Foundations for relativistic quantum theory. I. Feynman s operator calculus and the Dyson conjectures

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1 OURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 1 ANUARY 22 Foudos for relvsc quum heory. I. Feym s operor clculus d he Dyso cojecures Tepper L. Gll ) Deprme of Mhemcs d Elecrcl Egeerg, Howrd Uversy, Wshgo, DC 259 d Deprme of Physcs, Uversy of Mchg, A Arbor, Mchg 4819 W. W. Zchry Deprme of Elecrcl Egeerg, Howrd Uversy, Wshgo, DC 259 d Deprme of Mhemcs d Sscs, Uversy of Mryld Uversy College, College Prk, Mryld 2742 Receved 1 Augus 21; cceped for publco 16 Ocober 21 I hs pper, we provde represeo heory for he Feym operor clculus. Ths llows us o solve he geerl l-vlue problem d cosruc he Dyso seres. We show h he seres s sympoc, hus provg Dyso s secod cojecure for quum elecrodymcs. I ddo, we show h he expso my be cosdered exc o y fe order by producg he remder erm. Ths mples h every operurbve soluo hs perurbve expso. Usg physcl lyss of formo from experme versus h mpled by our models, we reformule our heory s sum over phs. Ths llows us o rele our heory o Feym s ph egrl, d o prove Dyso s frs cojecure h he dvergeces re pr due o volo of Heseberg s ucerly relos. 22 Amerc Isue of Physcs. DOI: 1.163/ I. INTRODUCTION Followg Drc s quzo of he elecromgec feld 1927, 1 d hs relvsc elecro heory 1928, 2 he equos for quum elecrodymcs QED were developed by Heseberg d Pul 3,4 he yers see Mller 5 d Schweber 6. From he begg, whe reserchers emped o use he srghforwrd d physclly uve me-depede perurbo expso o compue physcl observerbles, umber of dverge expressos ppered. Alhough ws kow h he sme problems lso exsed clsscl elecrodymcs, ws oed by Oppehemer 7 h here ws fudmel dfferece he quum problem s compred o he clsscl oe. Drc 8 hd show h, he clsscl cse, oe could ccou for he problem of rdo reco whou drecly delg wh he self-eergy dvergece by usg boh dvced d rerded felds d prculr lmg procedure. Erly emps o develop subrco procedures for he dverge expressos were very dscourgg becuse hey depeded o boh he guge d he Lorez frme, mkg hem pper mbguous. Alhough he equos of QED were boh Lorez d guge covr, ws geerlly beleved h, src sese, hey hd o soluos expdble powers of he chrge. The hkg of he mes ws clerly expressed by Oppehemer 9 hs 1948 repor o he Solvy Coferece, If oe wshes o explore hese soluos, berg md h cer fe erms wll, ler heory, o loger be fe, oe eeds covr wy of defyg hese erms; d for h, o merely he feld equos hemselves, bu he whole mehod of pproxmo d soluo mus ll sges preserve covrce. The soluo o he problem posed by Oppehemer ws mde depedely by Tomog, 1 Schwger, 11 d Feym. 12,13 These ppers my be foud Schwger. 14 To Elecroc ml: gll@howrd.edu /22/43(1)/69/25/$ Amerc Isue of Physcs

2 7. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry mog roduced wh s ow kow s he erco represeo d showed how he pproxmo process could be crred ou covr mer. Schwger developed he geerl heory d ppled o my of he mpor problems. Feym ook holsc vew of physcl rely hs developme. He suggesed h we vew physcl eve s occurrg o flm whch exposes more d more of he oucome s he flm ufolds. Hs de ws o del drecly wh he soluos o he equos descrbg he physcl sysem, rher h he equos hemselves. I ddo o solvg he problem posed by Oppehemer, Feym s pproch led o ew perurbo seres, whch provded esy, uve, d compuolly smple mehod o sudy ercg prcles whle gvg physcl meg o ech erm hs expso. Sce Feym s mehod d pproch ws so dffere, ws o cler how reled o h of Schwger d Tomog. Dyso, 15,16 mde mjor corbuo. Dyso relzed h Feym d Schwger were boh delg wh dffere versos of Heseberg s S-mrx. He he formlly roduced me-orderg d provded ufed pproch by demosrg he equvlece of he Feym d Schwger Tomog heores. Ths pproch lso llowed hm o show how he Schwger heory could be grely smplfed d exeded o ll orders of he perurbo expso. Dyso s me-orderg de ws cully obed from dscussos wh Feym, who ler explored d fully developed o hs me-ordered operor clculus. 17 A. Bckgroud Afer he problem proposed by Oppehemer ws resolved, udes owrd he reormlzo progrm d quum feld heory could be clssfed o hree bsc groups. The frs group cossed of hose who were olly dsssfed wh he reormlzo progrm. The secod group cosdered he reormlzo progrm erm sep d beleved h he dvergeces were dco of ddol physcs, whch could o be reched by prese formulos. The frs wo groups wll o be exesvely dscussed hs pper. However, we c ssoce he mes of Drc d Ldu wh he frs group, d Sk d Schwger wh he secod. See Drc, 18 Sk, 19 Schwger, 2 d lso Schweber. 6 The hrd group ws more posve, d dreced s eo owrd vesgg he mhemcl foudos of quum feld heory wh he hope of provdg more orderly pproch o he reormlzo progrm ssumg h he heory proved cosse. Ths dreco ws clerly jusfed sce pr of he problem hd bee cossely blmed o mhemcl ssue, he perurbo expso. Ideed, he whole reormlzo progrm crclly depeded o he expso of he S-mrx powers of he couplg cos. Ths cocer ws furher suppored sce emps o use he expso whe he couplg cos ws lrge led o megless resuls. Addol uese could be rbued o he fc h, h me, o much ws cully kow bou he physclly mpor cses where oe ws delg wh ubouded operorvlued fucos dsrbuos. Reserchers workg o he mhemcl foudos of quum elecrodymcs d quum feld heory doped he me xomc feld heory srg he 195s. These reserchers focused o ryg o fd ou wh could be lered bou he exsece of locl relvsc quum feld heores bsed o cer url ssumpos whch cluded he posules of quum mechcs, locly, Pocré vrce, d resoble specrum. Ths pproch ws ed by he work of Wghm, 21 d Lehm, Symzk, d Zmmerm. 22,23 Here, he quzed feld s erpreed mhemclly s operor-vlued Schwrz dsrbuo. Explc use of he heory of dsrbuos ws mjor sep, whch helped o prlly mke he heory mhemclly soud by smoohg ou he felds loclly. The rece pper by Wghm 24 provdes spred roduco o he hsory of Heseberg s erly observos o he ler cocep d s reloshp o he dvergeces. 25 The xomc pproch proved very fruful, provdg he frs rgorous proofs of umber of mpor geerl resuls, d rced my ble reserchers. The fvored me ody s lgebrc quum feld heory. The books by os, 26 Sreer d Wghm, 27 d Bogolubov d Shrkov 28 re he clsscs, whle more rece work c be foud Hg. 29 See lso he book by Bogolubov, Loguov, d Todorov, 3 d he rece revew pper by Buchholz. 31

3 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 71 For umber of resos, mos obly lck of orvl exmples, he xomc pproch evolved umber of drecos. Oe mjor dreco s clled cosrucve quum feld heory. Here, oe focuses o emps o drecly cosruc soluos of vrous model feld heores, whch eher hve exc operurbve soluos, or hve sympoc perurbve expso whch c be summed o he exc soluo. I hs pproch, sed of formulg he heory Mkowsk spce me, oe psses o mgry me d formules Euclde spce de whch frs ppered Dyso 15. Ths leds o formulo erms of Schwger fucos, lso kow s Euclde Gree s fucos. The dvge of hs pproch s h hyperbolc equos re rsformed o ellpc oes, d Guss kerels, for whch very rch se of lyc ools hs bee developed, replce Feym kerels. The oupu of hs eerprse s ruly mpressve. Cosrucve soluos hve bee obed for umber of mpor models. Furhermore, hs pproch hs gve us clerer pcure of he problems ssoced wh he rgorous cosruco of relvsc quum feld heory d provded ew mhemcl mehods. A erly summry of hs pproch my be foud he lecure oes, 32 whle more rece progress s coed he lecure oes, 33 boh eded by Velo d Wghm see lso Refs. 34 d 6. The books by Glmm d ffe 35 d Smo 36 gve dffere flvor d po of deprure. Alhough gre del of work hs bee doe cosrucve feld heory over he ls 3 yers, my dffcul problems sll rem. For exmple, he pperce of dffcules wh he cosrucve pproch o polyoml ypes of feld heores s dscussed he pper by Sokl He cojecured h he 4 heory 4 four or more spce me dmesos s geerlzed free feld, where s he couplg cos. Ths heory represes self-ercg boso feld. The cojecure ws prove by Azem d Grhm 38 d Fröhlch. 39 Three yers ler, Gwedzk d upe 4 proved h, f we chge he sg of he couplg cos, he soluo exss s empered dsrbuo d he perurbo expso s sympoc o he soluo. Ths se of ffrs led Wghm Ref. 33, p. 1 o lme h, We do o kow wheher he lck of exsece heorem for soluos wh he rgh sg reflecs he oexsece of soluos or merely he lck of echque o cosruc hem. Thgs re furher complced by he fc h he 4 4 heory hs perurbve soluo! Ths led Gllvo 41 o sugges h cosrucve pproches oher h he ferromgec lce pproxmo, used by Azem d Grhm, d Fröhlch, my be requred. The mos well-kow mehod for quum feld heory clculos s perurbve reormlzo heory. Ths pproch s dscussed mos sdrd exs o quum feld heory d hs eresg hsory h s bes old by Wghm. 42 The frs book o clude Dyso s reformulo of he Feym Schwger Tomog heory s he clssc by uch d Rohrlch. 43 Erly work he perurbve pproch focused o he developme of dffere reormlzo mehods wh he hope of defyg hose for whch rgorous mhemcl mehods could be used. The mehods geerlly cossed of wo prs. Frs, he Gree s fucos were regulrzed relvsclly d guge vr mer 28,34,42,44 o yeld well-defed empered dsrbuos, eve o he lgh coe. The ppropre couer-erms were roduced so h, he lm, whe he regulrzo ws removed, he vrous dvergeces of he S-mrx were lso removed. I ws foud h ll reormlzo procedures re equvle up o fe reormlzo cf. Refs. 42 d 34. Tody, heores re clssfed s reormlzble or ureormlzble ccordg o wheher he umber of reormlzble coss s fe or fe, respecvely. Some model heores less h four spce me dmesos cosdered cosrucve feld heory belog o specl subclss of reormlzble heores clled super reormlzble, for whch he reormlzo process c be crred ou whou usg perurbo heory. 32,33,35,36 For hese heores, he reormlzed perurbo seres c be show o be Borel summble o he exc operurbve soluo. A ce summry of hese developmes ws gve by Glmm d ffe. 35 O he oher hd, cosrucve models of he Gross Neveu ype re reormlzble bu o super reormlzble see Ref. 33. Feldm e l. 45 hve suded he mhemcl foudos of quum elecrodymcs from he perurbve po of vew see lso Rose Ref. 33, p. 21. Here, reormlzed forml

4 72. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry power seres reormlzed ree expso s obed for mesure o he spce of felds wh he Euclde formulo of QED. The ree expso mehod s ougrowh of Wlso s 46 reormlzo group pproch s dslled by Gllvo 41 d co-workers. I s he show h QED four Euclde dmesos s loclly Borel summble. Ther work s ruly remrkble d represes he frs forml proof h Euclde quum elecrodymcs c be reormlzed usg guge vr couererms. However, geerl, s orvl problem o reur from he Euclde regme o Mkowsk spce. The reur rp requres pplco of he Oserwlder Schrder recosruco heorem see Ref. 32. Ths heorem plces codos o he Euclde Gree s fucos whch guree lyc couo bck o he rel-me vcuum expeco vlues. Whe hese codos re fulflled, he Lehm, Symzk, d Zmmerm 22,23,32 reduco formuls my he be used o ob he S-mrx. For echcl resos, hey were o ble o drecly pply he Oserwlder Schrder heorem. They could sll ge bck o QED Mkowsk spce me by followg he mehods of Hepp 44 d Lowese d Speer. 47 However, ohg could be sd bou he covergece properes of her seres. B. Purpose I s cler h Dyso s use of me orderg ws he fudmel cocepul ool whch llowed hm o rele he Feym d Schwger Tomog heores. Ths ool hs ow become url pr of lmos every brch of physcs d s eve used prs of egeerg. Is mporce o he foudos of quum feld heory led Segl 48 o sugges h he defco of mhemcl meg for Feym s me-ordered operor clculus s oe of he mjor problems. A umber of vesgors hve emped o solve hs problem. Mrker d Wess 49 showed how he Feym orderg process could be doe formlly usg he heory of Bch lgebrs. Nelso 5 used Bch lgebrs o developed heory of opers s lere forml pproch. Ark, 51 moved by he work of Fujwr, used Bch lgebrs o develop ye oher forml pproch. Fujwr 52 hd erler suggesed h he Feym progrm could be mplemeed f oe used shee of u operors every po excep me, where he rue operor should be plced. Mslov 53 used he de of T-produc o formlly order operors d developed operol heory. Aoher mpor pproch o hs problem v he de of dex my be foud he works of ohso d Lpdus, see lso ohso, Lpdus, d DeFco. 57 Ths pper s pr of ew vesgo o he physcl d mhemcl foudos of relvsc quum heory. Our overll gol s o cosruc self-cosse relvsc quum heory of prcles d felds. For hs pper, we hve wo specfc objecves. Our frs d mjor objecve s o cosruc physclly smple d compuolly useful represeo heory for he Feym me-ordered operor clculus. A correc formulo d represeo heory for he Feym me-ordered operor clculus should les hve he followg desrble feures: 1 I should provde rspre geerlzo of curre lyc mehods whou scrfcg he physclly uve d compuolly useful des of Feym. 2 I should provde cler pproch o some of he mhemcl problems of relvsc quum heory. 3 I should expl he coeco wh ph egrls. I he course of hs lyss, ufco, d smplfco of he Feym Schwger Tomog heory, Dyso mde wo mpor suggesos cojecures. The frs cojecure cocered he dvergeces QED, whle he secod ws cocered wh he covergece of he reormlzed perurbo seres. I ddressg he problem of dvergeces, Dyso cojecured h hey my be due o delzed cocepo of mesurbly resulg from he fely precse kowledge of he spce me posos of prcles mpled by our Hmlo formul-

5 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 73 o whch leds o volo of he Heseberg ucery prcple. Ths po of vew c be rced drecly o he Bohr Rosefeld heory of mesurbly for feld operors d, ccordg o Schweber, 6 s ougrowh of Dyso s dscussos wh Oppehemer. I ddressg he reormlzed S-mrx, 16 Dyso suggesed h mgh be more resoble o expec he expso o be sympoc rher h coverge d gve physcl rgumes o suppor hs clm. The lck of cler mhemcl frmework mde mpossble o formule d vesge hs suggesos. Schweber 6 oes h Dyso mde wo oher well-kow cojecures. The overlppg dvergeces cojecure ws proved by Slm, 58 Wrd, 59 Mlls d Yg, 6 d Hepp. 61 Dyso s cojecure h cer Feym egrl coverges, ecessry for showg h he ulrvole dvergeces ccel o ll orders, ws proved by Weberg. 62 Our secod objecve s o provde proofs of he bove wo cojecures uder geerl codos h should pply o y formulo of quum feld heory whch does o bdo Hmlo geerors for ury soluo operors. The proof of he frs cojecure s, o some exe expeced, d s prl vdco of our belef he cossecy of quum elecrodymcs he sese h he ulrvole problem s cused by effec h s bsclly smple. Such resul s prly cped sce he effec c be mde o dspper v ppropre cuoffs. We lso defy specl codos uder whch he reormlzed perurbo seres my cully coverge. A proof of he bove-meoed cojecures s mplc, d s oe of he mjor chevemes of, cosrucve feld heory for he models suded. I fc, hese heores verfy sroger verso of he secod cojecure sce, s oed erler, he reormlzed perurbo seres s summble o he rue soluo. The work hs pper s boh geerlzo d smplfco of erler work h s eser d requres he wekes kow codos. We cosruc ew represeo Hlber spce d vo Neum lgebr for he Feym me-ordered operor clculus. I order o mke he heory pplcble o oher res, we develop usg semgroups of corcos d he Rem egrl. A corco semgroup o Hlber spce H c lwys be exeded o ury group o lrger spce H. Thus, for quum heory we my replce he semgroups by ury groups d ssume h our spce s H whou y loss udersdg. The Rem egrl c be esly replced by he operor-vlued Rem-complee egrl of Hesock 66 d urzwel, 67 whch geerlzes he Bocher d Pes egrls see Gll 63. Ths egrl s eser o udersd d ler compred o he Lebesgue or Bocher egrls, d provdes useful vrs of he sme heorems h hve mde hose egrls so mpor. Furhermore, rses from smple rspre geerlzo of he Rem egrl h ws ugh elemery clculus. Is usefuless he cosruco of Feym ph egrls ws frs show by Hesock, 68 d hs bee furher explored he rece book by Muldowey. 69 I Sec. I D we provde bref revew of he ecessry operor heory order o mke he pper self-coed. I Sec. II we cosruc fe esor produc Hlber spce d defe wh we me by me orderg. I Sec. III we cosruc me-ordered egrls d evoluo operors d prove h hey hve he expeced properes. I Sec. IV we defe wh s me by he phse sympoc he sese of Pocré for operors, d use o prove Dyso s secod cojecure for corco semgroups. We he dscuss codos uder whch he perurbo seres my be expeced o coverge. I Sec. V we ke phoogrph of rck lef by elemery prcle bubble chmber s prooype o coduc physcl lyss of wh s cully kow from experme. Ths pproch s used o rederve our me-ordered evoluo operor s he lm of probblsc sum over phs. We use o brefly dscuss our heory reloshp o he Feym ph egrl, d show h provdes geerl d url defo for he ph egrl h s depede of mesure heory d he spce of couous phs. The resuls from Sec. V re ppled o he S-mrx expso Sec. VI o provde formulo d proof of Dyso s frs cojecure. I prculr, we show h, wh our formulo, he ssumpo of precse me formo over prcle s rjecory roduces fe mou of eergy o he sysem ech po me. We use Dyso s orgl oo prly for

6 74. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry resos of oslg, bu lso o po ou wh we re o ble o expl wh our frmework. Also, sce ll reormlzo procedures re equvle, here s o loss. C. Operor heory I hs seco we esblsh oo d quoe some resuls from operor heory used he pper. Le H deoe seprble Hlber spce over C complex umbers, B(H) he se of bouded ler operors, d C(H) he se of closed desely defed ler operors o H. Defo 1.: A fmly of bouded ler operors U(,), defed o H s srogly couous semgroup (or C -semgroup) f 1 U(,)I, 2 U(s,)U(,)U(s,), d 3 lm U(,),H. U(,) s corco semgroup cse U(,)1. If we replce 2 by (2) U(,) U(,s)U(s,), s, he we cll U(,) srogly couous evoluo fmly. Defo 1.2: A desely defed operor H s sd o be mxml dsspve f ReH,, D(H), d R (IH)H rge of (IH). The followg resuls my be foud Goldse 7 or Pzy. 71 Theorem 1.2: Le U(,) bec -semgroup of corco operors o H. The 1 Hlm (U(,))/ exss for dese se. 2 R(z,H)(zIH) 1 exss for z d R(z, H) 1/z. Theorem 1.3: Suppose H s mxml dsspve operor. The H geeres uque C -semgroup U(,) of corco operors o H. Theorem 1.4: If H s desely defed wh boh H d H* dsspve, he H s mxml dsspve. II. INFINITE TENSOR PRODUCT VON NEUMANN ALGEBRAS I hs seco we defe me-ordered operors d cosruc he represeo spce whch wll be used Sec. III o develop our heory of me-ordered egrls d evoluo operors. Much of he merl hs seco ws developed by vo Neum 72 for oher purposes, bu s perfecly sued for our progrm. I order o see how url our pproch s, le H ˆ sh(s)deoe he fe esor produc Hlber spce of vo Neum, where H(s)H for s,b d ˆ deoes closure. If B(H ) s he se of bouded operors o H, defe B(H())B(H )by BHHH ˆ s I s H s I s,hbh, 2.1 where I s deoes dey operor, d le B # (H ) be he uform closure of he vo Neum lgebr geered by he fmly B(H()),E. If he fmly H()E s B(H), he he correspodg operors H()EB # (H )commue whe cg dffere mes: s HHsHsH. 2.1b Defo 2.: The smlles spce FD H whch leves he fmly H()E vr s clled Feym Dyso spce for he fmly. Ths s he flm. We eed he followg resuls bou operors o H. Theorem 2.1: vo Neum Ref. 72 The mppg T : B(H) B(H()) s somerc somorphsm of lgebrs. We cll T he me-orderg morphsm. Defo 2.2: The vecor s s s sd o be equvle o s s d we wre, f d oly f s s, s s

7 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 75 Here,, s s he er produc o H(s), d s udersood h he sum s megful oly f mos couble umber of erms re dffere from zero. Le H cl 1,, N closure, H, d le P deoe he projeco from H oo H. The spce H s kow s he complee esor produc geered by. The dels o complee esor produc spces s well s proofs of he ex wo heorems my be foud vo Neum. 72 Theorem 2.3: The bove-defed relo s equvlece relo o H d 1 f s o equvle o, he H H (.e., H H ; 2 f s s occurs for mos fe umber of s, he s s s s ; 3 f TB # (H ), he P TTP so h P TB # (H ). The secod codo Theorem 2.3 mples h, for ech fxed s s, here s ucouble umber of s s equvle o, whle he hrd codo mples h every bouded ler operor o H resrcs o bouded ler operor o H for ech. We c ow cosruc our flm FD. Le e N deoe rbrry ordered complee orhoorml bss c.o.b for H. For ech E,N, le e e, E E e, d defe FD o be he complee esor produc geered by he vecor E. Seg FD 1 FD, wll be cler Sec. III h FD s oe of fe umber of he url represeo spces for Feym s me-ordered operor heory. I should be oed h FD s oseprble Hlber spce budle over,b. However, s o hrd o see h ech fber s somorphc o H. I order o fcle he proofs Sec. III, we eed explc bss for ech FD. To cosruc, fx d le f deoe he se of ll fucos j()e mppg E N such h j() s zero for ll bu fe umber of. Le I( j) j()e deoe he fuco j d se E I( j) E e, j() wh e, e d j()k e,k e k. Theorem 2.4: The se E I( j) I( j) f s (c.o.b) for ech FD. For ech, F, se I( j),e I( j), b I( j),e I( j), so h I( j) I( j)f E I( j), b I( j) I( j)f E I( j) d, I( j) I( j)f b I(k) E I( j),e I(k). Now, E I( j),e I(k) e,i( j),e,i(k), uless j()k(), E, so h, I( j) I( j)f b I( j). We eed he oo of exchge operor. Theorem 2.6 s Ref. 63. Defo 2.5: A exchge operor E, s ler mp defed for prs,,b such h: 1 E,:B(H()) B(H()) oo, 2 E,sEs,E,, 3 E,E,1, 4 f s,, he E,H(s)H(s),H(s)B(H(s)). Theorem 2.6: 1 E, exss d s Bch lgebr somorphsm o B # (H ). 2 Es,sE,E,Es,s for dsc prs (s,s) d (,) E.

8 76. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry III. TIME-ORDERED INTEGRALS I hs seco we cosruc me-ordered egrls d evoluo operors for fxed fmly H()EC(H) of geerors of corco semgroups o H. We ssume h, for ech, H() d H*() re dsspve so h he fmly s mxml dsspve for ech. Ihe followg dscusso we dop he oo: e.o.v.: excep for mos oe s vlue; e.f..v. excep for mos fe umber of s vlues; d.s.c.: lmos surely d he excepol se s mos couble. The s vlue referred o s our fxed ervl E. For he gve fmly H()EC(H), defe exph() by exph ˆ I s exph I s, s[b,) s(,] 3.1 d se H z ()zh()r(z,h()), z, where R(z,H())(zI H()) 1 s he resolve of H(). I s kow h H z () geeres uformly bouded corco semgroup d lm z H z ()H() for D(H()). Theorem 3.1: Suppose for ech, H()EC(H) geeres srogly couous corco semgroup o H. The H()H z ()H z ()H(),D,where D deoes he dom of he fmly H()E, d 1 The fmly H z ()E geeres uformly bouded corco semgroup o FD for ech d lm z H z ()H(),D. 2 The fmly H()EC(H ) geeres srogly couous corco semgroup o FD so h H()EC(FD ). Proof: The proof of 1 s sdrd. Noe h H z ()z 2 R(z,H())zI d R(z,H()) 1/z, soexpsh z () expszexpsz 2 R(z,H()) 1. Now recll h lm z zr(z,h()),fd, so h, for D, we hve h lm z H z () lm z zh()r(z,h())lm z zr(z,h())h()h(). To prove 2, frs recll Gll 73 h esor produc orm,, s uform f, for ˆ se T s B(H ), ˆ T s T s. se se 3.2 Usg he uform propery of he Hlber spce esor produc orm, s esy o see h exph() s corco semgroup. To prove srog couy, we eed o defy dese core for he fmly H() EC(FD ). Le D 1 deoe he ordered esor produc of he doms of he fmly H()EC(H), so h D 1 D D 1 sedhs 1 s s DHs,sE. s 3.3 I s cler h D 1 s dese core H,soD D 1 FD s dese core FD. Usg our sdrd bss, f,d, I( j) I( j) E I( j), I(k) b I(k) E I(k) ; he, sce (exph()i ) s vr o FD d I s he dey o FD, we hve exphi, I( j) I(k) I( j) b I(k) exphi E I( j),e I(k), 3.4

9 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 77 d exphi E I( j),e I(k) e s, j(s) s,e s,k(s) exphie, j(),e,k() exphie, j(),e, j() e.o.v. exphie,e e.f..v., 3.4b exphi, I( j) I( j) b I( j) exphie,e.s.c.. 3.4c Sce ll sums re fe, we hve lm exphi, I( j) b I( j) I( j) lm exphie,e.s.c.. 3.4d The f d oly f pr s ow cler. Sce exph() s bouded o H d he bovemeoed lm exss o D whch s dese FD, we see h exph() exeds o corco semgroup o FD. Now use he fc h, f bouded semgroup coverges wekly o he dey, coverges srogly see Pzy, 71 p. 44. We ow ssume h he fmly H()EC(H) hs wek Rem egrl Q b H()dC(H). I follows h he fmly H z ()EB(H) lso hs wek Rem egrl Q z b H z ()db(h). Le P be sequece of pros of E so h he mesh (P ) s. Se Q z, l1 H z l l, m Q z,m q1 H z s qs q, Q z, l1 H z l l, m Q z,m q1 H z s qs q, d Q z Q z, Q z,m, Q z Q z, Q z,m Le,D ; I( j) I( j) L E I( j), L M I(k) b I(k) E I(k). The we hve: Theorem 3.2 frs fudmel heorem for me-ordered egrls : Q z, I( j) I( j) b I( j) Q z e,e.s.c Noe: The form of 3.5 s que geerl sce Q z c be replced by oher erms whch lso gve rue reloshp. For exmple, s esy o show h he fmly H z ()E s wekly mesurble, wekly couous, wekly dffereble, ec., f d oly f he sme s rue for he fmly H z ()E. Proof: we om he upper lm. Now Q z, I( j) I(k) I( j) b I(k) Q z E I( j),e I(k)

10 78. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry Q z E I( j),e I(k) l1 l1 q1 l1 m l H z le I( j) l l e, j() s q s q m,e I(k) q1,e,k() e, j() l H z le l, j( l ) Q z e,e e.f..v.. H z le l, j( l ),e,k() s q H z s qe I( j) H z s qe s q, j(s q) m,e l, j( l ) q1,e l,k( l ),E I(k),e s q,k(s q) s q H z s qe s q, j(s q),e s q, j(s q) Ths resul leds o 3.5. Theorem 3.3 secod fudmel heorem for me-ordered egrls : If he fmly H z ()E hs wek Rem (Rem-complee) egrl, he 1 he fmly H z ()EB # (FD ) hs wek Rem (Rem-complee) egrl. 2 If, ddo, we ssume h for ech wh 1, sup E (Hz s 2 H z s, 2 )ds, 3.6 he he fmly H z ()E hs srog egrl Q z, H z (s)ds whch geeres uformly couous corco semgroup o FD. Noes: 1 I s suffce h sup E (H z (s)e 2 H z (s)e,e 2 )ds for ech. 2 Codo 3.6 s ssfed f H z (s)e 2 s Lebesgue egrble for ech. I hs cse, we replce he Rem egrl by he Rem-complee egrl. 3 I geerl, he fmly H z ()E eed o be Bocher or Pes egrl, s s o requred h H z (),H z (), be squre Lebesgue egrble. I s possble h b H z () 2 d d b H z (), 2 d, whle 3.6 s zero. For exmple, le f () be y obsoluely squre egrble fuco d se H z () f ()I. The he bove-meoed possbly holds whle (H z (s) 2 H z (s), 2 )ds for ll E. Proof: The proof of 1 s esy d follows from 3.5. To see h 3.6 mkes Q z srog lm, le D. The Q z,,q z, I( j),i(h) I( j) ā I(h) k,m k m H z s k E I( j),h z s m E I(h) I( j) I( j) e sm, j(s m ) I( j) 2 km,h z s m e sm, j(s k ) 2 I( j) k k m H z s k e sk, j(s k ) k 2 H z s k e sk, j(s k ),e sk, j(s k ),H z s k e sk, j(s k ). 3.7 Ths c be rewre s

11 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 79 Q z, 2 I( j) 2 I( j) Q z, e,e 2 k k 2 H z s k e 2 H z s k e,e 2.s.c The ls erm c be wre s k, k 2 H z s k e 2 H z s k e,e 2 M sup E H z se 2 H z se,e 2 ds, where M s cos d s he mesh of P, wh s. Now oe h H z ()E H z ()e d H z ()E,E H z ()e,e e.o.v so h sup H z se 2 H z se,e ds 2 E We c ow use 3.6 o ge Q z, 2 I( j).s.c.. I( j) E sup H z se 2 H z se,e 2 ds.s.c.. 2 Q z,e,e 2 Msup H z E 2 H z E,E 2 ds. Thus, Q z, coverges srogly o Q z o D d hece hs srog lm o FD. To show h Q z, geeres uformly couous corco, suffces o show h Q z, d Q z *, re dsspve. Le be D, he Q z,, I( j) I( j) b I( j) Q z e,e.s.c. d, sce Q z,, s dsspve for ech, we hve Q z,e,e Q z,,e,e Q z,q z,,e,e Q z,q z,,e,e. Leg, we ge Q z,e,e, so h Q z,,. The sme rgume pples o Q z *,. Sce Q z, s dsspve d desely defed, hs bouded dsspve closure o FD. I should be oed h he heorem s sll rue f we llow he pproxmg sums for codo 3.6 o dverge bu order less h 1, 1, h s, wh sup H z E 2 H z E,E 2 ds, k, k 2 H z s k e 2 H z s k e,e 2 M. We lso oe h Q z, 2 I( j) I( j) 2 Q z e,e 2.s.c. 3.9

12 8. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry eher of he bove cses. Ths represeo mkes esy o prove he ex heorem. Theorem 3.4: 1 Q z,sq z s,q z,(.s.c.), 2 slm h (Q z h,q z,)/hslm h (Q z h,)/hh z ()(.s.c.), 3 slm h Q z h, (.s.c.), 4 slm h expq z h,i (.s.c.),. Proof: I ech cse, suffces o prove he resul for D. To prove 1, use Q z,sq z s, 2 I( j) I( j) I( j) 2 Q z,sq z s,e,e 2.s.c. I( j) 2 Q z,e,e 2 Q z, 2.s.c.. To prove 2, use 1 o ge Q z h,q z,q z h,(.s.c.), so h lm Q zh, h h 2 I( j) I( j) 2 lm h Q zh, h e,e 2 H z 2.s.c.. The proof of 3 follows from 2, d he proof of 4 follows from 3. Theorem 3.5: Suppose h lm z Q z,,q,, exss for dese se H (wek covergece). The: 1 Q, geeres srogly couous corco semgroup o H, 2 lm z Q z,q, for D d Q, s he geeror of srogly couous corco semgroup o FD, 3 Q,sQs,Q,(.s.c.), 4 lm h (Qh,Q,)/hlm h (Qh,)/hH() (.s.c.), 5 lm h Qh, (.s.c.), d 6 lm h expqh,(.s.c.),. Proof: The proofs re esy. For 1, frs oe h Q, s closble d use Q,, Q z,,q,q z,,q,q z,, d le z. The do lkewse for,q*, o ge h Q, s mxml dsspve. To prove 2, use 3.9 he form Q z,q z, 2 I( j)) I( j) 2 Q z,q z,e,e 2.s.c.. s Ths proves h Q z, Q,. Sce Q, s desely defed, s closble. The sme mehod s bove shows h s mxml dsspve. Proofs of he oher resuls follow he mehods of he prevous heorem. Sce Q, d Q z, geere corco semgroups, se U,expQ,, U z,expq z,, for E. They re evoluo operors d he followg heorem s slgh modfco of resul due o Hlle d Phllps, 74 kow s he secod expoel formul. Theorem 3.6: If Q,wQ, s he geeror of srogly couous corco semgroup, d U w,expwq,, he, for ech d D(Q,) 1, we hve (where w s prmeer)

13 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 81 U w, I k1 wq, 1 ww!! Q, 1 U,d. 3.1 Proof: The proof s esy. Sr wh U z w,i w Q z,u z,d d use egro by prs o ge h U z w,i wq z, wwq z, 2 U z,d. I s cler how o ge he h erm. Flly, le z o ge 3.1. Theorem 3.7. If b, 1 lm z U z,u,, FD, 2 / U z,h z ()U z,u z,h z (), FD, d 3 / U,H()U,U,H(), D(Qb,)D. Proof: To prove 1, use he fc h H z () d H() commue log wh U,U z, 1d/dse sq[,] e (1s)Q z [,] ds so h To prove 2, use so h, 1 se sq[,] e (1s)Q z [,] Q,Q z, ds, U,U z,q,q z,. U z h,u z,u z,u z h,iu z h,iu z,, U z h,u z, h U z, U zh,i. h Now se z U z, d use 3.1 wh 1 d w1 o ge: so h I follows h U z h, z I Q z h, 11U z h,q z h, 2 d z, U z h,i h U zh,i h z H z z z H z z Q zh, h z H z z 1 1U z h, Q zh, 2 h z d. Q zh, h z H z z 1 Q zh, 2 2 h z.

14 82. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry The resul ow follows from Theorem d 3. To prove 3, oe h H z ()H()zR(z,H())zR(z,H())H(), so h zr(z,h()) commues wh U, d H(). Now show h H z U z,h z U z,u z,u z,hzrz,h zrz,hh, z,z, so h, for DQb,, H z U z, HU, U,. The prevous heorems form he core of our pproch o he Feym operor clculus. Our heory pples o boh hyperbolc d prbolc equos. I he coveol pproch, hese wo cses requre dffere mehods see Pzy 71. I s o hrd o show h he requremes mposed hese cses re sroger h our codo of wek egrl. Ths wll be dscussed ler pper devoed o he geerl problem o Bch spces. IV. PERTURBATION THEORY Defo 4.1: The evoluo operor U w,expwq, s sd o be sympoc he sese of Pocré f, for ech d ech D(Q,) 1, we hve lm w w (1) U w, k1 wq, k k! Q,1 1!. 4.1 Ths s he operor verso of sympoc expso he clsscl sese, bu here Q, s ubouded operor. As oed erler, Dyso 16 lyzed he reormlzed perurbo expso for quum elecrodymcs d suggesed h cully dverges. He cocluded h we could, bes, hope h he seres s sympoc. Hs rgumes were bsed o o compleely covcg physcl cosderos, bu o precse formulo of he problem ws possble h me. However, he clculos of Hurs, 75 Thrrg, 76 Peerm, 77 d ffe 78 for specfc models ll suppor Dyso s coeo h he reormlzed perurbo seres dverges. I hs rece book 79 pp , Dyso s vews o he perurbo seres d reormlzo re reered:... spe of ll he successes of he ew physcs, he wo quesos h defeed me 1951 rem usolved. Here, he s referrg o he queso of mhemcl cossecy for he whole reormlzo progrm, d our bly o relbly clcule ucler processes quum chromodymcs. For oher dels d refereces o ddol works, see Schweber, 6,8 Wghm, 81 d Z-us. 82 The geerl cosruco of physclly smple d mhemclly ssfcory formulo of quum elecrodymcs s sll ope problem. The ex heorem esblshes Dyso s secod cojecure uder codos h would pply o y fuure heory h does o requre rdcl deprure from he prese foudos of quum heory ury soluo operors. I lso pples o he reormlzed expsos some res of codesed mer physcs where he soluo operors re corco semgroups. Theorem 4.2: Suppose he codos for Theorem 3.5 re ssfed. The: 1 U w,expwq, s sympoc he sese of Pocré. 2 For ech d ech D(Q,) 1, we hve

15 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 83 w k k1 ds1 s 1ds2 s k1dsk Hs 1 Hs 2 Hs k ww d ds1 s 1ds2 s ds1 Hs 1 Hs 2 Hs 1 U s 1,, where ()U w,. Proof: From 3.1, we hve 4.2 U w, k wq, 1 ww Q, 1 U!!,d, so h w (1) U w, k wq, k k! Replce he rgh-hd sde by 1 w 1! w(1) w d U,Q, 1. I 1 w 1! w(1) w du z,u,u z,q, 1. Now, expd he erm U z, wo-erm Tylor seres bou zero o ge U z,i Q z,r z. Pu he bove I, compue he elemery egrls showg h oly he I ozero vlue of 1/(1) whe w. The le z o ge erm gves lm 1w (1) w dw U,Q, 1 Q, 1. w Ths proves h U,expQ, s sympoc he sese of Pocré. To prove 4.2, le D(Q,) 1 for ech k1, d use he fc h Dollrd d Fredm 83 Q z, k k Hz sds k! ds1 s 1ds2 s k1ds H z s 1 H z s 2 H z s k. Leg z gves he resul. Our codos re very wek. For exmple, he rece work of Tg d L 84 requred h H() be Lebesgue egrble. There re well-kow specl cses whch he perurbo seres my cully coverge o he soluo. Ths c hppe, for exmple, f he geeror s bouded or f s lyc some secor. More geerlly, whe he geeror s of he form H()H ()H (), where H () s lyc d H () s some resoble perurbo, whch eed o be bouded, here re codos h llow he erco represeo o hve coverge Dyso expso. These resuls c be formuled d prove our formlsm. However, he proofs re esselly he sme s he sdrd cse so we wll prese hem ler pper devoed o he operor clculus o Bch spces. The rece book by Egel d Ngel 85 provdes some ew resuls hs geerl re. 4.3

16 84. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry There re lso cses where he reormlzed seres my dverge, bu sll respod o some summbly mehod. Ths pheomeo s well-kow clsscl lyss. I feld heory, hgs c be much more complced. A good dscusso, wh refereces, c be foud he revew by Wghm 81 d he book by Glmm d ffe. 35 V. SUM OVER PATHS I hs seco we frs revew d mke dsco bewee wh s cully kow d wh we hk we kow bou he foudos for our physcl vew of he mcro-world. The objecve s o provde he bckgroud for umber of physclly moved posules h wll be used o develop heory of mesureme for he mcro-world suffce for our purposes. Ths wll llow us o rele he heory of Secs. III d IV o Feym s sum over phs pproch d prove Dyso s secod cojecure. Ths seco dffers from he prevous oes h we shf he oreo d perspecve from h of mhemcl physcs o h of heorecl physcs. I spe of he eormous successes of he physcl sceces he ps ceury, our formo d udersdg bou he mcro-world s sll rher meger. I he mcro-world we re que comforble wh he vew h physcl sysems evolve couously me d our resuls jusfy hs vew. Ideed, he success of couum physcs s he bss for lrge pr of our echcl dvces he weeh ceury. O he oher hd, he sme vew s lso held he mcro-level d, hs cse, our poso s o very secure. The bly o mesure physcl eves couously me he mcro-level mus be cosdered belef whch, lhough covee, hs o plce scece s pror cosr. I order o esblsh perspecve, le us cosder hs belef wh he coex of ssfcory, d well-jusfed heory, Brow moo. Ths heory les he erfce bewee he mcrod he mcro-worlds. Some preseos of hs heory he creful oes mke dsco bewee he mhemcl d he physcl foudos of Brow moo d h dsco s mpor for our dscusso. Whe Ese 86 beg hs vesgo of he physcl ssues ssoced wh hs pheomeo, he ws forced o ssume h physcl formo bou he se of Brow prcle poso, velocy, ec. c oly be kow me ervls h re lrge compred wh he me me bewee moleculr collsos. I s kow h, uder orml physcl codos, Brow prcle receves bou 1 21 collsos per secod. Weer ook he mhemcl sep d ssumed h hs me me bewee collsos could be mde zero, hus provdg mhemcl Brow prcle. Ths correspods physclly o he ssumpo h he ro of he mss of he prcle o he frco of he flud s zero he lm see Weer e l. 87. From he physcl po of vew, use of Weer s delzo of he Ese model ws o ssfcory sce led o problems of ubouded ph legh d odfferebly ll pos. The frs problem s physclly mpossble whle he secod s physclly uresoble. Of course, he delzo hs ured ou o be que ssfcory res where he formo requred eed o be deled, such s lrge prs of elecrcl egeerg, chemsry, d he bologcl sceces. Orse d Uhlebeck 88 ler cosruced model h gves he Ese vew sympoclly bu, smll-me regos, s equvle o he ssumpo h he prcle rvels ler ph bewee collsos. Ths model provdes fe ph legh d dfferebly. The heory ws ler delzed by Doob. 89 Wh we do kow s h he very ure of he lqud se mples collecve behvor mog he molecules. Ths mes h we do o kow wh ph he prcle rvels bewee collsos. However, sce he ools d mehods of lyss requre some form of couy, some such bewee observo ssumpos mus be mde. I s cler h he eed for hese ssumpos s mposed by he vlble mhemcl srucures wh whch we mus represe physcl rely s model. Theorecl scece cocers self wh he cosruco of mhemcl represeos of cer resrced poros of physcl rely. Vrous reds d phlosophes h re prevle he me emper hese cosrucs. A cosse heme hs bee he ques for smplcy. Ths requreme s bor ou of he url eed o resrc models o he mmum umber of vrbles, reloshps, cosrs, ec., whch gve ssfcory ccou of kow expermel resuls

17 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 85 d possbly llow he predco of hereofore ukow cosequeces. Oe mpor oucome of hs pproch hs bee o mplcly elme ll referece o he bckgroud wh whch physcl sysems evolve. I he mcro-world, such co co be jusfed whou pror vesgo. We propose o replce he use of mhemcl coorde sysems by physcl coorde sysems order o prlly remedy hs problem. We deoe physcl coorde sysem me by R p 3 (). Ths coorde sysem s ched o observer cludg mesurg devces d s evsoed s R 3 plus y bckgroud effecs, eher locl or ds, whch ffec he observer s bly o ob precse del expermel formo bou physcl rely. Ths ur ffecs our observer s bly o cosruc precse del represeos d mke precse predcos bou physcl rely he mcro-world. More specfclly, cosder he evoluo of some mcro-sysem o he ervl E,b. Physclly hs evoluo mfess self s curve o X, where E R p 3 X. Thus, rue physcl eves occur o X where cul expermel formo s modfed by flucuos R p 3 (), d by he erco of he mcro-sysem wh he mesurg equpme. Bsed o he success of our models, we kow h such smll chges re he ose rego, d hey hve o effec o our predcos for mcro-sysems. However, here s o physcl reso o beleve h he effecs wll be smll o mcro-sysems. I erms of our heorecl represeos, we re forced o model he evoluo of physcl sysems erms of wve fucos, mpludes, d/or operor-vlued dsrbuos, ec. There re hus wo spces, he physcl spce of evoluo for he mcro-sysem d he observer s spce of obble formo cocerg hs evoluo. The lck of dsco bewee hese wo spces seems o be he cuse for some of he cofuso d lck of physcl clry. For exmple, my be perfecly correc o ssume h prcle rvels couous ph o X. However, he ssumpo h he observer s spce of obble formo cludes fesml spce me kowledge of hs ph s compleely ufouded. Ths leds o our frs posule: Posule 1: Physcl rely s couous process me. We hus ke hs vew, fully recogzg h experme does o provde couous formo bou physcl rely, d h here s o reso o beleve h our mhemcl represeos co precse formo bou he couous spce me behvor of physcl processes hs level. Sce he dve of he specl heory of relvy, here hs bee much dscusso bou eves, whch geerlly mes po R 4 wh he Mkowsk merc. I erms of rel physcs, hs s fco whch s frequely useful for resos of preseo bu so wdely used h, o vod cofuso, s ppropre o defe wh we me by physcl eve. Defo 5.1: A physcl eve s se of physcl chges gve sysem h c be verfed drecly by experme or drecly v subseque chges, where coclusos re bsed o pror greed-upo model of he physcl process. Ths defo correspods more closely o wh s me by physcl eves. I explcly recogzes he evoluo of scefc ferece d he eed for geerl greeme bou wh s beg observed bsed o specfc models. Before coug, wll be helpful o hve prculr physcl pcure md h mkes he bove-preseed dscusso explc. For hs purpose, we ke hs pcure o be phoogrph showg he rck lef by -meso bubble chmber d ke serously he mou of formo vlble. I prculr, we ssume h he followg reco occurs:. We furher ssume h he oreo of our phoogrph s such h he -meso eers o he lef me d he rcks lef by he -meso dspper o he rgh me T, where

18 86. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry FIG. 1. Idel pcure of he reco. T s of he order of 1 3 s, he me exposure for phoogrphc flm. Alhough he euro does o pper he phoogrph, we lso clude rck for. I Fg. 1 we prese smplfed pcure of hs phoogrph. We hve drw he phoogrph s f we couously see he prcles he pcure. However, experme oly provdes us dvdul bubbles, whch do o ecessrly overlp, from whch we mus exrc physcl formo. A more ccure hough sll o relsc depco s gve Fg. 2. Le us ssume h we hve mgfed poro of our phoogrph o he exe h we my dsgush he dvdul bubbles creed by he -meso s psses hrough he chmber. I Fg. 3, we prese smplfed model of djce bubbles. Posule 2: We ssume h he ceer of ech bubble represes he verge kowble effec of he prcle symmerc me ervl bou he ceer. By verge kowble effec, we me he verge of he physcl observbles. I Fg. 3, we cosder he exsece of bubble me j o be cused by he verge of he physcl observbles over he me ervl j1, j, where j1 (1/2) j1 j d j (1/2) j j1. Ths posule requres some jusfco. I geerl, he resoluo of flm d he relxo me for dsc bubbles he chmber vpor re lmed. Ths mes h f he -meso crees wo bubbles h re closely spced me, he bubbles my colesce d pper s oe. If hs does o occur, s sll possble h he flm wll record he eve s oe bubble becuse of s bly o resolve eves s such smll me ervls. Le us ow recogze h we re delg wh oe phoogrph so h, order o ob ll vlble formo, we mus lyze lrge umber of phoogrphs of he sme reco obed uder smlr codos pre-prepred ses. I s cler h he umber of bubbles d he me plceme of he bubbles wll vry depedely of ech oher from phoogrph o phoogrph. Le 1 deoe he verge me for he pperce of bubble he flm. Posule 3: We ssume h he umber of bubbles y flm s rdom vrble. Posule 4: We ssume h, gve h bubbles hve ppered o flm, he me posos of he ceers of he bubbles re uformly dsrbued. FIG. 2. More ccure pcure of he reco.

19 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 87 FIG. 3. Hghly mgfed vew showg dvdul bubbles. Posule 5: We ssume h N(), he umber of bubbles up o me gve flm, s Posso-dsrbued rdom vrble wh prmeer. To move Posule 5, recll h j s he me ceer of he jh bubble d 1 s he verge expermelly deermed me bewee bubbles. The followg resuls c be foud Ross. 9 Theorem 5.1: The rdom vrbles j j j1 ( ) re depede declly dsrbued rdom vrbles of expoel ype wh me 1, for 1 j. The rrvl mes 1, 2,..., re o depede, bu her desy fuco c be compued from Prob 1, 2,..., Prob 1 Prob 2 1 Prob 1, 2,..., We ow use Theorem 5.1 o coclude h, for k1, Prob k 1, 2,..., k1 Prob k k1. 5.1b We do o kow hs codol probbly. However, he url ssumpo s h, gve h bubbles pper, hey re eqully uformly dsrbued o he ervl. We c ow cosruc wh we cll he expermel evoluo operor. Assume h he codos for Theorem 3.5 re ssfed d h he fmly 1, 2,..., represes he me posos of he ceers of bubbles our flm of Fg. 3. Se d defe Q E 1, 2,..., by Q E 1, 2,..., j1 j1 j E j,shsds. 5.2 Here,, j (1/2) j j1 for 1 j, d E j,s s he exchge operor defed Sec. II. The effec of our exchge operor E j,s s o cocere ll formo coed j1, j j. Ths s how we mpleme our posule h he kow physcl eve of he bubble me j s due o verge of physcl effecs over j1, j wh formo cocered j. We c rewre Q E 1, 2,..., s Q E 1, 2,..., j1 j 1 j j E j,shsds. 5.2b j1 Thus, we deed hve verge s requred by Posule 2. The evoluo operor s gve by U 1, 2,..., exp j1 For FD, we defe he fuco UN(), by j 1 j j E j j1,shsds. 5.3

20 88. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry UN,U 1, 2,..., N(). 5.3b The fuco UN(), s FD -vlued rdom vrble, whch represes he dsrbuo of he umber of bubbles h my pper o our flm up o me. I order o rele UN(), o cul expermel resuls, we mus compue s expeced vlue. Usg Posules 3, 4, d 5, we hve Ū,EUN, EUN,NProbN, 5.4 EUN,N d 1 d d U,..., 1 Ū,, d ProbN exp.! 5.6 The egrl 5.4 cs o dsrbue uformly he me posos j over he successve ervls, j1,1 j, gve h j1 hs bee deermed. Ths s url resul gve our lck of kowledge. The egrl 5.4 s of heorecl vlue bu s o esy o compue. Sce we re oly eresed wh hppes whe, d s he me umber of bubbles he flm me s, we c ke j ( j/), 1 j j / for ech. We c ow replce Ū, by U,, d wh hs udersdg, we coue o use j, so h U,exp j1 j1 j E j,shsds. 5.5b We defe our expermel evoluo operor U, by U, expu!,. 5.4b We ow hve he followg resul, whch s cosequece of he fc h Borel summbly s regulr. Theorem 5.4: Assume h he codos for Theorem 3.5 re ssfed. The lm Ū, lm U,U,. 5.7 Sce 1, hs mes h he verge me bewee bubbles s zero he lm so h we ge couous ph. I should be observed h hs couous ph rses from vergg he sum over fe umber of dscree phs. The frs erm 5.4b correspods o he ph of -meso h creed o bubbles.e., he phoogrph s blk. Ths eve hs probbly exp whch pproches zero s. The h erm correspods o he ph of -meso h creed bubbles, wh probbly () /!exp, ec. Before dervg physcl reloshp, le P;s, fs d, for s, defe s

21 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I 89 s P;s,e k k, 5.8 k! where s s he grees eger s. We c ow wre U, s U, lm ds P;s,U s s,, 5.9 s U s s,exp j1 j E j j1,uhudu. Equo 5.9 mes h we ge boh sum over phs d probbly erpreo for our formlsm. Ths llows us o gve ew defo for ph egrls. Suppose he evoluo operor U, hs kerel, x(),;x(),, such h 1 x(),;x(s),s R 3x(),;x(s),sx(s),s;x(),dx(s), d 2 U, R 3x(),;x(),dx(). The, from Eq. 5.9, we hve h s U, lm ds P;s, j1 s x R 3 j, j ;x j1, j1 j j1 dx j1. Thus, wheever we c ssoce kerel wh our evoluo operor, he me-ordered verso lwys provdes well-defed ph-egrl s sum over phs. The defo does o drecly deped o he spce of couous phs d s depede of heory of mesure o fe dmesol spces. Feym suggesed h he operor clculus ws more geerl, hs book wh Hbbs 91 see pp VI. THE S-MATRIX The objecve of hs seco s o provde formulo of he S-mrx h wll llow us o vesge he sese whch we c beleve Dyso s frs cojecure. A he ed of hs secod pper o he reloshp bewee he Feym d Schwger Tomog heores, he explored he dfferece bewee he dverge Hmlo formlsm h oe mus beg wh d he fe S-mrx h resuls from reormlzo. He kes he vew h s cors bewee rel observer d fcous del observer. The rel observer c oly deerme prcle posos wh lmed ccurcy d lwys ges fe resuls from hs mesuremes. Dyso he suggess h... The del observer, however, usg o-omc pprus whose loco spce d me s kow wh fe precso, s mged o be ble o dsegle sgle feld from s ercos wh ohers, d o mesure he erco. I coformy wh he Heseberg ucery prcple, c perhps be cosdered physcl cosequece of he fely precse kowledge of prcle loco llowed o he del observer, h he vlue obed whe he mesures he erco s fe. He goes o o remrk h, f hs lyss s correc, he problem of dvergeces s rbuble o delzed cocep of mesurbly. I order o explore hs de, we work he erco represeo wh obvous oo. Replce he ervl, by T,T, H() by (/)H I (), d our expermel evoluo operor U T,T by he expermel scerg operor S T,T, where S T,T 2T exp2ts! T,T, 6.1

22 9. Mh. Phys., Vol. 43, No. 1, ury 22 T. L. Gll d W. W. Zchry S T,Texp / j1 j1 j E j,sh I sds, 6.2 d H I () R 3H I (x(),)dx() s he erco eergy. We follow Dyso for cossecy see lso he dscusso, so h mc 2 s he mss couer-erm desged o ccel he self-eergy dvergece, d H I x,ea x, x, x,mc 2 x,x,. 6.3 We ow gve physcl erpreo of our formlsm. Rewre Eq. 6.1 s S T,T 2T! exp / j1 j1 j E j,sh I si ds. 6.4 I hs form, s cler h he erm I hs physcl erpreo s he bsorpo of phoo eergy of mou ech subervl j, j1 cf. Mo d Mssey 92. Whe we compue he lm, we ge he sdrd S-mrx o T,T. I follows h we mus dd fe mou of phoo eergy o he mhemcl descrpo of he expermel pcure ech po me order o ob he sdrd scerg operor. Ths s he ulrvole dvergece d shows explcly h he rso from he expermel o he del scerg operor requres h we llume he prcle hroughou s ere ph. Thus, ppers h we hve, deed, voled he ucery relo. Ths s furher suppored f we look he form of he sdrd S-mrx: T ST,Texp / HI T sds, 6.5 d oe h he dfferel ds he expoe mples perfec fesml me kowledge ech po, srogly suggesg h he eergy should be olly udeermed. If volo of he Heseberg ucery relo s he cuse for he ulrvole dvergece he, s s vrce relo, wll o pper frs order perurbo bu should show up ll hgher-order erms. O he oher hd, f we elme he dverge erms secod order, we would expec our mehod o preve hem from pperg y hgher order erm of he expso. The fc h hs s precsely he cse quum elecrodymcs s cler verfco of Dyso s cojecure. If we llow T o become fe, we oce g roduce fe mou of eergy o he mhemcl descrpo of he expermel pcure, s hs s lso equvle o precse me kowledge fy. Of course, hs s he well-kow frred dvergece d c be elmed by keepg T fe see Dhme e l. 93 or roducg smll mss for he phoo see Feym, 12 p If we hold fxed whle leg T become fe, he expermel S-mrx kes he form: S,exp / j1 j1 j E j,sh I j1 sds, j1, j,, j Ths form s eresg sce shows how mml me elmes he ulrvole dvergece. Of course, hs s o uexpeced, d hs bee kow les sce Heseberg 94 roduced hs fudmel legh s wy roud he dvergeces. Ths ws prelude o he vrous lce pproxmo mehods. The revew by Lee 95 s eresg hs regrd. I closg hs seco, we record our exc expso for he S-mrx o y fe order. Wh ()D(Q,) 1, we hve

23 . Mh. Phys., Vol. 43, No. 1, ury 22 Foudos for relvsc quum heory. I S, k k ds1 s 1 ds2 s k1dsk H I s 1 H I s 2 H I s k d ds1 s 1 ds2 s ds1 H I s 1 91 H I s 2 H I s 1 S s 1,. 6.7 I follows h heorecl sese we c cosder he sdrd S-mrx expso o be exc, whe ruced y order, by ddg he ls erm of Eq. 6.7 o gve he remder. Ths resul lso mes h, wheever we c cosruc exc operurbve soluo, lwys mples he exsece of perurbve soluo vld o y order. However, geerl, oly prculr cses c we kow f he seres some whou he remder pproxmes he soluo. VII. CONCLUSION I hs pper we hve show how o cosruc url represeo Hlber spce for Feym s me-ordered operor clculus. Ths spce llows us o cosruc he me-ordered egrl d evoluo operor propgor uder he wekes kow codos. Usg he heory, we hve show h he perurbo expso relev o quum heory s sympoc he sese of Pocré. Ths provdes precse formulo d proof of Dyso s secod cojecure 16 h, geerl, we c oly expec he expso o be sympoc. Our vesgo o he exe h our couous models for he mcro-world fhfully represe he mou of formo vlble from experme hs led o dervo of he meordered evoluo operor more physcl wy. Ths pproch mde possble o prove h he ulrvole dvergece s cused by volo of he Heseberg ucery relo ech po me, hus prlly cofrmg Dyso s frs cojecure. We used Dyso s orgl oo so s o explcly exhb he couer-erm ecessry o elme he self-eergy dvergece h occurs QED. Ths dvergece s o ccoued for d s ousde he scope of he curre vesgo. Thus, wh our prese frmework, we co sy h ll he dvergeces rse from our dsregrd of some smple physcs, d re o he resul of deeper problems. Thus, Dyso s cocers bou he mhemcl cossecy of quum elecrodymcs, d quum feld heory geerl, s sll ope problem. Alhough we re o workg he frmework of xomc feld heory, our pproch my mke some uesy sce Hg s heorem suggess h he erco represeo does o exs see Sreer d Wghm, 27 p Hg s heorem ssumes, mog oher hgs, h he equl me commuo relos for he cocl vrbles of ercg feld re equvle o hose of free feld. I ryg o expl hs uforue resul, hese uhors po ou h see p Wh s eve more lkely physclly eresg quum feld heores s h equl me commuo relos wll mke o sese ll; he feld mgh o be operor uless smered me s well s spce. The work Secs. V d VI of hs pper srogly suggess h here s o physcl bss o ssume h we kow yhg bou cocl vrbles oe s me see posule 2 d he followg prgrph. Thus, our pproch cully cofrms he bove-meoed commes of Sreer d Wghm. ACNOWLEDGMENTS Work for hs pper ws begu whle T.L.G. ws suppored s member of he School of Mhemcs he Isue for Advced Sudy, Prceo, N, d compleed whle vsg he physcs deprme of he Uversy of Mchg. 1 P. A. M. Drc, Proc. R. Soc. Lodo, Ser. A 114, P. A. M. Drc, Proc. R. Soc. Lodo, Ser. A 117, W. Heseberg d W. Pul, Z. Phys. 56,