565

bullAi K6aVTIKai rE()~ETpial TUnOU Riemann B KivlolC nupfjvoc

0to6wpou Xp XWl(ou

I rCYIKa

Elt 0 npOOTOV ~euroPOC [I]e) fie 6~OTlAOO )leuroAeuro Tlt )lOO poundltPriPshyIWOO tciC K~avnK6c r euro OJ)uTpiaC 1(11(00 Hi e rnann de lIlv neshyP101POtp1KiW IdVllO1V tOi) HAEKTpoviou Ittp tOu npUltoviou )lpound tAlip trIltuxiav rHt() 9euro ooprltlIcilc Kat itClpUIJOtlKnc 0716shyEWe

Etc TO napov Jltpoc Tile jJdpoundTllC lUUTJ)t 86 ttpap)loa8tt ~ iSia 1tEP10tpOlplKti KivllOIC TOO nupflvoc lntO nv poundrtiopaolV loij crtaTII(oi) nUPllVIIO[) rmSi ou htpwv nupljvwv

Opl v pound~euroTOcrelj iJ roc avoo KivlOIC Ul1eurov8Ujll(W o n 0 ttvPTlV

9euro wpetCII 6n Cxeurol spin -t Kal on ovfjKCl de EVa xwpovshy

Xpovov TWV 5 SlaaTaowv be UgtV onoiulv ~ neurojJ1ttrl cival )l i a 1t pay 4J a T J( 1) t ( ltp U A L a )l t v 1) Eo I 6 a 1 a a 1 0

A) 0 txlpU) IOjJpoundVTW OIOOtUOW (nc 111V X ~) 9EOOPW tKEivTv ~ fU Pian Tilc A PX il~ t a u Anpoo15lOpiotOU TOU Heisen belg( I) 6KO)ou9Ei tac OXEOE1C (2)

DX X~D = - iljS~ (1 )

(8) D= agt xmiddot ~ dT~a X 0 D 0 (2)

(A)

tt1 AClOlj 10 crw~HlT~15IOV elc lilv reooJlspiav (B) cLvO npOOKsshyKOAA1ptvov s tc TOV xwpov - XPOVOV TOO E instein (Xamp = Otashy8epov) t(OV QrtpOaOtOPIOTov nooo11TO KIVtlOSWC (D~ = 00) Ked ouveftG) npooAa~j3avEt Ta~ HAetCTPIK(lC tOl01Jme wi) xoopoumiddotxPOvou TOC Einile in ncxpoum6(ov TOU10XPOvltuC JIIOV OUVCllllV KOla lJiV nEJIrtTJV ouicrTOOlv (b~ 0) -OOOV OIpOshyPQ poundIe Tllv rew~ETpLOV (A) TO Ooo)lCltiolOv KIVsiWI)Ie Ota9s pav noo-OHltCl KlvJioewc (D~ OTCla ) KOt poundIVClI QrtPOOOIOPIOtOV hi tfiC nEJIrttlle oloOtOOsoo (X~ = co)

TOIOUTOTponWe Ii ~h6OTOOlC X OEV KrlStOtatat QVflAl1nTrl 8Ul tfiC KlvftOetiM )IClC en Climle Jx toOt Q Ou~pctiVeuroI ~pound TaC cruvrshygelc 8IuOtaopoundtlt X X~ X J aAA6 rtUpOuOlu(el t6c crvveltSlOe Tip ultliptwc T11C Etc TOV xamppov-xpo vov tOi) Eins tein KUta tJiv Onoto)lov tVQAAcrrT)v TroV lEWIJetpl(A)V (A)(B) [J]

Touto ())IOIOSeL npoc t~v t~ rl()6vlmgt JotiC 8WOtoOEwc Etc TOV El)lrhi810v yamppOV Ehe 81el TI) QKIVTJcr[O~ )IOC tnt aVTije Oa) EhE 010 TijC KIV~OEWC en aotiic j1STa ota9upfic TOXUtT1TOO fl 1[OOOlTjTOC KIVrtOeuroCIX UP) (n( tile X ~ 0 6)

dx l 2 + d ~~ + d x3~ - dx~ - _ cd f

clx~ = icdl d x i =- 0 (3a

dx~ 1 dx~ = lt yenr -u- d e (3y)

Eivctl o~l()OTjILE(OtOV Ott ~ EOlt9 tfiC EKIpUAI(T~euroVJC blaOtOOEw to e Ep~OnUPlVlKa (j)OIVOj1EVD ouvavtQl n)ijpwc va EtlYllGwOl [2J PI olmi 0 nupJiv rpt p1 ~AEK1PIK6v lpoptiov(IO) OID1i 0ql[~ OTaV1Ui at ~(lyv11nKot Qvwj1aAiol TOW nuprtvwv() Blad to eEpj100UVO~IKQ qgtC(VO~EVIl Elval QAAIl)pound v15ETooC OUVOEOEj1EVQ j1ETa t(i)v JtUPTjV1KWV 10100tOOV Kai TEAOC Oillt crooj1oTiolO IJEtCt ow~opugt b qgtUAlOj1 euro VWV OIOOtacrEoov (KCti OuvEncllC 7tOPOUOIOshy~OVTU oop6pouC n010TI KOC 8wlpo pae) napouOa~o vtol EXOV1Cl IJOVOV ouvijaElc 8Iocr1ciOEIe 8llAooT tOC 8 10crtltioE I~ t oO XWw poumiddot xpovou roO Einstein (Xl X2 Xa Xmiddot = jcl) Kal Ol)vrpshyYO(OIJEVa )lEta nj)v HAeKTPIKWV OwIJOTlbioov AOYQJ tfjc rEWshyIJEtpiac (8)

a J DI + t~D2 + etD a + aD~ + et~D ~ - imc 1~o = 0 (~)

al- ai + a et)l=2 8)l1 1--gt - 123 45 (~a)

B) LuvEnampc T6 oWj1otiOlov ttOv JtEVtE OIOOtaoeuro oov KOT opxt)v eo tKcppa(poundtCl1 ono fjc (4) oui 1T)V m~pintOOOlV EUKAE10[OU 7wpoo(40) dIU tngt ftEpintwOl v t(A)V OIClO6crEwv Xl X~ X~ ft (4) yiVEtOl we i (5) j1Eta t(A)V ~Tj1PGlV 0 a 0 6 (6)

IalDI + o4D~ + algtD~ - imc I -0 = 0 (5)

ct I = 01 = Pili - p~ 0 + p)r 4

0 = a~ = PIAr + PlB - paiL

a~ = a~ = PIBr - PiA -- PaBLi l (6) A = cos heD iB __ sinhltIgt r = sinS Li = cosO I

AI TI~toi tampV JIJTp6)v (6) 8 loouOI TOUe KlaquoTW91 taVUoTac (7)

- el~ cQshkR2A = FI 41 =2Fl u = 2 - - - gt

)C n

s i nh~ R (7 )

It

2F5~1 =- 2F II = F~H=ik centl=kR e =~ 1_ 1nRz (7laquo)Ic

FL=o F5~-l - middotF~=2F~i =- 2F~)l ft4 = F~5 =0 (7~)

Qe Elval Ipovepov 01 tavuotai A4 Kal A~ OKOAou901)01 tTv yVWOt~V crxEltJv tOu )lEOOVIKOO ItEoiou (8)

I d 2 2 d 1 IdR+ 1f CfR -k l A~=O ~ = 45 RO (8)

Et aHou elva OOTOVOntOv OTt it (5) oi15EI TaC OXEOhC T(A)V TOVUOtampv (9) (9a)

(~) Opa tIIv l-I~kttlV 1-101 ErrorllH1 If thl Quantllm Theory or FIelds 1969 ITayoOlllQUAOIi A (Ci)vo(ClvlIvou 6 - AOivo bull

bull ileiCIfOOXO( til AVGDII I~OAII Mlxavol4TCI)I- HUICpOIOTQlv tfle poundllOsO( ~uuampuv(il toG OTE n O~I[Y Texvl1llo ~~uAlaquo tl toO EMn (192) il-tltbot T~lllilroPXI middotpoundIIOIoVIIC6)v Epdlv(i)v til EJlltpolllK npotJrt9tligtv 4TJIlooiou ICgl AvullpOf6poo 06tl AvUIIPOIshyPo6I0I)cyPU9PK Y1llTJjHal9 roO NgUfl1ll00 Ko91T1~ HuPIOIlOO sp~ EHlvIf AOtpoll)tnI1ll1l ElOlpdg ~rto(JIIIlOt en6 I(TXIC6aTlW Ix Ngun Ill oorIIlWI- Mlxov ilv TxVlICO( IIi~(Sgu)O( liJo9~ XPOUWl IIO)~ 1IIIXltOfII11tlt ll6Mff~ QVfI~p6Uv dO to rUDIlIJPfl(Ct-shypoIV EIolPCUraquoV a] Y1IITJtKOld)V ApxITO( toO nlpoa(SsoluoO ~oOlt gtII1J(J6~ OVfl)il~ 011 (KfktvtICal rUDpctpo IC)II)

S66 ItmtjJ ~PIOC 1910

l)4 ~ i C cos lilcR bull u ~ C silllLkR 8ta iu4 laquo)1

D~ i ( me + ~ ) oshkR D ~ me sin hkR bull

amp iD laquo0 E laquome ( ~ )

II Kivlou TOO llUp~yOC

en clamptacreil n KlVTlcrU ojtn El~ to ncpl1trwoEu lliC eb9uw YPC(jJjlOIJ to IVil OEOOC Ka( til c 1tpoundP1CfpOljH I( le rotnHIl t

A) EoOirYP0l-lfIOC Ki vllmc

H (6) Sibel tac KCtrWet nocr6tl1tnC

J) Err tllv IEUljeurotpiov (8) rj (5) aVtlKofltOrarm un6 nj C (10) Ole vd

l alDI +tX~D4-im c l of~--- O u~ =ia~c=O (10)

OpoundIXO) t6 poundKIPUAIO)leVQV tf) c 1tt~HmlC SUOtUltTftl (20) n toshyZl)tll~ u = 0 dVQI Jl n8Evlld Kcr i cruvt1Hil 010011)0 k = 0 ~lllScvi~Ql jlE OUVEltEIQV niv ~ U()UVlOW nC a bullbull 011 )0011 T6 crlJl ~latl l5 l ov fpQiveurol(u we vi Elva l lV llAEKrpIUjltvov rTCllotimiddot blOV (P) E~ ilUou )614) ti1c lt rrpoablo(llfJtiuC ti) t Do ~ Xl il nO(l6rll ~ KlVfo-tmc U Aa~lpuvcl uvaAoyou nIHl c t va noshyPUUOlUCCT(U 6 OEp~IOlt1Jon 1VlKOV rpyo v K(LHI n~ 6Jfo6~louC CW1Uuyoe liLlv rElupEtpul)v (A) (B) we KUl rrEpaH~p(O adshyKVUtUI

OUtW EXO)LEV file KCtTW91 TI)tlle ere 1nV rEW)LElpiov (B)

dO - shyF I --a- = F lu Dumiddot +FI1~ 0u6+F14 Du +

+ [i uDut- o () (l1)

dO - shyF = d --4 = FI1 Du l + 116 Du L + FSH DoU (11a)

U t~ iC1lJ)QI lt K1VncrF(I) t 20u JJa9)100 Aa)L~CtvEI T1)V )LOpq)v ( J2) Iby(p TOU k = 0 ul

J D+ D + m~cmiddot l 1 = 0 ti I P~ - 2il)]1 0 4 - 2mE I o~I =-= 0 D~ ~ (me + E)i

~ I p - 211l( F + ~ ) I I J ~ 0 P =- i~ ~ (2) O ilTJ) CUptQKOPrV On to aW~lUTlolOV OKO)OUOCpound T00 (HolmwOe tipOjloult fillt yvwot1c tel0WcrrU)~ TOO chrodinger

2) Ei TilV rf())~ f tpiav (A) t6 crID)LOSlOV h Cl n)v jlOPltP1iv Iilli (5) file 6no(~ to tKIPOAICHltvov tKCP P(z~CTC1 uno 11C D ashy

= Oto6ep6v Al ouvuJlel~ kO)L~civoual nie TL)LUe (13)

dOl - 1 -0 -4F t = --r- = F II

DU F Du + FH

D 6u+F Du=(+-)u n II

_ I - 0 1

= FH D4+ F DIu a - 2 Ttl (13)

(13)

C) middotO)II rJlI(QI (2611)

(-) 0 1 tipOI fi 1 D ~u F~III IJ IIIWd ~1I16cVll(ot 6tuv -u 0)1 6p(liolV ~ CNIIIJJo lcllT lliol irUpOIul 11111 CUatlllJt ll ot IQY 1I[t IXlIl~lOIJGI ttfllOfpU cnJIGI6 IfUPOAt flloV tcu ~tlajl1 t fK A tlI-fl iI~ Enlloyflo II) 1) In ll 1 ~ f1loiL rraquov (II) OJ) UlllPiJvm 10v Opou fOU1 UQC w~ u V41p6y dyen

(13~1

6nia51) ~ crmOEp6tne [ ile Us (t3j3) Kni taOtOXp6VOlt 10 5uvashy~IIJo6v tlir (12) )6Ytfgt [ ic (13) bllUloAoyoUVtal

ncruUTCrll 61tHlr ide Tmiddot~V nlpin[wTl 100 HhKTpnOO crW~I (It lshy~i(lU [II 11 OUVUJlI t F I uvacpalVELfll Elf -nv (A) r eOlJlnp(o v 0no SlrrAouv ptYEO~ ~vQ) oI8uvtij1I1r Ii F s~ n )v(B) r cU)shy~lctpiav arrO omk v 6)1o(0)c UJlf)v

I [ t~iOO)crl C Klvticrcm de t1)v (A) nCllJlEtp(nv b El tllV JlOPrplv rf)r (J4u) AUJlPUVOj1tvLle un OWL v llic APlfl1i t1~ Ern)o-y 1rl I1

0 1+ ot+ D2+mlc+ laquo l ct~ D1D-DDll + ctla l(D1D +1

+ DD) I F~~O (11 )

~ lo6yll ~ (9a) ~v Kilamp

Or Ip - R m - 2m EcoshkH - 2c~ctla (F~1) 1 + F~DI) +

F D) IImiddotA = 0u 0

~ ow 1614gt ~ (15) T~V (14)

P - 2mcoshkR ( E + Hcoihk R ) +

2 ~k (H)+ nckrn + K ~ I = 0

V 0114 =- ia 0ltilX1 gto _ - _ 0

o 0 lt t4D + aOe - imt gto ~ 0 (15)

fl (14) npooEyyI~6vtmc )O)Jp(tVEL I1)V )JopQl1iv t1i~ (6)

P IkO(-E _ ) + k kmiddotI1 -0II I - 2 rncos I[+ Reos hUt e 111 ()shy

(16)

3) At auvt-rrIgtIQl tl~C (16) ampival u Ey ( 0 t 1 e a 1t 0 l) Sal 6shyTTl tO

a) l-Ilivcu tWV OpltJlV (L iL a ttluCOOt [01) Sdl1odi ngcr (17) rr pt1tc l va [uullcr9tl ~It tilv (12) lllOUO u~Hp6tCPaL t~tppushy~OU(jj TV

p - 2m CO hkR( E + RC~ hkR ) 1 ~t - 0 (17)

auto ltptllv6JlEVOV eit lUC [ cw)Iftpia (A) Koi (B)(~)

Olhw tva TO OUVQ)LIK6v eft- o6Vatai va UcplcrTUtUi e(c tnY (17)

to OWjJOtLSI OV 90 npt7tEl va ti piUKEtaL el~ onocrtCOl v n 1(18) bc toO KpoundVtPOlgt lOO 1tUpnVI KOU rreo iou ( ) OVC1tWC QVllltPalvlshytal to

eR R -V=---- --~ 1 = cos1hkR H cOS hkH

OOVa)llKOV tOO Juknwil uno ItV 1I0PlP1lv rfc (18)

a eov va crn)JElw9i) ~vlCl09a 6n t (1 IJfytall ei c n)v ItmjJcmiddottpluv (A) 6vwp CPOvlCtI d~ XlilpOVmiddotXVOvov s 31aOtUOO)V 5ui ~vnOlV toO (J(J)Jlutd5iou (116 trlOeE poV 1toa6tllUl KV11(fCrot D bull LUVIshy1[amp tic ov allf~ltvo v XU)POV - XP6vov (QU Ei nstein to JIImiddot

tiOq (Iilv 5 OIQUlcicrECUV lt peuron IS L Vt~ 8 ICllprOtrlal Sui LOO l)pou CQslkR(1 9) rtO PEJlql( POUc 7tP~ ov 6pov (I + u ) tlil (3y)

(D+ mmiddotc) =-- ml c1 p si nmiddot hkR) _ mt rcos1ILkH (19)

Ev rAAm~ I ~ampOlV 0 6po( co~lh kR 1tp~7tel vci eTvm atoOeurop6e OJUip KUt 1tQxtSEx61-1E9a

I

561 Itmt~ppLDC 1970

Kat aKolou9iav tva hWJtEV to OUVOJtLKOV (I 8) o~oiwt nppoundrtEL

e (201

R cos hkR I

va ~XW~EV tliv Sova~lv (20) Ett t6v X~pov-XPovov tOU Einshystein

4) EuvEnQ)C Etc t6v xoopov-Xpovov tOU Einstein ~~(crWcrlt KtVaEWC (16) Aa~jOVfL trv pO(HPilv (21)()

--p~ lt ck IPA=O _2m (E + -- --) -Ishycos hkR Reo1 hkH cos hkR 0

(21 1

Evta()9a lEVVOtCIl to ~PcllTTlIlCI toQ rrpoaoloplaJtoC tfjc ~ t6 onotov rrpClYJtatEU6lle9a rrEpaLtEpw

5) A IEOal t1Jtal t~v rEWJtelpt()V (A) (8) Elt tOY XGgtpov-Xp6shyVOV tellv 4lhacrtoaEwv OUVE1tOOt 9u ElvelL

FJ = dd~1 - COSI~kR [F~l DIJ t + F1 DIV6] =+

e+ u - =---~= (22)R rQshkR

F dO 1 [ 0 t U 0 0 4-J = - d- - -- k- F + I IU

l cos h H U Ie

1231

a) nC poundIvClI cpavEp6v at OOV6~EIC (22) (22a) (22P) UKOAOu9000l tiv 6pxilv tij c 61CltllPi)crEoo~ tfllt 9EPlOSUVClJtIKft~ ct- (24) SEIKVOpound1

dOdW + dQ-dE =0 dQ = -ctf dx (2 4)

Ev9a Q ~ Q1fOPpocpOU1JeurovYj 6epf16Yj~

V 10 d1foPP0loUflpoundlov Epyov E ~ euroOwepo~ Elipypound~x ou OuO1~1J( I o

OLhco jEllatoOwl Otl iJ 9EPPOOUVClJtIK Kai ta rrUpT]V1KO ~aLmiddot V6)lEVo poundIvai a) I)MVOEtCOlt auvoESpoundptvCl ro~ Kai oUaxoO ~v AErttOjLEpELQ (3) ~XUl cr1)IlEHlapoundL

P) H pound~icrwcrlt tf]e KIVrcrEro (23) we dVCll lpovpoundp6v (I) ItXEL to((IutllV jLOpcpi)v we Mv i rpoundoo)lEtpio (8) eIXE tilv crovn9n oulcrtocrlv Xlgt (5pa nEpaltepoo)

B) n EpOtpoqmct VTlCUe nllpfvolt

H npoundp1tfwolt alhn 96 pound~poundfClcrel KClTO T6v rrpocrolopla~6v

tf]e BE tellV (MOlXeCOV N N~ N N

1) H IlEcrl) ~~icrroOle KlviJaEpoundaX (23) Eft tilv rtEp(rrtW01V tne neplcrtpoqHI(~c KlvftOEwc Kat St HAWVW molou llapot1)tOt (U) [1] 1tPOIpClvWc AaJJPovpound1 tliv 1l0PPilv Tfj (25) Ola 1=0

(~) ~IC -aeo~ d -amp ~tAt(Olt ~OIJ [2H11I1)c gt- TlcpGfa 10 v - _~I~_llcoslvR

tUVl7llllt 01 2J (51 va olop9coOOol paOel 11lt lopoUolt

( ) EI tilY 1tsphuugtotv fOO KsSioIJ PUPVftO l)OlltW ga~ - I go- =lt

00 5) )6YC) If toCCPIJ)IO~~VI X OIOOlaoEco

p 2ij 1 0 [ _

i----- - - ------- -2m E+ cos l hkR R cos~hk R oR

e ] hckm I B (251+ -Reossiiln + 2costhkR IYo = 0

H tt[aWOlt aUtll 6le Kat n(23) n potpXEtal h ti~ urr0geoECJJlt on al (A) Kat (8) rfco)lEtPial av1KouOlV Ele X(i)pov - Xpovov 5 auvi)9wv SLClataaEwv QPoC elval f ~taTl n~t a utGgtv (KU( 6Jhamiddot cp6proc Mv poundX1) -r IVEI ~ olaipEale SUl cos hkR)

OUtW poundXO~EV oUl tilv KatOatoOLv ~ tTiv BE (26)

2~ 0 R c-o-sh k R- -o-R- +

2me Ijck Rcos hkR 0 + 0 4cos hkR 0 (261

Ij - imc10 10 dv = -- 0lt gt 0

cxo = me f 0

uvpoundrr(i)e otav il ntjJnTTl Stacrtome euroIvai )lla hcpuAla)JeuroVTl TOlshyToHTl ticp eurov6t )lpoundV dt Tilv rpoundOgtIlpoundTpiov(A) 0 6por cos 2hkR rtptrrEl va 9EWPTl9t1 ro~ otCl9poundpo KOt [aw rrpoe cos 1hkoo KClt rrUov YEVIKGgtt npoc cos 1hkan ( ) alp tttpou os El~ ttv nooshyIlpoundtptav (B) ~ BE tftt pound~la6JaEroc toO Schrodinger npErtEl va

poundlval ia1rtP6~-2 1~ (12) Ouroo ~ auvtnEla to)twv dv(u a

1) (25) dC t6v XGgtpovmiddotXP6vov TOQ Einstein 1tptnEl va AaPu T~V TL~~V (27) (6pa Eunl~p~a (8raquo

2~ a ( - e) Ilckm I BIP~ - R ilH - 2m E+-i- + 2coShkR Yo=O

(271

61(ou Y~ = ~ lvwatli Kat6crraolC ToO Schrodinger Yo (Ij

=Y~ (27laquo)

OlStoo fl Qnapaltntoe otCl9poundPoTI1~ tfe costhkR tcpapjJ6lEtUl KClta tTiv OlClltPOPlOIV Kai 6AOKAi)pOOcrlV H SICl(pEaL~ Ol autoC SIOEL t6 6uVoIlLKOV Kal to~ OUVU~poundLC toO MpoundOOVIKOO molou dC t6v XropovXp6vov ToO Einstein ElS1K6lTEPOv il 6PXr tfe 6LQt1)PT1crEoo~ ttC tVfPYEiolt npoooLOpl~EtClI rot poundav t9EOOpEftO 6 X()po~-Xp6voc twv 5 Slacrtocrpoundoov Iwi jJtt6 va OLQ1PE9fj OlO tOO 6vnatolxoOVtOC crta8poundpoO 6pou cos 2hkR tni nopaotiY)lQn iJ olacpopo OUVCljJl KOO de TOy XUpov-Xp6vov tGgtv 5 Olo(MOcrpoundoov

-kll - ~~ de T6v XQ)pov tttlV 4 oLaaToapoundwv y(VpoundtO l

e1 Rl cos~ hkRJ - c1 R cos hkRbull

2) H (27) ocpopQ t1lV 1(Eplatpocpfv toO 1tOPfvolt rrtp~ ~v6e KEVtPIlCOO nUPllvlKoQ nEbiou univoc R Ei tilv rtpoundp[1tlOJaLV tGgtv atoLxpoundiwv TO rtUP1VIKOV rrE8iov rr poepXEtal 6rt6 liJAOU~ rrupftvac 1ttPlcrtPtltpO)l~VOUC 6110 (ooe LuvEnampe t6 SU VClIl IKOV ell e t l ell t-If rrptrrEl va uVtlKotOOtO i olu toO Z - R ~ upoundIKVUpoundtOl

n EpolTEpw

3) napClOE x6)lE9a nlv un69Ecrlv tOO Cammer(middot) 1(09 ~v OOlashy1p6poor toO crl)IlElOU tOOV nuP1lvlK6v Poptlwv toO P Kat N KClt auOtpOlpfte Cl OUVU)lpoundI~ t(moll Coulomb fTvaL tAKTIKClt Kat iaal ota N gt 2 00too 1 (27) Aa)laaVpoundl nlv )Jop~rv tij (28) (UpoO

I 2~ a (- e )P - - --- -2m E+Z- +I R oR R

hckm BI (281+ 2coshk R Yo = 0

( bullbull) R - all = oCd~ 100 B oh r

C-) -16e Mpound)hTlv lou Erroneous Is the Quantllm lbeory or Field s

bull bull

568

4) n magepac() (lilv rruplWIKlUV OIDJlall~HOlV AajJlOYOpev

() 1jck = 205Me V me ~ 940A-h V 01 Ei 3e

E ~ olt E gto 0 en o =~ =- I~ (29) 2 2gt0

fvOc eL TO lTUP1)VL)(OV ql0pT(OV dI(aTo~x() v TOU HAtshyllt Tp tzoi) OLOUTOU f

5) H BEE EupicKEtal (00 Mv 1iro EV HeKlplK6v rrr(i[ov() ~v TO 8EE1 J1EtCr tnV (~haipEcrlV OICt foslh2kR( U ) IUVIl1t(O c

rXO jJEV Kal OU1 toile mJ p1vo N N) N~ N I EtKovlOPCshy

voue d tel ox~Jlatn tfK ( I) rue KUtroat tl~IIh toov R EE (n - 1 1= 0)

Cal

a) To) N (IgtX Ja) A( olvajJpoundtc teal to LuVQlJlI(6v de tOV

Xtllpov middot XP6vov tLlV JrFVtamp SlI10l0ltnfOV aa elval at (30) Kat cr)shyvun6lc (JOu) U1tO tOUe 6pouc lOIYtOUC 5Uitl tllv B E avO rrushyp~vatoU N

el 1 BVF _ _ ___= _ ___ 4RI 2 a R

(30)

Z 6 - hck - E = - mc - - ~O 511 MeV2 J)lct - bull E 1 = 4cos hk R shy

50 0 (30a)eli c06 h18 GIl

I a7 1 1 3~ nc 1 ~ck Ket=---- = ---= uli ---- (30~)6 Z g Z e ~ Z mc

AoOavtot lSn -t6 spin toD N~ dVal S JI 1 6lBetal [51(middot) jlICl Ii shyttlil t llJlj ron QUIljJlKOO Kat crOVpoundmll t n avO ttupftva B E toD

N d vC E _ - 102 ~vavn tftt E - - 108 MeV ro 1tEIM

pUPUtOc

~) ToO N (LX lP) uql(ataVtol Qt KlLtw81 tl~lai

1+ 2 fJshy bull 2R

(31)

3 3 Klaquo = - Z- = J11

50 2 e

B --CO - 37 -iiiIIl 021 MeV bull Eel -= T Ta

omiddot (k [31a)- To 6H = 0216 MeV

E=E + E1 + E= -70

(H) TD ~ua6vtov k ItUIJX EI tlllO 1oI 0000000TnK~ oO)IIIgtII TOO 1IIgtPi1CU Kill tvl f p hl II (6po () ad I IiW 11 txt L t tl 6v1xwlifl

( ) Ol MllllflV all Nrroneoull 11 Ih6 QuantuUl llI r ory or 1lelu ~middot

( bullbullbull ) Opu lIjILrrlp~u - B-

n)middotaSn poundt0tJllV B E dvu mJP~VU 1CUlpa)H1Wlt - E == 705 1eV

Atov VI OlllnICJlO~ ~vta)Oa o n 1 Eel (1I)eKfpLlotl u1riiOcmd poundV(PYCIQ urroAoyi~ltal ~U(JE~ Ti)~ urroorucreurolV R I(ul eX I Ttl t

It 6 bull 2R I cos h2kR uQ)oCl 16 H IfKtPlt v tpOptlO el~)cr t

tUl btl nov j1Eoovl c1 K (t6 ~I1toi(l lPCflflviCovtut 1l Cii tOY j1 ClI10Xml(JTl(J~t6v tul (A)(B)(middot) (opu at crUjJ7tATU)Wj11l - B- )

() To) N (IX ly) T BE avO 1t1)f1fiVQ poundiV(1l - 27 5 M~V f~

0 1- - V _-__ U R

-pound = 7 6 - ~pound5 2i5 lcN 3 l ll

(31)

E 50 50 ooon v 1 = cos h 57 r 22560 ~ c

VaVtl tli TOU 1teurolpaj1Qro lt - 28 MeV

~Ocrov altpop1) tilv B E lOO N~ E)p(crKCtal va EIvUl - 2595 r-loV ivavtl tt toD 7tc1p0j1QtO - 256 MeV tuv AUPWj1ampV TTlV TAeurolCtpt(ltC cl1r(i)ot~ tVEPYEtoV 0205 (MeV)

e l e l

Eel I - ----=- = ~ -r- 1 __ e 0205 ih V (33)2 2 ~ck

3 I B~ 9 a )e ~

3) TOLOU10lfl0ltOOlt 13Ul tU)v npoll you)JEVfJlV 1tajJuBfIYJlitOlV ~M1toj1ev JlLUV no))) ~1Ipliv OIQrpOPIIV (til v 6nolv Mv BIE~ PlUviluv) ~lcto1u taw BE Beoopiuc Kut 1tflpaIlUlOC ittC bOllshyyEt d~ to Oupmq)Qa)Ja

middot0 t l 11 nap a 0 a X 11 t f ~ (28) ~ ~ Kat t ft lt nap 0middot

)lOLooaEUJlt t~r 1tEPlcrtpoQ)fi~ to) 1t Upf vor 1[ p 6 lt t 1 v t a N tp OpT I a)l t v 0 U 1I I a lzfl J It tgtshy

PllVIIiOU tpoptloo elval crU)llpWVO C npor to Jtllpa)lU fur Kat 1tPOr tur AflXtl tGgtv K 13 a v t I II OJ V )l our E (())l E t P I lil v

III Iut1ntpoOla

To crOjl1t(pua~lfl troy 1tpOTlYOlljltvwv t(~Urjlor(i)v ( ) e~l elvm

I) H 1tE)l1ttTl1tparJlCltltC1i fKQ)uAtaJlvl1 SIUshyotaat e 5 uvatal yo h(OOl] )liuv 1tA11Pl t l t)shyrnOtV t li) v Q)alvD)liv wv Ti)r nUP 1V1K 1tC $UOI K f1 c paos TOi) t vlul o) AO(IOpO) twv K~aVtl J((i) v rSUJ)lttplWV tunou Hi c IIlann 1I l (1 bull ti 6 It a a Q)cnv6pfva 11 tVf01woa ltDumKt) 6poundv 6iJVQtat

va o~crl) 611lYilOElt Kat h lOfI t B uuv N N N N~ ICl(VltlKIi 6V(r)~tO)ialt tGlv PN ampIKQ10AO)[U tOO TlAEltPl-lCOU CPIlpt(OU Ui)v 1tUPllvwv alu)lIIO~ lIPIO)lO(j( n 1 KA1t]

B) -r it e t p )l 0 5 U V Q J I tC U Ip Q I V 6 ~1 Sv a 6 lP amp i A 0 vshyt Q I d t Ii v It ( )l7t t II v to K Q) U A I a Jl t v 11 v S I UshyOtrtalV Ial lil auto e(vat aAATAEvlle1w c a v 0 F 5 E P t v n 1t p 6 t 1 n Jt up V I tC U Q) 0 I V 6 jl c va OOtW AUJlPctV0JlfV TTiV A PX-t1v SIIltlP110EWlt rl r Oe p)JG-o 6IlVU)lH bull r (I) Kai tJ)v tOHlI)tl t fle AUlTiaEwr T f~ Evq)o1ti uC )liacp t~ 1tron~lliOt(t)t teov 1[UrJivrov vii lPortIC(U vtCu ~i geshyweo TtA eKt~HaJ-Ltvov [31 11((11 i)~a Uege neral ~c1 Dilnollsions alld th e Lr g (h(lup S(i [31 nuvay61tllUAalt 1969 ky KwvshyOlaVTlOU 4 - AOllvatJ

n n ( p f X 1I Uall Ii v p e f u 6T 11 t ex a I a t Ii v t shyplUvav Kut AUCTIV ltiAAcoV 6epll0TtUrlVlt (i) v tpnlvo~tvIDv ~ta~ t~ ~ n fflnTnC t~~u l (j )l t v i1 r ti l a cr t u CT co (n l tOY KOPCOJ()V l(i)v ltUPI1VIshykuV ouu~ttUJ v 6J K ATt)

8) M C 0 II) r ~ i 6 f a c t fu v t Ie ltp A I cr )l t v III V 1) I ashy

(0) -Opu MdltlV ~IO) Urro neo us 11 the Qllanl tHTI TllfOty or Fie ld bull

Itmt~ppIOC 1970 569

ot60Ewv (tnV Gnap~lv tG)v 6no1oov ut KAUOshyOIKai rEwllEtpial bev ouvavtOl 110 olKaloshy 0 Y Ii 0 W 0 I I T a ql 0 1 V 6 Jl E V U bull fi lt Z W0 I C middot YA I ~ o u v 0 VTal t n i 0 11 lt V a b I K a lOA 0 Y n6 amp 0 1 v 6vuQgtEp6JlEVO Ele xG)pov - lp6vov 00 Einshystein ilic ciAAaxoD poundxoo O11IlElcOOEI [3]

IV IU~TT-rp(JJlQ B

An6 KAaoOlKlie an6IVEWlt 010 110 napouOlaCEtal 6 6UVoJ1lK6v

i Oiov I (11[60toollt llE1a~U nupftvoc Kal dvtpou TOi) nushy

PIlV1KOO 1[EOlOU va eIVUl 1 R bull - - - Rhk- R Koi 6x n R chltCOS I

np6y~aTl OU~~oIVEL Toi)to OIlWlt OIKatoAoYEiTUL AOylP tli~ ashynpoooLOpunialt Tftlt Damp (Ere tliv (B) rEWjJEtpiav) KU tije xr (dr niv (A) rEwjJpoundtpiIlV) ciVQntUcrOEtQI KCllo trv Qn6tojJov evaAAoyl)v tG)v (A) Kat (B) r W)jlEtpubv 6EPjlOrruPIlvLK6v EP-

ldD ~ e yov - d l dx onEp l(a9crtj tl)v BE va ElVCll -~ KUtshy

2R lOt ~ on601aOIe Elval R Kat 6v R l

~H aAutEpOV Kai (JaQgteuroatEPOV H BRE~ civaQ)tpEtOt pound1lt TiJv

)H~Ol1V tvtpYElav toO 6UVOlllKOI) -i - TOU xropou nov ItEVtE

olaOtacrEwv (12)(13) tvCil it BEE1 ltvaQ)tpE1ClI pound1lt tTtv ~tollv

bull tvtpYElav 100 6UVOjlIKOO Rcu~hkH toO xwpou tli)v tEOOUPOOV

(HaataOE6JV (18) EnElor 61lWlt t6 nEipQ)Ju mOtOnolEl 1l6vov t6 AuvaJlIK6v nnou 1 ukawa (24) EiJlE9a UR0lPeuroClljltVOI va napooEx6li)jlEV middot 0 tin 6 E P Jl 0 0 u 110 jll ICI) (24) Kal tv ytvEL anaool at tvtpYElal avaqlEpovshyT a I E l e T 6 v X (JJ p 0 v - X P 6 v 0 v ToD E ins Le i n 6 61ofoC 0 U ~4 v E I ft Jl E I 00 V E I T iJ v S v t p Y E I 6 v t 0 U I( a L a val 6 y 00 C t ij BEE_ ToOto e~ tiAAoU empaUeuro-

Tal Kal tK Myoov AnEIPlO)IOD [3] tlqloD ) xtUpoc ToD Einshystein dVCll ne1tEpaa~toc EVq ) 1010D10e jJpound10 EKcentlUAI(JJlEvrov olacrttlOEwv dvat 6lTPOOOl)pI01OC

B) AuvCl)JE6a 110 EXUljlEV 1~ auteu 1tEpfttOU JleuroOW BE Mv nopaOEJ6ampjJEV 6n a) To nUPllvll(OV cpoptlov Etval u2ltmiddot e l bull

P) H ~v BE bulln r~poundtpia (Al (16) Wpopfi tOv lvDytshy11011 XGlpov 1amp11 TEocr6pwv OIO(naoEoo 11 oe BE tf)~ reroshyVEtpae (8)(12) tOY Xli)pov TOO OOOjJQ u6iou o1Aaol tOV xtllshypo TampV 5 olacrtacrEoov

At napaooxai aihal OIEUKpwLSoUOl I( a A ) 1 E P 0 V TO p6shyMV EKatEpae tampv rEoo)1eurotplWV CA) (B) de 1Ttv btaJl6pCPO)Olv tfie jltOl1lt BE Wlt pound1lo11lt t rv (270) AAAa nEpl aU1oO 6))otB

n Atov vd l111jJElooll~ Chl at ecentlOpjloyai Tfe nopOOO1C jlEApoundtll~ QvaqlpoundpwVtal Jl ) v 0 v EI C T 11 v It E P It TWO l toO

D4 = alaS 6 tEat ~~ ~~ el v 0 I cr U v apT ncr e I ~ Il 6 v 0 T ft ~ R H rEVtKr ItEp(mwcne toD D4 = lJmiddotua6Allt)v

OtE o~ o~ Elva I cruvaptijaEle tampv Xt X21 X oev UEshyTQ~Etot EIC tl)v napoOOav ~EUt1V

Blp-loypa+ta

(II T C 8r01(05 AI Kl3ovmu ruollpoundrplQI tillroo Rlenumn (A H)ushy1poloo IYTloo) T X 8 - 1970

12 AI K(luvl ool r SIDIJttpIQ1 1aV HboPIKdlv KO nupl]shyVIKaV OCl)IJQ1s(CI) TX 1-196J

Stu 6 rpoundQlJhpnlt TXE 1967 bull H dU71fl] s6otao T X2 1962

nUpI]VIIlO 7lEsiov Kol t~(oQ)Olt fOO SchrOdluger T X - 1961

161 0 opeou6~ fa nUPTlVIdrv AuvoIJ6UlY T X )-1963

tnlJtiruOllt

T X = TeXVlllaquol XpomiddotU(o T E E Tpound(vIKOV Eu1poundln1i)plov T~lt pound116500

The Riemann Quantum Geometries B NUCLEON 11OTlON

By Theodore Chr Siokolmiddot

Summary

The conclusion of th e present sLudy will be

A) The fifth real degenerated dmen~iO~ Hay gipe a complete eCplanalion to (he Phenomena of Nuclear Physics Oft

the basis of the U flified calculus of Riemann Quantum Geometries to wAch phenomena the Presenpound Ph-ysics do nOl give a complete explanation and solution (fi the BB of N~ N N~ N magnetic annomalies or P N the justification or the existence of Electric charges on the nucleons atomic number I n and etc)

D) The Therornodynamics Phenomena are due 0 the fifh degenerated dimension and that is why are inshyseparably bound together with the IULclear Phenomena Thus we can take the Energy conserPolion Princip~ as well as the one of increase of Entropy (Through

which nucleons hav e to be charyed wiLh the positive charge 10 give P steady corpuscles) ()

C) I gives the certainty to the research and solution at other thermonuclear phenomena through the fifth degenerated dimension ((i the nuclear (orces saturation etc)

D Through the conception of the degenerated dimensions (tAe existense of which the classical Ceometriee cannot justified) the phenomena of Living Mauer may also justified repfJrted to the Einstein space-time as ] have reported elsewhere [3

( e ) Ste my studies Erroneous Is the Quantum Theory or Flelus Dlgenerated DlmellSlono and tile L IE group S U (3 ) Panagopoulos Aghlou ConstunLinou Atlleno-Oreece

e Degr~ In Mecbanltal and Elrttrical Bnglneerlnjf NatlonaJ Technlshy Cblef of tbe Jolrem~ Corps of Clreece DIrector 01 th e aTE and clIl University 01 Athens (191) He haamp been Mectlon clllef of Sclenshy o OIS Y Tecbnlcal Counsellor or tbe Public Provisions Committee Uflc IUdlo - lelegupblc Reteuchu of the Services olthe Navy and Its Vice-President Vice-PresIdent of tbe Aa1tonautlcal Greell Proluor uf tbe poundlectrlclLy at Ule Sebool of Navy Cadets In ingishy SocietyneerlnSmiddot Tecbncal Counsellor t)f dtrrerent Societies and Sen ices