vendredi 28 décembre 2012

Athene's Theory of Everything




A theory of everything (ToE) or final theory is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle.[1]
Many candidate theories of everything have been proposed by theoretical physicists during the twentieth century, but none have been confirmed experimentally. The primary problem in producing a ToE is that general relativity and quantum mechanics are hard to unify. This is one of the unsolved problems in physics.
Initially, the term 'theory of everything' was used with an ironic connotation to refer to various overgeneralized theories. For example, a great-grandfather of Ijon Tichy — a character from a cycle of Stanisław Lem's science fiction stories of the 1960s — was known to work on the "General Theory of Everything". Physicist John Ellis[2] claims to have introduced the term into the technical literature in an article in Nature in 1986.[3] Over time, the term stuck in popularizations of quantum physics to describe a theory that would unify or explain through a single model the theories of all fundamental interactions and of all particles of nature: general relativity for gravitation, and the standard model of elementary particle physics — which includes quantum mechanics — for electromagnetism, the two nuclear interactions, and the known elementary particles.

Historical antecedents

From ancient Greece to Einstein

Archimedes was possibly the first scientist that is known to have described nature with axioms (or principles) and then deduce new results from them.[citation needed] He thus tried to describe "everything" starting from a few axioms. Any "theory of everything" is similarly expected to be based on axioms and to deduce all observable phenomena from them.
The concept of 'atom', introduced by Democritus, unified all phenomena observed in nature as the motion of atoms. In ancient Greek times philosophers speculated that the apparent diversity of observed phenomena was due to a single type of interaction, namely the collisions of atoms. Following atomism, the mechanical philosophy of the 17th century posited that all forces could be ultimately reduced to contact forces between the atoms, then imagined as tiny solid particles.[4]
In the late 17th century, Isaac Newton's description of the long-distance force of gravity implied that not all forces in nature result from things coming into contact. Newton's work in his Principia dealt with this in a further example of unification, in this case unifying Galileo's work on terrestrial gravity, Kepler's laws of planetary motion and the phenomenon of tides by explaining them with one single law: the law of universal gravitation.
In 1814, building on these results, Laplace famously suggested that a sufficiently powerful intellect could, if it knew the position and velocity of every particle at a given time, along with the laws of nature, calculate the position of any particle at any other time:
An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
— Essai philosophique sur les probabilités, Introduction. 1814
Laplace thus envisaged a combination of gravitation and mechanics as a theory of everything. Modern quantum mechanics implies that uncertainty is inescapable, and thus that Laplace's vision needs to be amended: a theory of everything must include gravitation and quantum mechanics.
In 1820, Hans Christian Ørsted discovered a connection between electricity and magnetism, triggering decades of work that culminated in 1865, in James Clerk Maxwell's theory of electromagnetism. During the 19th and early 20th centuries, it gradually became apparent that many common examples of forces – contact forces, elasticity, viscosity, friction, and pressure – result from electrical interactions between the smallest particles of matter.
In his experiments of 1849–50, Michael Faraday was the first to search for a unification of gravity with electricity and magnetism.[5] However, he found no connection.
In 1900, David Hilbert published a famous list of mathematical problems. In Hilbert's sixth problem, he challenged researchers to find an axiomatic basis to all of physics. In this problem he thus asked for what today would be called a theory of everything.
In the late 1920s, the new quantum mechanics showed that the chemical bonds between atoms were examples of (quantum) electrical forces, justifying Dirac's boast that "the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known".[6]
After 1915, when Albert Einstein published the theory of gravity (general relativity), the search for a unified field theory combining gravity with electromagnetism started again with renewed interest. At the time, it seemed plausible that no other fundamental forces exist. Prominent contributors were Gunnar Nordström, Hermann Weyl, Arthur Eddington, Theodor Kaluza, Oskar Klein, and most notably, Albert Einstein and his collaborators. Einstein intensely searched for such a unifying theory during the last decades of his life. However, none of these attempts were successful.[7]

Twentieth century and the nuclear interactions

In the twentieth century, the search for a unifying theory was interrupted by the discovery of the strong and weak nuclear forces (or interactions), which differ both from gravity and from electromagnetism. A further hurdle was the acceptance that in a ToE, quantum mechanics had to be incorporated from the start, rather than emerging as a consequence of a deterministic unified theory, as Einstein had hoped.
Gravity and electromagnetism could always peacefully coexist as entries in a list of classical forces, but for many years it seemed that gravity could not even be incorporated into the quantum framework, let alone unified with the other fundamental forces. For this reason, work on unification, for much of the twentieth century, focused on understanding the three "quantum" forces: electromagnetism and the weak and strong forces. The first two were combined in 1967–68 by Sheldon Glashow, Steven Weinberg, and Abdus Salam into the "electroweak" force.[8] Electroweak unification is a broken symmetry: the electromagnetic and weak forces appear distinct at low energies because the particles carrying the weak force, the W and Z bosons, have non-zero masses of 80.4 GeV/c2 and 91.2 GeV/c2, whereas the photon, which carries the electromagnetic force, is massless. At higher energies Ws and Zs can be created easily and the unified nature of the force becomes apparent.
While the strong and electroweak forces peacefully coexist in the Standard Model of particle physics, they remain distinct. So far, the quest for a theory of everything is thus unsuccessful on two points: neither a unification of the strong and electroweak forces – which Laplace would have called `contact forces' – has been achieved, nor a unification of these forces with gravitation has been achieved.

Modern physics

Conventional sequence of theories

A Theory of Everything would unify all the fundamental interactions of nature: gravitation, strong interaction, weak interaction, and electromagnetism. Because the weak interaction can transform elementary particles from one kind into another, the ToE should also yield a deep understanding of the various different kinds of possible particles. The usual assumed path of theories is given in the following graph, where each unification step leads one level up:
 
 
 
 
 Theory of Everything
 

 
 
 
 
 
 
 
 
 

 
 Gravitation
 
 
 
 
 Electronuclear force (GUT)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 Strong interaction
SU(3)
 
 
 
 
 
 Electroweak interaction
SU(2) x U(1)Y
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 Weak interaction
 
 
 
 
 Electromagnetism
U(1)EM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 Electricity
 
 
 
 
 Magnetism
 
 
 
 
In this graph, electroweak unification occurs at around 100 GeV, grand unification is predicted to occur at 1016 GeV, and unification of the GUT force with gravity is expected at the Planck energy, roughly 1019 GeV.
Several Grand Unified Theories (GUTs) have been proposed to unify electromagnetism and the weak and strong forces. Grand unification would imply the existence of an electronuclear force; it is expected to set in at energies of the order of 1016 GeV, far greater than could be reached by any possible Earth-based particle accelerator. Although the simplest GUTs have been experimentally ruled out, the general idea, especially when linked with supersymmetry, remains a favorite candidate in the theoretical physics community. Supersymmetric GUTs seem plausible not only for their theoretical "beauty", but because they naturally produce large quantities of dark matter, and because the inflationary force may be related to GUT physics (although it does not seem to form an inevitable part of the theory). Yet GUTs are clearly not the final answer; both the current standard model and all proposed GUTs are quantum field theories which require the problematic technique of renormalization to yield sensible answers. This is usually regarded as a sign that these are only effective field theories, omitting crucial phenomena relevant only at very high energies.
The final step in the graph requires resolving the separation between quantum mechanics and gravitation, often equated with general relativity. Numerous researchers concentrate their efforts on this specific step; nevertheless, no accepted theory of quantum gravity – and thus no accepted theory of everything – has emerged yet. It is usually assumed that the ToE will also solve the remaining problems of GUTs.
In addition to explaining the forces listed in the graph, a ToE may also explain the status of at least two candidate forces suggested by modern cosmology: an inflationary force and dark energy. Furthermore, cosmological experiments also suggest the existence of dark matter, supposedly composed of fundamental particles outside the scheme of the standard model. However, the existence of these forces and particles has not been proven yet.

String theory and M-theory

List of unsolved problems in physics
Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything", or just a blind alley?
Since the 1990s, many physicists believe that 11-dimensional M-theory, which is described in many sectors by matrix string theory, in many other sectors by perturbative string theory, is the theory of everything. However, there is no widespread consensus on this issue, because M-theory and superstring theory is not a completed theory but rather an approach for producing one. All these theories attempt to deal with the renormalization problem by setting up some lower bound on the length scales possible.
String theories and supergravity (both believed to be limiting cases of the yet-to-be-defined M-theory) suppose that the universe actually has more dimensions than the easily observed three of space and one of time. The motivation behind this approach began with the Kaluza-Klein theory in which it was noted that applying general relativity to a five dimensional universe (with the usual four dimensions plus one small curled-up dimension) yields the equivalent of the usual general relativity in four dimensions together with Maxwell's equations (electromagnetism, also in four dimensions). This has led to efforts to work with theories with large number of dimensions in the hopes that this would produce equations that are similar to known laws of physics. The notion of extra dimensions also helps to resolve the hierarchy problem, which is the question of why gravity is so much weaker than any other force. The common answer involves gravity leaking into the extra dimensions in ways that the other forces do not.[9]
In the late 1990s, it was noted that one problem with several of the candidates for theories of everything (but particularly string theory) was that they did not constrain the characteristics of the predicted universe. For example, many theories of quantum gravity can create universes with arbitrary numbers of dimensions or with arbitrary cosmological constants. Even the "standard" ten-dimensional string theory allows the "curled up" dimensions to be compactified in an enormous number of different ways (one estimate is 10500 ) each of which corresponds to a different collection of fundamental particles and low-energy forces. This array of theories is known as the string theory landscape.
A speculative solution is that many or all of these possibilities are realised in one or another of a huge number of universes, but that only a small number of them are habitable, and hence the fundamental constants of the universe are ultimately the result of the anthropic principle rather than a consequence of the theory of everything. This anthropic approach is often criticised[10] in that, because the theory is flexible enough to encompass almost any observation, it cannot make useful (i.e., original, falsifiable, and verifiable) predictions. In this view, string theory would be considered a pseudoscience, where an unfalsifiable theory is constantly adapted to fit the experimental results.

Loop quantum gravity

Current research on loop quantum gravity may eventually play a fundamental role in a ToE, but that is not its primary aim.[11] Also loop quantum gravity introduces a lower bound on the possible length scales.
There have been recent claims that loop quantum gravity may be able to reproduce features resembling the Standard Model. So far only the first generation of fermions (leptons and quarks) with correct parity properties have been modelled by Sundance Bilson-Thompson using preons constituted of braids of spacetime as the building blocks.[12] However, there is no derivation of the Lagrangian that would describe the interactions of such particles, nor is it possible to show that such particles are fermions, nor that the gauge groups or interactions of the Standard Model are realised. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.[13]
This model leads to an interpretation of electric and colour charge as topological quantities (electric as number and chirality of twists carried on the individual ribbons and colour as variants of such twisting for fixed electric charge).
Bilson-Thompson's original paper suggested that the higher-generation fermions could be represented by more complicated braidings, although explicit constructions of these structures were not given. The electric charge, colour, and parity properties of such fermions would arise in the same way as for the first generation. The model was expressly generalized for an infinite number of generations and for the weak force bosons (but not for photons or gluons) in a 2008 paper by Bilson-Thompson, Hackett, Kauffman and Smolin.[14]

Other attempts

Any ToE must include general relativity and the standard model of particle physics.
A recently very prolific attempt is called Causal Sets. As some of the approaches mentioned above, its direct goal isn't necessarily to achieve a ToE but primarily a working theory of quantum gravity, which might eventually include the standard model and become a candidate for a ToE. Its founding principle is that spacetime is fundamentally discrete and that the spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between relative past and future distinguishing spacetime events.
Outside the previously mentioned attempts there is Garrett Lisi's E8 proposal. This theory provides an attempt of identifying general relativity and the standard model within the Lie group E8. The theory doesn't provide a novel quantization procedure and the author suggests its quantization might follow the Loop Quantum Gravity approach above mentioned.[15]

Present status

At present, no convincing candidate for a ToE is available. Most particle physicists state that the outcome of the ongoing experiments – the search for new particles at the large particle accelerators and for dark matter – are needed in order to provide theoretical physicists with further input for a ToE.

Theory of everything and philosophy

Main article: Theory of everything (philosophy)
The philosophical implications of a physical ToE are frequently debated. For example, if philosophical physicalism is true, a physical ToE will coincide with a philosophical theory of everything.
The "system building" style of metaphysics attempts to answer all the important questions in a coherent way, providing a complete picture of the world. Plato and Aristotle could be said to have created early examples of comprehensive systems. In the early modern period (17th and 18th centuries), the system-building scope of philosophy is often linked to the rationalist method of philosophy, which is the technique of deducing the nature of the world by pure a priori reason. Examples from the early modern period include the Leibniz's Monadology, Descarte's Dualism, and Spinoza's Monism. Hegel's Absolute idealism and Whitehead's Process philosophy were later systems.
Other philosophers do not believe their techniques can aim so high. Some scientists think a more mathematical approach than philosophy is needed for a ToE, for instance Stephen Hawking wrote in A Brief History of Time that even if we had a ToE, it would necessarily be a set of equations. He wrote, “What is it that breathes fire into the equations and makes a universe for them to describe?”.[16]

Arguments against a theory of everything

In parallel to the intense search for a theory of everything, various other scholars are debating the possibility of success.

Gödel's incompleteness theorem

A number of scholars claim that Gödel's incompleteness theorem proves that any attempt to construct a ToE is bound to fail. Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.
Fr. Stanley Jaki, in his 1966 book The Relevance of Physics, pointed out that, because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.[17] In a later reflection, Jaki states that it is wrong to say that a final theory is impossible, but rather that "when it is on hand one cannot know rigorously that it is a final theory."[18]
Freeman Dyson has stated that
“ Gödel’s theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. [...] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them. ”
—NYRB, May 13, 2004
Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable.
“ Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. ”
Jürgen Schmidhuber (1997) has argued against this view; he points out that Gödel's theorems are irrelevant for computable physics.[19] In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo-randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal ToEs describable by very few bits of information.[20]
Related critique was offered by Solomon Feferman,[21] among others. Douglas S. Robertson offers Conway's game of life as an example:[22] The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.
Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", most physicists argue that Gödel's Theorem does not mean that a ToE cannot exist. On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.[23] This definitional discrepancy may explain some of the disagreement among researchers.

Fundamental limits in accuracy

No physical theory to date is believed to be precisely accurate. Instead, physics has proceeded by a series of "successive approximations" allowing more and more accurate predictions over a wider and wider range of phenomena. Some physicists believe that it is therefore a mistake to confuse theoretical models with the true nature of reality, and hold that the series of approximations will never terminate in the "truth". Einstein himself expressed this view on occasions.[24] Following this view, we may reasonably hope for a theory of everything which self-consistently incorporates all currently known forces, but we should not expect it to be the final answer.
On the other hand it is often claimed that, despite the apparently ever-increasing complexity of the mathematics of each new theory, in a deep sense associated with their underlying gauge symmetry and the number of fundamental physical constants, the theories are becoming simpler. If this is the case, the process of simplification cannot continue indefinitely.

Lack of fundamental laws

There is a philosophical debate within the physics community as to whether a theory of everything deserves to be called the fundamental law of the universe.[25] One view is the hard reductionist position that the ToE is the fundamental law and that all other theories that apply within the universe are a consequence of the ToE. Another view is that emergent laws, which govern the behavior of complex systems, should be seen as equally fundamental. Examples of emergent laws are the second law of thermodynamics and the theory of natural selection. The advocates of emergence argue that emergent laws, especially those describing complex or living systems are independent of the low-level, microscopic laws. In this view, emergent laws are as fundamental as a ToE.
It is not clear that there is any point at issue in these debates. A well-known one took place between Steven Weinberg and Philip Anderson[citation needed]. Possibly the only issue at stake is the right to apply the high-status term "fundamental" to the respective subjects of research.

Impossibility of being "of everything"

Although the name "theory of everything" suggests the determinism of Laplace's quotation, this gives a very misleading impression. Determinism is frustrated by the probabilistic nature of quantum mechanical predictions, by the extreme sensitivity to initial conditions that leads to mathematical chaos, by the limitations due to event horizons, and by the extreme mathematical difficulty of applying the theory. Thus, although the current standard model of particle physics "in principle" predicts all known non-gravitational phenomena, in practice only a few quantitative results have been derived from the full theory (e.g., the masses of some of the simplest hadrons), and these results (especially the particle masses which are most relevant for low-energy physics) are less accurate than existing experimental measurements. The ToE would almost certainly be even harder to apply for the prediction of experimental results, and thus might be of limited use.
A motive for seeking a ToE,[citation needed] apart from the pure intellectual satisfaction of completing a centuries-long quest, is that all prior examples of unification have predicted new phenomena, some of which (e.g., electrical generators) have proved of great practical importance. And like in these prior examples of unification, the ToE would probably allow us to confidently define the domain of validity and residual error of low-energy approximations to the full theory.

Infinite number of onion layers

Lee Smolin regularly argues that the layers of nature may be like the layers of an onion, and that the number of layers might be infinite.[citation needed] This would imply an infinite sequence of physical theories.
The argument is not universally accepted, because it is not obvious that infinity is a concept that applies to the foundations of nature. The results of quantum theory strongly suggest that nature is not infinite in its foundations, because space and time have been shown to break down at smaller quantities than the Planck units.

Impossibility of calculation

Weinberg [26] points out that calculating the precise motion of an actual projectile in the Earth's atmosphere is impossible. So how can we know we have an adequate theory for describing the motion of projectiles? Weinberg suggests that we know principles (Newton's laws of motion and gravitation) that work "well enough" for simple examples, like the motion of planets in empty space. These principles have worked so well on simple examples that we can be reasonably confident they will work for more complex examples. So a ToE must work for a wide range of simple examples in such a way that we can be reasonably confident it will work for every situation in physics.

See also

Portal icon Physics portal

References

Footnotes

  1. ^ Weinberg (1993)
  2. ^ Ellis, John (2002). "Physics gets physical (correspondence)". Nature 415 (6875): 957. Bibcode 2002Natur.415..957E. doi:10.1038/415957b.
  3. ^ Ellis, John (1986). "The Superstring: Theory of Everything, or of Nothing?". Nature 323 (6089): 595–598. Bibcode 1986Natur.323..595E. doi:10.1038/323595a0.
  4. ^ Shapin, Steven (1996). The Scientific Revolution. University of Chicago Press. ISBN 0-226-75021-3.
  5. ^ Faraday, M. (1850). "Experimental Researches in Electricity. Twenty-Fourth Series. On the Possible Relation of Gravity to Electricity". Abstracts of the Papers Communicated to the Royal Society of London 5: 994–995. doi:10.1098/rspl.1843.0267.
  6. ^ Dirac, P.A.M. (1929). "Quantum mechanics of many-electron systems". Proceedings of the Royal Society of London A 123 (792): 714. Bibcode 1929RSPSA.123..714D. doi:10.1098/rspa.1929.0094.
  7. ^ Pais (1982), Ch. 17.
  8. ^ Weinberg (1993), Ch. 5
  9. ^ HOLLOWAY, MARGUERITE (2005). "The Beauty of Branes". Scientific American (Scientific American) (October 2005): 38. Retrieved August 13, 2012.
  10. ^ Smolin, Lee (2006). The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. Houghton Mifflin. ISBN 978-0-618-55105-7.
  11. ^ Potter, Franklin (15 February 2005). "Leptons And Quarks In A Discrete Spacetime". Frank Potter's Science Gems. Retrieved 2009-12-01.
  12. ^ Bilson-Thompson, Sundance O.; Markopoulou, Fotini; Smolin, Lee (2007). "Quantum gravity and the standard model". Classical and Quantum Gravity 24 (16): 3975–3994. arXiv:hep-th/0603022. Bibcode 2007CQGra..24.3975B. doi:10.1088/0264-9381/24/16/002.
  13. ^ Castelvecchi, Davide; Valerie Jamieson (August 12, 2006). "You are made of space-time". New Scientist (2564).
  14. ^ Sundance Bilson-Thompson; Jonathan Hackett; Lou Kauffman; Lee Smolin (2008). "Particle Identifications from Symmetries of Braided Ribbon Network Invariants". arXiv:0804.0037 [hep-th].
  15. ^ A. G. Lisi (2007). "An Exceptionally Simple Theory of Everything". arXiv:0711.0770 [hep-th].
  16. ^ As quoted in Artigas, The Mind of the Universe, p. 123
  17. ^ Jaki, S.L. (1966). The Relevance of Physics. Chicago Press. pp. 127–130.
  18. ^ Stanley L. Jaki (2004) "A Late Awakening to Gödel in Physics," pp. 8–9.
  19. ^ Schmidhuber, Jürgen (1997). A Computer Scientist's View of Life, the Universe, and Everything. Lecture Notes in Computer Science. Springer. pp. 201–208. doi:10.1007/BFb0052071. ISBN 978-3-540-63746-2.
  20. ^ Schmidhuber, Jürgen (2002). "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit". arXiv:quant-ph/0011122.
  21. ^ Feferman, Solomon (17 November 2006). "The nature and significance of Gödel’s incompleteness theorems". Institute for Advanced Study. Retrieved 2009-01-12.
  22. ^ Robertson, Douglas S. (2007). "Goedel's Theorem, the Theory of Everything, and the Future of Science and Mathematics". Complexity 5 (5): 22–27. doi:10.1002/1099-0526(200005/06)5:5<22::aid-cplx4>3.0.CO;2-0.
  23. ^ Hawking, Stephen (20 July 2002). "Gödel and the end of physics". Retrieved 2009-12-01.
  24. ^ Einstein, letter to Felix Klein, 1917. (On determinism and approximations.) Quoted in Pais (1982), Ch. 17.
  25. ^ Weinberg (1993), Ch 2.
  26. ^ Weinberg (1993) p. 5

Bibliography




Publié par à

Aucun commentaire:

Enregistrer un commentaire

Remarque : Seul un membre de ce blog est autorisé à enregistrer un commentaire.

Inscription à : Publier les commentaires (Atom)