Entangled Photon-Electron States and the Number-Phase Minimum Uncertainty States of the Photon Field

Transcription

1 Vaó: Eagld_Poo-Elco_Sa_A Eagld Poo-Elco Sa ad Num-Pa Mmum Ucay Sa of Poo Fld S Vaó Rac Iu fo Sold Sa Pyc ad Oc H-55 Buda POBo 9 Hugay E-mal: vao@mal.f.u Aac. T ac aalyc oluo of gy gvalu quao of ym cog of a f lco ad o mod of quad adao fld a ud fo udyg ycal mag of a cla of um-a mmum ucay a. T a of mod c mm ucay oduc of oo um ad Sud ad Glogo 96 co oao av oad y ac 968. Hov a av o fa mad m mamacal couco ou ay ycal gfcac. I ovd a mo fudamal aco quaum lcodyamc amly aco of a f lco a mod of quad adao fld lad qu aually o gao of mod mmum ucay a. I o a fom agld oo-lco a dvlog fom a gly cd um a du o aco a Gaua lcoc av ac mmum ucay a of ac y ca coucd. I lco' cooda ao ycal mag of ao coffc of a a jo oaly amlud of mulaou dco of a lco ad of a df um of oo. T oo occuao oal a v fucoal fom a m la u y dd o locao ac-m of dcd lco. A aaly of aglm o dvd fom oo um duo ad fom lco dy oao gv. Kyod: Eaglm Sog-fld oo-lco aco Quaum mcacal av ac Num-a mmum ucay a. PACS: 3.65.Ud.5.H Dv. Ioduco Eaglm ad o-localy quaum mcac av f dcud y E Podoly ad Ro 935 ad ma cocluo a a quaum mcac o a coml oy cau o vy lm of ycal aly av a coua oy. A Bom 95 oo a gg of Sco.5 T cm a fac o o ujufd [ Bo 935] ad ad o aumo cocg au of ma c mlcly coadc quaum oy a ou. Movad y aov o of E Podoly ad Ro EPR Scödg 935a--c d a dald udy of cocual ac of quaum mcac. I of a oducd famou Scödg ca ad coc of aglm Vcäug Scödg mology. I oo Sco.5 Bom 95 aaly EPR-aado dal y codg a dgao of a quaum ym a molcul avg ally o agula momum cog of o - aom ad dm colao of dco ovd a aally aad dco. T f lal m ood y Wl 96 co cad ou y Wu ad Saov 95 c y maud cocdc coug a a dff lav amu of olaao of o gamma ay mmg fom lco-oo a alao ad dcd y o oog cllao cou. Ty foud a coug a of dcula olaao o m lag a of aalll olaao. I ocal gm f mal alao of E-Podoly-Ro-Bom Gdam av acvd muc la y Ac al. 98a-. Ty maud la-olaao colao of a of oo

2 Vaó: Eagld_Poo-Elco_Sa_A md a adav cacad of calcum ad foud cll agm quaum mcacal dco ad ga volao of gald Bll quval a a m. Cocg Bll quval.g. fc Ac al 98a- ad Wg 97 ad fc. I mam ud ou a aglm lay a cucal ol oaday adly dvlog ac of quaum yc ad fomac amly quaum fomao oy.g. Al al Boum al ad Solm ad Suom 5 ad quaum comug ad quaum commucao.g. Wllam 999 ad Nl ad Cuag. I aov-mod aml agld acl a of am o. I a all dcu aglm oo ad lco. I ll u ou a agld oolco a o coucd Sco av a clo coco ccal a oducd y ac 968 c mm a um-a ucay oduc of oo fld. Ta y cocg olm of a oao of a mod of quad lcomagc adao a fo a of coml of a ucv o gv a f ummay of ac fc dalg ujc. I a-ag a o quaum oy of mo ad aoo of adao Dac θ 97 oducd oo aoo ad mo oao fom N ad N θ cvly oao mod d Plac coa dvdd y π ad do ma cojugao. T um aco oao N ad caocally cojuga agl oao θ a aumd o afy Hg commuao lao ad a a coquc. W o a a all u follog oao fo o mod: A A u [ A A ] AA A A. T ola dcomoo ud y Dac lacd y lao A EN ad A N E a ll dcud mo dal Sco. I am ya Dac mod a aad Lodo 97 uld udy o agl vaal ad caocal afomao quaum mcac. H ovd a oug ladd oao E ad E ±Φ av a ll-dfd ma ao y cao d a a oal of fom Φ ould a ma ma. I u a Dac a aa of dcacy. Accodg o oda 97 a covao m Dac mad a oly o dv may coc ul y ug fomal lao [ N Φ] com fom a coc lao [ E N] E a mlcly ud fac ad of fom o all dvao of ul. Fomally coc lao ENE N ad E NE N ca alo oducd y aumg E Φ ad Φ E a ma Φ afyg commuao lao [ N Φ]. A o l u o a aov-dcud olm of quaum a vaal do o o u ca of quaao of caocally cojuga a agl ad oal momum of a laa moo cau cum do o ma a o agula momum a lluad v ad ooug udy y Kau 6 aad cly. T o-c of a ma a oao of a amoc ocllao a dcovd y Sud ad Glogo 96. Ty oducd ma co ad oao o ac o ll fly ummad Sco of a. I v v a o a ad agl vaal quaum mcac Cau ad No 968 dvd a coul of uma ucay lao y ug co ad oao ad ac 968 coucd a ccal a c mm o of ucay oduc. Gao ad Wog 97 coucd a quaum aalogo of clacal odc a fuco a-oo c af Hg commuao lao um oao o a d of Hl ac of ocllao. Moov y av coucd ga of odc a oao. I ou oo a f mamacally coc aoac oad oluo of ogal olm of quaum a. Paul 97 a ood a alav dco of a of a mcococ lcomagc fld ad dcud ol of maum. A mu a gv o udy of quaum a olm af a of Pgg ad Ba 989 aad. Ty ucad a ac of amoc ocllao ad al o couc a ma a oao o f-dmoal Hl ac. W mo a

3 Vaó: Eagld_Poo-Elco_Sa_A 3 oly of ug a f-dmoal ucad Hl ac co a alady dcud y oda 97. T aoac of Pgg ad Ba 989 a fd y Poov ad Yau 99. T lm ma lm of a oao um ao a l dmo of Hl ac gog o fy oad y auo av alady d y Wyl 93. W o a mgly o of aov auo ulg a c of la cuy ad o aou fudamal aly o of Lodo 96 ad 97. T a duo of a glyqud a a dmd y Sclc al 989 fc o Lodo o f aad mod a y ug quaum a-ac duo Wg fuco of quad mod alo oo y Sclc acula Ca 8 ad 3. T olm of quaum a maum a dcud y Sao ad Sad 99 aly o a of omalal a a. T quo of oao of a a oougly aalyd y Bgou ad Egl 99 o fom fomal o of v ad fom ycal o of v. I a of a No al 99 99a- ad 993 av udd o ocally ad mally quaum a do o a of oaoally dfd co ad oao. I cm dfo a ad o maud oo um cou a g-o fom. Fyg ad Sclc 993 av fomd a aaly of a mla a oao alog m y No al 99 y ug adally gad a-ac duo. I co alo oougly dao y Fyg 99 ad fc. I mam a aml lau a accumulad cocg quaum a olm. Fo fu adg ad fc f ad o ocal u of Pyca Sca dd y Sclc ad Ba 993 c alo om ocal ac a ummad y No 993. S alo ccal v y Lyc 995 ad oo y Přová al 999 o dco of a oc. Cocg c dvlom of coc of quaum a of a la ocllao ooug gou ocal ud y Kau 3 6a ad 7 c a guly aoac o olm a od ou. I a ovd a mo fudamal aco quaum lcodyamc QED - amly aco of a f lco a mod of quad adao fld - lad qu aually o gao of aov-mod um-a mmum ucay a. W m a a mly dalg o-lavc quaum mcac aco of lco quad mod d y mmal coulg m a f cagd acl ad a ocllao. T aaly o d cd o udy of aco of o Scödg lco o quad mod of adao fld. To glc aco o mod jufd y a aum a vy gly occud gl mod. Tu fac a o ug coml fld oao ud vy quaum lcodyamc. I Sco fly umma ac o of Sud ad Glogo 96 co ad oao ad gv aocad um-a ucay lao ad ccal a foud y ac 968 c mm o of ucay oduc. I Sco 3 ac aoay oluo of oo-lco ym c aco a o accou u o f od. I Sco all couc agld oo-lco a o a of aoay a. I ll o a agld oo-lco a dvlog fom a gly cd um a du o aco a Gaua lcoc av ac av am fucoal fom a ccal a dvd y ac 968. I Sco 5 dv ducd dy oao of oo ad of lco. O a of ducd dy oao vaou aglm o a calculad. I Sco 6 a o ummay clo ou a. T mamacal dal of dvao of ou ul a d Adc A ad B.. T um-a mmum ucay a of ac T um-a ucay oduc coa o uual Hg ucay oduc c a vald.g. fo vaac of Caa como of momum ad oo of a acl ΔN ΔΦ? cao av a ll dfd mamacal mag fo a gc a of a quad mod of lcomagc adao. T cau Φ lf cao d y a ma o oao a Lodo 97 a alady o log ago. Equao ould vald f ould a Hg commuao lao [ N Φ] fo um oao N ad fo a oao Φ c o

4 Vaó: Eagld_Poo-Elco_Sa_A ca. Ta ao fo y Cau ad No 968 ood o ucay oduc gv m of C co ad S oao oducd y Sud ad Glogo 96 C E E S E E c a ll-dfd oao. H E o-calld oal a oao dfd y ola dcomoo of oo aoo oao A c ould quaum aalogo of ola dcomoo of a coml um ϕ : A E N N A A E E A N E. 3 W o a oal a oao ladd oao E a ogally oducd y Lodo 96 ad ud y oda 97 oo. I Eq. 3 {...} coml ooomal of ga of oo um oao.. N vg a a coual a of Hl ac of mod ud dcuo.. dy oao. T og o quao [ E N] E ad [ E N] E follog commuao lao ca dvd fo N C ad S : [ N C] S [ N S] C [ S C] P P ojco of vacuum a of mod fo c A. A co ad oao C ad S cvly do o commu cau y cao ±Φ d m of oal of a commo ma oao Φ fom. T ao fo a oal a oao E oducd Eq. 3 o uay u oly alf-uay calld aally omc mamacal mology.g. R ad Sőfalv-Nagy 965 Sco 9 ad. Rally EE old u o o ad E E P ad moov a a coquc of alf-uay oy of E um of qua co ad oao o qual o uy C S P. W mo a fo lag co cao of mod mom of C ad S av a mla fom of mom of oday c-um co ad fuco. W av o o a Kau 6a a cly ad ou ojco aga u of Sud ad Glogo co ad oao dco of quaal a o of la ocllao. O a of aaly d Ca 5 of a coclud a Lodo-Sud- Glogo oao C ad S a o aoa fo maug agl o of a a!. W ould l o ma a udy a o cocd quo oao C ad S dfd Eq. a ual o o ual o caac quaal a o. W mly o a a of ally am mamacal ucu a a of mmg a coucd y ac 968 may gad ouav oo-lco aco og fld gm. Tu all o dcu quoal o o-g ycal lvac of C ad S mlv co of olm of quaal a. T ucay oduc aocad o aov commuao lao Eq. a follog Cau ad No 965 ad 968 U Ψ ΔN ΔC S U Ψ ΔN ΔS C. 5 I aov quao Ψ f o a of quad mod of lcomagc fld ud dcuo ad Δ N Δ C ad Δ S a vaac a a. ac 968 av coucd a ccal a c mm f of ucay oduc U Ψ Ψ a κ I γ 6 ν I a modfd Bl fuco of f d of od dfo Gady ad Ry fomula ad κ a omalao faco. T aam ν N do ma oo

5 Vaó: Eagld_Poo-Elco_Sa_A 5 um γ S ad ca c C Ψ C Ψ a codd. T ao coffc a av dmd fom cuo lao ν a γ a a comg fom mmg codo y ag uday codo a o accou. T la qum c quval o quao I γ foc ν o afy < ν < ν. W av foud a qum a coquc of cod om of Hu o o of Bl fuco Wao 9 Sco T a c allo U Ψ o ac ¼ ca alo coucd y ug am mod. ac 968 a od o a gv y Eq. 6 a ufoualy a do o m o av ay ycal gfcac. I a ll o a a of am ucu a a of Ψ aually aa o-uav aaly of ml aco of QED amly aco of a f lco a quad mod of lcomagc adao. Tu o a of ou aaly may ay a a o coucd lo av a fudamal gfcac. 3. Eac gy ga of acg oo-lco ym I od o ma ou a lf-coad co fly umma ac oad dmao of ac gy ga of acg oo-lco ym. W mo a aco of lco a quad lcomagc fld a coducg clou a ad y Sm 96 u ud uao oy ad a quao o a g od oc. I og o o coco of commucao oy ad quaum yc Gao 95 alo udd a mla ym a of lco a av gud oug ud mclacal uao oy ad a dff gomy. L u cod gy gvalu quao of jo aco of a quad mod of adao fld a Scödg lco. Fo a of mlcy a fo mod a cculaly olad la av dol aomao. I ca do o g qug aoay a cau aco comg fom A m of Hamloa dagoal. T coml dcuo fo a Scödg lco ad fo a Dac lco av uld y Bgou ad Vaó 98a- ad y Bo 98 log ago ad av ald o dm o-uavly coco of muloo Bmalug ad muloo Como cag. Cocg quo of qug oo-lco ym.g. Bgou ad Vaó 98a B-Ay ad Ma 985 ad Bc al 987. W ll cod oly ml Scödg ca ad udy aco a cculaly olad mod dol aomao. T gy gvalu quao o ad A H f E 7 m c vco oal gv a * 3 A a εa ε A a πc ωl 8 adε ε ε y g coml olaao vco fo g ccula olaao fld 3 aumd o dcula o -dco ω ccula fqucy of mod ad L quaao volum. H A f ω A a fld gy. m ad c av uual mag; lco cag ad ma ad vlocy of lg vacuum cvly. do Plac coa dvdd y π. I Eq. 7 a ac aoay a of acg oolco ym caacd y o quaum um lco momum a o-gav g c y cg-off aco duc o al oo occuao um. E a coodg gy gvalu. T Hamloa o lf ad d of Eq. 7 ca a a 3 H Ω A A εa ε A Ω ω ω ω ω π ml. 9 m mc

6 Vaó: Eagld_Poo-Elco_Sa_A 6 Noc a ω fomally og l u lama fqucy fo a lco dy L. I oag Eq. 9 av a o accou a ε ε ε ε ad ε ε. T la aco m o g ad d of Eq. 9 ca aly afomd ou fom gvalu quao Eq. 7 y alyg follog dlacm oao a oly coo aam D[ ] [ A A ] a mcω ε. W o a dlacm oao of fom dlayd y Eq. av a moa ol quaum oy of ocal coc ad co a a a f o y Glau 963a- a-ag a. Suc dlacm oao alo ud muc al y Bloc ad Nodc 937 fudamal udy of olm of fad dvgc QED od o afom ou aco m. By alyg dlacm oao cv a afomd Hamloa c dagoal o lco ad oo vaal c goluo ca do a ml oduc of y a momum ga of lco. Accodgly oa goluo of ogal Hamloa Eq. 9 fom D[ ]. Equao o a aoay a of oo-lco ym a oduc of momum ga of lco ad gald co a of oo. If oluo av ucu a oday co a. Tu o may ay a a la accodg o vy mlfd dco lf adao fld of lco a co a. T coml aoay oluo g oluo of m-dd Scödg quao of jo ym ad [ ] E gy gvalu ca oug o fom ω ω E Ω m m. 3 m m ω ω I Eq. 3 av ud av como y co χ χ of lco momum. I g o o a av ma m gv cod quao of Eq. 3 ca cl gav f ω ω > u oal gy of ym ca alo gav ca aam ag c ould ma a o of aacv aco oud a of mod ad of lco. O o ad accodg o dfo of o-lco lama fqucy Eq. 3 9 fo a lag oug quaao volum L ω << ω u m accally qual o ogal a ma m. W all o dcu quo ay fu a. Fo mlcy follog ll alay aum a ω ω < u av ma m ov. I cla a f ao ω ω aoac fom lo m ca muc lag a ma m of lco. Fo la covc Eq. fom Ω m D[ ]. 5 m I od o mlfy oao Eq. 5 ymol y a ud fo av momum of lco.. y co χ χ co χ χ. W o a og o uay of dlacm oao Eq. ac oluo gv y Eq. fom a coml oogoal o oduc Hl ac H oo Hlco aocad o 3

7 Vaó: Eagld_Poo-Elco_Sa_A 7 acg oo-lco ym. T oo ac of gald co a of y D [ ] gv o g ad d of Eq. govd y ma lm!! L c D[ ] 6!! L < L do gald Lagu olyamal fo dfo of m.g. Gady ad Ry fomula To ou oldg ma lm of y gv Eq. 6 a f uld o y Bloc ad Nodc 937 c av alady quod fo. La Scg 953 dvd uc ma lm o of famou of a o oy of quad fld ad y alo aa udy o Boa moo of a quaum ocllao Scg 96. Fo om fu dal.g. Bgou ad Vaó 98a-. T cao valu of oo um ad vaac ca calculad o o ad dcly fom Eq. 6 o o o ad y ug dlacm o c D A AD D AD A ad Δ D A D A. 7. Eagld oo-lco a I co ovd a aco of a f lco a mod of quad adao fld lad o gao of um-a mmum ucay a dcud Sco. I o a agld oo-lco a dvlog fom a gly cd um a du o aco a Gaua lcoc av ac av am fucoal fom a mmum ccal a foud y ac 968. I lco' cooda ao ao coffc of a a d m of modfd Bl fuco of f d a a o Eq. 6 o agum o dd o lco' cooda. T oo ac of a v fucoal fom a m volv u occuao oal dd o aomoal oo of lco dco. W o a o ujc lmay ul av alady log d y u Vaó u av o uld m ul o. Accodg o Eq. ad oly av moo of lco coul o adao fld u logudal moo mly a f oagao. I follog all o dcu ay fu logudal dyamc u a coca o udy of av a of av ac dyamc c ou ca aco of lco ad quad mod of adao fld. T agld oo-lco a dvlog fom a um a du o aco a lcoc av ac av fom d g g g π 8 g a cald o a Gaua g fuco ad a oducd Eq. 5. I Eq. 8 av oducd av d of lcoc av ac lco am. T ycal uao o c a gv y Eq. 8 may aocad follog. L u mag a a lco jcd o a cavy a m oug a mall ol of. O a of ou al udy of u al valu olm Bgou ad Vaó 98a c a udd coulg of lco gly occud cavy mod ul c fomao of a dfd y Eq. 8. I a c ou aaly o udy of ao-moal voluo of aoma a c a agld alady a. Og o uay of dlacm oao D Eq. 5 uoo dfd y Eq. 8 a omald a oduc ac of oo-lco ym. I od o av a lc fom of a lco cooda ao ad a am m ad m of oo um ga d. 9

8 Vaó: Eagld_Poo-Elco_Sa_A 8 T ummao d aov quao a fd mly fo a of la covc. T omalao codo Eq. 9 follo fom o omalao ad fom coml lao oo d lco oo ad lco do u oao o Hl ac H oo ad H lco of quad mod ad of lco cvly. T cala oduc f quao of Eq. 9 ca d a dg. π dχ co χ ϕ D[ ] π m I Eq. 9 ad ad ϕ do adal ad agula av oo of lco cvly.. coϕϕ. T ycal mag of ma lm gv y Eq. a y a jo oaly amlud of mulaou dco of a lco a oo ad a of m ad of a df um of oo. A o Ad A fo lag valu of a aymoc o ca calculad fo ma lm of dlacm oao Eq. 6 quao ladg o Eq. A.5. W o a gal ov lco momum Eq. ca valuad acly fo a aay o caly a lag valu of ac aalyc o Eq. A.7 u cfo a all oly dcu ca of lag valu ad u aomao mmg fom Eq. A.5. Af gao c o amu agl χ momum ac oa ϕ π A O 3. d m mcω I Eq. av oducd quay A c ω πρω c fomally qual o amlud ω ω 3 of clacal vco oal Acl A ε ε aocad o oo dy ρ L f ma dfcao u E cl π ρω. H u do gy dy of mod Ecl Acl c ω c A ε ω ε y coω g lcc fld g. Accodg o Eq. A.8 oa fom Eq. lm fom ca of g al occuao um ϕ 3 I O. π I a modfd Bl fuco of f d of od ad A F 9 I E m m mc mcω c Ω D λ π. 3 I Eq. 3 av dfd dmol y aam o umcal valu ca m of y I of mod of adao fld maud Wcm ad of oo gy E maud V. W av alo oducd amlud of lcc fld g F ω c A πρω ad avlg λ of adao. T aoma qual Eq. 3 a vald fo lag L. If l o ad L gog o fy uc a ay a oo dy a fd aam la m o g ad d of Eq. ca ud ad ca fomally aocad o a clacal lcc fld of amlud F. T Eq. λ π com ju amlud of ocllao of a clacal lco ud aco of lcc fld of adao E F ε ω ε coω. T ca aly o y olvg No quao m& E ad y

9 Vaó: Eagld_Poo-Elco_Sa_A 9 m& y E y. Tu dmol quay λ π ao of amlud of clacal ocllao of lco o al av d a of lco ac lco am. W ma a aov ma mad mly o oul a oug cu od o gv a ycal acgoud of aam oducd Eq. ad 3. Of cou a o ayg a a clacal lcc fld ca aocad o a v vy gly occud um a. T ca coly do y ug Scödg-Glau co a Glau 963a-. Ayay ou lmay vgao o la ujc claly o a aam of a mla o aually aa oo u aam a allod o ud alc umcal ma. T m cal aam dfd Eq. 3 ca lad o od T π ω of adao fld oug a m cal aam m [ ω ω ω ω ] m m T av ma ω mc ωπ λ. m dfd Eq. 3 ca cl muc lag a a ma m f ω ω aoac fom lo. Coquly av adg of lcoc av ac ca cl ducd du o aco lcomagc adao. Fom Eq. 9 y glcg m of od Eq. av follog aoma fom fo 3 ϕ g I [ γ ] 5 g γ. 6 π I ca ovd y lc calculao dvao of Eq. A. a lm ad 3 L u L ρ fd aoma a gv y Eq. 5 a alo oly omald l ac a Eq. 9. By ug d afomao oa a alav fom of Eq. 5 g ϕ ϕ ϕ I [ γ ]. 7 Aa fom faco fo γ al oo a um c o o g ad d of Eq. 7 a am fucoal fom a um-a mmum ucay a Ψ Eq. 6 dvd y ac 968. Noc a quaum um coodg o aam ν ac oluo a g um ou ca coa o ν c alay av o av a o-vaug facoal a. T o dffc a omalao coa κ Eq. 6 dmd y quao κ I γ u ou ca g I [ γ ] ma om Hl uac of quad mod. Fo oo a of a Eq. 7 a mla omalao o a of ac a ca acvd y qug κ I [ γ ]. ν A d of Sco gv om umcal lluao of ao-moal avou of jo oal o a of aalyc o Eq. foud lag oo cao lm. T a oal of o mulaou v lco dcd a oo ad oo a aod o md a om oo c d o caly am a a of lco a fo accal ao ould dff. I umcal aml ll alay aum a avlg of quad lcomagc adao

10 Vaó: Eagld_Poo-Elco_Sa_A of od of λ cm.. oo gy of od of ω V. I ca dmol 9 y aam oducd Eq. 3 mly d a I I do y of oo fld dvdd y o Wa qua cm. Bd all alo aum a avlg aam oducd Eq. 3 o a good aomao cocd λ π. T ma accodg o dfo of Ω Eq. 9 a o-lco lama fqucy ω aumd o muc mall a ω fqucy of ocal fld. I fgu o ao-moal duo of jo oal fo om gv -valu. Fgu. So ao-moal duo of jo oaly comg fom Eq. o Eq. 5 fo dff valu of um of md c oo. I ac fgu av ud a umcal omalao faco od o av mamum valu of vcal cooda ougly uy. T faco a follog: a: 3 fo : 9 fo 8 c: fo 5 d: 8 fo 5. Bcau of ymmy of modfd Bl fuco c o cag of g of od am duo ul fo oo aoo.t ufac fgu llua lco dco oaly a adal oo ad a a of m f o fo cay a oo av md o aod dcd y a aally aad cou. H av a c cood o a y Wcm. T ca fom Eq. 3 y aumg 3 a λ π. Fo a ocal fld λ of od of - cm accodgly of od of -8 cm. A ca of al y a codg lac cal ad o-oo cal ± doma ad g od cal > av muc l jo oal. A fgu fo mama of doma lo od jo oal ± o u a omald adal oo λ π c quay ju ao of amlud of lco ocllao o aal d of lcoc av ac. T avou ca lad o a of fucoal fom of oo ao of agld oo-lco a gv y Eq. 5 ad 6. Ou aml fgu llua jo oo duo aoud cal lag al oo um fo dff ao of ad.. o ao-moal oo of lco dco a gv aam ac fgu. H aumd ad. 5 5 T oal a omald o mamum valu c a. a 9.5

11 Vaó: Eagld_Poo-Elco_Sa_A.97 c ad.86 d. A ag va fom. 3 o. 5 duo udgo a qualav cag. T moooc duo lluad y a go ov o ocllag duo a o y c ad d...8 HaL HL HcL HdL Fgu. So c oo um duo aoud cal lag al oo um fo dff ao of ad. T ma a - ddc alog l o la a lod fo dff - valu. T ag a:. 3 a. 6 c ad. 5 d. Fom fgu ca coclud a ca go o la lco g dcd oaly duo of mulaou dco of oo av qualavly dff a. I cla fom fucoal fom of oal dducd fom Eq. 5 ad 6 a fgu a a modfd Bl fuco avou ad o o ad fgu c ad d cou oday Bl fuco avou. I f ca duo a a mla fom a { I } a al um. I la ca duo av mla fom a { } fo dff al valu of ad duo oclla.. aa local mma ad mama a va. W av umcally udd a of jo oo um duo ad locad go of la a of duo a qualavly dff. T ul a cal ca dlayd fgu 3. A fgu 3 aum c cood o a y W cm ad λ π. T ag of lo l ad u l a. ad. 8 cvly. I ca f a lco dco ag lac ao-moal ag <. o >.8 oo um duo a o-ad moooc duo l fgu a. O o ad ag dfd y lao >. ad <.8 oo um duo a ocllaoy.. jo oaly duo av val local mma ad mama. Of cou ao fom moooc gm o ocllaoy gm o a a a fgu ould ugg a f glac.

12 Vaó: Eagld_Poo-Elco_Sa_A 5 moooc ê ocllaoy 5 moooc 5 5 ê Fgu 3. So cmacally ac-m go a of jo oaly duo a qualavly dff. 5. Rducd dy oao ad aglm o L u f calcula dy oao P of quad mod aocad o agld a oducd Eq. 8. By ag aal ac dod lo y T of dyad c o lco vaal av P T { } d l d g l. 8 D[ ] l D[ ] { } I oag Eq. 8 oogoaly of av momum ga a ud δ. A o Ad B gal o g ad d of Eq. 8 ca aalycally valuad yldg ac oo um duo gv y Eq. B.3. I follog all o dcu gal duo u a all udy ca of g al oo cao. Fo lag valu of ducd dy oao P ca oug o fom dvao ladg o Eq. B.7 3 P O q I q 9 q λ π 3 ad qua ad av alady dfd Eq. 3. A ovd Ad B of g { } oly omald... Og o oy I I duo gv Eq. 9 ymmc o c ma a g of -oo aoo a am a a of -oo mo. W o a fac { } gov u oo um duo a a ao coffc of oad fom Eq. ad ud

13 Vaó: Eagld_Poo-Elco_Sa_A 3 Eq. 5 ad 7. T la ao coffc a jo oaly amlud of dcg a lco a a oo a a a of m ad a am m dcg oo..5 HaL HL HcL HdL Fgu. So u oo um duo { } dvd fom ducd dy oao ad gv y Eq. 9 fo fou q y valu amly fo q. 5 a q 5 q 5 c ad q 5 d. 3 If aum λ π l fgu q valu ud oag fgu cood o of.5 W cm.5 W cm.5 W cm ad.5 W cm cvly. T mology u oo um duo a ug fo { } ca jufd y a ul u fom dagoal lm of dy oao of oo fld Eq. 9 c of cou do o coa lco vaal c la o av acd ou. I fgu claly a a y cag g od aoo ad ducd mo v com mo ad mo doma ad d of duo a comg lag ad lag. No a ucd ul. L u o a ul ad o ou aaly do o coadc o famou am accodg o c a f lco cao ao o m a oo. T am c ca foud ay of ac o QED l o uao oy of S-ma aoac dalg aymoc comg ad ougog la av g lco ad oo. T aco of lco a og la am fac a may-ody aco a am ca codd a a uoo of la lcomagc av oagag dff dco ad ag a g-od ducd oc. T quo av log dcud.g. y Bgou al 983 o ud a lavc mclacal dco. T udy of uc mo gal olm ou of co of a. H a ug a vy mlfd cm o-lavc dco of lco co o o mod aco dol aomao c a o o ad ll jufd ag of aam a ou umcal aml lo. Ou goal mly o o om ac caacc of agld oo-lco ym. 3 3

14 Vaó: Eagld_Poo-Elco_Sa_A S aocad o duo { } T vo Numa oy oo ca codd a o of aual mau of dg of aglm of oo-lco ym. By ug Eq. 9 oa S [ P] T[ P log P] S oo oo [{ }] log q I q log[ I q ] I q log[ I q ] q 3 q a dfd Eq. 3. Accodg o Eq. 3 oy of quad adao fld do o dd o m. T cau agld oo-lco a oducd Eq. 8 a a aoay a oug coa lcly m vaal a comlcad ma a o y aalyc fom gv y Eq. 5 ad 6. T a Eq. 8 o a oluo of a u al valu olm ould av aumd a ally o-acg ym d y a oduc a ad c o aco a om ay. W lav udy of la olm fo a aa o og Vaó 7. I Fgu 5 llua y ddc of vo Numa oy of oo fld. I aam ag a codg oy cuv o Fgu 5 y ug log-la cal com a ag l af y a ad valu W cm. T ma a oy S oo [{ }] ca logamcally y. A o y oy va cau aco of oo ad lco glgl ca c foo dy o..5 Eoy of Poo Fld W ê cm D Fgu 5. So y ddc of vo Numa oy of oo fld dfd y Eq. 3. I oag fgu 5 av aumd a dd vaal q S oo [{ }] d umcally a q [ I W cm ]. T ma accodg o Eq. 3 a λ π aumd.. avlg of adao fld ougly ouad m lag a al av of lcoc av ac. Fo a ocal fld av λ 8 cm. cm o ca a codg 3

15 Vaó: Eagld_Poo-Elco_Sa_A 5 No l u dv ducd dy oao P of lco aocad o agld a c a oducd Eq. 8. By ag Eq. 5 o accou aal ac dod y T of c o oo vaal ad { } [ ] ] [ ] [ D D m g g d d T P. 3 T ma lm of P momum ac ca calculad y ug Eq. B. ad 6 yldg ]} Im[ ]{ [ L m g g P P 33 L do Lagu olyomal of od. T dagoal ma lm of P momum ac a mly gv y modulu quad of g fuco g dfd Eq. 8.. g P Π Π π. 3 I Eq. 3 av oducd dmol momum vaal ad dy fuco Π. Accodg o Eq. B. ad B.5 ma lm of ducd dy oao P oo ac ca d a cala oduc of oo ao of agld oo-lco a oducd Eq. 9 ; 3 O P 35 a dfd Eq. 5 ad 6. A o Ad B ca of vy g oo cao mo accualy lm dy fuco Eq. 35 com ; ; F P ; π I F. 36 T dagoal ma lm of lco ducd dy oao a dmd y dmol dy fuco c call u oo duo of lco c oo vaal av acd ou. W oa ; π I F P. 37 T duo P omald o uy fo ay a of m. T ca o y ug a mla ocdu ald oof of Eq. A.. Accodg o Eq. 9 dy oao of oo fld dagoal u al o do mmdaly lc fomula Eq. 3 fo vo Numa oy. A fom Eq. 33 ad 36 lco dy oao P Eq. 3 momum ao o oo ao dagoal. I od o calcula vo Numa oy of lco f av o dagoal P c a mom m o u a ol a. I od o avod dffculy a udy o-calld la oy H c a a clo coco cod od Réy oy. T dfo of H ad H T H ρ log ρ T H 38 H cod od Réy oy ad ρ om gc dy oao. T la oy a ud y val auo.g. Zu al 993 ad oo al 3 cau muc a o

16 Vaó: Eagld_Poo-Elco_Sa_A 6 calcula c do o d dagoalao of ρ ad o o ad a good alav o vo Numa oy a a mau of aglm. Rally H va fo a u a ad mamum gvalu of ρ a dcal c ca of mamum mg. Ao uful quay o caac aglm a o acl ym Scmd um K Nl ad Cuag o dfo K [ T ρ ] ρ do ducd dy oao of o of o acl. T Scmd um a vly ud o caa coouvaal aglm y Fdoov ad coo Fdoov al 5 6 ad 7 ooug aaly o av ac dyamc au oc l oao of aom ad docao of molcul acula Fdoov al 6 a ufyg ovv of adly aag ym d. L u f calcula la oy of oo fld aocad o duo { } gv y Eq. 9. T dal of calculao ca foud Ad B. Accodg o Eq. B. av H oo q q [ P ] TP H [{ }] I q I q oo. 39 I od o calcula la oy of lco d f a lc o of P c ca oad fom Eq. B.9 y a agfoad calculao * P d d d 3g g g 3 [ 3 m ]. D [ ] D[ ] D [ 3 ] D[ ] 3 T ac of P ca calculad aalycally u ca dv a ac o fo la oy of lco a o Ad B y Eq. B.8. I lm ad a am m 3 3 quaao volum L uc a oo dy L g fd accodg o Eq. B.9 oa q TP d [ ] I q q. By ug Eq. ad gal dfo gv y Eq. 38 la oy of lco com q H lco [ P ] I q c cocd la oy of oo fld gv y Eq. 39 q H lco [ P ] H oo [ P ] I q. 3 Equao 3 a maal cocy ou calculao ladg o aalyc ul gv y Eq. 39 ad. Rgadl of ug dc oo um duo { } gv y Eq. 9 o ug doul gal of dyad aamd y o couou momum vaal of lco Eq. 3 d u am ul fo aglm o. Af all dy of o o mu qud fo a agld ym cog of o uym. I Fgu 6 coma y ddc of vo Numa oy S oo [P ] ad of dcal la o H [ P ] H [ P lco oo ] of lco ad of quad mod gv y Eq. 3 ad 39 cvly. W lod cuv y ug log-la cal a lag y ag a codd fgu 5. A fgu 5 av aumd a dd vaal q 3 d umcally a q [ I W cm ].. av a λ π. T logamc ca of vo Numa oy y cuv S claly fgu. T la oy cuv H alay mall a vo Numa oy ad ca muc lo a la o. T ca of ac mau of aglm y cag y af all o a ucd ul c aco of oo ad f lco comg og ad og a oo dy gg lag.

17 Vaó: Eagld_Poo-Elco_Sa_A 7.5 Eo of Poo Fld S H W ê cm D Fgu 6. So a comao of y ddc of vo Numa oy S oo [P] ad of dcal la o H [ P ] H [ P lco oo ] gv y Eq. 3 ad 39 cvly. 6. Summay I a av dcud aglm oo ad lco. W av o a agld oo-lco a oducd y u av a clo coco ccal a oducd y ac 968 c mm a um-a ucay oduc of oo fld. T a a of ally am mamacal ucu a a of ac a ad aually aa o-uav aaly of ml aco of QED av codd amly aco of a f lco a quad mod of lcomagc adao. O a of ou aaly av gv a ml ao of a of ac y u may ay a la a av a ycal gfcac a a g m mamacal couco a y ogally oug of. Bd av dvd ac aalyc o fo ducd dy oao of oo fld ad of f lco ad dmd vo Numa oy of oo ad la oy of oo ad of lco. I oduco gav a f ocal ovv of dvlom of coc o aglm ad o lad f ac m. Moov cd mo moa aoac o olm of quaal a of la ocllao o of a quad mod of adao fld. O uo quod aly fc oo o a d ad ca ac of voluo of coc o quaal a fom vy gg. I Sco av ummad ac o of co ad oao of quaal a oducd y Sud ad Glogo 96 ad d ccal a coucd y ac 968 c mm ucay oduc of um oao ad co oao. O a ou al o Bgou ad Vaó 98a Sco 3 dmd ac aoay a of acg oo-lco ym. T a a ml oduc a o oo a a gald co a ad lco a a momum ga. Sco a dvod o couco of agld oo-lco a c a dfd a Gaua uoo c o lco momum vaal of aoay a dcud Sco 3. A alady mad agld a dfd y Eq. 8 a o oluo of a u al valu olm ould av aumd a ally o-acg ym d y a a oduc a ad c o

18 Vaó: Eagld_Poo-Elco_Sa_A 8 aco ay a om ay. W lav udy of la olm fo a aa o og Vaó 7. I Ad A av gv a ac aalyc o fo ao coffc of agld a c o um a a of oo Hl ac ad oo ao lco Hl ac ad ma udd o of aocad oaly duo fo vaou aam valu lag cao lm. T ao coffc of agld a oad fom Eq. ad ud Eq. 5 ad 7 a fac jo oaly amlud of dcg a lco a om oo ad a a a of m ad a am m dcg ca df um of oo. T ac fau of jo oal av lluad Fgu --3. I Sco 5 d ducd dy oao of oo fld ad of f lco ad l of m u oo um duo ad lco momum ad oo duo av calculad. T ac o av dvd Ad B ad ma av aalyd caacc of duo lag cao lm. A mau of aglm vo Numa oy of oo fld ad la o of oo fld ad of lco av alo calculad acly ad clod aalyc fom fo m gv lag cao lm. W av ovd y a lc calculao a la o qua cocd. Ou ul a dlayd y Fgu -5-6 c o u oo um duo y ddc of vo Numa oy of oo fld ad comao of y ddc of la oy ad of vo Numa oy cvly. Fally o a may m o a ou co o cof ou o-uav udy o aaly of aco of a f lco oly o quad mod of adao fld. I aly of cou lco ac ol amly of mod du o.g. coday oaou mo oc fo ac ca of Como cag. T udy of aco oly o mod ca jufd f mod a vy gly cd a a a moly aumd a. I cal ca c o o ad of ga moac yc of ola oc ag lac la-ma aco aco o mod o om o d ody ca ad a mall uao. I co.g. o of Bgou ad Vaó 98a-. I od o av a ma fo magud of cao dg alc la ym ca u Eq. B.3 of Ad B c gv a umcal fomula fo ma oo occuao um. I Tal B av ummad umcal valu of aam a d fo d of la adao. I a fo maagal oaday ma occuao um ca omouly lag. Of cou a c-um lcc fld g cao aocad o v a vy gly occud um a a c. T aocao ca coly do fo ac y ug co a of Scödg-Glau y Glau 963a-. W la o udy of co uoo of agld oo-lco a l. Acoldgm. T o a uod y Hugaa Naoal Scfc Rac Foudao OTKA Ga No. T83 ad K7378. Ad A Dvao of lc fom of agld oo-lco a I ad o ac ladg o ac aalyc fom of ma lm Eq. dg A. π dχ co χ ϕ D[ ] π m ad dv aoma fom Eq. of m. W ov a aymoc a Eq. 5 a oly omald. I od o a f gv a alav fom of ao coffc of gald co a dlayd Eq. 6

19 Vaó: Eagld_Poo-Elco_Sa_A 9 <!]! [!]! [ ] [ L L D c A. accodg o Eq. mc a χ ε Ω mc a Ω ad. A.3 T qua a Ω ad av oducd Eq. 8 ad 8 ad dmol vaal ll ud o calcula adal gal Eq. A.. T gao c o amu agl χ momum ac Eq. A. ca cad ou y ug aco-ag fomula fo lco la av co ϕ χ ϕ χ l l l l A. Gady ad Ry fomula ad lmay lao d πδ χ π χ. T fom Eq. A.- oa <!]! [!]! [ L L y d ϕ π A.5 av oducd oao y ad m. A.6 do oday Bl fuco of f d of od. T adal gal Eq. A.5 ca d a aalyc fom y ug fomula 7.. of Gady ad Ry yldg <!!!! y L y L y y π ϕ. A.7 I od o udy aymoc avou of aov ac o fo lag valu u lm fomula Edély 953 fomula.36 ] lm[ L A.8 fo gv ad valu. Moov a lm uc a ay a oug o ad quaao volum 3 L a gog o fy oo dy 3 L ρ codd a a fd aam. Accodg o dfo of ad gv y Eq. A.3 ad A.6 cvly av mc A π λ Ω c ω π ω 3 L c A mc A ad. A.9

20 Vaó: Eagld_Poo-Elco_Sa_A T ycal mag of amlud A ad dmol y aam dcud Sco of ma. T agum of Lagu olyomal ad o o fo ad cvly y y A. dfo of y Eq. A.6 a alo a o accou. Tu o a of Eq. A.8- f l o g ad d of Eq. A.7 av [ ]...!! I L y L y A. ζ I a modfd Bl fuco of f d of od. I oag fal ul Eq. A. av ud lao ζ ζ I vald fo g ad fo a aay coml um ζ Gady ad Ry fomula T am o a Eq. A. com ou fom cod l o g ad d of Eq. A.7. T faco ca do follog alav fom A. ll-o lao lm a ud. Fom Eq. A.7 A. ad A. y ag oal y alo o accou fally oa lag lm I π ϕ. A.3 I ol o gv a alav ad o dvao of ul Eq. A.3 y ag a aymoc fomula alady fo ma lm Eq. A. ad of ug ac gal Eq. A.7. I od o do a l u aly follog fomula of Hl y Edély 953 fomula.5 ] [! 3 Γ O L ν ν A. vald fo > ufomly < < K ν. By alyg Eq. A. av 3 ] [ O D χ. A.5 T gao c o amu agl χ momum ac Eq. A. ca aga aly cad ou y ug aco-ag fomula Eq. A.. Tu o a of Eq. A.5 oa 3 Ω O mc A m d ϕ π. A.6 By ug W cod oal gal Wao 9 fomula 3.3 [ ] c a I c a c a d c A.7 codo ag π < c o afd fally oa 3 O I π ϕ A.8 c o quval o a gv Eq. A.3.

21 Vaó: Eagld_Poo-Elco_Sa_A A d of ad ov y a dc calculao a aoma agld oo-lco a dfd Eq. 5 of Sco ma oly omald lm. Rally af ϕ -gao av ] [ ] [ γ γ I I d d. A.9 No og o fomula ad of Gady ad Ry I ad ξ ξ ξ A. cvly oa ] [ γ γ γ γ I I A. av alo a o accou dfo of γ gv y Eq. 6. O a of Eq. A.9 ad A. omalao gal com d d A. u o omalao uvv aomao lm. I oag ul Eq. A. av ud W f oal gal c a c a d c A.3 ag π < c ad a a aay coml um Wao 9 fomula 3.3. Ad B Dvao of ducd dy oao ad of aglm o I ad f calcula dy oao P of quad mod aocad o agld a oducd Eq. 8. Accodg o Eq. 8 of Sco y ag aal ac dod lo y T of dyad c o lco vaal av { } { } ] [ ] [ D l D g d l d T P l. B. By ug Eq. A. ad A.3 af avg fomd gao c o amu agl χ fom Eq. B. oa [ ] [ ] [ ] <!!!! L L d P B. dfo of Eq. A.3 ad oao av ud. T gal Eq. B. ca aalycally do o a of fomula 7..3 of Gady ad Ry

22 Vaó: Eagld_Poo-Elco_Sa_A P P α [ ] [ ] P P B.3 P < P a aco olyomal of od. Fom Eq. B. cla a cau of uay of dlacm oao D[ ] ad c ofl fuco g omald duo { } P dfd Eq. B.3 omald o uy. Bcau > fo o > ad < < agum of aco olyomal aov quao ou of val [ ] o of c all o a locad. Tu g P cao a o o valu. Moov ou of val [ ] olyomal a all ov c y ay o qud fom a u oaly duo. I lm g P ca aomad l of aymoc fomula Edély 953 fomula.8 lm α α α P B. ou ca o α ad fom Eq. A.9 α q q. B.5 W l of Eq. B. ad B.5 g Eq. B.3 ca aomaly d m of modfd Bl fuco P q I q. Hov aoma duo o omald oly oly gv lav oo occuao oal. I od o dv a oly omald aoma duo lag lm dcly fom Eq. B. a o ocdg dffly a fo. By ug Hl fomula Eq. A. gal Eq. B. ca aymocally d a q [ ] d [ ] I q d B.6 av ud W cod oal gal gv y Eq. A.7. Tu fo lag valu of ducd dy oao P ca oug o fom 3 P O I [ ][ ] I q B.7 q λ π ad qua ad av alady dfd Eq. A.9 ad Eq. 3 of Sco ma. By ug aco-ag fomula Eq. A. ad lao I ca aly ovd a I c of g { } q oly omald... T vo Numa oy S oo aocad o duo { } dfd Eq. 9 ca codd a o of aual mau of dg of aglm oo-lco ym. By ug Eq. B.7 oa S oo [{ }] log q I q log[ I q ] I q log[ I q ] q B.8 aam q a dfd af Eq. B.7. No l u dv ducd dy oao P of lco aocad o agld a oducd Eq. 8. By ag Eq. 5 o accou aal ac dod y T of c o oo vaal ad

23 Vaó: Eagld_Poo-Elco_Sa_A 3 { } [ ] ] [ ] [ D D m g g d d T P. B.9 T ma lm of P momum ac ca calculad y ug lao ] Im ][ [ ] [ ] [ D D D B. ad Eq. 6 ]} Im[ ]{ [ L m g g P P B. L a Lagu olyomal of od. Fom Eq. B. cla a dagoal ma lm of P momum ac a mly gv y modulu quad of g fuco g dfd Eq. 8.. g P Π B. av oducd dmol momum duo Π Π π. B.3 T ma lm of ducd dy oao P oo ac ca dmd y ug d { } T P ;. B. By comag faco o g ad d of Eq. B. cooda ao of agld oo-lco a dfd Eq. 9 al a fac ma lm a d y follog cala oduc ; 3 O P B.5 a dfd Eq. 5 ad 6. I ca of vy g oo cao mo accualy lm dy fuco Eq. B.5 com ; ; F P ad ; π I F. B.6 I oag Eq. B.6 av ud fomula ad of Gady ad Ry c alo ud dvao of Eq. A. I B.7 ad lc fom of γ gv y Eq. 6. T dagoal ma lm of lco ducd dy oao a dmd y dmol dy fuco ; π I F P. B.8 T dy fuco P omald o uy fo ay a of m a ca o mlaly o oof of Eq. A..

24 Vaó: Eagld_Poo-Elco_Sa_A Accodg o Eq. B.7 dy oao of oo fld ud ou o dagoal u al o do mmdaly lc fomula Eq. B.8 fo vo Numa oy S [ ] [ oo P T P log P] S[ { }]. B.9 A fom Eq. B. ad B.6 lco dy oao P Eq. B.9 momum ao o oo ao dagoal. I od o calcula vo Numa oy of lco f av o dagoal P c av o al o do y o. I od o avod dffculy udy o-calld la oy H c a a clo coco cod od Réy oy. T dfo of H ad H T ρ H H logt ρ B. H cod od Réy oy ad ρ om gc dy oao. L u f calcula la oy of oo fld aocad o duo { } gv y B.7. H oo q q [ P ] TP H [{ }] I q I q oo B. q a dfd Eq. B.5. I dvg Eq. B. av ud ummao om of Bl fuco alady gv Eq. A.. I od o calcula la oy of lco d f a lc o of P c ca oad fo Eq. B.9 y a agfoad calculao * P d d d 3g g g 3 [ 3 m ]. D [ ] D[ ] D [ 3 ] D[ ] 3 B. T ac of P d a TP d d g g D [ ] D[ ]. B.3 By ug Eq. B. ad A. aov o ca oug o fom [ L ] TP d d g g. B. W l of follog afomao of gao vaal y y y y B.5 gao c ad ca aad TP π π d B.6 d [ L ] av ud lc fom g π of momum ofl fuco oducd ma Eq. 8. Af avg fomd -gao ad oducg dmol aam ad gao vaal av fom Eq. 6 [ ] TP d L [ ] P B.7 P a Lgd olyomal of od. I oag Eq. B.7 av ud dfo Eq. A.3 ad fomula 7.. of Gady a Ry. Accodg o Eq. B.7 ad gal fomula gv y Eq. ac o fo lco la oy ca a [ a H lco P ] P B.8 mcω

25 Vaó: Eagld_Poo-Elco_Sa_A 5 dfo Eq. A.9 a alo ud. I lag lm ca u Hl fomula Eq. A. ad aly dcly gad Eq. B.7. W av 3 q 3 TP d [ ] O I q O q. B.9 I oag aov aalyc ul av ud W cod oal gal fomula dlayd alady Eq. A.7. By ug Eq. B.9 lco la oy lag lm u ou o q [ P ] I q H lco B.3 c cocd la oy of oo fld gv y Eq. B. q H lco [ P ] H oo [{ }] I q. B.3 A d of ad gv a ma fo avag occuao um of oo fld d m of amlud of lcc fld g F o quvally m of y I of a qua-moocomac adao. I f ac -dmoal aal mod 3 dy a fqucy val ω ω Δω ca d a Z ω Δω Zω ω π c cal mod dy. W qua m avag of gy dy F π of adao mod dy Z ω Δω m avag occuao um of mod mulld y cal oo gy ω.. 3 π c F Zω Δω ω F π. B.3 ω Δωω π By oducg dmol comao of aam oa fom Eq. B.3 π c ω π mc F mc ω B.33 6 ω Δω ω 6α ω mc Δω α c 37 do f ucu coa ad F mcω dmol y aam oducd alady Eq. 3 of Sco. T cod faco o g ad d of la quao Eq. B.33 qua of ao of lco-oo a cao gy 6 mc V o oo gy ω. Fo ocal fld faco of od of cau ca ω V. T d faco ω Δω v lav add of adao c ca of uld la of od of ul T ul ul duao ad T od of cal cal como. O a of Eq. B.33 umcal valu of ca calculad accodg o fomula 5.7 ω Δω I B.3 E I do y dvdd y W cm ad E oo gy maud V -. I Tal B umma fo d of la umcal valu of oo gy v add ad coodg avag oo occuao um d m of dmol y I o a of Eq. B f T : Sa la 5. Nd : Gla la CW H-N la E ω Δω I 6 - I 3 I Tal B. So fo d of la umcal valu of oo gy v add ad coodg avag oo occuao um.

26 Vaó: Eagld_Poo-Elco_Sa_A 6 Rfc Al G B T Hoodc M Hoodc P Hoodc R Röl M Wfu H W R ad Zlg A Quaum fomao: a oduco o ac ocal coc ad m Sg Tac Mod Pyc Sg-Vlag Hdlg. Ac A Gag P ad Rog G 98a Emal alao of E-Podoly-Ro-Bom Gdam : A volao of Bll qual Py. Rv. L Ac A. Dalad ad Gag P 98 Emal of Bll qualy ug m-vayg aaly Py. Rv. L Bc W Wódc K ad Zuay S 987 Squg of cavy vacuum y cagd Py. Rv. A B-Ay Y ad Ma A 985 Poduco of qud a aco lcomagc adao ad a lco ga Py. Rv. L. 5 - acl Bgou ad Egl B-G 99 Oao of a. Fudamal Aal of Pyc N. Y Bgou ad Vaó S 98a Nola cag oc c of a quad adao fld: I. No-lavc am. Py. A: Ma. G. l69-l8 Bgou ad Vaó S 98 Nola cag oc c of a quad adao fld: II. Rlavc am. Py. A: Ma. G Bgou Vaó S Faa Gy ad Fdoov M V 983 Aoo ad ducd mo of quaa a al omogou lcomagc fld y a f lco ZETF Bo I Ya 98 Smclacal aomao fo mulad Bmalug Sov. Py. ETP Bloc F ad Nodc A 937 No o adao fld of lco. Py. Rv T a a d Scg 958 Edo Slcd Pa o Quaum Elcodyamc. Dov Pulcao N Yo 958 Bom D 95 Quaum oy. Pc-Hall Ic. Eglood Clff N.. 95 Bo N 935 Ca quaum-mcacal dco of ycal aly codd coml? Py. Rv Boum D E A ad Zlg A Edo T yc of quaum fomao: quaum cogay quaum loao quaum comuao. Sg-Vlag Bl Cau P ad No M M 965 Co a ad um-a ucay lao Py. Rv. L Cau P ad No M M 968 Pa ad agl vaal quaum mcac Rv. Mod. Py. - Dac P A M 97 T quaum oy of mo ad aoo of adao Poc. Roy. Soc. A 3-65 T a a d Scg 958 Edo Slcd Pa o Quaum Elcodyamc. Dov Pulcao N Yo 958 E A Podoly B ad Ro N 935 Ca quaum-mcacal dco of ycal aly codd coml? Py. Rv

27 Vaó: Eagld_Poo-Elco_Sa_A 7 Edély A 953 Edo Hg acdal fuco. Vol. II. McGa-Hll Boo Comay Ic. N Yo 953 Fdoov M V Efmov M A Kaaov A E Ca K W La C K ad Ely H Pac aog ad quaum aglm oooao ad oodocao Py. Rv. A Fdoov M V Efmov M A Kaaov A E Ca K W La C K ad Ely H 5 Soaou mo of a oo: av-ac ucu ad aom-oo aglm Py. Rv. A Fdoov M V Efmov M A Volov P A ad Ely H 6 So-ul o og-fld au oc: a ou o udy agld av ac. Py. B: A. Mol. O. Py. 39 S67-S83 Fdoov M V Efmov M A Volov P A Mova E V Sau S S ad Kul S P 7 Aoocally ad g aglm of oo a gad oaou aamc do-covo Py. Rv. L Fyg M ad Sclc W 993 Poo coug quaum a ad a-ac duo Py. Rv. A 7 R3-R33 Fyg M 99 Ü d a d Quamca PD Dao Faulä fü Naucaf d Uvä Ulm Gmay 99 Gao D 95 Commucao oy ad yc Pl. Mag. XLI 6-87 Gao C ad Wog 97 Caocally cojuga a ucay lao ad a oao. Ma. Py. -9 Glau R 963a T quaum oy of ocal coc Py. Rv Glau R 963 Co ad co a of adao fld Py. Rv Gady I S ad Ry I M Tal of gal ad oduc S Edo Acadmc P Sa Dgo. ac R 968 Mmum ucay oduc um ucay oduc ad co a. Ma. Py oo H Z D Kf C al. 3 Dcoc ad Aaac of a Clacal Wold Quaum Toy Sg-Vlag Bl 3 oda P 97 Ü u Bgüdug d Quamca. II Zcf fü Py -5 S ag 3. T a a couao of a al a: oda P Ü u Bgüdug d Quamca Zcf fü Py Kau H A 3 Quaao of ocal a ac S {φ mod π I > } m of gou SO Foc d Py S alo dd vo of a aad a a - : Kau H A 6a Quaao of ocal a ac S {φ mod π I > } m of gou SO. axv: qua-3769v Dcm 6 Kau H A 6a Quaao of ocal a ac S {φ mod π I > } m of gou SO axv: qua-3769v Dcm 6 Kau H A 6 Quaao of caocally cojuga a agl ad oal momum. Py. Rv. A

28 Vaó: Eagld_Poo-Elco_Sa_A 8 Kau H A 7 A loo a quaum mcac of amoc ocllao Aal d Py Lodo F 96 Ü d acoc Tafomao d Quamca Zcf fü Py Lodo F 97 Wlvaal ud aoc Tafomao d Udulaomca Zcf fü Py 93- Lyc R 995 T quaum a olm: a ccal v Pyc Ro Nl M A ad Cuag I L Quaum comuao ad quaum fomao Camdg Uvy P Camdg Eglad No M M 993 Quaum a ad quaum a oao: Som yc ad om oy Pyca Sa T8 5- S alo o a aad ocal u of Pyca Sca T8 993 dd y Sclc W P ad Ba S M 993 No W Fougè A ad Madl L 99 Maum of quaum a y oo coug Py. Rv. L No W Fougè A ad Madl L 99a Oaoal aoac o a of a quaum fld Py. Rv. A 5 - No W Fougè A ad Madl L 99 Fu vgao of oaoally dfd quaum a Py. Rv. A No W Fougè A ad Madl L 993 Maum of oaly duo of oaoally dfd quaum a dffc Py. Rv. L Paul H 97 Pa of a mcococ lcomagc fld ad maum Foc d Py Pgg D T ad Ba S M 989 Pa o of quad gl-mod lcomagc fld Py. Rv. A Přová V Luš A ad Přa 999 Pa oc Wold Scfc Sgao 999 Poov V N ad Yau V S 99 Quaum ad qua-clacal a of oo a oao. oual of Mod Oc R F ad Sőfalv-Nagy B 965 Lço d aaly focoll. éd. Gau-Vlla Pa ad Aadéma Kadó Buda 965 Sclc W P Quaum oc a ac Wly-VCH Wm Sclc W P ad Ba S M 993 Edo Scal u o Quaum a ad a dd maum Pyca Sca T8 Sclc W Hooc R ad Vaó S 989 A fucao a oaly duo of a gly qud a Py. Rv. A Scödg E 935a D ggäg Suao d Quamca D Naucaf 3. ag. Hf

29 Vaó: Eagld_Poo-Elco_Sa_A 9 Scödg E 935 D ggäg Suao d Quamca. Foug D Naucaf 3. ag. Hf Scödg E 935c D ggäg Suao d Quamca. Scluß D Naucaf 3. ag. Hf Scg 953 T oy of quad fld. III Py. Rv T a a d Scg 958 Edo Slcd Pa o Quaum Elcodyamc Dov Pulcao N Yo 958 Scg 96 Boa moo of a quaum ocllao. Ma. Py. 7-3 T a a d Flao M Fodal C ad Mlo K A 979 Edo Slcd a of ula Scg D. Rdl Pulg Comay Dodc 979. Sao H ad Sad S R 99 Quaum a maum: A ym-oy cv Py. Rv. A Sm L P 96 Quaum ffc aco of lco g fqucy fld ad ao o clacal oy Py. Rv Solm S ad Suom K-A 5 Quaum aoac o fomac o Wly & So Ic. Hoo N y 5 Sud L ad Glogo 96 Quaum mcacal a ad m oao Pyc 9-6 Vaó S Eagld oo-lco a I oo of aac of 7 Cal Euoa Woo o Quaum Oc Balaofüd Hugay 8 Al- May a 6 Vaó S 7 aao Wao G M 9 A a o oy of Bl fuco Scod Edo Camdg Uvy P Camdg 9 Wyl H 93 T oy of gou ad quaum mcac Pag 36. Dov Pulcao N Yo 93 Talad fom cod vd Gma do y Roo H P Wl A 96 A. N Yo Acad. Sc. 8 9 Wg E P 97 O dd vaal ad quaum mcacal oal Am.. Py Wllam C 999 Edo Quaum Comug ad Quaum Commucao Vol. 59 of Lcu o comu cc Sg-Vlag Bl 999 Wu C S ad Saov I 95 T agula colao of cad alao adao Py. Rv Zu W H Ha S ad Pa P 993 Co a va dcoc Py. Rv. L

30 Vaó: Eagld_Poo-Elco_Sa_A 3 Cao o Fgu So ao-moal duo of jo oaly comg fom Eq. o Eq. 5 fo dff valu of um of md c oo. I ac fgu av ud a umcal omalao faco od o av mamum valu of vcal cooda ougly uy. T faco a follog: a: 3 fo : 9 fo c: fo 5 8 d: 8 fo 5. Cao o Fgu So c oo um duo aoud cal lag al oo um fo dff ao of ad. T ma a -ddc alog l o la a lod fo dff -valu. T ag a:. 3 a. 6 c ad. 5 d. Cao o Fgu 3 So cmacally ac-m go a of jo oaly duo a qualavly dff. Cao o Fgu So u oo um duo { } dvd fom ducd dy oao ad gv y Eq. 9 fo fou q y valu amly fo q. 5 a q 5 q 5 c ad q 5 d. Cao o Fgu 5 So y ddc of vo Numa oy of oo fld dfd y Eq. 3. Cao o Fgu 6 So a comao of y ddc of vo Numa oy S oo [P ] ad of dcal la o H [ P ] H [ P lco oo ] gv y Eq. 3 ad 39 cvly. Cao o Tal B.

31 Vaó: Eagld_Poo-Elco_Sa_A 3 So fo d of la umcal valu of oo gy v add ad coodg avag oo occuao um.