Field theory in SO(2,3)* model of noncommutative gravity

Abstract

Arguably the greatest challenge of contemporary theoretical physics is to understand the profound interplay between Quantum Mechanics (QM) and the General Theory of Relativity (GR). To solve the conundrum of Quantum Gravity" (QG) one has to transcend some deeply rooted assumptions on which we are accustomed, in particular, at very short length scales we might have to abandon the notion of a continuous space-time and the associated mathematical construct of a smooth manifold that describes it. Field theory on noncommutative (NC) space-time is one distinguished approach to QG, and the one that will be advocated in this thesis. NC field theory is based on the method of quantization by deformation, originally developed for the purpose of establishing phase-space quantum mechanics. One speaks of a deformation of an object/structure whenever there is a family of similar objects/structures of which the distortion" from the original, undeformed one can be somehow parametrized. In physics, this so-called deformation parameter is usually related to some fundamental constant of nature that measures the deviation from the classical (i.e. undeformed) theory. To deform classical space-time, one introduces an abstract algebra of NC coordinates, denoted by ^ xµ, that satisfy some non-trivial commutation relations. The simplest case of noncommutativity is the so-called canonical (or θ-constant) noncommutativity, [^ xµ; x^ν] = iθµν ∼ Λ2 NC, where θµν are components of a constant antisymmetric matrix, and ΛNC is a hypothetical length scale at which NC effects become relevant. Instead of deforming abstract algebra of coordinates, one can introduce space-time noncommutativity in the form of NC products of functions (fields) on commutative space-time. These products are called star products (?-products). In particular, canonical noncommutativity is effected by the Moyal ?-product. During the previous studies of the theory of NC gravity, it was found that NC corrections to GR can be obtained by canonical deformation of anti-de Sitter (AdS) gauge field theory. Starting with an action of the MacDowellMansouri type, invariant under SO(2; 3) gauge transformations, one obtains the Einstein-Hilbert action with cosmological constant term, after choosing a certain gauge. NC deformation is based on the Seiberg-Witten approach to NC gauge field theory, and the first non-vanishing NC correction is quadratic in θµν. This model also predicts a non-trivial NC deformation of Minkowski space and offers an explanation for the apparent breaking of diffeomorphism invariance in the NC theory. Namely, the structure of the NC-deformed Minkowski metric suggests that, by assuming canonical noncommutativity, we implicitly choose a preferred frame of reference - the Fermi inertial frame...

Jedan od najve´cih izazova savremene teorijske fizike jeste usaglaˇsavanje Opˇste teorije relativnosti (OTR) i Kvantne mehanike. Da bismo razreˇsili problem kvantne gravitacije" neophodno je da prevazid¯emo neke duboko ukorenjene pretpostavke na kojima se zasnivaju sve naˇse dosadaˇsnje teorije. Jedna od njih je i pretpostavka da je struktura prostor-vremena kontinualna na svim skalama i da shodno tome odgovara matematiˇckom konceptu glatke mnogostrukosti. Teorija polja na nekomutativnom (NK) prostor-vremenu je jedan dobro definisani pristup problemu kvantne gravitacije, i taj pristup ´ce biti zastupljen u ovoj disertaciji. NK teorija polja poˇciva na metodu deformacione kvantizacije, originalno razvijenom radi zasnivanja kvantne mehanike u faznom prostoru. O deformaciji nekog objekta/strukture govorimo onda kada postoji familija srodnih objekata/struktura kod koje se odstupanje od nedeformisanog originala moˇze na odred¯eni naˇcin paramatrizovati. U fizici se ovaj takozvani parametar deformacije javlja u vidu neke fundamentalne konstante prirode i predstavlja meru odstupanje od klasiˇcne" (tj. nedeformisane) teorije. Da bismo deformisali klasiˇcno prostor-vreme, uvodimo apstraktnu algebru nekomutativnih koordinata, u oznaci ^ xµ, koje zadovoljavaju neke netrivijalne komutacione relacije. Najjednostavniji primer je takozvana kanonska (ili θ-konstantna) nekomutativnost, [^ xµ; x^ν] = iθµν ∼ Λ2 NC, gde su θµν komponente konstantne antisimetriˇcne matrice, a ΛNC hipotetiˇcka skala duˇzine na kojoj efekti nekomutativnosti postaju znaˇcajni. Umesto deformisanja apstraktne algebre koordinata, nekomutativnost moˇzemo uvesti u vidu nekomutativnih proizvoda funkcija (polja) obiˇcnih komutativnih koordinata. Ovi proizvodi se nazivaju star-proizvodi (?-proizvodi). Konkretno, kanonskoj nekomutativnosti odgovara Mojalov ?-proizvod. Tokom prethodnih istraˇzivanja teorije NK gravitacije, ustanovljeno je da se nekomutativna verzija OTR moˇze dobiti kanonskom deformacijom antide Siter (AdS) gradijentne teorije gravitacije. Predloˇzeno klasiˇcno dejstvo Jang-Milsovog tipa, invarijantno na lokalne SO(2; 3) transformacije, se pri odred¯enom kalibracionom uslovu svodi na standardno Ajnˇstajn-Hilbertovo dejstvo sa kosmoloˇskom konstantom. NK deformacija je sprovodena slede´ci Sajberg-Vitenov pristup NK teoriji gradijentnih polja, i ispostavlja se da je prva nenulta NK korekcija kvadratna po θµν. Ovaj model takod¯e predvid¯a netrivijalnu deformaciju prostora Minkovskog i pruˇza objaˇsnjenje porekla naruˇsenja opˇste kovarijantnosti koje je prisutno u NK teoriji. Naime, struktura NK-deformisane metrike Minkovskog ukazuje na to da, uvode´ci kanonsku nekomutativnost, mi implicitno prelazimo u odred¯eni referentni sistem - onaj koji odgovara Fermijevim inercijalnim koordinatama duˇz geodezika...