Bethe-Salpeter Equation
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- Jeffrey Spencer
- 5 years ago
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1 Behe-Slpee Equo No-elvs Fomlsm Behe-Slpee Equo: ouo o he op. Dgesso Seo Quzo. Dgesso: fs quzo s movo fo seo quzo. Quum Fel Theoel Hmlo Seo Quzo. Shöge Equo. Equo of Moo. Shöge Fomulo. Behe-Slpee Equo fo he Two-Boy Sysem. Dffeel Fom. Iegl Fom. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
2 Behe-Slpee Equo: ouo o he heme. We ose Bou Ses of My-Ple Sysems. I o-elvs quum mehs: ou ses ese y omlze wve fuos soluos of he myple Shöge equo. Relvs sysems: he oveol ppoh woul e o seek fo ov geelzo of he Shöge wve fuo he Behe-Slpee wve fuo soluo of he Behe-Slpee equo. O ss o he es of A.Fee D.Wlek Quum Theoy of My-Ple Sysems MGw-Hll N.Y. 97 D. Lué Ples Fels Iesee Pulshes N.Y. 968 we eve he o-elvs Behe-Slpee equo ssfe y he eg wo-oy Behe-Slpee mplues. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
3 Seo Quzo: movo I elvs heoy: he oeps of seo quzo e of fumel mpoe o ese he eo hlo of ples. I o-elvs heoy: seo quzo sogly smplfes he eme of my el eg ples. I hs se he seo quzo ppoh smply epeses efomulo of he ogl Shöge equo fo he my-oy sysem. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
4 Eve oelvs heoyseo quzo ehs my s vges whe ompe o he ogl ppoh. Fo se se of usg vey omple epessos wh symmeze -symmeze pous of sgle-ple wve fuos s he ogl Shöge fomulo seo quzo llows muh moe smple ssl eme of he wve-fuos of my-oy sysems: hs hppes euse he seo quzo opeos opoe ose osse wy hough he ommuo o -ommuo elos he Bose- Ese o he Fem-D sss. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Seo Quzo: movo 4
5 Seo quzo pems moeove whe ome wh he mehos of quum fel heoy o pu eo oly few m elemes of he heoy. These m elemes e lke o egees of feeom of selee su-sysem of ples se of elg wh he my-ple wve fuos ooes of ll he ples of he sysem. Fo e.: seek fo he physl ehvo of oe ypl ple of he sysem y lulo of he oe-ple Gee fuo. Seo quzo whe ome wh he Gee s fuo ppoh he lguge of Feym ules eomes poweful ose sysem meho of eme of my-oy eg sysems. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Seo Quzo: movo 5
6 Seo Quzo: vey mpo seme. Fo evey soluo o he ogl me-epee my-ple Shöge equo hee ess se of epso oeffes. A gve hs se s possle o osu soluo o he polem seo quzo.. Covesely f he polem s solve seo quzo we eeme se of epso oeffes whh he yel soluo o he ogl me-epee my-ple Shöge equo A.Fee & J.D.Wlek Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 6
7 Quum Fel Theoel Hmlo Seo Quzo. wh wo-oy eos H T kl V kl l k Shöge Equo h Ψ H Ψ Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 7
8 H T kl V kl l k h Ψ H Ψ H : I hese epessos: Opeo he s oupo-ume spe euse epes o he eo esuo opeos. ; ˆ : All sss opeo popees e oe he efos lge popees of he eo esuo opeos. T V kl : : M elemes of he ke eegy wo-oy poel eegy ke ewee he sgle-ple egeses of he Shöge equo fs quzo meely omple umes mulplyg he opeos seo quzo. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Quum Fel Theoel Hmlo Seo Quzo. Shöge Equo. 8
9 Dgesso: fs quzo he movo fo seo quzo. Ou sk: o use fs quzo o move fo he fom of he seo quzo Hmlo. Ĥ [] ef [N] ef... [] ef [] ef Cose ypl No- Relvs My-Boy Hmlo Fs Quzo Ĥ [] [] ef [] ef Is Epeo Vlue s ef [N] ef... 4 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 9
10 I hs epesso we hve he followg ems: Two-Ple Opeo [] My-Boy Hmlo Epeo vlue Epeo vlue Oe-Ple Opeo [] Epeo vlue Thee-Ple Opeo [] Epeo vlue Ĥ [] [] ef [] ef ef [N] ef... 4 Fs Quzo N-Ple Opeo [N] Epeo vlue Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 0
11 Usg sgle ple ple wve fuos: Ĥ ef [] [N] ef... [] ef [] ef 4 [ ] [ ] [ ] [ ] [ N ] [ N ] ef ef The epeo vlues of hose opeos e: My-Boy Hmlo - epeo vlue fs quzo: ττ τ τ ττ τ τ τ τ τ τ τ e τ τ τ [ ] τ τ e τ τ f τ τ τ τ τ f τ [ ] τ τ τ ττ τ τ τ τ τ 5 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
12 Now se of fs quzo ose sml epesso fo he Seo Quzo epeo vlue of Hmlo: Ĥ [] [] ef []ef ef [N ]ef 6 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
13 Ĥ ef [] [ ]ef ef [ ] [ N ]ef 6 Fel Fel Opeos Opeos :: ˆ :: ˆ Vey mpo: o mke he so fom fs o seo quzo eple sgle ple wve fuos y fel opeos whh epe o eo o esuo opeos he sgle-ple wve fuos hemselves. I hs fomulo e seo quzo opeos s oeffes of he epso. 7 llow eo esuo of ples Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
14 Comg 6 7 se of τ τ τ [ ] ] PQ τ [ ] ] ττ τ [ ] ] FQ ττ τ τ τ [ τ we hve fo se fo he epeo vlue of oeoy opeo seo quzo: A se of [ τ τ [ ττ τ τ we hve fo he epeo vlue of wo-oy opeo [ ] ] Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 8 9 τ [ ττ τ τ Dgesso: fs quzo he movo fo seo quzo. 0 4
15 A se of ττ τ [ ] ] FQ τ [ ττ τ τ we hve fo he epeo vlue of hee-oy opeo Flly se of τ τ τ [ ] ] ef [N]ef FQ ττ τ τ [ ττ τ τ τ [ N ] ττ τ τ τ τ e τ f τ we hve fo he epeo vlue of N-oy opeo: 4 ef [N ]ef e f ττ ef [ N ] ττ τ τ τ τ e f τ τ τ τ 5 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 5
16 6 f e ef f e ef [N]ef f e ef f e ef []ef [] [] [N] [] [] [] τ τ τ τττ τ τ τ τ τ τ τ τ τ τττ τ τ τ τ τ τ τ τ ττ τ τ τ τ τ τ τ τ 6 Comg ll hese ems we hve: Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
17 Replg he pevous epessos: 7 We ge fo he sum of epeo vlues of my-oy opeos: ef [] [] []ef ef τ τ τ τ τ e f [] τ τ τ τ ττ τ ττ [] τ ττ τ τ τ τ τττ [] τττ τ τ e τ f ef [N]ef ef e f ττ τ τ τ τ τττ [N] τττ τ τ e τ f 7 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 7
18 Replg he pevous epesso τ [] τ [] τ τ [] We o: ττ [ ] [ ] ττ Usg he ohoomlzo oo... [] τ τ τ 8 τ τ τ τ τ τ τ τ τ τ ef ef [] [] []ef [N]ef [] ef [] τ τ [] τ ef τ τ e [N] f τ τ τ ef τ τ τ e τ f τ τ τ τ τ τ... τ τ τ τ τ τ 9 τ τ τ Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 8
19 The y usg he ohoomlzo oo of he sgle-ple Shöge wve fuos we ge ef ef 0 [] [] []ef [N]ef [] ef [] [] ef e [N] f... ef e f Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 9
20 Ĥ ef [] [] [] [] ef [] ef e f [] [N]ef Fom epesso 0 ome wh 6 we ge M ef ef [] [] []ef [N]ef [] ef [] [] ef e [N] f ef e f 0 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 0
21 I he epesso fo he seo quzo Hmlo we efy omo of opeos -umes: -Opeos Ĥ [] [] [] []ef [] [N]ef ef [] ef e f -umes Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
22 I he followg we ose fo smply oly wo- oy eos. Quum Fel Theoel Hmlo Seo Quzo: H T kl V kl l k Covesely we o egl fom fo epesso. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
23 Ĥ H V T k l kl kl By oug omplee ses of sgle-ple ses epesso we o: l T k V kl l k M Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo.
24 Ĥ V k l kl l T k Ioue he fel efos he ees: ˆ ˆ 4 T T δ V V δ The esuls... δ 5 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Dgesso: fs quzo he movo fo seo quzo. 4
25 We he o he followg egl epesso fo he Quum Fel Theoel Hmlo: Ĥ m ˆ ˆ ˆ Two-Boy Poel: ˆ V ˆ ˆ V v The me seo quzzo ws spe y suh epesso: looks lke he epeo vlue of he ke eegy poel opeos ewee wve fuos fs quzo. Howeve hs se we e elg wh fel opeos se of wve fuos. Coluso: seo quzo opeos e epesee y fels poel ke ems e omple oeffes. 6 7 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Quum Fel Theoel Hmlo Seo Quzo. Shöge Equo. 5
26 : Fel opeo ssfyg ommuo Boso opeo o ommuo Femo opeo ules: smplfe oo [ β ] m δ β δ [ ] β m [ β ] m 0 0 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Quum Fel Theoel Hmlo Seo Quzo. Shöge Equo. 6
27 Equo of Moo: H m V 6 & [ H] m smplfe oo z [ z z] z z A [ z zv z z z z ] Poof: B Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Equo of Moo. 7
28 m z We ose fs epesso A. By usg he ey we hve fo he keel of hs egl fo oh Boso Femo opeos: [ [ z z] z z] δ z z A [ ABC] [ ABC ] BCA [ ] { AB} CB{ CA} Comg hese equos epesso eues o m z [ z z ] m 4 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Equo of Moo. 8
29 z z [ z z v z z z z ] 7B [ z zv z z z z ]] δ z z v v ] 5 v 6 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Equo of Moo. 9
30 Equo of Moo omg he pevous epessos : & [ H] m v m v Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos Equo of Moo. 0
31 Tsk: we wsh o eove he oy Shöge espo of -ple sysem eg hough wo oy foes. Segy: osu he -ple wve fuos show hey ssfy Shöge equo. N N :Tol ple- ume opeo [ NH] [PH] P :Le momeum opeo P Poof: Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
32 [ ] [ ] [ ] V z z z z z z m H N [ ] [ ] [ ] { } { } A C B C B A A C B C B A BC A 9 [ ] [ ] [ ] [ ] z z z z z z z z z z z z z z z z z z z z z z C B A C B A C B A C B A v v δ δ δ 40 9A 9B 9A 4 Come 9A wh 4: Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
33 [ ] [ ] 0 z z z z z m z z z z z m z z δ δ v [ ] [ ] [ ] z z z z V V z z z z V V z z V z z C B A C B A C B A B 4 4 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
34 4 [ ] [ ] [ ] [ ] [ ] [ ] z z z V V z z z z z V V z z z z z z z z V V z z z z V V z z V z z C B A C B A C B A m m δ δ ± δ δ ± ± ± 44 Come 9B wh 44: Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
35 ± z V ± V z δ z δ z δ z δ z z [ { } V { } 0 V 45 Comg he pevous epessos we hve: Smlly we hve: [ NH] [PH] Fom he ommuo ules: :oss of moo. N P [ NH] 0 P 47 0 N 46 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 5
36 Fom he ommuo elos he ommuo of he ume opeo wh Boso o Femo fels gve geel: [ N] [ N] Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 6
37 [ N] N f N N : Boso o Femo Ceo opeo. 50 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 7
38 [ N] N N 5 Boso o Femo Desuo opeo. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 8
39 We efe he vuum se s: Thus f follows fom he pevous esuls efos h: ; H 0 P 0 N 0 A lolze omlze -ple se s efe s:! Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 9
40 Nomlze poly mplue ofguo spe wve fuo fo he -ple sysem: Φ ke ; ; ke! 0 ke 55 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 40
41 ; Φ ke Ssfes he -oy Shöge equo: Φ ke v ; m Φ ke 55B ; Φ ke 55A ; 55 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 4
42 Poof: Come Φ ke! ;! 0 0 ke {[ ] [ ]} ke 56 Wh [ H] m v 55 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 4
43 Φ ke! ;! 0 0 ke {[ ] [ ]} ke m v Φ ke 0! ; 0! m m {[ v ] [ v ] [ v ] 4 [ v ]} ke ke 57A 57B 57 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 4
44 ! 0 ke m m 57A 0! {[ ] v [ ] v [ ] v 4 [ ]} v ke 57B Use ommuo ommuo elos: [ ] [ [ β β β ] m δ β ] δ m m 0 0 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 44
45 Φ 0{[ v ]! [ δ [ ± v ] ] [ δ [ ±δ ± v ] ] ke } 0 ke m! v 0 ke! Thus Φ ke ke ; m ; m Φ 0 ke! ke ; v Φ ke ; Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 45
46 46 Φ ke 0! ke ; ; ke 59 Two-ple wve fuo: Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos ; v ; m ; ke ke ke Φ Φ Φ
47 47 ; v ; m ; ke ke ke Φ Φ Φ Equo of moo fo he wople wve fuo: 60 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos ; v ; m ; ke ke ke Φ Φ Φ
48 Fozo of he epeee of he wve fuo o he ee of mss ooe: Φ ke!!! wh 0 e P. 0 0 ; e P. [ e P. ] e ke ke P. e k. e e k. P. ke 6 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 48
49 We we epesso 60 he fom: Φ ke whee X wh Φ ke ; / ;! V e k.xe Φ v 0 ke 0 v ke 6 V Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 49
50 50 χ E k T 0 ke T { > ± > Behe-Slpee Wve Fuo: 6 6 Φ ke 0! ke ; ; ke Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
51 5 ] [ } { T θ χ E k T 0 ke m m Apply Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
52 5 v m m ke ke χ δ χ Behe-Slpee Equo fo he Noelvs Two-Boy Polem: / X whee v e V E k k.x ke χ χ Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos
53 A smple movo: we w o osu seo oe o me evoluo equo fo he wo-oy polem ffee mes keepg usly oss vey useful he elvs se eplly ov. Cose fs m wh v [ H ] m v v v 69 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 5
54 Now ose pous of he sgle ple opeo fo he wo ples sysem m v 70 lf he equl-me eso of he Shöge ppoh keepg usly oss; we ge he opeo v usl v os m m usl os 7 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 54
55 To eove epesso 67 we ose he seous eo s spel se: v v usl δ v v os v usl os 7 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 55
56 Bou Ses. Behe-Slpee Dffeel Equo. The so ewee see ou ses ouum o seg ses emeges upo oveg he ffeel equo fo he wve fuo o egl equo. m δ v m χ ke χ ke 67 Homogeeous Behe-Slpee Iegl Equo: 4 χ ke δ v χ Fou-mesol oo. 4 ke S F S F Sgle-ple Fee Popgo 7 Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 56
57 I f euse he o-elvs popgo fuo ssfes SF 0T 0 m S F δ Usg S F o ege 67 we hve: χ χ ke ke δ v χ ke 4 4 S F S F 76 Whee eoes he homogeeous em. 0 χ ke 57
58 Poof h fo ou ses 7 ssfes 67 :: pply o he lef of 67 he opeo: Ths gves m m χ ke m m SF δ v χ m δ v χ ke ke 4 4 m S F 77 No oly euse: S F δ m χ ke Bu lso euse vshes he se of ou se hvg see eegy level E<0. Behe-Slpee Equo No-elvs Fomlsm. Ces A.Z. Vsoellos 58
59 Refeees.. D. Lué Ples Fels Iesee Pulshes N.Y A. L. Fee J. D. Wlek Quum Theoy of My-Ple Sysems MGw-Hll N.Y
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