The Dirac Equation. Elementary Particle Physics Strong Interaction Fenomenology. Diego Bettoni Academic year

Transcription

1 The Drac Equaton Eleentary artcle hyscs Strong Interacton Fenoenology Dego Betton Acadec year -

2 D Betton Fenoenologa Interazon Fort elatvstc equaton to descrbe the electron (ncludng ts sn) Conservaton of robablty lnearty n the te dervatves elatvstc nvarance lnearty n the sace dervatves x x x t 3 3 t x x x t 3 3 j j j

3 Massless Ferons Ter wth s absent j j j aul atrces t two-coonent snor D Betton Fenoenologa Interazon Fort 3

4 D Betton Fenoenologa Interazon Fort 4 Ferons wth non-zero Mass The sallest atrces for whch there s a soluton are 44 atrces

5 D Betton Fenoenologa Interazon Fort The Matrces ; g t Snce we have 4x4 atrces wth 6 eleents, there are 6 ndeendent ones We can wrte the results n ters of any set that satsfes the condtons n slde A convenent set s gven by the so-called atrces

6 D Betton Fenoenologa Interazon Fort 6 nuber 3 scalar n sn sace, transfors as a four-vector trasfors as a scalar trasfors as a seudoscalar (odd under arty) trasfors as an axal four-vector (even under arty) s of the for

7 D Betton Fenoenologa Interazon Fort 7 roertes of the Matrces g

8 D Betton Fenoenologa Interazon Fort 8 Currents x t x t Drac Equaton Hertan Conjugate j Conserved Current For exale, for electrcally charged ferons: e j electrc

9 Consder as an exale a free artcle of four-oentu, descrbed by a lane wave The Drac equaton becoes: Hertan Conjugate We obtan: Interretng as a robablty densty, s what one would exect for a free-artcle current D Betton Fenoenologa Interazon Fort 9

10 D Betton Fenoenologa Interazon Fort Free artcle Solutons We assue where e are two-coonent snors

11 (a) These equatons can be wrtten as: (b) Solutons exst for ostve or negatve and can be nterchanged by - (c) If = the two equatons searate Snce clearly easures the coonent of sn along the drecton of oton, s the large soluton for and ostvo or and negatve, wth havng the ooste corresondence helcty For assless or relatvstc artcles ˆ reresents a left-handed ostve energy soluton, a rght-handed one For a assless or relatvstc artcle, for a left-handed state wth > << (d) If the two equatons do not searate In artcular we wll see that a ass ter n a agrangan can be nterreted as an nteracton between and D Betton Fenoenologa Interazon Fort

12 It s conventonal to searate the sace-te deendence x ue o t u satsfes the sae oentu sace equatons we have been wrtng, snce we were lctly assung that we were worng wth energy egenstates Noralzaton: uu Antartcles: the negatve energy solutons of the free-artcle equaton corresond to ostve energy antartcles In a Feynan dagra an ncong (outgong) artcle can always be relaced by an outgong (ncong) antartcle D Betton Fenoenologa Interazon Fort

13 D Betton Fenoenologa Interazon Fort 3 eft-handed and ght-handed Ferons Even when artcles are not assless t s useful to searate the uer and lower arts of the wave functon are rojecton oerators U U U u u U U U u u

14 Helcty The helcty of a assve feron can be changed by a orentz transforaton, so t s not a quantu nuber that can be used to label the syste Stll left-handed and rght-handed ferons are treated dfferently n the Standard Model Under arty the sn does not change, therefore there s a change n sgn n the Drac equaton and the two solutons go nto each other Thus f nature s nvarant under the arty oeraton we exect both solutons to exst Neutrnos: only left-handed neutrnos exst Electrons: both left- and rght-handed electrons exst, but they nteract dfferently: e can nteract drectly wth a neutrno, but e cannot D Betton Fenoenologa Interazon Fort 4

15 D Betton Fenoenologa Interazon Fort Useful elatons 4 V-A One art transfor as a vector and one as an axal vector 4

16 The Drac agrangan agrangan of a sn ½ feron Used n the Euler-agrange equatons t yelds the Drac equaton D Betton Fenoenologa Interazon Fort 6