The Classical Wave Theory of Matter

Transcription

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2 Th Css W Thoy of M Dyn Inpon of Ry, Qunu Mhns, nd Gy nd Df Rob. Cos, PhD ISN: Copygh 4 Rob. Cos. ghs sd. No p of hs pubon y b podud, nsd, o nsd n ny fo o by ny ns, on o hn, whou wn psson fo h uho o pubsh (Vu Vs Pss), p fo bf ps n onnon wh ws o shoy nyss. Psson y b sough onn fo pubshng@vuvs.o

3 Tb of Conns Gossy... 9 Chp. Rw of Css Physs..... s Ids..... Css hns Conson Lws Spn Von Mhods W Equons Es ws Sss-Engy Tnso Spn ngu Monu Dnsy Rgd Roon Roon Ws Eogns W Engy Dnsy Hon Rpsnon of Ws Spon of h W Equon Pops of Ws W popgon Dspson nd Goup Voy Infn nd Dffon Dopp shf Unny Pnp Suy Chp. M Ws nd Sp Ry Inoduon Msuns wh ws T don Lngh onon Lngh nd sndds Dopp shf M ws nd gh Soon ws Engy nd onu Tnsfoon of oy Th wn pdo n npons... 7 Chp 3. Es Ws nd Qunu Mhns

4 3.. Inoduon Toson Ws On-Dnson S Ws Spnos nd spnos W Voy Th Dnson S Ws Roon of Gdn nd Voy Suss Roons Eu ngs Eps W Funon Fs-Od W Equon Vo Ws Roon of Poon Foon nd Fs-Od W Equon Conon nd Roon Ws n n Es Sod s ssupons Equon of Eouon Dyn Vbs Eon Ws F Eon Equon Mss, Conon, nd Roon ngu spon Voy Roon nd Mss W Infn nd Pons Lon Fo Mgn Mon Spn Ws Msun Coons Qunu Mhns Fons nd osons Po Knowdg nd Sss Hydogn o Sys Sp nson T s Cobnd Tnsfoons Mh nd Phys Pops of Spnos Spnos nd Inn Podus W Pops of M Suy Chp 4. W Rfon nd Gy Inoduon W popgon n non-unfo du Dspson Ron nd M Fos

5 4... Ron bwn oponns Th gon pon Consquns of gy Nwonn gy ndng of gh Cuu of sp k Hos Gogns Suy... 5 Epogu ppnd : ngu Egnfunons ppnd : Lon Tnsfoons

6 Pf Rson nd f nquy h ony ffu gns gns o. Thos Jffson, Nos on h S of Vgn, 784 Ths book s h sond df of n p o bdg h onpu gp bwn ss nd odn physs. odng o ss sonng sp s Eudn, s ndpndn of poson, gh ws us hough h, nd dyn syss dns. Modn physs sss h sp s ud, s dpndn on poson nd oy, nd fundn phys posss pobbs. I s os unsy bd h pn nd ho dopns of h h nuy no ony dspod spf ss ods, bu n f nd h possby h ny hns od oud popy dsb nu. Th of hs book w ky s sf-ondoy o physss usod o spon bwn ss nd qunu physs. How, h d w s h hns ods n n f b usd o pn fundn phys phnon h h hho bn supposd o b byond h of ss physs. In odn wh oon usg, I us h ss physs o f o ny hns dspon of nu phnon whh psus h sn of onnuous d, Eudn sp, nd bsou. Th of hs book fs o nw ppons of ss ds nd no o nqud hos of h ps. I off wo guns fo onsdon of hns ods of fundn phys phnon. Th fs son s h n sp of ps fus, s s possb h sub hns od n b found. fo ss physs n b jd ogh, us b pon h hns ods yd pdons nonssn wh obsons of nu. Sn h nub of possb hns ods s nfn, s possb o j of h unss hy n b pon o sh so no fu. s h d w s, fus h h bn psud o f hs on h bn noy npd. Th sond son fo sudyng hns ods of h uns s spy o bud nuon. On n hdy p o s h subs of ws o qunu fds n h ysous uu whou fs bng b o undsnd h bho of unduons n sp s sod. Y hs s psy h on und whh physss h bod fo h ps nuy. Hsoy, hns ods h fgud ponny n h pogss of physs [s.g. Whk 95]. Th oony s h nodud by Js MCugh n 839 podd hns od whh ws onssn wh of h known pops of gh nudng poon, fon nd fon, nd ys-ops. W Thoson (Lod Kn) dsbd how suh hn du oud b d. Josph oussnsq hd s suss n 867 by ssung n odny s sod h h ps s w s py sp. Js Ck Mw s hso fouon of h quons of ogns n h y 86 s d on hn od of h h onssng of s s nspsd wh ong ps. 6

7 Ths s p s puy sgnfn n gh of h s of y hoy h bgnnng of h h nuy. Th Lon nsfoons ng spo-po oodns of obss n oon wdy bd o b nonssn wh ss noons of sp nd. Y Mw s quons, whh on wh sp o Lon nsfoons, w dd fo ss hn od. How n ss od of hn h b onssn wh h Pnp of Ry? Ths book nsws hs quson by showng h h ws of Sp Ry onsqun of h w nu of. Wh sp o qunu hns, uy sudns nodud o h subj sudy of h non-s Shödng quon. Ths quon no ony os h pnp of y, bu so dus h fou-oponn D spno of odn qunu hoy o sng s b. Spy pu, h non-s Shödng quon pss so of h h bu dsds of h physs. On shoud no p h phys npons of Shodng s quon ppb o nu syss. Rn dns n ss physs h d so physss o d ou psn hnkng bou qunu hns s nfsd wh h dps sonpons [Gu 993]. Ths book ps o dsp suh sonpons by skng, hns pnons of nu phnon. In hs book h oupd quons dsbng n on dd fo sp od nd npd qu nuy s dspon of h popgon of ss ws yng ngu onu. Th w-p duy of s pnd by h f h ss ps soon ws (od osons). Ths book s wn h of sond- o hd-y unsy physs ous. I s ssud h h d hs dy sudd h bs pops of ws, s f wh phys onson ws, nd hs s bs undsndng of w nyss usng Fou nsfos. Ths book ddsss s bs physs qusons. So ps : Wh oon oon s ssod wh spn ngu onu? Cn ngu onu b dfnd ndpndny of h ho of ogn? Wh s h dyn npon of spno? Why ny h ods (nudng Mw s) onssn wh Sp Ry? Why do nd n bh k o gs of h oh? Why s gy so uh wk hn oh fos? How do ws popg n n s sod? Th "" s usd n h os gn sns, nudng nfsons of ngy nd no y hos wh ss. Sn phoons n obn o fo ons nd posons, s h ss nd non-ss ps shoud b gdd s dffn ods of sng phys phnonon. Th "n dnsy" s usd n p of "ss dnsy" n h od of h uu sn "ss" s popy of h hn popy of h uu sf. Mos of h hso nfoon n hs book hs bn gnd fo h n wok of S Edund Whk, Hsoy of h Thos of h nd Ey, Vos. I nd II (Nw Yok: Phosoph Lby 95 nd 954, spy). Th sho synopss psnd h no nndd o b op n ny sns. Thy spy pod hso on fo whh 7

8 n qusons bou h nu of os nd w nswd. Nuous sgnfn onbuons h nssy bn od fo h sk of by. Rhd Fynn on kd h f h oud pn hs wok o h g pson woudn' h bn woh Nob P. How, Ensn s w ws h you do no y undsnd sohng unss you n pn o you gndoh. Ths uho gs wh Ensn, nd I hop h hs book w hp physss o undsnd nu n nn h n b nngfuy shd wh h s of huny. Rob Cos Pond, 4 8

9 Gossy dspn o gn o pon gn fd j sn nso E fd ε ngy dnsy F fo oon ng Φ pon ngy o pon H Honn H Honn dnsy s un gny ~ psudos un gny j L s ngu onu dnsy (ob spn) J L S ngu onu (ob spn) J un dnsy k w o K kn ngy L ob ngu onu dnsy (ssod wh w popgon) L ob w ngu onu L Lgngn L Lgngn dnsy s ss µ s sh oduus p w onu (dnsy) P onu q ρu onu dnsy of du q hg Q ngu pon Q! spn ngu onu poson o ρ n dnsy s spn ngu onu dnsy (ssod wh oons of du) S spn ngu onu τ oqu dnsy T oqu Θ oon ng u du oy U pon ngy 9

10 p o w oy w Θ ( u) oy (ngu oy) of du ω ngu fquny ω ngu oy Ω Θ oson oponn Θ s oson ξ oodn b D bspno w funon Th y b pons o h bo dfnons, bu I h d o kp h o nu. Rfns Gu S, Lsnby, nd Don C 993 Igny Nubs no R - h Go gb of Sp Whk E 95 Hsoy of h Thos of h nd Ey, o. (Ednbugh: Thos Nson nd Sons Ld.) Whk E 954 Hsoy of h Thos of h nd Ey, o. (Ednbugh: Thos Nson nd Sons Ld.)

11 Chp. Rw of Css Physs If you woud b sk f uh, you us s on n you f doub, s f s possb, hngs. Rné DsCs, Dsous d Méhod (637)... s Ids of physs s h possb o. I s possb un you undsnd, nd hn bos. Ens Ruhfod Pogss n sn ss fo ps o on obsons nd pdons. Fs, phnonon s dsbd n d on h bss of obsons o suns. Sond, s of snf pnps, o ws, s nnd o pn h obsons. Ths s of ws s d hoy. snf hoy us b pb of ydng fb pdons (ohws h hoy s no snf, hough gh s b o). Sos up hos yd h s pdons of obsd phnon, n whh s h sps hoy s onsdd o b h bs (Okh s o). I ofn hppns h hoy pds phnon h h no y bn obsd nd dsbd. Ths ds o nwd ffos of obson nd dspon. If nw obsons no opy pnd, hn h y of pdon nd obson s pd. Fgu. so ( C) Fgu. so s Uns ss p of hs poss s h dopn of h w of gy on h bss of sono obsons. Th oon of ss nd pns hs bn obsd nd sudd sn

12 h dwn of hun on. hn bss fo hs oons ws dsbd by so [Fgu.] ound 35 C n hs s On h Hns. so supposd h hns o onss of onn sphs no whh s objs w ffd [Fgu.]. Th ouos, o py, sph onnd uud of ss. Oh sphs onn ony sng s obj (sun, oon, o pn). Th sph h ws h n (hough h Pyhgons bd ohws). Ths od pods good pnon of h oon of dsn ss (whh w now bu o h s oon), bu offs no phys pnon fo h ppn gu oon of pns wh sp o h h. Fgu.3 Poy (D 7-45) n pon on so s od of hny oon ws pod by Cudus Poous (Poy) [Fgu.3] n h sond nuy.d. Poy s od [Fgu.4] shfd h h fo h n of h ob o pon d n n, whh ws oupd wh n qun pon off-n n h oppos don. Cu oon ws bud o oon of sph, d dfn. Non-u oons w odd by ddon sphs, d pys, ong bou pons on h dfn. poonus of Pg (Pgus) hd pon h p oon oud b dsbd n hs wy. Fuh oons o hs od w d by png pys on pys. In pnp, ny pod oon oud b dsbd by suss pubons of hs od. How, h opy of hs hod pody d Kng fonso X of Spn o opn, If h Lod ghy hd onsud bfo bkng upon on, I shoud h ondd sohng sp.

13 Eh Cn Equn Cs Obj Sond Epy Fs Epy Dfn Fgu.4 Ep of p (fs py) nd ogd (sond py) obs podud usng pys. Nous Copnus [Fgu.5] spfd hs od n 54 by sonng h h sun, h hn h h, s h n of h ong sphs [On h Rouons of h Cs Sphs]. Ths nw ho dopn dud h nub of pys nssy o opu pny oon. In pu, h ppn ogd oon of pns ws d onssn wh gu u obs. Fgu.5 Nous Copnus ( ) In 65, Johnns Kp [Fgu.6] nd h nd fo up pys by png pns n p obs wh h sun on fous of h ps, h hn h n. H so ddud h n fo h sun o pn swps ou qu s n qu s, 3

14 nd h h squ of h ob pod of pn s popoon o h ub of h ngh of h s-jo s. Fgu.6 Johnns Kp (57-63) Fgu.7 Is Nwon (643-77) Kp s h ws of pny oon gy spfd h opuon of pny obs. Nonhss, Is Nwon [Fgu.7] found n n sp pnp o pn h obs. Eh pn s d o h sun by gon fo popoon o h ns squ of h dsn fo h sun. Ths gon fo no ony pnd pny oon, so pnd h on of s objs owd h h. Y s o u suns w d, n hs gn hoy dd no pn of h pny oons. In pu, h phon of h ob of Muy dnd whh oud no b pnd by h nfun of oh pns. b Ensn s hoy of gn y fny pnd hs nos of pny oon [Fgu.8]. In hs hoy gon fd s popoon o gdn n h spd of gh (hs popy of gn y s dsussd n Chp 4), whh s dud by h psn of ngy. Th hoy so pds h gh ws fo dsn ss f owd ss objs suh s h sun o pns. Ths pdon hs bn fd by sung, dung so pss, h posons of ss whos gh popgs os o h sun on h wy o h. N bk hos, h fon s so song h h gh nno sp. 4

15 Fgu.8 b Ensn ( ) Thus h opd okwok of Poy s sphs hs bn pd by h phys pnp h gy sus fo h psn of ngy n sp. Modn physs now hs s own son of Poy s sphs n h of p physs. n p hod, qunu fd hoy, hs bn found o uy opu ss ouos of pns. Th hod woks bu s on yd p onsns nd hs dfd pnon s o why nu shoud bh hs wy. Lk Poy s sphs wh pys on pys, pdons of qunu fd hoy opud usng suss ppoons o pubons. Eh oon s psnd by Fynn dg, whh s npd s psnng n non of ny ps [Fgu.9]. noon podu djuss non snghs n od o podu h o fn sus. T γ γ Sp Fgu.9 Rnoon n qunu odyns: n non bwn wo ons ( ) s odd fo f by () u phoon (γ ) hng, () ddon nd on nd dsuon of on-poson p (, ), nd (3) ddon u phoon (γ ) hng. ddon posss onbu d nfnu o h non. In h s of Poy s sphs, h dffus os bus sonos kd n undsndng of gy nd pd o dsb s obs n s of u oon. 5

16 Th hod ws suffn h fo s pupos of pdng fuu posons of s objs n h sky. Is jo fw ws unnssy opy. Eh pn hd s own s of py ps o dfn h ob. Nwon s (nd Ensn s) hoy of gy unfd s obs und sng phys pnp dpndn ony on h sss of h objs n h sky. In odn hos of ny ps, h yp of p s ssgnd unqu s of ps (ss, spn, sospn, hg, wk hg, song hg,.). In f, h p hs s own quon of ouon, whh nuds nons wh oh ps. Th og of ny ps nd h ssod onsns, n obnon wh h opuon hods of qunu fd hoy, s d h Sndd Mod of p physs. Th Sndd Mod hs pon o b y u n y s whh hs bn sd. Is fw s no nuy bu opy. Equons fo bs phys quns suh ngu onu h no sng psnon bu dpndn on whh ps psn n sp. Qunu nubs d up, down, sngnss, buy, h, nd uh h bn nnd o dffn ps on h bss of unpnd pops. Sp phys qusons suh s Wh oon s ssod wh spn ngu onu? f unnswd. No on hs bn pd o pn why ps shoud hb h w-k hss h obsd n pns. nd gy hs no obous on o h Sndd Mod. Eny ps, dsp h n, no ub. Fo p, wo phoons n od nd nsfo no n on-poson p. Th ns nsfoon s so possb. Thus s obous h phoons, ons, nd posons no uy ny ps bu nsd psn dffn ss of shd phys poss. Oh ps undgo s nsfoons of dns. Hn h s son o b h ny ps dffn nfsons of sng phys poss, nd gh b dsbd s ods o pubons of h uu. Th ps of hs book s h h onpu dffus of qunu hoy s bus physss p o dsb s ds ndpndn ps h hn s onnuous ws. W w d h bs dyn nd ss pops of fo sp w od, wh ss ps npd s soon ws. How, w w no d h nu pdons of h hoy s hs ns n unsod pob. Ou ops w nud sp y, p popgon, nons, spn, sss, nd gon on. Th d s ssud o h po undsndng of w posss h undgdu. Rds of hs book w bo qund wh h hs of qunu hns nd y. Thy w hn b f o fuh h undsndng of odn physs h by fuh ss nyss o by o don nus of sudy. W bgn wh w of ss physs... Css hns Tuh s by nu sf-dn. s soon s you o h obwbs of gnon h suound, shns. Mohnds Gndh Css hns ypy bgns wh Nwon's ws of oon: () n obj n oon os n sgh n onsn spd unss fo s upon (onu ponsn f fo F) 6

17 () n obj's onu hngs popoon o h fo on h obj (Fdp/d) (3) ny on on n obj sus n n qu nd oppos on fo h obj. In oh wods, h fo ( F ) whh obj s on s qu nd oppos o h fo ( F ) whh s on. uy, h fs nd hd 'ws' n b gdd s sp ss of h sond w. Th fs w spy dsbs h s of o fo: F. Th hd w n b dd fo h sond spy by onsdng h obnon of wo objs nd s sng 'obj'. In h bsn of n fos (F) h fs w qus h h o onu s p p onsn. Tkng h d yds: dp dp d d (-) Ths ps h ny hng n onu of obj us b opnd by n qu nd oppos hng n onu of obj. Th hd w s obnd by subsuon of fos fo h s of hng of on. ngu onu L s dfnd s h oss podu of h onu wh dspn o : L p (-) Th ng of fo s dspn s d wok: W F d Ths s h hng n hn ngy whh sus fo ppon of fo. In dffn fo, hs on n b wn s: (-3) W F (-4) In h s of fo ppd o soy obj, Nwon's sond w of oon yds: dp W d No h h wok pfod n ng n obj s ngy whh s nsfd o h obj. If w dsussng so of 'pon' ngy U whh s dpd o on obj (o nsd by don), hn w nd nus sgn: dp U d... Conson Lws In osd sys,.. on fo whh w nud sous of fo, h o fo us b o, nd hfo h o onu s onsn. Ths w s known s onson of (-5) (-6) 7

18 onu. o bs n of sonng fo onu onson gos k hs: If n obj (o goup of objs) w o n fo on sf (hss), hn h ngy of h obj woud dpnd on poson. On woud h o do wok o o h obj gns h sffo. Consy, f ngy s ndpndn of poson hn h s no sf-fo nd onu s onsd. Wh hs n of sonng, h w of onu onson foows fo h ssupon h ngy s ndpndn of poson. Ths s n p of phys syy. Fgu. Ey Noh (88-935) Th hn Ey Noh [Fgu.] pod h h syy n phys sys ps h sn of onsd quny [Noh 98, Godsn 98]. Th ons s so u. So f sys nd h ospondng onsd quns : Syy nson oon shf sp nson Consd Quny onu ngu onu ngy py Th s of hs y b nw o so ds. Py s h fo (P) whh opns nson of sp oodns ( ). Th oodn bs hss obousy h ng py (P). Monu nd oy so h ng py. Sp nson s qun o o gng foowd by 8º oon bou h oodn s ppndu o h o. Hn op spnnng okws hs spy nd g whh so spns okws. Thfo ngu onu hs pos py (P). Mo pony, h quons gonng ngu onu no hngd n h oodn-nd sys (.g. L p). In oh wods, h quon fo ngu onu s sy wh sp o oodn nson. Py onson ps h h quons dsbng phys sys 8

19 sy (unhngd) und oodn nson. Sn oodn nson hngs f- nd gh-hnddnss, oud so b bd 'hnddnss onjugon'. Inuy, wh w s n o ss jus s physy s s wh w s dy, so bfo h d-95 s snss gny ssud h py s onsd n ny phys posss (hough no osopy sn so ous suh s DN found usy n h gh-hndd fo n nu). How, pns h shown h n phys posss nnsy f- o gh-hndd fo n ps. If h ps hss (.g. poons, nuons, nd ons) ssud o b h own o g, hn hs sus py py oon n phys ws nd no jus n sy dsbuons of f- nd gh-hndd objs. Insngy, n- ps bh y k o-gs of. Th sps pnon of hs o-syy s of ous h n- s h o-g of. Sngy, ns physss h jd hs sp pnon bus s nonssn wh ho ssupons bou how o u o gs of qunu hn w funons. W w ddss hs ssu n Chp Spn Th spn hs wo nngs n physs. Th fs nng s h odny on, ny oon bou o s. If gd op, wh n of ss nd n on I, os wh ngu oy ω s bou s own s nd so os wh onu p ong sgh n, hn h ngu onu of h spnnng op s: J p Iωs L S (-7) Th fs s sos d h ob ngu onu nd h sond s d h spn ngu onu. How, hs spon s sowh f sn h spn ngu onu fo op wh unfo ss dnsy ρ n so b wn s: ( ) ρ( ) 3 S d whh hs h s fo s h ob ngu onu. In qunu hns, h spn ngu onu s no py d o oon oon (hs y b h fu of nu o h su of ou gnon bou nu; w w gu fo h ). Th sond nng of spn s o h nsfoon of b oponns und oons. In hs on h spn s h o of π dns ddd by h ng bwn ndpndn ss. s fd s dsbd h pon n sp by sng nub ndpndn of onon. Thfo s fds h spn o (nfn ng bwn ndpndn ss). o fd s dsbd h pon n sp by h ndpndn oponns wh n ngu spon of π dns bwn ny p of ndpndn oponns (.g. oodn s). Thfo o fd hs spn on. o nsfos und o nfns oon δ s: (-8) δ δ (-9) 9

20 No h oon of by δ s qun o oon of h oodn s by δ. Fo p, h oponns nsfo und oon of h -s s: ʹ ʹ ʹ y y os sn sn os (-) Th dffn fo of hs quon s: ( ) ʹ ʹ ʹ y y y y (-) Th nus sgn s spd fo h by onnon. Th ( ) s d spn, nd n b gdd s on of h oponns of spn o: ( ) ( ) ( ) y (-) Th supsp () s usd o dsngush hs spn- s fo oh s nodud. Usng hs s h dffn hng n h o undgong by oon s; ( ) y y y y y y y δ δ δ δ δ δ δ δ δ (-3) Th fn fo spy ss h oponns of h u opo: δ δ (-4) Th bo nyss s fo oon bou o s,.. n oodns wh. Fo oon bou n by s w us nud h ffs of h hng n du o oon: ( ) ( ) ( )Α δ δ δ δ δ δ δ (-5) In h fn fo bo h fs s d h ob oponn nd h sond h spn oponn of h oon nsfoon. Of sp ns s h s wh h dyn ngu onu J s sf pssd s oon nsfoon. W w s bow h h dyn ngu onu n b pssd s d of s ( L ) wh sp o ngu oy:

21 J L! (-6) In hs s h s no dsnon bwn h wo dfnons of spn. hough s hs spn of o, w w s h nono spn dos s whn h funon L s pssd n s of oh bs..3. Von Mhods n o dos no bo uh by son of upd popgon, no dos uh bo o bus nobody ss. Mohnds Gndh Fo sp syss s sy o d quons of oon spy by dnng h fos nd ppyng Nwon s w (F). How, n ny suons suh n nyss n b h dous. Insd, physss h dopd on hods whh u ngy h hn fo. Th bs d of on hods s h wh h u ouon of h sys ssfs n quon, oh fous ouons n b pd by h dffns, o os, fo h u quon. Th o quon of ouon n hn b gdd s h on whh ns hs os. Consd, fo p, b ong hough h und h nfun of gon fo: dp F d In ong bwn by pons nd, h quon of oon n b ngd: dp d F d ΔW d wh w h dnfd h gh-hnd sd wh h wok don by h fo ng on h b. Rngng yds: dp d ΔW d Wng h ph ngh n s of oy ( d d ) yds: dp d ΔW d Fo b of ss, h onu s p nd h quon dus o: (-7) (-8) (-9) (-) d d ( ) d ΔW Th fs psns h hng n kn ngy K, so w h: (-)

22 ΔK ΔW (-) Ths quon psss onson of ngy. Dfnng pon ngy U s h sou of wok ( U U W ): K U E onsn (-3) wh E psns h o ngy. Th quon of oon sus fo non of h dffn bwn kn nd pon ngy. Dfn h Lgngn funon s: L (, ) U( ) (-4) W qu h h ph ng b sony (o fs od hng) wh sp o ons of h Lgngn ong h ph: δ L (, ) d Us h fs-od pnson: δl L (, ) δ δ δ δ L Ingon by ps yds: L δ L δ L L L δ d Th ndpons ssud fd, so h fs s o. Th nng ng us b o δ. Ths ondon yds h Eu-Lgng quon: fo by hngs ( ) L L Th quny (-5) (-6) (-7) (-8) L L! s d h onjug onu (o onu onjug o h oodn ). Fo h Lgngn n (Eo! ookk no dfnd.), h onjug onu s: L p! Ths s of ous h usu dfnon of onu. Th ngy y b obnd fo h Lgngn usng h podu: E p L (, ) U( ) (-9) (-3)

23 Usng onu nd poson s h ndpndn bs, h funon fo of h ngy s d h Honn: p H K (, p) U(, p) U( ) Th quon of oon s hn: dp H d whh oups ny wh h quon fo oy: d H d p Th bs nd p n hs p d onjug bs. (-3) (-3) (-33).4. W Equons Edn s b hn hoy ( 論より証拠 ). Jpns pob Hsoy, hs bn hough of n s of ds ps. Whn w ook sn w nsny sgn no ds objs. Y s now h on subo h bho of s gond by w-k quons. In gn, ws gnd by h pubon of onnuous du whh hs wo pops: () ssn o hng o n, nd () y o hng o song fo..4.. Es ws Fo p, onsd dspns of s gon n h no of sod. us h gon onns ss, hs n. W h dnsy of n ρ. Nwon's fs w of oon ss h h gon w no hng s oon unss d upon by n n fo. us ns of sod bound ogh by n s on, ny shng w su n song fos whh oppos h shng [Mos nd Fshbh 953]. W w us h shohnd noon ( ; ) o dno ds of fd bs. Th o d s: d d u w wh u s h oy nd w ( u) s h ngu oy of h du. Th n on ( F I ) o hngs n h dspn s gn by Nwon's sond w: (-34) F I dp d ρ %& u u u ( w ϕ)u' ( d 3 ρ % &!! ' ( d 3 (-35) 3

24 wh w h usd h dffn ss ρ d. Th song fo s h su of sss (o nson) n h du. don of h on bwn sss nd sn n b found n Th Fynn Lus on Physs, Vo. II [R. P. Fynn, Rob. Lghon, nd Mhw Snds, (ddson-wsy, Rdng, 963), Chp 39]. W w g ony suy h. Th sn n du psns h (sp d) whh o dspns d fo n qubu poson. Fo p, f n obj s shd fo qubu n, h sn s gn by: h -don by n oun ( ) 3 (-36) No h f nghbong pons od by h s dsn (oy gd dspn wh u ) hn h s no sn bwn hos pons. Th fs subsp nds h don of dspn. Th sond subsp nds h don of on. If gon s od ounokws bou h -s by n ng, h dspns : ʹ os ysn yʹ y os sn Th dspns n b wn s: y ʹ yʹ y y [ os ] y sn [ os ] sn No h fo fn oons h dspn hs dgn ( ) n hough h oon s nopssb [Fgu.]. 8 Roon (-37) (-38) / Fgu. Dg of hf-dspns (/) fo 8 oon, donsng non-o dgn. 4

25 Inopssb oon qus ony h h oy h o dgn. Sn w png o opu sss (nudng opsson) n s of dspns, w fod o oons o b nfns n od o od noous dgns. Fo s oons h fs-od psson s: y y (-39) Of ous, pu oon s gd nd dos no nodu sn no h du. Ths sn-f ondon s ssfd f: y y (-4) Sh sn os bou whn h quny bo s no o, nd n b gdd physy s h don fo gd oon. Th sh sn ssod wh oons bou h -s s dfnd s: y y y y No gn h hs quny s o fo pu nfns oons, bu no o n gn fo fn oons. Indny, h ospondng oponn of h oon s ppod by: ω y ω y sn y y [ ] (-4) (-4) W n opu h oon of nghbong pons by obnng h sn nd oon: j j ω j (-43) So f w h dfnd h oponns of h sn nso:, y, nd y. Oh oponns of hs nso sy obnd spy by usng h ppop nds. Th song fos whh s n spons o sn opud fo sss nso ( S j ). Th sss nso onns h fo p un of ond whh woud su f s bok w u ou of h sod ( d [ d j dk ] n ) bu psd s shp. Fo p, pos u of S ns h h upp suf (d) woud h pos fo n h - don wh h ow suf woud h ng fo n h -don. If h ssss qu boh sufs hn h s no n fo. nd whn h bok s nsd h sod h suf fos nd by fos on h djonng sufs, whh dff ony by h oppos onon of. How, f h sss s non-unfo hn n fo p un ou s gn by h d of sss ong h don of on: 5

26 F S j j j Ths psson s d n whn h bok s pd bk no h sod, sn s fo on h ou nd no on h suf. ssung n on bwn sss nd sn yds: S j (-44) Cjkk k, (-45) Th offns C jk d h nso of sy. In n soop sod h n b no don dpndn n h sy offns, so ny -dpndn offns us b ss. Th fs s d h sh oduus (µ) nd s onnony upd by wo o yd h sy offn: S j ~ µ j Th sond offn s o opsson. Copsson s psnd by Th sung sss s popoon o h opsson nd qu n dons: S ~ λ j kk k δ j u wh h Konk d s dfnd s δ f j nd δ ohws. Th gn psson fo sss n n soop sod s hfo: S µ kk δ j j j λ k j j kk k Th psn of sss dos no nssy py song fo. f, w n sh ubb bnd nd hod s so h h n fo on ny n of h bnd s o. Rsong fos s whn h sss s non-unfo: F S j j j Puggng n h pssons fo j yds: (-46) (-47) (-48) (-49) F µ O: F µ j j j λ [ µ λ] [ ] [ ] Mupon by fd un os yds h o fo: (-5) (-5) 6

27 F µ λ [ µ ] [ ] W oud p h fs usng h o dny: [ ] [ ] o obn: [ µ λ] [ ] [ ] (-5) (-53) F µ (-54) No h n hs fo h fos sp no whh dpnds on dgn ( ) nd whh dpnds on oon ( ( ) ). Ths fo nfos h phys npon of sh sn s h don fo gd oon. Sng h song fo qu o h n on yds h quon fo dspn n n s sod: ρ!! µ [! ]! [ λ] [ ] µ [ ] Th do podu of ngu ds y b wn s u ([ ] y ϕ,.): (-55) ρ!! µ [! ]! [ λ] [ ] µ [ ] Th onon nd oon s usuy (hough no oy!) gnod: ρ µ [ λ] [ ] µ [ ] W w so ng onon nd oon fo now, bu w dsuss h. (-56) (-57).4.. Sss-ngy nso Th pon ngy dnsy s (Mos & Fshbh p.3): U S j λ j j [ ] µ y λ k µ y kk y j y [ µ ] j y y Th Lgngn fo onnon s ws s gn by: (-58) 7

28 8 [ ] k kk j j j j j S U K L λ µ ρ ρ (-59) Epy, hs s: [ ] y y y L y y y µ µ λ ρ (-6) Th non onu psns h onu dnsy of h du: ( ) L ρ p (-6) Th sss-ngy nso (o onu-ngy nso) s dfnd s: ( ) j j j L L W δ (-6) Th oponns y b wn s: W j H I I I 3 P W W W 3 P W W W 3 P 3 W 3 W 3 W 33 " # $ $ $ $ $ % & ' ' ' ' ' Th Honn psns h o ngy: H W L ( ) L K U (-63) Th w onu dnsy s gn by: P W L ( ) ρ (-64) Th nus sgn s nssy (bu sos gnod!) bus w ong n h pos - don hs d oppos o h sgn of h sp d. No h h don of non onu s dnd ony fo h d of dspn whs h w onu so nuds sp ds. Fo sh ws h w onu s ppndu o h don of du oon. Th w nnsy (I) so nuds po nd sp ds, nd psns fow of ngy:

29 I W L (-65) ( ) L Th Eu-Lgng quon s: L (! ) ( ) whh yds: " ρ % $ ' δ # & δ (boundys), λ [ ] " µ % $ ' " % y $ ' " % / 6 4. $ ' # & # y & # & 4 3 7, µ " y % y " $ ' # & % " $ ' y # & 4 % / 4. 4 $ ' # y & 84, " λ [ ] µ ˆ $ % ' ŷ " % $ ' y ẑ " % / 6 4. $ ' # & # y & # & 4 " $ ˆ y ŷ %" ' # & y 4, % y " $ ' $ ˆ # & ẑ %" ' # & %/ 4 3. $ ' 7 4 # & µ " $ ŷ ẑ %" ' y # y & % 4 4. $ ' # y & 8 4 Ths quon n b spfd by ngng s: " ρ % $ ' # & 3λ 3 3 3µ - 4, λ [ ] µ - ˆ $ ( ) ˆ $ y " % ' ŷ " % " $ ' y ẑ % $ ',- # & # y & # & " # y [ ] µ [ ] µ,( ). { /} Usng h o on: [ ] [ ] yds h n fo: % " ' ŷ$ & #. / y y % " ' ẑ$ & # y %. 3 y ' 3 &/ 7 (-66) (-67) (-68) (-69) 9

30 ρ {[ λ µ ] [ ] µ [ ] } If w onsd ony sh ws: ρ { µ [ ] } I s nsng o no h hs quon y so b dd fo h Lgngn: L ρ µ [( ) ] (-7) (-7) (-7) s sp p, onsd nss pn ws popgng n h -don wh dspn n h y-don. Th Lgngn s: L ρ y ( ) µ # ( y) $% Th Honn psns h o ngy: H W ρ y ( ) µ " ( y) #$ & '( (-73) Th w onu dnsy s gn by: P W ( y )ρ y % &' (-74) ( ) (-75) No h w wh oppos sp nd po ds hs onu n h pos -don. Th w nnsy (I) s: I W ( y )µ y ( ) w P (-76) wh w µ ρ s h w spd. Th Lgngn fo sh ws s ssny h fo usd by MCugh n 837 o dsb ss gh ws. Epy sp ws odd s oony-s sod (d h h). h, ws hough o h dnsy of h h, so non of h opsson ngy ws hough nssy o pn oupng o ongudn ws h nf. L, oussnsq poposd h h pops of h w ndpndn of h psn of, hby owng h h o b gdd s n odny s sod. Ths onuds h onnon nyss of sh ws n n s sod. N w w d n psson fo ngu onu dnsy Spn ngu onu dnsy R h onnon Lgngn fo sh ws: 3

31 L ρ µ [ ] ( ) ( ) (-77) Th pon ngy ssod wh sh ws n b npd s popoon o h squ of h o oon ng: [( ) ] Θ U µ µ (-78) Ths popy suggss h sh ws y b dsbd ny by oon bs. Consd oy gd oon wh oy w w, wh w s h o ngu oy. Ths psson dpnds on h non-o quny. How, h on bwn oy nd ngu oy n b wn n o fo s d d w. Fo p, f w s n h -don hn d w d nd d w dy. Thfo w y ( ) y d d d dy. Hn h dffn quon ospondng o w s w ( ). W ds s o spn ngu onu dnsy whos u s popoon o n onu. sd on h quon L p, w woud p h onshp o b ρu ( ) s. How, h ngu onu dnsy us b h s sgn s oy n od o h pos kn ngy dnsy ( )w s. I us so f o o nfny n od o h fn o ngu onu. Ths ondons qu n ngu onu dnsy h s of oon nd dsng wh nsng dus. Ths qus ds ~ d ρu, o: ρu ( ) s (-79) Ths sgn s oppos o h pon bsd on L p, bu y b undsood n gh of h f h onnon ngu onu dnsy s o h s nd nss ouwd, whs ou spn ngu onu dnsy ss o nfny nd buds up nwd. Sy, h d of hs onshp ps h fo nd oqu dnss d by: f ( ) τ (-8).4.4. Rgd oon Consd gdy ong ynd of dus R. onssn s of bs s: s ẑρw " # R $ % o R, hn o u ρ s w ˆϕ o R, hn o (-8) w u ẑ "# w w Rδ ( R) / $ % o R, hn o No h f w hd dfnd s ρu hn woud h -dpndn s. Sn w usng ynd oodns: 3

32 w ϕ Th oy s dfnd so h h oy dops o o R: ϕ d!! w w Rδ (! R) / #$ % ( * & ) * w fo < R fo > R Th o ngu onu p un hgh s: S sd 3 π dρw # $ R % & # % πρw ' R 4 R4 ( πρw R4 MR $ & w Iw wh I s h on of n p un hgh. Th kn ngy n s of ngu onu dnsy s: 3 3 K ρu d [ s] [ s] d 8ρ,* -.* (-8) (-83) (-84) W n ng by ps o pss kn ngy n s of oon bs, ssung h ds o nfny: K [ s] [ s]d 3 s j s j s j j s 8ρ ' ( ) * d 3 '( s j s j s j j s ) * d 3 ' ( s s s s 8ρ 8ρ 8ρ s { [ s] }d 3 w sd 3 [ ] ) * d 3 No h hs su qus w ( 4ρ ) s, onfng ou ho of sgn n ng oy u o ngu onu dnsy s. Fo h gdy ong ynd h o kn ngy s: K w sd 3 π dw ρw $ % R & ' $ R 4 πρw & ( R4 % 4 ' ) πρw R 4 MR 4 4 w Iw (-85) (-86) Th d funon n oy dos no onbu bus h ngu onu dnsy s o h. No h gn pon, kn ngy dnsy pssd n s of oy s no qu o h kn ngy dnsy pssd n s of oy. In hs sns ny hoy of spn ngu onu dnsy s nono. Spn ngu onu shoud b gdd s pon h y b usd o dn h o oy. 3

33 .4.5. Roon ws Th s sh fo quon: ( ρu) µ [ ( ) ] µ (-87) bos: ( s) µ W n w h w quon s: { s τ} ( ) 4µ Θ (-88) (-89) wh h oqu dnsy τ 4µΘ s popoon o h o oon. Now dfn b Q whos Lpn s qu o h oon ng (o onon): Θ 4ρ Q (-9) W ssu h h du hs n spons o oon, so h h oqu s: τ 4µΘ Q (-9) wh µ ρ. Thfo, fo nfnsy s oon (ngng ononu Q! ), h b s Q! psns ngu onu dnsy nd w h: s Q u ρ s ρ! Q τ Q w u 4ρ & ' Q! ( ) (-9) ρ Q W so h fo nfns dspns: Θ 4ρ [ Q ] 4ρ Q Equng h of hng of nns ngu onu dnsy o h oqu dnsy yds w quon fo Q: (-93) 33

34 Q Q (-94) Fo nfns oons, h u of hs quon s h f sh s fo quon (wh ): % Q ( & Q) ' * (-95) Ths quon s o gny d hn h pous on. Fo p, onsn gd oon dos no ssfy (-94). How, (-94) s suffn fo dsbng oon ws. Fo nfns dspns, h Lgngn dnsy n ngu bs s: L!! s µ (-96) Ths s h usu dffn bwn kn nd pon ngy. In hs psson w us gd s s funon of!. Thfo h onjug onu o s h ngu onu dnsy: p Θ j δl δ Θ! δ j δ Θ! j Θ sd! 3 δ 8ρ δ Θ! j [ s] [ s]d 3 8ρ [ s δ ] δ Θ! [ s]d 3 δ s Θd j δ Θ!! 3 sδ 3 ( )d 3 j wh 3( ) δ s h h dnson D d funon. Th Eu-Lgng quon s: δl δ (! ) δ ( ) δl whh yds s bo: s( ) (-97) (-98) s p 4µ p (-99) s w sw bo, h fo quon osponds o h u of hs quon. hough no opy gn, hs quon s d fo w souons. Now w nd o nud h ffs of fn pud. Fn oy ndus onon, nd fn oons n su n nsnnous oons whh no p o h ngu onu. ddng hs ffs yds: Q Q u Q! w Q! (-) If w ssu u Q! w Q! M Q hn w obn nss Kn-Godon quon: Q Q M Q (-) 34

35 Ths yds h s ngy-onu on bwn gnus: E p M (-) En f M, h onon nd oon s y onbu sgnfny o h phys dspon of h w. W w dsuss oon ws fuh n Chp Eogns Th pnn of onnon physs ws h dopn of op hoy of ogns n 865 by Js Ck Mw [89]. I ws by hn w undsood h objs wh hg woud f h hgs w oppos nd p f h hgs w h s sgn. d hgs sud n h sson of ogn ws whh d h s spd s gh ws. Hn ws oy ddud h gh s fo of ogn don. In odn noon Mw's quons n b wn s: E 4πρ E E 4π J (-3) wh E s h fd, s h gn fd, ρ s h hg dnsy, nd J s h un dnsy. Th fo d by h fds on p wh hg q nd ong wh oy s gn by h Lon fo w: F q E (-4) Fo ong ps h un dnsy s J ρ. Th fds E,, ρ, nd J n b dd E Φ 4πρ Φ 4π J Φ [ ] fo pon fds nd Φ s foows: (-5) 35

36 36 No h h dnsy nd un ssfy onnuy quon: J ρ (-6) Th pons n un n b dd fo sup-pon G: (-7) Ths s of pons s d h Lon gug. ny w oud pk ou on don ŝ fo dgn-f fd whh obys w quon: (-8) Ths s of pons s d h Couob gug. Sn ons (nd poons) do h nsoopy du o spn, hs dfnon of pons s no s ond s gh pp. Consd h s of ogn fds n uu wh ρ nd J boh o. If w k h u of h hd of Mw's quons nd obn wh h d of h fouh quon w obn: [ ] G G G G G G Φ Φ Φ 4 4 J π πρ G G G G Φ Φ Φ 4 4 G G G s s s s s s J π πρ

37 E (-9) Eh of hs quons s hoognous o w quon. In uu, boh E nd h o dgn, so hs quons h h s fo s h onnon quon fo sh ws W Engy Dnsy Th of wok pfod by ogn fds nng wh hgd ps n ou V s: W d V 3 E J E y subsung fo J nd usng so o dns w n obn (s Jkson p.36): (-) W 3 d 4π V [ E ] ds E 4π S (-) Cy h fs on h gh-hnd sd psns h ngy ssod wh h fds n h ou wh h sond psns h fu of ngy hough h suf of h ou. Thfo h ngy ssos wh n ogn fd n ou V s: U d V 8π 3 [ E ] Ths quon n b wn n s of pons s: U d 8π V Φ Hon Rpsnon of Ws Th foowng dsusson s bsd on Jkson [975 pp. 99-3]. Th pn w souon of h on-dnson w quon fo s s: (-) (-3) ( kω ) k, ω (-4) 37

38 oh k nd ω nno b gdd s ndpndn bs sn h w spd s gn by ω k. Usng ω ω( k), gn souon n b fod by ngng o possb us of k: [ ] kω ( ) ( ) ( k, dk k ) π (-5) Th noon fo s by bu onfos o sndd Fou nyss. Fo hon ws wh h ponn ( p ( k ω) ), h ds nsfo s ω nd k. So h ngy dnsy of ogn ws hs h dpndn: ω U ~ k Φ Α k Α Sn ωk w h ω U ~ k kˆ Φ Α kˆ Α ~ kˆ Φ Α kˆ Α. (-6) In h s of sh ws, h ngy dnsy s popoon o h sss s h sh ( du F d ). W won' d hs py, bu o show h s sonb onsd h foowng popoons: Sj u U ~ f d ~ du ~ dsj ~ dsj j ~ S nd sn S j, j j j, j j, j u ~ j ~ w onud h hon ws h U ~ u ~ k u ~ ω u j j j (-7) (-8) (-9) So fo boh ogn ws nd sh ws h ngy dnsy s popoon o h squ of h fquny (o w nub). Of ous, f on ks h d of h w pud o b h u pud (.g. usng E nd nsd of Φ nd ), hn h ngy dnsy bos ndpndn of ω nd k. How, h s no onnon w fo whh h ngy dnsy oud b wn so h s popoon o n odd pow of ω o k Spon of h W Equon In ny phys pobs souons of h w quon n b found by h hod of spon of bs. Fo p, ws nsd ngu nosu gh b ssud o h h fo: (, y,, ) ( ) ( y) C( ) D( ) Th w quon s hn: 38

39 39 D C y CD y CD D C (-) Dson by CD hn yds: C y C y D D (-) Eh n hs quon s funon of ony on b. Thfo h us b onsn, nd h su of h onsns s o. Fo p: y y k k k k C C k y k D D ω ω (-) ny funon wh k ω nd of h fo: ( ) ( ) k ± y ω,,, (-3) s y souon. Th gn souon s su of hs bss funons. Th w quon s so spb n sph oodns (h foowng s bsd on uko [968]). ssung souons of h fo ()()C()D() w h: sn sn sn C C D D (-4) Ths yds h foowng sp quons:

40 4 sn sn sn C C D D ω λ λ ω (-5) No h h po souon () dpnds on h u of h uh spon onsn, nd h d souon () dpnds on h us of h po ( ) λ nd po ( ) ω spon onsns. Th po souons h h fo: ( ) D D ω ± (-6) Th uh souons h h s fo: ( ) C C ± (-7) Du o h pod nu of h b, h u of s qund. If s n ng hn C() s sng-ud (C(π)C()). Hf-ng us of ow fo doub-ud funons (C(π)C()). No h doub ud funons C() sng-ud fo C(). N w onsd h po quon. Mupon by sn yds: sn sn sn λ (-8) Lng os nd [ ] sn d d d, w obn h ssod Lgnd Equon: [ ] ( ) ( ) [ ] ( ) b b b λ (-9) wh ( ) ( ) ( ) b. No h ± sngu pons. W qu h h souons b fn hs pons. Lng ( ) [ ] ( ) u b s yds h nd quon: [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) 4 u u u u s u s u s u s λ (-3) ± w h:

41 4 4 s (-3) whh ps h s±/. Sn ou pupos h s o k h souons fn, w hoos h pos u nd ssu >. Th quon fo u() s now: [ ] [ ] [ ] u u u λ (-3) W w u() s Fobnus ss: ( ) n n n u (-33) whh yds h quon: [ ] [ ] [ ] [ ] n n n n n n n n n n n n n n n n n λ (-34) Ts fo h pow of us dd o o. Ths ds o h un on: [ ] [ ] [ ][ ] n n n n n n n λ (-35) No h s n h un ons bos n n. Fo < h pow ss ongs sn n s n. How, h ss us n n od o h fn souon. Ths ony hppns fo sp us of h spon onsn λ : [ ] [ ] [ ][ ] n n n n n λ (-36) Whn s n ng s usoy o w h spon onsn s [ ] λ wh. Whn h spon onsn s of hs fo, h funons u() poynos, nd h funons b() d h ssod Lgnd Poynos ( ) P : ( ) ( ) ( ) ( ) ( ) ( ) P P! (-37) wh. So uhos o h fo of ( ) n hs dfnon. fu! Th obnd ngu souons ( ) ( ) C d sph hons. In nod fo nd owng fo ng h sph hons : ( ) [ ] ( ) π P Y os 4,!! (-38) fw sps foow (fo Jkson [975] p.99). Fo :

42 (, ) Y Fo : Y Y (, ) 4π 3 os 4π 3 8π (, ) sn Fo : Y Y Y (, ) (, ) os 4π 5 8π (, ) sn Fo 3: 5 sn os 8π (-39) (-4) (-4) Y Y Y Y (, ) (, ) sn [ 5os ] (, ) os 4π 4 4 4π 5 sn π 35 4π 3 os os 3 3 (, ) sn 3 (-4) In pnp on oud h hf-ng us of nd, bu hs no gny bd o b usfu sn h ng-ndd Y s dy fo op s. Th w quon y hfo b wn n s of gnus: [ ] ω,, ω Ths s d h Hho quon..5. Pops of Ws ( ) (-43) 4

43 n n Fgu. Dg of gh y popgng odng o Sn s Lw..5.. W popgon Mny ds pobby f wh Sn's w fo w fon boundy bwn wo d: n sn n sn (-44) wh n s h nd of fon whh s dfnd s h o bwn h uu w spd nd h w spd n h du: n (-45) Th ngs sud fo ppndu, so h w don s wys os o ppndu n h gon wh sow spd (hgh n). In oh wods, h w bnds owd h gon of sow spd. I uns ou h Sn's w, whh pps ony o dsn bounds, s sp s of o gn fouon d F's pnp: d δ (-46) Ths ondon ns h h ph of h w bwn ny wo pons s suh h h ng of ngh ddd by spd s n u (u, nu, o nfon). S hngs n h ph do no hng h u of h ng o fs od. Fo h s of ws popgng bwn pons (, y) nd (, y) nd nsng h boundy bwn gons nd h pon (,), F's pnp bos (s Fgu. nd Jnkns nd Wh p. 4-8): 43

44 d δ d d Th d yds: d δ [ ] ] y [ ] [ ] y δ (-47) [ ] [ ] y ] [ ] [ ] y ] Ths us b u whn δ, pyng: sn δ sn δ (-48) sn sn (-49) Mupon by h onsn yds Sn's w..5.. Dspson nd Goup Voy W h dy sn h souons of h w quon h h fo: ( ) f ( ) f, ± (-5) Th Fou doposon dpnds on (,) hough h phs fo k ± ω. Sng h phs o onsn u yds k ± ω. Dffnon yds h phs oy: p d ω d k (-5) If h phs oy s wh fquny, h du s sd o b dsps. Infn bwn dffn w fquns sps h w no pks whh h g osons wh h phss of dffn fquns gnd nd s osons wh h phss of dffn fquns sgnd. Th spd of hs w pks s d h goup oy, nd n b dd fo sp p onssng of wo fquny oponns (s.g. Chn p. 69-7): [ ] ( ) [ k dk ] [ ω dω], ( ) [ kdk ] [ ωdω], [ ] Th su of h wo oponns s: (-5) 44

45 [ ] [ kdk [ ] [ ωdω] ] ] [ kω ] [ dkdω] [ dkdω ] ( ) ( ) [ k dk ] [ ω dω],, [ ] [ kω ] [ os ( dk dω ) ] Th pdy yng phs of h ponn fo popgs wh h phs oy (ω/k), wh h sowy yng phs of h osn fo popgs h goup oy: g dω dk (-53) (-54).5.3. Infn nd Dffon Ws fo dffn sous, o ws fo sng sou bu foowng dffn phs, w no gny h h s phs h pon wh hy obn. Ths phs dffns su n h phnonon of nfn. Consu nfn ous wh h ws n phs, so h h puds ddd. Dsu nfn ous wh h ws 8 dgs ou of phs, so h h puds subd. Dffon s fo of nfn whh sus fo h w popgng ps s obj o hough s opnng. Dffn ph nghs fo ws popgng ps h obj (o opnng) fo on sd o h oh podu h nfn pn Dopp shf Fquny (ωʹ ) do ong wy fo fd sou (ω): ( ) ω ʹ ω (-55) Fquny (ω) fd do fo dng sou (ωʹ ): ω ωʹ ( ) (-56).5.5. Unny Pnp Now onsd w pk whos nnsy hs Gussn shp wh sndd don : ( ) [, ] ( ) [ ], 4 (-57) 45

46 Th Fou psnon s: ( ) [, d ] k π k π π d k k d 4 k [ ] 4 k [ ] ([ ] k ) wh h s sp os fo opng h squ n h ponn. Ths n b sod usng u h k: d Thfo: (, ) k (, ) k d [ y ] π d π k dy k k y ddy (-58) (-59) (-6) No h h shp of h nnsy pof (pow spu) n h Fou don s Gussn wh op pud nd sndd don k. I uns ou h h Gussn shp ns h podu of h sndd dons, so w n w h ss unny on s: k (-6) wh nd k now psn h sndd dons of by w nnsy pofs n h sp nd Fou dons. Thus f hs unny on s spy h popy of Fou nsfos. Wh ks physy nsng s h f h h ws so ssfy h w quon, n whh s h w nub k s nsy popoon o h wngh. In h dnsons, h o k nds h popgon don nd s hfo d o h w onu. In pu, f h w popgs hough s of wdh Δ hn h unny n h - oponn of h w nub s k [ Δ]. Hn h unny n w popgon don nss s h wdh of h s dss. S unny ons n b dd fo oh onjug bs suh s (ω,). n nsng s s h of ngu bs. Rqung h w pud o b pod n ps h h ngu w nubs qund n ng sps: f ( ) F( ) (-6) 46

47 Th Fou nsfo s: F ( ) f ( ) d (-63) Th ngu dsbuon nno b sp Gussn bus of h pody onsn. Fo uson of h unny pnp fo ngu dsbuon, w w pk dsbuon fo whh p of h Fou nsfo (h osn nsfo) n b pfod nyy. L: f f ( ) [ ] os ( ) s hs dsbuon bos shpy pkd π. Th p of h Fou nsfo s: f π ( ) f ( ) os d f [ ] f π os ʹ [ ] d ʹ f π os ʹ π os os ( ) d (-64) (-65) s h Fou oponns dop off sowy. Hn h spd, o unny, of -us nss s h spd of ngs dss..6. Suy W h wd h bs onps of ss hns, nudng h phys nd h pops of ws. Now w dy o ppy hs knowdg o hos of. Rfns uko, E. Mh Physs, (ddson-wsy: Rdng, Msshuss, 968) pp.37ff. Fynn, R. P., Lghon, R.. nd Snds, M. Th Fynn Lus on Physs, Vo. II (ddson-wsy: Rdng, M, 963), Chp 39. Godsn, H. Css Mhns, (ddson-wsy: Rdng, Msshuss, 98) pp Jkson, J. D. Css Eodyns, Sond Edon, (John Wy & Sons: Nw Yok, 975) L, T.D. nd Yng, C.N. 956 Quson of Py Conson n Wk Inons. Phys. R. 4, Mw J. Ts on Ey nd Mgns, Thd don, 89, Ofod Unsy Pss, os. (pnd by Do, Nw Yok, 954). Mos, P. M., nd Fshbh, H. Mhods of Tho Physs (MGw-H ook Co., Nw Yok, 953), pp. 4ff. Noh, E. 98 Nhhn Gs. Wssnshf. Gongn, p. 35. Fgus 47

48 Th foowng fgus bd o b f of opygh son, nd w obnd fo h sous sd. Oh fgus h ogn woks o d n h fgu pon. Fgu. so ( C). Sou: hp:// Fgu. so s Uns. Th Chsn son osos, ngng fo P pn's Cosogph, 54. Fgu.3 Poy (D 7-45). Sou: hp://byss.uogon.du/~js/gossy/poy.h Fgu.5 Nous Copnus ( ). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/copnus.h Fgu.6 Johnns Kp (57-63). Sou: hp://www-hsoy.s.s-nd..uk/ogphs/kp.h Fgu.7 Is Nwon (643-77). Sou: hp://www-hsoy.s.s-ndws..uk/hsoy/pdspy/nwon.h Fgu.8 b Ensn ( ). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/ensn.h Fgu. Ey Noh (88-935). Sou: www-gp.ds.s-nd..uk/~hsoy/ogphs/noh_ey.h 48

49 Chp. M Ws nd Sp Ry Ignon s pfb o o; nd h s ss o fo h uh who bs nohng, hn h who bs wh s wong. Thos Jffson, Nos on Vgn (Quy VI).. Inoduon Sn s h bf n h gnon of ps. Rhd Fynn [969] Ey ps w hoy of gh psud h gh ws popg hough uns du n h s nn s sound ws hough. Ths du ws dubbd h unfous h. Chsn Huygns [69] [Fgu.] pubshd n pnon of fon nd fon bsd on h pnp h h suf of w-fon n b gdd s sou of sondy ws. Huygns so dsod h bfngn yss n sp gh ys no wo dsn oponns (poons). Is Nwon, ong ohs, doubd h w hypohss n p bus oud no pn hs popy of poon. Nonhss Nwon dd p sy bwn oo nd h bons whh podu sound ons. Fgu. Chsn Huygns (69 695) In 675 Of Ro bud ons n h obsd ob pods of Jup s oons o b gh popgon dsn bwn Jup nd Eh. Ths npon, obnd 49

50 wh Gonn Dono Cssn s p dnon of npny dsns n 67, dnd h spd of gh o b bou. 8 /s (n suns pu h u /s). us gh, unk ps, popgs hs spd, Thos Young [Fgu.] ws onnd h gh onsss of ws. H donsd hs w nu by podung nfn fngs fo gh pssng hough wo now ss. Thn n 87 h pnd poon by poposng h gh ws onss of nss bons suh s ou n n s sod. ugusn Fsn [Fgu.3] dopd Young s d of nss bons nd dopd hghy sussfu hoy whh pnd dffon nd nfn n ddon o fon nd fon. H supposd h h o ss dsoon n h s nn s s sod whos dnsy s popoon o h squ of h f nd. Fgu. Thos Young (773 89) Fgu.3 ugusn Fsn (788 87) onpu pob wh sod h s h quson of how odny n os nd o fy hough. Gog Gb Soks [Fgu.4] poposd h h h ws nogous o hghy sous fud o w: s fo pd bons bu fud-k wh sp o sow-ong. 5

51 Fgu.4 Gog Gb Soks (89 93) Fgu.5 Js MCugh (89 847) o d dffuy wh h sod h od ws h dnsy ons (.g. h nf bwn uu nd du) d o oupng bwn nss nd ongudn ws, phnonon no obsd fo gh ws. Js MCugh [839] [Fgu.5] odd hs pob by poposng oony s h whos pon ngy Φ dpnds ony on oon (ppod by u of dspn ): Φ µ ( ) Th sung w quon s: ρ µ ( ) whh s spy h quon of s sh ws whh w dd n Chp. M ws now psud o h sy of h h h hn s dnsy. Ths od sussfuy ound fo of h known pops of gh. Josph oussnsq [868] [Fgu.6] poposd h h h oud b gdd s n odny d s sod whos phys pops (dnsy nd sy) unhngd by non wh. Th op pops of w hus ny du o h nn n whh ns wh h h. Wh hs ppoh ny ss op phnonon oud b onssny odd spy by fndng h ppop non. 5

52 Fgu.6 Josph oussnsq (84-99) Fgu.7 W Thoson (Lod Kn, 84 97) In sp of hs susss, snss onnud o pusu hos of fud h hough whh sod oud popg. W Thoson (Lod Kn) [Fgu.7] pd o od h h s o spong : fud fu of s-s os wh ny ndo onon. H gud h hs sys oud suppo nss ws nogous o hos n n s sod. Js Ck Mw [86,b, 86,b] [Fgu.8] odd h h s nwok of ong s s nspsd wh ong sph ps n od o d h quons of y nd gns. Hs sun quons fo gh ws qun o hos of MCugh. Fgu.8 Js Ck Mw, Fgu.9 b Mhson,

53 Sn ws psud o o hough h h s ps ong hough fud, ny ps w d o dy su h oon bwn h h nd h h. Th os nob of hs ws n pn fs pod by b Mhson [Fgu.9] n 88 nd subsquny pod [Mhson nd Moy 887]. Infn fngs w fod by obnng wo bs of gh whh popgd ong ppndu phs. If h h os wh sp o h h hn gh popgng bk nd foh ong ph gnd wh h h s oon shoud h sghy sow g oy hn gh popgng ppndu o h h s oon. Thfo h fngs shoud shf f h ppus s od so h gn b s ny p nd ppndu o h don of h h s oon. How, no suh ff ws obsd n hs o oh h-df pns. O Lodg [893] donsd h h oy of gh s no noby ffd by nby ong, ndng h h s no dggd ong wh. Gog FGd poposd h h nby o su oon o h h oud b pnd f ons ong h don of oon hough h h [Lodg 89]. Josph Lo [9] nod h n ddon o h shonng of ngh, ong oks shoud so un sow. Hndk Lon [94] [Fgu.] obnd ngh onon nd don o obn h op oodn nsfoons. Hn Poné [94] [Fgu.] g h n Pnp of Ry o h don h bsou oon s undb. H so ddud h n nss wh oy nd h no oy n d h spd of gh. b Ensn [95] foud y wh h o pos sson h h spd of gh s uns onsn ndpndn of obs oon. Fgu. Hndk Lon (853 98) Fgu. Jus Hn Pon (854 9) On dffuy wh h ss hoy of gh ws k of suss n dsbng don fo y fd pu ( bk body ). M Pnk [9] [Fgu.] dd h o fou fo bkbody don by supposng gh o b d by bos whos ngy ε nh ν n! ω s n ng up n of onsn h upd by h fquny ν (o up of! h π s h ngu fquny ω πν ). b Ensn [95b] usd h d h don onsss of ds qun n od o pn h phoo- ff, n 53

54 whh h fquny of gh us d n hshod n od o b ons fo. Ns oh [93] [Fgu.3] usd qunon of ngu onu nd ngy o d ngy s nd sp fquns of h hydogn o. Fgu. M Pnk ( ) Fgu.3 Ns oh (885-96) Rognng h qunon s ofn ssod wh ws nd bons, Lous Vo d og [94] [Fgu.4] poposd n hs doo hss h ons h w-k h wh ngy popoon o fquny ε! ω nd onu popoon o w o p! k. oh s qunon of ngu onu s hn qun o h qun h sb on obs onn n ng nub of on wnghs. W Esss [95] suggsd h hs w popy of ons gh pn nd n n h ngu dsbuon of ons sd fo pnu p n pns pod by Cnon Dsson nd Chs Kunsn. Th w nu of ons ws onfd n 97 whn on dffon by yss ws y donsd n pns by Dsson nd Ls G [97] [Fgu.5], nd ndpndny by Gog Thoson nd. Rd [97]. S hp://onn..og/physsb/onn/phypb/ssonnos/dunu/dsson_g.sp fo opson of dffon usng -ys nd ons. 54

55 Fgu.4 Lous Vo d og, Fgu.5 Cnon Dsson nd Ls G n 97 Th dsoy of h w-k popgon of uy sos h hso d of how n o fy hough sod h. In ddon, h s du sf nd no hng h nf bwn uu nd, hus pnng h k of oupng o ongudn ws. Th w nu of so ds dy o h Pnp of Ry whou ny odfon of h ss Gn w of Eudn sp nd bsou, s w b shown bow. How, hn odng of fundn phys posss ws no ong n ogu h of hs dsoy. M ws w no gdd s odny ss ws... Msuns wh ws "If w o h sus n bfo opshd, w us poy hods n bfo pd." Fns on Th fs p of h foowng dsusson osy foows Ensn s pnon of sp y bu wh dffn on [Ensn 956]. L us onsd h nsfoons bwn oodns of y ong obss who su dsns by ng how ong ks fo ws o popg bk nd foh bwn wo pons. Th dfnng quon woud b: ( ds ) ( Δ) ( Δy) ( Δ) ( ) 3 Δ p (-) wh ds s h sp dsn bwn wo pons fd, s n by onsn, nd s h woud k o popg w fo on pon o h oh f hy nd p sony. Wh hs dfnon of dsn, h onsn s spy sng fo whh s h uns of dsn o h uns of. Ths dsn osponds o h usu dfnon of dsn f s h spd of h w usd n h sun. 55

56 Now suppos w onsd popgon of w fo pon P o pon P. In fn f n whh h pons sony, Eq. - hods. n obs n dffn n fn f usng h s dfnon of dsn woud h: 3 ( Δʹ ) ʹ p Th quny ( ) ( ) ( ) p (-) Δ Δy Δ s hfo o fo boh obss. owng fo n by offs, h nn of hs quny fo dffn obss s psy h ondon whh Lon usd o d h s nsfoons. Th quny ( y ) s sos d h spon. Fo p, suppos subn ngo s usng son boh o su nd o d fsh n h w. Th sos us sp son oks whh su by yng sound w puss bk nd foh oss fd dsn n h w ppndu o h don of oon. Eh y of w nssson, fon, nd don h ogn s of nssson onsus k of h ok. In hs nyss w w ng ny ffs of dspn of w by ong subns. n nd psnon of hs nyss y b found hp:// Son ok Sony sub Rsng fsh ʹ γ Mong sub ʹ ʹ ʹ ( ) ( ) γ Fgu.6 T Don: Th ok on Oʹ ks sow hn h ok on O by h fo s bus ws fh bwn nssson nd don. oh O nd Oʹ su h s nub of ok ys fo w o popg fo h own sub o h fsh nd bk. Hn hy g on dsns ppndu o h don of oon. 56

57 ... T don If boh h sub nd h fsh s n h w, sound w fd fo h fsh dsn woud un f s, wh s s h sound spd. Th dsn o h fsh s hfo kn o b s. Suppos now h h sub nd fsh ong ogh n h w wh oon spd ppndu o h ogn don of w popgon (Fgu.6). Th ph of h son ok ws fos wo sds of ng fo h y. s ng s fod by h w popgon o h fsh nd bk. Thfo h nub of ok ks whh ou dung w popgon o h fsh nd bk s ndpndn of spd. If h ngo dosn h sh s ong, sh woud ssu h s on bwn dsn nd : ʹ s ʹ. Th ngo of sond subn sng s n h w woud obs h w popg o dsn: d s Subsung s ʹ nd song fo ʹ yds: (-3) ʹ s (-4) Ths quon y psss h f h h ok on h ong subn ks o sowy h h sony ok bus h ws h fh o bwn ks. Hn h () sud by h sony obs s ong hn h (ʹ ) sud by h ong obs. Ths phnonon s fd o s don. I s obous h f h unpd obs s uy sony wh sp o h w, hn h ong ok dos n f k o sowy. Ths s no y n uson. Wh s nsng s h h w suns pfod by hs subns nsuffn o dn whh sub s uy ong wh sp o h w. Thfo h ong sub woud np h sony ok s unnng sowy, nd n hs s h ff s n uson. Ths pon w b dsussd bow n onnon wh Dopp shfs. Sn h sony ngo ss h fsh (nd fs sub) o dsn wh h w s popgng, h bo quon n b wn s: ʹ ( ) s s s s (-5) whh s h Lon nsfoon of bwn wo obss, wh h pd obs ong n h -don wh oy wh sp o h unpd obs.... Lngh onon Sn boh obss su h s dsn ppndu o h oon us b spy: yʹ y ʹ ʹ, h nsfoon of oodns 57

58 Son ok Sony sub Rsng fsh γ Swng R spd ( - ) fsh ʹ Mong sub R spd ( ) ʹ ( ) ( ) γ ʹ ʹ γ Fgu.7: Lngh Conon: Th u w popgon fo h o-ong sub nd fsh s. Sn h ong ok uns ong hn fo h sony sub nd fsh by h fo ( ) s sow, h pd popgon s ong ony by h fo. Hn h sony s sub obss sho ngh hn h ong sub. Now suppos h h fs sub nd fsh ong o h sond sub p o h don of w popgon [Fgu.7]. s sn by h sony sub, h fquny of h son ok on h fs sub s sow odng o Eq. -4 sn h sud ʹ s popoon o h ong ok fquny ωʹ s h bsou : ωʹ ω s (-6) Th bsou dsn bwn h fsh nd sub ns onsn. How h spd bwn h ougong w nd h g fsh s (-) whs h spd bwn h sub nd h nong w s (). Thfo h popgon s: ( ) ( ) s s ( ) s s (-7) Of ous h ong sub s uss h on ʹ s ʹ. Subsung h po on s ʹ yds h on bwn nghs: ʹ s s s (-8) Th sony obs sus sho ngh hn h ong obs. Ths phnonon s known s ngh onon. In hs s h ong obs sun s fy ong du o h f h h u sound oy o h obs s no h s fo h ougong nd nong dons. Sn h w popgs fo ong n h don of sow oon, h ff s n ppn ns n ngh o sony 58

59 obs. gn, how, s pon o h h w suns on do no dn whh obs s ong. s nod pousy, h ogn of h ong f osponds o n h sony f. Thfo h oodn nsfoon s obnd by ʹ ʹ nd : ʹ s whh s h Lon nsfoon of poson ong h don of oon. I s usoy o us h dfnons: γ s ( ) usfu dny s: s ( ) γ γ Usng h bo pssons, h Lon nsfoons bo: ʹ γ γ s ʹ γ γ yʹ y ʹ s s wh subsps usd o phs h w dsussng sound ws. Th ns nsfoons y hng h sgn of (o ): γ ʹ γʹ s γʹ γ ʹ y yʹ ʹ s s (-9) (-) (-) (-) (-3) Thus w s how Lon nsfoons n b obnd by usng son o ny oh yp of w o su nd dsn. Lon nn s no popy of nd sp p s. Rh sus fo h hods usd o su nd dsn. If h bo-nond sos w o ndous o sh h d nd so odk, hy gh onud f fw dnks h bsou nd sp n ong undw fn fs d by Lon nsfoons usng h spd of sound n w. f sobng up, how, hy woud h son s no h ony wy o su nd dsn nd h h suns no dn of ny non-ss pops of undw sp Lngh nd sndds Th son ok gh s k n odd so of ok, bu onsd h sndd dfnon of sond, whh s 9,9,63,77 pods of h don ospondng o h nson bwn h wo hypfn s of h gound s of h su 33 o [Tyo 995]. If w gd 59

60 h su o s knd of op y whh sons h psbd fquny, hn hs s qu s o ou son ok. Consd so h h sndd dfnon of h s h ngh of h ph d by gh n uu dung n of / /99,79,458 of sond [Tyo 995]. So w do n f qu ngh wh w popgon jus s ou hypoh sos do, nd h quny s nohng o hn un onson fo...4. Dopp shf Thus f w h shown h whn ws usd o su dsn nd, h sp oodns nsfo bwn y ong obss odng o h Lon nsfoons. Tnsfoon of oh dyn bs s sghfowd. Th phs of pn w s gn by: k ω (-4) Ths quny s ndpndn of obs oon. Thfo: k ʹ ʹ ωʹ ʹ k ω Fo oon ong h -s w n pug n h ns nsfoons fo nd o obn: kʹ ʹ ωʹ ʹ k kʹ yʹ k y kʹ ʹ k y y ( γʹ γ ʹ ) ω( γʹ γ ʹ ) s Th offns of ʹ us b qu on boh sds of h quon, nd kws fo h offns of ʹ. Thfo: ωʹ γω γ kʹ kʹ y kʹ kʹ γ k k k y s k γ ω s Lng, h nsfoon fo by don of oy s: ωʹ γ kʹ γ ( ω k) s sk k ( ω ) s (-5) (-6) (-7) Hn h spo-po fquny oponns ( ω,k ) nsfo n h s nn s h oodns (,). Quns whh nsfo odng o hs Lon nsfoons d fou-os. Eh fou-o hs h sp oponns nd po oponn. Oh ps of fou-os (wh sp o gh ws) nud: γ, γ Fou oy ( ) ( E γ, p γ ) Engy, onu ( ρ, J) Eogn hg, un 6

61 No h fo gh ws k ω. Hn h fquny nd w o nsfoons fo oon p o k n b wn s: ωʹ γω kʹ γ k ( ) kʹ k ( ) ω (-8) Th fs of hs quons s h s Dopp shf fou fo gh ws. Th s Dopp shf hs sp npon. Fs, onsd h ss Dopp shfs s shown n Fgu.8 bow. Fgu.8 Css Dopp shfs fo ong (ppohng) sou nd do dff by fo of [ ][ ] γ. Ths fo s no ffd by s of h oy don. Consd sony obs O n ghhous whh puss wh ngu fquny ω. n obs Oʹ os wy fo h ghhous sng n spdbo. s ong do, Oʹ s ssy Dopp-shfd fquny of ω( ). How, Oʹ s ok s unnng sow by h fo γ bus h bo s ong. Hn Oʹ ps h ndn w fquny o b hgh by h fo γ so h ωʹ γω( ). Th sony obs O woud g wh hs o dspon of ns. No h obs O n su h spd of obs Oʹ by sung h of fgh of d puss whh f off of Oʹ nd bk o O. Suss puss spd by nssson n τ T w b d wh dy τ τ τ. n τ R τ T ( ), ydng ( R T ) T 6

62 Rd (R) O Rd ( R ʹ ) τ T O ʹ O R ʹ R τ T ( τ R ) τ R O ʹ O R τ R R ʹ τ T τ R O ʹ Fgu.9 Voy Msun: Rd sgns sn sunousy by O nd Oʹ w so b d sunousy f fon. hough Oʹ s ok ks sowy, h popoony bwn d pus popgon nd o psd s h s s fo O. Thfo boh O nd Oʹ su h s oy. Consy, h obs Oʹ noy bs h h s sony nd h O s ong. Oʹ sus h spd of sson of h ghhous d. Th u popgon of h h pus s h s s sud by O (s Fgu.9 bo). Th f h Oʹ s ok s unnng sowy dus of hs sud s by h fo γ, bu hs dos no ff h popoony bwn h nssson n nd h pon n. Thfo Oʹ ss O d wh spd. Obs Oʹ obss h ghhous gh fuu wh fquny ωʹ γ ( ). Ths fou ouns fo sowng of h ong ok nd Dopp shf h ong (dng). Oʹ psus h dd fquny o b ssy Dopp shfd h sou by fo of ( ). Cong fo hs Dopp shf yds ω ʹ ( ) fo h o-ong sou fquny. Sn Oʹ hnks h O s ok s sow, h oon fo γ s gn nodud o obn h fquny pd h sou. Ths ds o: ω γωʹ ( ) ωʹ whh s of ous h ns fquny nsfoon. No h Oʹ noy bus h Dopp shf o ong sou h hn ong do, sung n n onous fo τ R (-9) 6

63 γ. How, hs sk s y opnsd by h f h Oʹ noy bs h O s ok s unnng sow by h fo γ whn n f s unnng fs by h fo γ. Oʹ skny ups by γ whn h shoud h ddd by g o of ( )( ) o fo h dffn ok s (n onous fo of γ ). Th onous fos of γ nd γ n nd Oʹ oy ddus h fquny ω fo h sony sou O. Ths non of os nds possb h dnon of oon o h du whh s h w. I s h u of sp y. If h oon s no ong h n of spon hn h Dopp shfs dpndn on ng. Nonhss, on n o fo hs ngu dpndn o dn h hd-on Dopp shf onssn wh h nyss bo..3. M ws nd gh I s b o gh on s nd hn o us h dknss. ( 與其詛咒黑暗, 不如然起蠟燭 ) Confuus ( 孔夫子 ) On on of h bo dsusson s h sound ws n w oo sp o s s od of. Th son ok hd o b ond ppndu o h don of oon so h s ppn ngh ws ndpndn of oy. noh pob s h sound ws s ws, dsbd by sng nub (.g. pssu) h pon. o nsng du o onsd s n s sod, whh n suppo sh ws whos pud (dspn o oon) n h up oponns. Ws whh nud sgnfn oons spy of ns bus hs ows fo nns, o spn, ngu onu n ddon o h ob ngu onu ssod wh popgon of h w. Th bo sus show h h quons of sp y ppb o wd y of w phnon. Th Lon nsfoons w suns d n dffn fs of fn. I s w-known (nd sy fd) h ny w quon of h fo: M f (-) wh nn s M s nn und Lon nsfoons wh w spd. In oh wods Lon nn s gn popy of ws nd no spf o ogn ws. Now w n poson o pp wh s sp bou gh. Odny w do no su dsns nd s by popgng ws bk nd foh. Insd w us oks nd us. Th ng hng bou oks nd us s h h sung dsn nd suns nsfo wh y h s Lon nsfoons s woud b obnd f h suns hd bn d by popgng gh ws. In oh wods, bhs s f onsss of ws whh popg h spd of gh. Sn n pp o b sony, w us suppos h h ws sohow popg n y phs n h s f. Suh ws oony fd o s soon ws. 63

64 Hsoy, h quons of y w dd fo h obson h bsou oon s undnb. Ensn foud y on h bss h h spd of gh s ndpndn of obs oon. Y now w h sp n posu fo sp y: onsss of ws whh popg h spd of gh. Ths phys pu suggss h nd n- n nnh no phoons nd s bus phoons nd spy dffn pks of h s yp of w. W w s h ou nw hypohss s so onssn wh h D quon fo h on, n whh h oy opo hs gnus of gnud. Mss s ssod wh duon n goup oy whh y b bud o oon of h w popgon don. Wh sp o h-df pns suh s pfod by Mhson nd Moy, s h f ws h h s spd s gh ws hn ny ff of h s popgon hough h uu woud quy ff h gh ws nd h ppus usd o su h. I hs ong bn ognd h Lon nn of s qud o pn h nu su of suh pns. Wh hs no bn gny ognd (hough h nuous pons) s h h w nu of pods h bss fo y nd s ny onssn wh ss noons of bsou sp nd..3.. Soon ws L psn h hs spd of nss ws n n s du. Th quon of ouon of h w pud (,) s: u! w! (-) ssu h h onon nd oon s du o onsn offn of, so h h oponn ssfs: ω ( M ) I s oon o us Fou doposon so h h w quon n b wn s: ( k M ) (-) wh (k,ω) s h Fou nsfo of h w pud (,). Th w goup oy u s gn by: dω k u dk ω ( ω M ) ω k ω k u yds: Song fo ( ) (-3) k u M M u γ u wh w h usd h f dfnon of γ o obn h psson on h gh. Subsuon no h w quon yds: (-4) 64

65 65 M u M γ γ ω (-5) If w dfn! M hn w obn h qunu hn ons: [ ] [ ] [ ] [ ] [ ] [ ] [ ] k u u u u u k!!!! ω γ ω γ (-6).3.. Engy nd onu sp popy of on ws, whh w b dsussd n Chp 3, s h h ngy s popoon o fquny ( ω! E ) nd onu s popoon o h w o ( k p! ). Cssy, h quny " us psn h ngd w pud. W ssu h ws h s popoons, hough phps wh dffn ngd w puds. Usng hs subsuons yds n h bo quons yds h s ons: 4 p E E γ γ u p (-7) Ths s quon, gn h fs wo, y psss h uoogy: ( ) γ u u u (-8) Fgu. Ths s h Pyhgon on fo gh ng wh sds (,, ).

66 Th hyponus, whh osponds o ngy, nds h h dsubn os wh spd. Th oy osponds o onu nd nds popgon n h don of h w o. Th oy osponds o ss nd nds popgon ppndu o h w o (o s ndpndny fo h w o: h Pyhgon on so hods, on g, fo yod oon,.g. u ˆ u os ˆ ( u sn u ) wh u nd u ). Sn h popgon ssod wh ss dos no yd ny n nspo of h dsubn, us b s ppoy pod, nd h sps ssupon s u oon. Th gn popgon of h w woud hn b h o yod (o n bwn). Hsns [99] hs so poposd h oon of ny ps. Mupyng h sd of h bo oy ng by γ yds h ngyonu ons. Fgu. Tngu onshp bwn s ss, onu, nd ngy. If h sony fquny of n ny p s y ssod wh u oon hn w n opu h dus of h oon. Fo ons w h:! R ω (-9) No h hs quny s dffn fo h oh dus ( R! 5.98 ) whh s h ss dus of h on ob n h gound s of h hydogn o. Th o bwn hs wo dsns s d h fn suu onsn: R R!

67 67 Th dfnons of E nd p d dy o h quon of oon p E k ω n h Fou don. In h sp don hs s h ss onshp bwn kn ngy nd onu: p E (-3).3.3. Tnsfoon of oy Th psson fo goup oy n b obnd wh h nsfoon ws fo fquny nd w o o wok ou h nsfoon pops of h oy. Fo oon p o h oy ony h oponn k s ffd: u u u u u u k k k u ʹ ʹ ʹ γ γω ω γ γ ω (-3) Ths s h nsfoon w fo oy p o h don of oon. Fo oon ppndu o h oy h ony hng s o ω: γ γω ω u k k u ʹ ʹ ʹ (-3) Fo n by don of oon, w us u u o obn h nsfoon ws fo oponns of oy p ( u ) nd ppndu ( u ) o h don of oon: [ ] ʹ ʹ k k u u k u u u u k u u u k γ ω ω γ γ γω γ γω ω γ γ ( ) u u u u u u ʹ ʹ γ (-33).3.4. Th wn pdo On supposdy non-nu onsqun of y s h wo wns n hng h g hough oon. If on wn (ThoO) ns sony wh h oh wn

68 (PoOʹ ) ks hgh-spd jouny hough sp, hn h wn who d w un young h h wn who syd ho. o oon nfson of hs phnonon s h hgh-ngy os y ps whh oo o h s spds h ong fs hn ohws dn sow-ong ps. hough h ff of oon on y s os g, h pnon s y qu sp. Consd ok whh ouns h nub of u obs ud by n on w. ny ok d of ws w k popoon. Wh h sony on us u ph, ong on us sp (o yod) ph wh h s bsou spd. Sn h ong on s fh hn h sony on dung h oon y, ong on ok ( ʹ ωʹ τ ) w k o sowy hn sony on ( ωτ ). Fo nson oy of, h spd of uon s: ( ) γ ʹ nd hfo h ong ok ks o sowy ( ʹ ʹ ʹ γ ʹ < ) by h fo: (-34) (-35) Ths s qun hy nd s physy o h don bo of don fo sound ws n w. Hn h ong Po w g ss hn h sony Tho. Sony uon d S π Tnson d T H o Cyod d Fgu. T Don: Mong ws popg fh hn sony ws dung h y. Thfo ong oks k o sowy hn sony oks. d S dsn d n on y of sony w, d T nson dsn. Th dsn fou fo h yod s ony fo n ng nub of ys. π ( π) ( ) γd S W h sd bfo h w suns nno dn bsou oon o h du. Thfo Po shoud nd up young hn Tho n f hy ny ong wh sp o h du. Suppos h h wo wns Po nd Tho ny ong ogh wh oy n h don. sony obs ss Po sow o sop, w fo T, hn o spd o h up wh Tho T T T. In hs s Po s uy gng o pdy hn Tho fs, bu hn gs y sowy wh yng o h up. No h: 68

69 69 ( ) ( ) T T T T h h wns up gn, Tho hs gd by γ T sn hs ok s unnng sow hn sony ok (usng ( ) γ ). u Po hs gd by γ T T ( ) T γ. Th dffn n h gs s hfo: P γ γ T T T o Tho (-36) To sond od n / s, hs dffn s: P T T T T o Tho (-37) wh h nquy ss fo h f h. Mo gny, w n y o n h g dffn wh sp o (fo gn T nd ). Th non ondon s: γ γ d d (-38) whh yds f gb: γ γ (-39) Subsuon of hs psson no h dffn yds: P γ γ T T T o Tho (-4) Sn γ h nquy n b wn s: P γ γ γ T T T T o Tho (-4) sn γ. Hn h wn who os wy nd os bk wys gs ss hn h wn whos oon ws onsn. Ths s sp onsqun of h w nu of.

70 .4. n npons n y gn hngs h fs, bu h n ony undsnd hngs h u, fo f h hngs b fs, h pphnson of h s no undsndng. Is Nwon Th d shoud b wnd h h sp npon of y psnd h s no gny undsood. Sn s npon h dwn of h h nuy, h Pnp of Ry hs bn npd s phys w h hn s puy h onshp bwn sp nd suns. I s bd h go onshps bwn suns uy psn h goy of phys sp. Suh n npon ssus h suns of dsn nd n ppoh pfon. Th fou-dnson sp- h ssfs h pnp of y s usuy fd o s Mnkowsk sp. odng o ou pon of w, Mnkowsk sp s h sp of suns d wh ws popgng n Gn phys sp-. I hs ong bn ognd h opn wh h Pnp of Ry qus ws o b Lon on. How h ons og hs bn gy gnod. Lon on s popy of ws, nd h w nu of ps h Pnp of Ry fo ss Gn sp-. Thus hough bsou oon nno b sud usng gh nd ws, h s no son o psu h bsou oon hs no nns nng. Indd, f noh yp of w oud b sud (.g. gy ws) hn y b possb o dn bsou oon wh sp o h h. Th npon of y s phys popy of sp- s phosoph pfn h s n no wy jusfd by dn. I s ofn phsd h bsou oon nno b dnd. Ths s uy doubfu, sn oon o h os bkgound ow don n b dnd. Mo pony, s possb o dn bsou on. Two obss undgong hng n oy n dn whh of h s ng bus ony h ng obs w pn fo. If h n (onsn oy) obs ss h n d obj hs hngd s ngh nd ok, h n sonby onud h h on usd hngs o h obj. Conssny hfo dnds h h d obs shoud bu ny obsd hngs n ngh nd ok of dsn objs o hngs n hs own d us nd oks. Sp y s ny onssn wh h odny ons of sun n Eudn sp wh bsou. Ths sp f pns why ss ods of dsubns n h h h hsoy podud phys quons onssn wh h Pnp of Ry. Rfns oh N 93 On h Consuon of os nd Mous Ph. Mg. S. 6 6(5):-5 oussnsq J 868 Théo nou ds onds unuss J. d Mhéqus Pus ppqués Sé. II, 3:33-339,

71 d og L 94, Rhhs su Théo ds Qun, PhD Thss (Ps: Unsy of Sobonn) Cos, R 4 Th W ss of Sp Ry (Pond : Vu Vs, ISN: ) Dsson C nd G L H 97 Dffon of Eons by Cys of Nk Phys. R. 3:75 4 Ensn 95 Zu Ekodynk bwg Köp nnn d Physk 7:89 9 Ensn 95b Üb nn d Eugung und Vwndung ds Lhs bffndn husshn Gshspunk nnn d Physk 7:3-48 Ensn 956 Th Mnng of Ry Ffh Edon (Pnon: Pnon Unsy Pss) pp Esss W 95 kungn u Qunnhnk f Ekonn Nuwss. 3:7 Fynn R 969 Th Physs Th, 7 Spb, 33-3 Hsns D 99 Th Zbwgung Inpon of Qunu Mhns, Found. Phys. ():3-3 Huygns C 69 T d uπ Lo J 9 h nd M (Cbdg, Unsy Pss) Lodg 89 On h psn s of knowdg of h onnon bwn h nd : n hso suy Nu 46:64-65 Lodg 893 bon Pobs Ph. Tns. 84:77-84 Lon H 94 Eogn phnon n sys ong wh ny oy ss hn h of gh Po. d. Sn sd 4: MCugh J 839 On h dyn hoy of ysn fon nd fon, Po. Ish d. :374-9 Mw J C 86 On phys ns of fo. P. Th hoy of ou os ppd o gn phnon. Ph. Mg. :6-75 Mw J C 86b On phys ns of fo. P. Th hoy of os ppd o uns. Ph. Mg. :8-9, Mw J C 86 On phys ns of fo. P 3. Th hoy of os ppd o s y. Ph. Mg. 3:-4 Mw J C 86b On phys ns of fo. P 4. Th hoy of os ppd o h on of gns on pod gh. Ph. Mg. 3:85-95 Mhson nd Moy E W 887 On h R Moon of h Eh nd h Lunfous Eh. J. S. (3d ss) 34: Pnk M 9 Ub ds Gs d Engung Nospu Vh. dush. phys. Gs, :-4, Poné H 94 L'É u 'n d Physqu héqu, Confén ué 4 spb 94 u Congès d' ds Sn d Sn-Lous (Th psn nd fuu of h physs, 4 Sp. 94 u o ongss of s nd sn S Lous, U.S..) u. ds Sns Mhéqus, duè Sé, oé 8:3-34 Swnn W F G 94 Ry, h Fgd-Lon Conon, nd Qunu Thoy, R. Mod. Phys. 3:97-3 Tyo N 995 Gud fo h Us of h Innon Sys of Uns (SI), NIST Sp Pubon 8 (Ghsbug, MD: Non Insu of Sndds nd Thnoogy) ppnd Thoson G P nd Rd 97 Dffon of hod ys by hn f Nu 9:89-5 7

72 Fgus Th foowng fgus bd o b f of opygh son, nd w obnd fo h sous sd. Oh fgus h ogn woks o d n h fgu pon. Fgu. Chsn Huygns (69 695) Sou: hp://www-hsoy.s.s-nd..uk/hsoy/pdspy/huygns.h Fgu. Thos Young (773 89) Sou: hp://www-hsoy.s.sndws..uk/mhns/young_thos.h Fgu.3 ugusn Fsn (788 87). Sou: hp://www-hsoy.s.s-ndws..uk/hsoy/pdspy/fsn.h Fgu.4 Gog Gb Soks (89 93). Sou: hp://www-hsoy.s.s-ndws..uk/pdspy/soks.h Fgu.5 Js MCugh (89 847). Sou: hp://www-hsoy.s.sndws..uk/hsoy/pdspy/mcugh.h Fgu.6 Josph oussnsq (84-99). Sou: hp://bfn-.og/hyperl/people/bouss.h Fgu.7 W Thoson (Lod Kn, 84 97). Sou: hp://www-hsoy.s.s-ndws..uk/hsoy/pdspy/thoson.h Fgu.8 Js Ck Mw, Sou: hp://www-hsoy.s.s-nd..uk/hsoy/mhns/mw.h Fgu.9 b Mhson, 85-93). Sou: hp://nobp.og/nob_ps/physs/us/97/nd.h Fgu. Hndk Lon (853 98). Sou: hp://www-hsoy.s.s-ndws..uk/hsoy/pdspy/lon.h Fgu. Jus Hn Pon (854 9). Sou : hp://www-hsoy.s.s-nd..uk/hsoy/pdspy/pon.h Fgu. M Pnk ( ). Sou: hp://www-gp.ds.s-nd..uk/~hsoy/ogphs/pnk.h Fgu.3 Ns oh (885-96). Sou: hp://www-gp.ds.s-nd..uk/~hsoy/ogphs/oh_ns.h 7

73 Chp 3. Es Ws nd Qunu Mhns n on hs n o dy h psson h h on s d of ws hn h s d of w. hu S. Eddngon 3.. Inoduon w us hng on o h bs ds of og oss. Pu dn Mu D [989] Th ho dopns dsussd n hs book w opnd by yd pn dsos, os noby n h boos of J. J. Thoson [Fgu 3.] nd hs sudn (nd susso Cbdg) Ens Ruhfod [Fgu 3.]. J.J. Thoson s sudy of hod ys d o hs dsoy of h on [897]. Ruhfod [9, 94] obsd h bs of ph ps osony s g ngs fo hn g. Ths obson d h o popos h os onn posy hgd nuus of y s s (of od dus) suoundd by uh g ou (of od 8 dus) of ngy hgd ons. Th Ruhfod o od b h bss fo fuu hos of o suu. Fgu 3..J. Thoson (856-94) Fgu 3. Ens Ruhfod (87-937) W h dy nond h bgnnngs of qunu hoy n h noduon o h pous hp. Now w w dsuss ns whh d o h dopn of w quon fo h on. Ths synopss s bsd gy on Whk [954]. 73

74 : odng o oh s o od [oh 93] h on ngy s n hydogn 4 π W h n 4! n R n (3-) wh R s d h Rydbg onsn nd! h π. Rdon s d whn n on dops fo hgh ngy (g n) o ow ngy (s n), nd h fquny of h don s popoon o h dffn n ngs. Fgu 3.3 nod J.W. Sofd (868-95) W Wson [95] nd nod J. W. Sofd [95, 95b, 96] [Fgu 3.3] ognd oh s qunon of ngu onu of u obs (ydng ngy qunu nub n) o b sp s of qunon of on: p dq h, wh q s oodn b nd p s h ospondng onu. Sofd pnd uh of h fn suu of hydogn sp ns by gnng oh s u obs o pss, nudng s n oons nd nw uh qunu nub k. Th s oon o h ngy s of hydogn-k os s: ΔW 4! R n 3 k Z 4 ( k ) (3-) Th fn suu onsn,! 37, psns h o bwn h oy of h fs oh ob nd h spd of gh [Whk 954, p. ]. K Shwhd [96] nd Pus Sophus Epsn [96] usd on qunon o d h sp n shfs fo hydogn n song fd (Sk ff). Sofd 74

75 [96b] nd P Dby [96] pnd h spng of sp ns n song gn fd (Zn ff) by usng h qunon ondons: ngy (n), gnud of ob ngu onu ( k n ), nd oponn of ngu onu p o h ppd gn fd (). No h. Qunon of sng oponn of ngu onu, d sp qunon, ws fd whn O. Sn nd W. Gh [9] sp b of s os no wo ds oponns spy by ppyng nonunfo gn fd. Pnp sp ns of k ns (.g. N) doubs whh oud no b pnd by h fonond qunu nubs. Vous shs w poposd o nud n ddon ngu onu qunu nub whh ws gny supposd o b ssod wh h o o. Wofgng Pu [Fgu 3.4] dspud hs dnfon of o ngu onu n p bus d o 3 Z dpndn n h s ngy shfs. H nsd bud h qunu nub j o h dn on whh possssd ssy non-dsbb wo-udnss. Pu [95] so obsd h son of h s of qunu nubs n, k, j, nd o sng on (h uson pnp ) ws onssn wh h noon of on shs (poposd by Edund C. Son nd J. D. Mn Sh) whh os whn of h qunu nubs fo gn u of n fd by ons. Rph Kong d h sf-oon of h on wh ngu onu of! woud pn h 4 Z -dpndn of h doub ngy shfs, bu sn hs uon of h ngy s ws off by fo of wo h dd no pubsh hs d. Uhnbk nd Goudsd [95] dd pubsh h d of on ngu onu of!, bu unsussfuy pd o whdw h pp f ng h fo of wo dspny. hs Lwyn Hh Thos [96, 97] sod h fo of wo dspny by pubshng pp whh donsd h h (ss) s psson of h on gn on n h nn o gn fd, nd hn h spng of ngy s, hd bn opud noy. Hn h on s spn ngu onu of! ws sbshd. Fgu 3.4 Wofgng Pu (9-958) Fgu 3.5 Wn Hsnbg (9-976) 75

76 Wn Hsnbg [95] [Fgu 3.5] poposd h nsons bwn sony ss (.g. nd n) oud b psnd by n y of ns (.g. n) whos pud s d o h khood of h nson. M on [95] nd Psu Jodn quky dopd hs d no op fouon of hns n whh ouon us pd on ngs s h bss of qunon (.g. qp pq! wh q s oodn nd p s h onjug onu). Lous d og [94] poposd no pnon fo oh s qunon us. H poposd h hs wk h wh ngy popoon o fquny ε! ω nd onu popoon o w o p! k. Th pod ondon fo w of wngh λ popgng n u ob of dus : π nλ (3-3) ps qunon of ngu onu: p n! (3-4) Fgu 3.6 Ewn Shodng (887-96) Ewn Shödng [96] [Fgu 3.6] subsquny pubshd dffn w quon bsd on d og s ws. Fo non-s p of ss n pon V(,), h ngy s gn by: p E V Th ospondng dffn quon fo d og ws s d h Shödng quon:!! V (3-5) (3-6) 76

77 77 wh h w funon s op s. Fo Couob pon ( V ) hs quon yds ngy gnus qu o oh s ngy s. Shödng ny npd h w funon o b d o hg dnsy, bu M on s [96] npon of * s pobby dnsy ws soon wdy pd. pobby onson quon n b obnd by upyng * nd ddng h op onjug: [ ] [ ] { } * *! (3-7) Th Shödng quon hs h ss Honn fo (s.g. Godsn [98]): H! (3-8) wh! psnng Hon s pnp funon whos gdn s h onu p. Th dffn quon ospondng o h s ngy-onu on 4 p E s d h Kn-Godon quon (o s Shödng quon): 4!! (3-9) Inpon of hs quon pod o dffu hn Shödng s non-s quon. I dos no h h ss Honn fo wh fs-od d. Th sung onson quon s obnd by upyng * nd subng h op onjug: [ ] * * * *!! (3-) Th dnsy n hs quon (h fs squ bks) n h h sgn, kng pob s n psson fo pobby dnsy. Nonhss h Kn-Godon quon nuy b pd s dspon of ps wh o spn. Shödng subsquny donsd h Hsnbg s ouon u! pq qp foows dy fo h dfnon of onjug on s ds:!!! q q q q (3-) Pu [97] upd Shodng s w funon by wo-oponn fo (d spno) o od h wo-ud sp qunon du o on spn. Mup opos on Pu spnos n obnons of ndpndn s whh by onnon :,,, y I (3-)

78 Th s h of hs s fo o (.. nsfo s o und oons) nd d h Pu s. Fgu 3.7 Pu D (9-984) Pu D [98] [Fgu 3.7] fny dd d s w quon by ndng h w funon o fou oponns nd usng offns. Th D w funon hs fou op oponns whh n b wn s: 3 4 [ ] T 3 4 (3-3) Suh w funon s d D spno o bspno. D spno n b doposd no f- nd gh-hndd Pu spnos whh h h wo op oponns. D s quon dsbng n on n n ogn pon s:!! ( Φ ) wh nd h s: (3-4) 78

79 ; y ; (3-5) D so donsd h qunu hn quons oud dsb up ps by nodung nw w funon whos ngd squ gnud s kn o b h nub of ps. Ths podu s d sond qunon (s.g. [Toong 974]). D dopd hs hod fo bosons by ssung h s puds ( ) of ous ss (k) o b opos whh ssfy h ouon on k k k δk. Th podu k k hn hs non-ng ng gnus nd psns h nub of ps n h s. Jodn nd Eugn Wgn [98] dpd hs d o fons by usng n n-ouon on k k δk. In hs s h podu k k hs gnus of o nd on, onssn wh Pu s uson pnp. D s sh d h o b n h sn of n h: If on ns h quson n gh of psn-dy knowdg, on fnds h h h s no ong ud ou by y, nd good sons n now b dnd fo posung n h. [ D 95]. Hs on ws h n h oy ws qud fo sng up Honn fouon of h on pnp. D s quon ns h foundon fo dsbng ws. Th Sndd Mod of p physs sss h h n h uns s d up of ny fons nng hough fds, of whh hy h sous. Th ps ssod wh h non fds bosons. [Congh nd Gnwood 998]. Th w funons gdd s dnsonss quns whos gnud ny pon psns pobby dnsy fo h psn of on o o ps. So ffos w d o fou ss npon of h w funon (noby by d og [98] nd Dd oh [95], s.g. Godsn []) bu non ws sussfu n h h nuy. Th h nd go pops of spnos w fs sudd by h hn É Cn n 93 (s.g. Hdk [999] fo h nyss of spnos). Th gb of spnos s osy d o h of qunons, whh w nnd by S W R. Hon ound 843 s gnon of op nubs o hgh dnson. Qunons onss of fou oponns. Thy n n f b wn n fo wh bss os I,,, nd. y Spnos h hsoy bn gdd by hns s opos (n psnons of oon goups) nd by physss s bs quns wh no ss npon. How, Dd Hsns [967] dopd sp- gb whh pods 79

80 go npon of h D quon. Th w funon dsbs gnd Lon oon (sp oon nd oy boos) n ddon o n pud nd on ddon p whh pps o nsfo bwn nd n-. Th h bn sussfu ps o fou h D hoy n s of ons bwn o phys obsbs [Tkbysh 957, Hsns 973]. Th D quon unquy dns h ouon of o dyn quns suh s ngu onu dnsy, n onu dnsy, nd ngy dnsy. In oh wods h D quon s dns wh sp o dyn quns. In hs hp w w d D quon o dsb oon ws n n s sod. W w gd ps s soon souons. W w hn d nuous pops of ny ps fo hs od. 3.. Toson Ws...h usns n whh hs w podu sus whh no on hs y bn b o undsnd n ny d fshon. n p s h D quon, whh pps n y sp nd bufu fo, bu whos onsquns hd o undsnd. Rhd P. Fynn, Rob. Lghon, nd Mhw Snds [963] Qunu hoy dopd fo n n ss pu of s ps. Y w h sn h sp y s nu onsqun of h w nu of. Thfo h ss hoy whh osponds o qunu hns us b w hoy. On hso d of qunu w hoy s h k of n obous phys npon of h w puds. M on suggsd h h w nnsy b npd s pobby dnsy, bu h phsd h "...h pobby sf s popgd n odn wh h w of usy" [on 96]. Wh h s no doub h h qunu w funons n pd h khood of pn sus, h ouon nds us h hn sohs nons. uy, h dyn npon of h w funons n b sod by sp dnson nyss. In s of D spnos, h -oponn of spn ngu onu dnsy s s: s! 3 d wh s h 4-oponn op w funon wh nd s h -oponn spn ngu onu : Th dng fo n Eq. 3-6 s spy onsn whh sbshs uns. 3 (3-6) 4 (3-7) 8

81 Consuon of ss w hoy of us hfo bgn wh ws yng ngu onu. Cssy, ngu onu s ssod wh oons of n bods. Ws of ngu onu qu no ony n bu so oqu whh sss oons. Gnon of oqu n spons o o oons ps sy. Thfo h ss od of ws onsss of oons n n s sod (oson o sh ws). W dy know h h s sod ws h bss fo ss w hos of gh, so w n pod wh so onfdn. Fs onsd oson n on dnson, suh s on oson w hn o shdou ubb bnd [Fgu 3.8]. oson w hn hs s on ngung p wh p physs. If on os sng od n h n of h w, gh-hndd ws popgs n on don nd f-hndd ws popgs n h oh don, nogous o h poduon of ps nd n-ps. In y known phys poss, n- bhs k o g of. noh nsng popy of -D oons s h h s nu dsnon bwn oons of odd nd n ups of π, nogous o h dsnon bwn odd (fons) nd n (bosons) ups of h un ngu onu!. Th noon h oson shoud b ssod wh s n f wdy pd [Kn 989].Thfo h s son o b h h nyss of oson ws gh pod so us o h npon of qunu hns. Ths nogy ws pod by Cos []. Rgh-hndd Lf-hndd Fgu 3.8 Roon of sng b on oson w hn sus n o-sy ws popgng n oppos dons. Ths s on-dnson nogu of poduon of ps nd nps. M nd n- sy podud n ps, nd bh physy s o gs of on noh. If h on of n p un ngh s I, nd h oson spng onsn of h w (o ubb bnd) s K, hn h w quon s gn by: Θ I (, ) Θ(, ) K (3-8) wh Θ (,) s h onon poson nd. Th w spd s gn by K I. 8

82 s wh dspn ws, unqu fquny nd wngh nno b dfnd fo oson ws unss ny ys podud n susson. If on nd of h w hn s od onsn ω, h oson ws popg ong h hn wh unfo wngh λ ω. Eh od ong h hn os wh h onsn dng fquny ω. Th ngu onu p un ngh s hfo Iω Ik IΘ. Th ngu onu s hfo popoon o h sp d of h ng. Th ngu onu of ws fo o Θ n b obnd by ngng o ng: L ( ΘΘ ) ( ΘΘ ) Θ dθ dθ I d I d IdΘ IΘ ( Θ ) d ( Θ ) d (3-9) Thus w s h h o ngu onu of ws s popoon o h oon ng nd ndpndn of fquny. ws popgng wh onsn wngh hs no oqu, so h kn nd pon ngs n onsn s h w popgs. Th kn ngy p un ngh s ω I / nd h pon ngy p un ngh s K ( Θ / ) π K / λ Iω /. Ingon fo o Θ yds fo h o ngy: ( ΘΘ ) ( ΘΘ ) Θ dθ dθ ε Iω d Iω d Iωd Θ IωΘ Lω ( Θ ) d ( Θ ) d (3-) Th w ngy s qu o h w ngu onu s h ngu fquny. Ths s nogous o h ngy qunu of! ω. hs pon w k h dnfons: L I Θ K L I Θ so h h w quon s spy: Θ (, ) Θ(, ) (3-) (3-) Indny, hough w h bn dsbng oson ws ong hn w, h quon s d fo oson ws n hk ynd od (s.g. Fynn. [963b]). Th s of hn s od hs bn sudd by Msun nd Tsuu [99], who npd nonn ws s fons. W w so fon npon of nonn ws whn w sudy n nfn 3-D s sod bow. Now w w k ook h ss w quon o s f n b ppd o h sudy of. W w s wh on-dnson ws s bo, hn gn o h dnson s nd o ws. 8

83 3.3. On-Dnson S Ws "I h dp fh h h pnp of h uns w b bufu nd sp." b Ensn Consd s quny () whh ssfs w quon wh w spd () n on sp dnson (): Ths quon n b fod: [ ][ ] (3-3) (3-4) Th gn souon s supposon of fowd ( F) nd bkwd ( ) popgng ws: ( ) ( ) ( ), F (3-5) Ths fo of h souon o h on-dnson w quon n b found n ny ny book on ws. W n w h quons fo fowd nd bkwd ws n fo: F ( ) ( ) Th sp ds d o h po ds: L F ( ) ( ) F( ) ( ) F ( ) ( )! nd ʹ. W now dfn w funon n s of h ds:! Ψ! F ( ) ( ) Th w quon fo h fowd nd bkwd ws s now:! Ψ! F ( ) ( ) ʹ ʹ F ( ) ( ) W h now dud h sond-od w quon o fs-od quon. (3-6) (3-7) (3-8) (3-9) Spnos nd spnos If w gd h -s s on of h ohogon s, hn h wo ndpndn oponns! F nd! dff by 8 dg oon. Ths s h dfn popy of ndpndn ss n spn on-hf syss. Unfouny, hs popy s d-phsd (o n unognd) n h physs u n fo of h o o popy h op spnos hng sgn upon 36 dg oon. Ths popy dos no ppy o phys obsbs whh opud fo bn podus of spnos. How, h spon of ndpndn ss by 8 dgs dos ppy o w oy, pyng h souons of h w quon gny 83

84 84 fo spn on-hf syss. No h unk pos nd ng ss o o oponns (whh n so b pssd s bn podus of spnos), ws wh pos nd ng oy no d by up fo of nus on. Th fowd nd bkwd ws ndpndn ss [Fgu 3.9]. Th h bss of hs popy s h w oy s popy of h funon guns nd s no spy n pud. Fgu 3.9 Ws popgng n oppos dons ong n s ops ndpndn ss spd by 8 oon. Ths s h bss of hf-ng spn. Th onshp bwn ws nd spnos n b d p s n Cos () by fuh doposon no pos-dfn oponns ( ) F F,,,!!!! o ( ) ʹ ʹ ʹ ʹ F F,,, psnng pos () o ng () onbuons o h w ds: ( ) ( ) ( ) ( ) ( ) F F,!!!!! (3-3) nd ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ʹ ʹ ʹ ʹ ʹ F F F F,!!!! (3-3) Fo h on h funon guns w no b wn py. No h h posdfn oponns y h dsonnuous ds wh h ogn sgnd quns pss onnuousy hough o. Fo p, o k h ds onnuous qus hng ondons fo! : F F F F F F F F F F F F!!!!!!!!!!!! (3-3) S ons hod fo h bkwd w oponns. Suh dsonnus do no ff h dy of h fs od quons. How, hgh ds y b undfnd so pons. Sn h oponn hs unqu sgn, w n pss! nd ʹ n spno fo wh h on-dnson w funon (h subsp fs o h oy s): T F F T F F!!!!!!!!! F 8º

85 85 T F F T F F ʹ!!!!!!!! (3-33) wh h supsp T nds nsposon of h oun nd h bus h fowd nd bkwd os (): F F!!!! (3-34) Ths w funon s on-dnson bspno. In on dnson h oponns of h bspno y b kn o b nd pos-dfn. Enson o h dnsons qus op oponns. Chngng h od of s n h w funon s d hng of psnon. fw pon pons :. Th oponns of h oun w funon nd pos-dfn.. Ony on fowd oponn nd on bkwd oponn n b non-o ny gn nd p (fo on-dnson ws). 3. Th spo-po on of h oponn us b onssn wh s oon n h oun. Sn so of h oponns us b o, F δ nd δ b h o o on. Thn h w funon s: [ ] [ ] [ ] T F F F F δ δ δ δ!!!! (3-35) Usng Lon booss, h w funon n b wn s: ( ) [ ] [ ] [ ] p T F F δ δ δ δ! (3-36) Ths fo hs wo ndpndn onnuous ps nd wo bny ps. Th quon of ouon of h w oponns s: (3-37) Ths s h on-dnson D quon. Ths quon n b npd s on d wh wo oppos os psnd by h. Th on bwn on dnson bspno quons nd s w quons s sud n Tb 3-I.

86 Tb 3-I Cospondng spno nd S W Equons n On Dnson spno Equon T T [ ] [ ] S Equon T T [ ] [ ] F F T T [ ] [ ] [ ] [ ] T T [ ] [ ] F F W Voy Th n oy () of h w s popoon o h o bwn h dffn nd su of h fowd nd bkwd oponns [Cos ]: T T! F!!! Sn!! F F!! F nd! pos-dfn, w n dfn h by h on:! p p ( ) ( ) so h ou dfnon of oy s: ( )! p( ) ( )! p( )! p nh! p If w s fo o-oy s wh! F!!, hn w n hng h oy p ): usng h Lon boos opo ( ( ) T [ p( ) ] [ p( ) ] p( ) p( ) T p( ) [ p ] p( ) p( ) [ ] ( ) No h suss booss ps h fo of h opo: ( ) p( ) p( [ ] ) nh (3-38) (3-39) (3-4) (3-4) p (3-4) Ths popy nbs us o o h s quon fo ddon of p os: 86

87 87 ( ) nh nh nh nh nh nh nh (3-43) Ths su s noh p of how h ws of sp y ppy o ss ws n odny Gn sp-, s dsussd n Chp. Usng Lon booss, h w funon n b wn s: ( ) [ ] [ ] [ ] T δ δ δ δ p F F! (3-44) Ths fo hs wo ndpndn onnuous ps nd wo bny ps Th Dnson S Ws "... n qunu phnon on obns qunu nubs, whh y found n hns bu ou y fquny n w phnon nd n pobs dng wh w oon." Lous d og [963] Roon of Gdn nd Voy Th sp d gns n h dnsons o by don, wh h nd () psns n by don. W oy s dfnd o b p o h gdn. Sn h s ssod wh pu s, us b on oponn of o. W n h 3 nd dfn h gdn oponns s:, ~ ~ ~ ~, 3 (3-45) Th sybo ( ~ ) psns un psudos gny whh s odd (hngs sgn) wh sp o sp nson. Ths popy s nssy bus oy s po o nd: 3 ~ (3-46) W us now ow h w funon o h op oponns. Ths s h ouon ons qun o h Pu s: j j j δ ; k jk j j ε ~ (3-47) n gn wy o w hs ouon ons s: j j j ~ (3-48) wh:

88 j ~ j [ ] j [ ] j j j (3-49) Hn w n gd hs s s bss os whos ouon ons pss h onon. Ths d s h bss fo h h fd of go gb. No h h un gny now hs go npon s h podu of h ohogon un os (.. n ond un ou): ~ ~ (3-5) [ ] [ ] ~ Th oon opos fo hs sp h h fo: ~ ~ ~ R ( ζ ) p( ζ ) j p( ζ ) j osζ [ j j ] sn ζ j (3-5) whh n b wn n o fo: ~ ~ R ( ζ) p( ζ ζ ) p( ζ ζ ) osζ [ ζ ] sn ζ (3-5) To nud oons, h on-dnson d T us b odfd o nud onon. Ths onon s opud o h 3-s. Usng h dfnons: p [ p( ~ ζ ) p( ~ ) ] ( ~ 3 ζ ζ ) p( ~ T ζ ) (3-53) Th w funon now hs op oponns. Th oon opo ~ R p ζ ppd o h on-dnson w funon ns h oon of ζ ( ) ( ) h bss os so h h d n b ud usng h on-dnson -ud 3 nd w funon. Th sp d s: T 3 (3-54) Sn h b s uuy ohogon, h oponns of ppndu o us b o. Thfo h h dnson gdn s: T ˆ 3 (3-55) 88

89 3.4.. Suss Roons Suss oons n b pfod usng h fd s o bddd s. Th su of suss oons bou fd s dpnds on h od n whh h oons kn. Fo p, suss oons of π bou h - nd -s o 3 o h o, dpndng on h od. Hn: p( ~ π 4) [ p( ~ π 4) 3 p( ~ π 4 ) ] p( ~ π 4) ~ π [ ~ π 3 ~ π 4) ] p( ~ π 4 ) (3-56) H h psson nsd h squ bks s ud fs, foowd by ppyng h oon opo ousd h squ bks. If w np hs oon opos s ng on ~ ~ spnos hn h od pps o b bkwd. Th psson: p ( π 4 ) p ( π 4) psns spno oon of π bou h -s foowd by oon bou h -s Eu ngs W n pu h opons bk n od f w onsd h sond oon opo o h bn od ong wh h w funon by h fs on: ʹ R ( Θ ) ( ) ( ) ( ) p( ~ ) p( ~ ) p( ~ R Θ R Θ R Θ Θ Θ Θ ) (3-57) Two suss oons yds: [ ] R( Θ ) ( ~ Θ ) p( ~ ) ( Θ ) R( Θ ) R( Θ ) R( Θ ) R ( Θ ) R ʹ p Θ (3-58) s whh od ong wh h spnos d bddd s. Roon ngs whh f o bddd s d Eu ngs. W us ps o dno oons bou bddd s. Th Eu oon opo Rʹ ( Θ ) n b npd s foows: Fs, o h spno bk o s ogn onon. N, o h spno bou h fd s ospondng o Θ. Fny, o gn bou h bddd s ospondng o Θ (h ogn s now od by Θ ). Th quon ss h oon by Θ Θʹ foowd by oon bou h fd s Θ s qun o oon fs by Θ foowd by oon by Θʹ bou h bddd ˆΘʹ s. In h bo p, oon by π bou foowd by π bou (o yʹ ) s qun o oon by π bou foowd by π bou y (o ʹ ). Th ngu d of h w funon s: ~ ~ ~ ~ ~ ~ [ p( ) ] p( ) p( ) p( ) ʹ I s usoy n qunu hns o dfn h ngu d o b: ~ Ths on s ony d f h ng ϕʹ s sud wh sp o h bddd s. (3-59) (3-6) 89

90 uud oons n b opud fo suss oons bou bddd s. Gn oon w () wh sp o bddd s, h uud oon opo s: R ( ) R( Θʹ ( ) ) p( ~ Θʹ ) p( ~ d wʹ ) (3-6) Eps L us fy hs psson wh p ps. Fs, w opu h gn psson fo oon bou wo suss bddd s: Ro by ng ʹ bou n s ʹ foowd by ʹ b bou ʹ b. Th oon opo s: R R Θʹ p ~ ʹ p ~ ʹ ( ) ( ) ( ) ( ) ( ) b b bʹ ~ bʹ ʹ ~ ʹ os b sn os sn bʹ ʹ bʹ ʹ ~ bʹ ʹ os os b sn sn os sn bʹ ʹ b sn os R h b b ~ b. W onsd wo sp ss. Fs, f ʹ nd p hn: R ( Θʹ ( ) ) ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ b b b b os os sn sn os sn sn os ʹ ʹ ʹ ʹ b b os sn (3-6) ʹ b (3-63) whh s obousy o sn p ngs dd. N onsd wo ppndu s wh b ~ : R ʹ ʹ ~ ʹ ʹ ( ʹ b b b b Θ ( ) ) os os sn sn os sn sn os ~ ʹ ʹ Fo h sp s wh boh ngs π hs yds: R ( Θʹ ( ) ) π ~ π ~ π π π π os sn os sn sn os 4 4 b ~ b ~ b os( π 3) sn( π 3) 3 b ʹ ʹ (3-64) (3-65) Ths osponds o oon opo fo π 3 dns bou h s [ ˆ ˆ b ˆ ] 3. Th dy of hs su n b fd by pung n qu ng wh ons on h s qudsn fo h ogn. Cy oon by π 3 bou h n of h ng y pus h posons of h s, whh s of ous wh hppns whn ong by π ound suss ohogon s. No so h h syy of h fn su ps h: 9

91 p ( ~ ) ( ~ ) ( ~ ) ( ~ y π 4 p π 4 p π 4 p y π 4) p( ~ π 4) p( ~ π 4) (3-66) ( foowd by yʹ, y foowd by ʹ, foowd by ʹ ) whh s onssn wh ou pnon of h sondy oon opo bo W Funon In h dnsons h gdn n b dfnd s on-dnson d od by ng ζ o nw s ˆ. L: p p ( ~ ζ ) 3 p( ~ ζ ) ( ~ ζ ) (3-67) Roon by ng ζ s dnod R ζ nd dfnd o dfu onon ong h 3 s. Th h-dnson gdn s: Rζ ˆ T ˆ Wng oun s h nspos of ow, h od w funon s:! p ζ ( ~ ) p( ) δ δ [ δ ] [ δ ] 3 T [ ] F How, n h dnsons h onsn oun whh psns 3 ss y h nono oy ppndu o 3. Ths s ndd h s fo T T [ ] nd [ ] obnd by oon of oy fo: T [ ] F (3-68) (3-69). Th nng ss wh o oy (3-7) Ths s hs o d bu nono gdn. Whn Lon booss ppd boh h d nd oy n b non-o. Th fn fo of h w funon s hus: ~! p( ζ ) p( 3 ) (3-7) Ths s h gn fo of h s w funon. Th onsn s upd by fos psnng n pud, -D oy boos, nd gn oon n oy sp (wo ngs o dn oy don pus oon bou h oy s). Cy fou ps ndd o dn nd. Th sgnfn of oon bou h oy s w b dsussd bow Fs-Od W Equon Th d of (3-67) yds h fs-od quon: 9

92 9 ( ) ( ) ~ p ~ ~ p ζ ζ ζ (3-7) H w n s h ff of oon bou h oy s. Roon of h f-hnd sd nos ony d oon of h w funon, bu oon of h gh-hnd sd so nos oon of h ngu fquny ζ. Roon bou h oy (o gdn) s n hng h don of hs ngu fquny. Ths s h sgnfn of h ffh p n h foon bo. Inng h oon fo yds h on-dnson w funon, whh ssfs h on-dnson w quon: [ ] ( ) [ ] ~ p ˆ 3 ζ (3-73) Ds of h ponn fos : ( ) ( ) [ ] ( ) ( ) [ ] ζ ζ ζ ζ ζ ζ ~ p ~ p ~ p ~ p (3-74) Subsung ( ) ( ) ζ ζ ~ p ˆ ~ p 3 no (3-73) yds: ~ ~ ζ ζ (3-75) Ths quon ss h h on d s nono ony du o (on) oon of oy don. Th quon of ouon of h s w pud s obnd by upyng nd ddng h djon: [ ] [ ] [ ] ζ ζ ζ ζ ζ ~ ~ ~ ~ (3-76) In s of h s poon, hs quon s: [ ] ζ (3-77) Th ons bwn oon ngs nd oy un os :

93 ζ ˆ ˆ ζ ˆ [ ˆ ] ζ ˆ [ ˆ ] [ ˆ ] ˆ [ ζ] [ ˆ ][ ˆ ] [ ˆ ] [ ˆ ] So h h bo quon s ndd qun o h on-dnson w quon: [ ˆ ][ ˆ ] If w wn o obn h onnon 3D s w quon: Thn h sps ospondng fs od quon s: (3-78) (3-79) (3-8) (3-8) 3.5. Vo Ws "Qunu hns s ny posng. u n nn o s h s no y h hng. Th hoy sys o, bu dos no y bng us ny os o h s of h "Od On." I, ny, onnd h H s no pyng d. Ws n h-dnson sp whos oy s gud by pon ngy (fo p, ubb bnds)..." b Ensn, 96 [Ensn nd on 5] N w onsd o ws (po o os). n by poon o n b dsbd by s pud nd h oon ngs. Sn s ws qu f ps, w p o ws o qu gh ps. s wh oy oons, ony wo ngs nssy o dn h don of poon, bu hd ng s nssy fo o dspon of hngs n h poon don Roon of Poon R h h s poon s T! T. W now gd hs s on oponn of o:! 3 3. Th o oud b po o, bu w w ssu n o (psudoo). Th h ohogon poon s :,, 3 (3-8) Th sybo ( ) s un s gny whh s n und sp nson sn h spn s psudoo. 93

94 Ths s h h s ouon ons s h Pu s: δ j j j ε j j jk k ; Th oon opos fo hs sp s o h oy oon opos: R j ( ξ ) p( ξ ) j p( ξ ) j osξ [ j j ] snξ W oud spy gn h w funon o b: ~! p ξ p ζ p ( ) ( ) ( 3 ) W gh hn p h npon: (3-83) (3-84) (3-85) S j j (3-86) How, h nn nso oponns (pus h oponns of h d) nd ony gh ndpndn oponns of h bspno. Thfo hs npon s no ssfoy unss ddon onsns posd. Insd, w w ssu sng oon opo fo boh w oy nd poon. Sn h on-dnson oy s 3 3, h h-dnson oy fo o ws s. Th s whh dsbd oy fo s ws now psn 3 dons o oy, wh 3 psnng h p don. Ths noon s d h h psnon of oy. ny, w oud sso ny of h s wh oy by ong n h -oy sp of s. Suh oon s d hng of psnon. Th fo hs h fo ogny usd by D, nd w w us hs s fo oy. Hsoy, dffn noon hs bn usd fo h s. Insd of (, ), hs s h bn d 5 5 ( γ, γ γ, γ ), 3. How, w w onnu o h s p whn opng wh sndd sus fo oh u Foon nd Fs-Od W Equon Th h-dnson bspno w funon y h Lon boos wh by gnud nd don, nd y so b od by n by ng ξ. Ths opos onnd n h foon:! ( ξ ) p( ) p (3-87) Th w funon hs sn f ps: n pud, h oon ngs, nd h oy ps. Th s on ddon dg of fdo whh dns h dfnon of h dons nd 3. Ths dfnd wh sp o h oy s by h ~ opo p( ζ ), so h h w funon s [Hsns 967]: ~! p( ξ ) p( ) p( ζ ) (3-88) 94

95 Now w woud k o know h quon of ouon of h w funon. Gnng h s w quon (3-73) o nud by gdn don yds: (3-89) Ths s oun fo w popgon n n by don. To s h w quon n s of obsbs, upy quon o obn h d of h poon: [ ] [ ] [ ] j nd dd h nspos Th s n hs quon nuy ssod wh spnos by h foowng dfnons: j [ j ] j[ ] j [ ] { } ε { } j jk Ths dnfons yd h w quon: k k (3-9) (3-9) (3-9) [ ] [ ] (3-93) Ths on s sy dd fo quon (3-89). so fo (3-89): [ ] [ ] (3-94) Ths s h qunu hn onnuy quon. Ths s h h dnson gnon of h -D quon: F F (3-95) Conon nd Roon ddng s fo onon nd oon o h bspno w quon yds: u w (3-96) Fo h w foon w n subsu nus h ngu d (fo pss oon) n h fn : u w (3-97) 95

96 To s h w quon n s of obsbs, upy o obn h d of h poon: # $ % & # $ % & ε # jk $ k k % & ê j u # $ % & w # $ % & j nd dd h nspos quon Ths dnfons yd h quon of w popgng n ong du: (3-98) u! w! (3-99) Usng quon (3-97) now yds dffn onnuy ondons: [ ] [ ] u [ ] (3-) Conssny wh ou dfnon of bs qus h: [ ] u (3-) so fo (3-97): [ ] [ ] u [ ] (3-) Th onnuy quon now nuds n ddon onon. N, w w np h w poon Ws n n Es Sod "I n onn un I h onsud hn od of h subj I sudyng. If I sud n kng on, I undsnd; ohws I do no." W Thoson (Lod Kn) 94 In hs son w w ny oon, o oson, ws n n d s sod. Ths son s bsd on pousy pubshd wok by h uho. Th bs ds w pubshd n Cos [8] nd h fu Lgngn nd dyn opos w pubshd n Cos [] s ssupons W k h foowng bs ssupons:. Th s sod s hd by n n dnsy ρ nd offn of sy µ, wh hs w spd µ ρ.. Th s n spons o ons of onon ng Θ o qubu. Ths ns h n n s pubon (wh oy u) woud yd h spons: ( f u ) Θ Θ (3-3) 96

97 3. Th oy fd u hs no opsson: u. Thfo h oy y b wn s h u of o fd: u ρ S [ ] (3-4) Th o fd S s d h spn ngu onu dnsy. I dffs fo h onnon dfnon of ngu onu dnsy ρu n h s ndpndn of h ho of ogn nd n h by don ny pon. If u fs o o suffny pdy owd nfny, hn kn ngy y b pssd s: K d 3 ρu d 3 w S wh w u s h ngu oy, o oy (s Chp ). ddon ssupons w b nodud n od o spfy h hs, nd hs y h gny of h sus Equon of Eouon Sng fo (3-65), w dfn n ngu pon Q suh h: Q 4ρΘ (3-5) Th s ondon fo Q s: { Q Q} ( f u ) (3-6) Dfn h spn ngu onu s: S Q (3-7) Th s ondon s hn: { S Q} ( f u ) (3-8) Whn oon s psn, onbus o h d ony hough onon ( u S ) nd oon ( w S ). Ths ssus h h no oy-dpndn fos suh s fon dpng. Fo h on, w w onsd ony w-k souons ssfyng: S Q u S w S (3-9) Fo osoy souons o hs quon, h fs wo s wys n phs ( Q Q), whs h nonn y h dffn phs. How, f h nonn s no o hn us h h s phs s h n s: u w S Ω ( )Q (3-) wh Ω ( ) s so funon of poson (o gny, ( ) fo h oponn of Q). Subsuon yds: Ω oud h dffn us 97

98 ( ) Q Q Q Ω (3-) If ( ) Ω s onsn nd pos, hn hs s h Kn-Godon quon, whh s odny ssod wh bosons. Th w quon (3-9) n b wn n s of fou-oponn op D bspno ( ) usng h foowng dnfons: Q j [ j ] [ Q] [ ] j { Q} ε { } Th s j j j jk k k h D oy s, o onnony dnod s γ 5 j. (3-) Th bo dnfons pod 7 onsns on h 8 f ps of h D bspno. In s of bspnos, h oon w quon (3-Eo! ookk no dfnd.) s: " # j $ % " j # $ % ε jk { k k } u " # j $ % (3-3) ε kjw k " # $ % Epndng h ds yds: % j u w & ' ( ) * h.. (3-4) wh (h..) psns h Hn onjug. Th Hn onjug w funon y b gdd s n ndpndn b (h ndpndn nd gny ps of h w funon n obnons of ns of nd ). Vdy fo by qus h s n bks o su o o. Ths yds h D quon: u w χ (3-5) wh χ y b ny opo wh h popy: R( j χ) (3-6) Sn χ dos no onbu o ngu onu dnsy, w ssu o b o. Fo onnn, w upy h D quon by h un gny s h djoun w funon: 98

99 u w (3-7) Now w onsu Lgng dnsy. Lgng s quons of oon fo fd b : L L " # $ j % j " # j $ % L (3-8) s quon hods wh png. I s possb o onsu Lgngn wh no ds of, so h h quon of oon s spy L. Th nonn s (wh u nd w) onn wo fos of. In h oon, hs y b nhngd usng ngon by ps. Thfo hs qus fo of on-hf n h Lgngn. In h onon, how, ngon by ps yds onnng u. Sn hs s o, h fo of onnd n dos no onbu o h Eu-Lgng quon. Hn w obn: L u w (3-9) Ths Lgngn s no, bu w y k h p s psnng phys quns. No h h fn psns nus h kn ngy, so hs s no h ss fo (K- U). Ths w qu so wh h sgns of onjug on. p Th onjug onu o h fd s p : L [ ] W ogn h s n h Lgngn s h kn ngy dnsy K. onssn (whn ngd) npon of h oh s s: R{ u } w E p u p u q (3-) (3-) wh u nd q h oy nd onu of h du, spy, wh nd p h oy nd onu of h w, spy. Th s no q, psuby bus h w s nss (w oy ohogon o du oy). Th w ngy nd onu dnsy kn o b: { } { } E R p R (3-) 99

100 Ths w b dd bow. Sn h p nos ony sp ds of h w funon, s o ppop o np s s pon ngy dnsy U: E U u p K (3-3) Th onon nspos onu nd ngy, bu w hypohs h ngs o o, hby hng no ff on h o ngy Dyn Vbs Engy nd onu Th Honn s: H p L u w U u p K (3-4) Hon s quon fo h w funon s: H H p " # $ % * u w -., / (3-5) W n so dfn Honn opo wh H (s n qunu hns): H u w T ν µ Th Honn s sp s ( T ) of h ngy-onu nso: L µ ν Lδ ν "# µ $ % (3-6) (3-7) Th dyn onu dnsy s: p T L [ ] (3-8) Ths s dn o h onu dnsy of s qunu hns. Th dyn ngu onu s sy: L L [ ] ϕ (,) (,) [ ] (3-9) ϕ Fo h o onu nd ngu onu, w us dd h onbuon of h oon of h du:

101 P p q J L S (3-3) oons dsbd by h opo U(ϕ) [Shff]: ϕ U ϕ (,) ( L S ) U ϕ, ( ) p L S ( ( ) ϕ), ( ) (3-3) In suy, w h donsd h oon ws n n s sod dsbd by Lgngn wh y h s dyn opos (whn noon fo) s found n s qunu hns Eon Ws g sp woud b d whn w shoud b b o sy of y h whh w sy of gh, n syng h onsss of unduons. S Gog Gb Soks, F Eon Equon Th bspno quon fo ngu onu dnsy s: u w (3-3) fo souon s: ( ) p - d ' u w &,,. /. % (. * )., ( ) (3-33) Mss, Conon, nd Roon D s don of h ss spy qud h h oponn of h w funon ssfy h Kn-Godon quon. On possb fouon woud b: µ [ ˆ 3 ] (3-34) Th sond-od quon s: [ µ [ ] ] ˆ 3 (3-35) whh s qun o Kn-Godon f h w funon s n gnfunon of h opo ˆ. Th qun ss quon s: [ ] 3

102 µ [ ] [ ] ε [ ] [ ˆ ] 3 jk whh n b qun o Kn-Godon f µ [ ] Ω k ˆ 3 k ˆ. j (3-36) D Equon D s ho of ss dffs fo h on bo: Ω3 (3-37) wh M!. Oh psnons of hs quon : Ω (3-38) In qunu hns, Pnk s onsn! pps py n h opos nd h w funon s nod o on fo h pupos of opung oons. How, physy s o snsb o no h w funon o! so h s h h w funon dsbs h ouon of ngu onu dnsy. On n s opu oons, of ous, s w w s. Fo onssny wh don qunu hns, w w nud h fo of! n ou quons. Th quon fo spn ngu onu dnsy s spy: [ ] [ ] ε [ ] jk k k (3-39) whh w np s n odny w quon (h onon nd oon s psud o n): [ Q] [ Q] Q Q (3-4) D s ho of ss ns h ss fo h sond-od w quon. On onsqun of hs ho s h h on fo qunon soon ws s os. So wh D s quon n b usd n dsbng p oon nd nons, nno pn h sn of ds ps. D so ssud h sony ss h h fo: E (3-4)! whh hs h fo souon: γ γ γ γ 5 E! (, ) p ( ) ( ), ( D's ogn noon ) Ω ( Rs qunu hns noon ) µ µ (3-4)

103 Ths souon s pung bus h phs on psnd by h ngy gnu E dos no ospond o ny u oson n sp. Th phs spy ns ou whn opung obsbs. o sonb sng pon woud b o ng gdns n (3-3) o g:, (3-43) (, ) p w ( ) ( ) If h w funon s spn gnfunon ( s ) s, wh gnu s, hn h ponn n b d s s, s n qunu hns. Th ngy gnu woud hn psn w h oon ngy ( E w S ), onssn wh n qupon of ngy bwn kn nd pon ngy. In hs s h woud so b no oson. How, w n k hs su snsb by ssung o b n ppoon. W suppos h h w funon s no y n gnfunon of spn, so h h osons n sp. Fo p, h spn don y o s opd o h gnud of ngu oy. Fo p, on n nson onn sph shs wobbng gdy so h h op nd boo pons fo h qubu poson o n s bou h -s, ydng n g ngu onu. u w ssu h h ppoon of spn gnfunons s d fo h puposs of opung gnus nd oons bwn ss. Consdng h k of oson n onnon qunu hns, s nsng o no h physss n h nnnh nuy, d by W Thoson (Lod Kn), poposd od of uu s onssng of fud fd wh os. Ths od s d h o spong, nd s s ns ody. Th od so hs n o h bho of qud hu. Ths od woud n h qun of oson, sn sdy fows possb n fud. Th od n so podu sh ws popgng ong h os. u h od s onpuy o op h h s sod, so w w no pusu h. If w ng gdns n h on quon, w h: E 3!Ω whh hs souons: [ ] T nd [ ] T [ ] T nd [ ] T fo E! Ω (3-44) fo E! Ω, nd. Fo h sgn of E, h wo souons dff n h sgn of h 3 -oponn of spn. Ths souons fd o s spn-up nd spn-down souons. Th pos nd ng sgns of E ssud o ospond o nd n-, spy. W w now n h onshp bwn nd n fuh ngu spon R D s quon fo f p: Ω3 (3-45) Th opo n b fod: 3

104 4 [ ] L (3-46) Th wo-oponn ngu souons of h gnu quons ( ) κ Φ, L nd ( ) [ ] κ Φ, L w known (.g. (jokn nd D 964)), nd dd n ppnd. Ths wo ngu souons d by ( ) ( ) Φ Φ,, nd yd oppos gnus of h py (sp nson) opon. Ths ngu souons y b obnd o fo wo ndpndn w funons: ( ) ( ) ( ) Φ Φ F G,, ~ o ( ) ( ) ( ) Φ Φ G F,, ~ (3-47) Voy Roon nd Mss I s nsu o opu h ff of ss on h w oy: ( ) ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) Φ Φ Ω Φ Φ Ω Φ Φ Φ Φ Ω Ω Ω FG FG FG d d,,,,,,,, δ (3-48) Th ss psns d on of h w, whh s nwd podd h h ppop sgn s hosn fo Ω. Ths su ps u popgon, onssn wh h pnon of h s ss-ngy on gn n Chp W Infn nd Pons N w nsg h ogn of ogn pons. Cn obsbs (ss nd os) shoud b dd whn wo ws supposd. Ths ps h whn wo ws nd supposd, h o w T hs h popy h: T T G G G (3-49) fo so n Hn opo G. If w spy ddd h wo w funons, w woud h nsd: [ ] [ ] G G G G G (3-5)

105 Th ddon s y no o n gn. How, hy n b fod o o by nodung phs shfs o h w funons. Usng subsp o o psn h w funon n h bsn of nfn, : p p ( [ δ π ] ) ( δ ) (3-5) Th phs shf π oud b dsbud bwn h wo ws o nopod no δ nd δ, bu w w s h s w nd s h sou w nd qu h ondon bow o hod n wh δ nd δ qu o o. Ln ddon of h obsb G qus: G G If h w funon s n gnfunon of so dd obsb suh s spn ( G λ o G λ fo so s λ ), hn hs su dus o: In s of h unpubd w funons: ( [ π δ δ ] ) p( [ π δ δ ] ) p If w np h quny yds: [ ] (3-5) (3-53) (3-54) s wo-p s, hn nhngng h wo ps (3-55) Ths ns h h wo-p s s n-sy wh sp o hng of ps. Ths syy s d h Pu Euson Pnp bus pohbs wo dn fons fo bng n h s s ( yds ). Thus h Pu Euson Pnp sus fo h by spon of h op w funon no wo ndpndn ps. In qunu hns, h wo-fon s s ypy onsud s:, so h h Euson Pnp s uoy ssfd. Th onsn phs shf π hs no ff on dyns. How, so obsbs opud fo hs ndpndn w funons y dff fo hos of h f p w. Fo p: [ p( δ ) G p( δ )] Gʹ G (3-56) (3-57) 5

106 Hn h ff of w nfn s o hng h opo fo w pk fo G o Gʹ : ( δ ) Gp( δ ) Gʹ p (3-58) ppyng hs u o h opos nd H yds: { [ p( δ ) p( δ ) ]} p( δ ) H p( δ ) H! w Subsung h gn fo of h Honn nd owng fo onon: " % # $ δ & ' ( " % δ # $ & ' ( " u u % δ # $ & ' ( ( w w ) ( (3-59) (3-6) (3-6) Subsung h ss fo h f on: " % # $ δ & ' " δ % # $ & ' " u u % δ # $ & ' [ w ] Ω 3 (3-6) Sn w nsd n h ffs of h phs shf, w w ng h s whh ndpndn of δ (whou p jusfon). W hn dfn h ogn pons s: δ! Φ δ u!! hough h o pon s gdn, s u (h gn fd) y b nono bus δ s phs ng whh y b u-ud. Fo p, h u-ud δ n hs gdn oponns: funon ( ) δ δ [ ] ( ) [ ] ( ) (3-63) (3-64) Th u of hs gdn s non-o (, )(,). S Kn [7] fo dsusson of u-ud pons n ogns. 6

107 Wh hs dfnons, h on quon n h psn of noh w bos: Φ Ω!! (3-65) 3 Hn ogn pons su fo w nfn und h ssupon h dffn w pks ndpndn. Th bo nyss s no y ps, how, s w ngd hngs n du oy nd oy, nd dd no spfy whh obsbs shoud b dd (o onu dnsy nd o ngu onu dnsy shoud boh h hs popy). op nyss of p nons woud qu knowdg of h soon w funons of h p. Sng! H, h odfd Honn s: H Φ Ω3 (3-66) Mup sou ws y b d squny, s s fs ppoon. Fo gn s w, k ndpndn of h fs sou w s bo. Thn k h odfd s w nd k ndpndn of h sond sou w. Rpon of hs poss fo sou ws sus n h ddon of phs shfs o quny, h ddon of pons. M nd n- souons ssud o yd oppos sgns of phs shf. On y so nf h soon ws wh dn ong-ng (ogn) pons (.g. posons nd poons) so h dn bspno w funons g dsns fo h ns. In qunu hns, s nssy o ous w pks s ndpndn ps. How, wh ss w hoy of y b sp o so h sng quon fo h o ngu onu dnsy, hn dopos h souon no soon ps fo opson wh pn. H! Lon Fo In s of ogn pons, h odfd Honn s: qφ q u w qφ q Ω3 (3-67) Rng h u-dpndn of qφ nd w (nd ou hng of sgn of H), h onjug onu fo s now: p δl δ H q q u p q [ u] [ u] ρ δ δ wh p { } s h f p w onu. Th d of ny obsb Q s: [ Q ] [ ] Q Q [ Q] [ H, Q] Q (3-68) (3-69) n p of hs s h fo dnsy. Subsung h n w onu fo Q yds h Lon fo w: 7

108 8 Φ Φ E p q q q q q q q q q (3-7) wh E nd h usu nd gn fds, spy. Hn h Lon fo hs sghfowd npon n s of ss w nfn Mgn Mon Th quon of ouon n ogn fds s: [ ] 3 Ω Φ q q (3-7) Usng wo-oponn spnos wh [ ] T,, hs quon n b spd no wo oupd quons: [ ] [ ] [ ] [ ] Ω Φ Ω Φ q q q q (3-7) L ( ) p χ Ω nd ( ) p χ Ω. Subsuon yds: [ ] [ ] [ ] [ ] Φ Φ χ χ χ χ q M q q q (3-73) N, ssu h [ ] χ χ Ω << Φ q. Ths yds: [ ] [ ][ ] Ω Φ χ χ q q q (3-74) Ths s h Pu quon, whh ws h fs quon o nopo on spn. [ ] [ ] χ χ Φ Ω q q q (3-75) Usng h ouon ons fo h Pu spn s: [ ] [ ] [ ] [ ] [ ] [ ][ ] { } [ ] { } χ χ ε χ χ q q q q q q q q q j j k jk j j j (3-76) Subsuon yds: [ ] χ χ Φ Ω q q q!! (3-77) Ths quon s of ous spy n ppo quon fo wo oponns of h D w funon. Nonhss, s of hso pon bus ws usd by Pu o nud

109 ffs of on spn. Whou h spn, h sun s quon s h on Shödng fs usd o opu h hydogn ngy s: [ q ] p!! ( q!) Φ qφ (3-78) Ω Shödng s quon s uny h onnon sng pon n h sudy of qunu hns. hough sp hn h D quon, s f ss nu. oh Lon nn nd h onnon wh spn ngu onu h bn os. In wk, unfo gn fd wh, w n ng! χ!! q q Ω [ ] Ω ([ ] ) q qφ χ! qφ χ! q o obn: [ L s] qφ χ (3-79) Th fn fo wh h spn ngu onu opo ( s ) s obnd by opson wh h ngu onu opo (3-Eo! ookk no dfnd.). Ths su s sgnfn bus shows h, n hs ppoon, h offn of spn ngu onu s w h offn of ob ngu onu n h on gn on: q µ Ω [ L S] f on wh h oh gnon (3-8) q, L, nd S, hs gn on qu (whn.%) o 5! 5.78 V/T Spn Ws Consd h quon fo h ouon of spn (3-9): S Q u S w S (3-8) If w ng h sp gdns, w h: S w S (3-8) Th oy s gn by: ( ) w u R S ρ (3-83) Kpng ony h nong spn yds: S 4ρ 4ρ [ S] S S S (3-84) 9

110 Ths quon dsbs h sps fo of spn w, whh s oony obsd n fogn s Msun Coons In 935 Ensn, Podosky, nd Rosn suggsd hough pn nndd o dons h qunu hns ws no op hoy. Th d ws h ps gnd n ps oud b subjd o ndpndn suns h no qunu hny owd on sng ps (du.g. o h unny pnp). How, whn u pns w pfod hy suppod h qunu hn w h phys quns do no h spf us un hy sud. I s wdy bd h h oons bwn poon suns of nngd ps nno b pdd ssy. Ths bf s bsd on oon pdons usng n quon of h fo: P (, b) (, λ,..., λn ) ( b, λ,..., λn ) ρ( λ,..., λn ) dλ... dλn (3-85) wh ρ λ,..., λ n s h pobby dsbuon of hs bs, nd b h sud poon dons fo h wo nngd ps, nd h ho ouos of h sun (±), nd P(,b) s h oon. λ psn bs whh dsb h s of h sys, ( ) John [964] pod h qunu oons nno b psnd n hs fo. In pu, h pod h fo h dffn suns (, λ,..., λ n ), ( b, λ,..., λ n ), nd C(, λ,..., λ n ): ( b, ) P(, b) (, ) P P (3-86) Ths ondon s od by qunu hn (nd physy obsd) oons, whh n b sud usng wo o o ps whos spns onsnd. Fo p, f p of spn ½ ps s podud wh oppos spn, h oon bwn h spn suns by dos ond wh ng ϕ s: P ( ϕ) osϕ (3-87) p Ths oon os s ondon. Fo p, f h dos, b, nd ond ngs, π 4, nd 3π 4, spy, hn: P P P (, b) P( π 4) ( b, ) P( π ) (, ) P( 3π 4) P( b, ) < P(, b) P(, ) (3-88) Th f h u suns o s ondon ps h so sp of hs don dos no onfo o y. On p ssupon s h h s no ounon, nsnnous o ohws, bwn h wo dos. Mny h npd oon of s nquy s dn of nsnnous ounon bwn h dos (nonoy), bu h s no d dn h nfoon n b nsd nsnnousy.

111 W w hs ssu unsod, bu pon ou h h ng (3-85) s qusonb. Fo p, f so ps op nubs hn h ng s onou-dpndn nd y b -dfnd. Fuho, h w od of s so nono nsof s h p s uy spy ndd w pk. n n fouon us ondon pobbs. L j psn h fou possb sun ouos dos nd :,,, nd. Dfnng P s h pobby of, h oon y b wn s: ( ) (, b) P( ) P( ) P( ) P( ) P j j j j j j (3-89), j, j W w us hs fo nd ssu h h ondon pobby P ( ) P( ) j j s popoon o h squd oon bwn spno gnfunons spd by h oon ng bwn h sud ss. To opu h oon bwn wo bspno w funons, onsd h foowng pops: Fs, h gnud of h w funon us b sond-od n h of h oponns nd pos-dfn. Thfo:, j (3-9) Sond, phys bs bn n h w funon. Thfo s h squd gnud h s of phys ns. Th un-nod oon P bwn wo funons us b dfnd n suh wy h h squd no s h sf-oon: P (, ) (3-9) Ddng by h gnuds of h w funon yds h nod oon C: P ( ), P ( ), (3-9) Th oon bwn ss d by oon ( ) s: ( ) R bou n s ppndu o h spn R os ϕ P (3-93) Th oon fo ng ( π ˆ ) s os [( π ϕ) ] sn [ ϕ ].

112 ssung h spn suns ondn o n-ondn n popoon o h oons bwn h spno w funons, h oon P s bwn spn suns spd by ng s: ϕ ϕ P s ( ϕ) P ( ϕ) P ( π ϕ) os sn osϕ (3-94) In h s of p poduon n EPR-yp pns, h spns of h wo ps oppos (hngng ϕ o π ϕ bo), hby hngng h sgn of h oon. Hn w b o d h qunu oons fo so sp ssupons, hough hs s by no ns dfn souon of h EPR pdo Qunu Mhns In h pdng son w opud h oon bwn wo ss d by oon., nd R ( ϕ ) (,). Th oon gn poson, nd Th wo ss y b dnod by ( ) nd s gn by (3-93). o gob oon bwn wo w funons ( ) (,) gn s obnd by ngng o sp: d P 3 d d 3 3 d 3 d d 3 3 (3-95) Th oon bwn spn on-hf ss s non-ng, nd h oon of w funon wh sf s uny. Ths pops pod h bss fo pobbs npon of h w funons. gn w funon y b doposd no up w funons (ss), nd h oon bwn h w funon nd h s y b opud. In qunu hns, hs oon s npd s h pobby of dng h s wh sun. Ths ns h oons bwn phys ss (s opposd o suns) qu o h squ of op pud. Ths fundn popy of qunu hns hs ysfd gnons of physss. Y w n now s y h hs popy of s du o h sp f h ndpndn w ss 8 dgs p. Tpo ouon of h w funon s pssd s:, (, ) p H(, ) d ( ) Thfo h oon bwn n n s ( ) nd fn s nd ( ) P ( ( ) ( )) p H d 3 (, ) d d d 3 3,, s: (3-96) (3-97)

113 In qunu hns, h ss nod o on: ʹ 3 d Doppng h ps, h oon ngs hn wn n h fo: (3-98) P( ( ) ( ) ) p (, ) H d (3-99) In qunu hns, hs oon psns h pobby dnsy fo h n s o o no h fn s. ny phys poss n b nyd ssy n s of op puds suh s dsbd bo. If dffn possb fn ss dsngushb, hn h jon pobby s obnd spy b ddng h of h sp pobbs. How, f dffn possb fn ss ndsngushb (.g. on on o dffn on hng do), hn h jon pobby s opud by ddng h puds nd ony hn opung h gnud. In h w od, hs u s pnd by h f h ndsngushb ps (.g. wo ons) w pks wh h s fquny hss. Non-dn ps, whh h dffn fquny hss, h b phs bwn h wo ws nd hfo ny nfn bwn h wo ws woud g o o Fons nd osons Ps whos oons opud odng o h bo us d fons n hono of h physs Eno F. Fons onsdd o b h fundn ps of nu. Ths nud ons, poons, nuons, nunos, nd quks. R h h Pu uson pnp ws dd fo h ssupon h h p w funons w gnfunons of n obsb (.g. spn). If hs s no h s, hn h s no uson pnp. Ps whh n b supposd d bosons n hono of physs Syndnh os. Eps nud phoons nd π sons. Mup bosons y os wh h n y h s s (nd s poson). In qunu hns h boson wo-p w funon ssfs: Ths ondon s wys ssfd f, so h s no uson pnp fo bosons. (3-) To s how spn s d o sss, onsd ssss phoon whh n h pn w ppoon ssfs h quon: ( kˆ )( kˆ )Q Q Q (3-) 3

114 Eh ( ˆ k ) Q o ( ˆ k ) Q. In h s h o Q obys onon quon nd s hfo h quny usd o opu oons. Q s o, whh nsfos und oon wh spn on. Mup phoons n b supposd spy by ddng h Q us whou h nfn ssod wh spnos. Fo noh p, suppos fons nd sohow bound ogh wh jon w funon whh ssfs h uson pnp:,, (3-) If w us, o opu oons wh n dn p oposd of fons nd, w h:, ʹ, ʹ ʹ, ʹ, ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ Hn opos ps fod fo wo fons bh ssy k bosons. (3-3) In h Sndd Mod of Physs, h wpon s h fundn ps fons whh n hough fds, nd ps ssod wh h fds bosons. W h sn how hs npon n s fo n undyng ss w poss Po Knowdg nd Sss Inpon of qunu sss n b onfusng. Consd h s of Shodng s. Th s pd n bo whh onns do n, don do, nd posonous gs. If h do s ggd by do dy hn w n un gg h s of h poson nd hby k h. odng o qunu sss, ny gn h s no y hn h h w b dd o, bu h h dspon nos op pud fo h possby. Jus s on sss w dsbd bo by op supposon of spn up nd spn down ss, h s f s dsbd by op supposon of nd dd ss. Physss hfo pd o sy h h s n supposon of ng nd dd ss, whh s h bsud. Th dffn wys o so hs pdo, bu h sps souon s o sy h h y s h dd o, nd no boh. Th op pud y nds ou knowdg (o k of knowdg) of h suon. Physss h gny jd hs og bus hy n d h ss sss (.g. h pobby h h s dd) shoud b opud n y h s nn s h qunu sss. Th Copnhgn npon of qunu hns poss h h ss npon of h op w funon s so h phys npon (.. h no dns phys bs bus f h w hn h oons woud b opud dffny). How, w n obn h s oons whou h b npon h h s py nd py dd un w opn h bo. 4

115 Hydogn o Th poon podus Couob pon ( Φ Z ). Ngng h o pon n h ogn on quon (3-65) yds: Φ Ω3! (3-4) ssu s bfo po gnu E, nd ssu h h ngu gnfunon ( Φ ), hs n py nd ( Φ ), odd py. w funon of h fo ( ) yds h oupd d quons: E Φ Ω G! E Φ Ω F! κ F κ G (3-5) Souons o hs oupd quons obnd s foows (.g. Shff 968): Fo g h sypo quons : [ E Ω] G F [ E Ω] F G whh obn o yd: [ Ω ] F F (3-6) E (3-7) W skng bound s wh p wh [ Ω E ] s F ( ). E < Ω. Thfo h sypo bho Now : F ( ) f ( ) p( ) ( ) g( ) p( ) G Th oupd quons bo: E Φ Ω g! E Φ Ω f! κ f κ g ssu h f nd g n b wn s pow ss: (3-8) (3-9) 5

116 f g ( ) fν ν gν ν ( ) s ν s ν L γ Z! nd h pows of : [ E Ω] gν fν γgν [ s ν κ] fν [ E Ω] f g γf [ s ν κ] g ν ν ν W n n h ( ν ) s o g onshp bwn f ν nd g ν. {[ E Ω] [ s ν κ] γ } g [ E Ω] γ [ s ν κ] f ν ν { } ν whh fo g n bos [ E Ω] gν fν. Fo n: γ g γ f [ s κ] f [ s κ] g (3-) (3-) (3-) (3-3) Th dnn fo hs oupd quons us b o. Ths ondon yds souon fo s: s ± κ γ (3-4) R h h u w funon onns n ddon fo of /. Thfo w hoos h pos sgn h so h h souon s gu (o ony sghy dgn f s <) h ogn. Usng h on bwn offns dd bo, h sypo bho fo g ν s: fν νfν gν νgν Th o bwn suss s hs h Tyo ss pnson fo p(): p ν ν! ( ) [ ] ν (3-5) (3-6) If h ss pods o nfn n hn h w funon woud b nfn g us of. To k h w funon fn, h ss us n so fn u of ν. Cng hs u n', Eq. (3-) yds h on bwn h hghs offns: g n ʹ [ E Ω] f n ʹ (3-7) Cobnng hs on wh q. (3-) yds n psson fo h hs fquns: [ s nʹ ] [ s nʹ ][ Ω E ] Eγ (3-8) 6

117 Song fo! E : { γ [ s ʹ ] }! E! Ω n (3-9) Ths h ds ngy s of n on n Couob pon. Th fo of!, whh s ngy nd fquny, s ssud o b h ng of h squd w funon. Dno h ngy by ε! E nd ss by! Ω. Ths ngy s w uy dd by Sofd [96] usng h od of s p popgng n p obs. Th wo n sous of dspny fo h u hydogn ngy s. Fs, w ssud s pon, pyng h h nuus s unffd by h psn of h on w. y nogy wh ps w n po h uons by png h on s ngy! Ω wh h dud ss ngy! Ωʹ [ ], wh p s h poon ss. Sond, w h ngd ny ffs of h gn o pon. Th ngy s ypy ssfd usng pos ng pnp qunu nub n nd pos hf-ng ngu qunu nub J κ : n J nʹ In s of hs qunu nubs h ngy s : ε! Ωʹ n J γ [ J ] γ p p (3-) (3-) Tb 3-II ops sud ngy s ( o h gound s) wh ngy s ud usng hs fou. Th onfguon b (nl) nuds h pnp qunu nub n foowd by od fo h ob ngu onu L: s, p, d, f3,. No h h fou bo dos no dsngush bwn dffn L us fo h s n nd J. Wh h gn wh pn s good, us b nod h h ssud Couob pon s spy p (s s so n onnon qunu hoy). Fo op hoy h pons of h nuus shoud b dd fo s f p w funon. Confguon J Msud L (V) L Copud fo (3-) s / s / p / p 3/ s / p /

118 3p 3/ d 3/ d 5/ s / p / p 3/ d 3/ d 5/ f 5/ f 7/ n Tb 3-II Msud nd opud hydogn ngy s. Rhnko, Yu., Jou, F.-C., Kh, D.E., Kd,.E., Musgo,., Rd, J., Ws, W.L., nd Osn, K. (7). NIST o Sp Dbs (son 3..), [Onn]. b: hp://physs.ns.go/sd3 [7, My 8]. Non Insu of Sndds nd Thnoogy, Ghsbug, MD Sys I nno b h God s wk f-hnd Wofgng Pu Sp nson Sp nson (onnony d h py opon, P, hough w w us h M fo ong) s h poss of nng h h sp s. Ths opon osponds o o g foowd by 8 dg oon bou h s ppndu o h o. Sn oon dos no ff ny phys ws, w w sos subsu h o g fo sp nson whn fng o gn phys onsquns. Py onson s gny kn o n h whn sp nson s ppd o ny phys poss, h sung poss s quy fqun n nu. Py oon ns h poss nd s o g no quy ky, nd py oon ns h sp nson of phys poss yds poss wh no phys npon. In hs hp, w no nsd n h fquny of oun of ns nd h o gs. W ony onnd wh h quson of py oon: Is h o g poss possb n nu o no? W w f o py oon s o syy, nd sn of o g poss s o syy. 8

119 Fgu 3. dy of 6 Co (f) hs o g whh s onssn wh b dy of s n ounp 6 (gh). Th b S psns h nu spn. Co Whn wd n o, known phys posss pp o pod s f nd n- w hngd. n p s h b dy of 6 Cob shown n Fgu 3.. Th sps pnon fo hs obson s h sp nson hngs nd n-. Th h bss fo hs pnon ws dd by Cos [b] s foows. L us onsd how h w funon hngs und sp nson. Connon py opo D s ogn quon fo f p hs h fo: Ω3 (3-) wh Ω!. Th -s y b kn s: 9

120 (3-3) Wh ~ s h psudos gny, s w b sn bow. Th spn s u u s gny ( ): (3-4) Mupyng h D quon by nd ddng h Hn onjug quon yds onnuy quon: [ ] [ ] (3-5) Ths onshp s suffn o sbsh h pobby dnsy ( ) nd un ( ) s h oponns of Lon fou-o. hough h bo nyss s ssfoy, s uny fshonb o us h noon: γ γ γ γ γ ; ; (3-6) nd upy h n h ogn D quon (3-) by γ o obn: γ γ γ µ µ Ω (3-7) Ths podu nno h ny ff on h nsfoon pops of h D s. Th onnon py opo P s ssud o h h fo: ( ) ( ) U P. I s dd fo h qun h h D quon n h fo (3-7) b nn wh sp o h nsfoon: ( ) ( ) ( ) Ω γ γ U U U (3-8) Inng h py opo yds: ( ) ( ) ( ) Ω γ γ U U U U U U (3-9) Equn wh h ogn D quon qus: ; ~ ~ ~ ~ ; 3 ; ; 3

121 U U U γ U γ U γ U γ (3-3) Ths ondons ssfd by Uγ. Whn n by phs fo h onnon py opo s hfo: ( ) γ ( ) ( ) P (3-3) 3 Th wo pobs wh hs don. Fs, h fo ( ) U ( ) P s no h os gn possb opo. Fo p, h onnon hg onjugon opo nuds op onjugon. Sond, h γ 3 s no nd bus s psud o psn po oponn of fou-o. Ths uson s nnd by wng h pobby dnsy nd un oponns s γ nd γ, spy, wh γ. Ths hng of noon dos no hng h f, how, h h pobby dnsy s ndpndn of γ. Th ssod wh h po p of h pobby un 4- o s h dny, no γ. Ths s n pon fw n h onnon don of h py opo. Sn h 4-o (, γ 5 ) s ndd Lon-nn, h s bsouy no bss fo h h γ s po oponn. On h ony, w w show h γ s goy d o w oy nd y qu sonby b nd by sp nson. W w s h h sung sp nson opo ns of h s n h odfd D quon (3-7). Nw sp nson opo In dsussng sp nson, w b nssy o dfn wo dffn un gny nubs. s dfnd bo, h podu of spn s s u s wh sp o sp nson: 3 (3-3) Th -s no nod n sp nson, whh ns h w oy bu no h spn. How, w n dnfy h s ssod wh po os whh h h s gb s h -s. Th s dfn dons o h oy o 5 γ, wh h bks nd pon u. On n so dfn bsou os (,, 3 ). If h w funon s n gnfunon of oy gnd wh sp s so h, hn (usng ):

122 3 [ ] 3 [ ] 3 [ ] [ ] (3-33) Ths sus foow fo h f h s fon opo fo boh nd 3, nd h ony nub qu o s ng s o. Thfo nd ndd ppndu o oy 3 fo oy gnfunons. Fo p, n ou noon h w funon ( ) T s sunous gnfunon of,, nd 3 3. Thfo h h os,, nd 3 uuy ohogon os (f-hndd) n h dnson sp, s fo oy gnfunons. Th o s p o ˆ. Roon of h o by 9 dgs bou h o yds 3, whh s p o ˆ 3. Ths s of ous h s s oon of ˆ by 9 dgs bou ˆ, whh s ssod wh h. I s hfo h fo oy gnfunons, h os psnd by (,, 3) goy qun o h bsou os psnd by (,, 3). W ssu h h os,, nd 3 po os so h h o sp (,, 3) dos no h d py. Th fo γ 3 n h onnon py opo psns oon by 8 dgs bou h 3 s ( ˆ 3 n ou p). Ths opon ns ony wo of h h ohogon os ssod wh oy. Cop hs suon wh ss nss ws n sod. W oud dfn n opo (nogous o h D P opo) whh fs h qubu poson of h pon n h sod, nd so fs h w oy don. W so n o dspns nd os ong on of h wo s ppndu o h w oy. Th sung fd w woud popg ong jus s on woud p fo h spy nd w. u of ous h opo w dfnd s no h sp nson opo, bus w fd o n on of h s of h o dspn nd oy of h sod du (n o w nd wo of h h o s, ospondng o 8 oon bou h hd s). Sy, h D P opo ns h w (o p ) oy don, bu ns ony on of wo oh quns whh goy d o h w oy ( 8 oon n h oy-psnon sp). W w d nw sp nson opo whh ns h os,, nd 3 ssod wh oy. Th spn s oponns of psudoo nd shoud no b nd. Thfo h sp nson us b opshd by nng h h s (,, 3 ). Ths qus h h ssod gny ~ b psudos, s ssud bo. Th un gny ssod wh ss s ssud o b psudos sn s upd by ( 3) n h ogn D quon. Th os of h dffn gns n b fd by fong h D w funon n nn s o h of Hsns [967]:

123 ( ) p( ϕ ) p( ) p( ζ ) ~ (3-34) I s h ε jk j k s ssod wh oon n h pn ohogon o h ~ s. Sy, 3 s ssod wh oon n h oy-psnon sp. N w dfn nw w funon n whh gny psudos fos nd: # ~ ~ ( ) ( ). Ths psudos onjugon opon dffs fo op onjugon, whh ns boh s nd psudos gns. Psudos onjugon ns sn: # # # # # [ ] [ [ ] ] [ ] (3-35) Th sp nson (o ong opo M) whh ns of h oy os, s hn (whn n by phs fo): # ( ) ( ) ( ) M M (3-36) Ths opo ns obsbs opud fo,, nd 3 ndpndny of h hng n sgn of. Th D quon fo p n ogn pons s: [ ~ Ω Φ ] 3 (3-37) Whn ppd o hs quon, h py opo ns 3,, ~, nd (h s nd bus hy n-ou wh ). Dnong spy nd quns wh subsp M, h spy nd D quon s: [ ~ # # Ω Φ ( )] 3 M M M M M M (3-38) W ssu Ω M Ω. Th nsfod quon hs h s fo s h ogn D quon p fo h sgn of h o pon. Ths sgn hng s nssy fo onssny wh gug nsfoons. Th gug nsfoon Φʹ Φ ʹ ( χ ) ʹ p χ χ (3-39) suggss h h s pon y b gdd s d nd h o pon y b gdd s sp d. Tkng Φ g nd G g woud h fo of h quon nn: [ ~ # # Ω g ( G g )] 3 M M M M M M M (3-4) 3

124 Th s nd o pons us h oppos sp nson gnus. W w ssu h: M M # [ Φ( ) ] M Φ M Φ( ) # [ ( ) ] ( ) M Th nsfod D quon s hn: M [ ~ Ω Φ( ) ( ) ] 3 M M M M (3-4) Wh hs nsfoon pops, w w show h h nw py opo s onssn wh n hng of nd n-. Egnfunons nd gnus N w onsd h ff of h nw py opo on h gnu quon. Fo spy w ssu h o pon o b o. ssung po dpndn p E, h gnu quon s: ( ) [ E Φ ] Ω Th opo 3 ~ n b fod: L [ ] Th wo-oponn ngu souons of h gnu quons ( ) [ ] κ (3-4) (3-43) ( LΦ ) κ nd, LΦ, w known [jokn nd D 964]. Ths wo ngu souons d by ( ) ( Φ Φ ) nd yd oppos gnus und oodn,, nson ( ). Ony h u s gny n pp whn hs funons. Dno wo w funons s: ~ ( ) ( ) GΦ, ( ) ( ) FΦ, o ~ ( ) FΦ, ( ) GΦ, (3-44) Eh of hs s n gnfunon of h onnon py opo, bu hy hngd by h nw sp nson opo: M M ( ) 4 ( ) ( ) # ( ) ( ) ( γ ) ( ) ( ) 4 ( ) ( ) # ( ) ( ) ( γ ) ( ) Usng ( ) n h (ogn) D quon yds h oupd d quons: (3-45) 4

125 [ E Φ Ω] G κ F κ [ E Φ Ω] F G (3-46) ( ) yds s oupd quons wh oppos sgn of E nd Φ, s pd fo hng of nd n- (on npon s h nps psn pos-ngy hos n s of ng-ngy ps. If w wn wo pos ngy souons d by sp nson hn w nd o us dffn gnfunons). Th ngy s fo ( ) n ng Couob pon hfo qu nd oppos o h ngy s of ( ) n pos Couob pon. Th nd fo hs su ws h son fo ssung h h py Φ. opo oy ns h s pon ( ) Wk nons Th pojon opo fo f-hndd spno oponns s: L ( I ) (3-47) Th un I s s nd s psudos. How, h pojon opo dos no o o syy so ong s h fd ounp R ( I ) M s s physy pusb s h ogn pojd w funon. Sn h nw sp nson opo hngs nd n, of h ny ps nod n h wk non do n f h spy fd ounps n nu (ons nd posons, f-hndd nunos nd gh-hndd n-nunos,.). Th h fo of h wk fo s ny onssn wh o syy. Copson wh onnon PC Th onnon PC opo s: * 5 * ( ) γ γ ( ) γ ( ) PC (3-48) Ths dffs fo ou sp nson opo by n by phs fo, h fo of γ nd onjugon of h s gny (dnod by *# ). Th fo s, whn phs fo, spy oon by π bou h s: p( π ). Cop onjugon of h s gny ns h spn oponn S: PCS * * * [ ] [ [ ] ] * * S (3-49) Thfo h n ff of *# s o n h spn. Th ddon fo of γ ns oy by oon of h oy-psnon s. ppd o h nd n- gnfunons, s qun o nng h 5

126 sp guns n h w funons. Thfo h onnon PC opo, hough hngs nd n-, dffs sgnfny fo h nw sp nson opo M T s Physy, s us n h d opo, oy, nd spn ndpndny of h hng n gun. On of h ogn pons us so b nd. Voy nd spn boh nd by h nsfoon: ( ) ( ) *# ( ) (3-5) Th oy-psnon sp 5 4 (, γ, γ ) γ s unffd by hs nsfoon. y ons, h onnon s opo * T ( ) ( ) ns 4 γ bu no oh s of oypsnon sp. Ths suggss h h onnon s opo s so no. How, unk h onnon py nsfoon, h s no p dn o d hs. ppd o h D quon, h nw s opo yds: 5 {[ ~ 5 γ γ Ω Φ γ ] } 5 [ ~ *# *# 5 γ γ Ω Φ γ ] (3-5) W o h ogn fo of h D quon f Ω Ω (.. Ω s n gnu of n opo whh nsfos k d) nd h pons npd s ds. W ssu h pons o nsfo s: *# [ Φ( ) ] Φ Φ( ) *# [ ] ( ) odng o ou npon of nd n- s o-gs, s dos no hng h wo Cobnd Tnsfoons Th obnd M nsfoon s: 4 * * (, ) γ (, ) γ (, ) (3-5) M (3-53) Ths s osy d o h onnon hg onjugon nsfoon C : 4 * * (, ) γ (, ) γ ( ) C, (3-54) Th onnon hg onjugon opo ns h spn nd oy n p, whou nng h sp o po oodns. In s of dyn bho, hg onjugon hs h s ff s nng h sgn of h ogn pons n h D quon. Th onnon PT nsfoon s: 6

127 * (, ) γ (, ) PT (3-55) Ths dffs fo h nw M nsfoon by h fo 5 γ, whh os h oypsnon sp by 8 dgs. Th onnon PCT nsfoon s: 5 (, ) γ (, ) PCT (3-56) Ths nsfoon s h onnon ho on bwn nd n. Copd wh h M opo, dffs ony by hg onjugon (whh hs s ff o song h pons nd by M) nd by h fo 5 γ Mh nd Phys Pops of Spnos ou psn hnkng bou qunu hns s nfsd wh h dps sonpons. Sphn Gu, nhony Lsnby, nd Chs Don [993] Spnos nd Inn Podus n undsndng of so h pops of spnos w b usfu. Epssons fo phys quns (.g. Q) opud fo opos (.g. Q) n h fo: Q [ ] [ ] Q Q [ Q ] [ Q ] [ ] (3-57) Sn h djon of s s s op onjug, h phys quny Q s -ud. Whn ngd o sp, suh pssons k h fo of n nn podu: Q 3 d [( f, g) ( g, f )] [ f g g f ] Th quny <Q> s h ngd u (o pon u n QM). (3-58) op sp of funons wh n nn podu ssfyng so sp pops (.g. ny) s d Hb sp. I suffs fo ou puposs o sy h h nn podu dfnd bo ssfs of h nssy. [No: h nn podu s ofn dfnd usng ony on of h s n h ngnd bo (whou h fo of on-hf). Wh hs dfnon o dnss y b op n hough h ng s.] Th nn podu bwn wo spno funons s nogous o h do podu bwn wo os o h oon bwn wo s funons. Th nn podu of spno funon wh sf s s pos-dfn gnud: f 3 ( f, f ) f f d (3-59) 7

128 In s of oponns hs s: * 3 ( f, f ) f f d Th o pojon p ( ) of on funon ( ) ono noh funon ( ) p ( ) Φ Φ Φ ( ) ( ) ( ) Φ( ) Φ Φ ( ) Th gob pojon Φ ( ) ( ) ( Φ, ) P Φ( ) Φ Φ Φ s dfnd s: P of on funon ono noh funon Φ s dfnd s: (3-6) (3-6) (3-6) Th pojon by sf gny fs o h gob pojon n h u. Fo opson, h pojon of o ono o b s h oponn of h s p wh b: b Pb b [ bˆ ]b ˆ b (3-63) If n opo hs Hn ( H H ) nd n-hn ( ) ps, hn ony h Hn p onbus o h phys u: [ [ H ] ] [ H ] ] [ H H ] [ ] H Q (3-64) Fo hs w n onud h h ondon fo -ud nn podu s h h opo s Hn ( Q Q). Fo p onsd h sp d j j : ( f, f ) f j Th djon s: j f d ( f f, ) [ f ] j j 3 f d Ingon by ps yds: j 3 [ f ] f d f f d [ ds f f ] d f [ f ] j j j j (3-65) (3-66) j (3-67) W ssu h h spno funons f o o po o hng h boundy of ngon (.. h h boundy s suffny f h h s no onbuon o h ou ng ousd h boundy). Ths ssupon ows us o dsd h boundy, bu s ou by o g phys npon o h o funons. ssung h boundy onbuon o b o, w h: 8

129 9 ( ) ( ) f f f f j j,, (3-68) Hn h sp d s n n-hn opo (nus sgn h hn pus sgn). Cy hs popy hods fo oponns of h gdn, so w n w: ( ) ( ) f f f f,, (3-69) Whh ds o h h obous psson fo h ngd u: [ ] d f f d f f d f f f f Ths onshp n opo fo s: [ ] [ ] f f f (3-7) No h h fo of h gdn opo s no hngd by h djon opon ( ). Th sgn hng os fo nsposng h opo fo h f o h gh sd ( ngon by ps). No h: [ ] [ ] [ ] f f f f f f (3-7) Ths psson s obousy no o n gn, bu s ou ng s o s ong s h funon f fs off suffny pdy n h ngon bounds. I s sp o onsu Hn opo fo h gdn opo by upyng wh h un gny: ( ) ( ) f f f f,, (3-7) M gb fo podng fuh, w b usfu o bu so onshps bwn s. y,, 3 (3-73) In h D psnon of qunu hns hs s psn ( ) 5 5,, γ γ γ γ, spy. In sph oodns h sg s :

130 3 sn os os sn sn os os sn os sn sn os os sn sn os (3-74) Th opo hfo yds: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] sn sn os os sn sn os os sn os sn sn os os sn sn os (3-75) In ynd oodns (,, ) h s :

131 os y sn sn y os (3-76) W Pops of M W h shown h ss w hoy n dsb Fon dyns. Ths su nds suppo o n ffos o h ss h (o h) s du of popgon of ws. Duffy (6) hs suyd odn h hoy. Th od of uu s n d s sod ws qu sussfu n pnng ss pops of gh n h 9h nuy (s.g. Whk (95)). Qunu ffs ony ppn n nons wh, whh gh b npb s ss soon ws. psn h pps o b no ssfoy dspon of oon ws n n d s du. Kn (989) pd o nud oons n h s ngy bu ws opd o nodu nw s onsns dpndn on n by s ngh. Cos () showd h oson ws (wh oon s p o w oy) n b dsbd by D quon. In hs book w us w quon wh onon s s h ss bss fo h qunu hn onu nd spn opos. Sh (9) ny donsd h u od n yd h s goup suu s h Sndd Mod. Ths od s sonshngy s o h ong s od whh Mw usd o d h quons of ogns (hough h ong ps bodng Mw s s w pd by n unspfd bwn h s). Mny phys pops of n b dd fo w od of. Th Unny Pnp pps o ss ws nd psns bs popy of Fou nsfoons. Lon nn s so popy of ws, nd Sp Ry s hfo onsqun of ny w hoy of. Fo p, h s phnonon of don s spy pnd by h f h sony soon ws u pod obs (.g. s) whs ong soons u obs whh h ong w phs n h y (.g. sp o yod). Hn ong ok whh ouns soon w obs ks fs hn s ong ok. bsou oon wh sp o h h woud no b 3

132 db bus whou po knowdg of bsou oon s unknown whh sgn s Dopp shfd h sou o h, o boh. Th hs bn onsdb ns n dsbng ny ps s soon (o pk) w souons of nonn D quon. S Rñd (983) fo sho w. Mo n woks nud Fushhyh nd Zhdno (997), Gu (998), ohun nd Coopsok (999), nd M (6). Ths ffos suff fo bnss n h ho of nonny. Idnfon of h D quon wh sond-od ss w quon pods sp ns fo npng, y o nogousy, ny non-n s. Th Kn-Godon (o s Shödng) opo n b fod no podu of wo D opos ng on h w poon (o pud) : { M } { γ γ M}{ γ γ M} wh h ouon ons : γ γ γ γ γ γ γ δ j j ε jk γ k (3-77) (3-78) Th quns γ µ nd un gny () h dony bn gdd s s, bu hy n so b npd goy usng u os [Hsns 967, 973, 99]. Th w poon s ss 3-o n Gn sp-. Th Mnkowsk of y s nodud hough h opos. If w dfn w funon: Ψ { γ γ M} (3-79) hn h sun fs-od D quon s qun o h ogn Kn-Godon quon: { γ M} Ψ γ (3-8) In h bo s h wo D opos h dffn sgn fo h ss. Rownds [998, 5, 6] nd Rownds nd Cun [] usd obnon of u 4-os nd qunons o w h D quon n npon fo n whh h wo suss D opons dn. Ths fouon yds n gn ssfon of p ss whn h Sndd Mod. Sndd souons of h Kn-Godon quon yd dffn ngy gnus hn h D quon (s.g. Shff [968]). Ths su s qu pu gn h f h h oponn of h D w funon uy ssfs h Kn-Godon quon! Fong h Kn-Godon quon nno hng s gnus. Th pob s h n h usu nyss of Kn-Godon, h ngu funons hosn o b gnfunons of h squd ob ngu onu L, whs n h nyss of h D quon h ngu funons gnus of h squd o ngu onu J. Th dffn s no n h quons, bu n h ho of ngu gnfunons. Th usu nyss of h Kn-Godon 3

133 quon ngs h spn onbuon fo oon of w oy. Ths souons psn bosons wh o spn. Souons obnd by usng ngu gnus obnd fo D hoy psn fons wh spn on-hf. In h n hp w sh s h s gon fd nd s ff on h sp- y b npd s spy yng gh spd. S Whk (954) fo h hso dopn of hs d whh ogns wh Ensn (9, 9) nd hs so bn nsgd o ny (d F (97), Ens ()). Ths npon s onssn wh gn y, whh so pds on of gh spd popoon o h gon pon (Ensn 956). In n s sod h, opsson o ons n sy py b w spd nd hn pod sonb phys od fo bs gon ffs. 3.. Suy En f you noy of on, h uh s h uh. Mohnds Gndh In hs hp w np h D quon s ss sond-od w quon fo oon ws n n s du. Th fs od sp nd po ds psnd by bspno w funon. Hf-ng spn s bub o h o-sn of ws ng n oppos dons ong h gdn s. Th w funon n b fod no onsn, sng pud, h-dnson Lon oy boos, oon, nd n by hng of psnon. W nfn yds boh h Pu uson pnp nd h Lon fo. Th ogn pons psn w nfn. Inpng h ss bspno quon s dsbng n on, s found h h ss s ssod wh dy nwd on of h w, suggs of soon. Th ss hoy s onssn wh py onson. Hn pps h ss w hoy onsus n ngb bss fo h phys bus of. Rfns JS 964 On h Ensn Podosky Rosn Pdo, Phys. :95 jokn, J.D. nd D, S.D. 964 Rs Qunu Mhns, (Nw Yok: MGw-H ook Co.) p 53. oh D 95 Suggsd Inpon of h Qunu Thoy n Ts of Hddn Vbs, I nd II, Phys. R. 85:66-93 oh N 93 On h Consuon of os nd Mous, Ph. Mg. S. 6 6(5):-5 on M nd Jodn P 95 Zu Qunnhnk, Zs. Phys. 34: on M 96 Zu Qunnhnk d Sossogng, Zs. Phys. 37: d og LV 94 Rhhs su Tho ds Qun, PhD Thss, (Ps: Unsy of Sobonn) d og L 98 L énqu onduo, Rppo u 5 Cons d Physqu Soy. usss, 97 (Ps: Guh-Vs) d og L 963 (hp://www-goups.ds.s-nd..uk/~hsoy/rfns/og.h) Cos R Toson Ws n Th Dnsons: Qunu Mhns wh Tws, Found. Phys. L. 5:

134 Cos R 6 Pogss Towd Css Thoy of M, Po. Phys Inpons of Ry Thoy X, d M C Duffy (London: sh So. Phosophy of Sn), o b pubshd. Cos R 9 Css D Equon, n: Eh Sp- & Cosoogy, Vo 3 (Phys uu, y, nd qunu physs) ds. M.C. Duffy nd J. Ly, pon, Mon, Cos R E Dspon of Roon Ws n n Es Sod, d. pp. Cffod gbs :73-8. Cos R b Th Mo Syy of M nd n, d. pp. Cffod gbs : Congh WN nd Gnwood D 998 n Inoduon o h Sndd Mod of P Physs (Cbdg Unsy Pss) p Dsson C nd G LH 97 Dffon of Eons by Cys of Nk, Phys. R. 3:75-4 Dby P 96 Qunnhypohs und Zn-Effk, Phys. Z. 7:57 D P M 98 Th Qunu Thoy of h Eon, Po. Roy. So. (London) 7:6-64 D P M 989 Mhods n Tho Physs, Fo Lf of Physs (Sngpo: Wod Snf) D P M 95 Is Th n h? Nu 68, pp D P M 95 Rpy o L. Infd. Nu 69:7. Duffy MC 6 Th Eh Conp n Modn Physs, Ensn nd Poné: h phys uu, d. V. V. Dogo (Mon: pon). Ensn 9 Üb dn Enfu8 d Shwkf uf d usbung ds Lhs, nn. Phys. (Lpg) 35: Ensn 9 Lhgshndgk und Sk ds Gonsfds, nn. Phys. (Lpg) 38: Ensn 956 Th Mnng of Ry (Pnon: Pnon Unsy Pss) Ffh Edon, pp Ensn nd on M 5 Th on Ensn Ls : Fndshp, Pos nd Physs n Unn Ts (Nw Yok: Mn) p 88 Epsn PS 96 Zu Tho ds Skffks, nn. Phys. (Lpg) 5:489-5 Ens JC, sng PM, Gog S, nd Nnd KK M ws n gon fd: n nd of fon fo ss ps n gn y,. J. Phys. 69:3- d F F 97, On h gon fd ng s n op du, Gn. R. G. : Fynn RP, Lghon R, nd Snds M, 963 Th Fynn Lus on Physs, Vo. I (Rdng: ddson-wsy) p -6. Fynn RP, Lghon R, nd Snds M, 963b Th Fynn Lus on Physs, Vo. II (Rdng: ddson-wsy) p 38-8 Gh W nd Sn O 9 D Epn Nhwss d Rhungsqunung Mgnfd, Zs. Phys. 9: Godsn H 98 Css Mhns (Rdng nd Mno Pk: ddson-wsy Pubshng Co., In.) p

135 Godsn S ohn Mhns, Th Snfod Enyopd of Phosophy (Wn Edon), Edwd N. Z (d.), URL <hp://po.snfod.du/hs/wn/ns/q-boh/> Gu YQ 998 So Pops of h Spno Soon, dns n ppd Cffod gbs 8() 7-9 Gu S, Lsnby, nd Don C 993 Igny Nubs no R h Go gb of Sp ( Hsnbg W 95 Üb qunnhosh Uduung knsh und hnsh hung, Zs. Phys. 33: Hsns D 967 R Spno Fds J. Mh. Phys. 8(4): Hsns D 973 Lo obsbs n h D hoy, J. Mh. Phys. 4(7): Hsns D 99 Th Zbwgung Inpon of Qunu Mhns, Found. Phys. ():3-3 Hdk J 999 Spnos n Physs (Spng, Nw Yok). Jodn P nd Wgn E 98 Üb ds Push qunbo, Z. Phys. 47: Kn H 989 Gug Fds n Condnsd M o II (Sngpo: Wod Snf) p 59 Kn H 7 Muud Fds n Condnsd M, Eogns, nd Gon (Sngpo: Wod Snf) p L TD nd Yng CN 956 Quson of Py Conson n Wk Inons, Phys. R. 4:54. Msun S. nd Tsuu H., Phys on bwn qunu hns nd soons on hn s od. Phys. R. 46 (99), Mhson nd Moy EW 887 On h R Moon of h Eh nd h Lunfous Eh,. J. S. (S. 3) 34: Mos PM nd Fshbh H 953 Mhods of Tho Physs o I (Nw Yok: MGw-H ook Co.) pp Pu W 95 Ub dn Zusnhng ds bshusss d Ekonnguppn o d Kopsuku d Spkn, Zs. Phys. 3: Pu W 97 Zu Qunnhnk ds gnshn Ekons, Zs. Phys. 43:6-63 Rñd F 983 Css Nonn D Fd Mods of Endd Ps, Qunu Thoy, Goups, Fds, nd Ps d O u (sd: Rd) pp 7-88 Rownds P 998 Th phys onsquns of nw son of h D quon, n G. Hun, S. Jffs, nd J-P. Vg (ds.), Cusy nd Loy n Modn Physs nd sonoy: Opn Qusons nd Possb Souons (Fundn Thos of Physs, o. 97, Kuw d Pubshs, Dodh) Rownds P nd Cun JP Th onnon bwn h Hn-Nbu quk hoy, h D quon nd fundn sys Nu Phys. 684, Rownds P 5 Rong dundny n s qunu hns, Ppn X:physs/5788 Rownds, P 6 kng h D Cod, n Po. P.I.R.T. X., M. C. Duffy, d..(london) o b pubshd Ruhfod E 9 Th Sng of nd Ps by M nd h Suu of h o, Ph. Mg. (S. 6) : Ruhfod E 94 Th Suu of h o Ph. Mg.(S. 6) 7: S 969 Wk nd Eogn Inons, Eny P Thoy, d. N. Sho (Sokho, qs nd Wks) p

136 Shff LI 968 Qunu Mhns (Nw Yok: MGw-H, Thd Edon) pp Sh I, 9 Condnsd M Inpon of SM Fons nd Gug Fds, Found. Phys. 39():73 Shodng E 96 Qunsung s Egnwpob. (Es Mung.) / Qunon s n gnu pob (fs p) nn. Phys. (Lpg)79: Shwhd K 96 n S. 548 Shff LI 968 Qunu Mhns (Nw Yok: MGw-H, Thd Edon) pp Sho S nd Tng HZ 5 Inon fo h soy ws of nonn D od, Phys. L. 345(-3):9-8 Sofd JW 95 Zu Tho d shn S, Sungsbh d Münhn kd d Wssnshfn, Physksh-hsh Kss Sofd JW 95b D Fnsuku d wsssoff- und wsssoffhnhn Lnn, Sungsbh d Münhn kd d Wssnshfn, Physksh-hsh Kss Sofd JW 96 Zu Qunnho d Spknn, nn. Phys. (Lpg) 5:-94, 5-67 Sofd JW 96b Zu Tho ds Zn-Effks d Wsssoffnn, n nhng ub dn Sk-Effk, Phys. Z. 7:49-57 Tkbysh Y 957 Rs hydodyns of h D, Supp. Pog. Tho. Phys. 4():-8 Thoson J J 897 Chod Rys Ph. Mg. 44:93-36 Thos LT 96 Th Moon of h Spnnng Eon, Nu 7:54 Thos LT 97 Th kns of n on wh n s, Ph. Mg & J. S. 3:- Thoson GP nd Rd 97 Dffon of hod ys by hn f, Nu 9: Toong SI 974 Spn w guu / Th Soy of Spn (U. of Chgo Pss, nsd by Tksh Ok) p Uhnbk GE nd Gouds S 95 Esung d Hypohs o unhnshn Zwng duh n Fodung bügh ds nnn Vhns jds nnn Ekons, Nuwss 3: Wnbg S 967 Mod of Lpons Phys. R. L. 9:64 Whk E 954 Hsoy of h Thos of h nd Ey, o. (Ednbugh: Thos Nson nd Sons Ld.) Wson W 95 Th qunu hoy of don nd n sp, Ph. Mg. 9:795-8 Wu CS,. 957 Epn s of py onson n b dy, Phys. R. 5:43-45 Fgus Th foowng fgus bd o b f of opygh son, nd w obnd fo h sous sd. Oh fgus h ogn woks o d n h fgu pon. Fgu 3..J. Thoson (856-94). Sou: hp://nobp.og/nob_ps/physs/us/96/hoson-bo.h Fgu 3. Ens Ruhfod (87-937). 36

137 Sou: hp://nobp.og/nob_ps/hsy/us/98/uhfodbo.h Fgu 3.3 nod J.W. Sofd (868-95). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/sofd.h Fgu 3.4 Wofgng Pu (9-958). Sou: hp://www-hsoy.s.s-ndws..uk/hsoy/pdspy/pu.h Fgu 3.5 Wn Hsnbg (9-976). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/hsnbg.h Fgu 3.6 Ewn Shodng (887-96). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/shodng.h Fgu 3.7 Pu D (9-984). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/d.h 37

138 Chp 4. W Rfon nd Gy I s ony h on of h gnud o h nsun h w su, nd f hs on s d, w h no ns of knowng whh s h gnud o h nsun h hs hngd. Hn Poné, Sn Méhod Inoduon "Gy s pobby du o hng n h suu of h h, podud by h psn of. Gog Fns FGd 894 Is Nwon [Fgu 4.] pubshd hs hoy of gy n Pnp n 687. Nwon d h fo popoon o h ns squ of h dsn bwn wo sss woud yd p pny obs wh h sun on fous of h ps. H onjud h h gon fo gh psn ndny of o o fo dns o gons of h h. Tss of Nwon s hoy w sos dffu nd qud pny obson d uud o ong pods of. Fo p, n 784 P-Son Lp [Fgu 4.] dnd h h ppny su (non-pod) oons of Jup nd Sun w uy pod wh pod of 99 ys, h fquny ospondng o h dffn bwn f pods of Sun nd wo pods of Jup. hough Nwon s w nuy sudd n pnng os sono obsons, fw obsons ssd npon. Ths nudd h of oon of h p s of Muy. Fgu 4. Is Nwon (643 77) Fgu 4. P-Son Lp (749-87) 38

139 Lóánd (o Rond) Eöös [89] [Fgu 4.3] pod pn sus ndng h n ss nd gon ss y qu. b Ensn [97] [Fgu 4.4] hn poposd h Pnp of Equn bwn n ng fn f nd gon fd. H so ddud h h spd of gh us y n gon fd [Ensn 9, 9]. Fgu 4.3 Lóánd on on Eöös (848-99) Fgu 4.4 b Ensn ( ) Hy n [99] obsd h h ondon fo popgon of gh: d d dy d (4-) dos no hod n gon fd. Insd ondon of h fo: ds gµν dµ dν µν (4-) dsbs h popgon of gh n gon fd whh s hd by h offns g µν. T s dnod by nd h offn g s qu nd oppos o h sp offns g n h bsn of gy. b Ensn nd M Gossnn [93] poposd h p oon n gon fd s dsbd s gods n sp- dnd by h on quon: δ ds (4-3) wh ds dfnd s bo. Cobnd wh n quon ng h offns wh h ngy nso of, hs d fod h Gn Thoy of Ry [Ensn 95,b,]. Dd Hb [95] [Fgu 4.5] showd h h n hoy oud b foud usng 39

140 on pnp. K Shwshd [96] [Fgu 4.6] found souons fo pon ss. Fgu 4.5 Dd Hb (86 943) Fgu 4.6 K Shwshd (873-96) Mny pdons of h Gn Thoy h bn sussfuy dd by pn obsons. In ddon o h usu on bwn ss objs, h hoy so uy pds dfon of gh ys ound ss objs, dons fo sp p pny obs, nd non-eudn uu of sp. Th hoy so pds h sn of bk hos: gons wh gy s so song h gh nno sp. Th s now y song sono dn of bk hos, nudng on h n of ou own gy. In hs hp w w op w fon wh Gn Ry. In pu, w w us h nogy of opsson of w-yng du, suh s n s sod. hough b Ensn s sos dd wh nng h nd fo n h o y gh ws, hs own w ws h odng o h gn hoy of y sp s ndowd wh phys qus; n hs sns, hfo, h ss n h [b Ensn, 9 Ldn Lu]. u Ensn dd no b h h h ws subsn whos oons oud b kd. D, how, onudd h I s nssy o s up n on pnp nd o g Honn fouon of h quons sub fo qunon puposs, nd fo hs h h oy s qud. [D 95]. Oh nsgos h pd o od h uu s n s sod. Two n ffos hos of Hh [99] nd Ksn [998]. Gy hs bn npd by ny s fon du o b nd of fon of sp [sng., nonyous, Cosn 997, Ens., d F 97, Ps 974]. hough ny physss b h gy shoud h qunu hn dspon, h ss dspon dquy pns wd ng of gon phnon. 4

141 4.. W popgon n non-unfo du "I s woh nong h, sy spkng, h nno b ny pon ps n gn y. Thy h o b uh g hn h Shwhd dus..." Hgn Kn [989] Sn s ws yd bspno quons s o h quons of qunu hns, s nu o quson whh s ws n podu gy. sp hns s h wsng of h du n gn nson whh uss h du o opss. Ths ff n b sy obsd usng ubb bnd. Twsng h ubb bnd shs, hby gnng nson whh pus nwd fo h nds. Th squ of h w spd s nsy popoon o dnsy nd hfo dss s on ppohs h gon of nsd dnsy. Sn ws f n h don of dsd w spd h s uu on bwn oon ws [Fgu 4.7]. Ths hns s onssn wh h wknss of gy wh sp o oh fos ( s sond-od ff), nd so pods n pnon fo gy bng n h hn pus fo. Fgu 4.7 Ws f owd h don of sow w spd. Th ys ppndu o sufs of onsn phs Dspson Ron nd M Fos Now onsd h popgon of s ws n n s d s du wh nonunfo dnsy. Fo soon ws h dspson on n b wn s: ω k M (4-4) 4

142 Th dspson on s h ous sous of phs shfs n h w ( ds nd sp ds). Th ss psns h onbuon of onon nd oon o h fquny, whs k psns h onbuon of song fos o oqus n h du, sung n w popgon h dud fquny ( ) ω ʹ k ω M s of h dud fquny h dspson on pps o psn odny w popgon:. In ω ʹ k (4-5) Th ondon of onsn phs s: ω ʹ d k d (4-6) Ths quon dsbs w popgon spd (sn d d ω ʹ k ) wh d p o k, s dsn fo onon nd oon. In oh wods, dsubn os o du o onon nd oon of h du (sung n ss) nd w popgon (onu). Th w popgon ous wh h hs w spd s dsbd bo, bu onon nd oon ns h fquny, hby sng h phs oy nd dung h goup oy. I s usoy o ssu pos fquny, n whh s h sgn of h w o y nd o b d: ω ʹ d ± k d (4-7) Th phs oy ( Φ ) s: Φ d d ω k ω ( ω M ) Th goup oy ( G ) s: (4-8) G dω k dk ω ( ω M ) ω (4-9) Usng h popgon ondon, w n dfn phs spon d χ fo by sp- phs whh sus h don fo h popgon ondon ( ω ʹ d ± kd ): dχ ωʹ d k d Ths quon oud b odfd by upyng ny non-o b on boh sds. Fo p, w dfn h dffn spon ds by h on: Eo! Objs nno b d fo dng fd ods. n n fouon ydng F s pnp n b found n Ens. []. Ths dffn spon shoud b o fo h u popgon ph. Th ngd spon s: Eo! Objs nno b d fo dng fd ods. (4-) 4

143 If h spd of gh s b, hn nghbong sp- phs us s yd qu phs shfs n od o nn h nss onon of h w. Ths ondon yds h quon of gods: δ ds δ d d (4-) Ths psson s qun o Ensn's fouon of gn y [Ensn 956 p. 78] f w ssu dgon nso wh: g g g yy g So h h gods quon s: µν δ g dµ dν µν Wh d d. (4-) (4-3) Hn Ensn's fos n b npd qu spy s h nod us of h squd w nub nd (dud) fquny. Ensn's fouon s b o gn n h ows fo non-soop s, nd h bo fou n b sy gnd o ow fo ndpndn ons of k k, k y k y, nd k k (wh ppop dspson on). I s no h hs gnon s pon n nu, so s no pusud h. To spfy h ng, w nodu p τ nd w h gods quon s: µν dµ d δ g ν µν dτ dτ d τ δ f Th Eu-Lgng quons : f d τ d ν d µ dµ g ν g f dτ ν dτ ν dτ Sn h nso s sy, hs yds: µν d ν d ν g dµ d g ν dτ ν dτ µν dτ dτ µν g dµ dν µν dτ dτ W wn o so d ν dτ n h bo quon. Ths ks onsdb ffo o obn: (4-4) (4-5) (4-6) 43

144 d ν dτ γ γ ν g g g d dγ g γ dτ dτ (4-7) Fo os s opd wh h spd of gh τ nd d dτ, so o ows od: d d g (4-8) Th gh hnd sd y b npd s gon on nd s qun o Ensn s psson [Ensn 956 p.89] p fo dffn sgn onnon (Ensn uss gny, hby hngng h sgn of h po oponn) Ron bwn oponns Fo s hngs n h spd of gh wh : Δ Δ In s of oponns: (4-9) Δ Δ g Δg Δg Δg (4-) yy Ths quy of fs-od hngs n oponns s n gn wh Ensn s su [Ensn 956 p.89] p fo h po sgn onnon. I s d o h f h h Ensn nso hs o dgn. d d Usng hs su n h psson fo gon on yds: 4 (4-) Hn, o fs ppoon, h gon on s dy popoon o h gdn of h squd spd of gh. Th gon pon s: U Δ Δ 4 4 (4-) wh Δ s h dffn n h squ of h spd of gh fo s unpubd u. Ths psson fo h gon pon s onssn wh Gn Ry [Ensn 956 p ]. On y wys offs hs pon by onsn o k h us pos Th gon pon 44

145 Th os nophnsb hng bou h uns s h s ophnsb. b Ensn Th hng n spd of w n n s sod y b bud o h hng of dnsy o hng of s onsns. In hs son w suppos h h hng s du o opsson. Th ρ ρ quon of opsson ws wh spd n n s sod s: ssung h dnsy o b sowy yng ows h ds o b ngd: (4-3) ρ (4-4) Mny g ss objs ny sph n shp, pyng ony d dpndn: ρ whh hs h souon: η ρ ( ) ρ wh ρ nd η onsns. Th spd of nss ws s gn by: ( ) µ µ ρ η ρ wh µ s h sh oduus. Th fon on of s gn by: δ ( ) [ η ] η η O (4-5) (4-6) (4-7) (4-8) Hn h hng of w spd dffs fo h (/) dpndn of h ss gon pon by h ddon of hgh od s. How, n n h dg of h sun h 6 on s ony δ, so h sond od dffn s y s. Th hng n h spd of gh s dny usd by h psn of ss (M) nd fs off nsy popoon o dsn () wy fo sphy sy dsbuon of ss (p fo y s dsns). Th psson fo h Nwonn gon pon s: 45

146 ( ) U GM (4-9) Wh 3 G kg s s h gon onsn. No h h gon pon hs uns of oy squd Consquns of gy Nwonn gy Gn h fo of h gon pon nd h psson fo on n s of ons n h spd of gh, w n pss h gon on of n obj n s of h pon: d d 4 U (4-3) Th on s spy qu o h gdn of h gon pon, s n Nwonn gy ndng of gh Fo popgon of gh ws, w n no ong ng hngs n poson o hngs n. Tk h oy n h 3 don o b nd h gdn n h spd of gh o b ong s n Fgu 4.8. Eh Lgh fo s Sun M oo n Th on s hn: Fgu 4.8 ppn poson of s nd h ph of gh y ps h sun. 46

147 d d g g g g 33 Ths s w h Nwonn on. U Ingng o ph wh / gon pon yds: d d U d d 3 µ 3 ( ) d 3 µ 3 µ Th gon offn fo h sun s: 3 3 (. kg) µ (4-3) (4-3) 3 3 GM 6.7 (4-33) kg s s Ths yds ppndu oy of: ( s ).9 8 ( 3. s ) 4µ s d d Jus ousd h dus of h sun, d d 3.9 s : s Th ng of dfon s h o bwn h dfon oy nd h spd of gh: dns dgs.8' ' (4-34) (4-35) (4-36) Ths dfon ws fs obsd dung 99 so ps [Dyson, 9]. Mo n suns us do ws, whh do no qu wng fo pss [Lbh. 995]. Sn h gh sows down n h sun, h s so dy n h sgn s opd wh popgon n f sp. Ths dy hs so bn sud nd s n gn wh pn [Shpo. 977, o. 3] Cuu of sp On supposdy b pdon of gn y s h sp s ud. Wh hs ns s h suns of go shps no onssn wh Eudn goy. Fo p, suppos w su h ufn of of dus R by shnng gh ps 47

148 ss of os obng n sp s shown n Fgu 4.9. Fo spy, w w h h s pon-k sou of gy. R R Fgu 4.9 Dsn suns n gon fd. W k h spd of gh o b n ppoon of h fo dd bo: ( ) [ η ] / η Ngng ny dy dung h fon poss, h gh popgs wh onsn spd o dsn π R, so h h popgon s: πr π R η Sn on nno dy dn h bsou spd of gh, h sud ufn s: L π R η (4-37) (4-38) L (4-39) Th ufn of sond wh dus y b sud sy. To od ffs of dffn ok spds, h ns n b sud usng h ok R by sndng sgns whn h gh w s nsd nd whn s d by h s R. Th sud ufn of h R s: L π R η Th of fgh of gh bwn h wo s s: (4-4) 48

149 D R d R ( ) R η d R R R Ths ns h h sud dffn n d s: Δ η R R R n R D η R n R odng o Eudn goy, h wo ufns shoud b d by: L L πδ. Insd, w h: πδ L L η n[ R R ] ( R R ) > (4-4) (4-4) (4-43) Copd wh Eudn goy, h sud ufn s s hn pd fo h sud d. Ths s h nng of ud sp. How, h ppn uu s uy bub o h on n h spd of gh, whh dsos h sun of dsns k Hos W sw bo h gh s dfd whn psss by ss obj suh s h sun. If h gdn n h spd of gh s g nough, hn h gh n bo ppd. n obj whos gon fd s song nough o p gh s d bk ho. Fo h goy dsbd bo n Fgu 4.8 wh b pd by h pon of oss ppoh, h np on ondon fo ppng gh s: d d ( ) In s of h gon pon, hs ondon obnd wh (4-) nd (4-3) yds: U ( ) 4U In s of h ss of h bk ho: GM 4GM Song fo : GM GM (4-44) (4-45) (4-46) (4-47) 49

150 5 Ths dus s d h Shwhd dus. ny gh whh hs hs pon fo h ousd w b ppd. k hos w on onsdd n bsudy, bu h s now wh of dn fo h sn n h uns Gogns Thus f w h gdd h gon pon, nd h s h, s bng s bu b n sp. Ths pon of w nno b d fo obss wh by oon. If h gon dsubn popgs hough sp wh spd dffn hn h of gh, hn oud b possb o dn h bsou s f of h h fo h don hngs n h ppn oy of h gon dsubn (gy ws). How, f h gon dsubn popgs wh h s spd s gh ws, hn w onfo o h odny Lon nsfoons nd h woud b no wy o dn n bsou fn f. W w ssu h hs s h s. R oon of h sou of gon dsubn (hng of s h) dos ff w popgon. Jus s h gn fo n b bud o h Lon onon of ong n of hgd ps, gn oos fo y b bud o Lon onon of ong. S fo fn f wh s gon sous: Δ Δ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ y y g g g g g g (4-48) ppyng Lon nsfoons o h sond nd yds: Δ Δ Δ Δ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ g g g g g g g g g g g g γ γ γ γ γ γ γ γ γ γ γ γ (4-49) nd ppyng Lon nsfoons o h fs nd hn yds:

151 5 ( ) ( ) ( ) ( ) γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ O g g g O g g g O g g g O g g g Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ (4-5) Mong gon sous hfo nodu non-dgon oponns of h nso. Fo oon n h -don, h oponns o fs od n : g g g g g g yy Δ Δ Δ Δ Δ Δ Δ Δ (4-5) Fo p, h gon pon of n nfn od ong wh spd p o s s y pp o b s, bu ny pubon woud uy popg ong wh h od. Th su s h h gon dfon of n obj s nsd f h oy bwn nd h gon sou s nsd. Ths s d h gogn, o fdggng, ff. Th sn of hs ff hs ppny bn onfd n pns, bu doubs bou uy n [Ioo 9] Suy Th bgg hy h hd hy f. nonyous Th bo nyss donss h gy n b npd s w fon n nonunfo du. Unk qunu hos n whh gy ws ssgnd spn of, h psn od s gy s s ssod wh hngs n h dnsy o sy of h sod h. Th s bsouy no phys dn ndng h gy shoud b qund. Copsson ws n sod n n pnp popg spd qu o g hn h spd of nss (o oson) ws. Thfo s qu possb h gy ws popg spd g hn. If h s h s hn h sud spd woud so b don

152 dpndn du o h h's oon o h uu. Suh dffn n w spds woud so b dn n gogn ffs. In suy, gy y b npd s dspon of w fon du o dsd oy of gh n h ny of. If h h s kn o b n s sod, hn h on n gh spd gh b bud o opsson o hng n sy. Th sp oponns npd s h o bwn h squd w nubs dffn posons. Th po oponn s npd sy s h o bwn squd fquns dffn posons. Conson of ngu onu nd ngy yd h o on bwn sp nd po oponns. Th dd fo of h gon pon fs off s / fo g dsns bu so nuds hgh-od s. Gon dfs gh n odn wh h ws of w fon. I so ks sp pp o b non-eudn. k hos bnd gh ys so songy h h gh bos ppd. of hs ffs sy undsood usng h ss od of n s sod h. Rfns sng PM, Ens JC, nd Nnd KK Th Phs of Qunu Mhn P n Cud Sp, Gn. R. G. 33: nonyous Rfons on Ry Rfons on Ry (hp:// Son 8.4 n H 99 Th Tnsfoon of h Eodyn Equons Po. London Mh. So. 8:3-64 o, Iss, nd Too 3 Nu 45: Cosn EE 997 Th Rf Ry Thoy u.. Phys. So. 46:K.4. D P M 95 Rpy o L. Infd. Nu 69:7. Dyson, F. W.; Eddngon,. S., nd Ddson C. 9 dnon of h dfon of gh by h Sun's gon fd, fo obsons d h o ps of My 9, 99 Phos. Tns. Roy So. London : Ensn 97 Üb ds Räspnp und us dsbn gsognn Fogungn. Jhb. Rdok. Ek. 4:4-46 Ensn 9 Üb dn Enfu8 d Shwkf uf d usbung ds Lhs nn. Phys. (Lpg) 35: Ensn 9 Lhgshndgk und Sk ds Gonsfds nn. Phys. (Lpg) Ensn nd Gossn M 93 Enwuf n gnn sho und n Tho d Gon Zs. Mh. u. Phys. 6:5-6 Ensn 95 Zu gnn Räsho S. b. Puss. kd. Wss. (n) 47: Ensn 95b Ekung d Phbwgung ds Mku us d gnn Rsho S. b. Puss. kd. Wss. (n) 47:799-8 Ensn 95 Zu gnn Räsho S. b. Puss. kd. Wss. (n) 47:

153 Ensn 956 Th Mnng of Ry (Pnon: Pnon Unsy Pss) Ffh Edon. Eöös R 89 Üb d nhung d Ed uf shdn Subsnn Mh. N.. Ungn 8:65 68 Ens JC, sng PM, Gog S, nd Nnd KK M ws n gon fd: n nd of fon fo ss ps n gn y. J. Phys. 69:3- d F F 97 On h gon fd ng s n op du Gn Ry nd Gon : Hh R 99 Esp fo Ensn (Wngon, Cfon: Kn Co.) Hb D 95 D Gundgn d Physk (Es Mung) Gö. Nh. p Ioo L 9 Rn ps o Msu h Gn Rs Lns-Thng Eff wh Nu nd f ods n h So Sys X:95.3 Ksn U 998 Skh of M Mod n n Es Uns (hp://ho.onn.no/ ~uksn) Kn H 989 Gug Fds n Condnsd M Vo II, P I (Sngpo: Wod Snf) p 387 Lbh. 995, Phys. R. L. 75: 439 Ps PC 974 Ind of fon fo s, ogn, nd gon ws n wk gon fds Phys. R. D 9:7-8 Shwshd K 96 Üb ds Gonsfd ns Mssnpunks nh d Ensnshn Tho S. b. Puss. kd. Wss. (n) Shpo. 977 J. G. R. 8:439 Fgus Th foowng fgus bd o b f of opygh son, nd w obnd fo h sous sd. Oh fgus h ogn woks o d n h fgu pon. Fgu 4. Is Nwon (643 77). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/mhns/nwon.h Fgu 4. P-Son Lp (749-87). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/lp.h Fgu 4.3 Lóánd on on Eöös (848-99). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/ogphs/eoos.h Fgu 4.4 b Ensn ( ). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/ensn.h Fgu 4.5 Dd Hb (86 943). Sou: hp://www-goups.ds.s-nd..uk/~hsoy/pdspy/hb.h Fgu 4.6 K Shwshd (873-96). Sou: hp://www-goups.ds.snd..uk/~hsoy/pdspy/shwshd.h Epogu 53

154 W h sn h oon, o oson, ws n n s sod oby h s bs quon s ws: h D quon. Wh ppop dnson sng of h w funon, dyn quns (onu, ngu onu, nd ngy) h onssn h pssons fo boh oson ws nd ws. hough w h no y found spf soon souons fo ny ps, w y on h gh k. En f h spf od poposd h pos o b ndqu, shoud b possb o odfy h hoy o h gn. Thfo h od of h uu s n s sod, n f nop o nu, shoud s b ub oo fo dnng h u suu of h uns. Toson soons s o f h quns of w-p duy. If h nn suu s unhngd by n non (s oson) hn h w pk n b d s pon 'p'. How, hy pu ws nd hn subj o unny pnps. Th hoy pds h sn of boh nd n- sn ny souons us o n ps whh d by onjugon of hnddnss h pon. R h nu pps o b sy wh sp o onjugon of boh f nd gh hnddnss nd nd n-, bu no h on spy. Th oson w npon of h D quon py bus hnddnss o nd n-, hby song h nu pon of o syy of phys phnon. W h sn h soon w oons opud n h s nn s fo qunu hns. Th obnon of soon ws s y f n nu: os fod by obnng poons, nuons, nd ons. Qunu hns (nd qunu fd hoy) hs ydd kb suss n pdng h khood of ous ouos fo p nons. How, qunu hoy nno b gdd s op unss n dsb wh hppns dung n non. popu oon by Sdny Hs [ shows pofsso dng phys su on n quon-fd bkbod. In h dd of h don s wn, "hn ous. hough nndd s jok, hs oon py dsbs h psn s of qunu hoy. phoon ppohs n o, hn dspps s h o jups fo on ngy o noh. No on n sy wh ws hppnng wh h phoon ws dsppng nd h o ngy ws jupng. Th hoy s nop nd us y on. Roon, o oson, ws n n s sod shoud h souons whh ssfy Kn- Godon quon n h dnson, wh s ny s h dffn sss bud o sng soon souon. Th sn of h oupd quons suggss n pnon fo ous yps of ny ps: pons wh on ss, sons wh wo ss s of qu u, nd byons wh h ss s (quks). Phps hs supf nogs, bu w won know un h ss souons sudd. W sw h nons of oson soons no -ud pons. Ths s osy nogous o h Sndd Mod, whh "sss h h n h uns s d up of ny fons nng hough fds, of whh hy h sous. Th ps ssod wh h non fds bosons" [Congh nd Gnwood 998]. In sndd qunu hns h pon s ssud o b dy ssod wh ps. In h oson npon, h 'ps' ssod wh h pons psn w pks whos bsopon o sson hng h soon s of h 'sou p'. 54

155 Th s n og pob wh h d h fons h 'sou' of boson fds, fo n f h fons nno b spd fo h suoundng fds. In pu, sn n on wys hs ong-ng ogn pons ssod wh, h s no og jusfon fo ssng o b 'pon p'. On h ony, ss o snsb o sb of h pops of h on o h D w funon fo whh h pons dd. W now undsnd h h ws of sp y spy onsqun of h w nu of nd no dn of ny nns go onshp bwn sp nd. Th ppn onsny of h spd of gh wh sp o ong obss s du o h sp undspud f h ws h s spd s gh ws (gnus of h D oy opo h gnud ). Th ng su of h Mhson-Moy pn dd no dspo h noon h h uu s du fo w popgon. I spy onfd h gh nd ws h h s w spd. Th quons of gn y so y wh w woud p o fnd f ws f n n nhoognous du. Insd of syng h sp s ud, w n wh qu dy sy h us hng ngh n gon fd. Sn w p oson o b opnd by sgh opsson, h oson npon of offs sp pnon of why h psn of woud ow h w spd n s ny. On ky pdon of h oson hypohss s h h fquny nd w nub y nsy (h fon hngs qu nd oppos fo s ons), n gn wh gn y. On pnp n foung snf hos s h ppon of "Okh's o". ny unnssy opy shoud b u ou of h hoy so h s s sp s possb. If h wo pnons fo so phnonon, h sps on s gny pfd. Modn physss h d g sds owd dsbng nu. Ensn's hos of y oy pd h fon of gh n ss objs nd h nfun of gy on pns. Th 'Sndd Mod' ss o off h dspon of p nons whh s onssn wh pn sus. Nw hos of sngs nd supsys hod g pos fo nopong gy no h qunu hn fwok. u h s pob wh odn physs. Th pob s h wh wok pogssd on opd pobs of non-eudn goy nd hgh-dnson sps, sp pobs w f unsod: Wh s h ss psson fo w ngu onu? Wh y pnp woud su f onssd of ws? Wh s h h psson fo oson? How suns of d by h f h h nsuns of sun so onss of? Ths bs qusons shoud b nswd bfo dng no h op hs of odn phys hos. Gn h h phys pops of h uu unknown, on n hdy p o undsnd ws whou fs bng b o undsnd ws n sp du wh known pops: unfo, soop s sod. Ign wh ou npon of odn physs woud b f Pu D hd undsood oson ws n 98. Dsong o oby h s quons, wh h s dyn opos, h woud h hd y son o hnk h onsss of oson ws. ny suggson h ws w uy 'pobby ws' wh no oh phys npon woud h bn pposous. En whou suh knowdg, D ws onnd h h Honn nu of h quons qus h sn of n h. 55

156 I s ofn sd h odny s qu odny dn. No on oud sousy o h n npon of qunu hns whou podung n u don of so bs phys quns suh s h fn suu onsn o sss of ps. hs no suh don hs bn d, nd h popos h onsss of oson ws us b gdd s unpon. Nonhss, w h shown h hs sp od yds h quons whh nno b dsngushd fo h quons fo ps whou fuh dd nyss. wd ng of 'non-ss' phnon n f opy onssn wh ss noons of physs. Wh hs og fwok n hnd w n now hop o uy undsnd h uns, nd no y dsb. 56

157 Rfns Congh W. N. nd Gnwood D.. 998, n Inoduon To Th Sndd Mod Of P Physs (Cbdg Unsy Pss) ppnd : ngu Egnfunons Iso h ngu ds: [ ] [ ] sn (-) Spon of h quon qus: K sn (-) wh K s onsn. L: (-3) so h h wo-oponn w funon ssfs: sn (-4) Th wo-oponn s n po oodns : sn os os sn os sn sn os (-5) No h:

158 58 os sn (-6) Epnd (-4): sn sn os os sn (-7) En h dpndn fo [ ] T g f, : g g f f f g f g sn os sn os (-8) Fo n ponn dpndn ( ) f p ~, h wo-oponn w funon us h h fo: ( ) ( ) ( ) ( ) [ ] ( ) Φ Φ p p f (-9) L: ( ) ( ) ( ) [ ] ( ) Φ Φ p p (-) Thn: [ ] [ ] sn os sn os Φ Φ Φ Φ Φ Φ Φ Φ (-) Rgoup: [ ] [ ] [ ] [ ] sn os sn os Φ Φ Φ Φ (-) Subsu:

159 59 [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] sn os sn os sn os sn os Φ Φ Φ Φ (-3) Th quon fo Φ s: [ ] [ ] [ ] sn os sn os Φ Φ Φ Φ [ ] [ ] { } sn os sn os Φ Φ Φ Φ (-4) [ ] sn sn os Φ Φ Φ Φ (-5) L: [ ] [ ] [ ] sn sn os Φ Φ Φ Φ Φ Φ Φ (-6) In s of : [ ] [ ] Φ Φ Φ (-7) whh s h Lgnd s ssod dffn quon. Souons wh by pud boundd n h n y b dnod: ( ) [ ] [ ] ( ) d d P Φ! (-8)

160 6 hough h d shoud b wy of dffng sgn onnons. Th nod sph hons : [ ] [ ] ( ) ( ) π P Y p os 4!! (-9) Th quon fo Φ s obnd by png wh (): [ ] [ ] sn sn os Φ Φ Φ Φ (-) O n s of : [ ] [ ] [ ] Φ Φ Φ (-) Th souon wh by pud b s: ( ) [ ] [ ] ( ) d d b bp Φ! (-) Th sond nod sph hon s: [ ] [ ] ( ) [ ] ( ) π p os 4 P Y!! (-3) Th w funon y now b wn s: ( ) by Y f, (-4) Th o of h wo souons n b dnd fo h ogn quons: [ ] [ ] [ ] [ ] [ ] [ ] 4 4 sn os P b P π π!!!! (-5) Copson wh h dny: P P d d P sn os (-6)

161 6 yds: [ ] b (-7) Th nod ngu funons : [ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] ( ) [ ] ( ) [ ] ( ) π π p p p 4 p 4 P P N P P Y Y!!!! (-8) Ths souons upd by h n h bspno quon. [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ] [ ] ( ) p sn os p sn os p p os sn sn os P P P P P P N (-9) Cop hs wh h dns: [ ] P P P sn os (-3) nd: [ ][ ] sn os sn P P P (-3) whh obn o yd: [ ] [ ] sn os P P P (-3) Ths dny pps o h op n. Fo h ow n us h dny: [ ] os sn P P P (-33) Th su s:

162 6 [ ] ( ) [ ] ( ) [ ] [ ] [ ] ( ) [ ] [ ] [ ] ( ) π π p 4 p 4 p p P P P P N!!!! (-34) In s of Y s: [ ] [ ] [ ] [ ] Y Y (-35) Lng yds: [ ] [ ] ( ) [ ] ( ) p p P P N Y Y R (-36) Ths souons h h popy: [ ] [ ] ( ) [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ] { } ( ) [ ] [ ] [ ] { } [ ] ( ) [ ][ ] [ ] { } ( ) [ ] [ ] { } [ ] ( ) [ ] { } ( ) { } [ ] ( ) N P P N P P P P N P P P P P P N P P N P P N ʹ ʹ ʹ ʹ p p p sn sn p sn sn sn p sn os sn p os sn sn sn p p sn sn os os sn p p sn sn L (-37)

163 63 [ ] [ ] ( ) [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ] { } ( ) [ ] [ ] [ ] { } [ ] ( ) [ ][ ] [ ] { } ( ) [ ] [ ] { } [ ] ( ) [ ][ ] { } ( ) [ ] { } [ ] ( ) [ ] N P P N P P P P N P P P P P P N P P N P P N p p p sn sn p sn sn sn p sn os sn p os sn sn sn p p sn sn os os sn p p sn sn ʹ ʹ ʹ ʹ L (-38) Lng κ, w n w h onon s: [ ] [ ] κ κ G G G F F F (-39) O, o onsy: [ ] κ 3 (-4) ppnd : Lon Tnsfoons L: ρ J ʹ (-) Lon boos ppd o h w funon hs h fo: ( ) p ʹ (-)

164 ppd o h pobby 4-un : ( ) [ osh snh ] ʹ ρ p ρ osh J snh ʹ J p J osh ρ snh Jʹ p J ( ) p( ) [ osh snh ] ( ) p( ) [ ( osh snh )] (-3) Nos: Roon: ϕ j U R ( ) ϕ j R R Pss Roon: ( ) j ( ) ϕ j ( ) j ϕ j ( ) ϕ j ( ) j ( ) ( ) j ϕ j ( ) ( ) & w S w ϕ ' ( ) * w & k k ) k ' ( * & wk ) ' ( k * ( ) ( ) 64

165