that is to say in which one can calculate exact processes or physical quantities without resorting to the theory of perturbations. Unfortunately, their number is very limited and they remain only models, which do not describe the reality of Nature.
Edward Witten shows that some theories with a strong coupling constant can be linked by mathematical transformations, called dualities, to other theories, called dual theories, in which the coupling constant is very weak. In the latter, thanks to the theory of perturbations, we can thus calculate various physical quantities. The idea is then to exploit the duality to export these quantities to the initial theory of strong coupling constant where the perturbation theory can not be used.
What is duality?
This duality, called S, exists between the theory of the strings of type I and the heterotic theory SO (32). The theory of the strings of type IIB is self-dual: by a transformation of duality S, it is transformed into itself.
Dualities between string theories and relation to M-theory.
These theories are defined in a 10-dimensional space-time. If we add an additional dimension, we can establish new links between them but also with the theories of 11-dimensional supergravity. This suggests that all these theories are included in an even more general theory, which has received the name of M-theory. The letter M would probably come from
Magic or
Mystery .
If the theory is compactifies M the 11 thdimension on a circle, the string theory of type IIA is obtained. If we compactify it on a segment, we get the heterotic theory E8xE8. The theory M is of a richness without common measure and contains other mathematical objects having p dimensions called p-branches, which generalize the particles and the strings. Thus the 0-branes are the ordinary particles of dimension 0 and their trajectory in space-time is thus a line, the 1-branches are strings of dimension 1 and their trajectory in space-time is thus a two-dimensional surface , the 2-branches are two-dimensional membranes and their trajectory in space-time is therefore a volume, and so on. A special class of p-branches is the Dirichlet-branes (or D-branes). For reasons of mathematical coherence, an open chord of type I must have at its two ends certain conditions: either its ends are free and the rope does not exchange energy by its extremities, condition called von Neuman, or the rope can exchanging energy but its ends are constrained to move on surfaces, condition called Dirichlet. These are the so-called D-branches.
A closed string interacts with a D-brane and turns into an open string whose ends rest on the D-brane.
What is D-brane?
The D-branes can be of exceptional importance in the explanation of elementary particles of the Standard Model. However, there is still a strong connection between the M-theory and the Standard Model.
One of the last tests to describe the elementary particles and the fundamental interactions of the Standard Model: the particles would be open strings, their ends being on D-branches.
M-theory is very complex and physicists are still far from having understood all its facets, which are currently the subject of continuous and intense research.
