Elementary Particles

BY ENRICO TERMI

NEw FtvEN: YALE t NIvn asr?r Pntss

UNIVERSI?f PNESS

-

Cawilltt, 1951,hu lot. Lalnvi\ Pt.tiPr'ii..l in lE United Statu aJ Anr.tit:u

FiBt PahlilL.l, Mat!, 1951

SuantJ Printrts, /1xsrtt, M)e

A tislts tsetre'l Thi{ baa| nds tat l)r r.-

r,odked, it uhoL. nL ytt. in LL! Tnnr

lotuptltl tri.1trttsJot th. pnblit p ti),n:nh

att ritbr pd"tirian ,onL the PrbliiLo!

TED EILLIMAN FOUNDATION

In the sear t88s a leltdc! oJ eishty thousand Aoll,rs

aas leJt to ttrc Presit:lent and Fellaus d Y&le ao\ese in the cila ol

Nen Ilauen, to hc hekl in trust, as a silt lrcm her chiLbm, in memary

ol their belo\ed, and hanored mathet, LIrs. He?'a Elv Silliman.

On tlis loudation Yale Collese uas req\ested anA

.lirected to esldblish '1n annllal alse o! Iectves desisned ta iLlu'trak

the tfesencc antl, prol)iLenc., the uis|lo'r. ar^,] goodncss al God' as

naniJesled in thc naLural and narul uo d These uere to be de$isnaled

s.s tlw Mts. Hepsa Dt! Siltirnan Menorial Lectutes ll uas the beL;el

oJ the testatar that an! or(lerl! presentatian al tlrc lacb ol nature at

historu cantributed to Lhe enl oI tlis tounLlattion nrore ellectiuela tlnn

dr! attempt to emphdsi1e the clenents oJ doctrine or aJ crecd; antl he

thcreJore prarided that lrctwes on dogmatic ar palemical theolosu

shouLl. be enludett Jram the scope al this launalatian, and tlwt the

subjccts shaltld, be sclecle.l rather !rcn the donains oJ nalwal science

and. histar!, giting $pecial praminP'rLce tn astronoma, chenishv, geol-

It uas Jltflher d;rected that edch annual aLrsc should

be w1d.e th! basis o! a talune Ia lann r)ot ol tr scties canstitut;ns a

llut&rial ta Mrs. Silhnan The memarial lund came intn the pcl'

sessian oJ the CorporaLion oJ ydle Ltnil)ersila in lhe lear 19a1; and

the prcscnt lnorh con:tlitutes the thirtietk Mlume published on lhis

-v

Foreword

Wiih nunerous additions and subtractioDs thc presentbook contains the matorial presented by ilre author in the Sillimarl-€ctures at Yale Unir-crsity in lpril, 1950. Six l€ct:ures for thogen€ral public snd six lcctures lor ttle physics students weregiven. In this rewritins ihe subject matter of tlre latier has beenemphasized and amplincd.

IIany of the lheoretical papers on the subject of el€mentawpariiclc; and of Ureir intcra(tions are 1'ery dimcult reading erceptfor r small, highly specializcd sroup of theoreticnl physlcists. Ttisbook is not $'ritten for thrit sroup. It attempls insiead to makeaccessiblc to a larg-"r mrnber ol shrdenis and, I hope, e larae frac,tion of experjmLjrtal phtsicists so c ol the most signilicant rosultsof tbe 6eld ihoodes of elemeniiary particles that can be understood, at leasii in s semi quantitaiive wry, \Lithout e{cessive mrthematical apparal,us- A rcader $ho has lo o$'€d

^nd understood a

sood Etandard univcrsity course in quanlum neclanics shouldnot firld sedous djfrcutiics in the follo{'ing putsris.

I have not been able io sivc jn the text rdeqrLaic references totbe rerjr extensive originr,l litcrs,ture. I am afruid, also, that inse\''€ral instarccs I lnav noii ha\,c succeeded in givins du. cr€ditto Ure origin{tor of a paticuhr iden. I rpologize for the omissions.

l'ortunately rn €xcellent "cu e to tlle l-iteraturc ol ElcmentaryParticles Physics" has been prcparert recently by Tlomno andwl€eler (,,lnen.lr?' Sci.n,ist, s7, Apdl and Juty, 1919), ond ttereader is relerred to it lor much dctail. A list of books rnd reliewaruclcs in {'lich cxknsive additional information can be found isappendcd here.

II. n. Beihc, lllementrry Nuclear Theory, Ncrv York, 1917.

87, 1932.'l\'. Hejtler, Qu{ntum Theory of Ra.li:rtion, Oxford Univ. l,reEs,

1944_

E. J. Konopinski, "Bela Dccay," R€r. Motl. Ph!'i. 15,209, t943.

V

W. lauli, I{€son l'heorY,Thcories of Elementary1941.

FOEEI''OND

Nex' York. 1946i "Relativistic FieldPs,rti0les." Rrr. Moal. PhtJs. 1s' 203'

r Ro.cn"e'o. \u.t,er fo-... \r\v Yorl,, lqrS

.r s"t'*1"*.-. QL r "m Ih ''Jtnani'r" Pi's f'' ); r439'

1948: *. 65I, 1949i7n,790, 1949

J Tiomno and J A \lheelcr' "spcctrum of Eleclirons from l4eion

Decrv. ' /?nr. f,fnd. Phlr' 21, 1+s, 1q79

c \\enL)e].a,n|mT\'""Jo'f 'ld'. \'$ Y rk ll19: Be'"nr

Rcsear0h in tr{eson Thcorv," Rt,. Mod Ph't/s 19' I' l9!\7

Chi.cago, Seplember '

l95A

Contents

One. Qu&nta or o Field as P&riicles r

1. IntrcdLclion I2. The ELectranagnctic Field 4t. S.al6 Field uih Mass a

4. Fiekl oJ Ch setl Scalar Patticles 10

5. Particks Abeging th. Pauli Ptinciple 12

TYo. lni€ra.tioD ol i\e Ficlds 16

6. Gen al TlpB aJ Interd.tion !t)7. Con*Naton aJ Monentw n8. Ylkaba ltutatdcti.ans 2\L Ather Intetuclions 26

!0. Cdlcxlation aJ 'lransitial Eatet 3011. Der.l,.pnent Patmelers 32

TLree, 'l'he Itrteioction Consl,aris 36

1t, glecttonasnctic md yLkdud Intarar:tia Canstdntsts. DecaiJ aJ the Pion ds Dtrrdr Pr...ss 3814. 'l lLe BeLa Interaclion 391n. Spa an.aLs Dq:all aJ the lrhlar 411A. lrarcetl D.ca! aJ the MLon 4t"

Four. Pions, Nucleons, .rd dnti-nucl-"oN a0

17. ylkaua'1hcoru aJ Ntclaat Fotces ,A18. PntlLc[ion aJ Pians ilL Nrdah Colliia$ 5519. Ef.4.t ltn NL.leafl Bonn 5E

10. P ad a. n at r 'r. h, Go1^o \'o . t.ti1 . Captue aJ i-csati"?, Pions b! Pratons 6622. Scatterins oJ Pians by Nucl.nns 1-0

23. 'l'he Anti, tLucleolt,. ,\nniliLatiaL 73

,4. Decalt aJ lln l:ethal Pion in ) Tat PholoB 7695. StaListl.dl. Tlbotlt aI Pian Pradtctiatu 79

doN?tv?a

26, Collisions oJ lrdtelnel! EiqlL Eners! Patticl$ 81

1. O)antizdLian oJ thc naAi66or lrield 9L

,. Seond Qumtization ui L PdLLi I'rin iple 9a

L LI.a |abiliti! o! the Fi.lds 101

,1. Re atitisti. In\atidn e fi66. Eelaliorships betueen Intera.tiatu Cor.stants rca

ltQT,lTlO^'8

This is a collcction of ihe notations most frcouen vLspd :, r\is boot \ -l-cn

I l...enF h,,Fr i. use, ror,.t % -n, n er;-jnss the I'arious meaninss sre listed, somerjmes \r,ith indicatioD

d, d* I)estruction and creation operators.o Bohr mdius.,{ \rector potential.B Plane wave spinor.c Yelocity of light.D Deuteron wave function.€ ElemeDtary €lect c chaqe., SFnbol for el€ctron. Also amptitude of electron field.€' Yukawa bteraction conStant.€j Interaction constanij of pion decay./ Composite interaction constant for pion production (secs.

18, 19,25).ft Constant of the beta interaction.,r htcraction constant for spontaneou6 muon d€cay.t, Intcraction constant for forced muon de.'v,i. ft PleD.L co,r.renr , ,d p

"n,.t cor.,er . di!i;d b, 2"l/. li Hrnrlrorian or .'Fra. ion Hari' on,rD/ Culrert densiiy.a l-agrrngian density.nl Mass of a pa:rticle, pariicularly electroD mass.M Mass of a nrcleor

lr'equeDtly used as number of particles in stat€ s.llrequently used as number of stnteE.Strnbol for the neutroD. Also amplitude ot neut.on field.S]'nbol lor the anti-neutron.Mom€ntumSymboi for ihe pmton. Also anplitude of the proton lield.Symbol for the anti proton.Relalive momentum (secs. 18, 19).Yolume of momentum space.

t/

ivpP

q

a

NOf.ITIONS

r Vector of a pojnt in splce.s Frequcntly used as index for a particlc state.

7 Kinei,ic enersy.1,(r) Iiisenfuncijon of a partjcle.U Potential of nuclear forces.

I Velocity, somelimes speed.

Y. Yo Volumes (sccs.25,20).u Energy of e pariicle including rcst mass.

I,II Total ene.sy ol rhe system.Energy of bombftrding nucleon in labomtory syslem.

a Dirac rnatriccs (space componcnts and vecto.).0 Dirac matrix (fourih component).

Symbol for s. pholon.Pohrization unit vector.De llrosljc $,ave length, ordinnry and dn cd by 2tr.

SJ'nbol lor the muon. Also rrnptitude of mLLon fieldp }ilass of pjon-p1 l,Iass 0f muon./ Slnnhol Ior tire neut no. Also ampliiudc of neutino field.

t Slmbol for t]]e anti nelrtrino.. Symhol for pion. Also nmplitude of pion field. Cross section.o I'auli spin operators.

" Lilelrime or invcrse jiransition rate./ TFmfFr|r. r' pr a-gJ u' r- '$ . 26\.

e Scftlar lielil amplitude.

'y' Fieki amplitlrde for particl.s $'ift Pauli principle

o Angular lrequency.o Nomalizationl'olune.

CEAPTER ONE

Quanta oJ'a Field as Particles

1, INTIIODUCTION

Po.haps the most central problem in theoreticalphysics during the last iwenty years has beel Ure sesrch for adescdption of the elencntrry parriclcs and ol iheir interaotjons.The radiation theory oI Dnac aDd tho subsequent devetoprnentof quantum clectrodynamics form thc present basis tor our under-standing of ihe eleciromasnetic fleld fld iis associarcd partictes,the photons. In paflicular, this theory is caplble of €xplaining iheprocesses of crcation of photons when light is emitted and ofdestruction of phoions \r'hen tighi i6 absortred. Thc field theoriesof other elementary particl€s are patterned on ihat of the phojron.The assumption is mado that for esch type of el€nrcntsry parj,ictethere exists an associatcd field of which the particles e the quanta.Tn additionto the electromsgnetic ficld an elecijror positron field, anucleon frcld, several mcson fields, etc., llre also introduoed.

The Nla"xwell equaiins that describe t}le macroscopic bebaviorof the electromagnetic lield ha1'e bcen knos'n for a long time.It is therefore natural to rssume ihat the6e are the basic €quationsone should !.ttempt to q ontize in constructing a quaniLrm electro-dlnanics. 'fhis has bccn done n'iilr { consjderable mccsure ofsuccess- Inthe prst twoorth.ee y*irs ihclast r€maining djflicultiesassociated with the inlinitc value of thc clectmmagnetjc mass andthc so-called vacuum pohdzal,ion hnvc been targely rcsolvedthrough the $'ork of Bethc, Soh$'inser, Tononaaa. Feynman, andothers. They hrvc been able to inieryrei satisfactority i,hc t,ambshiJt in the finc structure ol hydrogen ond tlie anomaly of the in-trinsic magneiic moment of the elecuon rs due to the interactionwith tihe radiaiion ficki.

Next to the photons tlL€ partid€s ii'hich are bcst known experi-menially rnd besi, undorstood ileoroticellv llre ihe clectrons and

l

7

r,i

QUANTA Otv t FILt,D ls PAnTlCLnSpositrcns. In the field theory of €leciroxs and positroDs the reta-iivisiic cquarions ot Dirac rre irken as rhe tiel,l eq.aij,rns of;h;€lectron-posirron field. .IIle proccdure ot quanlizntion rhis casernust, LoweverJ Lc of a r1'pe sucli as to f.ietd th€ prrl p;;"i;i;for electuons and posirlons nrstead of ure Bose litnstein srat;si;csthat lpplies ro rhc phoions. This can be,lotr","nh th" "";;,;qllantization procedure ot Jordan aml \\iiener.

L:- .ol,\, ,c," r,. I....0pr.., "..m t ,. . tp. .ip, :!, ,.r i.r.t"$, h.\pmld ldr i. ""\rFll".. at L,;qhdap.r

"oi.,1e}J rFu,,urq $ni,.r ,i.i rh.,pJ...,rorF. _b.J rh,.px,.tiprinciple alrd have spin 1/2 !,re rrsl,,rity atso dcscribediy a Diraceo',j1r,io j,t,t u.^.,,t.,.,u.c'.,.,ni"ur p."*n, pxt,F,.upni.Lr

:::l:l"i::."'- n' no!n'r"eri *iun' 'h' r'"'og''.,.r Ji..of"r. J \pi,r prt a!..:u .i ncut,;n,

1:,^n--1.::.-, rL,- rn.,r,.,i_,, p,, i.,".-,",,",.,,,,,, 01,-or nF pLp' rror . Ilr,.!n n.Irru.. d T".. . - ^l hc n,r|r,.r ir r-:rr,its intrinsic mrsrctic monent is dirc(red parallel t" th;;;i;:ll.l, ',1 1"" ,rn.. in. -J or i r,.r"j t,"r .,s :, is r^, h. u.d , ryn"r'r^1. Al o. Jd;rion,, ."mpti-r oD ..r pr, ,.nrp.4d,p.,use

h, rrp",f,, or hF Lr "b,,,t lr1 Ln4 .,ouJJ p\j,F.tthe masneiic n)(,nent of ihc proion to bc 1 nuctcrr magncton andthrt of the-neutron ro bc 0. .the laci rhar Ule proron has irNiead a

lllTrl : r;{,c.,,: | " n,,r,r.,.. or .i.Jrnr 'u.r-j I mj F.F. .nss. rFIllj .rL,.r,,J , ha, ri,n" .t.,n".., .,..1qrrr.,.no Djl]" "", 1 I rr r' . , i"a is -. m,,,"o \F ,r ,..t ru,r ,.or,.r .:iuo,' 1"" ,.,, .rr nr,ion .rn.l I . r, ,ctr mo,{',. ,t .r,,ru jp., .,."..1"y.-F,,

"r., 1J,- t", ".;:"i;.

":1,. *h",".i..:,": or.,.snroi nlr. r..op..r:.. lrp,iF i i'.D.,..t,".a. ,.,..":.,,..,^"". ,.1o\,

' o. ...-p Ld ,no \,o r t,.. p ..pi . rp | .,.rp.1t .rres

. The existcnce of rhe r.ui,rino hls bcen suqgosicd by l)auli as an

:._:l:lr..,r.pr"m.,, rr,.,: .r "n.F ,.,, " ",.a,r,,,,.\rFn pt j-'l., /t .. r'. n.r I s . p-i."pv.q .. r,..,r . Ir. .p,n isLrlie! d t^ be r 2; it. m,renciit rrumeni eitt,r' th" th""; ;i ;i,;;;;;;;;,' * Hfif l"'::il1;:T";"f#l

jD tons ol r Dirac {rqurlion {hrt gives r\ro r,lpcs ot nentrino,ncLrtrino proper rnd anLi neLioino, rctrl,ed ro €xch other lit<c thcelfflrui rnrl tlrc posiirl)n. l'his is nr)r tlc ontr narhemrricalporsibiljt]'. -ltiotir one hds boen slqgcste.:t Ly ilajrlrari in $.tictrlhere ls !o ioti nouirino. In the alp]nra on to the b{jta iheoi.

resulLs &s thc l)itrr iheory crccpi iD Lhir ctlsc ot ilre very inprobable dooblc bcta r)roc6sres rctxrl]I irr..ligrte,:t Ly l'irerrdn_ Thebelx ra\. i|ix)ry brldl on tlic cur,l.nro ttli)o resjs has hrd somesuccels ;n cxr)i[n]iDs the serei[l li]ahu e-! of ihe beia disinregrationsand in piri,icll,|r ihe (]nu.gJr djsirib|tion ol the emjri.!t etectrons.On the ol|cr h!nd. LLDr,il nor!,ro rerr y conrincirg torm ot ttristL{xDv lllls Lc(, disc.jv.red. Tnstc ot oDc srristictuv ])eta rtr€oryrl.r '-. \" to ., n,,. .p1..t op

r\ sre'lt dc:'] of {ork 1]l1s bi:rn devoted to the tidd theorv ot.p, \ y,....,i Li.,rrr.r , ,"",1 ,""r"i,.,,

lor ccs. 'lhe me-a )n ol YoLrrrl] sh ou td b€ irlcrrii fiecl wiih the /-mesonol Powcll (bricfll.,nlled lrere pn't. Ihe prrson ot porve (caltedh"'r'n. r.,..' ',,',.t

,;4g.,, , p o,. o h. o,on, outrx'eihll. ]nrked i,o ihe nuchlns ard rlicretor.e ot lit e impotance inthe expliDralion oI nn.rlc.lr lorfes.'fho yukxv''a rlieorvhas proved averv Yrlu able gui<l: in erpcrimenlat rcscrirch and probabty {xDtxinsmanv corr(jOt le.rds io a firiure theorl. In pa.tirrlar, it is plri,lyresponsiLl. l0r the (lisc,:,rery of Ure productim ot nesons in thecollision ol hst nucl(xms. Or tlrc orhcr hrnd, Lhe lttemprs to puttlis theory in a quan{itrti\ c lorm l1arc h&d 1,ely ncdiocre success.Olton r pond(,ous nrrthemrti(jrl apparatls js used in rtcrir-ingrcsults ihal ere no bctier tl'xtr coutd be obiaincrl bv r skeiehv. . rtrr' , or . r'" ..1r.. en. ,r I I r ri. r., ..r.,,," ,,",vill p( rcps iDprore only rhen nore erp(jrimenist informrtionbecomcs a1-ailablc io point the s'ry tr) {| corrcct under;tanding.

'l'he r)rrrpose ol this djsc ssion is noi to aiicDrpi a m!,itrematjcaltreooneot of tlio ioid r:hdn.ies Llrr rxther ro erenplity semi_qlrrnrill1iire ploc(Irres lhnt are sjmltc rnd mry be helptul in theirterpreiation of rixpe Dre ts. Tnere gre serer cases jn i,hictlnot rnucl irould he gained bv a morc elaboftte marticnaticaltrcrtD en i. since a convnrcing ir(jatment hlt,norycir ireend;scoleredIn othei chcs the quditrtjrc dis(jussion prosr,"nrcd trere mev scrve3s0|i rodu,ion ,t"ra". u1,o.l " I,""uLj,.,.r

9. IIl E LLECTROII'IGNlITIC IlLLD

^s r firsi (ixarnple ol the quantizatio of a iekl l,he

c{Sc ol the electromeltnciic field aDd of ilis photons wiil be discussed

l:nfortunately tlie electromqnetic field lics r rathff complicatedstNcture sjnce ii is slre('ificd rt each pojnt hv ihij clecLri0 rnd ihemqnctic veciors. On the olihcr hlnd, il is thc mosi. lrmiliu field

rnd its qrlanlum propertics arc nost clerrlv undostood.Iri qtr{rtum mechanifs ol)scrvrblc physical qurnlil,ies rr{i de-

soibcd by opelalors obeying r non commrhtil.c Li$r' .,1multilnica-ttun. 'lhis is true, lor e\:Lmplc, r)l thc coodirrtcs :ind the xr-poDents ol ille mc,menlurn of a mlrs-* point. It is t e also ol oiheriypcs ol obscrlables lihe, for crnnple, anl' (nlponent of theeleotric ficlcl rt a given point of sprc.. h qurntum elccirodvn{micsthe comrJrtrrcnts ol i,he field or thc potcntials 0t a posiiion in sp.ice

rre cons crcd as oper:Ltors i{hich in senenl do not commute\ritli eAch oiher. ,\ field, ho$'e1-cr, is a system Nith an infinjtenumbcr ol degrees of {reedon and lrom tlii; hci mxny com

No aitcmpl rvill be madc here to gil'e c romplele dcscriptionol tLe quanlization pror:cr:lurc adopted; it is dcscribed in detnilh more speciaUzed publications. Orlv thc sinplesi idclrs of thorrri tion iheory rvill lx: ourlined. I"or {dditional exphnationssoe ,\ppendix l.

Since the early rttdnpts at seiting up thc staLjstic- oi radiatiolljt hm been custom y to ialk ol ndi:iiin,n oscillatoL. 'fhe ele0tro-

magneljc fie].1 cutoscd irr a cavit"v tr'ith perfecl,l"Y rellccting srllsis capable of os(iiLhiirg accordins to { number ot dji{orent modes

{'ith dilTerent clnrracicrjstic lrequen.ics. Each mod.r crn be e\citedindependently of ihe oLhers 1d hus properties qrrile similrr lothose of an $.itlrtor. In prrtnxtltir, one of the modes can t:rl(e up

an amount 0{ e crgv:

Q.Li.lNTA Ol A FiELD '15 PInTICLLS

fil

1'IIE 1I LECTNOM/1IiNETIC FIEI,D 6

$ritli this riirormalizrti(n the toial cnergy content of the rldiationfield rncy l,c writien

(2)

Ea{rh terrD ol the sum rcpresenis ilr{j coni,ribuiion to the roralencrgv of one vibrational mode.

In the langtrll]€ ol thc lhotons (2) nldjcatcs ihat rhere arc n,photons oI cncrsy id. . Erch one oI them is thought of as a rcrpuscle 11il,h a momenl,rin p" related io the \\'avc length I, by thcde Broglie rclotionslip I

r' - t i", ",

.

2rh h lb,l"d

(3)

\rbere @" is thc anslrllrr lrequency ol thc mode and n" : 0, 1, 2

The additioml tctn no./'2, thc so c.illcd zero poi t eDergv, can

be neglecterl c6 1l non-essentirl : ditiye constrnt to lhe energvl

1. Aciuellr-', llis cofftart is jtnnjLelv larsc 'l')e.umbo ol osoilhlorsol lrequ€.cr b.t{eer o rtd o + d- is giYcn by (7) OorsequertlJ lhe loialaoouDi or zerc poini, ene.gt i6

The rcpre"qcntrtion of thc elechom{snetjc [c]d jn terms otos0illators is i lmplete. It is suiiallc to reprcscnr radjarionplienonena bui itoes noi irxtude, for .xampte, :rn clectrcstatic1ield. ()tre can shoN, hoirevcr, that rhe railirrion phcnomena car betreated separatdy lrom ihu {rl€ctroslati( phenornc . r\ completedescription of cl{xrtrodyrrrnn,s iF obl,aiicd by consjdejng on orlclimd the radir,inD fiekt dLrc to the supqposjtion of transvcn{lwd\'es of all Ircquencjes and on ihc other hard lihe Coutombfor0es b€tseen rilcctric charscs. In thc prcsont discussion 11e shnube pdDrarily intcrcsted in thc behavior of the radiain,n fietrt andth.refore \ye shell linit oursclves to thc rninsl,ersal ll,!ves.

As tong as no pefturbalion .tisturbs thc elecLronqlnetic [e]dthc qLrantum nunhcls r" of thc radiation osci aiors r.il be con-siants and there will be no ch{nge jn thc mrnber of photons.A pcrturbation x.ill induce l,ransitjons \rhercbv the nurnber n. mrr-Fir', n,F,." ",..,i ,o ru";.,ora..."" 1,,'-sorption or desiructjon of qui1]rta). Tn ordcr to undcrsrm,:l ihislundamentrl point Ne shall discuss first an ordinary lirlc{r oscil-lafor. TLis lan be cx.ited to ihc ?rth qrur,ntum stlite. Thc cxcitatjonenerey of this slate cxduding tho rest energy is ion. Wc sav thatthe cx.it{tion amounts to,1 qua t{ ol €nergy I'o each. Thc nuntrer?1 oI qurntc Nill be a constarrL as long as rhe oscillotor is lcft rlon€.

This inLes.rl is obviously djvcrscrt rt,larsc trcoueocies. This is re 6rsibul noi lle vo.Bt exampi!^ ol innniti.e thsi one en;ounters in tield rheori€s.

IiTt-^

v t,xQOAN'I/I OF A FIELII AA I1R1:ICLL}J

Perturbations, however, may eil,her inffease or d(rcasc the .,,,n_ru n. nu',l r , \mu.linL 1,. .cn.,., .,..r..i.,t i q.,1|.1rmp- E.is t.rrrrlror .(it ,..,r.I I r:I ueof lhe quanium nunbor ?1 vhen flLe mrtrix ijterncnt ol re n.n.,u-hp'ror . onl 'on I n6 o ,, .n \:,l,rp ,.r t.",dJl-er l.,m1ro fu "r..rp'.., rrr"pF..u,,.,r.1 ..o"ot,.ri r.tlo h, 3br r.. r ul '\r o.. .t ,..,t poIpr,,r,.,rion.ri.reilosFb-\',,a.jt,a.,trto.q .h 1"r.. .i\

".F,,rr. ,Ir d,,..ru \rr:.h. lhF..m!ri{.-n r,"an.jt. .,. I n. Ft,r,4-r. v,, :'Ll . . ... J I "! br' .t ,{.r,. ,. i.., n ;,. o''11) J'r,r.r" i, o .re,,. .^rn",,.errnrM I I

L for ,, ,r ".. o- t, ,,,. oi n,s. a a , t r u.n,., . t,"

only non-vanishing mrtrjx €lenenis of rhc coordinate r rre

I1]L IILECTROMACNETIC 1'] EI,D

trrnsitio.s in wlich the quentum number?r. ofoneolihe oscillatorschanses to ,, + 1.'1

Note that whil€ .1(r) is en operrior, ihc ,ecror / rhar de6n€stle posjtion in sprce N,t which the isjtor potential is obscrved is&n odinary clcssical vector.

The rcturl valucs of the matrix ctemenis c&n be obtrincd fseeApp"n Ji\ r' o! "\tr, . rs,4 n r" rn. o InF.- tr " ..oora ,rtesand using (a). Ore findsihc ftnlowing resutr for themAtrjx etementsof the obucrvable,4(r) corrc$pondjns to rr:inSitioff in which onephoton id, is eiihcr oreai.ed or desirovedl

r,. ,." .." | _,,v li" , .,,. , ,.., tvlln-.(5)

x(n+n - t) - y'#ro'Here p, is the moneDturr of the plroton. Its masniiude js

Ip" : h."/. (6)

wliile €, is n unit vcctor perpcndicular ro p" and poiniing in thedirection ol the pol:r.rizrtior.

The formula givins the numbo of osciltarjort nodes offrequencybet\veen o nnd @ + dd $'ill also be gi'en I

'/f ': )'J' - 2Qa1t'' dP

- Er"i tt'

In lhe l.rst form ol lN thelrrtor 0 X 4rpi dp js the rotunlc elemcntof phasc sp:Lce. This, dnidcd by the cube ot phnck,s {jonsrrnt/l : 2ri, cives tie rllrmber ol morles excepr for l,he fsctor 2, due tothe t1l'o possible pol:rrization directions.

'fhe second of Lhc i\ro expressions (i) contains ihe tactor

u" : L "oao".,V!

\lhich iirn be resl1ded $ rhe cisenfunction 0t a photon of mo_meniun p" (iorm{lized plane wrve). T}ris is a pdticutsr case of a

2. ln lhe lhotrn largurse a chlnge l.onz. hra, l- l nLears r,j, ! phoion.l th. .o.respordins trcqocDcr lns been c.e,red, !nd a ctrarse tr.or; z" roz, - I Drcan3 tbat . lt'oioD hrs bc.n desrro,cd.

rth-n1r) | ;

(4

/ 'z';v n t t

The processes of creaiion and d€struciion of photons jn thel-d.ilion r',"or, drF.n.dtv,:pd ro hi. l,f^oc"J or rt"oeitr ur.l,Jp- J. h, rsu,riou 'pjd:i "o,ri\st r.r r,,,1 lr_"n,hyo|ir..roscilhtors. Transitions in w-liich rhe exc ation nunbc; n" of oneof Ure radiation os0iltators increftcs are proccsses jn rvhich Dtroions

" " .', ".,"d f .i..:" ^r r.J ., io.. r, ,nsi,ion: n "t,i.t," a.

creases describe ihc desrrLrcrn,n of photons (absorpUu" ot."airtio,rl.In_ r.orking out I quarllitative radiation tbeory orre tn.ls ;

simpler to dcscribe the fietd in rerms of rhc veci;r pot€ntial ,4rather ihsn of the elecrric rnd masnerjc frirds. As iong as ih;discussion is limited to the radiaridr thcory t]re scatar potcDtialcan rllvays be lssruned io be ,ero since nn elcctro-agneitc rvrve.ani"oF.riLFdb\ , or y. lhe.,, orpo,.oriatafr"i , to'n'" is ',F.,rr .:,i .. ot r\p .^.ro- po,.nriat, 4,rontributed bv tLe \.arious modcs.

. .,1".rcpresents the ficld oI the sth mode. Its masniiude is prcpor_

tional to the coodinate of rhc racliorion oscilaror numbel s.l-ihe lihis coodinate,4. lus nlltrix eicments indrcj4 rransitjonsI.an n, ?n,. - L T|, v"ru"o.,rFn,rdt I/ a r gr"r po.iriunD rp8c' rs'LF quT ot I p qJ!, F. ,4" /. .l\/ qi ,bq.torc

also be !n opel.alior h.r,nrg non vanishi.g mstdx etemeniis for

t-

3 QUANI'A OI .1 I'ITI'D AS PAR'I'IALES

'l'he rnatrix clcmenl lor ihe destruclion of a pnrticlc js propor'tiond io thc eipienlunction of the state ol ihe particle rhat isdestDycd.

A sinilar lrlle is:Thc nalriri element for the oeriiorl of I prrti0le is proporlional

to thc compler conjug,lte ol tho eisenfunction of thc sii1le ol Lhe

pfticlc ihal is created.This sccond rlrle is exemltificd b-v thc first frrmuh (5) s'hich

si\.es ihc matrix elerlent for tlie lllniion of li phon,n and isprop0rtionrl to

., l^-,,i,,,",.,v!t

3. SCALAN TlLLD IIIl']'H M,i.I]S

The photons behrve likc pfticlcs ol rcst mlss zcroas is indicrted by tlie relatiouship bot\v$D cniirsy lnd m(trn(inhrm

hu": cp" (8)

that holds for lhem. Thjs relalionship is o conscqucncc of thcD'Alenberi equaiion

v,.r * !44: o

so.tL.4n I7IELD VlrE MASS o

The snnplest field €quaiion for thc s(rhr pions is the ltlein-Gordon equotioD

v"o -!$ - r'r: o

which difr'ers only bythe lost term lrom thc D'rllcmbert equation.The field amplitude e is a scalar.s

Iiqu:Ltion (10) has planealrvc soluii,,ns of thc form

a cxp i(:r,i i o0.

Substitutins in (10) one obtains

(10)

(i1)

(13)

(14)

(e)

Is nr the o|ise of the elechomagrctili ficld, the vibrations ol thelield,, (nn be desoibed as a supcrposiiion ol fundamental nodeswith difierent dumcterjstic frcrlLLr:rrrirr o". I]aoh mode behareslilie n linear oscillallrl ot frc(luuxry o" . Omitting :lgaiD the zero-point cnergy, the enetgl levcls oI tlis os(iillntor lrill be no"n" . ,{lsoin this cose the jnlegral rurlnb{rr n, is niterpreted as lhe number ofquanta in the uode s, each hrlvins thc enersv rr" : td". Themomentum p. of each quanluln is si\1xi i)y the de Rroglie relation-ship p" : ii'i". Usins this vrlLrc ol p, and (ll) the follo]vinsrelrtionship is obtrjned bei\\.ccn ihc cnersv arld momeni,um ol a

-f+!-i":o

. : \/ c?f +.\'lor a particlc with r€st nass

1t" : h6" - \/ cil+jF c,Tx. (12)

Tbis isqluilrlent io tLe rehtj\ istic cncrs]. nom€ntum rehtionship

f|on which it follorys th:rt ilio photons tm\.el n'ilh thc \.clociit' otlight.

l or thG rc{son thc liel(] {au:rtn,n (C) crnnot be used lor thc pionliekl. The pions In1'c r rcst mNs dillorcnt from zero r d tralclwith r rc|)(iity lcss tlran t][i ol l]ght. l,ihe the phoions t ey rreb€lievcd to obey thc llosc Mrisiiiiri strtistics since rccodi g 1io theYuli:rN! thc(ny pions (frn l)c cmittcd or :lbsorled by r{clij1)ns.In this scctjrn rnd ir tlrc r(ixt r il'pc of lield Nill be djsclrsscd\\t!)sc (1trsntr hrlc r n(n vurishirrg rcst mass. T|is liekl mis|i bcarloptr:rl lor ihc dcscil)innr ol piorls. The clrse ol nenlr pjonsNill bc discuslcd in this scctio nnd lheL of olurscll piors in ihetrcxt. In both cases these parLicles sjll be rssurarll to have zerospin so that they mry be represenied try r scahr [e]d.

3. Scda. (luartitics rrc usrrllj- chssed rs serlrN lrrolcr and pscudoF.llrru. $rlLer ilie -spare coordnrats arc enccierl $iih respeei to theorisi! tscali. J.rnains an.liungcd rnLl a ls.ud.s.ah..hrnsed ils disD. A(oordnrsto th. lssunttiors rnj.lc ,,,1 llt behlvior ol e on. .,n ...sl,dr.l eiilie. Iscdu or a pscudos.rlD lh.oLT.l llre pioh. T|ere ii no dio.rcn.e b.t$ecnthcsc l,no llLelie! e\c€pt nr t[e{llo1t.b1c ftrdrs ol interrctiol ol t}]epiom$.iih ih. r .lcons. (!ec Sccrior 8.)

ro Q a1N71Ott,1 !1i!,t) 1S P-jR:]ICLLS

Ntost ol the discussjon ol Section 2 a d Appcndir I can berepeated for the scalar ficl(l e. In prrticr lr ihc anllitude e(Dof the lield rt n posii.io r ls al opemtor iLii.h miltrix clenentscorr€sponding lo lhe creatioD or to illc destl\rction ol a qruntun.Thc matr elenents ol e(rJ alc

h.I-- ^

P"J'\, li)v zydt

h.lt n ."-l \.., ,16l

\/ 2t)u

Tbc mairix elemenis (1;) &nd (lr-jl are quitc rftLlogous io (5).Siucc q is a scalar, ol ctnxse \re lL:Lve no pohrizrliioD vcci,or ..The dih'crlnre br l,hc irclor l/+f; is due to hI-ins u;ed llca.'ysjdcurils lor e rnd Dor r:lijrnlliz(l urits lor thO electrousnciiictield. Whdi the field is ilcltrdirl n,si{le:L lolume a onty a djsc,cicseries ol nomenluDr l.rlu($ is Nllo\\.rble.'flic lorirula givins 1,lri)

nuntrer of loNoble nlomentum vNhres bel\um t ond p + d? j-.

. -:t f't. ndnd\ _ .,.,,

_;_, 7\

This founuh comrnonly used io stitisiilrlll mechrnics js qritesimihr io (7) and diffeN floJl it onl)' l)y rr l!ctor 2, du{i io ihe r\\,o,li{, n1r to.. ':.:.o '. 1.,., r..

!j. IIELD Oti l:IIAnUED nCAL.4R P/1H,tICLES

Asnr b, p ticles Niihout elcctri(i.tnrge are(tuanrrof r ficld Njlh real componenLs. Ior e1:Llrlplc, rlLc ptroton! xrc itrquanfu ol the e]ecrron,gnetiN [dr]. Insrexd, jt i,c fouud thai afielrt irhose quanie rrc olerirically chxrsed h$ compLex anplit:rrir:.lllis js tNe also lor r rinrle! irhiirh, thoug| n(Itrrl, bale son{)elecl,romrguelic pnrt)cnir- lor e\mit)lc, the ncuiiur \ritlr its nxgnelic monenl. This t)ro|e(y is q ji{r senel3l and is r(rated ro ttrcrequircment of gaugc irrr-nirnce a

For tlris reiLson thc r{rrl fjeld e ol i|e l)rcr.b s secLion hrs neuirJpuliclcs trs iis phoions Dd therelore cou|l be rL €d xs iho lietd otl.he neutrtl pions. Paltli xrd Iieissltopl hrr(i shos,n, troNever,ihal, a sc.tlu roDrdex qLuniiiy e obe) iris {lrc lilcnr cordon equa-

.r Sc. Ior dxurjlle l,roli, -rler. oj itlod. PU$ , 13,2t3, t!4L

F11"1,1) OF CEIiGED SCAL/|L'? pARIICLES ll

lion mr.y be used to ctisrfibe :r Iirtd s'hose qMnt:r arc e]ectricaltychalged. I complex 1l{)l() ol tlir tl.pc s,ill be used for ihe chalgedpions. Th€ quanla ol th;s complcl fietd can l,e ch'rsed boihposiiively &nd negaiivcly. Itor eidl llloll,able vrluc p. ot rhemomc tum there rrny be rr.1 posiiivo and n; nesr"rive pcrtictes.Omittirs again ihe zcro poinl enersics, the iotal eners'- in ihefield rlill be

rr : t D.(nl +,r). (1s)

Both the complcx qucniii,y e ard 1ts compler conjngare e* $ill beinportant. The qur.iitie.r (.(r) rnd e+i.) r,c operai.o$ \'hosen{trix elelrcnis (onnecr. slalcs in $'hich ilrc rrrmljer ol prftictesobrnges by +1. l,Iore prcciscLl., I'ruli and l.i,.skopf havc toun,:ltl*Lt e(4 indorcs tmnsiiion! in whitlh eir,hr:r rii decreascs l,y oneuDit or 71; irr(J{}l;es by one unit. The oppo"cite hotds tor t}re con-jus:ite rielde+(r) \!hichprodu(s either ihe tr:insitioDs rl+ zl + I

1 ln other nbrds, e is the opcr:rtor tlar rcdures ttietotxl chargc by one unit, ciihcl bv destloyins a posirivo particteor by clel1iirlg , negative onc- p* has lhe opposite effecr.

Sirce as far as rve hnorv l,hc iilcctric chargc is ahv0\,s consaNed,we expecl theii neither e ror' e* \rill eyer nlpeor alone ;n rnyterm of the Hamiltonian. Thcy s'ill alwrJ,s b(i rs.qociaied ryithoth€r qrlantities so tllnt the clursc trill be conserved. Iior ex,mpl{r,erndp* olone1l'oultl be inoccep tallc. Ilur rjhejr produd e*(r)e(r) is

Exccpt for these difererces, Lhe matrix elemerts of e and !,* arequitc similar to (15) and (10). Thcy arc

(\/ nIa\', 4 n: t'

V2a'lr/i__ + r

i.u;-=.,- , tto'h"p'rt,) -,i = t)- ;- ,"'" :v 21rt, ) ,_

The formula siviru the number of alloirable valucs of the lno-mentun p" is identic{l io (17).

I P)R] /CLES 0Rlt'Ir"G t IIEP 1TiT,T PRI):'IPLE

'fll(i (rurntr ol lire eled|ornrgnetic ficl(i rnd thosc ofthc scrhr fiel(l. b.{h neutrtl .r d clisrsrd, obe}-tlr(i llose-Tiinsi(iin

sl.ali-.1ii:s. lhis is evide.t btt:luse l iriiegr rlurLber 2" of

p.rl.i( L{rj clln be iornd irr r silitc ol giren nNmentum l,-{ dillcrefl ilpri of iheorY is necded fr)r the nurr .l|meniuv

]ral1rrl{s ifrl cm l! o$n or b{tnrvcd lo ohcl Lhe l'tuLi prirciple.'lr, I '"" r.,rt' r'.torr' n

tnrl probehly muons ?l|rliltes ob{r) ing lhe l'aLlli €\clusion princjple

olso.rn bc desoiL(l in lems ol .r liek1 ilir;ory by thc so called

se0oDrl quantizotidr procedure ol Jordlm and \Vigner, {or Nhiciir.he nrrml)n ol p|lxtnilcs jr c qu riun] stttc cin be onlv 0 or i.

Ii$t, rclalirily .x!recLions xrd spir $-ill be resle(it{t1 On€

fin(h nr this crse tLari l]le rolc ol leld qutniir;y i.r pltlo(t bv tlleproi)rl)ili1y aDrflittrdc,y' obe-ving the limo dcFjndentS.ltrocdjrgerp1, ..,\',1'I,r.. lLFr '".rFnI ." -

QI].INTA O1' l NIELD 'I3 PAI]TICLLS

(20)

P,lRTlt:l,t:SOBEII,'C'lE|tP.!Li1,II'tlttitlpLt ls

ono nil, .ritlro fr.oot 1 rio 0 or lrom 0 l.o 1. One fiDds Gec ,\ppeuLli\2) the folL.,Njris mairft derne|ts:

t/ /rll 0 - v1.l|2l

I (r 0-,, v{tlhe -l or sisn shoul(l t)c ll.:lop{d accordnis to a s.n\trrt(onpiicai(id rulc,i s'hich n(!d noi (n.reln us liere sincc ir moslcLses Lhc sqrrle Dr)dulLN ol rhe nririr eleururr; Ni be u.rol.

It shout(l be noiql tlui'y' r.Lir lis a rl.stiucrioL opcrrror (iursition L - 0) :lnd,r'* rs a c(,arion opcrator GlrilsirnD 0 - 1).

$ilren no i(x.1's act on rlie p rir,les rhejr r.|r.s{xrt ion jrterms ol shies ol gi\.cl Nrmenirm (phnc 1\rvcs) i,. the mostconleriort. lvhcn ihe st)jn pl.opurijes :Lre rcg|rctcd this ler!:tsto thc mrhir .lonenis (21). 1\rh()1 e\tern:il fr)roes Nci on rheporliclcs, ho1rl]\'(T, ii, is prelel.|1blc ro annl].zrj in tcrms ot €iscl-lunclidrs ol lho plr.r.tic]cs nr the ldd ot U]e ext,(lrxil torres rdherthan in plcne war..is.

Tnsi(x of chrracterizirs I stalc t)v indical,i s ihe number '. oteleclrons of r gi1B. mollintu p., \{r drrlaci{Jizc ilie slxte b!i-cli- r qr pp rr., i- ,

of Ure de0i.ron in the cricrnal fntrl. ,\grin rhc l1urLi principtewilL limit the vrlu; ol tlic o.cxpr.ij(n numtrers ,'l, to onlr 0 or t.'fhe mrtri\ elerrcnt ol thc fttrl,y'(d rld of tne comtncx conjugrrequrnlity /*(f \,i11 (Dnrect strLtes in slnrtl,rr clmgcs L\. + l rsfoLlows (scc {ppendir ,)l

In the ordin J Srlurnxlirger ilr(r)ry lhis equttioo is construcd i,o

desoibc thc st!le ol t single ptrticLc (elertron) 'Ihe rmplirude olprobrrbilii:! lor obsrirr.rtion of thc electron {1i position I is then

t(r'). h the nel(t thco,t'ol tllc dccirorr: equttion (20) is;till LLsed,

lornrLly unchrLngcrl but lvith a vers dillerent mr:tnirtg ltrthe (iase ol thii lilciD-(loiln equrlion N'( ha1'e trcricd thefiel(l s ls ar op tlor whi(ih has mrlrix cl{rncDli givrl }rv (15)

and (1ti) conncdirs statcs i $l1ich tlt(, nuuber ol parlicles

chtrng.s by +l. I a similtr $.rv tfe ficl(l (|taniit)',y' rlso rvjll be

consirtued rs rn operalor.lior docLlons ol)cving ih€ Ilu iprincipl€'ho$1rvcr, the nrrml)er of ptrticlcs in .1 strtc ol gi\'en monlcnlum p"

crn bc on11'0 or 1 m lons $ the spirl is nogltcled ln olhcr ilods'th(i occuprtion unlbers n" cxn be only 0 or I ffc opefutorpr)r)crties olilr(i clcctron fittd nust be sI(ih.t-.lo lntd this prop tv(s(x: Appendi\ 2). As in thc crse ol thc s.tl t fietd s one finds thtthoth 9(r) rnd,1,+(,) arc oper:rtors \rhidr hrl'c mal ri elemdrts

connectin'. steles lor i'hich the ocorPrtion nurnber n. changr:s trv

jyhclc r.(r) is thc eisenflnn tion ol rh{) t l sratc ot ! ete(tronh ihcsir(! nnce firld. Notico tl,rt ilc mrt x eldnu,rs (!l) c,rpariicul crsc of (22) 11hrin no lorcs lcr on tL{) (t(jrron rrec:,us(jthe cxr)ressions rn the r.iglrr Land snhs of (21) rrc rtro pl:lne $rKjr

5. (he nNt ordc. llie srai.s a..rrding io s.m. r.biimrj.rul{r, tor crahpl., in.(!sirg vrlncs ol2. . Oi tlr slrles tlLri pr..cdc s r o.ririn nx,trlje.\Lill be oocut,icd.'l'lie sisn l,! be choscr is + or - r..ojdirs to rtrcrtrcrlhis runber is .v.,r .r o.id

t, (r) (i + 0) : +1r,(/)

,r'*(r)0+1):+trl(r)(.2))

QUTNIA Ot ,1 IIELI) '15 l A E',f ICI'88

that rcpresent lhe normaliz l ejgenhnci.ioDs ol an electroD orlvhich no forces lr,ct.

In mosi calcuhtions of orders-of mrgnitudc lornules (21) or

(22) are uscd. HoNcver, tllc case in Nhich thc relatifistic Dimcequation is acloptcd will also be bricnv ouilined. ln tlis case

instead ol (20) the licld eq,,Ation Nill Le llie Dilac equ{tior

1!-"rt,+ffa* (28)

ll'lrere a., d/, a., P ar€ the rvell kno$n llil:rc mairices ln this

clse th.,lield,y' js no lonser t scalar brrt js r four componcni' spinor'

Ior crch momcniuur p, there ue four staies corrcsponding

to the two spin orienlations and to positivc or neg:Ltivc energv:

PANTlCLES OBEYING TI]E PAULl PRINCIP].1' 15

I moit ol Ure :rppliittio.s ol iljs formula \t shal bc loncencdmercly with orde$ ol mqlnitude end shall rhcrcfore rijplace tl1ilh its order ol magnitu(]c 1. The malilix etements rhen beconeideniical i0 (21).

ID thc Dn&c hole thcory alL tlle posirjve-energy sl,ales in theva.,,,,o ' ,r1o-.D:^d .i i. rh",.. '...F.i. jy .,,nc d-4fuuy ociiupied by one elartroD per sirtc. Creation of a posit|onconespords to the deslirLietion oI :'n cl€otron in one ot thesenegaLive cDergy slaies. Consequenily the opeHtor / serves thedual purpose either ol d.istroying {n cleciron or of crcnting apositron. l'he operaior ,y'* hlrs the opposite efect.

b,: :tv t'p; + tn'c' (24)

The fo r componetrt \rave functioDs ot one of tlrcse siates can be

ryritte in the lorm of a planc wave sotulion ol the Dimc equntior:

J-- 31' ;"^""VO

\tLclc the lour vahed indr:r p specifies the spin rnd tlle sign of the

encr$'of the strLe, and ihe lourvrlucd indc\ I dcroles the fo r.omponents ofthe wave fu ciion Tlle qulnliiies Ij :'r'e dnncnsion-

less and are ol Lhe ord€r ol magnitude 1 "

In the electron-field ih€ory all loL1r componcnts ,h(,.) j1nd ihejrconplex conj,rsates 'r'i(r) are operttoN Thev induce destruclionrind crertiorl transitions ol the odlpatjon nunber 1'l'" {rom I to 0or from 0 to 1 $'ith m{trix elernents

lt^.r)Ll ,0/ - ' ;. E, "' "w!

Q;)l

l:Lt(o l) - + '^ Bl P '1'"'VV

6. Whcn the eleclron mo!€s f ilL velocitv smrll corrr)r.cd nith c o.enrds tbai t$o of thc lour quttrlitica A ar€ ol l,!e o er or 1lrd t\to.r€ol ihe orde. of masDilude ,/. For a losilivc cncr$ staie tlLe conilonenis3 rnd 4 ore larse and lh. cornporerls I ard 2 rre smdll, ud lor a siate oIregaii"e elergy lle otposiie is ilo crse

clt,lPf ER t lt'o

I nltraclion oJ' thr Iri,'/ds

ti. cttNERIL TfPES O! INTnnlATIOlf

The fieldr dijcusled in (i,{ptcr I $'i]l b(iuled in thc:1ppli0flinn$ to folloN lor tlrc inleryrotlrtirnr ol i nuilLcr of proper

lies ol tlio tr)llo$ins elomuitrry prri csl

'lhc photor (.qyrnhol'r) is describd bv itrc r.rdiation field

ol So.ti(D 2 s'iih amt)Lilude,l lhe l'e.ior pr)tc ii: .

The pitrrs, clarged t d neulml (indicticd by syrnboLs

n*, u , l'1, .r'itt tre assumcd lor simtli(itv to hlve spin zero.

Th€ clurscd pions rrc (lcicribed by a oonplca scrlar field as

in SectnD l rnd 1e ncutl.ll pions lrl'a rc.rl scalar field as nr

Se.inm 3. As a rule, tlru lctier indnrtinu ihc partjcles will bc

used to i dicite tho ficld amplitndc as well. lor pions, ftrrerrmt)lc, Lhe leller n n'il] denotc tlic .rmplitude of the fiold inp] ce of ihe letter e [sed in Scciidrs l] lnd 4.

Electrons Gvmhol r), prot!,r$ (symbol P), neutrons (synbol-\ ), mLrons (stmbol !), reut*ros (syorbol !) lrro {ll tssuncdio obey the Ilirrc licld €qurti(n ol Seclion 5. lt (istinalics oforder ol mngnitudo ihe spn!' ch acter rvill hc dlsrcgttdedand ihe simplilicd e\pressnns (21) for tlic metrix ddnenisof the fiel(l rml)lilrldes Nill Le applied.'l.hc synbols of tliepaticles $i11 bc used to rcprcsent eiiller ihe particle or the:unrlitudc oI the conespo di4 tield

InterlctioN bc{s'een vorious particles rrc responsible for a

variety ol phcnornena su(fi as ihe scrttcring of t\Yo colliding€lementrry prrticles and th.r more complcr cvents in ilhich some

prrticlcs djsapper,r *ud some are {rrcricd TNo Dorticl€s mry

cE);IR)t,'tf Pr.F oli t]*TtR.1at tntL

interaci only lcn llrey occupt ih(i slurc posilion in sp[(e (contactinterrciion) or \Lhen tlre]'' e scprratcd by r finitc rli,(lNnce. 1'lrisli'si .ise is us(ally id(irpreled noi :1s .iL acturl lcti(,r rt r dist.ncebtrt rs lhe eflccl, oI r field thit lunsnils ilru i)r(i) lrom oDc io thcotlier parliclc. h thc qlnnlurn illi.crprcl.til,i l.lis ficl(] r'ill lureits o\yn quml,a lnd the iDlc,racl,ioo bcl$.txr tlli tn( orislnxlprrlicles \rill bc dcsoibed as. proccss irr \lhnih I' qr!nturn ol tlListielcl js cmilftrl hy o.e ol lhe pxllicL.s rnd rlbsorlrcd by lhe oilLcr.l'l1is gircs riso to 0 muluxl cncrgl.ol illr t\'o prflirles thrl is 1lnnciirD ol ilrcn dislllnce m(l r{)trrsdrts ihc poti.jrtial ol the lorceIield. Thc tr.nd is to co|sid.r n1,1i sriisir(torr.l theorr' Njlh oolyconia,.i. iniorr(iions irhich is nr)rc olsily hrousht nrto :r rela-tivistically n1!rriant loml.

In:rlmost:Lll lield lheorics ilut ue at prcsent under.liscus-.i(rth{i inierrction €nergy ot i\Lo or morc ivpes ol ptrriicles is rtt)r(ls{xitcd rs ihe voluDre inl,.gral ol rn niierriion energJ. dcnsilt.lhis hlrs usurily the form ol .l frodu.t ol,rmt)liiudos ol ihe yxriousficltts.

tn thc 0!se of the nrtenciil:,r ol (t(xirons {nd tle furlirtion h(t(]th. sul()rl lorm of the clensiil ol irit(irrctior {rrcrrjv is suggcsicllby iis cx|r.ssion dr$sical el:c trorll rrrn ics. l his is siren bJ ifcscalar prrhrrt -1 ./ ol lhe lecl,or r)o{crlirl ,{ rrxl ihe c entdcnsiiy ,/ ol th. c|ritrons. Sincr: thr: scrler f,)t{)riirl rxrlshesin e radiatiui licLrl this is the onl) iuicriclio]l l'rrrn. OilierNise theprodler| ol cl(]cliriri dcn;it! and sc rr potenliJ also should beaddcd. Thr: crrint dcl1jit,r J crD Le e\pressed iL t.m: ol theampliiudc / ol l,hc cl{1.lircn neld. Negleciing spj .ind rditi!.itiv,one can exprcss ,/ .rs in ilie elernenluy Sdlroedi gcr theor). b-v

,1,

' - .,,",,,1 \,1 - lv/.ru' z,\

Thr: iriierLction ene|gy is tlrLrs \rriiten ir tlie lorm

T_ /,/it_ r,t Jl lt. :;ln order lo crlculiio iL. transitjr)ns due to the intcr:rcin)n (27)

ot cl$,irons rnd photrtrrs, its mrtrix clonents {ill bc n.cdtrl. Incont)utins lherrr rho list tenn (27) \.ill be conlid{)r(xt fi$r.11, niirins lhe opurtor ,{ which c,L (ir se the clestrLrcti('r orcrc:lii,n ol c pholon accortnrg 1,o (5), n d the opercrors,y' and r+

.

T3 INTEEAQTIO,' AI TEN FIELDS

which, according to (21), putu(ie respectively the desir ctionand tlie creation ol aD electron. Let p. be i,le momeDtum ol thephoton thai is emitted (or absorbcd), ft be lhc momentum of theelectron tliat is deslroyed, nnd p) tLe momentum ol the electronthrt is cr€ated. Destroying nn electron ol momentum Ir1 s.ndcre:itjng anotller onc oi momcnium pr is of couEe equivalent to atr,rnsition in which an clcciron changes its momenium from a tozi . Therefore the mrtrix clements of (27) will connect the inirialsicte to a find stftte ilrit djlTet's from it beoause a photon ofmomeDilun p, ciliher hls been created or destrcJed and at thesame iime an elcctron has ohanged iiis momentum from a to p: .

We considerfor exrrnple the case in iyhich tlr€ jnii,ial si,rte contajnsno pho ion of momentum p. I in other \vords, the initial valu€ of theoccupatior number u, of the photors in tlie given state is zero.Il a photon ol momentum p" is crcrteil the occupation nunber otthe final siat€ will be 7i" = 1. ,{ccordins io the firsii form la (5) theconesponding mnirix el€menii of the vcctor poteni,ial is

vanishes, bc(ruse otherwise rle interjrand avenges to zero. Thecondiiion (30) expie,caes lhe colserr-ation of momentum: Themalri\ clcrlerlt lanishes unl.ss the initial norncrfun p, of iheelectron is quil to the vecior sum of l,le mom€nts ?, ol ihephol,on nn{l ?! ol the elecLron i the finrl state. Il itre nomentum ;sconservcd tlre exponeni.ixl is equrl to I and its volune inLegralis 0. thc totil co.liibution to ilrc nrtrix elemenli ol the Iirsrterm ol (27) is then

GElit:RAL fYP}]S Ott Ii"TERICTION

ov-ar the larca r-olune a will be dificrent from

i"-i":0 (30)

2rhaq rPL €' r'

t

(3r)

Th€ second tcrm ol (27) lriekls r similtr coDtdbuliion wjth Z,changed into p!. Sin.e (27) is a on r.lativistic fornuln, theerpressions 21,i m rnd pr,/,r aretle velo.jtics,r and u ol thc€leiitronbefore end after the transition. The m{trix eleme.li is lh.,',lore

2.

Jii n" . ,,,^,"" "'" t-ffi' (28)

(2e)

Accordnq h, (21) the operalor,, th"t cruses the destruction ol .rnelectron ofmomentum pr 11ill contribute to ihe matrj\ elemcnt th€t^cLat +etthb" /y'e. Accoriling to (27) we need ihe crdicnt ofthis factor. This gives merely an ildditional facior tr'ln. Fnially,the operator \r+ thrt is responsible lor the ueation of an el,"ctron ofmomenlum p, contributes in (27) l,hc frotur +c t'tt)'.'/\/s.

In conclusion thc intcsr:md ol th€ fir$t tcm of (27) contribuicsto ihe malrir elemcnt 1ilrc term'

(32)

Ib can be proved in a sinil&r ryay thaii thc matri\ elementcorrcsponding to the desLruclion of a phoron also is sil'en by (32),plovided thc momcntun is conscrled between initi:rl and finalBtates. Othe visc lihis mairir element al6o vrnishes.

'fh€ t{'o processes which have becn considered correspond to iheeni$ion or Lhe abEorption of light rccompenied by , chrnse ofmoment:um of the eledron in comlormit)' x.ii.h ttre momentumconsen':rlion. It shotrl(l be noticed, however, that ihe flot thaithese processes have a non vanishins mjitrix element does notmeen thAt the processes ,.iulllly happen. It is well hnown that:rfrce clcct.on cannol emil radiation bec&usc in this case iL is imposEible to sntisfv aL the ssme time Ure conserr-stion of momentumand ol en€rgy. Tn Seclion 10 the methods lor c&lcularing rhcprcbibilities ol l,ransiiion and in perticuhr lihe role ol the conservrtion ol energy wjll be dlscussed. (Soc also AppeDdix 4.)

Eornuln (32) sives the correct order of magnitude of the mairixelemert also iD the rclativistic case in which the elcctrons obev the

ehL2it c \/A

2-a(yt e,) rttt h) t"'-' "' "1'

I

JN tpr.")1:!! x

'l'his expression rnust be ini.rsrried over l,he normalizrtion volume0. 1he integraiion is immcdirte, sinre the onlv space-deperdenlifactor in the abovc expr{)ssion is the e{porentiil. Its inl,€gral

1. Tle sigtr ol uris cruessio! is eorectlr gire! iD (29) iD ih. pDciicollyimloi.ni crs. tlat onc c!.ctrD only is irvolled.

9*

!o I\IE&ICIIO\" OT IEE FIEJ,DS

Dirrc equaii.,D. l'he expresslons for the currenr densfty a d theinterl1cti(D ene|gJr arc then

yuKAtyA IN't ER,lCl IONA 91

ments.]'he vohme irtlegrrl ofthe encryy densiry invoh-es thereforcmerely the iDtegnl over ihc spoce 0 oI r producij.,i cxponentjalswhich mrv be Nr;iienr - -e{,av

x: I t.raa: -" ! t"v tan I t*'*'-* -' '

Thjs intcsral is dilTu.eni lrom zero only $4ren the vector

+it+i,+.+i,:0 (37)

(34) (3ri)

vhere ,y' is thc i,rinsposod conjuglte ot rhr tour-conponeni Dtrac

In the relrtirisiic cllse both the Dirr(i mltriccs a and thcqurniiiies, of (25) hrye ihr r)rder of ron!{nirude 1. lrljne (5)orrc fi!ds ihalithe mat.ix elemu t for rh e crerrioD or ihc desiructionol ,. single photoD is ol the ordcl ol mlgniiude

(35)

When thc relocitics src clos(i to c rhis exDressio difi€rs fron32 or r Iv. rJ, .rur,r,).Iror particLcs obeyi s th," Bosc ltinsiej st:riisrics, like the pions

for exampl., the exprcssioD of thc ounent donsii,y an.i consequcn yalso the lom ol Lhc cle0i,romqtnei,ic into.rction are soneNh4tdilTerenii fron (27) or (3r). Orc finds, troll.,er, rhrri jn rhe simpleprocesses wLnjl] (e arc going tio discuss rhc order ot murnibuitc

"l l' ^r,,i, pio..." o1 "r..';onanJ I-.riu",orof r phoion is gir.en also by (32).

f, CONSEN|ATION OF MOMEN TU|I

It hrs l)een secn thc1, ihc 0onserQtion ot mom€ntumin the casc ol ihc €lectronrisneijic interaciion i! a necess!.rycondjtion jn order to harelr, non-v:l ishing m.t x etcmeni. Thesamc is lru. also for rlie oth€r field inreraciions rlt we shallencounter. It will be scen ihaji nll inier.rctions :ire volumc integratsof tcrms contrjnins fr.tors equ to thc rmplii,rdes of the va;ioustields (or sonxjtimes of tlieir d{rrjl,ativcs). rii is scen from (5), (15),(19), (21), (2;) thai lor rlt iypes ot ltdds ijtre space cl{rpendenccol the mrtri\ clements ol the n ous ficld components is containedin an c\poneniirl lacior exr) \+ip.t lh) \.|rcrc p js the momenrumof th€ puticle th!.t is crc.Lted or destroycd_ Thc sisn is alilays +for the destuction matrix elemcnts and for the crearion ele_

becausc oiher$ise the intosrrnd avcfuses ro zero. When thisconditidi is fr []lcd the inkisrand is I and rhe irrcgr becomesequal lo the nornrlizrtion rolune O. Thc above equr.iion expressesrhp tu^r p um 1"... r. ir " r,,, r L. " ot ,heparl,iclcs desj.roycd appcsr witli ihe plus Eign and those of rhepariiclcs creaied oppear $'ith the minrs sign. Ilencc rhe vectorsum of the lrlom(lri& of thc particles crcrted equats the \'$jtor sumof iihc momeni{ of the parti(iles destroy(jd. One obrri$ theretor€ihe foll(xvnrg thcor€n and the follorving rule:

Theorcn a! thc nonL.ntun conseruition. Two sl,r.ies that arcconnected Ly I non vlnishing matrj: etement of ihe inter-action eneryy opellLor hrve equal momenta.

RuIe Jot culxularinlt N)hme irLrestuts. rn calcr rrins thevollrme iniegrrl of thc en€rgv dcnsity the momcntum con-scNation is r{rquircd. 'lhe jntegral is then obiaincit by sub-stitutins the volume [] for the volumc integral of ttrc prcductofalltlre exporlcntjal iactors. These prope.ties greal,lv simptjtvrh" rurpurr r,' a rrp nJ,i\ F.FmF.lrj.

For fLuiiher discu-$ion ol thc en€lgy-momcntum conservaiion jnfield theories see Appendix 4.

3. YU K,,)IT',,) INTENAqTIONS

In ilis section and the next tLe rnain interactionsbets'een elcmeniary p:triricles llill be enumcrated. We begin viththe Yuhawa interaciion belween pions and nucl€ons. Thc assump-

--

.e lNTl:R,1ufION AF T E I.tt:LDS

tion jsmade thal lhere.remNirix elem.nis seneratins the foli{) \.ins

fIi KlIf1 INTERACTTANS

possibilitv for the irlerrcinm brm in the Lrsrrnsiin densiiy is

therefore tlrc HcrmjiisD opc[io]

e;(11P,9",",".,\ + rriri-rd,a:d!P).

ITem ei is a coefiicicnt lhlrl deicnnines the Etrcrlith ol the inler-actrn and ha! ihc dimensionB ol ar elecliic ch{rsc If is found

ih,t the abole tcrm in the Lrsfungixr le&ds to " Hrmilionjandensity eqrl.1l to it crcepi lor the sign. 1Ve ceD th.*forc wrile i,he

(38)

In the originrlYuhria theory only the reactnlrs jnvolrjrLg chareed,r:idF r.t |p.cLr,.ite,.p rn,t F-. t.,.,,.e,rain

that a neutral pion also eriists, rnd xccordinsly Lhe r\ro rexcl,iolls(38c) and (38d) havc been oddcd. tl ttre r(a'rtions (38) arc correctone \rould cxpect thc pion io obcy Rose stntistics and ha1.e inl,egml spin. If the spj iE 0 a repr(isc.taLion $ith a sc:Ju fi{rld rs discussed jn Sociions 3 And 4 is plausible. tI rhc spin \\'.t? I :r, l.ectorfield shol,l(l be adopied insic&d. tn \.liat ft,lo$s a sp 0 wjlt be

Tlis stiLl learnjs op€n the possjbilii,jes ihrli the lletd amptjtude,tr'hich \1ill be denote,:l b], Ii, may be a scuhr or a pseudosr,lar.It is lashionable io rssume the latter sircc onc linds rhar i1i lcds tonuclear forces less 1n disagreemcnt irith (jxperimeni. 1V. slrallloUow ihis fashbn.

Whcn II reprelcnts a neuirAl pion {icld jt rrill be r rc{t qurnritvu.'r. r:c",ior ?. Io.:,\...n, a".l.. Lr,d

nrskad by a complcx nel{l. Onc shoutd ot corrse use ll diflcrcntnotalion tor the ivo cxses. In order ro keep our founulas snnpldthis $ill not be done.'rhe poinL will Lr€ ctarificil $henever necessrr.v.ll-p dis.u.,:no r'!r ",llu\- sr r,.1., o rr",.\rn ntF,t..,r.q;dp.ors ror q Lr, l- n is a .uTp ar fFlJ

From (19) and (21) it follows il,rt the reacijons (3S:r :lnd b)can be producc{l by terms in the intcrx.tion enersl cont!in;rs r,here.r',,.I V P,,ndnP \ ln.onLurr un oIo|ar.,ir.j-r 'u-"tbe intemci,ior cnergy density {jU Le trl(en proporijonrt tr) these

For lihis parlicul case, hoirever, it nay be inslr.uctil,c to tmksonewh&t nlorc closely iDio the xciual fo.n of the possibtc inter-action tcrm.l Sinca n and II* hive LeeD $sumed io L{j r)scrdo-loalars, the simplest iDvarirnt combiDations ol the desjrld typeare obtrin€d by issocioting l,hepion rmplii.udcs n ith a pseudoscrhrexpresslon p&tterned accordnrs to (3) in Appendix 1. .uhe simllcst

2 TILo reod.r inr.ro6ted iir this l)rrl, ol the,lir. sj.r sl,oulrL tjrsr udApDerdir 4. Otl)€rsis. Eo r\,er to lormnh l.1l).

(39)

This is not the only possibilil,l', becouse ii n is a pseudoscalar

its gladient is r p;eudorecior. Thcrefore an jnrllri'rnt expression

crn Le lormed l.ls l,he for.rr dinensionrl ve.tor producl of the

srrdieni of n or II+ and a fsord.,veclor frrmed wil,h P and lr'iccordina to thc p:ittern of (5) i Appendix l This yields in theH'nillonian r tdrn

:t' = "l / trrPl","^,'v + r*-Vrja,d,d,P) .10

c,,n'pt. "o.'1.)

an {+o)

a:"/{nr-.r-+n-rv)ao (r1)

(a) 1'= li + II+

(, P.1P + f(1,)NcP+r(l) N,rn +no.

:ll'' -*4 I(-.,r qi".\'+where p is thc pirn rnas:. The factor n,/!. h$ been writteo e \plniitlvjn the cocHicic l, !1, thal the consturt e!' again will ha.vc tllednnensions ol rn €lectric chlrlge.

In oomputmg odcrs ol mrgritudr: snnplihed erprejsions $'itl be

userl. Instr:ed ol Lrsing (39i or (10) or d co,nbnltiidr of tLen ooe

can eimpLily thc crlculaLiions by Dcglcciirg thii spln propcrties

of tlrc nlrcleons ard by writiDs thc inleraclion tc.m jn the form

where P, /'*, N, N* hrvc mahir elemenis conptted tccodingto (21).

1'hc errolls itltlodurcd by adopiing (1I) .rre varions Firsi ofall, lle resulis aie nol rd|1iivistically inrtriani. Therclore one

crnooi e\leci tlie bcllrjor ot pi li.les wiih enersl' rery hrscconprlrcd lo lheif rc-ll elt1gy i.o be rct)reseolerl properly ilsoai l,nv 1'elociri€s errors mny be exDecictl. Althongh l,l1e Dirrcmlllrices rnd tLr.rir producls hrve elcrncdts of thc order oi uDitv,factors ol tlrc orcler ?,/c ffequcDtly appeu thejr expccl.iiion

Y

I Ii 1'F)RAI]TlON OT IHE NILLDS

values. TLe snmc is true ol tllo quantity (r,/pc)V .ppearnrg to".'xamplc in (40).

The mrin features thai are losi in substiturins (.11) for (39)o- ru rr. l,p quari, rr\",r1,1"r. . dirrg.,r r r" pir.l"l"n r.r.cof the nlrclexr lorces trnri of the contribution of rhc pion fietd tothe magn.itic moment ol the nucleons. 'tliese eflcclis ot coursc\rould not be obtain:rble from ! spintcss expressnni litre (11).

T F ,'ij. rtar'r1ts.. ,.rp irr t. r nt,, -1,"J , r h" er. ,

' mplr.Iy ,.f he . l, -. ion t",-pd o.. \j.., ulc .\., rl.lobservc that a convincing form of the inicracijons is nor known.ericcpt in the case ol the intem.liion bei$ren elecirons !,nd etccrro-masn{rtic field. Indced, no expcrimental c'idence crn be quorcd topror-c thot inieractions lilte (39) or (a0) cre corrc.t. This uns{tis-fach)ry silurtion is probably due in p t to rhe t:ici thar no Dropermath.im&lics h:rl been evolvcd cap{ble ot 11:1 dling inier.r{rtionterms umlnbiguously. The prccedurc most us&t currentlv is to'eB.'I I'cin.r,.riuI rF ,'.,s16ru|l .rjon.rrI i.omp prpprol)rbiliiies ol lirsnsition to th€ lo$,est rpproxjmation rhlt vi€ktsthc desired result. Iiven \.ithin tliis limited scopc one trcqucntlycncounters diversent c\pressions ihai {re esiinated bv cur otT

nr'" " lu,." I lrr.'o1. 1r'..6,,; r'Under these conditn,ns one wonders wh{)ther ;i plvs to spend

time and efiort h computing some terms exaclittr $,hile other'lirerg, r'r re"ms r. 1FC ".r.J I l,F r".r rh., Fptd r..or:F. otrengive results jr:rsreem€ni $'ith expe ment ar leasr as to the order ofmagnitude mrkes it appcrir likely that the linai theory IriI beiror.,".iTijri'\ r.,.l6pr.n. a:.nr1,r . t'"r rp.rt, ,1,o.t

",,.nof a finite sizc of the €l.meniary plriid€s or evcn a granulcrgeomctrt such as is suggested by I{eiscnberg :rnd Snyder may bcclucs to lle solLrtior.

Th{rse sonxj\vL:r.t disprr:rging rcmarks as ro tlie shhrs ot Dresenl,.hj iFl , rr,,,," o., ", ";t,, o r,..' ,, .t . ,t-\, ni..This disciplinc hlls progrcssed to r poinr Nhere it givcs usR (tetajtedundcrstLnclirp of the plroton-elc(jiron irrcmcliion. cr.*Li shideshrvc heen nulc in ii i the lNt l€$, yea$. Sj.ce, ho$.{x,!I, iheree\ist .xcellent monogrrlhs on qurntun eledrodvnanics, $,e wjllnol girrc il mucL &l,tenti(D here bnt reter rhc reader to morespecializ$t publicstions.

The Yukrwa pro.csses (3E) h:rve a vcry imporianli consequence.

YUKATYA INTENACTIONS 95

According to (38b) lor exlmple, a neutron is converiiLrlc into a

proton lrnd a negatire pion. llroln the existence of { strorlg inter-action bctwccn these two possible strtes it {olloNs, m $ill be

discusscd in more detail in Section I1. thrt tlicre is d coniinuousinterch&ngc Loineen these lrvo forms of the neutron. fhc ncutronspends prrt of its time as ner.rlron proper cnd prrt ns r protonwith o ncg:rtirc pi(]II noarby. Similarly, p:Lt of the iiimc:r protonis a Droton p(4xr and prri of rle liDre it is a neutron \vith n

positive pion jn thc vicinity. The anomrlies ol thc masncticmom€nt of thc n clcons alrc:rdy mentioted are nrturyrcicd :1s

b€ing due to thc mas ()iic ellecl,! of ],he pioDs thrt {r(i lound apart ol the time in t}lc l.ici il,x ol 1,!e nucleon. UnfortunAtcty illeattempts to e\phin thc mtignctic aDomaljes quantitttil.cly in thismanrer li&\'e t'rihl, snd nothing more tlian cn qrccmurt jn theorders ol mosnitud.i hrs bcc ilchieved.

lrormula (11) rppLnis io ihe clse of chrrged pions. fhc ilssump-tion js usurll]' madc ttltl, the irl.eracljon of tlic ncutr pions

with the nr0lei is simihr to il. In ihis case, lio\re!.r, II is a realquaniity:rnd th{:rofor(i lr and rl* arc thc stmo. The present

evidenc€ seems to indicrte Lhai, lhe couplins constitDts r! haveappro\imntcly equill values for lhe inter$.tions ot thc cbargedand thc ncutral pio s. In tlis booh ttte t$'o t{rrAction consLants

will bo i{kcn l1! (r|rlr].Wc conclLldc illis seclion by compuiing thc mairix elements

ol the intc ction (11) accordins to the lcs ai lhe end ol Section7. lhcse lomulas vill later be usnl in ihc applications. fhematrix rloncnis fol aU pro!€,.ses (38) in which r pion js created ordestroycd are obitrin€d fro,r tlle formul$ (19) lor crlculrtnrs thomal,d\ €lernenis of iI and (2I) nrr ciiculating the rnatrix elemcntsof P and -\t. The inl,egral over thc cxponerttial lrciol is replr(€d bl'o as explained in Secl,ion 7. The rcsult is

hc I 1 t'hc.., -

O:-::"Vz!)u, VA Va V 2ttu"(42)

{'here u" is the totil cnogt of the Dion ihaL has been either 0t criedor desiroyed. In computirg ihese formulas ii hm bceu msnntedthat the tmnsiinm oi ihe occupaLion number of thc pii[ was

cither fron 0 to 1jn a crealjon process orfrorn I io 0 ir a dcslruclionpro(rss. Othcrlvisc ihe factors.v/n. -l- 1 or Vn. appcaring in (19)

sbould be iDcluded.

-trF

9. 01' II ER ] N T ER AC T IO N S

It is knoNn €riperincnially th$t the pion is anunslablc psrticle and thrt it dec:,ys sponi.ancoLrsly vith a tifctimeof abort 2 X l0'3 rcconds into a rnuon. Powc]l, $,tro firsr otrscn-edlhis decay, fouDd that the muon decayirig trom a pion at restalways hll6 a consirnt r&nge ot about {j00 micrors in ihc photo-gr*phic emulsion. From considerutions ol enersv momentum con-scrv&iion one conclLxle6 that, in th€ pioD docav o neurrat partjctemu5t also be emittcd which lcr\'es no photosrcphjc tracc. It isconsislent $ith tbe rrther $idc limits ol experimeni.:it error roassume ihat tbis neutral pariicle has niss zero. Tbc or.renrasslrmption l,hat se shilL loll(,w is i,hat it is a nentrino. Ii is,ltowever, more convenieDL to cail this parrjcle an anii neutrjno(symbol t), which ol oourse nakcs titLte diiTercnce. t.hercft,re weshali postulcte the rc{etions

lNTLnlCftON OF TIIt) F I lt /,DS

IIt=p'+t ind 11:A +,. (43)

"" | 0u*, + n*,*u) a9 (44)

oTul t lN't LRACTION I r7

inv|1rnDt and should be replaced by more eiaborrte exprcssionssomewhat on the liDcs folloircd in discussing vrrious lorrN of theYulia$o jnieractio . Sincc orLr oim is ncrcly to cdcula|e orders ofmagniiude \ve shalt turi:go thcse conplicaiions in lhe inier€st ofsimpliciL]'.

f'hc mrtrix €lemcnis of tlle interaciioD (aa) can be calculai€dwiih lhe Lsual proccdure by combining the matrix elements ofII, p, r ohtnincd r.spci]tircly lron (19) atrd (21) and applyinsihe rulc of Scctiol 7. One {inds t}r&ii thc mtrix elements for aliproccsscs (13) Lave equal values,' nrncly:

\/ 2a,"(45)

(46)

An inier:r,ction term suiiable to causc tlre first ol thcse iransiiionsshould include lhe operotor II, s.hidr acmdins ro (tg) desiroys sposiiivc pion, ard the operators p+ and /, vhicL r(jcording to(21) crc.ric a posiiisc nuon and destroy a neutrino, i.e., crcaie rn&nti-neutdno. The samc combinatior ol ope.atoN wilL also causethe jnvcrse of i,he seud reaclion (13) because the operaior trcan creaie ! legnlive pioD.'fhe opemnrr !* ma) crc,te e positiremuon of ihc negative Dir^c seo of ihesc prrrictes, thrr is, desi,ro)-ancsative muon. Simlhrll.ihe operlror / mrv desl,roy a neutrjno.Since the hierartion erreryy is lL real qLL.tntitv (Ilerlnirir operaror)its complex conjug:Lie u+/*/ should bij riilded to the prcdous term.'lhis complex conjugai,e is snitrble to prodrrce the second rc.cr,ion(43) nnd ihe inlerse ol l,he first. The sinplcsr form ot intcuciionenersy suil0.ble to produce the relrctioos (.13) ard rhejr inverseswould thereloro be

s'h€re u, is ttLe energv of the pion. ]'hii formula "'ill be used inSection I;l when the lifetime ol the pion is dis(,ussed.

Th€ interaction constrnt betweer elecirors and the mdictionneld is the elecironic 0harge

':. The interaction consliants e1 of the

Yuk!$'a theo.y (formuia (11)) snd the constant e3 lor ihe process(a3) all hrve tlie dnnensiors of an electric cbarse. Theil aciualvahes, liorvever, are quite di8'erent, as $'ill be seen in the next

rve shall procced now to dis{iussthree other el€mentary processcs\ hr' r \. .' .,,' '" :u .n-,.r.* " l",liTr,'"r..rsc / ,mo.'I'hesc const{nts will hc dcnoi.ll by sj , t' rn{l ra .

rhc first ol tlcse int{rrrctnms responsiblc lor the bcta reydni-.sion. This proccss h.is long sincc bccn intcrprctcd as due to thedcmcniery'prrticle reaction

The formulas becone more symmetric by assuming thai an anti-neutrino and not a neutrino is associated wiih the emission ol aneleciron, and the reaction (ad) has been rvitten accordinsly. Atransition of this type might be due to an iDteraction tenn in theEcniltonion of the snnplilied lorm

rh€reca,is r suitable nrteraction {xrnFtxnr hrr,ing ihe dimensionsofalr cleclric c]]!rg€.1'lc interactiol iun (+ l) is not relaririsinja y

s, I e.N"., + N*Pre) do. (.47 )

3. ^s

ir ih€ crse ol tlre similar lorN ! (42) the assumpiion has beerrntde Urri ihe occuJJilior Jrurtrbe.s ol lle J]ioos chsnse either lron0 io I o.rrom I to 0. othowis. tlc Imtor y'i + I oL V; sotrld &ppcar in (aI).

-INTERACTION OF IIIL IIELDAIn this casc rll loul paticles arc of the I'ruli tvpe anrl (21) rvillrruly. \\ r'- 1,, r, ps o c", .,o ; L, m. ., " "nl, ,,, , on."i. ,i .rine,o rle i1 .r1. rior, , i7 L rrmpd:r .t-, .d ,

''G)"":3 (48)

This formula, horverer, corrcsponds to a rranEilion in which $llfour p:rrticles ol reaciion (16) rre in siates $.irh de lnirc rnomentumvalues (planc wavcs). In rhc applictrions atso ttrc toll1lutas .or_rcsponding to the eLse in Nhioh thc states of rhe noltron, proton,and_clectm! ll,re not reprcscntabtc !s ptanc rLal es de treq[en yused l'hesc formul:is coutd be obr$nred by us;ng tLe erprcssions(22) irstead of (21) lor ihemlirix cl{rmenrts ofihen(td conlrronenis.Th ,-.p.,di,c o,mul.s, a. \1tts.rha, l'1!Fl...rm:,.|p nonJFrrod.,"...ir r,"n.,.,e-.-n ot he

"eru r.yirr"r.cri,.r.inr, .r.d""r h,r, i. q.c;,\ I ""iTl.:rFdtj'pe (47), $'ill be meni,i.,ncd in Scction 11.'lve procccd to discuss rnother ctement:ut jnremotion wtrich isrcsponsiblc for i|c spontaneous dccay ot ttrc uruon. t.he mLlon isan unstaLlc partide. It has a mcrn life oI 2.15 1 t0 'sccondsand emits an electron. The me{sLrre ents ot Sr,.nrbergcr rndAndcNon indjcai,e that ihc energy of tte clecrror (lnir,terl is noralways the srme. This fact has bccn inicrpreted br: Si.eilbereer1rd \V'p"l r rta :t,e mu n dF- .. nrLo rl,eepr"i1...e|la r"r'.Ip1r:.,..,"-,"... .""to be ncutrims. Or this assrLmption rhe reaciion $.outd be

(1C)

This reacinrn has been witten for the case that one of th€ t$,oneuirinoE is an anti-neutri o, a ma ftcr of li fttc prac ticjil importrnce.-4. sinplilied interacliioD term leadi4 to (a{l) coutd te

(50)

Also in this case the four particles rr€ all of the pnuti tJ,pc aDd (21)n'ill be used for caloularirls rhe mririx ctelnents of rhc fie1d cornpoDents. Onc finds as in ihc case ot the beta irtemctim rh,i ih{lmat x el€ments for the proce,.ses (d9) are

0

OTEER 1!t7 ER'|aTION t 99

The spontrDeous de(ny (a9) js noi i,he only procesd by rvhlch a

nuon mtyvanish. Thr:ri: is d idcrlce thal anegative muon crpLuredDear r nucl€us mrx disrppear without ernitting rn clccLron. ,lr*r.tion ot ihc typc

r+t+N+r (52)

is consisient with thclinown facts. ,\s soon o$ tLc muon is capturedneu { nucleLrs ili ryould rcact with one of tlrc nuclcri protons andthe t\{o ptrrticLcs rvould change into a neutron and ft neut no.

One leoture o{ tlris hypothcsis js that lrom it lolldrs thai iheDudeus in $.hnti thc rclciior takes place would not be slronglyexcited sinc€ nost ol t]rc mass energy ot the muon r1'ould be losijDto neuirino energy. This w{,uld erplajn why thc process as arule does rot l€ad to st:ir lormalion.

In the usual simplificd lorm ar jntemction tern suiLable tocause (52) may be $'rittcn

s, I @-P,.p* + P*Np/) .Jr. (53)

The mrtrix elemcnt corresponding to this inleractioD is quitesimilar to those of thc prcvious two iDterrNliions and is given by

(5,1)

The three constants sL , s: , s3 ol (a7), (s0), and (53) rll h{vc thesame dimeDsi{,ns, crgs X cm3 = ,'M? '. Ii is x reurLablc lactthat thcir rctld values seem to be quite close, ntmcly abouiltl " ergs x crnr. This faot w&s noticed indeperdently by Thmnoard lvlicel(ir, Lee, Rosenbluth and Yang, aud othcr luthom, andprobably is !ot a cojncjdenc€, {lthough at prcsent iis sigrtilicsnce js

unknoirn. Perhaps ii may bc (orrelaled to the somc\thal similarfact tL{t thc electric chargrs ol all ijhe elemontffy prrticles are

equ0l. Thcir common talue e phyr ihe role of i"t.r^.1id. .nnstrntbetwecn thc clec tronagnetic ficld:!nd lhevariouskinds ofparticles.

It shonlri be noticed hrrthcr that some ol the interactionsdislusscd iri thc prel.ious scctions may not bc primary but may bc0onstrucd is a conse.Ludrc. of oihers. Tlis possibilily, i'hioh iserenplilicd by the Yuliilwa theory ol betll disjntegntnrns, is

discussed jn Appendix 5.

0j1'

s" J a.*." + c+ptt) iot.

(5t)

Vll. 'IN7'E]tAI,'TIAN OF TIl

': FIT:LDS

IA. CALQT]LATION O]I 1,IIAN8lTION EA|,ES

In practicrl computations it is cusrornarv to hrn.lt.,le in -rr.r,,. ,Frns h,,nFFL i6tuq t.(e I,d. r,, ,,;o". - .,.,'pnr'a1ri,, lar,-.ty'rr rn"po,.r,,.-io. ,- " -"*t.L,rofj bF., "" .. ,d ^,ti,"rg,nr .t. . r rncxcepi.ion. lvhethcr ultjmjitety a math€maljcaL procedure \vilt bcL]+el ,ryu ,. oacl, .,'h,r , rgr."pr^,tFm.,,.'hF',.rn^ " 1,r" .,LJ ,,,oifi.J ,r1....h, rpdInriF.\i ,,

.0lled fo. is not lino$n ar prcscnt.Tn this clem€nt{ry discussic,n $e shdL avont the difiicllries l,\.rFrn,n.:rg hcrt,oo,ia- o. I.o,!q. .."o".. ,.n_r..,i,:r

"e"rl :. 1.1,i"?d..nJ:r .or."..qc. t.J .r, rr..r: r.m- rlrr lil".,.nrrib'||ion. ul"ar.. o. \"Jrgr r. ,r"rl n ., .t, acsenerrillv responsible for lilre divcrsencjcs.

Extcnsive use witl be made of I geneful formutaa which gir-es rherate at r.hich the transitions trom arr initiat staie zer; inro acontinuum of states ?i t!.ke place. (lrjg. 1_)

a(0+z): ($)

CALCULATION OF TRANSITION RATES 3I

ijnuun. This nrfinity, however, is compensated for by ihe fsctthal :1t;,,, is inlinitesimal.

A conver$nt pror:edure lor avoiding mistnkes is to quantizein a finiie |olumc a \rhich is later on allo$'ed to increase to -.trVhen ihis proccdurc is followed the number ot finrl stntes will befrcquently exprcs-qiblc by a lomula lihe (17)

le.+apl

= 4n r'j

an -! +i

d,," -- -Ly1 1t,

at'.ii

dN:ltlrwhere p is the momertum of the particle crcat€d in the reaclion.Botli in relativistic lr,nd in cl{ssical meoliaDics the energy If andmomentum p ire rehtcd for s single pafiicle of l.elocity 1, by

dW : I d,p. (57)

(56)

_F

:aTl,'79

STATE O

R'0-,' . rl,erruu.iri,nrJ ".rh.r ,,,"oo. :ti., uf ,r. - :o,,

P4r ur . rm...^m. rrna. rTp"otr ,'y I L r-d .1,. uioorLrl \ o

".nsi r". tC f rh.mf i.,tpm, oIr-.De . .".,n,".po,.r tFl^, ,p, !1.i.u 1, i. jr..TFd t,1 i, I.t,..1" J",r,.rtemean square modulus :lC,J for a sroup ot si:lres ?, cDerserjc,ltv.lurp osErFz.'o. lb!"\pr, !'orJ\ a ,.p,,.".,.r"nu,,,e ott:o.rl .re,ps p.r uni ..r" sy nrer.,t ror pnFrsr .to.. o ,.. I oistate zero- StricUy specking, dlldtt/ $.iI be infinitc for a con_

. 4 Spp ^r...r. "1., fi.t/ o,.1 tt..,rr?r. \.u \,r... t.{,,.-. .,,J.

)",,1,1-"," ...or. n,...,nsl.esdy been rlis.ussed itr $eciio! 7. Th€ inositio! .lisous6ed L,.e krFo''.-r,o,^, "s'.."?.,o . ..a,p .i.1,.e, *,,,-..,,. .,-",,.;i,;"0. 9. . q ,r- s ,," /" o 1 ,.,s ,, 1.r.,, o,,",.h ",i ..^nyi\a,i ot " ..s|h

'h.r,.ro..,,,,,r "h.n:-..^,. ".. *.,...":'... .. :, . r ..-F , i I o,r. .."."".. L1 , "r

.-o .. ,.i..\bi.h ma\ /iT"r I " ,A

".,,o., ,s rror. ,tp, o --,, "n, tsv,0., L r., \. .,.J

]'rc. 1 TrDsiiiors lrom lhe sirtc zcro irio ihe.ortinuum of ststes a

When a sinsle p ticlc is produced one will haye

dN Opr

lvhen ihe reactjon yields tx'o psrticles of equal and oppositemomentum, p and -p, and speeds u and r,, formuia (56) ismchanscd but one finds insNeld ol (57)

JW:(r,+Ddp. (5e)

In the parenthesis tlie 6um of 1,he magnitud€s of the velociiiesand not their vector sum appears. In ihis cr,se one obtains

!)p"

(58)

dNtlIV 2n'1hxlh + u')

(bo)

-A2 |NTEN|CTION OI' TEE ILIILD'The cnse x.heD three prrtictes cmerge olrt of rhc reacrion wil bediscussed in the srxrcial applioarions.

Fornula (55) for the tri1nsition rrte is obrr,ined fron rhc ouaD-r.nmalar '.-'l p"r 'rrbr .,1 al.tr,\i.'ld:o,..\,..'Jrdirts ro i, 5l,ouu p " l,t,"p .,,t..- .h" n. ri\,lam" or,r..,i-sr\p \o"r1p, FJir,";,.tr,mzero tnrn.rv

o .. na ,""",,,, , , ;,; ,.":.,;"; ",..-,,;"even wben th€ matrix €lemcnt vanishes. In rerrns oI the perrurba_

tjon ihcory this is due to a p.ocess of highcr approximarion. i.het.aDsitjon occuls tluough rhe jntermerliary of onc or more staresconnected by non,vsnish;is mairix etements borh to srat. zero&nr:l to the strtes n. As a rule these intermcdiate strtes havc energyquite dilTerent fr.,n sts,re zcroJ and therefor€ no permane t transi_tion of the system into ihcm is possibte o sccount ot the consewa-

One rpplics in such cas€s ihe strndard rules of jitre perturbationtheorys (second approxjmiion). The rr:rnsitjons from zcro to,io.cur ns if l,he tso st:rtes w.jre connected dir€orty by an ,,cfi€ctive

a: tv,- tr, ^(.t1)

The sum is cxtendcd to all rhe intermcdiate sraics ??1 conneciect toslatcs zero rnd n by non vanisbing mrtrir elcnenrs. In pracri.esuch suns rre frcqu€ntty diverseni,. When rhis is ihc inse we6hall .dopi crude cut-ot1 procedures in ordel to obrain so catled"plausible" results.

DEYELOP}TENT PANAMET'ERg 99

obviursly spoil ihe argLrment. In any case, the srrlatl vrlue ot l,hedevelopmeni prmmeter may bc tahen as a indjcaiion of the.eliability ol thc approximatior procedure.

,A,s rn e:rample of ihe meanirlg of the development paramerersappropri:rte io the va ous forms ol field iniGricrions \'€ shaudjscu;s ji$t ihe Yuka$a interaction (a1). The st$te represcnrins 11

proton P ai rcst i6 connectcd hy a non-vanishing mardx element(a2) tu r sialie (X + I1 )! reprcscnting a posiiir-e pion and a neuj.ronwith €qual nnd opposiie rnomenta p and -p. !-rom thc pefiurba-tion theorl. we hnox'lhat ihe unpefiurbed sliete (P) will be mixedwith siates (N + n)! i the l €or conbhrtion

(P)+Xk(N+n), (62)

nhere .AlIr is lho anergy dificrence bet$een thc two statcs. In rhis:lpproximatior the probability of firlding the stsrem h the staie(rV + n), js :hj,,, ',/Alt/!and ihe prcbrbitiiy ot findins the svsieminrnyoneof lhcsi,it€s(N+ l]),is>! j1!. i,A /!whcrerhesumshould be rahcn Ior all allowablc v ues of p. In order ro cv{luatethis sum one r.phoes the matrix element:h- bv iis vatue (42).1 \ '/.1 .r,.^, -\r' " |,r' - Ip.n,rqJ o|1,, p,on.Sirut tlLe pion muss is smat compiu€d to that of the nuctcon theencrsy difierence bcli$'een the statc (rV + n), rnd the unperturbedsiatc (/) caD be tak{r equal to u. Tho sum cln rhen be convertedinto an integral by mtltiplying l,he snmmand Ly rhe expression(56) ol lihe nunber of sraics beiNeen p rnd p + dp rnd integratinsto all vrlres of p. Introd cins as a virirlte insread of p the newvadaLlc, = tipc, onelinds

",, r ,d,' \rl - trh Jo ; t 1rr''

U lorhrnatelJ' the int{rgral has 11 losariilnnic dir,'ergcnce at,? : o. We gei arol1lld ihis dilticultr by one of thc clude cur ofiproccdrros Nlready nlultlorcd. Instead of integrarjng up to infidtevalues ol ihe momerrLum w(i limir thc integral io rnaximumrnomcrltrun ol the ordel p., lihrt is, to 1'alrrcs of r7 ol orrler onc. Ajusliticrijon h this cut-ofi is thc expecrtatirD ihat rhc eapressiorr(42) ol Lhe rmtrix elenrent max in frct breali d.nl.n at rclrrivisticvelocities of ihc pion. Th€ inteeral in tlie formul:r above bc(x)mes

1]. bEI'EI,O PJ4EN T P}lRAMETENS

^ \ ("ll bo\'F- ."J h.r i. t,., .h .n"-.,fl,.r1-".oi.r1nr ,'6, ll37 1.... r." ro,.orJ. "top1"ar p,nFF i,hJi.ri,, r'..,n"n" f . rpan. rj, I r r jps p\t"r:ion rnporrers of e obtainrd x.helr applt'ing rhe perturbari.D meihod

Jp. - .p n ord. I ot m. rnir .,h r. rr,+s.i\^pu p' ol Vp a. .08;. \. u.'llr, ml l" r. . .i rrLr ,ti-rensionallv, irrfinitc numcrical cocfiicicnrs ofren arc excountere.t which

5. See foi era.rlle: Sctjifr, emntlm ltect,anics, D. tg6.

INTER}ICTlON OF TEE IIELDAnow of the order oI magnftude of udty and the expr€ssion itreubecomes of tbe order of Dagn ude

DEVDLOPMENT PANAMETENS 36

Since e, ijuns out to be verl small i,his p3rametal is &lso quitesm l so that th€ pcrturbaiion method is presumebly reliable Thesame is true ol the development parameteB for lhe int€ractions(a7), (50), (53). These parameters ore siven byGh; (63)

whichjs the usual expression adopted tor the devetopmeni parameter of. the_Yuk$wa theory. The devetopmenji prraneler represents therefore ns order ol magnirude tlie traction of ihe states(..v, l,n \"di1 wi,l, P,.Trpnyurpi.ion.L,3 L n.ar"og,r"ol ,LFfroesru.,u,F.o,sranr oir " r,o",ior .h.".1 Tt,. ir^i,"4/ in the denominator is thcre beccuse;n r. yukawa ih€on, onerdopr, r, :on1l:Fd urir. Tl,F d"re opmcn 1,.r,r.."r 6r, ;"

"", .qmsll nraber. I i, d..i.Ll. ro a.c, in ro . a fr".F. \l! ,r,. sin.c romeson theory accounts for the nuctear torces $€lt enough topermit 6. determinr,tion of the coupling consiant. However, it isb€lieved ihat (63) nu.y bc of thc order of 1/a. Then r, wouid beabout twenty times the elcmentary etectric charge.

These lsrge velues of 4 and of (63) mean of coursethai d physicproton.is rath$ poorty reprcsenied by rhe methemrticaffy simpfe6late that $.e have indicaicd bjr (l'). _{cruattx, ir, js a rnixture ofthe state (j') rcpresenring D proion proper, nnd ot stahs ff + [represeriins a_neutron with a positive pion n€arby. The physicnlproioDis,btrFro " far hum, L n "1 J.y.Tl,Fu usIn11 o"a... i; ,;p.rha a ruar'u1 " o "rr 'h"r ! f,o,.,. o..!.ionellv pmir. a p,- ;..pion snd converts temporerily into a neurron. Soon atrcrs.ardit reabsorb the pion and goes i]'ctr io the proton forni. Thancurron h1c a .;n,r1r 'e\r\ "". I t" pr.,b, r,t,r! ot In, r,6 r, ^p-uron rn 'hr si ir \ fl ralre- lrrr r I " -r.r .. ,t i luireiarse, boing of the order of masniiudc ot ttr{r dev,toproonr, pa.r,,n_eiier, or perhrips 25 per ceni. When trro rLtcteons

"pp.o""t "u"r,other thci surrounding pjo. fields inicracr in a mrnner thai givesrise to the stronll short,rflnse foroes betileon nucteons m rvill teexplsined in Scction 17.

The developmeni parameters for the other typcs of intemctionarc obtained in s similar way. tror rhe interactjoir (a+) rhe pararneter b siven by sn expressioD like (63), naneiy

9mc (65)

where t sliould be repiaced in ihe tllree cases by 91 , s' , g' s.Ld m

should be rcpllced for tlie mses (a7), (50) by the m,ss of theelectron rnd for clse (58) by th&t ol the muon ln aL1 cr,r.scs theparanet€r (65) is qujle smail, ranging in value fro," to " to l0-"The approxi,ni1tion meihod lor thcsc cales should therefore b€

relirble prorjdcd lhe actunl foD ol the jnterottion ter!]! has

been gucssell correcl,ly.

"1 164)

=

CElPTER TIl RFN

T'he Interaction Constants

19. ELLC'l FOIl7t)N |t't,IC tlND I'OK,4WAI N I LRAQ 1' ION CONSII]\r?S

. Tn 11- 1,r'n".l,rg,rap,." 6i\ rarFr.,crion tro"".".sra\. I,Fpn , . r .,1 Tl-r do ro n,"r,l no. .hr ri,.. ThL.ecould be additiond iricrt.tions amons the iiementarJr partjclcs,and bcsides there ftre prrrilles hose 0xistence is either hnown or-(uspectcd drich we have teti out of consideratioD becruse lo.

"-l.n,r, orr'. tr,rp.rip. I.orF,. ,.f .p.\in..rcrionprocesscs ol Chapicr Il 1r. consr.lni, h{s bocn inliroduce{t irt,r.dci€rmlnes its strensth. Three ot rhem ha1,e the dimensions of anelclt.ic chargc lnd ihree have the dinensjons of energl,. X votume.The first tlircc dre

r-the elemenrary etectdc charge ihat deic.mjncs thestrength of the elcctronugnetic interaction_

cFthe interaction constrnt of thc yuk:iwa theorv .torer_mrling rle .."",.gr1 _t ,1,. r,r, rs.,i,.1 b" ."er 1.ior" rr.l

e-the consrant ot an intc.action ihAt has been postut$.tedto act bets.een pions, muoDs, aDd nentrinos. Nhich .ont,l h.responsjblc for the Epontancous dccay of the pion.

The threc constants $ith climensions cnergy X volumc &re

,1-ilre iltcraction constant of thc betli pmcesses.

,,-on jnter&ction that has bcen postutated to act ]retneenmuons, alectrons, andneuirinos and vhnjh coutd bc responsibtefor ihe lpontancous dc(jly of the nuon.

ELECTEOMAGNETIC AND YUEAI|A 37

t3 the interaction constant of a hypothetical process similarto the beta interaction excepi that the electron is replaced

Perhaps fubure developments of thc thcory will eniblc us toundeNtand ihe rcasons lor the e\isicncc nnd thc str€nsth ol thcsevadous interactidF. Ah prescnt, howcvcr, \1'c must trkc an em-piricd approach and detcrmine lihc values ol thc various coutArrtsfrom the intcnsity of the phcnomem lihnt arc c:ruscd by them.In Appendix 5 sono ol thc possiLle relationships bet$,een variousconstanis Are dis(usscd.

Thc ehctromisnctic intcr{.tion constent € is ofcoursc vcry $'€llknown. Its v ue is

a : 4.8025 x 10 " cm'/u sr'/'sec*'.It is a rcmrrkable laci thab itre valuc of thc clementuy chnrge isthe same for all chargcd pariiclcs: clcctrors, protons, muons, lndpions. This ideniity is pr.sumably relntcd to tlro conscNrtion ofcleclriciiiyand to ihe faci thalparticlcs can chango into erch other.For cxample, the bcta processcs in which a neuliron chrnges into eproton, an elcctron, rnd a neutrino $'ould be incomprtible $riththe conservalion of eleciriciijr il the absoluie valucs oI the chargeol the proton and the electron $'ere diflerert. $re $ill troi discussthe eleciiromagnetic processes here oicepi lor a few cases h whiclthey are necessary in order to undeniand oiher properlies of iheelementary particleE.

The Yukawa interaction is responsible lor the nuclear lorces,and the 1'alue of the corespondins interaction constant c' shouldbe adjusted to fit the expcrimcnt'rl vrlues of these forces. r\sketoh of the Yukor& theory ol Duclear forces will be sivenlater in Section 17. It wi be lound there tllAt in order to obtain anorder of maqnitude aerc€ment bctircen the observed andcalculatedstrcngth of thc nucl€sr forccs lihe mnstrnt c' must be taken aboutti.'€nty times hrger th$n the clemcntary elcctic charge. In thenumedcal c:ilculations we shall $dopb the value

r? : ru cm Ar sec (06)

It should be stresEed tlirt this 1'alue i6 merely an order of magni'tude. The quantitstive rgreement betlveen the Yukawa theoryof nuclear forces and experiment is so indeflnite that it haE not

v3A TEE INTENACTION CONS?,4NID'

been possibla 1io determi.e rhe precise fotm of rhc jni,€rlctionen€rsy and therclo.e thc value of ihe constnnii musr atso belell Lrncertain. In pnrticutar, rhe vatue of th€ spin irl rhe pioni.rot noro. I .^,rll Fnur Lorp"rt .psr...prD,.Fr r.:ErrF-r.L.p. In ou, "i1lpt r- I Jr".f;o" a cnr zi.., ,,q ...,.," t."",siLtucd.

13, DECAv OT THE PION,ISDIRLCT PROAESS

TIre spontaneous conversjon ol the pion inio a muons'ld i t'", ri.l di.. .."o i, cp.,:u1 eniJ t.- uu, .o I " Jirc(Lrdc., .l rh":r ."!.r,u ..1 . t hp L F nF to hi" I,role"-, $i.t no\lr ."l,, 1 eo ,{, rli.. 3. .n ,rio, , in , rd.r ,o ,"J_ce rr,,m rl.iuTtJ Unor rr,orJ ,n.r"\t"..n,.1 rr,,.v,1,, or,t" in.rcrron

-{ pion at rcst h$ rcsi energy pc'. Tl,is cnergy is convertedinio ihc energy of ile ncutrino oi momcntum p ancl tLe muon ofrnornFniLm -i,. C^.!p.r. ,u.. ot,.aFfg\ rpluil.-

ce + t/ dc1 +-Cp, = pcx (67)

,f EE BE'I'N INTENACTION SO

from the experimerltal value ', : 2 X l0 3 eeconds given by

Martinelli arrd Panofski we obtain t|e interacii.m consklnt

rvhere pr is the Frcm this cquaiion one computesthe momctrtum p of thc tFo particles produced in the rcaciio;. let'- be the lifetime of the pion. The ratc of thc pjon decay ft(cor.linrlto tlre reaction II - p + y is 1/r, and can be cllcutaterl v:ii,(55) {nd (60) usjns the rnatrix ctement (15). Sincc o pjon at rcsidisappears, its encrsv 1r. in (a5) \,iI be pcz. one finds

| _?".( :r- \ ap, _ ."p". h \\'/2a"-l 2nt u i. 2;t.,,, u,'

AI spins have bcen ncstecred, and ur and r,r are rihc sDecds of'hF In | -i!,. ,n I ,. ,h. -iL, r,. Orp hr. ,t"rFt,. p - ... t.evelocltt' 4 of the nuoD rnd the momcnrum p c:rn be compuiedfrcm the ron-"ervainrn ol onergy. ]\ssuming p : 276 m antt pt :,210

m, with m the eleciron mass, one finds by soh,ins (67) p :58.1 nc, florn $tich ii, Ioilorvs thai r]re neuirino canies N,vabout 30 Mev rvlLile the lrilx,tic encrsy of ihe muon ts onty a.I Mti.The velocity of ihe nuon is 0.27 c. $/irh thesc nunefcal 1,eljr.rs

In vritilrg this crpression the complex conjugate quaDtiiy has

been added sincc the Hamillorlian mttst bc Hcrlnitian. The interacijon (70) is callcd scllar' inteEction. Onc can forn four cdditionalrelativistio inruinrli lypes of intertctio by trsing the erpressions(3) to (6) of \ipendix a in a sinilarmcnrcr. They are called respec-

tively psetrdoscnlar, vector, pE.xxlolcctor rnd tensor intc.Actions.For example, the pseudovcctor intcr.rction would lead to n term oltho foliowing folm:

,".iac li,. i ..r , I d, f otdld, \..d 4"o3 t-J l. r

' "","U]. " ".)

It Nill be secn in Appendix 5 that this interrction may not bcpdmary. 1t mry Lc an indirect efect of other iDter;ictions thathsve been tostulatcd.

14, T E BNTA INTEAACTION

The interaction responsjble for the beta transitionshos aL€ady been givcn in a crude forn in Section g. Nexi to thecase ol ihe mdiation theory tbis is lhe process on \4ich Nc lnrvetlle greaiest amount ol etperimcnLal jnfonnriion. ln spitc of this{:rvorable situation thc attcmpis to subsiitute n,r (a7) a more

delftiled expression lnYe not lcd so lar to a linol {ronclusior. Nlosiof the attempts sre biscd on assuning an interscliion tenn in theLagrangian forned with cxpressioN p:rtternetl on those gjven inAppendjx 4. Since the tcrnl in Lhe Lrll{lansirn must L,€ a surl&r,one might for exrrnple proceed as folLos's: 'lhe expressior PPNjs a scalar c.ecording to (2) in Appendix 1; so is the exprersion e$v.

Therefore thef product multiplied by ;! arbitrrry jnteraction

constant sr \ri[ be an jnva cnt sealar that cou]d be added to iheLagrlnsian to represent the iDterrctior. In prssins from theLasrcngirn to the Ilamiltonicn thcrc is in ibis case only a chanseof sian and therefore a possiblcinlarirni inieraclion temr Fould be

cr : 1.1 I 10 15 : 2.4 ,( 10 6r. (63)

s, I Qi:NeS, t \trl)Pttsc) da. (70)

l " = 3r l0'ej. (68)

(71)

-10 lII D lNlER,lCttO^" CONSTltt,tE*hcrc dp, represonts tlre vrlue oI rhc inieraotion constrnt torihis c,rse. ,Uso, linc combirariions ot lihei;e \ ious inier.t{itionsmay be adopted. ditonpts hive becl made b cornpl1rij $,i rerpcrinenl ih€ coDscquenoes oI ihe v n,us ir|crection lorns inordcr to decide which if an]- ol rhenr sives aftepiabte rcsult!.Ii turDs out, unfortuutcly, i,hri lor rhc so caltcd Derniftcd r*insi-i".. I'F h t" ul h, l."r ,,by.,," 11,T i r.,. ..rp or rlt rynF.ol interacl,im. TIle cx(jelleni ssreemcni treiwcen the thir)rericaland erperim{rntol shrpe ol ihe fermjiied spoirr rherctore doesnot oller l1 clLre lor a decisior. Or tlic other txind. l,h(i difierenrtypes of interr(ition si|e diflerenr sclection tes. l1or €xarrlptc,t|e scrlar {nd tle v€cior irtcrocrions sjve rs s seleciiol rute torpennilted transitions I

\o clangc of nucletr spin, no clxnec ot pa iy. (/-scle.tion fttes.)The scle{,tion rules for the pscudovector or tcnsor inrc*r,ctiors

Chenge ol spin by 0 or +1, no clnngo ot paritv. 0 + 0 transjrionslorbidd€n. (Cz-selacrion rulcs.)

Ii appcrrs lhat experim€ntal evnlence tends i,o faror rhe Gfsdection ml€s, or perhaps r mixrure of thc IJ ancl 6'? nJes. Thiscvidence, hor.ever, is not quiie co.dllsive, sincc unforrunal,etylittle is l(]ro\i'n expedmenlally of ihc spin and parity of betn aciivesul)stances nnd ore is usuallyforced tosuess on the b$is ot nuctearnodels whioh, althoush plausible, e nor v€w wc esreblishecl.Il on€ assuDes the AI rulcs one {outd bc ted i,o rse eithcr Lhepscudovector interaction (?1) or rhc tunsor iniemcrjon. In mostcrlscs thesc ts'o inrtcftrltions gjve almoEt jdcnti0at rcsulis.

lvjsner ftnd CriLchnetd ture co$ntered in addiiion the pos,sibility ol a liDear conbinarion ot scr,tar, pscrrdovector anct pseudo_scahr inieraotions $hich can be expressed in a rcrnerkabtv svm,m,,r',., lorn. 1,. r.., . .h.$ j1n alrFpn,||,,,,h "tlr..ir;"n. n"$-orse than thri of other {hcories. Tlie resutts of the c.,mDlrisonof \.,J s,l.p{rprincnr .,n.€.'rrr Iiz,,t ov .vi.lsrt" ,tL"shap€s ol tlrc permitted spcctr,r and of sevcral forbjddcn speclraseem to folld{ the theoreticrl shape or shrpes wjth srrrprisinE:r.cuI r Or 'ce orr-r l,r,L o ll rnl I",,r .,..,.... f,,r ,cc;met until no$ in thc rttemprs to corretaic ihe inrensiiv ot ihehera r"asr'on. .Li \ rl,n mrrrir FtFm.., s.rt, .'r."o frorllunte-r

THE BET\ INIERACTIOtr" 41

nodels. Ol course, the nuclear rnodels used are very crude so tlatlbe tost is not vcry con.lusir-e.

'flr(i virious trpos of nrftraction lead to difierent prcdiciions as

to thc rnsular corelaiions of ihe dircctions ir Fhich thc clcctronand thc ncutdno sr€ emitted. Since, horrever, l,he obscrv{tionol tlre rc(ioil of the neutrino is not sumciently rccuratc ii h{s notprovert possible until noir to drcw reliable conclusions. In thepresent discussion ile sholl lirnit ou$elves to the use of i.hc simpli6ed interaction (47) and to ine dedvation of its most imnediato

ln calcr aiing ihe maLrix element ol a bela iransiriion one cannotapply $'ithout modificai,iorl ihe rules ol Secliioll 7 bccausc |hcncuLron r d l,hc prol,o cannoi be rcpr€senicd by planc wavcsbut tuc bound by slrorg torces i sidc the nrLclcrs. l'or this samcrerson no ixnscrvdtion ol noncnturn holds lor thc loL[ rnuticlcsjn cqurtion (40) sircc thc rcsidLrI] nuclcus crn trkc up rny arnount

fhe crltulation *'ill bc simplilicrt by disrcgrrdins thc Coulomhlorces e.rtins on tlrc {rlectron. ,{1so tho rrsLral {ssumption ii'ill bcna.de that the ivave lcnsths of the clcctron and thc ncutrino arclarse conparcd to th€ nuclear dimcnsions rnd thr:refore the phaso

factor can be omitted {,om]rutins thc .(nitributiorl i,o tlrc mri,rixelement ol the frctors €, e*, r, r* according to (21). Iisch of thcscerpressions cort butes ther thc mnstrnt frctor 1/\,/4, exceifor the sisn Nhi(h is of no rele\arncc. TL€ firs1i ic.n ol (17) is lihcone that, is r€sponsible lor the bclia proccssr:s with tnission ofnesatire €lectrons, whilc ihc sceond icrm produces thc reycrser%1.'rion pnJ rr" op I u''.. sr.\ I',.rrr"1 n' :u,

Since proton and D€utron a.e bound in the nurLeus tlicir rnn-tlibution to the natrix clemcnt of (17) should bc complrtcdaccording to (22) and not (21). Dcnotnu by N rnd.1'tlt(i \\a'cfunciions of the neuircn that is prcscnt in ihc nudqrs bcforc thctransformation and ol the proton that is th€rc aftcr\vird, onefinds ihe meiri\ e.lement

n". : ff/r-r'ao : ff,rn. (72)

This matrix elem{)nt can Le uscd lor c culaiing thc probabilityof the tr.insiiion froni the inil,ial si.ate to a siaie in wlich anelectron is emittcd $ith momentum bei\veen p and p + dp and

Y4E IEE INTDR,ACTIAN CONS?,1N7S

lihe neutriDo escrpcs Nith momentum q, carrJ,ins awaythe bahDceof the araril:ible energy. lf ?o is ihc maxjmum momcrtum wjthrvhich elecir.,ns crn be cmitt€d, the ndrtrino enerlrjr .q $,ill be rtrcdilTcrence beiwccn the energy of elecirons rvilh momenrrr p! and pl

TgE BIIT.I INIENACTION 43

io do much morc lhan guess the order ol magnjtude of9n. Therearc some fcrorable cises lor rliich orc has remon to bclieve thstthe neutron and tlie protm ]ur-c approxinliely e(1u41 wa1'c Iu c-tions. Such cases arc trorlucntly ertuuntered for lighh elelncnts,p:rrlicularll' for the so 0 |:rl \\'1gncr nuclei. If P rnd N in (72) arethe sa e {ave lunclions, thc intcsml r{rduccs to thc normdljzaiionjntegrd and is tnerelore e.tu{l io onc- In such ijtses, tir{]rclorc, thesimplified lheory lerds to the simple result 9)t : 1. Some{h$tmore c.,mplicaled reslrlt! are etperted, however, even in thissimplesi case il one oi i|e moLe elaborate forms of jnteractions

thrt hrvc already been discussed is adopted. In particular, ifonc uscs the G'l''scleciion rules, one finds for 9t the followjngexprcss n nlstcad ol (7:):

t:.1 : \//n/ +.i'pt ^,/m'4, + Cf (73)

We complrtc the prob&biliiy ol r:rinsition witt tormutr (551.lhF ra ,./\ r'lr'$il p.o uo " I ". l' ;a "'r' :l

for p the rnomenlun q of thc neutino and fo its vclocity, c.An additional l{ci)r represcnting tlrc nunber ot sratcs ot aneleclrod of nonenium belillt€n p and p + dp given Lj. (56) mustbe included. ]'he fina] result is

2n s,rL " ad ap" ,:tp silt)]t i, , , .

n s ,."ii t""h: : ,2rrt{c qP t1P

., rrh '71''r,or'V,.." r'; - 1/n; ,p.p'dp.

This fornula sivcs the shape ot rtrc spectrum ot the emifted berarays. lrrom it orc our also ralcutare the toiat prol)ebilirv ot rransi-rro. fo",1. r,.. ,.i.rnv rt ,.,.r'" en.r|v biinlesrating lrom zero to p0 . Inl,roducins for corrlenience the newvariAbles J? :rnd a0 repres{rniing rtrc momenra ni rurirs ?nc onefindr the lilcinne Ior the tnnsilion sir-en by Lhc forDluta

I _ /i 1 9[ s'l?s.r -,, - :,""n "n''

t'(n|t : I i,/ L + ti \/r + ri \, d,

(75)

. (70,0 1ta

; ,;'.!:, -;/ --;rn', /' ,r'\

The tcrrn i tll ]!i (75) js rerared to ihe Duclecr s.ave funcrionsby (72). Its celcula&,n requircs ttre.efore l(no\,terlgc ot rhe $?\.c{un0tioN of i,hc ntidrl and linnt strrc ot rLe nucleus. Since thescn' -e functions i! mosi crses ue noi knoi1.n, ir is usually impossiblc

i. Snr.c electror xnd aeuirino ue retri,il,isiic lu.tiot.s tle retaiivisrico ,ul . /i .t r,, .. L. \,r, rh-.."'1".'-.' mi!,L u..,p11' ..,,. too....\..I,,L.,ru,-.r-rtr, rr

(for G? rules) (77)

$'here o is the Pauli spir operator of lhe nucleors. Even with (77),however, 9r usually is ol ih€ order of nngnilude ol ah least 1iD the

An addjiional complication concerns the shape of i]]e spcctrumand the lorm of the funclion a(,r0). The aciiou of lhe Coulombfield of the nucleus on lhe eleclron llis b€en neglecied so lar. TheCoulomb field distorts lh€ electron rvi1.e lunctions and changesthe shape of the speclrum especially rt low electron energies.Sirnil:[1y, the function /(?t \fiI] be nore coDpliccted. The readeris releued to specialized xrtntcs lor a rnore elaborat€ discussjonof tlie beta rav theory and of it; comprison irith experiment.

The velue of th€ intcrurtion lonstrnt that sives the best aerce-ment with experimental drtr is f(,r c1l f(,ms of nrterootion of theorrler ol 10 " r;rgs X cm'. fho prccise valuc depcnds ol courseon ihc inicraciion lorm th,i is a$umcd. For the pse,.tdoscalar

intcructn,n (71) onc lirds for cxample

er : /P%N ,ro

rr - 2 X 10 a'erss X cm'. (78)

s' has the dimensions of enersy X 1'olume. In order to appreciateits order ol nasriiude we divnle it by tlre clasBical volume of the

z \M) = 9.4 X 1['3 cm.'.

-v-14 IllE INI.IIRACIION CANATAIITS

The rcsult is 2.1 X 10 u crss or aboui 1.3 ev. fhis mry sive anidtrr of ho$ ileali thc bell1 bteraction is. lr.or erampte, rhe int{}aclion potential beilrccn iwo nuctcons itrioh atso is somerimesrepr€sented l)r r poicntial hote ot radius equat ro re ctnssicrlcledron udtus has a depth oi 20 Mev, th&t is, abour ien mi ion

16. SPANfII"EOUS DEtatf OF IIIE MUON

The sponirrneous decry of ihc nuon Nith .rnissionol an elcctron h{s becn afirjbuied jn Secrjr)n 9 to ! pr.orcss p +! + 2, dLle to {n inicraciion rerm (50) w h mBtrjx etemenio,/{l siven by (51).

ot

lrc.2. I{onenl! oI th-" €t.oi.on {p) ,n,t ot rhc iso rcuirilos (p,,n.l p:)

Th€.ralculation ol tLetiferime ot rhcmuon due to this ;rteuctionis a liitle bit more comptical,(] un lihe simitar cil( jL l1 tion of thelifeiimc ol i,hc pion car ed orr jn.qoction t3 because in rhis.,ser'"e r,, i 1". Lr "rJ , ,s.,.,+ p, .t r.,J. I.rorn rr:. ,1 rol,,,srr l', p1"" ,, "e. .l ns n,.m h. I ,n,^r.,.ior

-p" (r l'1., " ""t, .p*,.,,., ,r" ,r"rimurelecllon energy is oLiained wh{rn tbe two ndrtrinos flv oui in ihF. n p J, ,,:on j,J i ..,.n"-rj t. ,t" m,.. .nrrAl t 1" .-on o..o'. Ll |p ir -r"n,.",. ,l ,.' ,,...1t p..rrrro!. ln rt,,cal0ulalion tlDi loliows ihis rvil be dore.

lvhen r muon at resr de(ays into ar olectron and r$,o ncut nostltc vecr.or sum oI the monleria ?, ZL , p, of these three p&rtjctes\vill vl1nisL.

The rai,e of iransitioD into a sirt€ in *{rich rhc elecrron escaDes$il, n omFn,u 1L,,\Fpr pl,.il p r dp .,.t.rx:r..t L1 mrt iplirrg

'?oN?-4XtOli'S DECAv 0tt lni Mu(ttt

ibe nunber of these clecttrn st.ttts siven bv (56) bv tlie rtte ol

trrnsiiion into one electron sirtc compuLed with (55) Nesle(itnis

,r," u. n of n" r..|l'...-. r, a-mLa- 'l ' g r" 1

nttop r.', t,t . !\'i'' 1. '{p1 :or, J" Q I r'- n '\ 'lFr''rr'one 6nd. i! iltrs iYal ihe transiiion ruie

2' l. \'t:P'dD !'t . ;lt \a / 2 ,'.|' l]r'The number oi neut no srci(is per unii cnersv intcrval' 'lN/'llr'still niust l)€ lrlculrted ln lisure 2 tre reprer:ented thc monrentun

, "t

tt " "t,,"t."o ""a

lhc rDomrntr pL ancl p! ol ihe t\to ndrLrinos

1i",," ,r." *"t- *-..anishes the ilrrce \tclo$ lorm ihe sidts ol a

triangle. The rest mdls of th€ dectron bclng ncglectcd' tho enersv

of thc ihrce eniited Pariicles is

It': c(l p, j + lP, L+ lP ) (80)

Tf 11r h rD the monentum p of ihe eleciro and ihe eners) Iy

constant, ihe sum ol thc rnru tudes ol pL and ?r,

lP'1+ 2' : tric-\P '

will also be corstnnt- From this it follos's thit hhe 'ertex

i ol

ii'". ,.ilrr'"]" wiu nro\rc on an cliipsoid Bv sinple geomerri'

con::]e.lor on.."n" .,,1 ,.;..\ lun,:

rl ll - rrl , tZ" "\!.j\.. ,. t, ).

Tl,, c"rresDon.lrls Ih. " 'pa " o'r,oe: r"1-od oltlerlore

^ ,'e.,, 'rr :,nd rlp \J,um' a Tr" run r'r V nr rau ro : ";;l l;;i"",;;;, ; : i" ot, ".u" " or r.d' 'o"r" r,/ii. i:a.^' '" ,*'1. ,rp-. ion ' rr" ''\ /rr-;'.io-,,,. r' i ".1ur - ,,". ll - r'" "hF" J i rlFauon r'la''

and \re find

,./\ a",:f3. dPrz {,). ,8t.dll lb!\

Srbstiiuting in (7!) oue obirins ihe rrte ol lrarNition

ri':c 11-nP +21\0",,r.48nr,{'i \ r,r !{ /

(82)

-ta TUL IT1.ERAC'| )^" CONs/,{l?sl'he to1,al rAte of iiransil,ion, rli{t is, itioinveNcot ihelifcrnne ot thcmuon, is obtained by integhting (E2) ro atl ratues of p, trolr]0 to pLcl2. TLa resull, is

roRcED DLC.lv Of ',t HE MOON 41

known only vcry cmdely. The thcorctjcnl spcctrum. horvever. isnoi, inconpaiiLle wilih ttrc meNuremeots of Sleinbcrgcr rnd Ander-son. It should be re icmbdrrl ihrL tbe prccise form of th€ inter-

'iciion (;0) js not knond. \hrious tvpes ol interacliojr 0ou1d be

Llscrl irr ilris c$e as $'e l1rro {lrcxdv seen lor the betr interaciion.Wheeler hrs mide aL e\iensifc study ol the elfe{ri that a spccilicchoice ol the inieraciion lorm hm on the shape of th€ spectrumand on the oiher lealures of disintesration of the nuon.

16. FORCED DDCAY OF TNE MUON

Thc mLron may decay also br a difiereni procesgwlich is calkrd forcr:tl rtcr.ry becnrse ii hlppens only when tllemuon is ciptur&l rcrr r nurtcus. l'he lorced decay is observedonly for negrti\-c nuons, pLsunably beoaus€ only muons of thissisn arc atiraclied n*r! lihc nu(tei bv electric forces. In Seclion Iforced decay has bccn rlitributed to the reaction (52) with m&irixelements (5a).

In order to estimate Lhc v ue of ihe coupling constant g? veshal assLlme thal n'hen a n€rtive muon comes to rest inside anaierial it is very rapidly capiurcd in the vicinity of one of th€nuclei in an orbiii $'hich is the andogllc of the loly€st Bohr orbit,only much smaller. Iis radius x'il bc obiained by dividing theBoh. radius o : 5.3 X 10 ! cm by a Iictor Z due to the nuclearchaqe and by a factor 210, ihe miio of the muon and the electronmasses- The radius of this Bohrlihe orbii is therefore

n' : (rtl210z) :2.t l10 "/Z cn. (85)

L: (83)

Subsiiluting this

7080rr n,

Expedmcntally ', : 2.15 X 10 0 seconds.

. Frc.3. llnergy sDccl,.um ", ,,," ",".,,.." r.",:tj" I::

value and p1 : 210n in (83) onc ot)trins iihe vrtuc of Lhe.orptins

9x=3.3X104! (8:1)

The closencss of the vrlues of sL rud s! i6 r.ery st kjns_Fomulc (82) sives ilre specrillll disrribuiion ot r|e ernji:r,ed

cLectrons. Its rcsults aro repreNrrte{t sMphjcalty jr Iisure J.Expedmeri,allv the slupc of ihe specirum ot rhese eltciroos is

The iirne required for ihe capture in this orbit js estimated to beof t e order ol 10 1' seconds so thrt the probability that the muonmay decay spontaneously during this time is neslisible. ,qjter thecapture haE tlhen place 1,he destruction of the muon may occureither by the spontaneous proc€ss of the prcvious section or by theforced prccess. f'he relative probrhility ol thcse t$'o mecharismsdepends on the nucleus tllat hlis capturcd iihc muon since theprobability of forced decay incrcases rripidly $'ith Z for tworeasons. For larse Z the radius a' ol 1,he o rih is smaler so that thernuon is found on the a1'erage closer to the nucleus. The nucleusfurthemorc contains c large number of proions that ma,y inducethe dccay. The experim€ntal rcsulis confijm this seneral picture

48 T D INI'ERACTION COIVS?INrSald dical,e lhat tlre proccss (52) is ncsligible lor

'ery ljglrr

ccpl,utng demcnts ard is clomlmnt f or hul,icr elemerl,s. A0cordingto the mcffiurcm€nls ol ficho thc rate ol trxnsirion for the rrvocompeting prolxjsses is alproxim.rtclv equrl lor Z : tl (sodium).Situie tbe sporrt:'neous dc.rr has .l tifetiDre of 2.15 X 10 i secondsn'c {ronolLlde thrt tle rate of for(ed demx ol a muon mpturcdby t nucleus wilih Z : 11 i6 also

1/(2.1i X 10 6) sec I (tot Z - 11).

$t) fir6t calcrlaic the rate ot rllnsjrion for rhe process (52)

"1,.r. . r\ orr" I ,.n rrJ ,,n" p.r ,r or 2,.,u pto.i,\. p prF.Fr,in a vohrme o. The rnairia elemenr accordi {l ro (51) is o3i,0. Thcral,e ol trcnsitiol car thur be cslcutated with (55) anct (60):

,u. o, r'.o"nin" - '' /t'\' tt/"t, \'.t / 2ah.+u.)

(8ti)

- :'' " !'^;The term . + r ihat appcars jn the dcnomjnrijor is the sun of thespeeds . ol the neutrino and I, of the ncutron. ?, is the maqnii,udeo r r. rur ard nf 'o-:r" I uT"1,r of \ !.r.r , i.8" r ,1 , a bpcomputcd frcm ttre conscrvrtion o{ crergy. II the sma m[Ssdifierence bei,wccn proton ilnd neutron is negtccii€d, the avrjliblecnergy is the rest cnergy of the muon, 210nd1. This energy appc:rlsrs liineiic cnergy oI the neutrino and of the nentron, both havlng amomenlum Z. Thc.efore

ztOnc':ce+{r. (87)

FORCED Dlic,tf Ot lAD MtiON 40

In calculating the prolubilitl' thai a muon capt{rcd in a Itohrlikc orbit near a nucl:us ol charge Z js destroycd by the lorced

decay process ire usc (88) sith the lollowing cb{ ecsi (a) multiplyby Z because thcrc t're Z protons tliet crn iifitct lle carrture;(b) substitute lor a the eileciive volume oi thc Bohrlike orbii,

of the muon of rdius r,', namely za"; (c) makc i1 lurlher cli{4ein the coefijcient of (88) in order to accouni lor lhe possibilitythat ihe process mry bc inhibiied by the I'ouli principle. In thcforced deca o rathcr llng energy neutron is produc€d and thcreliciion $ill be possil)lc only il the orbit ini,o which lhe neutronmry be crealed is not occupicd in ihe nuclcus. The reductnnrlactor would be diftiorlt lio calculaie accurately, but $'e eslimatei,hst A reduclion of the coellicicni from 11,400 to;,000 lnay be ofthc correci order ol mrgnitudc.

In conclusion, the probability ol iransjtion lor t]to lorced decay is

1 s000Q101' " m'cZa

Notice the proportionality to Za Fhichbc:rrs out thc greaii difier€n(cin il1e iDtensity oI this process for light arld heal'y nuclei.

From thc expcrimdtal fact prcviously quoied ihat,lat Z : ll,,, is about ertual to 2.15 X L0 'seconrts, one ulculaies the valueof the coupline constant si :

gg:1.3X10i' (s0)

It hos aheady been noiiciid that the three couplins constants,

91 , 9i , si are quit€ dos€ : 91 -2xL1-an,9t:33X104',s!:1.3 X 10 .]'his sinil,rdty probably js not an accidcnitl rcin-c ence t,ut has some dcep mctning vhjch, ho$'€vcr', is not under'

(8e)

.U : trucleon rnrss. lrom this equatjon onc fincls the momentum2 of the tn'o prrticles, p : l99,nc. The velociijy of the ncutronsiven by this momentum divided by ihe n€uiron mass is r,. :0.11c. From (86) one finds now

ra,te ot transitioD : 11,400 9.1-* . (ss)h4A '

TIF nrF ol ,ransi,:nr is io\"'b"tt p,.,ponio,rt ro O Lprau.n wphov rs.und hlr Ir,."i6on, roonrn hplot ,rneO,ldrrprFfo"cthe concentmtior of protons is jn\€rsctl' proportionat to A.