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Quantum Field Theory Applied in Compton Scattering

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Quantum Field Theory Applied in Compton Scattering Quantum Field Theory Applied In Compton Scattering Aristotle University of Thessaloniki Faculty of Physics Institute of Theoretical Physics Gkiatas Dimitrios Supervisor : Nikolaos D. Vlachos March 2016 Contents Abstract in English 4 Abstract in Greek 5 1 Classical Field Theory 7 1.1 Relativistic Notation . .8 1.2 Mathematical Background . 10 1.3 Lagrangian Formalism . 12 1.4 The Hamilton Formalism . 14 1.5 Symmetries and Conservation Laws . 16 2 The Klein-Gordon Field 21 2.1 Canonical Quantization . 21 2.2 The Real Klein-Gordon eld . 22 2.3 The Bosonic Fock Space . 29 2.4 The Complex Klein-Gordon Field . 31 2.5 Microcausality . 35 2.6 The Scalar Feynman Propagator . 37 3 The Dirac Field 40 3.1 The Dirac Equation . 40 3.2 Covariance of the Dirac Equation . 41 3.3 Classical Dirac Field Theory . 43 3.4 Fermionic Fock Space . 44 3.5 Canonical Quantization of the Dirac Field . 45 3.6 Plane Wave Expansion of the Field Operator . 46 3.7 The Dirac Field Feynman Propagator . 50 4 The Electromagnetic Field 53 4.1 The Maxwell Equations . 53 4.2 Plane Wave Expansion of the Electromagnetic Field . 56 4.3 Lagrangian Density and Conserved Quantities . 57 4.4 Quantization of the Electromagnetic Field . 58 4.5 Canonical Quantization of the Lorenz Gauge . 59 4.6 Feynman Propagator for Photons . 63 5 Interacting Fields 65 5.1 The Dirac Picture . 65 5.2 The Time-Evolution Operator . 67 2 CONTENTS 3 5.3 The S-Matrix . 69 5.4 Wick's Theorem . 70 5.5 Coupling Photons to Fermions (QED) . 73 5.6 Feynman Rules in QED . 73 5.7 Cross Section . 76 5.8 Spin Sums . 78 5.9 Trace Technique . 79 6 Compton Scattering 80 6.1 Feynman Diagrams . 80 6.2 Compton Scattering Dierential Cross Section . 82 6.3 Energy Shift Relation . 84 6.4 Klein-Nishina Formula . 85 Bibliography 90 Abstract in English This thesis, provides an extensive introduction to Quantum Field Theory with an application to Compton Scattering, studying for 3 chapters the canonical quantization of almost every free eld, i.e Klein-Gordon-Real and Complex-eld, Dirac eld and Elec- tromagnetic eld. Prior to the eld quantization throughout the 1st Chapter a short in- troduction of Classical eld theory from the aspect of Lagrangian formalism takes place. During the 5th Chapter we discuss interacting elds and more specically the coupling of fermions to photons (QED). In addition, we deduce the expressions of quantities such as the probability rate and scattering cross section which we will use in our application. Concluding this chapter, we state the Feynman translation rules and Feynman rules in momentum space and correlate them with Feynman diagrams and the probability ampli- tude respectively. In the 6th Chapter, we use the tools proven before to draw the Feynman diagrams and nd the Klein-Nishina formula-unpolarized scattering cross section- for the process of Compton Scattering 0 γ(k) + e(pi) ! γ(k ) + e(pf ) 4 Abstract in Greek H παρούsa πτυχιακή ergasÐa asqoleÐtai me tÐtlo ¨Κβαντική Jewria PedÐou me Efarmoγή sthn Skèdash Compton’ apoteleÐ mÐa eiσαγωγή sthn Κβαντική JewrÐa PedÐou melet¸ntac sta pr¸ta 4 κεφάλαια ta βασικά ελεύθερα pedÐa, en¸ sto 5o meletάμε thn allhlepÐdrash aut¸n katαλήγοntac sto 6o κεφάλαιo ston upologiσμό thc ενεργού diatομής gia thn skè- dash Compton. Αρχικά sto 1o kefάlaio gÐnetai mÐa eiσαγωγή sthn κλασσική jewrÐa pedÐou aπό thn πλευρά tou Λαγρανζιανού Formaliσμού, mèsw thc opoÐac εξάγοntai oi pediakèc exis¸seic oi opoÐec kai apoτελούν genÐkeush twn klassik¸n exis¸sewn Euler-Lagrange συναρτή- sei thc Λαγκρανζιανής πυκνόthtac. Sthn sunèqeia meletάμε ton Qamiltoνιανό formal- iσμό upologÐzontac tic αγκύλες Poisson metαξύ suzug¸n megej¸n kai τελικά εξάγουμε thn γενικόterh moρφή thc exÐswshc sunèqeiac από thn opoÐa kai προκύπτoun oi διατηρήσιμες ποσόthtec (je¸rhma Noether), anafèrontac κάποιες basikèc summetrÐec pou prèpei na υπακούει κάθε sqetikisτική jewrÐa (αναλοιωσιμόthta κάτw από qwrikèc metajèseic kai Lorentz covariance). To 2o κεφάλαιo asqoleÐtai tόσο me to πραγματικό όσο kai me to miγαδικό pedÐo Klein- Gordon. Pio sugkekrimèna, brÐskoume όlec tic pediakèc ποσόthtec pou anafèrame kai upologÐsame sto 1o κεφάλαιo (suzugèc pedÐo, Qamiltoνιανή πυκνόthta k.l.p). Katόπιn, kbant¸noume ta pedÐa antikajist¸ntac tic suzugeÐc pediakèc ποσόthtec me telestèc kai tic αγκύλες Poisson me metajetikèc sqèseic kai upologÐzoume touc antÐstoiqouc telestèc. Proqwr¸ntac, αναπτύσσουμε touc peδιακούς telestèc γύρω από klassikèc βάσειc epipèdwn κυμάτwn eiσάγοntac touc telestèc dhmiourgÐac kai katαστροφής αφού pr¸ta πούμε δύο λόgia gia ton q¸ro Fock ton opoÐo orÐzoun (αναπαράστash ariθμού swmatidÐwn). Tèloc brÐskoume κάποιες από tic διατηρήsimec ποσόthtec (Dianusma ορμής, Qamiltoνιανή kai FortÐo) kai upologÐzoume ton διαδόth Feynman gia to miγαδικό bajmwtό pedÐo αφού pr¸ta epibebai¸soume pwc h jewrÐa mac υπακούει sthn mikroaiτιόthta. Sto 3o κεφάλαιo meletάμε to pedÐo Dirac. Pr¸ta από όla κάνουμε mÐa eiσαγωγή sthn exÐswsh Dirac kai ta Dirac spinors pou apoτελούν tic kumatοσυναρτήσειc thc jew- rÐac από thn σκοπιά thc sqetikisτικής kbantομηχανικής. Epiplèon, apoδεικνύουμε thn αναλοιωσιμόthta thc jewrÐac κάτw από metasqhmatiσμούς Lorentz (Lorentz covarience). Metά suzhtάμε thn κλασσική jewrÐa pedÐou kaj¸c kai tic διατηρήσιμες ποσόthtec pou προκύπτoun από thn jewrÐa mac kai orÐzoume ton fermioνικό plèon q¸ro Fock ston opoÐo kai εργαζόμασte katά thn κβάντwsh tou κλασσικού pedÐou Dirac h opoÐa epiτυγχάνετai mèsw thc epiboλής katάλληλων antimetajetik¸n sqèsewn metαξύ twn suzug¸n pediak¸n telest¸n. Wstόσο, se αυτή thn perÐptwsh epilègoume na κάνουμε ανάπτυγμα twn pe- diak¸n telest¸n se epÐpeda κύματa pou eÐnai λύσειc thc ελεύθερης exÐswshc Dirac kai παρατηρούμε thn εμφάνιsh δύο eid¸n telest¸n dhmiourgÐac kai katastrοφής pou ermh- νεύουμε pwc antapokrÐnontai se fermioνικά σωμάτια kai antiσωμάτια. KleÐnontac, brÐsk- oume ton fermioνικό διαδόth Feynman. 5 6 CONTENTS Sto 4o κεφάλαιo, αναπτύσσουμε thn κβάντwsh tou ηλεκτρομαγνητικού pedÐou. Xek- in¸ntac από tic exis¸seic Maxwell apousÐa hlektrik¸n kai magnhtik¸n phg¸n tic opoÐec kai γράφουμε se antalloÐwth tανυστική μορφή συναρτήσει tou διανυσματικού kai bajmwtού δυναμικού, επικεντρωνόμασte sthn melèth thc summetrÐac bajmÐdac, pou paÐzei kajorisτικό ρόlo sthn κβάντwsh, anafèrontac tόσο thn bajmÐda Coulomb όσο kai thn bajmÐda Lorenz. Παρακάτw κάνουμε anάπτυγμα tou 4-διάστatou plèon διανυσματικού δυναμικού se epÐpeda κύματa orÐzontac tic tèsseric αρχικά katastάσειc πόlwshc pou apoτελούν touc εκάσ- tote βαθμούς eleujerÐac tou pedÐou. Tèloc kbant¸noume to ηλεκτρομαγνητικό pedÐo eiσά- gontac tic antalloÐwtec mpozonikèc metajetikèc sqèseic mèsw twn opoÐwn upologÐzoume ton τελεστή Qamilton, όpou εργαζόμενοι sthn bajmÐda Lorenz mèsw tou Gupta-Bleuler pe- rioriσμού exasfalÐzoume thn apousÐa αρνητικού mètrou ston q¸ro Hilbert epibebai¸nontac touc 2 εγκάρσιouc bajmoύς eleujerÐac tou fwtonÐou. H συζήτηση gia ta ελεύθερα pedÐa kleÐnei me thn exagwγή kai ermhneÐa twn διαφόρων όρwn tou fwtoνικού διαδόth (εγκάρσιoc, Coulomb kai upoλειπόμενος). To pèmpto κεφάλαιo anafèretai sta allhlepidr¸nta kbanτικά pedÐa, me stόqo thn εύρεση eνός susτηματικού τρόπου efarmoγής thc jewrÐac diataraq¸n, me σκοπό thn ex- αγωγή παρατηρήσιμων posoτήτwn kai pio sugkekrimèna tou pÐnaka skèdashc (scattering matrix). Από thn εικόna Heisenberg pou qrhsimopoiούσαμε mèqri t¸ra, metabaÐnoume sthn εικόna thc allhlepÐdrashc (Dirac). Sth sunèqeia, apoδεικνύουμε thn sqèsh tou Dyson kai suzhtάμε to je¸rhma tou Wick. 'vEpeita asqoloύμασte me thn Κβαντική Hlektro- δυναμική εξάγοntac thn Lagkranzianή πυκνόthta kai meletάμε thn diadikasÐa thc skèdashc. Katόπιn, brÐskoume thn èkfrash thc ενεργούς διαφορικής diatομής, kai parajètoume touc κανόnec metatόπιshc kai Feynman ston q¸ro ορμής, susqetÐzontάς touc me ta diagράmmata Feynman kai piθανόthta metάπτwshc antÐstoiqa. Sto teleutaÐo κεφάλαio, εφαρμόζουμε tic teqnikèc pou anaπτύξαμε ston σχεδιασμό twn διαγραμμάτwn Feynmam kai ston upologiσμό thc διαφορικής ενεργούc diatομής gia th skèdash Compton, h opoÐa περιλαμβάνει th skèdash fwtonÐou από hleκτρόnio 0 γ(k) + e(pi) ! γ(k ) + e(pf ) Chapter 1 Classical Field Theory In this chapter we will take the elds at each point as the dynamical variables and quantize them directly. This approach generalizes the classical mechanics of a system of particles and its quantization, to a continuous system, i.e the elds. A eld is a quantity dened at every point in space and time (~x;t). While classical mechanics deals with a nite number of generalized coordinates qr(t); r = 1; 2; ::::; n (1.1) in eld theory we are interested in the dynamics of elds denoted by the generic name 1 φr(~x;t) , r = 1; 2; :::; N (1.2) where both r and ~x are considered as labels. Therefore, each point of a (nite or innite) region in space will be associated with some continuous eld variable as I have stated above. This obviously constitutes a system with an innite number of degrees of freedom. Consequently, our dynamic variables are the values of the eld φ(x) at every point in space, instead of the nite coordinates qr(t). Next, we will introduce the Lagrangian (more specically, the Lagrangian density (L)) for which the eld equations follow by means of Hamilton's Principle. In addi- tion, we will also introduce the momenta conjugate to the elds and impose canonical commutation relations directly on the elds and the conjugate momenta. This process, provides a systematic quantization procedure for any classical eld theory derivable from a Lagrangian. Finally, we will derive the general form of Noether's Theorem, thus, calculating the current density and nding conserved
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