From Monopoles to Textures: A Survey of Topological Defects in Cosmological Quantum Field Theory

8/2/2019 From Monopoles to Textures: A Survey of Topological Defects in Cosmological Quantum Field Theory

1/7

Ebtrnmu`tenb

Tjct es tjg bcturg nf gopty spc`g>

@hcsse`chhy, wg `cb cbchyzg gopty spc`g ay `nbsemgrebdrgonvebd chh octgrech naig`ts frno tjg ubevgrsg. Tjctrgocebs es c jedjhy syoogtre` stru`turg nf pnebts bcogmGu`hemgcb Ppc`g, eb rgogoarcb`g nf tjg Drggl dgnogtgrwjn ferst hcem nut ets prnpgrtegs. ]jes pe`turg, tjnudjoctjgocte`chhy wghh mgvghnpgm, es c fcr `ry frno tjg vcùuo

eb rgchety.

]jg rgvnhutenb tjct jcppgbgm eb pjyse`s murebd tjg hcst`gbtury `jcbdgm tjg wcy wg tjebl canut gopty spc`g.Tjgrgcs Gu`hemgcb spc`g es es mgs`reagm ay tjg drnup nfesnogtregs, tjg rghctevety cbm qucbtuo tjgnregs cs`reag tngopty spc`g vgry meffgrgbt syoogtry drnups.

DQ tghhs us tjct spc`g cbm teog crg ebtre`ctghy rghctgm tn nbg

cbntjgr. Eb Gebstgebs nwb wnrms, Ppc`g ay etsghf, cbm teog

ay etsghf, crg mnnogm tn fcmg cwcy ebtn ogrg sjcmnws, cbm

nbhy c lebm ubenb nf tjg twn wehh prgsgrvg cb ebmgpgbmgbt

rgchety. Tetjeb tjg `nbtgxt nf tjes tjgnry, spc`gteog es

mgs`reagm cs c =-meogbsenbch Qgeocbbecb ocbefnhm. ]jg

Gu`hemgcb esnogtregs crg rgphc`gm ay tjg [neb`crg drnup cs

tjg syoogtry nf gopty spc`g. ]jes drnup nbsests nf fnur

trcbshctenbs, nbg tgopnrch cbm tjrgg spctech, cs wghh cs tjg sex

dgbgrctnrs nf tjg prnpgr Hnrgbtz drnup, tjg tjrgg rntctenbs

cbm tjg tjrgg annsts.

Sucbtuo og`jcbe`s `jcbdgs tjg wcy wg ubmgrstcbm gopty

spc`g ay `jcbdebd tjg wcy tjct wg hnnl ct octtgr. Tetjeb tjg

`nbtgxt nf tjes tjgnry wg spgcl canut ffighms tjct gvnhvg eb

spc`g. Jnwgvgr, mug tn tjg ub`grtcebty prebèphg, et es

eopnsseahg tn `nophgtghy cbbejehctg c ffighm. ]jg agst tjct `cb

ag mnbg es tn hnwgr tjg ffighm tn ets oebeouo vchug, lbnwb cs

tjg drnubm stctg nr vcùuo.

Eb feghm tjgnry, tjg gffg`tevg pntgbtechs feghms hevg eb crgtgopgrcturg mgpgbmgbt. Ct jedj tgopgrcturgs, tjgsg

pntgbtechs crg jedjhy syoogtre` cbm tjgrg es c sebdhg vcùuo.Cs tjg tgopgrcturg mg`rgcsgs aghnw snog `rete`ch vchug, tjgoebeouo nf tjg pntgbtech ag`nogs mgdgbgrctg cbm tjg

nredebch syoogtry nf tjg drnubm stctg es arnlgb. ]jg sgt nfmgdgbgrctg vcùc fnros c spc`g `chhgm tjg vcùuo ocbefnhm.

Tetj tjg vcùuo ocbefnhm mgfebgm, gopty spc`g `cb agmgs`reagm cs c `nbtebunus ocppebd frno rgch spc`g ebtn tjgvcùuo ocbefnhm. ]n gc`j pnebt eb spc`g, c pnebt nb tjgvcùuo ocbefnhm cb ag cssnèctgm.

Ppc`gOV 9p (p)Bnw tjes ocppebd sjnuhmt `crg canut tjg systgo nf`nnrmebctgs wg usg tn mgs`reag tjg vcùuo ocbefnhm. ]jesebmgpgbmgb`g ocbefgsts etsghf cs c dcudg trcbsfnroctenb nftjg vcùuo stctg chhnwebd us tn hn`chhy trcbsfnro tjg vcùuovchugs ct gc`j pnebt eb spc`g. Ef tjg vcùuo ocbefnhm jcs ctrevech tnpnhndy, tjgb chh pnebts eb spc`g `cb ag arnudjt tn tjgscog pnebt nb tjg vcùuo cbm bntjebd ebtgrgstebd jcppgbs.Jnwgvgr, ef tjg vcùuo jcs c bnb-trevech tnpnhndy, tjgb tjeses bn hnbdgr pnsseahg, cbm tjgrg es tjg pnsseaehety tjct tjg c`tnf argclebd tjg syoogtry nf tjg vcùuo wehh dgbgrctgnaig`ts chhgm tnpnhnde`ch mgfg`ts.

]jg og`jcbeso fnr tjg fnroctenb nf tnpnhnde`ch mgfg`ts eb c`nsonhnde`ch sgttebd wcs ferst gxphcebgm ay Leaahg. Tjgb tjgtgopgrcturg nf tjg ubevgrsg mrnps aghnw tjg `rete`chtgopgrcturg, tjg vcùuo nf tjg syoogtre` pntgbtech, chsnlbnwb cs tjg fchsg vcùuo, mg`cys ebtn nbg nf tjg bgw

Frno Onbnpnhgs tn ]gxturgs9 C Purvgy nf ]npnhnde`ch Mgfg`ts

eb @nsonhnde`ch Sucbtuo Feghm ]jgnry

Mcoecb Q. Pnwebsle

Mgpcrtogbt nf [jyse`s cbm Cstrnbnoy, Mcrtonutj @nhhgdg, Jcbnvgr, BJ ;0477

Ocr`j 33, 1;31

Castrc`t

]jgnregs wetj spnbtcbgnus syoogtry argclebd gxjeaet mgdgbgrctg vcùuostctgs. ]jes sgt nf stctgs `cb ag mgs`reagm ay c vcùuo ocbefnhm wjnsgbnbtrevech tnpnhndy mgtgroebgs tjg typgs nf mgfg`ts tjct `cb fnro eb tjg fcare` nfspc`gteog. Tg gxcoebg tjg seophgst `hcssgs nf mgfg`ts, cbm cbchyzg tjger

prnpgrtegs, eb pcrteùhcr tjger `nbtreautenb tn tjg gbgrdy `nbtgbt nf tjg Wbevgrsg,

cbm tjger gvnhutenb eb cb gxpcbmebd FQT spc`gteog.


2/7

vcùc. ]jg ebfnroctenb tjct tjg syoogtry jcs aggb arnlgbwehh trcvgh nutwcrm frno tjg ebetech pnebt wjgrg et n`ùrrgm,`rgctebd rgdenbs tjct crg chh eb tjg scog vcùuo stctg.Jnwgvgr, pnebts tjct crg nut nf `cusch `nbtc`t murebd tjgsyoogtry argclebd `cb fchh ebtn meffgrgbt vcùc. Cs tjgrgdenbs nf arnlgb syoogtry gxpcbm nutwcrm, tjg ocppebdebtn tjg bgw vcùc bggms tn ag `nbtebunus< tjes `nbstrcebtocy, jnwgvgr, bnt chhnw fnr chh pnebts eb spc`g tn fchh ebtn tjgbgw vcùuo. Qgdenbs nf fchsg vcùuo ocy ag squggzgm ebtntnpnhnde`chhy stcahg `nbfedurctenbs. ]jes prn`gss nf mgfg`tfnroctenb es lbnwb cs tjg Leaahg Og`jcbeso.

Tg crg frgg tn tjebl nf tnpnhnde`ch mgfg`ts, tjgb, cs rgdenbs nffchsg vcùuo tjct crg prgvgbtgm frno mg`cyebd cwcy mug tntjg tnpnhnde`ch `nbstrcebt eopnsgm ay tjg vcùuo ocbefnhm.Eb tjg heoet tjct tjg trcbsetenb rgdenb agtwggb vcùc ag`nogsebfebetghy tjeb, tjgsg mgfg`ts `cb ag ;-meogbsenbchonbnpnhgs, 3-meogbsenbch `nsoe` strebds, 1-meogbsenbchmnoceb wchhs, nr jedjgr meogbsenbch tgxturgs. Tg wehh fnùs

preocrehy nb tjg prnpgrtegs nf strebds cbm wchhs.

Fnr `nophgtgbgss, et sjnuhm ag bntgm tjct tjgnregs `cb ag

`nbstru`tgm eb wje`j sgvgrch syoogtry argclebd gvgbts n`ùrgbmebd eb tjg syoogtry drnup nf tjg stcbmcrm onmgh9

D J ...

... PW(0)` PW(1)E W(3)U PW(0)` W(3)go

Eb tjgsg s`gbcrens tjgrg es tjg pnsseaehety nf sgvgrch meffgrgbttypgs nf mgfg`ts fnroebd. Furtjgronrg, ebtgrc`tenbs chhnw fnrhnwgr meogbsenbch mgfg`ts tn fnro nb tjg anubmcregs nfjedjgr meogbsenbch nbgs. Ptrebds fnro nb tjg anubmcregs nfmnoceb wchhs, onbnpnhgs fnro nb tjg gbms nf strebds. ]jeshgcms tn c fcr re`jgr spg`truo nf pnsseaehetegs fnr tjg gvnhutenbnf tjgsg fcsèbctebd gbtetegs.

Agfnrg mghvebd ebtn tjg prnpgrtegs nf snog nf tjgsg naig`tswg wehh hnnl ct tjg ubmgrhyebd octjgocte`s tjct es rgquergm tnmgs`reag syoogtry argclebd cbm tjg rgsuhtebd tnpnhndegs nftjg vcùuo ocbefnhms tjct rgsuht. Tg wehh tjgb gxphnrg tjgmgfg`ts eb tjg bcrrnw auop heoet wjgrg tjgy `cb ag onmghgm

ay seophg dgnogtre` nbstru`ts.

Octjgocte`ch [rgheoebcregs

Eb tjes sg`tenb wg wehh gxcoebg snog ghgogbtcry drnuptjgnry cbm tnpnhndy eb nrmgr tn devg us tjg prnpgr tnnhs tnmgs`reag tnpnhnde`ch mgfg`ts. Tg wehh agdeb ay rg`chhebd snogdrnup tjgnry tn sgg jnw et pgrtcebs tn mgs`reaebd spnbtcbgnussyoogtry argclebd. ]jg acse`s nf jnontnpy tjgnry wehh tjgb

ag mgvghnpgm tn gxcoebg tjg tnpnhnde`ch stru`turg nf tjgrgsuhtebd vcùuo ocbefnhm. Eb pcrteùhcr, tjg jnontnpydrnups wehh phcy c `gbtrch rnhg eb tjg `hcssefe`ctenb nf tjgtypgs nf mgfg`ts tjct wehh ag chhnwgm tn fnro eb c devgbtjgnry.

@nbsemgr c tjgnry wjnsg vcùuo stctg jcs c syoogtry drnupD. Vec c og`jcbeso su`j cs tjg nbg mesùssgm canvg, tjessyoogtry wehh ag arnlgb tn c suadrnupJ. ]jg bgw vcùuowehh tjgb ag pcrcogtrezgm ay snog pcrcogtgr !. Hgt us ag c

aet onrg rednrnus.

]jg nredebch vcùuo trcbsfnros trevechhy ubmgr ghgogbts nfD9

d D d|;e ? |;e

Pyoogtry argclebd wehh dgbgrctg c sgt nf vcùc9

D J 9|;e |;,e

]jgsg bgw vcùc wehh trcbsfnro trevechhy ubmgr J, aut bntubmgr ghgogbts nutsemg nfJ9

j J j |;,e ?|;,ed /J d|;,e ? |;,;e

]jes ogcbs tjct cby pnsseahg trcbsfnroctenbs agtwggb tjgvcùuo stctgs es nf tjg fnrodj9

d D, j Jdj |;,e? d(j |;,e)

?d |;,e?|;,;e

Bntg tjct ghgogbts nf tjg fnro jd dgbgrctg tjg scogtrcbsfnroctenb nb tjg vcùc, sn J es c bnroch suadrnup.Jgb`g et es tjg `nsgts, dJ, gc`j ghgogbt nf wje`j dgbgrctgstjg scog trcbsfnroctenb, tjct prnpgrhy mgs`reag chh tjg wcyseb wje`j tjg vcùc `cb trcbsfnro agtwggb gc`j ntjgr. ]jg sgtnf chh nf tjgsg `nsgts fnros tjg quntegbt drnup, D/J9

D/J?{dJ|d D, JD}dJ|;,e ? |;,;e

Peb`g tjg quntegbt drnup `cpturgs chh tjg stru`turg nf tjgvcùc nf tjg arnlgb tjgnry, wg `chh et tjg vcùuo ocbefnhm,cbm mgbntg et9

OV ? D/J

Hgts tclg c hnnl ct c fgw seophg gxcophgs tn agttgr c`qucebtnursghvgs wetj tjg vcùuo ocbefnhms cssnèctgm wetjmeffgrgbt syoogtry argclebds.

Gxcophg9 Y1!3

@nbsemgr c rgch s`chcr feghm wetj c pntgbtech9

V() ?

o1

1 1 +

=!=

Jgrg o1 es pnsetevg. ]jes pntgbtechs vcùuo es ebvcrecbtubmgr tjg pcrety trcbsfnroctenb9

Jgb`g tjg syoogtry drnup nf tjg ubarnlgb tjgnry `nbsestsnf twn ghgogbts, tjg emgbtety cbm tjg pcrety fhep. ]jes drnup esmgbntgm9

D ? {3,3} ? Y1

Ef o1es chhnwgm tn `jcbdg sedb sonntjhy, tjgb tjg vcùuo nftjg tjgnry sphets ebtn twn, cbm tjgsg bgw vcùc crg bn hnbdgrebvcrecbt ubmgr tjg nhm syoogtry drnup.




3/7

VJH

Fed.3 Cs o1 vcregs sonntjhy frno pnsetevg vchugs,pcst zgrn, cbm ebtn tjg bgdctevgs, tjg pntgbtechmgfnros frno jcvebd c sebdhg oebeouo tn jcvebdtwn.

]jg vcùc crg ebvcrecbt nbhy ubmgr tjg emgbtety, sn tjgvcùuo ocbefnhm `cb gcsehy ag rgcm nff9

OV ?

Y1

3? Y1

]jes es cb gxcophg wjgrg tjg vcùuo ocbefnhm esmes`nbbg`tgm. ]jes es c dgbgrch `nbsgqugb`g nf cby mes`rggtsyoogtry agebd arnlgb, cbm wehh dgbgrctg 1-meogbsenbchmgfg`ts lbnwb cs c mnoceb wchhs. Tg wehh rgturb tn gxcoebgtjg tnpnhndy nf su`j c ocbefnhm nb`g wg jcvg mgvghnpgm tjg

prnpgr octjgocte`ch tnnhs. "

Gxcophg9 W(3)!3

@nbsemgr c nophgx s`chcr feghm wetj tjg pntgbtech9

V() ?

o1

1 ||1 +

=!||=

Nb`g cdceb wg jcvg o1pnsetevg, cbm bntg tjct tjg pntgbtech es

ebvcrecbt ubmgr c W(3)trcbsfnroctenb9

D? W(3) 9 ge

Ef wg chhnw fnr c og`jcbeso tjct sonntjhy `jcbdgs o1 tnbgdctevg vchugs, tjgb tjg pnebt vcùuo nf tjg tjgnry wehhgxpcbm ebtn c rebd cs tjg syoogtry es arnlgb.

Fed.1 ]jg pntgbtech fnr c `nophgx s`chcr feghm.Tjgb tjg syoogtry es ubarnlgb tjgrg es c sebdhgvcùuo ct tjg `gbtgr nf tjg pntgbtech. Tjgb o1 es`jcbdgm tn bgdctevg vchugs, tjg nredebch syoogtryes arnlgb.

]jg bgw vcùuo stctgs crg bnw nbhy ebvcrecbt ubmgr cbemgbtety trcbsfnroctenb, sn tjg vcùuo ocbefnhm es gcsehy`chùhctgm9

OV ?

W(3)

3? W(3) ? P3

Jgrg wg jcvg usgm tjg fc`t tjct tjg drnup W(3) estnpnhnde`chhy gquevchgbt tn tjg èr`hg, P3. Wbhelg tjg hcstgxcophg tjes vcùuo ocbefnhm es `nbbg`tgm. Et es bnt,jnwgvgr, seophy-`nbbg`tgm, cbm tjes wehh ag c `ruèchebdrgmegbt eb ubmgrstcbmebd 3-meogbsenbch strebd mgfg`ts. "

Pn fcr wg jcvg hnnlgm ct twn spgèfe` gxcophgs nf syoogtryargclebd, bcoghy Y1!3cbm W(3)!3. ]jgrg crg, nf `nursg,ocby ntjgr tjgnregs tjct `cb ag `nbstru`tgm tjct jcvg cspgèfe` fnro nf syoogtry argclebd. Eb pcrteùhcr, tjg

argclebd nf cb PW(1) syoogtry `cb hgcm tn ocby ebtgrgstebdvcùuo ocbefnhms.

Cby tjgnry wetj PW(1)!W(3) rgsuhts eb c vcùuo ocbefnhm

wetj c tnpnhnde`ch stru`turg gquevchgbt tjct nf c spjgrg, P

1

.]jgsg onmghs rgsuht eb pnebt-helg mgfg`ts `chhgm onbnpnhgs.Cb PW(1)!3 syoogtry argclebd hgcms tn c vcùuo wjnsgtnpnhndy es tjct nf c 0-spjgrg, P0. ]jgsg onmghs jcvg c vgryebtgrgstebd `hcss nf vnhuog mgfg`ts lbnwb cs tgxturgs. Tgwehh jcvg onrg tn scy canut jnw chh nf tjgsg mgfg`ts crg tjgrgsuht nf tjg tnpnhndy nf tjg vcùuo ocbefnhm cftgrmgvghnpebd c aet nf octjgocte`ch oc`jebgry `chhgm jnontnpytjgnry, wje`j wg sjchh bnw gbmgcvnr tn mn.

Hgt O ag c ocbefnhm. @nbsemgr tjg sgt nf `nbtebunusfub`tenbs ocppebd tjg jypgr-meogbsenbch ubet uags ebtn O.

Fb ?{f 9Eb

O}

Tg wehh bnw sjnw tjct tjes sgt `cb ag usgm tn `nbstru`t c

drnup stru`turg ay mgfebebd c wcy eb wje`j tn dhug tjgghgogbt fub`tenbs tndgtjgr. Fnr seopheèty wg wehh `nbsemgrtjg `csg nf b?3, tjg sgt nf fub`tenbs ocppebd tjg ubet ebtgrvchebtn nur ocbefnhm.

Hgtf3cbmf1ag twn su`j fub`tenbs su`j tjct tjg gbm pnebt nftjg ferst `nebèmgs wetj tjg stcrt pnebt nf tjg sg`nbm,

f3(3)?f1(;). ]jgb wg `cb mgfebg c nb`ctgbctenb prnmu`t nftjg twn fub`tenbs vec9

(f3 f1)(t) ?

f3(1t) 9 t \;,

3

1_

f1(1t 3) 9 t \3

1,3_

Ef wg rgstre`t nursghvgs tn tjg suasgt nf chh fub`tenbs wje`jstcrt cbm gbm nb tjg scog pnebt, e.g. hnnps, tjgb tjes npgrctenbwehh ag bcturchhy mgfebgm fnr chh tjg ghgogbts nf tjct sgt.

]jg hcst ebdrgmegbt wg wcbt tn eb`nrpnrctg ebtn tjes`nbstru`tenb es c wcy tn prnpgrhy mgfebg meffgrgbt hnnps ebnur sgt. Cs cb gxcophg `nbsemgr tjg tjrgg hnnps eb fedurg 0.




4/7

Fed.0 Hnnps mrcwb nb c ocbefnhm tjct crg bntjnontnpe`chhy gquevchgbt tn nbg cbntjgr. ]jg hnnps`cbbnt ag sonntjhy mgfnrogm ebtn nbg cbntjgr.

]jg hnnp nb tjg hgft es bnt wrcppgm crnubm tjg jnhg nrmecogtgr nf tjg tnrus. Jgb`g wg `cb eocdebg sjreblebd tjghnnp ebmgfebetghy, ubteh et `nbtrc`ts mnwb tn c sebdhg pnebt.]jg redjt hnnp es wrcppgm crnubm tjg mecogtgr, cbm bn octtgrjnw wg sonntjhy mgfnro et, et `cb bgvgr ag sjrubl mnwb ebtnc pnebt. ]jg scog dngs fnr tjg hnnp wrcppgm crnubm tjg jnhgnf tjg tnrus. Furtjgronrg, tjes hnnp `cbbnt ag mgfnrogm ebtntjg nbg tjct es wrcppgm crnubm tjg mecogtgr. Chh tjrgg nftjgsg hnnps `cb, frno c tnpnhnde`ch pnebt nf vegw, ag`nbsemgrgm mesteb`t.

Eb `nophgtg dgbgrchety, wg `cb febm cb gquevchgb`g `hcss nfhnnps quetg gcsehy. Mgfebg tjg gquevchgb`g rghctenb9

f3 f1 F Fb+3s.t.

F(;, t) ? f3(t) & F(3, t) ? f1(t)

]jg mgfnroctenb nf nbg ùrvg ebtn tjg ntjgr es c``nophesjgmay tjg jedjgr meogbs enbch sonntj ocppebd wjnsganubmcregs crg tjg twn hnwgr meogbsenbch ùrvgs. Et es gcsy tnsgg tjct tjes es cb gquevchgb`g rghctenb. Qgfhgxevety es sctesfegm

ay tjg emgbtety ocppebd. Pyoogtry es c``nophesjgm ay cocppebd tjct es tjg scog cs tjg nredebch aut tjg ferstcrduogbt,s, es rgphc gm ay 3-s. ]rcbsetevety `cb ag sjnwb ayusebd tjg scog `nb`ctgbctenb npgrctenb nb tjg jedjgrmeogbsenbch ocppebds tjct wg usgm gcrhegr.

]jg sgt nf gquevchgb`g `hcssgs nf hnnps tjct crg mgfnrocahgebtn nbg cbntjgr es tjgb iust tjg quntegbt spc`g nf tjg sgt nfocppebds onmmgm nut ay tjg gquevchgb`g rghctenb ebmu`gm aytjg nb`ctgbctenb npgrctenb9

Fb

?{\f_|\f_ ?{d|d f}}

Ay bcturchhy gxtgbmebd tjg `nb`ctgbctenb npgrctenb tn tjgsggquevchgb`g `hcssgs, wg jcvg c drnup stru`turg, wje`j wg

mgfebg cs tjg btj jnontnpy drnup nf nur ocbefnhm, O9

b(O) 9?

Fb

,

]jes naig`t wehh ag nf `gbtrch eopnrtcb`g eb gxcoebebd tjgvcùuo ocbefnhms. Ef c ocbefnhm jcs c trevech btj-jnontnpydrnup tjgb chh nf tjg b-meogbsenbch ùrvgs hevebd nb tjgocbefnhm `cb ag `nbtrc`tgm tn c pnebt. Ef c jnontnpy drnup esbnt trevech, tjgb tjgrg gxest ùrvgs tjct `cbbnt ag `nbtrc`tgm tnc pnebt, sedbefyebd tjg prgsgb`g nf c tnpnhnde`ch mgfg`t. ]jgfnro nf tjg ferst fgw jnontnpy drnups nf c vcùuo ocbefnhmwehh tjgb mgtgroebg gxc`thy wjct typgs nf mgfg`ts wehh agchhnwgm eb gc`j pcrteùhcr onmgh wg mgvghnp.

Hgts gxcoebg c fgw gxcophgs tn dgt c fggh fnr wjct tjg ferstfgw jnontnpy drnups crg nf sgvgrch meffgrgbt ocbefnhms.

Gxcophg9OV ?YB

Jgrg tjg vcùuo ocbefnhm `nbsests nf B pnebts. ]jg nbhybnbtrevech jnontnpy drnup es tjg ;tj. Et sjnuhm ag bntgm tjcttjg ;-jnontnpy es bnt c`tuchhy c drnup, seb`g c aebcrynpgrctenb `cbbnt ag mgfebgm< et es seophy c sgt wetj c sebdhgmestebduesjgm pnebt.

;(OV) ? {;, 3, . . . , B}"

Gxcophg9OV ?P3

Tjgb tjg vcùuo ocbefnhm es c èr`hg tjgb pnebts `cb chh agonvgm tn tjg scog phc`g, sn tjg ;-jnontnpy es trevech. Hnnpsjnwgvgr `cbbnt ag mgfnrogm ebtn nbg cbntjgr ef tjgy webmcrnubm tjg èr`hg c meffgrgbt buoagr nf teogs. ]jg webmebdbuoagr nf c hnnp mgtgroebgs ets gquevchgb`g `hcss, cbmdhugebd hnnps tndgtjgr rgsuhts eb c hnnp wjnsg webmebdbuoagr es tjg suo nf tjg mcudjtgrs. Jgb`g tjg 3-jnontnpy esseophy tjg cmmetevg drnup nf ebtgdgrs.

3(OV) ? Y

Et turbs nut tjct chh nf tjg jedjgr jnontnpegs, helg tjg ;tj, crg

trevech. "

]wn febch rgsuhts wehh jghp us prndrgss furtjgr. ]jg ferst es tjgfubmcogbtch tjgnrgo nf `nsgt spc`gs eb jnontnpy tjgnry. Etchhnws us tn rghctg tjg jnontnpy nf c bnroch suadrnup tn ets

pcrgbt drnup ef tjg pcrgbt drnup sctesfegs `grtceb `nbmetenbs.Ef tjg btj cbm (b-3)tj jnontnpy drnups nf c Heg drnup crgtrevech, tjgb tjg btj jnontnpy drnup nf tjg `nsgt spc`g`rgctgm ay onmmebd nut tjg nredebch drnup ay nbg nf etsbnroch suadrnups es esnonrpje` tn tjg btj jnontnpy drnupnf tjg bnroch suadrnup9

b(D) ? b3(D) ? 3 b

D

J

? b3(J)

]jg sg`nbm peg`g nf oc`jebgry es tjg fc`tnrebd tjgnrgo fnrcr`wesg `nbbg`tgm spc`gs. Ef Zcbm Ucrg spc`gs wetj trevech;-jnontnpegs, tjgb9

b(O3 O1) ? b(O3) b(O1)

Tetj tjgsg hetthg dgos stu`l eb nur acd nf octjgocte`choc`jebgry, wg crg rgcmy tn gxcoebg tjg oceb `hcssgs nftnpnhnde`ch mgfg`ts.




5/7

Onbnpnhgs

Onbnpnhgs rgsuht eb onst cby tjgnry tjct jcs c sgregs nfsyoogtry argclebds tjct gbm eb tjg febch drnup agebd c tgbsnr

prnmu`t wetj W(3)9

D . . . JLW(3)]jgsg tjgnregs crg nf bntg ag`cusg tjg hcst twn syoogtrydrnups eb tjg Ptcbmcrm Onmgh jcvg tjg scog fnro cs tjg gbmnf tjes sgqugb`g. Onst DW] tjgnregs nf ebtgrgst tjct argclebtn tjg Ptcbmcrm Onmgh sctesfy tjg `nbmetenbs nf tjgfubmcogbtch tjgnrgo nf `nsgts fnr tjg sg`nbm cbm jedjgrjnontnpy drnups. @nbsemgrebd tjg jypgr`jcrdg syoogtry,wg jcvg tjgb tjct9

1(OV) ? 1(D

J)

? 3(L) 3(W(3))? 3(L) Y6? 3

Peb`g tjg 1-jnontnpy es bnb-trevech vchugs nf tjg vcùuoocbefnhm ct pnebts nb c `hnsgm 1-surfc`g eb rgch spc`g prgvgbttjg surfc`g frno agebd `nbtebunushy mgfnrogm ebtn c pnebt.]jgrg gxests c sochh rgdenb nf hn`chezgm fchsg vcùuo, cbm ebtjg heoet nf agebd ebfebetghy sochh eb sezg wg `chh tjes`nbfedurctenb c onbnpnhg.

Onbnpnhgs jcvg c qucbtezgm `jcrdg, cbm cb gxtrgoghy jedjocss rghctevg tn tjg ocinrety nf stcbmcrm onmgh pcrte`hgs.Furtjgronrg tjgy crg prnmu`gm eb `npenus qucbtetegs murebdtjg syoogtry argclebd, wje`j hgcms tjgo tn que`lhymnoebctg tjg gbgrdy mgbsety nf tjg ubevgrsg.

Peb`g tjgrg jcs bgvgr aggb c `nbferogm mgtg`tenb nf conbnpnhg, tjg usuch wcy tjgy `cb ag gxphcebgm cwcy es

tjrnudj ebfhctenb. Peb`g tjgy crg prnmu`gm murebd DW]syoogtry argclebd, ebfhctenb `cb gcsehy mehutg tjg pnpuhctenbnf onbnpnhgs. C frc`tenb nf tjnsg tjct rgoceb wnuhm tjgbjcvg vghnètegs hnw gbnudj tn ag anubm drcvetctenbchhy tnstcrs. Ef tjgrg crg cby anubm tn tjg Pnhcr Pystgo, tjgb wg`nuhm nasgrvg tjger sedbcturg ag`cusg tjgy `cb `ctchyzg

prntnb mg`cy cbm prnmu`g Z-rcys. Cs nf tjes mctg, bnt su`jZ-rcy gvgbts jcvg aggb nasgrvgm.

@nsoe` Ptrebds

C vcùuo ocbefnhm tjct es gquevchgbt tn c èr`hg wehh rgsuht ebtjg fnroctenb nf tnpnhnde`chhy stcahg hnnps nf fchsg vcùuo`chhgm `nsoe` strebds. Eb fc`t, wjgbgvgr tjg 3-jnontnpy esbnt trevech tjes wehh n`ùr.

3(OV) 6? 3

Nbg `cb ubmgrstcbm tjes ay `nbsemgrebd tjg vchug nf tjgvcùuo nb tjg pnebts nf c `hnsgm hnnp eb rgch spc`g. Cs tjghnnp es `nbtrc`tgm, tjg vchugs nf tjg vcùuo ocbefnhm `cbbnt

ag `nbtebunushy arnudjt tn tjg scog vchug ef tjg ocppebd nftjg `hnsgm hnnp tn tjg vcùuo ocbefnhm jcs c bnb zgrnwebmebd buoagr.

]n agttgr ubmgrstcbm tjg gbgrdy `nbtgbt nf tjg hnnp, wg `cb`nbsemgr tjg csyoptnte` snhutenb nf tjg feghm gquctenb frno cseophg onmgh fnr c `nophgx s`chcr feghm9

H ?

3

1

o

1

1||1

=||=

Tg chhnw fnr tjg `csg wjgrg tjg ebetech syoogtry es arnlgbay hgttebd o1 `jcbdg sedb. ]jg csyoptnte` agjcvenr nf tjgsnhutenb tn tjg feghm gquctenbs `noebd frno tjes Hcdrcbdecbes, fnr webmebd buoagr b9

()

r

gb

]jg strgss gbgrdy tgbsnr nf tjes `nbfedurctenb `cb chwcys agarnudjt ebtn tjg fnro9

]

?f(, )

3 ; ; ;; ; ; ;; ; ; ;; ; ; 3

]jes snhutenb rgprgsgbts cb ebfebetghy hnbd strebd pcrchhgh tntjg z cxes. Bntg tjct tjg sedb fnr tjg prgssurg tgro esbgdctevg, eophyebd tjct tjgrg es tgbsenb chnbd tjg strebd,wje`j wehh phcy cb eopnrtcbt rnhg fnr tjg mybcoe`s nf tjgmgfg`t. ]jg fub`tenb f es rcmechhy syoogtre`, cbm jcs snogùtnff, ", agynbm wje`j et es vcbesjgs. Et `cb ag tjnudjt nf cstjg rcmeus nf tjg strebd, cbm es mgtgroebgm ay tjg mg Arndhegwcvghgbdtj nf tjg feghm qucbtc. ]jg hebgcr mgbsety nf tjgstrebd `cb ag cpprnxeoctgm ay `nbsemgrebd tjg `rnss sg`tenbfehhgm wetj tjg fchsg vcùuo9

V(;)1 o1

Ef tjg hebgcr mgbsety eb dgnogtrezgm ubets es drgctgr tjcb 3;-4,tjgb wg ocy ag cahg tn nasgrvg tjg gffg`ts nf strebd hgbsebd.]jes `nuhm `cusg ouhtephg eocdgs nf naig`ts, tjnudj tjgsgpcrctenb agtwggb eocdgs wnuhm ag nb tjg nrmgr nf 0-0;cr`sg`nbms. Eb 1;;: tjgrg wcs c hgbsgm qucscr mes`nvgrgmtjct oedjt ag mug tn tjg pcssebd ay nf c `nsoe` strebd,jnwgvgr sgvgrch ntjgr gxphcbctenbs put tjes `nb`husenb nbsjcly drnubm. ]jg sedbcturg nf c strebd oedjt chsn ag sggb ebtjg @OA, seb`g c onvebd nbg wnuhm `cusg c stgp fub`tenbhelg tgopgrcturg pgrturactenb mug tn tjg hedjt pcssebd eb frnbtnf cbm eb tjg wclg nf tjg strebd.

[grturactenbs nb tjg strebd `cb prnpcdctg chnbd tjg strebd,tjnudj tjgy mn bnt nagy tjg supgrpnsetenb prebèphg. Eb tjg`nbtgxt nf cb FQT `nsonhndy, onmgs wetj ùrvcturg rcmee

hcrdgr tjcb tjg Juaahg s`chg wehh ag frnzgb nut, cbm wehh bntgvnhvg. Cs tjg ubevgrsg cdgs onmgs hcrdgr tjcb tjg jnreznbwehh ag strgt`jgm ay tjg Juaahg fhnw. Jnwgvgr seb`g tjggbgrdy mgbsety nf c strebd dngs cs c-3, tjgsg onmgs wehhgvgbtuchhy gbtgr tjg jnreznb. Mug tn tjg tgbsenb chnbd tjgstrebd, tjgsg onmgs wehh tjgb agdeb tn mg`cy, `cusebd tjgstrebd tn cttgopt tn strcedjtgb etsghf nut.

Ebtgrc`tenbs agtwggb strebds rgquerg bnt nbhy ebfebetghy hnbdnbgs, aut hnnps cs wghh. C `rnssebd nf c strebd wetj etsghf nrcbntjgr strebd `cb `cusg c hnnp tn argcl nff. ]gbsenb wehh




6/7

`cusg tjg hnnp tn `nbtrc`t. C `nbtrc`tebd hnnp wehh dgbgrctgqucmrcpnhg drcvetctenbch rcmectenb, cbm mgpgbmebd nb wjgtjgrtjg s`chcr feghm `nuphgs tn tjg tjg pjntnb feghm, dcooc rcyscs wghh. ]jes `nuhm hgcm tn c sedbcturg eb tjg dcooc rcy`nbtreautenb tn #$. Wbfnrtubctghy tjes `nbtreautenb tn tjgdcooc rcy fhux es heagrchhy gsteoctgm ct crnubm 3;-5, wjgrgcstjg ùrrgbt nasgrvctenbs phc`g #$ct crnubm 3;-4. Jgb`g tjessedbch wnuhm ag truhy meffeùht tn gxtrc`t frno wjct wgchrgcmy sgg.

]jg mnoebcbt fnro nf octtgr eb tjg `nsons wehh chsn cffg`ttjg gvnhutenb nf c actj nf strebds. Ptrebds jcvg c s`chgebvcrecbt pnwgr spg`truo, sn tjg gxpcbsenb rctg nf tjgubevgrsg wehh jcvg cb eopnrtcbt rnhg eb mgtgroebebd tjg rctgct wje`j hnnps crg prnmu`gm. Murebd rcmectenb mnoebctenb tjgubevgrsg gxpcbms ct c shnwgr rctg, sn s`chg ebvcrecb`g `cbnbhy ag ocebtcebgm ay cb eb`rgcsg eb tjg rctg nf hnnp

prnmu`tenb. Nb`g octtgr mnoebctenb le`ls eb, tjg tgbsenb ebtjg strebds ag`nogs tjg mnoebcbt fc`tnr eb ocebtcebebd s`chgebvcrecb`g, sn hnnp prnmu`tenb wehh fchh mnwb mrcste`chhy. Efwg wcbt tn febm mg`cyebd hnnps nur agst agt es tn hnnl fnrtjgo eb tjg gcrhy ubevgrsg wjgb tjgy wgrg stehh agebd

`npenushy prnmu`gm.

Nb nsonhnde`ch teogs`chgs strebds ag`nog vgry eopnrtcbt tntjg gvnhutenb nf tjg Wbevgrsg. ]jger gbgrdy mgbsety gvnhvgswetj tjg s`chg fc`tnr cs c-3, sn tjgy gvgbtuchhy `nog tnmnoebctg tjg tntch gbgrdy mgbsety nf octtgr.

Mnoceb Tchhs

Mnoceb wchhs crg tjg rgsuht nf cby mes`rggt syoogtry agebdarnlgb. ]jg rgsuhtebd vcùuo ocbefnhm es tjgb seophy c sgtnf mes`nbbg`tgm pnebts, tjg buoagr nf tjgo mgtgroebebd tjgbuoagr nf pnsseahg typgs nf mnocebs. Tg `cb stctg tjessu`èb`thy cs c `nbstrcebt nb tjg ;-jnontnpy nf tjg vcùuoocbefnhm9

;(OV) ? 3, 1, . . . , B

Hgts `nbsemgr c seophg onmgh fnr c rgch s`chcr feghm ay`nbsemgrebd tjg Hcdrcbdecb9

H ?

3

1

o

1

11

==

Nb`g cdceb wg chhnw fnr c argclebd nf tjg mes`rggt syoogtrynf tjg pntgbtech ay hgttebd o1ag bgdctevg. Fnr tjes onmgh tjgvcùuo jcs mgdgbgrc`y twn, cbm tjg snhutenb tn tjg feghmgquctenbs tjct ebtgrpnhctgs agtwggb tjg twn vcùc es nf tjgfnro9

(z) ?

otcbj(

oz

)

]jes snhutenb mgs`reags c mnoceb wchh nregbtgm eb tjg xy-phcbg, wetj c tje`lbgss mgpgbmgbt nb antj tjg ocss cbm`nuphebd `nbstcbt nf tjg tjgnry. ]jg strgss gbgrdy tgbsnr fnrtjes snhutenb wehh sjgm onrg hedjt nb ets stru`turg9

?o=

1

3

`nsj(oz

)=

3 ; ; ;; 3 ; ;; ; 3 ;; ; ; ;

]jg spcèch ebme`gs `nrrgspnbm tn c bgdctevg prgssurg eb tjgxy-phcbg, sn tjg wchhs jcvg tgbsenb. ]jg surfc`g gbgrdymgbsety `cb ag fnubm ay ebtgdrctebd tjg teog-teog `nopnbgbtnf tjg strgss gbgrdy nvgr tjg tje`lbgss nf tjg wchh9

?

Y mz]

;

; ?11

0o

0

Tg bnw ebvnlg tjg Leaahg og`jcbeso tn mgs reag tjgtrcbsetenb frno tjg fchsg vcùuo tn tjg trug vcùuo. ]jgrgsuhtebd mestreautenb nf mnocebs `cb ag mgs`reagm ay c

pgr`nhctenb onmgh. ]jg buoagr nf mnocebs, bs, nf c pcrteùhcrsezg,ses tjgb9

bs s1

gs

0/1

Jgrg #cbm $crg onmgh mgpgbmgbt pcrcogtgrs tjct `nog frnotjg hctte`g stru`turg usgm tn seouhctg tjg vcùuo trcbsetenb.]jg mestreautenb nf mnocebs wehh ag mnoebctgm ay sgvgrchebfebetghy hcrdg mnocebs, wetj vgry ebtgrgstebd tnpnhndy, cbmocby sochhgr mnocebs.

]jg mnoceb wchhs tjct mgvghnp wehh, eb dgbgrch, jcvg hn`chùrvcturg. Eb cb FQT nsonhndy, urvcturg s`chgs hcrdgr tjcbtjg `cusch jnreznb wehh ag frnzgb eb. Tjgb tjgy gbtgr tjgjnreznb tjgy wehh mg`cy cs c-0/1< tjg tgbsenb eb tjg wchhs wehhcttgopt tn strcedjtgb tjgo nut `cusebd cby pgrturactenbs tndgt mcopgm nut. ]jrnudj tjes prn`gss, sochhgr mnocebs wehhgvgbtuchhy dgt squggzgm nut, ogcbebd tjct tjg nvgrchhmestreautenb nf mnoceb wchhs es s`chg ebvcrecbt.

Cbntjgr ebtgrgstebd prnpgrty nf mnoceb wchhs es tjct tjgergbgrdy mgbsety gvnhvgs wetj tjg s`chg fc`tnr iust helgùrvcturg, jgb`g tjgy `cb `nog tn mnoebctg tjg gbgrdymgbsety nf tjg ubevgrsg cftgr octtgr. Mug tn tjger `nbstcbtdrcvetctenbch feghms, wjgb tjgy mn sn tjgy chsn dgbgrctgjnreznbs< `cuschhy `nbbg`tgm spc`g `cb frc`turg ebtn

`nopnbgbts tjct `cb bn hnbdgr `nooube`ctg wetj nbgcbntjgr murebd tjg trcbsetenb ebtn wchh mnoebctenb.

Wbfnrtubctghy fnr tjgnrests, mnoceb wchhs jcvg bgvgr aggbnasgrvgm. Mngs tjes eophy tjct cby tjgnry tjct prgme`ts tjgergxestgb`g sjnuhm ag mes`nubtgm cs ubpjyse`ch> Bnt quetg,tjgrg crg sgvgrch wcys nut nf tjg bn nasgrvctenb `nbstrcebt.

]jg ferst wcy nut nf tjg prnahgo es tn pnset tjct tjg syoogtryargclebd gvgbt tjct hgcm tn tjger fnroctenb n`ùrrgm agfnrgebfhctenb. Peb`g tjg tje`lbgss nf c mnoceb wchh es vgry tjebrghctevg tn ets spcb, et es jedjhy helghy tjct ebfhctenb wnuhmtjgb jcvg seophy ahnwb chh tjg mnoceb wchhs fcr agynbm tjgùrrgbt `cusch jnreznb. ]jnudj c ùtg wcy nf dgttebd nut nftjg bn nasgrvctenbs nf jnreznbs prnahgo, et mngs sggo rctjgr

ubsctesfc`tnry.

]jg sg`nbm wcy nut es tn pnset c sochh aecs eb tjg vcùc nftjg tjgnry. ]jg cmmetenb nf c sochh hebgcr tgro eb tjgHcdrcbdecb `cusgs nbg vcùuo tn jcvg c shedjthy hnwgrgbgrdy mgbsety tjcb tjg ntjgr. ]jes ogcbs tjct tjct vcùuowehh ag gbgrdgte`chhy fcvnrcahg, cbm cby mnoceb eb tjctvcùuo wehh pusj nb mnocebs eb tjg ntjgr vcùuo onrgstrnbdhy tjcb tjg pusj ac`l. ]jg fcvnrcahg vcùuo wehhtjgb ag`nog tjg mnoebcbt mnoceb, scvebd us frno jcvebd tnsgg wchhs.




7/7

]gxturgs

]jg argclebd nf onrg `nophe`ctgm syoogtregs `cb hgcm tnbnb hn`chezgm tnpnhnde`ch mgfg`ts `chhgm tgxturgs. ]jgrg

prgsgb`g es syoptnocte` nf cby vcùuo ocbefnhm tjct jcsbnb trevech jnontnpy drnups nf nrmgr 0 nr jedjgr.

Mug tn tjger bnbhn`ch bcturg, c tgxturg `nuhm hgbs @OA hedjteb c vgry pgùhecr wcy. Eb 1;;4, c `nhm spnt wcs bnte`gm ebtjg @OA tjct `nuhm ag `cusgm ay su`j c mgfg`t. Futurg

pnhcrezctenb nasgrvctenbs nf tjg spnt `nuhm ag usgm tn`nbfero wjgtjgr et es c`tuchhy `cusgm ay c tgxturg. ]jes es

ag`cusg c tgxturgs teog mgpgbmgbt drcvetctenbch feghm wnuhmebmu`g bn pnhcrezctenb, wjgrgcs c stcteste`ch fhu`tuctenbwnuhm rgsuht eb c prgfgrgbtech rcmech pcttgrb.

@nb`husenb

Sucbtuo feghm tjgnry cbm dgbgrch rghctevety prnvemg tjg onstc`ùrctg mgs`reptenb tjct wg jcvg nf Bcturg, aut frno ctjgnrgte`ch pgrspg`tevg tjgy crg eb`nophgtg. ]jg PtcbmcrmOnmgh `cb ag goagmmgm ebtn c oyrecm nf hcrdgr syoogtry

stru`turgs< et wnuhm ag prumgbt tn gxcoebg tjg wcy eb wje`jtjgsg syoogtregs argcl ebtn tjg wnrhm tjct wg sgg crnubm ustnmcy.

]jes prn`gss bcturchhy hgcms us tn cb ubmgrstcbmebd nf goptyspc`g cs `cpcahg nf suppnrtebd tnpnhnde`chhy stcahg mgfg`ts.]jgsg mgfg`ts crg rghe`s nf fchsg vcùuo, wetj cb gbgrdymgbsety tjct `cb ag quetg meffgrgbt frno tjg octtgr cbmrcmectenb tjct jcvg mnoebctgm tjg gvnhutenb nf tjg Wbevgrsgup tn tjg prgsgbt mcy.

Eb pcrchhgh wetj rg`rgctebd tjg jedj tgopgrcturg gbvernbogbtnf tjg gcrhy Wbevgrsg eb pcrte`hg c``ghgrctnrs, et es prumgbt tngxphnrg tjg `nbsgqugb`gs nf syoogtry argclebd nbcstrnbnoe`ch cbm `nsonhnde`ch s`chgs. Mgtg`tebd tjg

sedbcturg nf tnpnhnde`ch mgfg`ts `nuhm jghp us `nbstrceb tjgonmghs tjct wg crg cttgoptebd tn mgvghnp tn gxtgbm tjgPtcbmcrm Onmgh, cs wghh cs tgst tjg syoogtry argclebds tjctjcvg chrgcmy aggb eb`nrpnrctgm eb et.

E wnuhm helg tn tjcbl antj Oe`cj Mnoarnwsle cbm DrgdnryFgemgb fnr steouhctebd `nbvgrsctenbs murebd tjg hctg bedjtjnurs wjgb E sggo tn mn onst nf oy wnrl. E wnuhm chsn helgtn tjcbl Ocr`ghn Dhgesgr fnr `nbvgyebd jes lbnwhgmdg nfSucbtuo Feghm ]jgnry tn og eb antj c `nbse`g cbm eb`rgmeahyehhuoebctebd wcy. Hcsthy E wnuhm helg tn tjcbl Mcrtonutj@nhhgdg, wetjnut wjnsg suppnrt E wnuhm bnt jcvg tjg teog tn

pnbmgr tjgsg fcsèbctebd tnpe`s.

Qgfgrgb`gs

\3_ Vehgbleb, Chgxcbmgr. "@nsoe` Ptrebds cbm MnocebTchhs." [jyse`s Qgpnrts313.7 (35:7). [rebt.

\1_ ]rnmmgb, Ocrl, cbm Pgcb @crrnhh. "]CPE Hg`turgs9Ebtrnmu`tenb tn @nsonhndy." Ogbmghgy Qgsgcr j Bgtwnrls.

Tga. 31 Ocr. 1;31. 8jttp9//www.ogbmghgy.`no/rgsgcr`j/

tcse-hg`turgs-ebtrnmu`tenb-tn-`nsonhndy/2.

\0_ Mg Hcex, Cbmrgw C., Hcwrgb`g O. Lrcuss, cbm]cbocy Vc`jcspcte. "Drcvetctenbch Hgbsebd Pedbcturg nf

Hnbd @nsoe` Ptrebds." [jyse`ch Qgvegw Hgttgrs 45.33 (3554)9

356:-543. [rebt.

\=_ Tcbd, @jej-Jubd, Jebd-]nbd @jn, cbm Uu-JugeTu. "Mnoceb Tchh Ppc`g-teogs wetj c @nsonhnde`ch

@nbstcbt."Crxev.nrd3;31.7171 (1;33). [rebt.

\7_ Hcrsnb, Pgacstecb G., Puaer Pcrlcr, cbm [gtgr H.Tjetg. "Gvcmebd tjg @nsonhnde`ch Mnoceb Tchh [rnahgo."

[jyse`ch Qgvegw M77.: (3554). [rebt.

\6_ Lcoenblnwsle, Ocr`, cbm Crtjur Lnsnwsly. "]jg@nosoe` Oe`rnwcvg Ac`ldrnubm cbm [crte`hg [jyse`s."

Cbbu. Qgv. Bu`h. [crt. Pè.=5 (3555)9 44-310. [rebt.

\4_ Lnhet`j, Pjcwb I. "Sucbtuo Mg`cy nf MnocebTchhs eb @nsonhndy. E. Ebstcbtnb Cpprnc`j." [jyse`ch

Qgvegw M76.: (3554)9 =673-661. [rebt.

\:_ P`jehm, Q., E. P. Ocsbycl, A. E. Jbctyl, cbm V. E.Yjmcbnv. "Cbnochnus Fhu`tuctenbs eb Nasgrvctenbs nf

S;574 763 C,A9 Ponlebd Dub nf c @nsoe` Ptrebd>"

Crxev.nrd/cstrn-pj (1;;:). Tga. 8jttp9//crxev.nrd/cas/cstrn-

pj/;=;6=0=v32.

\5_ @ruz, O., B. ]urnl, [. Veghvc, G. Ocrtbgz-Dnbzhgz, cbm O. Jnasnb. "C @nsoe` Oe`rnwcvg

Ac`ldrnubm Fgcturg @nbsestgbt wetj c @nsoe` ]gxturg."

]npnhnde`c3.3 (1;;:)9 ;;:. [rebt.\3;_ Dcbiue, Chgicbmrn. "]npnhnde`ch Mgfg`ts eb@nsonhndy." Crxev.nrd (1;;3). Tga. 8jttp9//crxev.nrd/cas/

cstrn-pj/;33;1:7v32.

\33_ Hchcl, Yydoubt. "]nwcrms Pnhutenb nf tjg@nsonhnde`ch Mnoceb Tchhs [rnahgo." Crxev.nrd. Tga.

8jttp9//crxev.nrd/cas/jgp-pj/54;1=;72.

\31_ Vehgbleb, C., cbm G. [. P. Pjghhcrm. @nsoe` Ptrebdscbm Ntjgr ]npnhnde`ch Mgfg`ts. @coaremdg9 @coaremdg W[,

355=. [rebt.

\30_ Mc Pehvc, Insebchmn O. @nsonhnde`ch@nbsgqugb`gs nf ]npnhnde`ch Mgfg`ts9 Mcrl Gbgrdy cbm

Vcryebd Fubmcogbtch @nbstcbts. ]jgses. Fcùhmcmg Mg

@ebècs Mc Wbevgrsemcmg Mn [nrtn, 1;;4. Mgpcrtcogbtn Mg

Fse`c. [rebt.

\3=_ Ygg, C. Sucbtuo Feghm ]jgnry eb c Butsjghh.[reb`gtnb, BI9 [reb`gtnb W[, 1;;0. [rebt.

\37_ [gbrnsg, Qndgr. ]jg Qncm tn Qgchety9 C @nophgtgDuemg tn tjg Hcws nf tjg Wbevgrsg. Bgw Unrl9 C.C. Lbnpf,

1;;7. [rebt.

\36_ Bclcjcrc. Dgnogtry, ]npnhndy cbm [jyse`s(Drcmuctg Ptumgbt Pgregs eb [jyse`s) (54:;:714=;57:)9Oelen Bclcjcrc9 Annls. Tga. ;4 Ocr. 1;31.



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From Monopoles to Textures: A Survey of Topological Defects in Cosmological Quantum Field Theory