Tekst źródłowy strony Wikiprojekt:SKFiz/brudnopis/Kwantowa grawitacja pętlowa

'''Pętlowa kwantowa grawitacja''' ('''LQG''' od ang. '''Loop Quantum Gravity''') to teoria próbująca opisać [[Kwant|kwantowe]] własności [[grawitacja|grawitacji]]. Jest to ponadto teoria kwantowej [[przestrzeń_(fizyka)|przestrzeni]] i kwantowego [[czas|czasu]] - jako że z [[ogólna teoria względności|ogólnej teorii względności]] wiadomo, że grawitacja wyraża się poprzez geometrię [[czasoprzestrzeń|czasoprzestrzeni]]. LQG jest wynikiem zastosowania metod [[kwantyzacja kanoniczna|kwantyzacji kanonicznej]] do ogólnej teorii względności wyrażonej w [[zmienne Ashtekara|zmiennych Ashtekara]].

Jednym z głównych wyników pętlowej kwantowej grawitacji jest obraz fizycznej czasoprzestrzeni, w której przestrzeń jest ziarnista. Ziarnistość jest bezpośrednią konsekwencją [[kwantowanie_(Fizyka)|kwantyzacji]] wielkości geometrycznych, takich jak powierzchnia czy objętość. Ziarnistość ta ma naturę analogiczną do cząstkowej interpretacji [[foton|fotonów]] w elektrodynamice kwantowej lub dyskretne poziomy [[energia|energii]] [[atom|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 [[sieć spinowa|siecią spinową]] s. Ewolucja takiej sieci w czasie nazywana jest [[piana spinowa|pianą spinową]]. Przewiduje się, że rozmiary tej sieci są rzędu [[długość Plancka|Długości Plancka]], czyli około <math>10^{-35}</math> metra. Zgodnie z LQG odległości mniejsze niż skala Placka nie mają fizycznego znaczenia - podobnie jak nie da się opisywać momentów pędu mniejszych niż <math>\hbar</math> lub ładunków mniejszych niż elementarny.

W ramach pętlowa kwantowa grawitacja prowadzone są obecnie badania w wielu kierunkach przez około 50 grup badawczych na świecie.<ref>{{cytuj stronę | url = https://maps.google.com/maps/ms?msid = 201675987190934929965.0004843830d27f3e6c50e&msa = 0 | tytuł = Loop Quantum Gravity - Google Maps | opublikowany = Maps.google.com | data =  | data dostępu = 2012-08-13}}</ref> 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 [[teoria pian spinowych|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 [[pętlowa kwantowa kosmologia|kwantową pętlową kosmologią]] (LQC, ang. Loop Quantum Cosmology). LQC bada fizykę bardzo wczesnych etapów rozwoju Wszechświata - okolic [[wielki wybuch|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. [[Wielkie Odbicie|Big Bounce]]).

== Historia ==
<!-- {{osobny artykuł|Historia pętlowej kwantowej grawitacji}} -->

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 [[Graph (mathematics)|graphs]].

In 1994, [[Carlo Rovelli|Rovelli]] and [[Lee Smolin|Smolin]] showed that the quantum [[operator (physics)|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 network]]s, which are [[Graph (mathematics)|graphs]] labelled by [[spin (physics)|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.<ref>{{cytuj stronę | url = http://inspirehep.net/record/889853 | tytuł = Zakopane lectures on loop gravity | opublikowany = Inspire-Hep | data =  | data dostępu = 2012-08-13}}</ref>  The finiteness of these amplitudes was proven in 2011.<ref>http://inspirehep.net/record/899209. http://inspirehep.net/record/1080875.</ref>  It requires the existence of a positive [[cosmological constant]], and this is consistent with observed [[Accelerating universe|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 [[model standartowy|Modelu Standartowym]] [[cząstki elementarne|cząstek elementarnych]] nie działają dla grawitacji, gdyż prowadzą do [[renormalizacja|nierenormalizowalnej]] [[kwantowa teoria pola|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 [[kwantowa teoria pola|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.

[[Plik:Spinnetwork.jpg|right|thumb|250px|Przykład prostej [[sieć spinowa|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 field|electric]] and [[magnetic field]]s.<ref>See {{Harvnb|Ashtekar|1986}}, {{Harvnb|Ashtekar|1987}}.</ref>
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.<ref>For a review, see {{Harvnb|Thiemann|2006}}; more extensive accounts can be found in {{Harvnb|Rovelli|1998}}, {{Harvnb|Ashtekar|Lewandowski|2004}} as well as (lecture notes {{Harvnb|Thiemann|2003}} and {{Harvnb|Rovelli|2011}}.</ref>

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<ref name="ReferenceA">See [[List of loop quantum gravity researchers]]</ref>).

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

# It is a [[perturbation theory (quantum mechanics)|nonperturbative]] [[Quantization (physics)|quantization]] of 3-space geometry, with quantized area and volume [[Operator (physics)|operators]]
# It includes a calculation of the [[entropy]] of [[black holes]]
# 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 theory|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, [[fermion]]s,<ref>{{cytuj pismo | rok = 1997 | autor link = John C. Baez | nazwisko = Baez | imię = John | last2 = Krasnov | first2 = Kirill | tytuł = Quantization of Diffeomorphism-Invariant Theories with Fermions | arxiv = hep-th/9703112 | bibcode = 1998JMP....39.1251B | doi = 10.1063/1.532400 }}</ref> an arbitrary [[gauge group]] (or even [[quantum group]]), and supersymmetry,<ref>{{cite arxiv|year=1999 |last=Ling |first=Yi |last2=Smolin |first2=Lee |title=Supersymmetric Spin Networks and Quantum Supergravity |class=hep-th |eprint=hep-th/9904016}} }}</ref> 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*-algebra]]ic quantization to a [[canonical commutation relations|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 [[holonomy|holonomies]]) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with [[topology|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 ===
{{osobny artykuł|Lorentz covariance}}

LQG is a [[Quantization (physics)|quantization]] of a classical [[Lagrangian|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{{fakt|date=August 2012}} 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 law]]s under arbitrary [[Coordinate transformations#Transformations|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 [[Relativistic dynamics|dynamics]] and the so-called "[[problem of time]]" in general relativity.<ref>See ''e.g.'' Stuart Kauffman and Lee Smolin "A Possible Solution For The Problem Of Time In Quantum Cosmology" (1997). [http://www.edge.org/3rd_culture/smolin/smolin_p1.html]</ref> A generally accepted
calculational framework to account for this constraint has yet to be found.<ref>See Lee Smolin, "The Case for Background Independence", in Dean Rickles, ''et al.'' (eds.) ''The Structural Foundations of Quantum Gravity'' (2006), p 196 ff.</ref><ref>For a highly technical explanation, see Carlo Rovelli (2004). ''Quantum Gravity'', p 13 ''ff''.</ref>

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally [[background independence|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 horizon]]s.  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]].<ref name="Bousso 2002 825–874">{{cytuj pismo | nazwisko = Bousso | imię = Raphael | rok = 2002 | tytuł = The Holographic Principle | czasopismo = [[Reviews of Modern Physics]] | wolumin = 74 | wydanie = 3 | strony = 825–874 | doi = 10.1103/RevModPhys.74.825 | arxiv = hep-th/0203101 | bibcode = 2002RvMP...74..825B}}</ref>
[[Plik:Black Hole Merger.jpg|thumb|right|275px|An artist depiction of two [[black hole]]s 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]]<ref>[http://www.technologyreview.com/blog/arxiv/25805/ "First Observation of Hawking Radiation"] from the ''[[Technology Review]]''</ref><ref>{{cytuj pismo | autor = Matson, John | tytuł = Artificial event horizon emits laboratory analogue to theoretical black hole radiation | czasopismo = Sci. Am | rok = 2010 | miesiąc = Oct. 1 | url = http://www.scientificamerican.com/article.cfm?id = hawking-radiation}}</ref> corresponding to a certain temperature (Hawking temperature).<ref>[http://www.charlierose.com/guest/view/6294 Charlie Rose: A conversation with Dr. Stephen Hawking & Lucy Hawking]</ref><ref>''A Brief History of Time'', Stephen Hawking, Bantam Books, 1988.</ref> 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<ref>{{cytuj pismo | imię = Parthasarathi | nazwisko = Majumdar | tytuł = Black Hole Entropy and Quantum Gravity | arxiv = gr-qc/9807045 | czasopismo = ArXiv: General Relativity and Quantum Cosmology | rok = 1998 | bibcode = 1999InJPB..73..147M }}</ref>:

:<math>S_{\text{BH}} = \frac{kA}{4\ell_{\mathrm{P}}^2}</math>

where '''A''' is the area of the event horizon, calculated at '''4πR<sup>2</sup>,'''  ''k'' is [[Boltzmann's constant]], and <math>\ell_{\mathrm{P}}=\sqrt{G\hbar / c^3}</math> 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 <math>A</math>. 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]].<ref name="Bousso 2002 825–874"/>

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 theorem|no hair]]"<ref name="arxiv.org">http://arxiv.org/abs/gr-qc/0702006 No hair theorems for positive Lambda</ref> theorems appeared to suggest that black holes could have only a single microstate.

In LQG<ref name="ReferenceA"/> 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.<ref name="Rovelli1996">{{cytuj pismo | nazwisko = Rovelli | imię = Carlo | rok = 1996 | tytuł = Black Hole Entropy from Loop Quantum Gravity | czasopismo = Physical Review Letters | wolumin = 77 | strony = 3288–3291 | doi = 10.1103/PhysRevLett.77.3288 | arxiv = gr-qc/9603063 | bibcode = 1996PhRvL..77.3288R}}</ref><ref name="Ashtekar1997">{{cytuj pismo | nazwisko = Ashtekar | imię = Abhay | nazwisko2 = Baez, John; Corichi, Alejandro; Krasnov, Kirill | rok = 1998 | tytuł = Quantum Geometry and Black Hole Entropy | czasopismo = Physical Review Letters | wolumin = 80 | wydanie = 5 | strony = 904–907 | doi = 10.1103/PhysRevLett.80.904 | arxiv = gr-qc/9710007 | bibcode = 1998PhRvL..80..904A}}</ref>  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.<ref name="Bianchi2012">{{cytuj pismo | nazwisko = Bianchi | imię = Eugenio | rok = 2012 | tytuł = Entropy of Non-Extremal Black Holes from Loop Gravity | arxiv = gr-qc/1204.5122}}</ref> 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 [[Jacob Bekenstein|Bekenstein]] and [[Stephen Hawking|Hawking's]] calculation of the [[black hole thermodynamics|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.<ref>http://inspirehep.net/record/940357?ln=en. http://inspirehep.net/record/1111991.</ref>  The result is the expected formula <math>S=A/4</math>, 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  [[Unified field theory|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 (quantum mechanics)|Hamiltonian]].<ref name="autogenerated2005">{{Cytuj | tytuł = Loop quantum gravity: an outside view | data = 2005 | czasopismo = Classical and Quantum Gravity | s = R193–R247 | wydanie = 19 (22) | doi = 10.1088/0264-9381/22/19/R01 | bibcode = 2005CQGra..22R.193N | arxiv = hep-th/0501114 <!-- nieprzetłumaczone parametry: | postscript = . | author3 = Marija Zamaklar | author2 = Kasper Peeters | author1 = Hermann Nicolai -->}}</ref> 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]]).<ref name="autogenerated2005"/> The fate of [[Lorentz invariance in loop quantum gravity]] remains an open problem.<ref>{{cite arxiv | eprint = 0908.1832  | title = Testing Einstein's special relativity with Fermi's short hard gamma-ray burst GRB090510 | author1 = Fermi GBM/LAT Collaborations | class = astro-ph.HE | year = 2009 }}</ref>

While there has been a recent proposal relating to observation of [[naked singularity|naked singularities]],<ref>{{Cytuj | tytuł = Quantum evaporation of a naked singularity | data = 2006 | autor = Goswami et al. | czasopismo = Physical Review Letters | wydanie = 3 (96) | doi = 10.1103/PhysRevLett.96.031302 | bibcode = 2006PhRvL..96c1302G | arxiv = gr-qc/0506129 <!-- nieprzetłumaczone parametry: | postscript = . | page = 31302 | author3 = Singh, Parampreet | author2 = Joshi, Pankaj S. -->}}</ref> 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 <ref>{{cytuj stronę | url = http://math.ucr.edu/home/baez/week280.html | tytuł = week280 | opublikowany = Math.ucr.edu | data =  | data dostępu = 2012-08-13}}</ref> and [[entropic gravity]].<ref>{{cite arxiv |eprint=1001.3668 |author1=Lee Smolin |title=Newtonian gravity in loop quantum gravity |class=gr-qc |year=2010}}</ref>

== See also ==
{{col-begin}}
{{col-break}}
* [[C*-algebra]]
* [[Category theory]]
* [[Double special relativity]]
* [[Group field theory]]
* [[Heyting algebra]]
* [[Immirzi parameter]]
{{col-break}}
* [[Invariance mechanics]]
* [[Kodama state]]
* [[Loop quantum cosmology]]
* [[Lorentz invariance in loop quantum gravity]]
* [[Noncommutative geometry]]
* [[Regge calculus]]
{{col-break}}
* [[S-knot]]
* [[Spin foam]]
* [[String-net]]
* [[Supersymmetry]]
* [[Topos theory]]
{{col-end}}

== Notes ==
{{Przypisy}}

== References ==
* Topical Reviews
** {{Cytuj | tytuł = Zakopane lectures on loop gravity | data = 2011 | autor = Carlo Rovelli | arxiv = gr-qc/1102.3660 <!-- nieprzetłumaczone parametry: | authorlink = Carlo Rovelli -->}}
** {{Cytuj | url = http://www.livingreviews.org/lrr-1998-1 | tytuł = Loop Quantum Gravity | data dostępu = 2008-03-13 | data = 1998 | autor = Carlo Rovelli | czasopismo = Living Reviews in Relativity <!-- nieprzetłumaczone parametry: | volume = 1 | authorlink = Carlo Rovelli -->}}
** {{Cytuj | tytuł = Lectures on Loop Quantum Gravity | data = 2003 | autor = Thomas Thiemann | czasopismo = Lectures Notes in Physics | s = 41–135 | doi = 10.1007/978-3-540-45230-0_3 | bibcode = 2003LNP...631...41T | arxiv = gr-qc/0210094 <!-- nieprzetłumaczone parametry: | volume = 631 -->}}
**{{Cytuj | tytuł = Background Independent Quantum Gravity: A Status Report | data = 2004 | autor = Abhay Ashtekar; Jerzy Lewandowski | czasopismo = Classical and Quantum Gravity | s = R53–R152 | wydanie = 15 (21) | doi = 10.1088/0264-9381/21/15/R01 | bibcode = 2004CQGra..21R..53A | arxiv = gr-qc/0404018 <!-- nieprzetłumaczone parametry: | authorlink = Abhay Ashtekar -->}}
** [[Carlo Rovelli]] and Marcus Gaul, ''Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance'', e-print available as [http://arxiv.org/abs/gr-qc/9910079 gr-qc/9910079].
** [[Lee Smolin]], ''The case for background independence'', e-print available as [http://arxiv.org/abs/hep-th/0507235 hep-th/0507235].
** [[Alejandro Corichi]], ''Loop Quantum Geometry: A primer'', e-print available as [http://arxiv.org/abs/gr-qc/0507038v2].
** Alejandro Perez, ''Introduction to loop quantum gravity and spin foams'', e-print available as [http://arxiv.org/abs/gr-qc/0409061v3].
** Hermann Nicolai and Kasper Peeters ''Loop and spin foam quantum gravity: A Brief guide for beginners.'', e-print available as [http://arxiv.org/abs/hep-th/0601129v2].
* 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]]''
** {{Cytuj | tytuł = The Complete Idiot's Guide to String Theory | wydawca = Alpha | data = 2008 | autor = George Musser | czasopismo = The Physics Teacher | isbn = 978-1-59257-702-6 | wydanie = 2 (47) | doi = 10.1119/1.3072469 | bibcode = 2009PhTea..47Q.128H <!-- nieprzetłumaczone parametry: | location = Indianapolis | page = 368 | authorlink = George Musser -->}} – 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 [http://arxiv.org/abs/gr-qc/0410054 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 [http://arxiv.org/abs/hep-th/0310077 hep-th/0310077] (2003)
* More advanced introductory/expository works:
** [[Carlo Rovelli]], ''Quantum Gravity'', Cambridge University Press (2004); [http://www.cpt.univ-mrs.fr/~rovelli/book.pdf draft available online]
** Thomas Thiemann, ''Introduction to modern canonical quantum general relativity'', e-print available as [http://arxiv.org/abs/gr-qc/0110034 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 [http://arxiv.org/abs/hep-th/0501114 hep-th/0501114]
** H. Nicolai and K. Peeters, ''Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners'', e-print available as [http://lanl.arxiv.org/abs/hep-th/0601129 hep-th/0601129]
** T. Thiemann [http://arxiv.org/abs/hep-th/0401172v1 The LQG – String: Loop Quantum Gravity Quantization of String Theory] (2004)

* Conference proceedings:
** [[John C. Baez]] (ed.), ''Knots and Quantum Gravity''
* Fundamental research papers:
**{{Cytuj | tytuł = New variables for classical and quantum gravity | data = 1986 | autor = Abhay Ashtekar | czasopismo = Physical Reviews Letters | s = 2244–2247 | wydanie = 18 (57) | doi = 10.1103/PhysRevLett.57.2244 | pmid = 10033673 | bibcode = 1986PhRvL..57.2244A <!-- nieprzetłumaczone parametry: | authorlink = Abhay Ashtekar -->}}
** {{Cytuj | tytuł = New Hamiltonian formulation of general relativity | data = 1987 | autor = Abhay Ashtekar | czasopismo = Physical Review D | s = 1587–1602 | wydanie = 6 (36) | doi = 10.1103/PhysRevD.36.1587 | bibcode = 1987PhRvD..36.1587A <!-- nieprzetłumaczone parametry: | authorlink = Abhay Ashtekar -->}}
** [[Roger Penrose]], ''Angular momentum: an approach to combinatorial space-time'' in ''Quantum Theory and Beyond'', ed. Ted Bastin, Cambridge University Press, 1971
** {{Cytuj | tytuł = Knot theory and quantum gravity | data = 1988 | autor = Carlo Rovelli; Lee Smolin | czasopismo = Physical Review Letters | wydanie = 10 (61) | doi = 10.1103/PhysRevLett.61.1155 | bibcode = 1988PhRvL..61.1155R <!-- nieprzetłumaczone parametry: | postscript = . | page = 1155 | authorlink2 = Lee Smolin | authorlink = Carlo Rovelli -->}}
** {{Cytuj | tytuł = Loop space representation of quantum general relativity | data = 1990 | autor = Carlo Rovelli; Lee Smolin | czasopismo = Nuclear Physics | s = 80–152 <!-- nieprzetłumaczone parametry: | postscript = . | volume = B331 | authorlink2 = Lee Smolin | authorlink = Carlo Rovelli -->}}
** [[Carlo Rovelli]] and [[Lee Smolin]], ''Discreteness of area and volume in quantum gravity'', Nucl. Phys., '''B442''' (1995) 593-622, e-print available as [http://xxx.lanl.gov/abs/gr-qc/9411005 gr-qc/9411005]
**{{Cytuj | tytuł = Relativity, Astrophysics and Cosmology | wydawca = D. Reidel | data = 1973 | autor = Karel Kuchař | s = 237–288 | isbn = 90-277-0369-8 <!-- nieprzetłumaczone parametry: | editor-first = Werner | editor-last = Israel | contribution = Canonical Quantization of Gravity -->}}
**{{Cytuj | tytuł = Loop Quantum Gravity: An Inside View | data = 2006 | autor = Thomas Thiemann | czasopismo = Approaches to Fundamental Physics | bibcode = 2007LNP...721..185T | arxiv = hep-th/0608210 <!-- nieprzetłumaczone parametry: | page = 185 | volume = 721 -->}}

== External links ==
*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/rovelli03.pdf "Loop Quantum Gravity" by Carlo Rovelli] Physics World, November 2003
*[http://universe-review.ca/R01-07-quantumfoam.htm  Quantum Foam and Loop Quantum Gravity]
*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html Abhay Ashtekar: Semi-Popular Articles . Some excellent popular articles suitable for beginners about space, time, GR, and LQG.]
*[http://www.edge.org/3rd_culture/smolin03/smolin03_index.html Loop Quantum Gravity: Lee Smolin.]
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=205615 Loop Quantum Gravity on arxiv.org]
*[http://sps.nus.edu.sg/~wongjian/lqg.html A list of LQG references catered to fresh graduates]
*[http://www.perimeterinstitute.ca/Events/Introduction_to_Quantum_Gravity/Introduction_to_Quantum_Gravity/ Loop Quantum Gravity Lectures Online] by Lee Smolin
*[http://jdc.math.uwo.ca/spin-foams/ Spin networks, spin foams and loop quantum gravity]
*Wired magazine, News: ''[http://www.wired.com/news/technology/0,71828-0.html Moving Beyond String Theory]''
*[http://www.sciam.com/special/toc.cfm?issueid=40&sc=rt_nav_list  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 ''[http://www.economist.com/science/displaystory.cfm?story_id=7963608 Looping the loop]''
* Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
*[http://uk.arxiv.org/abs/physics/0505042 Zeno meets modern science.] Article from [http://th-www.if.uj.edu.pl/acta/ Acta Physica Polonica B] by Z.K. Silagadze.
*[http://space.newscientist.com/article/mg19826514.300-did-prebig-bang-universe-leave-its-mark-on-the-sky.html?feedId=online-news_rss20 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)

{{Theories of gravitation}}
{{quantum gravity}}

{{DEFAULTSORT:Loop Quantum Gravity}}
[[:Kategoria:Theories of gravitation]]
[[:Kategoria:Loop quantum gravity]]

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== Przypisy ==
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