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Cordes Paraquantiques Dans Un Espace Temps Noncommutatif

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Cordes Paraquantiques Dans Un Espace Temps Noncommutatif Republique´ Algerienne´ Democratique´ et Populaire Ministere` de L'Enseignement Superieur´ et de la Recherche Scientifique UNIVERSITE´ DES FRERES` MENTOURI CONSTANTINE Faculte´ des Sciences exactes Departement´ de Physique No d'ordre : S´erie: These` Present´ ee´ pour Obtenir le Doctorat LMD Specialit´ e´ : Physique Theorique´ Cordes Paraquantiques dans un Espace Temps Noncommutatif Par : Mohamed Ali Seridi Devant le jury : Pr´esident : L. Chetouani Prof. Universit´eFr`eresMentouri Rapporteur : N. Belaloui Prof. Universit´eFr`eresMentouri Examinateurs : A. Bouda Prof. Universit´ede B´ejaia T. Boudjedaa Prof. Universit´ede Jijel K. Ait Moussa Prof. Universit´eFr`eresMentouri A. Benslama Prof. Universit´eFr`eresMentouri soutenu le : 20/01/2016 i Remerciements J e tiens tout d'abord `aexprimer mes vifs remerciements `amon directeur de th`ese le Professeur Nadir Belaloui, qui m'a soutenu attentivement depuis le Master et du- rant toutes ces cinq derni`eres ann´eesde Doctorat, pour sa rigueur scientifiques qui m'a guid´ee,ses conseils, ses encouragements, son aide et son support qui m'a permis de participer `abien des manifestations scientifiques r´ecentes de haut niveau. Je n'oublie- rai jamais sa personnalit´ecalme et gentille et le fait qu'il n'a ´economis´eaucun effort `a donner pour mener `aterme ce travail de recherche. J 'exprime tous mes remerciements `al'ensemble des membres du jury, sp´ecialement `amon professeur Lyazid Chetouani de m'avoir honor´e`apr´esiderce jury. Egalement, tous mes respects vont aux examinateurs h^otesqui ont accept´ede participer et donner de l'int´er^etpour discuter cette th`ese: Mr. Ahmed Bouda, professeur `al'universit´ede B´ejaia. Mr. Tahar Boudjedaa, professeur `al'universit´ede Jijel Un grand merci ´egalement`a mes enseignants et pour avoir honor´epar leur pr´esence la soutenance de cette th`ese: Mr. Karim Ait Moussa, professeur `al'universit´edes fr`eres Mentouri pour son aide consid´erable que j'appr´eciefortement. Mr. Achour Benslama, professeur `al'universit´e des fr`eres Mentouri pour les discussions fructueuses `ades moments critiques lors de la r´ealisationde ce travail. A ceux qui ont cru en moi, et qui n'ont jamais h´esit´e`ame donner tout le support dont j'avais besoin, mes parents, qui m'ont donn´ela vie, l'´education. Ma petite famille : je ne trouve pas comment remercier ma grande sœur Fairouz pour sa g´en´erosit´e,qui m'a soutenu jusqu'au bout et surtout support´edans tout ce que j'ai entrepris, ´egalementMouna qui a tout le m´eritede m'avoir orient´evers la physique et m'avoir aid´e`atrouver le trajet `asuivre. A mon fr`ere et ami Salah, `aqui j'exprime tous mes remerciements pour son int´er^etet son soutient moral qu'il m'a t´emoign´e`atout moment. Le plus grand remerciement `ama grande m`ere Laarfa pour ses pri`eres `aDieu `atoujours trouver des solutions `ames soucis. A nos anc^etres qui n'ont jamais ´et´eabsents de nos bons souvenirs. Un Sp´ecialRemerciement A tous ceux qui m'ont soutenu jusqu'au bout. Table des mati`eres Table des mati`eres ii Liste des tableaux iv 1 Introduction1 1.1 Th´eoriedes Cordes Bosoniques........................2 1.1.1 Particule Ponctuelle..........................2 1.1.2 Corde Relativiste............................4 1.1.3 Expansion en Modes d'Oscillateurs..................6 1.2 Quantification de la Corde Bosonique....................7 1.2.1 Jauge Covariante............................8 1.2.2 Jauge Transverse............................9 1.3 Paraquantification............................... 10 1.3.1 Oscillateurs du Type Bosonique.................... 11 1.3.2 Repr´esentation de Green........................ 14 1.4 Propos de la Th`ese.............................. 15 2 Cordes Parabosoniques Ouvertes et Param`etresde la Noncommutati- vit´e 18 2.1 Le Mod`ele.................................... 18 2.1.1 Alg`ebrede Virasoro.......................... 21 2.1.2 Spectre de Masse............................ 22 3 Cordes Parabosoniques Ouvertes entre Deux Dp-Branes Parall`elesdans un Espace Temps Noncommutatif 25 3.1 Alg`ebrede Virasoro.............................. 26 3.2 Spectre de Masse................................ 27 3.2.1 Red´efinitionde l'Espace de Fock................... 27 3.2.2 Op´erateurde Masse.......................... 28 4 Cordes Parabosoniques Ouvertes entre Deux Dp-Dq Branes Parall`eles dans un Espace Temps Noncommutatif 30 4.1 Relations Trilin´erairesModifi´eeset Configurations D-Branes........ 30 4.2 Alg`ebrede Virasoro.............................. 31 4.3 Spectre de Masse................................ 32 ii Table des mati`eres iii 4.3.1 Red´efinitionde l'Espace de Fock................... 32 4.3.2 Op´erateurde Masse.......................... 32 4.3.3 Discussion................................ 33 4.3.3.1 Mod`elesans Tachyon.................... 33 5 Cordes Parabosoniques Ferm´eesdans un Espace Temps Noncommu- tatif et R´eductiondu Spectre 36 5.1 Alg`ebrede Virasoro.............................. 37 5.2 Red´efinitionde l'Espace de Fock....................... 38 5.3 Spectre de Masse................................ 38 5.4 Op´erateurde Masse.............................. 39 5.4.1 Restauration du Graviton et R´eductiondu Spectre......... 40 6 R´esum´e 43 A Calcul de Quelques Relations Trilin´eaires 45 I J K A.1 Calcul de αm; αn; αl + ........................... 45 h I J K i A.2 Calcul de αm; x0 ; αl + ........................... 48 h I J K i A.3 Calcul de αm; x0 ; x0 + : .......................... 50 h I J K i A.4 Calcul de x0; αn; αl + : .......................... 51 h I J K i A.5 Calcul de x0; x0 ; αl + ........................... 53 h I J K i A.6 Calcul de x0; x0 ; x0 + ........................... 55 h i B Calcul de l'Alg`ebrede Virasoro 57 C Calcul de Quelques Masses 59 C.1 1er Niveau Excit´e............................... 59 C.2 2eme Niveau Excit´e............................... 60 C.3 3eme Niveau Excit´e............................... 61 Bibliographie 65 Liste des tableaux 2.1 Spectre de Masse : Cas Corde Ouverte..................... 24 3.1 Spectre de Masse : Cas Dp-Branes...................... 28 3.2 Spectre de Masse Modifi´e: Cas Dp-Branes.................. 29 4.1 Spectre de Masse : Cas Dp-Dq Branes..................... 33 4.2 Spectre de Masse Modifi´e: Cas Dp-Dq Branes................ 34 5.1 Spectre de Masse : Cas Corde Ferm´ee..................... 41 iv A` MES PARENTS . v Chapitre 1 Introduction En physique moderne, la g´en´eralisationdes concepts fondamentaux joue un r^oleim- portant dans la progression de la recherche et la d´ecouverte de nouvelles physiques. Diff´erentes ´etapes de l’unification ont conduits `aune meilleure maitrise des diff´erentes th´eoriesde la physique moderne. D`esle 19`emesi`ecle,les premi`erestentatives ont conduit `ala th´eoriede l'´electromagn´etisme´etabliepar Maxwell-Faraday. La constance de la c´el´erit´ede la lumi`eredans cette th´eories'av`ere^etrela pierre angulaire pour l'unification des notions d'espace et du temps par A. Einstein dans ses th´eoriesde la relativit´eres- treinte en 1905 et g´en´eraleen 1915. Dans le cadre de la th´eoriequantique des champs, la th´eoriede jauge nonab´elienne fond´eepar Yang-Mills en 1950 a pu d´ecrirel'ensemble des trois autres types d'interactions fondamentales. Elle est `ala base conceptuelle du mod`ele standard de la physique des particules, elle a r´eussi`aunifier l'´electrodynamique et la m´ecaniquequantique dans le contexte de QED et l'interaction faible dans la th´eoriedu mod`elestandard de Weinberg-Salam. Avant QCD, le mod`eledual ´etaitla seule th´eorie qui d´ecrivait l'interaction forte entre les hadrons. Il comportait cependant deux obstacles `asavoir : la th´eorien'est coh´erente du point de vue math´ematiqueque si la dimension de l'espace temps est sup´erieure`a4 en plus de la pr´esencedans le spectre d'une particule de spin 2 et de masse nulle qui n'avait pas sa place dans cette th´eorie.Une classification des hadrons par rapport aux spins et aux masses correspondantes des particules du spectre est possible dans la th´eoriede Regge qui regroupe les particules sur des trajectoires dans le plan `atravers la relation : 2 0 2 J = αM = α0 + α M : (1.1) o`u α0 : constante universelle 0:9(GeV ) 2 ' − α0 : param`etrequi d´epend de la famille de Regge choisie. 1 Cordes Paraquantiques dans un Espace Temps Noncommutatif 2 L'apparition de QCD a mis `amal le mod`eledual, ce qui a conduit `ason abondant jusqu'au moment o`uon d´ecouvreque son spectre peut ^etreretrouv´e`atravers l'´etude de la dynamique d'une corde et la possibilit´ed'associer la particule qui posait probl`eme au graviton en plus du fait qu'en s'inspirant de l'id´eede Kaluza Klein, le deuxi`eme obstacle s'av´ereraitplut^otun atout qui pourrait conduire `aune probable unification de la gravitation avec le mod`elestandard. Ce qui voulait dire que les cordes sont beaucoup plus ´el´ementaires (´echelle de Planck) que celles initialement construites pour relier deux quarks dans un hadron. C'est la naissance de ce qui a ´et´ebaptis´eecomme la nouvelle th´eoriedes cordes, d'abord, la th´eoriede la corde bosonique qui d´ecritaussi bien le photon que le graviton mais qui devait se propager dans un espace temps `a26 dimensions et que la pr´esence de tachyons dans le spectre en plus du fait qu'elle ne d´ecritque des particules bosoniques affaiblissait la th´eorie.Pour rem´edier`aceci, il a fallu introduire des degr´esde libert´esfermioniques. Un pas a ´et´efait en abaissant la dimension de l'espace temps de 26 `a10, mais ceci n'a pas directement r´esolule probl`eme,vu qu'il y a eu deux mod`elesdiff´erents, l'un ne d´ecrivant que des bosons ; c'est le mod`elede Neuveu- Schwarz, l'autre ne d´ecrivant que des fermions ; c'est le mod`elede Ramond.
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