Wikiprojekt:SKFiz/brudnopis/Kwantowa grawitacja pętlowa

Pętlowa kwantowa grawitacja (LQG od ang. Loop Quantum Gravity) to teoria próbująca opisać kwantowe własności grawitacji. Jest to ponadto teoria kwantowej przestrzeni i kwantowego czasu - jako że z ogólnej teorii względności wiadomo, że grawitacja wyraża się poprzez geometrię czasoprzestrzeni. LQG jest wynikiem zastosowania metod kwantyzacji kanonicznej do ogólnej teorii względności wyrażonej w zmiennych Ashtekara.

Jednym z głównych wyników pętlowej kwantowej grawitacji jest obraz fizycznej czasoprzestrzeni, w której przestrzeń jest ziarnista. Ziarnistość jest bezpośrednią konsekwencją kwantyzacji wielkości geometrycznych, takich jak powierzchnia czy objętość. Ziarnistość ta ma naturę analogiczną do cząstkowej interpretacji fotonów w elektrodynamice kwantowej lub dyskretne poziomy energii atomów. Jednakże w tym przypadku to przestrzeń (objętość) i geometria (powierzchnia) jest dyskretna.

Z punktu widzenia LQG przestrzeń to bardzo gęsto utkana tkanina - sieć, której oka (pętle) są jednakże skończenie małe. Ta sieć nazywana jest siecią spinową s. Ewolucja takiej sieci w czasie nazywana jest pianą spinową. Przewiduje się, że rozmiary tej sieci są rzędu Długości Plancka, czyli około $10^{-35}$ metra. Zgodnie z LQG odległości mniejsze niż skala Placka nie mają fizycznego znaczenia - podobnie jak nie da się opisywać momentów pędu mniejszych niż $\hbar$ lub ładunków mniejszych niż elementarny.

W ramach pętlowa kwantowa grawitacja prowadzone są obecnie badania w wielu kierunkach przez około 50 grup badawczych na świecie. Wszystkie projekty bazują na tych samych fizycznych założeniach oraz tym samym matematycznym opisie kwantowej przestrzeni. Na szczególną uwagę zasługują dwie gałęzie badań: tradycyjna kanoniczna pętlowa grawitacja kwantowa oraz kowariantna pętlowa grawitacja kwantowa, nazywana też teorią pian spinowych.

Badane są potencjalne fizyczne skutki pętlowej kwantyzacji czasoprzestrzeni. Wśród badań tych najbardziej rozwinięty jest model powstały przez zastosowanie LQG do kosmologi, nazywany kwantową pętlową kosmologią (LQC, ang. Loop Quantum Cosmology). LQC bada fizykę bardzo wczesnych etapów rozwoju Wszechświata - okolic Wielkiego Wybuchu. Najbardziej spektakularnym rezultatem jest to, że ewolucja Wszechświata może być przedłużona poza Wielki Wybuch. Sam Wielki Wybuch zastąpiony jest w LQC przez zjawisko nazywane "Wielkim Odbiciem" (ang. Big Bounce).

Historia

In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler-DeWitt equation, admitted solutions labelled by loops, when rewritten in the new Ashtekar language, and Carlo Rovelli and Lee Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Jorge Pullin and Jurek Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, and the theory should be formulated in terms of intersecting loops, or graphs.

In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. That is geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins.

The canonical version of the dynamics was put on firm ground by Thomas Thiemann who defined an anomaly-free Hamiltonian operator, showing the existence of a mathematically consistent background-independent theory. The covariant or spinfoam version of the dynamics developed during several decades, and crystallized in 2008, from the joint work of research groups in France, Canada, UK, Poland, and Germany, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, and this is consistent with observed acceleration in the expansion of the Universe.

Główne idee

Problem sformułowania kwantowej teorii grawitacji jest jeden z głównych nierozwiązanych zagadnień współczesnej fizyki. Standardowe techniki kwantyzacji pól, które świetnie działają w Modelu Standartowym cząstek elementarnych nie działają dla grawitacji, gdyż prowadzą do nierenormalizowalnej kwantowej teorii pola.

Źródłem tego problemu jest fakt, że grawitacja to geometria. Gdy brane pod uwagę są kwantowe efekty grawitacji, sama czasoprzestrzeń staje się obiektem kwantowym, a zatem metody konwencjonalnej kwantowej teorii pola, które korzystają z istnienia dobrze zdefiniowanej klasycznej geometrii, przestają działać. W tym miejscu do gry wchodzi pętlowa kwantowa grawitacja, której definicje nie zakładają istnienia klasycznego tła (dlatego mówi się, że LQG jest teorią 'background independent'). LQG dostarcza metod badania kwantowych pól, gdy nie ma dostępu do czasoprzestrzeni. Stany kwantowe w LQG, zdefiniowane są na sieciach spinowych (czyli odpowiednio pokolorowanych grafach), które nie są zanurzone w żadnej fizycznej przestrzeni, ale które są przestrzenią. Tak wyraża się kwantowa wersja teorii Einstaina: stany w tej teorii nie są polem grawitacyjnym w pewnej czasoprzestrzeni, lecz właśnie definiują czasoprzestrzenią, w której mogą żyć inne pola.

Przykład prostej sieci spinowej, używanej w pętlowej kwantowej grawitacji

The reformulation of general relativity introduced by Ashtekar, which goes under the name of Ashtekar variables represents the gravitational field using fields similar to the electric and magnetic fields. In the quantum version of the theory, the Faraday lines of these fields become the loops of LQG, and carry discrete quanta of geometry. A spin network can be seen as an ensemble of a finite number of (quantum) Faraday lines of gravity, which define a quantum space. These evolve in time in discrete steps.

Quantum matter is usually coupled to the quantum gravitational field as an independent field, in LQG. But there are also attempts to relate directly matter and geometry (by Renate Loll, Jan Ambjørn, Lee Smolin, Sundance Bilson-Thompson, Laurent Freidel, Mark B. Wise and others).

To date, the main successes of loop quantum gravity are:

1. It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators
2. It includes a calculation of the entropy of black holes
3. It replaces the Big Bang spacetime singularity with a Big Bounce

None of these results has so far received any direct empirical support. Therefore the theory must be considered only tentative for the moment, like all other tentative theories that go beyond the standard model, such as string theory or supersymmetry. The research community in fundamental theoretical physics is following different research directions, and this is of course a healthy situation. A lively critical and competitive attitude between groups following different research directions is also healthy, since science requires sharp critical thinking. Polemics between some partisans of the loop and string communities have been quite lively sometime ago.

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. Originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions, an arbitrary gauge group (or even quantum group), and supersymmetry, and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory.

In concise terms, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much-studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. Classical general relativity can be formulated as a BF theory with constraints, and one way of deriving covariant LQG is by adding appropriate quantum constraints to BF quantum theory.

Lorentz invariance

LQG is a quantization of a classical Lagrangian field theory equivalent to the usual Einstein-Cartan theory. It leads to the same equations of motion as general relativity. In general, there is no global Lorentz invariance in classical general relativity and therefore there is no global Lorentz invariance in LQG. However, there is "local" Lorentz invariance in classical general relativity. In covariant LQG, it is possible to show that the transition amplitudes are Lorentz invariant under local Lorentz transformations. Therefore there is no breaking of Lorentz invariance in LQG. This is consistent with recent observations that appear to be consistent with the expectation that Lorentz invariance is still valid at the Planck scale.

A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.

General covariance and background independence

General covariance, also known as "diffeomorphism invariance", is the invariance of physical laws under arbitrary coordinate transformations. An example of this is the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffeomorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called "problem of time" in general relativity. A generally accepted calculational framework to account for this constraint has yet to be found.

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent. The equations of LQG are not embedded in, or presuppose, space and time (except for its invariant topology). Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length.

Black hole entropy

In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. Much as the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.

An artist depiction of two black holes merging, a process in which the laws of thermodynamics are upheld.

The only way to satisfy the second law of thermodynamics is to admit that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.

Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. Bekenstein suggested (½ ln 2)/4π as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature (Hawking temperature). Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at 1/4:

$S_{\text{BH}}={\frac {kA}{4\ell _{\mathrm {P} }^{2}}}$

where A is the area of the event horizon, calculated at 4πR², k is Boltzmann's constant, and $\ell _{\mathrm {P} }={\sqrt {G\hbar /c^{3}}}$ is the Planck length. The subscript BH either stands for "black hole" or "Bekenstein-Hawking". The black hole entropy is proportional to the area of its event horizon $A$. The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.

Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "no hair" theorems appeared to suggest that black holes could have only a single microstate.

In LQG it is possible to associate a geometrical interpretation to the microstates: these are the quantum geometries of the horizon. LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes.

The problem is that a crucial free parameter in the theory, known as the Immirzi parameter, can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value. A prediction directly from theory would be preferable.

A recent success of the theory in this direction is the computation of the entropy of all non singular black holes directly from theory and independent of Immirzi parameter. The result is the expected formula $S=A/4$, where S is the entropy and A the area of the black hole, derived by Bekenstein and Hawking on heuristic grounds. This is the only known derivation of this formula from a fundamental theory, for the case of generic non singular black holes.

Problems and comparisons with alternative approaches

The theory of LQG is one possible solution to the problem of quantum gravity, as is string theory. There are substantial differences however. For example, string theory also addresses unification, the understanding of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries. Contrary to this, LQG is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction. On the other hand, the consequences of LQG are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.

Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections). Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian. Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of Quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see One-loop Feynman diagram in Feynman diagram). The fate of Lorentz invariance in loop quantum gravity remains an open problem.

While there has been a recent proposal relating to observation of naked singularities, and doubly special relativity, as a part of a program called loop quantum cosmology, as of now there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity). Because of the above mentioned lack of a semiclassical limit, LQG cannot even reproduce the predictions made by the Standard Model.

Current LQG research directions attempt to address these known problems, and include spinfoam models and entropic gravity.

See also

• C*-algebra
• Category theory
• Double special relativity
• Group field theory
• Heyting algebra
• Immirzi parameter

• Invariance mechanics
• Kodama state
• Loop quantum cosmology
• Lorentz invariance in loop quantum gravity
• Noncommutative geometry
• Regge calculus

• S-knot
• Spin foam
• String-net
• Supersymmetry
• Topos theory

References

• Topical Reviews
  • CarloC. Rovelli CarloC., Zakopane lectures on loop gravity, „{{{czasopismo}}}”, 2011, arXiv:gr-qc/1102.3660 .czasopismobrak strony (książka)
  • CarloC. Rovelli CarloC., Loop Quantum Gravity, „Living Reviews in Relativity”, 1998 [dostęp 2008-03-13] .
  • ThomasT. Thiemann ThomasT., Lectures on Loop Quantum Gravity, „Lectures Notes in Physics”, 2003, s. 41–135, DOI: 10.1007/978-3-540-45230-0_3, Bibcode: 2003LNP...631...41T, arXiv:gr-qc/0210094 .
  • AbhayA. Ashtekar AbhayA., JerzyJ. Lewandowski JerzyJ., Background Independent Quantum Gravity: A Status Report, „Classical and Quantum Gravity”, 15 (21), 2004, R53–R152, DOI: 10.1088/0264-9381/21/15/R01, Bibcode: 2004CQGra..21R..53A, arXiv:gr-qc/0404018 .
  • Carlo Rovelli and Marcus Gaul, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, e-print available as gr-qc/9910079.
  • Lee Smolin, The case for background independence, e-print available as hep-th/0507235.
  • Alejandro Corichi, Loop Quantum Geometry: A primer, e-print available as [2].
  • Alejandro Perez, Introduction to loop quantum gravity and spin foams, e-print available as [3].
  • Hermann Nicolai and Kasper Peeters Loop and spin foam quantum gravity: A Brief guide for beginners., e-print available as [4].
• Popular books:
  • Lee Smolin, Three Roads to Quantum Gravity
  • Carlo Rovelli, Che cos'è il tempo? Che cos'è lo spazio?, Di Renzo Editore, Roma, 2004. French translation: Qu'est ce que le temps? Qu'est ce que l'espace?, Bernard Gilson ed, Brussel, 2006. English translation: What is Time? What is space?, Di Renzo Editore, Roma, 2006.
  • Julian Barbour, The End of Time: The Next Revolution in Our Understanding of the Universe
  • GeorgeG. Musser GeorgeG., The Complete Idiot's Guide to String Theory, „The Physics Teacher”, 2 (47), Alpha, 2008, DOI: 10.1119/1.3072469, ISBN 978-1-59257-702-6, Bibcode: 2009PhTea..47Q.128H .??? – Focuses on string theory but has an extended discussion of loop gravity as well.
• Magazine articles:
  • Lee Smolin, "Atoms of Space and Time," Scientific American, January 2004
  • Martin Bojowald, "Following the Bouncing Universe," Scientific American, October 2008
• Easier introductory, expository or critical works:
  • Abhay Ashtekar, Gravity and the quantum, e-print available as gr-qc/0410054 (2004)
  • John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)
  • Carlo Rovelli, A Dialog on Quantum Gravity, e-print available as hep-th/0310077 (2003)
• More advanced introductory/expository works:
  • Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online
  • Thomas Thiemann, Introduction to modern canonical quantum general relativity, e-print available as gr-qc/0110034
  • Thomas Thiemann, Introduction to Modern Canonical Quantum General Relativity, Cambridge University Press (2007)
  • Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).
  • Abhay Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
  • Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press (1996)
  • Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view, e-print available as hep-th/0501114
  • H. Nicolai and K. Peeters, Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners, e-print available as hep-th/0601129
  • T. Thiemann The LQG – String: Loop Quantum Gravity Quantization of String Theory (2004)

• Conference proceedings:
  • John C. Baez (ed.), Knots and Quantum Gravity
• Fundamental research papers:
  • AbhayA. Ashtekar AbhayA., New variables for classical and quantum gravity, „Physical Reviews Letters”, 18 (57), 1986, s. 2244–2247, DOI: 10.1103/PhysRevLett.57.2244, PMID: 10033673, Bibcode: 1986PhRvL..57.2244A .
  • AbhayA. Ashtekar AbhayA., New Hamiltonian formulation of general relativity, „Physical Review D”, 6 (36), 1987, s. 1587–1602, DOI: 10.1103/PhysRevD.36.1587, Bibcode: 1987PhRvD..36.1587A .
  • Roger Penrose, Angular momentum: an approach to combinatorial space-time in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971
  • CarloC. Rovelli CarloC., LeeL. Smolin LeeL., Knot theory and quantum gravity, „Physical Review Letters”, 10 (61), 1988, DOI: 10.1103/PhysRevLett.61.1155, Bibcode: 1988PhRvL..61.1155R .
  • CarloC. Rovelli CarloC., LeeL. Smolin LeeL., Loop space representation of quantum general relativity, „Nuclear Physics”, 1990, s. 80–152 .
  • Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005
  • KarelK. Kuchař KarelK., Relativity, Astrophysics and Cosmology, D. Reidel, 1973, s. 237–288, ISBN 90-277-0369-8 .
  • ThomasT. Thiemann ThomasT., Loop Quantum Gravity: An Inside View, „Approaches to Fundamental Physics”, 2006, Bibcode: 2007LNP...721..185T, arXiv:hep-th/0608210 .

External links

• "Loop Quantum Gravity" by Carlo Rovelli Physics World, November 2003
• Quantum Foam and Loop Quantum Gravity
• Abhay Ashtekar: Semi-Popular Articles . Some excellent popular articles suitable for beginners about space, time, GR, and LQG.
• Loop Quantum Gravity: Lee Smolin.
• Loop Quantum Gravity on arxiv.org
• A list of LQG references catered to fresh graduates
• Loop Quantum Gravity Lectures Online by Lee Smolin
• Spin networks, spin foams and loop quantum gravity
• Wired magazine, News: Moving Beyond String Theory
• April 2006 Scientific American Special Issue, A Matter of Time, has Lee Smolin LQG Article Atoms of Space and Time
• September 2006, The Economist, article Looping the loop
• Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
• Zeno meets modern science. Article from Acta Physica Polonica B by Z.K. Silagadze.
• Did pre-big bang universe leave its mark on the sky? - According to a model based on "loop quantum gravity" theory, a parent universe that existed before ours may have left an imprint (New Scientist, 10 April 2008)

Kategoria:Theories of gravitation Kategoria:Loop quantum gravity

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