EP523 Introduction to QFT I

Transcription

1 EP523 Introducton to QFT I Toc 0 INTRODUCTION TO COURSE Deartment of Engneerng Physcs Unversty of Gazante Setember 2011 Sayfa 1 Content Introducton Revew of SR, QM, RQM and EMT Lagrangan Feld Theory An Introducton to Grou Theory Dscrete Symmetres and Quantum Numbers The Drac Equaton Scalar Felds Sn 1/2 Felds Sn 1 Felds Sayfa 2 1

2 Resources for the Course Course web age htt:// Course Book McGraw-Hll (2008) Davd McMahon Amazon: htt:// Addtonal Web Resources: quantumfeldtheory.nfo quantumfeldtheory.org Wkeda Sayfa 3 Lectures, Attendance, Exams Frdays three-hour lecture 09:00-12:00 Engneerng of Physcs S? Attendance You must attend all of the lectures Exams Wrtten Homeworks 70% Wrtten Fnal Exam 30% December 2011 Sayfa 4 2

3 Introducton Quantum Feld Theory (QFT) s a theoretcal framework that combnes: * Quantum Mechancs (QM) and * Secal Relatvty (SR). QM: s a theory descrbng the behavor of small systems, such as atoms and ndvdual electrons. SR: s the study of hgh energy hyscs, the moton of artcles and systems at veloctes near the seed of lght (but wthout gravty). Sayfa 5 QM There are three key deas we want to recall from QM hyscal observables are mathematcal oerators e.g. Hamltonan (energy) of a SHO the uncertanty relatons between the oston & momentum oerators and energy & tme the commutaton relatons. In artcular, Sayfa 6 3

4 SR Energy and mass relaton: E 2 mc f there s enough energy that s, enough energy roortonal to a gven artcle s mass then we can roduce the artcle. Due to conservaton laws, we actually need twce the artcle s mass, so that we can roduce a artcle and ts antartcle. So n hgh energy rocesses, Partcle number s not fxed. The tyes of artcles resent are not fxed. The last two facts are n drect conflct wth nonrelatvstc Quantum Mechancs! Sayfa 7 Schrödnger Equaton In Non-Relatvstc Quantum Mechancs (NRQM) we descrbe the dynamcs of a system wth the Schrödnger equaton, whch for a artcle movng n one dmenson wth a otental V = V(x) s We can extend ths formalsm to treat the case when several artcles are resent. However, the number and tyes of artcles are absolutely fxed. The Schrödnger equaton cannot n any shae or form handle changng artcle number or new tyes of artcles aearng and dsaearng as relatvty allows. Sayfa 8 4

5 Klen-Gordon Equaton Early attemts to merge QM and SR focused on generatng a relatvstc verson of the Schrödnger equaton known as Klen-Gordon Equaton (KGE): Schrödnger dscarded KGE, because t gave the wrong fne structure for the hydrogen atom. KGE aears to gve negatve robabltes, somethng that obvously contradcts the srt of quantum mechancs. KGE allows negatve energy states! Sayfa 9 Drac Equaton The next attemt at a relatvstc quantum mechancs was made by Drac. Hs famous equaton s Here α and β are actually matrces. Drac Equaton (DE) resolves some of the roblems of the KGE but also allows for negatve energy states. Sayfa 10 5

6 Why QFT? As we have mentoned, n QM we deal wth a sngle artcle such as an electron n a otental e.g. square well, harmonc oscllator, etc. the artcle retans ts ntegrty e.g. an electron remans an electron throughout the nteracton. there s no general way to treat nteractons between artcles e.g. a artcle and ts antartcle annhlatng one another to yeld neutral artcles such as: e there s no way to descrbe the decay of an elementary artcle e.g a muon decay μ e e γ e γ μ Sayfa 11 Q.F.T. rovdes a means whereby artcles can be annhlated, created, and transmgrated from one tye to another. s a relatvstc theory, and thus more all encomassng. s an extraolaton of non-relatvstc quantum mechancs (NRQM) to relatvstc quantum mechancs (RQM). Sayfa 12 6

7 QFT Aroach The roblem was to get a relatvstc wave equaton. Part of the roblem wth these relatvstc wave equatons s n ther nterretaton. In order to be truly comatble wth secal relatvty, we need to dscard the noton that φ and ψ n KGE and DE descrbe sngle artcle states. In ther lace, we roose the followng new deas: The wave functons φ and ψ are not wave functons at all, nstead they are felds. The felds are oerators that can create new artcles and destroy artcles. Sayfa 13 Posson Brackets Classcal artcle theores contan rarely used enttes call Posson brackets. Posson brackets are mathematcal manulatons of certan ars of roertes (dynamcal varables lke oston and momentum) Posson brackets for two obects A and B s reresented by { A, B} For oston x and ts conugate momentum x Followng rules are vald { x, x} { x, x} { y, y} { z, z} 1 x, } { x, } 0 { y z { y, x} { y, z } { z, x} { z, y} 0 0 x1 x2 x 3 x y z n general: and and and { x, } x y z Sayfa 14 7

8 Frst Quantzaton (Partcle Theores) Shortly after NRQM theory had been worked out, theorsts, led by Paul Drac, realzed that for each ar of quantum oerators that had non-zero (zero) commutators, the corresondng ar of classcal dynamcal varables also had non-zero (zero) Posson brackets. 1. Assume the quantum artcle Hamltonan has the same form as the classcal artcle Hamltonan. 2. Relace the classcal Posson brackets for conugate roertes wth commutator brackets: { x, } [ x, ] the classcal roertes (dynamcal varables) of oston and ts conugate 3-momentum become quantum non-commutng oerators. We wll see later Sayfa 15 Second Quantzaton (Partcle Theores) In QFT, snce we have romoted the felds to the status of oerators, they must satsfy commutaton relatons. 1. Assume the quantum feld Hamltonan densty has the same form as the classcal feld Hamltonan densty. 2. Relace the classcal Posson brackets for conugate roerty denstes wth commutator brackets: { x, } [ x, ] [ ( x, t), ( y, t)] ( x y) * (another feld) s the conugate momentum densty of the feld φ and lays the role of momentum n QFT. * x and y are two onts n sace. f two felds are satally searated they cannot affect one another We wll see later ( y, t) Sayfa 16 8

9 In QM, oston x s an oerator whle tme t s ust a arameter. In SR, oston and tme are on a smlar footng. In QFT, > Felds ϕ and ψ are oerators. > They are arameterzed by sacetme onts (x, t). > Poston x and tme t are ust numbers that fx a ont n sacetme they are not oerators. > Momentum contnues to lay a role as an oerator. We wll see later Sayfa 17 Comarson of QM, RQM and QFT In NRQM & RQM solutons are states (artcles such as an electron) In QFT solutons are oerators that roduce and destroy states and QFT accommodates mult-artcle states. QFT can handle transmutaton of artcles from one knd nto another (e.g., muons nto electrons, by destroyng the orgnal muon and creatng the fnal electron), whereas NRQM and RQM can not. The roblem of negatve energy state solutons n RQM does not aear n QFT. And n both RQM and QFT (as well as NRQM), oerators act on states n smlar fashon. Exected energy measurement s determented same way n both theores: E H * H dv Sayfa 18 9

10 Sayfa 19 Lagrangan Feld Theory In QFT, we frequently use tools from classcal mechancs to deal wth felds. Secfcally, we often use the Lagrangan: L T The Lagrangan s mortant because symmetres (such as rotatons) leave the form of the Lagrangan nvarant. V The classcal ath taken by a artcle s the one whch mnmzes the acton: S Ldt We wll see how these methods are aled to felds n later chaters. Sayfa 20 10

11 Man Am of QFT We want to understand Nature. To do so, we need to be able to redct the outcomes of artcle accelerator scatterng exerments and elementary artcle half lves. To do ths, we need to be able to calculate robabltes for scatterng, and decay, to occur. To do that, we need to be able to calculate transton amltudes for secfc elementary artcle nteractons. And for that, we need frst to master a far amount of theory, based on the ostulates of quantzaton. Sayfa 21 References [1]. Davd McMahon, Quantum Feld Theory Demstfed, (2008), The McGraw-Hll Comanes. [2]. Sayfa 22 11