k011. - Объединенный институт ядерных...
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18 K011
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KOTOPWC COOTBeTCTByut1iIe MaCCHlOCKHe Y1lSD1JOHIiJi JOIIIJlelll1J HMel()1 HeTpUshy
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Ho-moprenf leCIWe pornelille bull
MOTOnb onopalopHoll TCOPlIJI nOJIll KOTopble rT03ROJLfl1OT IlJ)OKBaHTOBaTIgt
CHCTeMY B OqlOCTlIOCTH luracCHleCKOIO pemeHJIR IlruiTII COCTOJIHID I
ClIeKTp KBaJTOBhlX 4urYKTyauKll xopomo U3BeCTllhl 1 Ilac oyneT 1IHTepecoshy
BaTT W1I 014 acrreKT TfOPIUI ]orna OCHOBHOC BIDIMrume ynellIeTCFI nOCTposhy
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B03lmlUlunHe oco6eHHOCTH I FIBJIjeTCR MOnOJrb )leilCTlHI18JrbHOro CKaJmpHOTO
leii3eu6eproBClltOTO nOM B lJBYMepHOM rrpOCTp8HCTHe-BpeMfIIM
(Igt (l) f ( J~middot )-IIV)1 IlJlOlltllaHTOBaHHOIO no 3aHOlIy
bull J ~Y-j t ~ I d o ) 0 I J
tr ( ~ bull I bull ( I ) bull - (- 0 ~ ( ) I ) ()
lpeDTIoJIo~HM iTO COOTBeTCTBYJOIIlee M 8CCHQeCKOe ypaBHeHue nBIDlteHHR
rJ l~ I (l] ) - 0 U1w1eeT HOTPI1D1HlJTbHOe peUiolUIc I HarrpllMep COJIHTOHHOro TlATI8
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PeweHle J ~eTCR TUlBCClflaCKOH COCTaBJUlJOIIleM nOJleBOli
lJyHKllJlI l onpeneJlRer npOCTpaHCTBelUlNe cBOilCTBa OCHOBHOro COC-
TOflHHfl CymecTByeT deCKOHe~Oe ~~nno pa~ KnaCc~eCKHX pemo~ xapaJToplI3yeMliX napBM9TTIOM TIJaJiCJIJlUill l J RMeDIIIU OnEO il TO lK9 SH8shy
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MTpmo poundm lQ)YJIJ fllnOh jlUiIOUKii ypaBHelflrii JiuIepa B03MOJilHO H8pyme1fi1e
II WYnlX CaMM8rpII CBilloHlI~X C rpynnoli nBHE9IDli 3qqeKTHBHblM MeTonOM
nUJIIiJ1lffiOIlKM paCCM8TpimaTh CKCT9MH C BIlPO~9reM bull RBJIR9TCfl M9rOll K8J10shy
1IleCKlIX npe06p1iOpoundlPHn~ HlI lorOm06oB8 5-7
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lltOo plUfHaT w B aB~ICiMOC Tll 01 ItWil1lOCKO~i c KTy8URK MO)l(HO BBeCTll TaKshy
w yrJlO BWe rrepsMemme K COOTBOTCT3YJOOIHeurol UlCTO JlopeHlJeBCKHM Bpa-
UGlliU1M nepeM8HHHe ~~ bull BBe)1eHKe KOJIJICI(TKBHIlX lOOPMH8T n03BOJIJleT peshy
~30BaTh Ha nOJIeBHX nepeM9ltHHX lleKOTOpoe TlJl9nCT8BIIJiIfI19 rpymw IIyaHK8shy
pc 4 lLnR 3TO ro C pDRTCil KHBapHaHTIill8 OTIlOCHTeJJbHO npeo6pa30 BaHHH
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l KOJlJleKTIIBHblX lltOOp)lIDl8T (y H RBJIOHHleCKH corrpmKeHshy
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JIOlKeHJl e 11 o JReHTal(l11O npOTRlIWHlIOro OObeKT8 B npoCTpaHCTBe
0600weHMe JIBYMepHOM Mone~ C~HOI~ nORfl BKnD~aeT B ceOH H3y shy
~eHHe MHOIOCOJIlTOHHOrO c e opa 48 pacIIIlfPeHHa l1ICJla H3MepeHIDi rrpoc shylTpaHCTBa MHlIKOBcoro 9 IO bull rroJIei1 60JIe e CJIOJllHOh 1 c~ JIopeHUeshy
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UanL naHHOH p aOO TIl COCTOHT B nocTpoeHHK B leTupe XMepHoM rrpOCTshy
paHCTBe KBa 3HIUIaCCHeCKoro rrpH~effilfl AHR ceKTopa TeopKH conepMashy
mero orum npoTIDKeHlIWt OObeKT OCHOBHOl-i 3anae~ RBJlReTCH H3yqeshy
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lil HllX ) OObeKTOB B leTup9XMepHoM rrpOCTp aHCTBe-BpeMeRlJ
fuaH H8illerO 3iOyeHliil CJIenYJOllllR B sect 2 omlcuaaeTCR M8TOll IUlHOH1shy
tteClillX rrpeoop a30Erumi HE borom060Ba lI)IR rpymm ITYaHKaje B sect 3 peshy
waeTCR sanal8 0 il O llfl I-IHBapHBlITHUX qJyHKIUtiI ~(L ) B le TupeuroXMOp shy
HOM rrpoCTpaHCTBC ( ( = middot1) i3 sect 4 B KBa3HKJIaCClfleCKOM rrpl6JrDemm
CrpOIlTCfl KOBapvaHTlioe JlOY-aJIbHOe nOJIe JIJIR MoneID ITpOTroKeHHoro 06beKTa
YlTOnl PI HUBOr IT) BnlHII UDle-ULI sect 5)
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B STOM pasneJIe MIJ paCCMOTpIDiI nonxon K pemeHlflO 3analH KB3HTOBashy
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lmleCKHX rrpeo6paSOBElHldt H H EoroJl1060Ba BUIJeJT9lme KO1leKTJlBHIIX KOshy
opIHHaT n03BOJlReT nopenncaTlgt KCXOJlHlle ypaBHeHRil lepe3 nepeMellHue RHshy
BapH8HTHbl9 OTHOCHTeJllgtHO npe06pa30BllIiKf1 rpyrrIJ] CllMMerpRH It CTPOro
Yl9CTL COOTBeTCTBYJOutI9 38KOHbI COXpaHeHl1R B JOOC50M nOpR)tKC l eOplH BO~gtshy
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HapyruelfHeM OID1MeTpl1Jl
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HapywelllOO TpaHCIUIIrOHHOH H JlopeHlleooii CaMMeTpl1i1 TO ooycJlODJIellO TeM
ITO B JUI8cclPlecKOiI MexazrnKe MarOpH8JThHOi TOUltH OOH3aTenJHO CJlenYCT
y-Ka3aTlgt B K81ltoki TOUte ITpOCTpaHCTBa II C I(Bl()ji CKopocn1O JIBI1liBTCR laCshy
TKl(a ECJm rrpoTfIlI(eHHhtii 06b9J1 crpeplfleCIJI HeCllMMelpI1l9H TO llOllO1lH1shy
reJlbHO CJIenyeT YKa3aTb ero 01l19J1r8UR1O B lI]10CTpaHCTBe
IIepelineM KAicCMOTpelUfJO rJxgtlUympOBlltH KOBapHBHTHoro aHaJI01a M9shy
Tona Eoromo60Ba Ha npHMepe nOJIeBoi1 MOlleJrn CaMoneiiCTBYJOI1ero CI(8]RJr-
Horo nOM c--t) B ( - MepHOM npOCTPaHCTIl8
Bu6epeM 1larpaHRaaH eopIDf B cpo~e
I( lA -= - J lt-~ ) Lmiddot ( ~ I)
r ne rJ = Jt bull = D I d - I ll C (W -CKaJlJJpHaRI ~ -
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pemeHl19 JI~ ()) ( He OORSaT8JThHO CTaTllleCRoe) C KOHellloll 3Heprliel1
J~aCCHqeCKOrO ypaBHeHRR A~emlR
I - ( - i-~ t r ( () ~ 0 (2 2 amp
npraquo K8HOilllfJCKOM KBaHTOBaHIDl cPY1i~ IX) C1aJIOBHTCR onepashy
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ypaJ3HeH1flO ( 2 ) 1 ltaH01UfleCKKM tOMM-yTauHOHHHM COOTTIOllleHIDlM
1 1 l [J J i f~) - ~ i A 7
1= 0 (23 )$ I c) ~ (~o)
3neCh Cf - pOpMamHIDi napaMeTp TeOpKH B03MYll1eillili bull
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ypanlleHlfil (n H CooTHowelUlM ( 23) He M9liffeTCR 1Iplf rrpe06pa30BlHlffiX
nOJIR
T ~ ~ i -l ) L -1 bull 1 J I J ) ( v I K ( N4)~ 1 I L = +( X I 1ltmiddotl v -==
reHepaTOpaMll OTRX rrpeo6pa30aaHHil RBJIRlOTCR onepaTopbl
v I I middot I bull l J J )j C - J t L - )JlI) ~ l ~ ~ ) r U (~ Ii J
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1 J middot Z J j
J
nOCTPOeHHbl9 lIO TeopeMe HeTOp H3 (2I)
KnaCCi1leCKOe pelUelUfe ypaIlH9Hldi (2 2 ) B006mo roBOPiI HellRBapUshy
aJlTHO OTHOCllTeJIbHO npc06p a30Brumi (21) Ii cIH3lfleCKaR CHCT8Ma ORa3Hshy
BUSTCR Bupo~eHHo~ TpYnHOCTK CBR3aHHHe C ~eM B~~eBRfl B
CHCTeMe CHIDllaJOTCfl 8CM BUllomUTb npeo6pa30BaIDe BoroJl06oBa K HOshy
BUM nepeMeHHhIM
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BOpmoT COOTHOIlJeIUUlM aJlrc6pu ITyaHKape I
1 10 0) 1- 0 [ l) II I - J ( 1
~ rI urLo I r ~ j ) - - lor id~ I
i 1 ~ 1 - I t ~ -- (t f middot1 L 1 I I I ~ j - l ltJI tmiddot I ~ - ~ 7 1_ bull 1 v I 7 j J J i I J ~
PaCCMOTpHM rrODCTaHOBKY
On-) = ~lS())) jJ i ~pl t lts~lx) l til-f) ( 2 6)
rne t lt -lllfC]lOBHe nepSMeHHIl8 IiHTerpepoBClHIDl a orrepaTopno 3HatJHHa
i[lYH~ C (1) fIBJIJlj)rCfi tlJYH~ ti=l MJ fi t bull IIpeIJJl0JI8rae1cJI
iTO BTOPIflliO KBaJ-ITOBDFIHClC 110ne i Ll) ]OMMYTHpyeT C onopaTOprum 113
aJIreOpli P a rrpoClP8HCTBO rrpCDCT8DJeHlUI rrOJIJ C(~) HBiLIleTCR 1Tp1lshy
MIlM npDM3BonelliHlM npoCTprulC IB ITpenCTaMeami rurre6pH P If rrOJIJ qLX)
ITepeMellHhle C ltl) OIlJ)eneJIeHH CBOHCTBOM
LJ ( - ) 4- bull I
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r e ~~~ 1-0 (- amp- - ~~ ( 2 7))
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llJfOHIillMH CBol1CTBaMH nOAA qgt( x ) _
IIoMe BbIllOJTHeHlU 3aMeHbI ( 26) ( 27) onopa T0lgtu ) bull I~I r v C1ashy
llOBilTCR reuepaTOpaMM rrpocrp8HcTueHHo- DlJeMeHlillx IlpI)06Ja~OBam[li OTOlKshy
neCTBJlCHH8 3TgtlX orrepaTopoB C DblpaKC lmHmiddot~ 1 COOTIleICTIJyY1Uij1X r eH9[JaTopoB
(2 5) nocTpoellliillC no rroJllO lJLl ) C nOMOll1hlO TeoperllJ leTep lae T one shy
paTOpHIle CBSl3H KOTopue 11 06eCrrelJffiaOT COXpaJlCHJ1e amp101lt1 H93llillCJLUIX
nep9MeHlillX llO 11 IlOCJIe npeo6pa 30BaHJl[R ( 2 6 ) II Clffi311 paCCMrlTpgtIB8JilTCR
Ha ReMTope COC1OflHlllil )~) M ID-leJOT mUI
bull j shy
) 0 I -- C iI M ) - l L r - [ l raquo) J l r - [ [ - I 4 _ ~ I bull I y- I - bull (2 8) I
ITpe0 6pa30B8JlIIe (2 6 ) C YCJIonPlfMJl c Efl3eJi (~o C ) Jo ]OMMYTaI(J10HIl1lMl1
COO THoweH1HIMVI DJIR nOM C x ) bull neperrncaHHbJM B HOBUX nepeMe HHblx pound -t)
bull ~ ) ~ j ) f ) ~ JL -Su h~ J - I cgt (2 9)
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pc1 1 n03BOJIRCf fi lUlHOM BVIDe peaJlJl30B8Th np0)1CTaBJIelUf8 lTOti rpymrn
JJnR TOlO 1m6b1 HDHUM 0 6pa30M BHITOmUfTb npeoops 30Bamre BaroJTI(-OBa CJJc nyeT IlIll1TH I1lIBaplIaHTHHe nepeMelllible ( J ) VI rrpoK3se CTK 3aMeshy
By nOlJeMeHHJiX B COOTIlOlUeHl1JlX ( 0) ( 9) lTO VI 6YJ16T CJ1eJI8HO B sect 3
II sect 1 COOTBllTC TIJeHIO
~ J 1 0ClpoeHliu P11lllapHaJfTIIIIX nepeMOIU11IX
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B IHrMHllTeJHMaJIbHO~i 11-OpMe -
1 ~
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AMee n TOM pa3J1eJre H8M 0 YDeT YllOOHO nOJIOJKlllTlgt I) 1
Cymec TBOBaHHe pelllemrn ypaBHeFlliH (3 I) H FIX mgtmm BlUl 3aBHClIT 0 T
Budopa KOHKpeTHoro rrpe~cTaaneFllilI anredpu rpyrrnu llyaHKape jO B Tepshy
MHHax KaHOmrqeCKFlX orrepaTopoB ( y p I 0 Po ) Budop lKe rrpencTaaneshy
IDlI orrpe~eIDeTClI lM3FIleCKIDIIlI CO 0 clpalKeFlliHMH bull l1epeMeHHHs S (X)
FlMeIDT CMHC~ ododmeHHHx KBaHTOBHX Koop~aT KOTopue B COCTOJIHHH rroshy
KOlI od~eKTa rrepexo~JlT B ~CCHieCKOM rrpen~e B rrpocTpaRCTBeHHO-Bpeshy
MeHHlle KOopmmaTu rrpoCTpaHCTBa MmlliOBCKOro 2- bull OTMeTflM lTO t gt
~O~ dUTb de3pa3MepHHMH Be~HBMH TaR RaK rrpeodpa30BaHHe ( 2 6)
He co~epJKHT lliC~OBOro pa3MspHoro rrapaM9Tpa IIpH 3TlIX ~orrO~Te~HHX
ifJH 3HlSCKFlX rrpenrrO~OlKelmRX ypaBHeFlliJI (3 I ) rrO~OCTbID orrpene~T d (cIshy-pa3MepHocTb rrpoCTPaHCTBa-BPSM8HFl ) H93aBHCRMllX HHBaPllaRTHliX OTHOCHshy
TenHO rrpeodpa30B8IUlli rpymm n yrurnape rrepeMSIDillX S-lt RaK lPyHKIUm aprYMeHToB X ~ bull J bull () bull tJ B WHRILHRX S rrpenrro~araeTClI orrpe~e~eHHoe yrropRUOleHHe H8JltOMMYTHPynttHX rrap orrepaTopOB Ii bull ~ H
I () () - HarrpgtIMep ( II Jj c D J)
I L~ I l l
PaCCMOTpgtIM Terrspb rrpOCTei1wee rrpenCTaBJISIDe aJIIsdpu F lepe3 Kashy
HOHWleCKH8 orrepaTOpu I il KOTOpoe peaIDl3ye TcH B rrpOCTpaHCTBe
QJyHKlUfi1 1 ) H 3MaIIO C rrOMOIllMl Coo TBolDemrli
p~== r ( 1v1r = ((~ J P - l( G Pf) (32)
Lp l~ ( shyI I ) L (1
B rrpOCTpaHCTBenBYX H3M8peHlll1 ( j = 2) ypaBHeFlliH (31) B rrpeJ]shy
CT~eHHH (32) gtlMeIDT CBOFlM pellleFllieM ~Ba ~HejiHllX rro 2 orrepaTopHo-
3HalHHX pemeHKH
toUJ~2~1 +f~Ir T-Y))~ JI r~ (33)
~ _ -ltv ) t~l(lf= cl (X7~r -= u ~ L v r flo1
B BHpaJKeHm ( 3 3 ) oIIepaTop JIOpeHIJeBCKIX BpameHJ-fi1 OL c ~ a orrepashy
TOp j)= lIc l 11 P pr(1 ~) gtlMSeT TaIDle lKe KOMMYTamlOHHue CBol-icTBa C
anre6poJ C2 IlTO H ohepaTOp pacTJIlKeHIDi
I1HBapHaHTHUe rrepeMeHllile S - (v-G = 0 I) ItOMMyrnpylOT MaJKny COshy
dOll FI rrOJIHOCTbID y~OBJIeTBOpHIDT BceM TpedOBaHllHMHeodxo)lHMHM ~ HCshy
IIOn30BaHHff 3THX ifurmami1 RaK apryMeHToB B rrpaBofI laCTH rrpeodpa30BaIllilI
(26) B nByMepHllX MO~eJIRX OrraCliBaIDmKX KBaHTOBOe peIDTllBHCTcKoe nBHshy
llleFllie OlJHoro rrpoTJIlKeHHoro od~eKTa2 4
KoMMyrHPyromne KOBapHaHTHe Koop~HaTu qrBHp8lKeHHl1e lSpe3 orrepa-
TOpH H3 aJIIedpH t(f2 ) D 1 6
l-iL ~Cl_~J il(~ - 2 D M ~ 2 L Me 1- M2 J j
i -=1 r pi l i if I ~ 2 lD tj 2 J -2 LMOL Mo J I
(34)
1v)=P -P~ C l )
JIBiIRKlTCR aHaJlorBMI1 KOopmmaT fu IOToHa--BITHepa B ~YMepHOM iIpOCTPaH-
CTBe
OleBHJHO lTO B rrpoC TpaHCTBe d raquo 2 H3Mep eHllM cywecTByeT Tonshy
KO OIHO JIlIHelHo e rro X pelliorme (31) Sf2) = 2~~j) 1I03Toshy
MY B 06mSM cnyqae rrpelLCTaBJI8flHe (32) HerrpIDoIlRO II)IR OIIHCaHHH peJlRshy
THBHCTCKOro IIpOTHlKeIDloro Od1eKTal1 CJI8Iye T paclllFlpHT1 KO lqmrypaIJHOHshy
Hoe rrpoCTPaBC TBO rrpe~CT~ermlI ~OrrOrrHffB ero YTJIOBHMH rrepeMeHHYMH
OlLHaKO eCJIB rrpoTHlKeHHilll od~eKT 06~aJLaeT CBO~CTBOM c~p~eCKOH
CI1MMeTplm TO OKa3UBaeTClI ~OCTaTOiHO nOCTpoHTb ~e IDmapHaHTHllX KOshy
opIJffiaTu BpeMeHHyIO H paJJlfanHyro
Sc()c) = (ctr~)rpr) l~ x)l coo LJlql() (35)
~TO HaxO~TCR B COOTBeTCTBllH C pe3YJIbTaTaMll pa60T9
KaK j~e OTMelaJIOCb Bume paCCMOTpeHH8 dOJIee od~ero CJIYiaJI MOshy
ue1IH cifgtepWleCKI1 HeCHMMe TpHlHoro od~eKTa TpedyeT rrepexoua K dOJIse
C~CraquoUUlM npelLCT~eH1UlM lJJm OrmCaHMH CTaTHleCKoro ~eiJoPMrrpOBaHHOrO
06beKra MH BH6epeM rrpe~CT~8ID8 anredpu lIyaHKape 13 iPJPMe
J C l e lL r I f f11S P = Igt IV) = (i f - ~I ) shyJ ) ) ( (J 1)
)-) K1 -- (C( pd - ~r~ ~ ~ - c - J
1) fj =t ~ )(36) j ~ - J
1 r~ [S ) f ~1 Li ( y) 7 ( - ( r 1 1= (C Coil S~
~ro npSuCT8BJI8HH8 peam3yeTcH B rrpoC Tp aHCTBe IWH1QWlli If yJ ~ r~e X - yrJill rrpoCTpaIlCTBCHHHX BpamellmL TIo ifJH3H1eCKOMY CMliCJIy KO[shy
JIeKTIlBHlle KOOPlUrnaTH r HI ol1IJe~eJIRlOT rrMOlKefDlle ~8txlpMllpOBaHHOshyr o od~eKTa 1lt1 ero opHeraquoTaIJJm B rrpo CTpaHCTB MaTp~ vl lre( YJ B (35 )
3aJLaIDT BHn r eHepaTopoB rpymru Bpame~ S t I l) pyt
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
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~~ ( bull l )
c )H Ij n i 1 I () ( 6)
~ )l i )1 rT ) 1) 111
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orrcpAToF3 ~
12
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AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
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JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
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ITPU)JCT8BI1M IToJIe ([II C ) R13Jrmomeecp lBWeHveM (4 5 ) H Blfne Pamp3JlomiddotshyteHJUl no nOJllioll CHCTeMe pem8H1f~ YP8BHSHIfl
J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
YCJWBI1H (4 8a) (4 8d) flCIltJ1X)laroT fl p a ilIoJtEilTllil J(OMIIOHGIITLI COOTlef shy
cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
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1 laquo) (I) (4 Ill)I ( ~
HrullleM fII 1(lMMY1lHlIICI II II II
11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
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nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
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-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
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1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
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(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
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pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
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lIWHJlOIO o01ewra
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u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
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npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
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5t BKJIlOliHT8JIbHO
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JI H T C I a Y P a
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Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
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T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
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J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
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ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
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npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
sectI B B e ~ e H 11 a
B HaCTOflllle~ 180010 Mb 118CCM01PUM 3analY 0 nOC1poellJm KOBapIWH1shy
HbI)( KBWITOBO- fLOJl013llX pomellltil B MOl(B1JlX npOTJlli(GI1HUX O(YL8KTOB raRllC 10shy
nllrm upimSITO paCCM8rI1iB8Th B CBfl31 C rrpo6JIOMOii rrOCTpoelUlfl MOneJlei1
3JIeMe HTapHWI lBCTmJ K lIMN OTllOCATCfI HrurPMM8j) MO)le1lH Maumoe CTJYH I
BO1TlKOB MIJ 6yneM paCCM8TlJJl138TT KnaHTOBUe JJeJIJIfl13f1C1CKIH MOneJlJl JlJlJi
KOTOPWC COOTBeTCTByut1iIe MaCCHlOCKHe Y1lSD1JOHIiJi JOIIIJlelll1J HMel()1 HeTpUshy
BJllalThHOB T e 3aBHcsnnOe 01 nallaMe1poB IpynnlJ CmAM8TIim MOneJlJl KOHelshy
Ho-moprenf leCIWe pornelille bull
MOTOnb onopalopHoll TCOPlIJI nOJIll KOTopble rT03ROJLfl1OT IlJ)OKBaHTOBaTIgt
CHCTeMY B OqlOCTlIOCTH luracCHleCKOIO pemeHJIR IlruiTII COCTOJIHID I
ClIeKTp KBaJTOBhlX 4urYKTyauKll xopomo U3BeCTllhl 1 Ilac oyneT 1IHTepecoshy
BaTT W1I 014 acrreKT TfOPIUI ]orna OCHOBHOC BIDIMrume ynellIeTCFI nOCTposhy
ermil JlORalIhHOIO KOB8PlUUlTllOTO rrOJIR2- 4
I1pocTefllllegt MoneJThHl KOTOJ8fl B nOCTaTOtlloH Mepe HJlJIIOCTPlAPY8T Bce
B03lmlUlunHe oco6eHHOCTH I FIBJIjeTCR MOnOJrb )leilCTlHI18JrbHOro CKaJmpHOTO
leii3eu6eproBClltOTO nOM B lJBYMepHOM rrpOCTp8HCTHe-BpeMfIIM
(Igt (l) f ( J~middot )-IIV)1 IlJlOlltllaHTOBaHHOIO no 3aHOlIy
bull J ~Y-j t ~ I d o ) 0 I J
tr ( ~ bull I bull ( I ) bull - (- 0 ~ ( ) I ) ()
lpeDTIoJIo~HM iTO COOTBeTCTBYJOIIlee M 8CCHQeCKOe ypaBHeHue nBIDlteHHR
rJ l~ I (l] ) - 0 U1w1eeT HOTPI1D1HlJTbHOe peUiolUIc I HarrpllMep COJIHTOHHOro TlATI8
A1 ilt lt --(_t lt ) I -Z
xlTe~H ITJOTRlI~IIHlIH o6bowr 3aMeR1OT 3neCh Te~ COJIHTOH TJaKTyet~ B 1ID11JOKOM CMHCIle
PeweHle J ~eTCR TUlBCClflaCKOH COCTaBJUlJOIIleM nOJleBOli
lJyHKllJlI l onpeneJlRer npOCTpaHCTBelUlNe cBOilCTBa OCHOBHOro COC-
TOflHHfl CymecTByeT deCKOHe~Oe ~~nno pa~ KnaCc~eCKHX pemo~ xapaJToplI3yeMliX napBM9TTIOM TIJaJiCJIJlUill l J RMeDIIIU OnEO il TO lK9 SH8shy
leHKe 3HeprIDI I bull rne i1 j 1( LYU)l- TalUIM 06pa30M OCHOBH08 COCTOIDUlO BIIPOJKn6110 OTHOCJlTeJThHO npe06pasoBashy
HlIil TpaHCJUIlUli a KJlaCCIflOCltO(gt pewemte HapymaeT TpaHCJJUlKOHHJ1O CHMshy
MTpmo poundm lQ)YJIJ fllnOh jlUiIOUKii ypaBHelflrii JiuIepa B03MOJilHO H8pyme1fi1e
II WYnlX CaMM8rpII CBilloHlI~X C rpynnoli nBHE9IDli 3qqeKTHBHblM MeTonOM
nUJIIiJ1lffiOIlKM paCCM8TpimaTh CKCT9MH C BIlPO~9reM bull RBJIR9TCfl M9rOll K8J10shy
1IleCKlIX npe06p1iOpoundlPHn~ HlI lorOm06oB8 5-7
OCHoBnoii l1lll)l)1 1~IJTona igtorom06ona JlaJUleTCfl nepexon 1lt HOBbIM mmashyMIIWC KHM ne peMtl 11111 1M C1IJIJJ KO 10plH HMeJOTCil )lOnOJI1llTElJI1gtHlie nepeMoHiffie
nBJIIDOlIHCCfl napaMl IlBMM rpYTmu (IMMU11I1J I1x npllllIiTo HaSliBaTL KOJIJIBKshy
1aHIIIlMH Koop)JHHaTaMK tl CJly48 lpy l1ltH MlDK9IDrn KIIOMO TpaHCJIJIDltO~UlJD(
lltOo plUfHaT w B aB~ICiMOC Tll 01 ItWil1lOCKO~i c KTy8URK MO)l(HO BBeCTll TaKshy
w yrJlO BWe rrepsMemme K COOTBOTCT3YJOOIHeurol UlCTO JlopeHlJeBCKHM Bpa-
UGlliU1M nepeM8HHHe ~~ bull BBe)1eHKe KOJIJICI(TKBHIlX lOOPMH8T n03BOJIJleT peshy
~30BaTh Ha nOJIeBHX nepeM9ltHHX lleKOTOpoe TlJl9nCT8BIIJiIfI19 rpymw IIyaHK8shy
pc 4 lLnR 3TO ro C pDRTCil KHBapHaHTIill8 OTIlOCHTeJJbHO npeo6pa30 BaHHH
FgtynI1H IIyaHl8pe ~YHKIJIDI OT npDcTp81CTB9HHO-BpeMOHHJD( KOop)UUlaT
l KOJlJleKTIIBHblX lltOOp)lIDl8T (y H RBJIOHHleCKH corrpmKeHshy
HlX 1m Be1llfmH B KJIaCCIPl eC1ltOM ITp9ne1l8 ipyluuum ~ xspaKTepH3YllT no-
JIOlKeHJl e 11 o JReHTal(l11O npOTRlIWHlIOro OObeKT8 B npoCTpaHCTBe
0600weHMe JIBYMepHOM Mone~ C~HOI~ nORfl BKnD~aeT B ceOH H3y shy
~eHHe MHOIOCOJIlTOHHOrO c e opa 48 pacIIIlfPeHHa l1ICJla H3MepeHIDi rrpoc shylTpaHCTBa MHlIKOBcoro 9 IO bull rroJIei1 60JIe e CJIOJllHOh 1 c~ JIopeHUeshy
Boi1 CTPYKTypII pemoffile npod1leMbl rrepeHopMlfpOBOI I2 If T bull n
UanL naHHOH p aOO TIl COCTOHT B nocTpoeHHK B leTupe XMepHoM rrpOCTshy
paHCTBe KBa 3HIUIaCCHeCKoro rrpH~effilfl AHR ceKTopa TeopKH conepMashy
mero orum npoTIDKeHlIWt OObeKT OCHOBHOl-i 3anae~ RBJlReTCH H3yqeshy
HIIe B03MOXHOCTH pacCMOTpeHKR ne~pMKPOa8HHllX ( cqepKqeCKH HeCI~eTshy
lil HllX ) OObeKTOB B leTup9XMepHoM rrpOCTp aHCTBe-BpeMeRlJ
fuaH H8illerO 3iOyeHliil CJIenYJOllllR B sect 2 omlcuaaeTCR M8TOll IUlHOH1shy
tteClillX rrpeoop a30Erumi HE borom060Ba lI)IR rpymm ITYaHKaje B sect 3 peshy
waeTCR sanal8 0 il O llfl I-IHBapHBlITHUX qJyHKIUtiI ~(L ) B le TupeuroXMOp shy
HOM rrpoCTpaHCTBC ( ( = middot1) i3 sect 4 B KBa3HKJIaCClfleCKOM rrpl6JrDemm
CrpOIlTCfl KOBapvaHTlioe JlOY-aJIbHOe nOJIe JIJIR MoneID ITpOTroKeHHoro 06beKTa
YlTOnl PI HUBOr IT) BnlHII UDle-ULI sect 5)
l
sect 2 Ilp eo6pasoBamlo HH EoroJIJ06oBa
B STOM pasneJIe MIJ paCCMOTpIDiI nonxon K pemeHlflO 3analH KB3HTOBashy
HRR rrOHR B OKpBCTHOCTH KJlaCCH~ecKoro pemeHKR C TIOMO~~ MeTOna KaHOshy
lmleCKHX rrpeo6paSOBElHldt H H EoroJl1060Ba BUIJeJT9lme KO1leKTJlBHIIX KOshy
opIHHaT n03BOJlReT nopenncaTlgt KCXOJlHlle ypaBHeHRil lepe3 nepeMellHue RHshy
BapH8HTHbl9 OTHOCHTeJllgtHO npe06pa30BllIiKf1 rpyrrIJ] CllMMerpRH It CTPOro
Yl9CTL COOTBeTCTBYJOutI9 38KOHbI COXpaHeHl1R B JOOC50M nOpR)tKC l eOplH BO~gtshy
MYIIeHlfR TauaR npouenypa YJFe B HYJIe130M nOpfllIRe TeOplrn BO 3MymeHru1
yltUJTllBa9T H9TpHBli8lThHYJO CrpyKTYJlY peweHHR CIlfI38HJljO co cnOHTaJIIIllM
HapyruelfHeM OID1MeTpl1Jl
HaJmUe B CHCTpoundJMe npOrf11tCHIlOro OOM I(T8 OOR3areJTbllO npHBOIUIl I
HapywelllOO TpaHCIUIIrOHHOH H JlopeHlleooii CaMMeTpl1i1 TO ooycJlODJIellO TeM
ITO B JUI8cclPlecKOiI MexazrnKe MarOpH8JThHOi TOUltH OOH3aTenJHO CJlenYCT
y-Ka3aTlgt B K81ltoki TOUte ITpOCTpaHCTBa II C I(Bl()ji CKopocn1O JIBI1liBTCR laCshy
TKl(a ECJm rrpoTfIlI(eHHhtii 06b9J1 crpeplfleCIJI HeCllMMelpI1l9H TO llOllO1lH1shy
reJlbHO CJIenyeT YKa3aTb ero 01l19J1r8UR1O B lI]10CTpaHCTBe
IIepelineM KAicCMOTpelUfJO rJxgtlUympOBlltH KOBapHBHTHoro aHaJI01a M9shy
Tona Eoromo60Ba Ha npHMepe nOJIeBoi1 MOlleJrn CaMoneiiCTBYJOI1ero CI(8]RJr-
Horo nOM c--t) B ( - MepHOM npOCTPaHCTIl8
Bu6epeM 1larpaHRaaH eopIDf B cpo~e
I( lA -= - J lt-~ ) Lmiddot ( ~ I)
r ne rJ = Jt bull = D I d - I ll C (W -CKaJlJJpHaRI ~ -
IPYHKl(HR orpaHWleHHM cIDI3y llonYCTlIM ITO cymccrByeT ReTplfBllaJILHOO
pemeHl19 JI~ ()) ( He OORSaT8JThHO CTaTllleCRoe) C KOHellloll 3Heprliel1
J~aCCHqeCKOrO ypaBHeHRR A~emlR
I - ( - i-~ t r ( () ~ 0 (2 2 amp
npraquo K8HOilllfJCKOM KBaHTOBaHIDl cPY1i~ IX) C1aJIOBHTCR onepashy
TopH03Hawoil CIUlUIpHoi JlOJltalIbHOn nOJleBOJt dlY1UUIKel YJlOBlIeTBOpmlUIol
ypaJ3HeH1flO ( 2 ) 1 ltaH01UfleCKKM tOMM-yTauHOHHHM COOTTIOllleHIDlM
1 1 l [J J i f~) - ~ i A 7
1= 0 (23 )$ I c) ~ (~o)
3neCh Cf - pOpMamHIDi napaMeTp TeOpKH B03MYll1eillili bull
~
TeopHH OrrpeDCJIeHHaR JIarpaOOaHOM (2 1) 0 tl9BHRO mmapRaHTHa
OTHOCBTcmHO rrpo06pa30Bamm rpyrrrnl IIYBJiKape lTO o 3 HaWeT bull lTO BllA
ypanlleHlfil (n H CooTHowelUlM ( 23) He M9liffeTCR 1Iplf rrpe06pa30BlHlffiX
nOJIR
T ~ ~ i -l ) L -1 bull 1 J I J ) ( v I K ( N4)~ 1 I L = +( X I 1ltmiddotl v -==
reHepaTOpaMll OTRX rrpeo6pa30aaHHil RBJIRlOTCR onepaTopbl
v I I middot I bull l J J )j C - J t L - )JlI) ~ l ~ ~ ) r U (~ Ii J
_ - L ( u
I I C ~ ~) ( -= - ) J
I J l L l ~ J=
- (25)1 I I ) I f ~ ) J f
1 J middot Z J j
J
nOCTPOeHHbl9 lIO TeopeMe HeTOp H3 (2I)
KnaCCi1leCKOe pelUelUfe ypaIlH9Hldi (2 2 ) B006mo roBOPiI HellRBapUshy
aJlTHO OTHOCllTeJIbHO npc06p a30Brumi (21) Ii cIH3lfleCKaR CHCT8Ma ORa3Hshy
BUSTCR Bupo~eHHo~ TpYnHOCTK CBR3aHHHe C ~eM B~~eBRfl B
CHCTeMe CHIDllaJOTCfl 8CM BUllomUTb npeo6pa30BaIDe BoroJl06oBa K HOshy
BUM nepeMeHHhIM
ITYCTb apmTOBli Ol1ep aTopll 1-13 MHO~e(rBa 5 1Pl J 1 bullbull 1 YDOBJIeT-
BOpmoT COOTHOIlJeIUUlM aJlrc6pu ITyaHKape I
1 10 0) 1- 0 [ l) II I - J ( 1
~ rI urLo I r ~ j ) - - lor id~ I
i 1 ~ 1 - I t ~ -- (t f middot1 L 1 I I I ~ j - l ltJI tmiddot I ~ - ~ 7 1_ bull 1 v I 7 j J J i I J ~
PaCCMOTpHM rrODCTaHOBKY
On-) = ~lS())) jJ i ~pl t lts~lx) l til-f) ( 2 6)
rne t lt -lllfC]lOBHe nepSMeHHIl8 IiHTerpepoBClHIDl a orrepaTopno 3HatJHHa
i[lYH~ C (1) fIBJIJlj)rCfi tlJYH~ ti=l MJ fi t bull IIpeIJJl0JI8rae1cJI
iTO BTOPIflliO KBaJ-ITOBDFIHClC 110ne i Ll) ]OMMYTHpyeT C onopaTOprum 113
aJIreOpli P a rrpoClP8HCTBO rrpCDCT8DJeHlUI rrOJIJ C(~) HBiLIleTCR 1Tp1lshy
MIlM npDM3BonelliHlM npoCTprulC IB ITpenCTaMeami rurre6pH P If rrOJIJ qLX)
ITepeMellHhle C ltl) OIlJ)eneJIeHH CBOHCTBOM
LJ ( - ) 4- bull I
~~ h t~ t- l L lor ~ 1- ( tr r l j
r e ~~~ 1-0 (- amp- - ~~ ( 2 7))
~ - JI ~ iL~) ( ) f ttf v shy yOlopoe r apaHTHpyeT COBMeCTHOC1h npe0 6pn30BaHlUI (~ 7) C Tjl8JICT1loprlttshy
llJfOHIillMH CBol1CTBaMH nOAA qgt( x ) _
IIoMe BbIllOJTHeHlU 3aMeHbI ( 26) ( 27) onopa T0lgtu ) bull I~I r v C1ashy
llOBilTCR reuepaTOpaMM rrpocrp8HcTueHHo- DlJeMeHlillx IlpI)06Ja~OBam[li OTOlKshy
neCTBJlCHH8 3TgtlX orrepaTopoB C DblpaKC lmHmiddot~ 1 COOTIleICTIJyY1Uij1X r eH9[JaTopoB
(2 5) nocTpoellliillC no rroJllO lJLl ) C nOMOll1hlO TeoperllJ leTep lae T one shy
paTOpHIle CBSl3H KOTopue 11 06eCrrelJffiaOT COXpaJlCHJ1e amp101lt1 H93llillCJLUIX
nep9MeHlillX llO 11 IlOCJIe npeo6pa 30BaHJl[R ( 2 6 ) II Clffi311 paCCMrlTpgtIB8JilTCR
Ha ReMTope COC1OflHlllil )~) M ID-leJOT mUI
bull j shy
) 0 I -- C iI M ) - l L r - [ l raquo) J l r - [ [ - I 4 _ ~ I bull I y- I - bull (2 8) I
ITpe0 6pa30B8JlIIe (2 6 ) C YCJIonPlfMJl c Efl3eJi (~o C ) Jo ]OMMYTaI(J10HIl1lMl1
COO THoweH1HIMVI DJIR nOM C x ) bull neperrncaHHbJM B HOBUX nepeMe HHblx pound -t)
bull ~ ) ~ j ) f ) ~ JL -Su h~ J - I cgt (2 9)
H83UBaeTCil IltaHOlUflleCKHM rrpe OOpS30BallliCM IJoroJT1OCio Ba DJIjl pPyntlH lIYaHlltashy
pc1 1 n03BOJIRCf fi lUlHOM BVIDe peaJlJl30B8Th np0)1CTaBJIelUf8 lTOti rpymrn
JJnR TOlO 1m6b1 HDHUM 0 6pa30M BHITOmUfTb npeoops 30Bamre BaroJTI(-OBa CJJc nyeT IlIll1TH I1lIBaplIaHTHHe nepeMelllible ( J ) VI rrpoK3se CTK 3aMeshy
By nOlJeMeHHJiX B COOTIlOlUeHl1JlX ( 0) ( 9) lTO VI 6YJ16T CJ1eJI8HO B sect 3
II sect 1 COOTBllTC TIJeHIO
~ J 1 0ClpoeHliu P11lllapHaJfTIIIIX nepeMOIU11IX
$lUUUeM c VlcTeMY YPaBlIOHlHI ( ~ ) onpeneJlR1OOllofx iJJyHKUKJol C J)
B IHrMHllTeJHMaJIbHO~i 11-OpMe -
1 ~
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AMee n TOM pa3J1eJre H8M 0 YDeT YllOOHO nOJIOJKlllTlgt I) 1
Cymec TBOBaHHe pelllemrn ypaBHeFlliH (3 I) H FIX mgtmm BlUl 3aBHClIT 0 T
Budopa KOHKpeTHoro rrpe~cTaaneFllilI anredpu rpyrrnu llyaHKape jO B Tepshy
MHHax KaHOmrqeCKFlX orrepaTopoB ( y p I 0 Po ) Budop lKe rrpencTaaneshy
IDlI orrpe~eIDeTClI lM3FIleCKIDIIlI CO 0 clpalKeFlliHMH bull l1epeMeHHHs S (X)
FlMeIDT CMHC~ ododmeHHHx KBaHTOBHX Koop~aT KOTopue B COCTOJIHHH rroshy
KOlI od~eKTa rrepexo~JlT B ~CCHieCKOM rrpen~e B rrpocTpaRCTBeHHO-Bpeshy
MeHHlle KOopmmaTu rrpoCTpaHCTBa MmlliOBCKOro 2- bull OTMeTflM lTO t gt
~O~ dUTb de3pa3MepHHMH Be~HBMH TaR RaK rrpeodpa30BaHHe ( 2 6)
He co~epJKHT lliC~OBOro pa3MspHoro rrapaM9Tpa IIpH 3TlIX ~orrO~Te~HHX
ifJH 3HlSCKFlX rrpenrrO~OlKelmRX ypaBHeFlliJI (3 I ) rrO~OCTbID orrpene~T d (cIshy-pa3MepHocTb rrpoCTPaHCTBa-BPSM8HFl ) H93aBHCRMllX HHBaPllaRTHliX OTHOCHshy
TenHO rrpeodpa30B8IUlli rpymm n yrurnape rrepeMSIDillX S-lt RaK lPyHKIUm aprYMeHToB X ~ bull J bull () bull tJ B WHRILHRX S rrpenrro~araeTClI orrpe~e~eHHoe yrropRUOleHHe H8JltOMMYTHPynttHX rrap orrepaTopOB Ii bull ~ H
I () () - HarrpgtIMep ( II Jj c D J)
I L~ I l l
PaCCMOTpgtIM Terrspb rrpOCTei1wee rrpenCTaBJISIDe aJIIsdpu F lepe3 Kashy
HOHWleCKH8 orrepaTOpu I il KOTOpoe peaIDl3ye TcH B rrpOCTpaHCTBe
QJyHKlUfi1 1 ) H 3MaIIO C rrOMOIllMl Coo TBolDemrli
p~== r ( 1v1r = ((~ J P - l( G Pf) (32)
Lp l~ ( shyI I ) L (1
B rrpOCTpaHCTBenBYX H3M8peHlll1 ( j = 2) ypaBHeFlliH (31) B rrpeJ]shy
CT~eHHH (32) gtlMeIDT CBOFlM pellleFllieM ~Ba ~HejiHllX rro 2 orrepaTopHo-
3HalHHX pemeHKH
toUJ~2~1 +f~Ir T-Y))~ JI r~ (33)
~ _ -ltv ) t~l(lf= cl (X7~r -= u ~ L v r flo1
B BHpaJKeHm ( 3 3 ) oIIepaTop JIOpeHIJeBCKIX BpameHJ-fi1 OL c ~ a orrepashy
TOp j)= lIc l 11 P pr(1 ~) gtlMSeT TaIDle lKe KOMMYTamlOHHue CBol-icTBa C
anre6poJ C2 IlTO H ohepaTOp pacTJIlKeHIDi
I1HBapHaHTHUe rrepeMeHllile S - (v-G = 0 I) ItOMMyrnpylOT MaJKny COshy
dOll FI rrOJIHOCTbID y~OBJIeTBOpHIDT BceM TpedOBaHllHMHeodxo)lHMHM ~ HCshy
IIOn30BaHHff 3THX ifurmami1 RaK apryMeHToB B rrpaBofI laCTH rrpeodpa30BaIllilI
(26) B nByMepHllX MO~eJIRX OrraCliBaIDmKX KBaHTOBOe peIDTllBHCTcKoe nBHshy
llleFllie OlJHoro rrpoTJIlKeHHoro od~eKTa2 4
KoMMyrHPyromne KOBapHaHTHe Koop~HaTu qrBHp8lKeHHl1e lSpe3 orrepa-
TOpH H3 aJIIedpH t(f2 ) D 1 6
l-iL ~Cl_~J il(~ - 2 D M ~ 2 L Me 1- M2 J j
i -=1 r pi l i if I ~ 2 lD tj 2 J -2 LMOL Mo J I
(34)
1v)=P -P~ C l )
JIBiIRKlTCR aHaJlorBMI1 KOopmmaT fu IOToHa--BITHepa B ~YMepHOM iIpOCTPaH-
CTBe
OleBHJHO lTO B rrpoC TpaHCTBe d raquo 2 H3Mep eHllM cywecTByeT Tonshy
KO OIHO JIlIHelHo e rro X pelliorme (31) Sf2) = 2~~j) 1I03Toshy
MY B 06mSM cnyqae rrpelLCTaBJI8flHe (32) HerrpIDoIlRO II)IR OIIHCaHHH peJlRshy
THBHCTCKOro IIpOTHlKeIDloro Od1eKTal1 CJI8Iye T paclllFlpHT1 KO lqmrypaIJHOHshy
Hoe rrpoCTPaBC TBO rrpe~CT~ermlI ~OrrOrrHffB ero YTJIOBHMH rrepeMeHHYMH
OlLHaKO eCJIB rrpoTHlKeHHilll od~eKT 06~aJLaeT CBO~CTBOM c~p~eCKOH
CI1MMeTplm TO OKa3UBaeTClI ~OCTaTOiHO nOCTpoHTb ~e IDmapHaHTHllX KOshy
opIJffiaTu BpeMeHHyIO H paJJlfanHyro
Sc()c) = (ctr~)rpr) l~ x)l coo LJlql() (35)
~TO HaxO~TCR B COOTBeTCTBllH C pe3YJIbTaTaMll pa60T9
KaK j~e OTMelaJIOCb Bume paCCMOTpeHH8 dOJIee od~ero CJIYiaJI MOshy
ue1IH cifgtepWleCKI1 HeCHMMe TpHlHoro od~eKTa TpedyeT rrepexoua K dOJIse
C~CraquoUUlM npelLCT~eH1UlM lJJm OrmCaHMH CTaTHleCKoro ~eiJoPMrrpOBaHHOrO
06beKra MH BH6epeM rrpe~CT~8ID8 anredpu lIyaHKape 13 iPJPMe
J C l e lL r I f f11S P = Igt IV) = (i f - ~I ) shyJ ) ) ( (J 1)
)-) K1 -- (C( pd - ~r~ ~ ~ - c - J
1) fj =t ~ )(36) j ~ - J
1 r~ [S ) f ~1 Li ( y) 7 ( - ( r 1 1= (C Coil S~
~ro npSuCT8BJI8HH8 peam3yeTcH B rrpoC Tp aHCTBe IWH1QWlli If yJ ~ r~e X - yrJill rrpoCTpaIlCTBCHHHX BpamellmL TIo ifJH3H1eCKOMY CMliCJIy KO[shy
JIeKTIlBHlle KOOPlUrnaTH r HI ol1IJe~eJIRlOT rrMOlKefDlle ~8txlpMllpOBaHHOshyr o od~eKTa 1lt1 ero opHeraquoTaIJJm B rrpo CTpaHCTB MaTp~ vl lre( YJ B (35 )
3aJLaIDT BHn r eHepaTopoB rpymru Bpame~ S t I l) pyt
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
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j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
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~~ ( bull l )
c )H Ij n i 1 I () ( 6)
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orrcpAToF3 ~
12
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AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
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JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
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J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
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cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
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1 laquo) (I) (4 Ill)I ( ~
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11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
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nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
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-[ [1 ~ (t 1 ~ ( T
L1i ~
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ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
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1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
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(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
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11
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pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
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lIWHJlOIO o01ewra
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5t BKJIlOliHT8JIbHO
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JI H T C I a Y P a
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2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
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TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
PeweHle J ~eTCR TUlBCClflaCKOH COCTaBJUlJOIIleM nOJleBOli
lJyHKllJlI l onpeneJlRer npOCTpaHCTBelUlNe cBOilCTBa OCHOBHOro COC-
TOflHHfl CymecTByeT deCKOHe~Oe ~~nno pa~ KnaCc~eCKHX pemo~ xapaJToplI3yeMliX napBM9TTIOM TIJaJiCJIJlUill l J RMeDIIIU OnEO il TO lK9 SH8shy
leHKe 3HeprIDI I bull rne i1 j 1( LYU)l- TalUIM 06pa30M OCHOBH08 COCTOIDUlO BIIPOJKn6110 OTHOCJlTeJThHO npe06pasoBashy
HlIil TpaHCJUIlUli a KJlaCCIflOCltO(gt pewemte HapymaeT TpaHCJJUlKOHHJ1O CHMshy
MTpmo poundm lQ)YJIJ fllnOh jlUiIOUKii ypaBHelflrii JiuIepa B03MOJilHO H8pyme1fi1e
II WYnlX CaMM8rpII CBilloHlI~X C rpynnoli nBHE9IDli 3qqeKTHBHblM MeTonOM
nUJIIiJ1lffiOIlKM paCCM8TpimaTh CKCT9MH C BIlPO~9reM bull RBJIR9TCfl M9rOll K8J10shy
1IleCKlIX npe06p1iOpoundlPHn~ HlI lorOm06oB8 5-7
OCHoBnoii l1lll)l)1 1~IJTona igtorom06ona JlaJUleTCfl nepexon 1lt HOBbIM mmashyMIIWC KHM ne peMtl 11111 1M C1IJIJJ KO 10plH HMeJOTCil )lOnOJI1llTElJI1gtHlie nepeMoHiffie
nBJIIDOlIHCCfl napaMl IlBMM rpYTmu (IMMU11I1J I1x npllllIiTo HaSliBaTL KOJIJIBKshy
1aHIIIlMH Koop)JHHaTaMK tl CJly48 lpy l1ltH MlDK9IDrn KIIOMO TpaHCJIJIDltO~UlJD(
lltOo plUfHaT w B aB~ICiMOC Tll 01 ItWil1lOCKO~i c KTy8URK MO)l(HO BBeCTll TaKshy
w yrJlO BWe rrepsMemme K COOTBOTCT3YJOOIHeurol UlCTO JlopeHlJeBCKHM Bpa-
UGlliU1M nepeM8HHHe ~~ bull BBe)1eHKe KOJIJICI(TKBHIlX lOOPMH8T n03BOJIJleT peshy
~30BaTh Ha nOJIeBHX nepeM9ltHHX lleKOTOpoe TlJl9nCT8BIIJiIfI19 rpymw IIyaHK8shy
pc 4 lLnR 3TO ro C pDRTCil KHBapHaHTIill8 OTIlOCHTeJJbHO npeo6pa30 BaHHH
FgtynI1H IIyaHl8pe ~YHKIJIDI OT npDcTp81CTB9HHO-BpeMOHHJD( KOop)UUlaT
l KOJlJleKTIIBHblX lltOOp)lIDl8T (y H RBJIOHHleCKH corrpmKeHshy
HlX 1m Be1llfmH B KJIaCCIPl eC1ltOM ITp9ne1l8 ipyluuum ~ xspaKTepH3YllT no-
JIOlKeHJl e 11 o JReHTal(l11O npOTRlIWHlIOro OObeKT8 B npoCTpaHCTBe
0600weHMe JIBYMepHOM Mone~ C~HOI~ nORfl BKnD~aeT B ceOH H3y shy
~eHHe MHOIOCOJIlTOHHOrO c e opa 48 pacIIIlfPeHHa l1ICJla H3MepeHIDi rrpoc shylTpaHCTBa MHlIKOBcoro 9 IO bull rroJIei1 60JIe e CJIOJllHOh 1 c~ JIopeHUeshy
Boi1 CTPYKTypII pemoffile npod1leMbl rrepeHopMlfpOBOI I2 If T bull n
UanL naHHOH p aOO TIl COCTOHT B nocTpoeHHK B leTupe XMepHoM rrpOCTshy
paHCTBe KBa 3HIUIaCCHeCKoro rrpH~effilfl AHR ceKTopa TeopKH conepMashy
mero orum npoTIDKeHlIWt OObeKT OCHOBHOl-i 3anae~ RBJlReTCH H3yqeshy
HIIe B03MOXHOCTH pacCMOTpeHKR ne~pMKPOa8HHllX ( cqepKqeCKH HeCI~eTshy
lil HllX ) OObeKTOB B leTup9XMepHoM rrpOCTp aHCTBe-BpeMeRlJ
fuaH H8illerO 3iOyeHliil CJIenYJOllllR B sect 2 omlcuaaeTCR M8TOll IUlHOH1shy
tteClillX rrpeoop a30Erumi HE borom060Ba lI)IR rpymm ITYaHKaje B sect 3 peshy
waeTCR sanal8 0 il O llfl I-IHBapHBlITHUX qJyHKIUtiI ~(L ) B le TupeuroXMOp shy
HOM rrpoCTpaHCTBC ( ( = middot1) i3 sect 4 B KBa3HKJIaCClfleCKOM rrpl6JrDemm
CrpOIlTCfl KOBapvaHTlioe JlOY-aJIbHOe nOJIe JIJIR MoneID ITpOTroKeHHoro 06beKTa
YlTOnl PI HUBOr IT) BnlHII UDle-ULI sect 5)
l
sect 2 Ilp eo6pasoBamlo HH EoroJIJ06oBa
B STOM pasneJIe MIJ paCCMOTpIDiI nonxon K pemeHlflO 3analH KB3HTOBashy
HRR rrOHR B OKpBCTHOCTH KJlaCCH~ecKoro pemeHKR C TIOMO~~ MeTOna KaHOshy
lmleCKHX rrpeo6paSOBElHldt H H EoroJl1060Ba BUIJeJT9lme KO1leKTJlBHIIX KOshy
opIHHaT n03BOJlReT nopenncaTlgt KCXOJlHlle ypaBHeHRil lepe3 nepeMellHue RHshy
BapH8HTHbl9 OTHOCHTeJllgtHO npe06pa30BllIiKf1 rpyrrIJ] CllMMerpRH It CTPOro
Yl9CTL COOTBeTCTBYJOutI9 38KOHbI COXpaHeHl1R B JOOC50M nOpR)tKC l eOplH BO~gtshy
MYIIeHlfR TauaR npouenypa YJFe B HYJIe130M nOpfllIRe TeOplrn BO 3MymeHru1
yltUJTllBa9T H9TpHBli8lThHYJO CrpyKTYJlY peweHHR CIlfI38HJljO co cnOHTaJIIIllM
HapyruelfHeM OID1MeTpl1Jl
HaJmUe B CHCTpoundJMe npOrf11tCHIlOro OOM I(T8 OOR3areJTbllO npHBOIUIl I
HapywelllOO TpaHCIUIIrOHHOH H JlopeHlleooii CaMMeTpl1i1 TO ooycJlODJIellO TeM
ITO B JUI8cclPlecKOiI MexazrnKe MarOpH8JThHOi TOUltH OOH3aTenJHO CJlenYCT
y-Ka3aTlgt B K81ltoki TOUte ITpOCTpaHCTBa II C I(Bl()ji CKopocn1O JIBI1liBTCR laCshy
TKl(a ECJm rrpoTfIlI(eHHhtii 06b9J1 crpeplfleCIJI HeCllMMelpI1l9H TO llOllO1lH1shy
reJlbHO CJIenyeT YKa3aTb ero 01l19J1r8UR1O B lI]10CTpaHCTBe
IIepelineM KAicCMOTpelUfJO rJxgtlUympOBlltH KOBapHBHTHoro aHaJI01a M9shy
Tona Eoromo60Ba Ha npHMepe nOJIeBoi1 MOlleJrn CaMoneiiCTBYJOI1ero CI(8]RJr-
Horo nOM c--t) B ( - MepHOM npOCTPaHCTIl8
Bu6epeM 1larpaHRaaH eopIDf B cpo~e
I( lA -= - J lt-~ ) Lmiddot ( ~ I)
r ne rJ = Jt bull = D I d - I ll C (W -CKaJlJJpHaRI ~ -
IPYHKl(HR orpaHWleHHM cIDI3y llonYCTlIM ITO cymccrByeT ReTplfBllaJILHOO
pemeHl19 JI~ ()) ( He OORSaT8JThHO CTaTllleCRoe) C KOHellloll 3Heprliel1
J~aCCHqeCKOrO ypaBHeHRR A~emlR
I - ( - i-~ t r ( () ~ 0 (2 2 amp
npraquo K8HOilllfJCKOM KBaHTOBaHIDl cPY1i~ IX) C1aJIOBHTCR onepashy
TopH03Hawoil CIUlUIpHoi JlOJltalIbHOn nOJleBOJt dlY1UUIKel YJlOBlIeTBOpmlUIol
ypaJ3HeH1flO ( 2 ) 1 ltaH01UfleCKKM tOMM-yTauHOHHHM COOTTIOllleHIDlM
1 1 l [J J i f~) - ~ i A 7
1= 0 (23 )$ I c) ~ (~o)
3neCh Cf - pOpMamHIDi napaMeTp TeOpKH B03MYll1eillili bull
~
TeopHH OrrpeDCJIeHHaR JIarpaOOaHOM (2 1) 0 tl9BHRO mmapRaHTHa
OTHOCBTcmHO rrpo06pa30Bamm rpyrrrnl IIYBJiKape lTO o 3 HaWeT bull lTO BllA
ypanlleHlfil (n H CooTHowelUlM ( 23) He M9liffeTCR 1Iplf rrpe06pa30BlHlffiX
nOJIR
T ~ ~ i -l ) L -1 bull 1 J I J ) ( v I K ( N4)~ 1 I L = +( X I 1ltmiddotl v -==
reHepaTOpaMll OTRX rrpeo6pa30aaHHil RBJIRlOTCR onepaTopbl
v I I middot I bull l J J )j C - J t L - )JlI) ~ l ~ ~ ) r U (~ Ii J
_ - L ( u
I I C ~ ~) ( -= - ) J
I J l L l ~ J=
- (25)1 I I ) I f ~ ) J f
1 J middot Z J j
J
nOCTPOeHHbl9 lIO TeopeMe HeTOp H3 (2I)
KnaCCi1leCKOe pelUelUfe ypaIlH9Hldi (2 2 ) B006mo roBOPiI HellRBapUshy
aJlTHO OTHOCllTeJIbHO npc06p a30Brumi (21) Ii cIH3lfleCKaR CHCT8Ma ORa3Hshy
BUSTCR Bupo~eHHo~ TpYnHOCTK CBR3aHHHe C ~eM B~~eBRfl B
CHCTeMe CHIDllaJOTCfl 8CM BUllomUTb npeo6pa30BaIDe BoroJl06oBa K HOshy
BUM nepeMeHHhIM
ITYCTb apmTOBli Ol1ep aTopll 1-13 MHO~e(rBa 5 1Pl J 1 bullbull 1 YDOBJIeT-
BOpmoT COOTHOIlJeIUUlM aJlrc6pu ITyaHKape I
1 10 0) 1- 0 [ l) II I - J ( 1
~ rI urLo I r ~ j ) - - lor id~ I
i 1 ~ 1 - I t ~ -- (t f middot1 L 1 I I I ~ j - l ltJI tmiddot I ~ - ~ 7 1_ bull 1 v I 7 j J J i I J ~
PaCCMOTpHM rrODCTaHOBKY
On-) = ~lS())) jJ i ~pl t lts~lx) l til-f) ( 2 6)
rne t lt -lllfC]lOBHe nepSMeHHIl8 IiHTerpepoBClHIDl a orrepaTopno 3HatJHHa
i[lYH~ C (1) fIBJIJlj)rCfi tlJYH~ ti=l MJ fi t bull IIpeIJJl0JI8rae1cJI
iTO BTOPIflliO KBaJ-ITOBDFIHClC 110ne i Ll) ]OMMYTHpyeT C onopaTOprum 113
aJIreOpli P a rrpoClP8HCTBO rrpCDCT8DJeHlUI rrOJIJ C(~) HBiLIleTCR 1Tp1lshy
MIlM npDM3BonelliHlM npoCTprulC IB ITpenCTaMeami rurre6pH P If rrOJIJ qLX)
ITepeMellHhle C ltl) OIlJ)eneJIeHH CBOHCTBOM
LJ ( - ) 4- bull I
~~ h t~ t- l L lor ~ 1- ( tr r l j
r e ~~~ 1-0 (- amp- - ~~ ( 2 7))
~ - JI ~ iL~) ( ) f ttf v shy yOlopoe r apaHTHpyeT COBMeCTHOC1h npe0 6pn30BaHlUI (~ 7) C Tjl8JICT1loprlttshy
llJfOHIillMH CBol1CTBaMH nOAA qgt( x ) _
IIoMe BbIllOJTHeHlU 3aMeHbI ( 26) ( 27) onopa T0lgtu ) bull I~I r v C1ashy
llOBilTCR reuepaTOpaMM rrpocrp8HcTueHHo- DlJeMeHlillx IlpI)06Ja~OBam[li OTOlKshy
neCTBJlCHH8 3TgtlX orrepaTopoB C DblpaKC lmHmiddot~ 1 COOTIleICTIJyY1Uij1X r eH9[JaTopoB
(2 5) nocTpoellliillC no rroJllO lJLl ) C nOMOll1hlO TeoperllJ leTep lae T one shy
paTOpHIle CBSl3H KOTopue 11 06eCrrelJffiaOT COXpaJlCHJ1e amp101lt1 H93llillCJLUIX
nep9MeHlillX llO 11 IlOCJIe npeo6pa 30BaHJl[R ( 2 6 ) II Clffi311 paCCMrlTpgtIB8JilTCR
Ha ReMTope COC1OflHlllil )~) M ID-leJOT mUI
bull j shy
) 0 I -- C iI M ) - l L r - [ l raquo) J l r - [ [ - I 4 _ ~ I bull I y- I - bull (2 8) I
ITpe0 6pa30B8JlIIe (2 6 ) C YCJIonPlfMJl c Efl3eJi (~o C ) Jo ]OMMYTaI(J10HIl1lMl1
COO THoweH1HIMVI DJIR nOM C x ) bull neperrncaHHbJM B HOBUX nepeMe HHblx pound -t)
bull ~ ) ~ j ) f ) ~ JL -Su h~ J - I cgt (2 9)
H83UBaeTCil IltaHOlUflleCKHM rrpe OOpS30BallliCM IJoroJT1OCio Ba DJIjl pPyntlH lIYaHlltashy
pc1 1 n03BOJIRCf fi lUlHOM BVIDe peaJlJl30B8Th np0)1CTaBJIelUf8 lTOti rpymrn
JJnR TOlO 1m6b1 HDHUM 0 6pa30M BHITOmUfTb npeoops 30Bamre BaroJTI(-OBa CJJc nyeT IlIll1TH I1lIBaplIaHTHHe nepeMelllible ( J ) VI rrpoK3se CTK 3aMeshy
By nOlJeMeHHJiX B COOTIlOlUeHl1JlX ( 0) ( 9) lTO VI 6YJ16T CJ1eJI8HO B sect 3
II sect 1 COOTBllTC TIJeHIO
~ J 1 0ClpoeHliu P11lllapHaJfTIIIIX nepeMOIU11IX
$lUUUeM c VlcTeMY YPaBlIOHlHI ( ~ ) onpeneJlR1OOllofx iJJyHKUKJol C J)
B IHrMHllTeJHMaJIbHO~i 11-OpMe -
1 ~
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AMee n TOM pa3J1eJre H8M 0 YDeT YllOOHO nOJIOJKlllTlgt I) 1
Cymec TBOBaHHe pelllemrn ypaBHeFlliH (3 I) H FIX mgtmm BlUl 3aBHClIT 0 T
Budopa KOHKpeTHoro rrpe~cTaaneFllilI anredpu rpyrrnu llyaHKape jO B Tepshy
MHHax KaHOmrqeCKFlX orrepaTopoB ( y p I 0 Po ) Budop lKe rrpencTaaneshy
IDlI orrpe~eIDeTClI lM3FIleCKIDIIlI CO 0 clpalKeFlliHMH bull l1epeMeHHHs S (X)
FlMeIDT CMHC~ ododmeHHHx KBaHTOBHX Koop~aT KOTopue B COCTOJIHHH rroshy
KOlI od~eKTa rrepexo~JlT B ~CCHieCKOM rrpen~e B rrpocTpaRCTBeHHO-Bpeshy
MeHHlle KOopmmaTu rrpoCTpaHCTBa MmlliOBCKOro 2- bull OTMeTflM lTO t gt
~O~ dUTb de3pa3MepHHMH Be~HBMH TaR RaK rrpeodpa30BaHHe ( 2 6)
He co~epJKHT lliC~OBOro pa3MspHoro rrapaM9Tpa IIpH 3TlIX ~orrO~Te~HHX
ifJH 3HlSCKFlX rrpenrrO~OlKelmRX ypaBHeFlliJI (3 I ) rrO~OCTbID orrpene~T d (cIshy-pa3MepHocTb rrpoCTPaHCTBa-BPSM8HFl ) H93aBHCRMllX HHBaPllaRTHliX OTHOCHshy
TenHO rrpeodpa30B8IUlli rpymm n yrurnape rrepeMSIDillX S-lt RaK lPyHKIUm aprYMeHToB X ~ bull J bull () bull tJ B WHRILHRX S rrpenrro~araeTClI orrpe~e~eHHoe yrropRUOleHHe H8JltOMMYTHPynttHX rrap orrepaTopOB Ii bull ~ H
I () () - HarrpgtIMep ( II Jj c D J)
I L~ I l l
PaCCMOTpgtIM Terrspb rrpOCTei1wee rrpenCTaBJISIDe aJIIsdpu F lepe3 Kashy
HOHWleCKH8 orrepaTOpu I il KOTOpoe peaIDl3ye TcH B rrpOCTpaHCTBe
QJyHKlUfi1 1 ) H 3MaIIO C rrOMOIllMl Coo TBolDemrli
p~== r ( 1v1r = ((~ J P - l( G Pf) (32)
Lp l~ ( shyI I ) L (1
B rrpOCTpaHCTBenBYX H3M8peHlll1 ( j = 2) ypaBHeFlliH (31) B rrpeJ]shy
CT~eHHH (32) gtlMeIDT CBOFlM pellleFllieM ~Ba ~HejiHllX rro 2 orrepaTopHo-
3HalHHX pemeHKH
toUJ~2~1 +f~Ir T-Y))~ JI r~ (33)
~ _ -ltv ) t~l(lf= cl (X7~r -= u ~ L v r flo1
B BHpaJKeHm ( 3 3 ) oIIepaTop JIOpeHIJeBCKIX BpameHJ-fi1 OL c ~ a orrepashy
TOp j)= lIc l 11 P pr(1 ~) gtlMSeT TaIDle lKe KOMMYTamlOHHue CBol-icTBa C
anre6poJ C2 IlTO H ohepaTOp pacTJIlKeHIDi
I1HBapHaHTHUe rrepeMeHllile S - (v-G = 0 I) ItOMMyrnpylOT MaJKny COshy
dOll FI rrOJIHOCTbID y~OBJIeTBOpHIDT BceM TpedOBaHllHMHeodxo)lHMHM ~ HCshy
IIOn30BaHHff 3THX ifurmami1 RaK apryMeHToB B rrpaBofI laCTH rrpeodpa30BaIllilI
(26) B nByMepHllX MO~eJIRX OrraCliBaIDmKX KBaHTOBOe peIDTllBHCTcKoe nBHshy
llleFllie OlJHoro rrpoTJIlKeHHoro od~eKTa2 4
KoMMyrHPyromne KOBapHaHTHe Koop~HaTu qrBHp8lKeHHl1e lSpe3 orrepa-
TOpH H3 aJIIedpH t(f2 ) D 1 6
l-iL ~Cl_~J il(~ - 2 D M ~ 2 L Me 1- M2 J j
i -=1 r pi l i if I ~ 2 lD tj 2 J -2 LMOL Mo J I
(34)
1v)=P -P~ C l )
JIBiIRKlTCR aHaJlorBMI1 KOopmmaT fu IOToHa--BITHepa B ~YMepHOM iIpOCTPaH-
CTBe
OleBHJHO lTO B rrpoC TpaHCTBe d raquo 2 H3Mep eHllM cywecTByeT Tonshy
KO OIHO JIlIHelHo e rro X pelliorme (31) Sf2) = 2~~j) 1I03Toshy
MY B 06mSM cnyqae rrpelLCTaBJI8flHe (32) HerrpIDoIlRO II)IR OIIHCaHHH peJlRshy
THBHCTCKOro IIpOTHlKeIDloro Od1eKTal1 CJI8Iye T paclllFlpHT1 KO lqmrypaIJHOHshy
Hoe rrpoCTPaBC TBO rrpe~CT~ermlI ~OrrOrrHffB ero YTJIOBHMH rrepeMeHHYMH
OlLHaKO eCJIB rrpoTHlKeHHilll od~eKT 06~aJLaeT CBO~CTBOM c~p~eCKOH
CI1MMeTplm TO OKa3UBaeTClI ~OCTaTOiHO nOCTpoHTb ~e IDmapHaHTHllX KOshy
opIJffiaTu BpeMeHHyIO H paJJlfanHyro
Sc()c) = (ctr~)rpr) l~ x)l coo LJlql() (35)
~TO HaxO~TCR B COOTBeTCTBllH C pe3YJIbTaTaMll pa60T9
KaK j~e OTMelaJIOCb Bume paCCMOTpeHH8 dOJIee od~ero CJIYiaJI MOshy
ue1IH cifgtepWleCKI1 HeCHMMe TpHlHoro od~eKTa TpedyeT rrepexoua K dOJIse
C~CraquoUUlM npelLCT~eH1UlM lJJm OrmCaHMH CTaTHleCKoro ~eiJoPMrrpOBaHHOrO
06beKra MH BH6epeM rrpe~CT~8ID8 anredpu lIyaHKape 13 iPJPMe
J C l e lL r I f f11S P = Igt IV) = (i f - ~I ) shyJ ) ) ( (J 1)
)-) K1 -- (C( pd - ~r~ ~ ~ - c - J
1) fj =t ~ )(36) j ~ - J
1 r~ [S ) f ~1 Li ( y) 7 ( - ( r 1 1= (C Coil S~
~ro npSuCT8BJI8HH8 peam3yeTcH B rrpoC Tp aHCTBe IWH1QWlli If yJ ~ r~e X - yrJill rrpoCTpaIlCTBCHHHX BpamellmL TIo ifJH3H1eCKOMY CMliCJIy KO[shy
JIeKTIlBHlle KOOPlUrnaTH r HI ol1IJe~eJIRlOT rrMOlKefDlle ~8txlpMllpOBaHHOshyr o od~eKTa 1lt1 ero opHeraquoTaIJJm B rrpo CTpaHCTB MaTp~ vl lre( YJ B (35 )
3aJLaIDT BHn r eHepaTopoB rpymru Bpame~ S t I l) pyt
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
IJ8CCMO TPJIM Hrl Atlno C()CTOIlHHJI coo rHuTC InYlI1li1 1I0 KOlweMYCJl OOmiddotH KlY
~~ ( bull l )
c )H Ij n i 1 I () ( 6)
~ )l i )1 rT ) 1) 111
jfl1 COOTIOUlelllR I ypan BlIUlR (44 ) Ilol11(CJ11JYJOT jO ~~i(lJ1W1e IlTU ra3ll0JWHlUI
orrcpAToF3 ~
12
M ~ -ll l U 1 (1 c -= t x ( (J ((L ) Mi - gt
AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
~ I ~ )CPl XJ _ 0II -_ ( lB )3t TA gt
JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
~ I ~ III( l (X JJ - J) ) G-I ~ = D I u ) x ( 1 ( u ) I L
ITPU)JCT8BI1M IToJIe ([II C ) R13Jrmomeecp lBWeHveM (4 5 ) H Blfne Pamp3JlomiddotshyteHJUl no nOJllioll CHCTeMe pem8H1f~ YP8BHSHIfl
J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
YCJWBI1H (4 8a) (4 8d) flCIltJ1X)laroT fl p a ilIoJtEilTllil J(OMIIOHGIITLI COOTlef shy
cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
IIIElnCll 18W(IlIWM YjlEllIllt llIlfl (1r) ~I 1I0)I IIIWIIlrj O yoqOllllHM (middotI tmiddot) I lIonyshyIJiM lU~fWlltf l lIll
1 laquo) (I) (4 Ill)I ( ~
HrullleM fII 1(lMMY1lHlIICI II II II
11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
(2 6) rrpli yllCpl8IiID B pa3Jolle lUIR BI1pall81ildi (2 3) CJIara ElMHX He Bhllll8
nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
- [ 6 Ip) R l~) lt t) I -COafeCrHOCTb pa3JOlfeIDIR (410) H COOTHOm8HHA (4 n) rrpImOllwr K rrpashyBwry KBaHTOBalliUl
-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
TaKHM o dpa30M B lCKOMOM rropU)(e T80Pruf B03MymCHHIl nOJIe ctshynpellCTaBHMO B ~e
1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
OJ I i ) JIBJIJIeTCR peWeHItOM KJIaCCIIlel)oro yIaBHeHIUI (45 ) ae BTOPtllUIO lltBaJiTOB8HHoe IlOJIe J1 1 1) ()rrpelleJIReTCR pa3JlOIIIBHl1eM
(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
Hmm 110 KlllUlJ I) ITO 111(IIpLH IJ83TIIIlJW( rtl)onCTalVlllI1H 8JIluOpU ITyaHIC1shy
pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
leCIWM nrllI~-1t 11m nocTllOOHll npouenypa KBaHTOllrumR nOKaJIbUOlO reli-30Ildl11JCIIIC JtOIO nOAA B MOlleJIH CTaTIPieCKoro neq()PltIiij)oBatlHoro II[oOTRshy
lIWHJlOIO o01ewra
C 3TOH ~eJnlO B sect C npOli3BSnSHO rrpeo dpa30Brurne K HOBUM Tlq)eM8Hshy
HLIM n03DOJUlOlltee cTporo yqecTb 3axOll~ COxpaH8lffiR II OTIleJUfTigt KHIJ8THshy
iecKYlO laCT 3anaTffi OT Jl1Ill8MJ~lecKOI1 11 pS3yJllii8T8 rrpo06pa30BalHfR apshyryMellTaMH nOJIR CTaHOBRTCR CneUKanbHHe HllBapHaHTH~e n eVeMeHHU8 ~TH
nepeMeHH~e ~TCR KOMdllll~eR reHepaToI~B rpynTIH IlBIIXeHHR II B
IltJIaCClfleCKOM npellene XapaKTepH3YJlT I(BIDtelrue npOTmteHHoro 06bampKT8 181
u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
OT B~dopa npencTaBJIeflliR aJIre6ptl nyruIKapo 110 B CBOID Olepellb onpeneshy
JmeTCR ~3lfleCKO~ CHTYauHeK B paCOTe paCCMOTpeHU rrpenCTaBJIe1~R 00shy
OTBeTCTBJ1OUU1e nBIDIteHIDO JUfIlffi onHo ro ITfJOTRZeHHoro Odb9YT8 _ 13 sect 1 IlJlH
pemeIHR JIRHaMHeCKOH laCTH 3alIa1Jll I1CnOlTh30DaHO lltBaSHKJlACCfImiddotreCKOO
npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
B03MYUleHIrt CTPOHTCR JOOIlI nO nepBoro IIOPRllKa pS3JlOlKemm nOAA no
5t BKJIlOliHT8JIbHO
ABTop 6JIarDll8PHT 11 II BorOJIlOdoBa K ACBellllliKOBa Ii H E CKaKoua 3a 0 6CYJKlleHHe BOnpOCOB SaTpOHYTUX B CTa fhe 11 p e syJIb TarO B paCOTti
JI H T C I a Y P a
I Cj llmiddot_~t J ~ I l cc T U I _middot~_middot bull _-f 1 -( 1J~~j O I il ~ t t - 1Lff
TOHUO d i ~J middotn middot ~J J~_ ulc IC f 1 I ~ - ~ -J4
Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
3 YM J)l3aB8 X llauyMoTo X TaTuKil - TCtMOIIOJl l Jlftrl Jl)1I1HMHI fJ It 1(1l11shy
neHc HpoBaIlHtle COC1OJIHllH -1gt1 1 HIL) l Jl I X
4 CBEUIHIgtIKOB ILA_ l1 I~L N ~ CI-i ll
i _ 1 ()1fM~J(nII 11 11 JlljlMrIMl 1 -~ I IA Ill 1 U-1 II1I1I11
T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
7 j r LI
_ I shy 11 141 bull I U 1 1 lHlt
I i
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
TeopHH OrrpeDCJIeHHaR JIarpaOOaHOM (2 1) 0 tl9BHRO mmapRaHTHa
OTHOCBTcmHO rrpo06pa30Bamm rpyrrrnl IIYBJiKape lTO o 3 HaWeT bull lTO BllA
ypanlleHlfil (n H CooTHowelUlM ( 23) He M9liffeTCR 1Iplf rrpe06pa30BlHlffiX
nOJIR
T ~ ~ i -l ) L -1 bull 1 J I J ) ( v I K ( N4)~ 1 I L = +( X I 1ltmiddotl v -==
reHepaTOpaMll OTRX rrpeo6pa30aaHHil RBJIRlOTCR onepaTopbl
v I I middot I bull l J J )j C - J t L - )JlI) ~ l ~ ~ ) r U (~ Ii J
_ - L ( u
I I C ~ ~) ( -= - ) J
I J l L l ~ J=
- (25)1 I I ) I f ~ ) J f
1 J middot Z J j
J
nOCTPOeHHbl9 lIO TeopeMe HeTOp H3 (2I)
KnaCCi1leCKOe pelUelUfe ypaIlH9Hldi (2 2 ) B006mo roBOPiI HellRBapUshy
aJlTHO OTHOCllTeJIbHO npc06p a30Brumi (21) Ii cIH3lfleCKaR CHCT8Ma ORa3Hshy
BUSTCR Bupo~eHHo~ TpYnHOCTK CBR3aHHHe C ~eM B~~eBRfl B
CHCTeMe CHIDllaJOTCfl 8CM BUllomUTb npeo6pa30BaIDe BoroJl06oBa K HOshy
BUM nepeMeHHhIM
ITYCTb apmTOBli Ol1ep aTopll 1-13 MHO~e(rBa 5 1Pl J 1 bullbull 1 YDOBJIeT-
BOpmoT COOTHOIlJeIUUlM aJlrc6pu ITyaHKape I
1 10 0) 1- 0 [ l) II I - J ( 1
~ rI urLo I r ~ j ) - - lor id~ I
i 1 ~ 1 - I t ~ -- (t f middot1 L 1 I I I ~ j - l ltJI tmiddot I ~ - ~ 7 1_ bull 1 v I 7 j J J i I J ~
PaCCMOTpHM rrODCTaHOBKY
On-) = ~lS())) jJ i ~pl t lts~lx) l til-f) ( 2 6)
rne t lt -lllfC]lOBHe nepSMeHHIl8 IiHTerpepoBClHIDl a orrepaTopno 3HatJHHa
i[lYH~ C (1) fIBJIJlj)rCfi tlJYH~ ti=l MJ fi t bull IIpeIJJl0JI8rae1cJI
iTO BTOPIflliO KBaJ-ITOBDFIHClC 110ne i Ll) ]OMMYTHpyeT C onopaTOprum 113
aJIreOpli P a rrpoClP8HCTBO rrpCDCT8DJeHlUI rrOJIJ C(~) HBiLIleTCR 1Tp1lshy
MIlM npDM3BonelliHlM npoCTprulC IB ITpenCTaMeami rurre6pH P If rrOJIJ qLX)
ITepeMellHhle C ltl) OIlJ)eneJIeHH CBOHCTBOM
LJ ( - ) 4- bull I
~~ h t~ t- l L lor ~ 1- ( tr r l j
r e ~~~ 1-0 (- amp- - ~~ ( 2 7))
~ - JI ~ iL~) ( ) f ttf v shy yOlopoe r apaHTHpyeT COBMeCTHOC1h npe0 6pn30BaHlUI (~ 7) C Tjl8JICT1loprlttshy
llJfOHIillMH CBol1CTBaMH nOAA qgt( x ) _
IIoMe BbIllOJTHeHlU 3aMeHbI ( 26) ( 27) onopa T0lgtu ) bull I~I r v C1ashy
llOBilTCR reuepaTOpaMM rrpocrp8HcTueHHo- DlJeMeHlillx IlpI)06Ja~OBam[li OTOlKshy
neCTBJlCHH8 3TgtlX orrepaTopoB C DblpaKC lmHmiddot~ 1 COOTIleICTIJyY1Uij1X r eH9[JaTopoB
(2 5) nocTpoellliillC no rroJllO lJLl ) C nOMOll1hlO TeoperllJ leTep lae T one shy
paTOpHIle CBSl3H KOTopue 11 06eCrrelJffiaOT COXpaJlCHJ1e amp101lt1 H93llillCJLUIX
nep9MeHlillX llO 11 IlOCJIe npeo6pa 30BaHJl[R ( 2 6 ) II Clffi311 paCCMrlTpgtIB8JilTCR
Ha ReMTope COC1OflHlllil )~) M ID-leJOT mUI
bull j shy
) 0 I -- C iI M ) - l L r - [ l raquo) J l r - [ [ - I 4 _ ~ I bull I y- I - bull (2 8) I
ITpe0 6pa30B8JlIIe (2 6 ) C YCJIonPlfMJl c Efl3eJi (~o C ) Jo ]OMMYTaI(J10HIl1lMl1
COO THoweH1HIMVI DJIR nOM C x ) bull neperrncaHHbJM B HOBUX nepeMe HHblx pound -t)
bull ~ ) ~ j ) f ) ~ JL -Su h~ J - I cgt (2 9)
H83UBaeTCil IltaHOlUflleCKHM rrpe OOpS30BallliCM IJoroJT1OCio Ba DJIjl pPyntlH lIYaHlltashy
pc1 1 n03BOJIRCf fi lUlHOM BVIDe peaJlJl30B8Th np0)1CTaBJIelUf8 lTOti rpymrn
JJnR TOlO 1m6b1 HDHUM 0 6pa30M BHITOmUfTb npeoops 30Bamre BaroJTI(-OBa CJJc nyeT IlIll1TH I1lIBaplIaHTHHe nepeMelllible ( J ) VI rrpoK3se CTK 3aMeshy
By nOlJeMeHHJiX B COOTIlOlUeHl1JlX ( 0) ( 9) lTO VI 6YJ16T CJ1eJI8HO B sect 3
II sect 1 COOTBllTC TIJeHIO
~ J 1 0ClpoeHliu P11lllapHaJfTIIIIX nepeMOIU11IX
$lUUUeM c VlcTeMY YPaBlIOHlHI ( ~ ) onpeneJlR1OOllofx iJJyHKUKJol C J)
B IHrMHllTeJHMaJIbHO~i 11-OpMe -
1 ~
I ) ) I ri J ~ -= lil f y ~~ J - Iv J~rJt (3 I) 1 I I I
AMee n TOM pa3J1eJre H8M 0 YDeT YllOOHO nOJIOJKlllTlgt I) 1
Cymec TBOBaHHe pelllemrn ypaBHeFlliH (3 I) H FIX mgtmm BlUl 3aBHClIT 0 T
Budopa KOHKpeTHoro rrpe~cTaaneFllilI anredpu rpyrrnu llyaHKape jO B Tepshy
MHHax KaHOmrqeCKFlX orrepaTopoB ( y p I 0 Po ) Budop lKe rrpencTaaneshy
IDlI orrpe~eIDeTClI lM3FIleCKIDIIlI CO 0 clpalKeFlliHMH bull l1epeMeHHHs S (X)
FlMeIDT CMHC~ ododmeHHHx KBaHTOBHX Koop~aT KOTopue B COCTOJIHHH rroshy
KOlI od~eKTa rrepexo~JlT B ~CCHieCKOM rrpen~e B rrpocTpaRCTBeHHO-Bpeshy
MeHHlle KOopmmaTu rrpoCTpaHCTBa MmlliOBCKOro 2- bull OTMeTflM lTO t gt
~O~ dUTb de3pa3MepHHMH Be~HBMH TaR RaK rrpeodpa30BaHHe ( 2 6)
He co~epJKHT lliC~OBOro pa3MspHoro rrapaM9Tpa IIpH 3TlIX ~orrO~Te~HHX
ifJH 3HlSCKFlX rrpenrrO~OlKelmRX ypaBHeFlliJI (3 I ) rrO~OCTbID orrpene~T d (cIshy-pa3MepHocTb rrpoCTPaHCTBa-BPSM8HFl ) H93aBHCRMllX HHBaPllaRTHliX OTHOCHshy
TenHO rrpeodpa30B8IUlli rpymm n yrurnape rrepeMSIDillX S-lt RaK lPyHKIUm aprYMeHToB X ~ bull J bull () bull tJ B WHRILHRX S rrpenrro~araeTClI orrpe~e~eHHoe yrropRUOleHHe H8JltOMMYTHPynttHX rrap orrepaTopOB Ii bull ~ H
I () () - HarrpgtIMep ( II Jj c D J)
I L~ I l l
PaCCMOTpgtIM Terrspb rrpOCTei1wee rrpenCTaBJISIDe aJIIsdpu F lepe3 Kashy
HOHWleCKH8 orrepaTOpu I il KOTOpoe peaIDl3ye TcH B rrpOCTpaHCTBe
QJyHKlUfi1 1 ) H 3MaIIO C rrOMOIllMl Coo TBolDemrli
p~== r ( 1v1r = ((~ J P - l( G Pf) (32)
Lp l~ ( shyI I ) L (1
B rrpOCTpaHCTBenBYX H3M8peHlll1 ( j = 2) ypaBHeFlliH (31) B rrpeJ]shy
CT~eHHH (32) gtlMeIDT CBOFlM pellleFllieM ~Ba ~HejiHllX rro 2 orrepaTopHo-
3HalHHX pemeHKH
toUJ~2~1 +f~Ir T-Y))~ JI r~ (33)
~ _ -ltv ) t~l(lf= cl (X7~r -= u ~ L v r flo1
B BHpaJKeHm ( 3 3 ) oIIepaTop JIOpeHIJeBCKIX BpameHJ-fi1 OL c ~ a orrepashy
TOp j)= lIc l 11 P pr(1 ~) gtlMSeT TaIDle lKe KOMMYTamlOHHue CBol-icTBa C
anre6poJ C2 IlTO H ohepaTOp pacTJIlKeHIDi
I1HBapHaHTHUe rrepeMeHllile S - (v-G = 0 I) ItOMMyrnpylOT MaJKny COshy
dOll FI rrOJIHOCTbID y~OBJIeTBOpHIDT BceM TpedOBaHllHMHeodxo)lHMHM ~ HCshy
IIOn30BaHHff 3THX ifurmami1 RaK apryMeHToB B rrpaBofI laCTH rrpeodpa30BaIllilI
(26) B nByMepHllX MO~eJIRX OrraCliBaIDmKX KBaHTOBOe peIDTllBHCTcKoe nBHshy
llleFllie OlJHoro rrpoTJIlKeHHoro od~eKTa2 4
KoMMyrHPyromne KOBapHaHTHe Koop~HaTu qrBHp8lKeHHl1e lSpe3 orrepa-
TOpH H3 aJIIedpH t(f2 ) D 1 6
l-iL ~Cl_~J il(~ - 2 D M ~ 2 L Me 1- M2 J j
i -=1 r pi l i if I ~ 2 lD tj 2 J -2 LMOL Mo J I
(34)
1v)=P -P~ C l )
JIBiIRKlTCR aHaJlorBMI1 KOopmmaT fu IOToHa--BITHepa B ~YMepHOM iIpOCTPaH-
CTBe
OleBHJHO lTO B rrpoC TpaHCTBe d raquo 2 H3Mep eHllM cywecTByeT Tonshy
KO OIHO JIlIHelHo e rro X pelliorme (31) Sf2) = 2~~j) 1I03Toshy
MY B 06mSM cnyqae rrpelLCTaBJI8flHe (32) HerrpIDoIlRO II)IR OIIHCaHHH peJlRshy
THBHCTCKOro IIpOTHlKeIDloro Od1eKTal1 CJI8Iye T paclllFlpHT1 KO lqmrypaIJHOHshy
Hoe rrpoCTPaBC TBO rrpe~CT~ermlI ~OrrOrrHffB ero YTJIOBHMH rrepeMeHHYMH
OlLHaKO eCJIB rrpoTHlKeHHilll od~eKT 06~aJLaeT CBO~CTBOM c~p~eCKOH
CI1MMeTplm TO OKa3UBaeTClI ~OCTaTOiHO nOCTpoHTb ~e IDmapHaHTHllX KOshy
opIJffiaTu BpeMeHHyIO H paJJlfanHyro
Sc()c) = (ctr~)rpr) l~ x)l coo LJlql() (35)
~TO HaxO~TCR B COOTBeTCTBllH C pe3YJIbTaTaMll pa60T9
KaK j~e OTMelaJIOCb Bume paCCMOTpeHH8 dOJIee od~ero CJIYiaJI MOshy
ue1IH cifgtepWleCKI1 HeCHMMe TpHlHoro od~eKTa TpedyeT rrepexoua K dOJIse
C~CraquoUUlM npelLCT~eH1UlM lJJm OrmCaHMH CTaTHleCKoro ~eiJoPMrrpOBaHHOrO
06beKra MH BH6epeM rrpe~CT~8ID8 anredpu lIyaHKape 13 iPJPMe
J C l e lL r I f f11S P = Igt IV) = (i f - ~I ) shyJ ) ) ( (J 1)
)-) K1 -- (C( pd - ~r~ ~ ~ - c - J
1) fj =t ~ )(36) j ~ - J
1 r~ [S ) f ~1 Li ( y) 7 ( - ( r 1 1= (C Coil S~
~ro npSuCT8BJI8HH8 peam3yeTcH B rrpoC Tp aHCTBe IWH1QWlli If yJ ~ r~e X - yrJill rrpoCTpaIlCTBCHHHX BpamellmL TIo ifJH3H1eCKOMY CMliCJIy KO[shy
JIeKTIlBHlle KOOPlUrnaTH r HI ol1IJe~eJIRlOT rrMOlKefDlle ~8txlpMllpOBaHHOshyr o od~eKTa 1lt1 ero opHeraquoTaIJJm B rrpo CTpaHCTB MaTp~ vl lre( YJ B (35 )
3aJLaIDT BHn r eHepaTopoB rpymru Bpame~ S t I l) pyt
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
IJ8CCMO TPJIM Hrl Atlno C()CTOIlHHJI coo rHuTC InYlI1li1 1I0 KOlweMYCJl OOmiddotH KlY
~~ ( bull l )
c )H Ij n i 1 I () ( 6)
~ )l i )1 rT ) 1) 111
jfl1 COOTIOUlelllR I ypan BlIUlR (44 ) Ilol11(CJ11JYJOT jO ~~i(lJ1W1e IlTU ra3ll0JWHlUI
orrcpAToF3 ~
12
M ~ -ll l U 1 (1 c -= t x ( (J ((L ) Mi - gt
AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
~ I ~ )CPl XJ _ 0II -_ ( lB )3t TA gt
JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
~ I ~ III( l (X JJ - J) ) G-I ~ = D I u ) x ( 1 ( u ) I L
ITPU)JCT8BI1M IToJIe ([II C ) R13Jrmomeecp lBWeHveM (4 5 ) H Blfne Pamp3JlomiddotshyteHJUl no nOJllioll CHCTeMe pem8H1f~ YP8BHSHIfl
J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
YCJWBI1H (4 8a) (4 8d) flCIltJ1X)laroT fl p a ilIoJtEilTllil J(OMIIOHGIITLI COOTlef shy
cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
IIIElnCll 18W(IlIWM YjlEllIllt llIlfl (1r) ~I 1I0)I IIIWIIlrj O yoqOllllHM (middotI tmiddot) I lIonyshyIJiM lU~fWlltf l lIll
1 laquo) (I) (4 Ill)I ( ~
HrullleM fII 1(lMMY1lHlIICI II II II
11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
(2 6) rrpli yllCpl8IiID B pa3Jolle lUIR BI1pall81ildi (2 3) CJIara ElMHX He Bhllll8
nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
- [ 6 Ip) R l~) lt t) I -COafeCrHOCTb pa3JOlfeIDIR (410) H COOTHOm8HHA (4 n) rrpImOllwr K rrpashyBwry KBaHTOBalliUl
-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
TaKHM o dpa30M B lCKOMOM rropU)(e T80Pruf B03MymCHHIl nOJIe ctshynpellCTaBHMO B ~e
1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
OJ I i ) JIBJIJIeTCR peWeHItOM KJIaCCIIlel)oro yIaBHeHIUI (45 ) ae BTOPtllUIO lltBaJiTOB8HHoe IlOJIe J1 1 1) ()rrpelleJIReTCR pa3JlOIIIBHl1eM
(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
Hmm 110 KlllUlJ I) ITO 111(IIpLH IJ83TIIIlJW( rtl)onCTalVlllI1H 8JIluOpU ITyaHIC1shy
pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
leCIWM nrllI~-1t 11m nocTllOOHll npouenypa KBaHTOllrumR nOKaJIbUOlO reli-30Ildl11JCIIIC JtOIO nOAA B MOlleJIH CTaTIPieCKoro neq()PltIiij)oBatlHoro II[oOTRshy
lIWHJlOIO o01ewra
C 3TOH ~eJnlO B sect C npOli3BSnSHO rrpeo dpa30Brurne K HOBUM Tlq)eM8Hshy
HLIM n03DOJUlOlltee cTporo yqecTb 3axOll~ COxpaH8lffiR II OTIleJUfTigt KHIJ8THshy
iecKYlO laCT 3anaTffi OT Jl1Ill8MJ~lecKOI1 11 pS3yJllii8T8 rrpo06pa30BalHfR apshyryMellTaMH nOJIR CTaHOBRTCR CneUKanbHHe HllBapHaHTH~e n eVeMeHHU8 ~TH
nepeMeHH~e ~TCR KOMdllll~eR reHepaToI~B rpynTIH IlBIIXeHHR II B
IltJIaCClfleCKOM npellene XapaKTepH3YJlT I(BIDtelrue npOTmteHHoro 06bampKT8 181
u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
OT B~dopa npencTaBJIeflliR aJIre6ptl nyruIKapo 110 B CBOID Olepellb onpeneshy
JmeTCR ~3lfleCKO~ CHTYauHeK B paCOTe paCCMOTpeHU rrpenCTaBJIe1~R 00shy
OTBeTCTBJ1OUU1e nBIDIteHIDO JUfIlffi onHo ro ITfJOTRZeHHoro Odb9YT8 _ 13 sect 1 IlJlH
pemeIHR JIRHaMHeCKOH laCTH 3alIa1Jll I1CnOlTh30DaHO lltBaSHKJlACCfImiddotreCKOO
npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
B03MYUleHIrt CTPOHTCR JOOIlI nO nepBoro IIOPRllKa pS3JlOlKemm nOAA no
5t BKJIlOliHT8JIbHO
ABTop 6JIarDll8PHT 11 II BorOJIlOdoBa K ACBellllliKOBa Ii H E CKaKoua 3a 0 6CYJKlleHHe BOnpOCOB SaTpOHYTUX B CTa fhe 11 p e syJIb TarO B paCOTti
JI H T C I a Y P a
I Cj llmiddot_~t J ~ I l cc T U I _middot~_middot bull _-f 1 -( 1J~~j O I il ~ t t - 1Lff
TOHUO d i ~J middotn middot ~J J~_ ulc IC f 1 I ~ - ~ -J4
Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
3 YM J)l3aB8 X llauyMoTo X TaTuKil - TCtMOIIOJl l Jlftrl Jl)1I1HMHI fJ It 1(1l11shy
neHc HpoBaIlHtle COC1OJIHllH -1gt1 1 HIL) l Jl I X
4 CBEUIHIgtIKOB ILA_ l1 I~L N ~ CI-i ll
i _ 1 ()1fM~J(nII 11 11 JlljlMrIMl 1 -~ I IA Ill 1 U-1 II1I1I11
T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
7 j r LI
_ I shy 11 141 bull I U 1 1 lHlt
I i
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
Cymec TBOBaHHe pelllemrn ypaBHeFlliH (3 I) H FIX mgtmm BlUl 3aBHClIT 0 T
Budopa KOHKpeTHoro rrpe~cTaaneFllilI anredpu rpyrrnu llyaHKape jO B Tepshy
MHHax KaHOmrqeCKFlX orrepaTopoB ( y p I 0 Po ) Budop lKe rrpencTaaneshy
IDlI orrpe~eIDeTClI lM3FIleCKIDIIlI CO 0 clpalKeFlliHMH bull l1epeMeHHHs S (X)
FlMeIDT CMHC~ ododmeHHHx KBaHTOBHX Koop~aT KOTopue B COCTOJIHHH rroshy
KOlI od~eKTa rrepexo~JlT B ~CCHieCKOM rrpen~e B rrpocTpaRCTBeHHO-Bpeshy
MeHHlle KOopmmaTu rrpoCTpaHCTBa MmlliOBCKOro 2- bull OTMeTflM lTO t gt
~O~ dUTb de3pa3MepHHMH Be~HBMH TaR RaK rrpeodpa30BaHHe ( 2 6)
He co~epJKHT lliC~OBOro pa3MspHoro rrapaM9Tpa IIpH 3TlIX ~orrO~Te~HHX
ifJH 3HlSCKFlX rrpenrrO~OlKelmRX ypaBHeFlliJI (3 I ) rrO~OCTbID orrpene~T d (cIshy-pa3MepHocTb rrpoCTPaHCTBa-BPSM8HFl ) H93aBHCRMllX HHBaPllaRTHliX OTHOCHshy
TenHO rrpeodpa30B8IUlli rpymm n yrurnape rrepeMSIDillX S-lt RaK lPyHKIUm aprYMeHToB X ~ bull J bull () bull tJ B WHRILHRX S rrpenrro~araeTClI orrpe~e~eHHoe yrropRUOleHHe H8JltOMMYTHPynttHX rrap orrepaTopOB Ii bull ~ H
I () () - HarrpgtIMep ( II Jj c D J)
I L~ I l l
PaCCMOTpgtIM Terrspb rrpOCTei1wee rrpenCTaBJISIDe aJIIsdpu F lepe3 Kashy
HOHWleCKH8 orrepaTOpu I il KOTOpoe peaIDl3ye TcH B rrpOCTpaHCTBe
QJyHKlUfi1 1 ) H 3MaIIO C rrOMOIllMl Coo TBolDemrli
p~== r ( 1v1r = ((~ J P - l( G Pf) (32)
Lp l~ ( shyI I ) L (1
B rrpOCTpaHCTBenBYX H3M8peHlll1 ( j = 2) ypaBHeFlliH (31) B rrpeJ]shy
CT~eHHH (32) gtlMeIDT CBOFlM pellleFllieM ~Ba ~HejiHllX rro 2 orrepaTopHo-
3HalHHX pemeHKH
toUJ~2~1 +f~Ir T-Y))~ JI r~ (33)
~ _ -ltv ) t~l(lf= cl (X7~r -= u ~ L v r flo1
B BHpaJKeHm ( 3 3 ) oIIepaTop JIOpeHIJeBCKIX BpameHJ-fi1 OL c ~ a orrepashy
TOp j)= lIc l 11 P pr(1 ~) gtlMSeT TaIDle lKe KOMMYTamlOHHue CBol-icTBa C
anre6poJ C2 IlTO H ohepaTOp pacTJIlKeHIDi
I1HBapHaHTHUe rrepeMeHllile S - (v-G = 0 I) ItOMMyrnpylOT MaJKny COshy
dOll FI rrOJIHOCTbID y~OBJIeTBOpHIDT BceM TpedOBaHllHMHeodxo)lHMHM ~ HCshy
IIOn30BaHHff 3THX ifurmami1 RaK apryMeHToB B rrpaBofI laCTH rrpeodpa30BaIllilI
(26) B nByMepHllX MO~eJIRX OrraCliBaIDmKX KBaHTOBOe peIDTllBHCTcKoe nBHshy
llleFllie OlJHoro rrpoTJIlKeHHoro od~eKTa2 4
KoMMyrHPyromne KOBapHaHTHe Koop~HaTu qrBHp8lKeHHl1e lSpe3 orrepa-
TOpH H3 aJIIedpH t(f2 ) D 1 6
l-iL ~Cl_~J il(~ - 2 D M ~ 2 L Me 1- M2 J j
i -=1 r pi l i if I ~ 2 lD tj 2 J -2 LMOL Mo J I
(34)
1v)=P -P~ C l )
JIBiIRKlTCR aHaJlorBMI1 KOopmmaT fu IOToHa--BITHepa B ~YMepHOM iIpOCTPaH-
CTBe
OleBHJHO lTO B rrpoC TpaHCTBe d raquo 2 H3Mep eHllM cywecTByeT Tonshy
KO OIHO JIlIHelHo e rro X pelliorme (31) Sf2) = 2~~j) 1I03Toshy
MY B 06mSM cnyqae rrpelLCTaBJI8flHe (32) HerrpIDoIlRO II)IR OIIHCaHHH peJlRshy
THBHCTCKOro IIpOTHlKeIDloro Od1eKTal1 CJI8Iye T paclllFlpHT1 KO lqmrypaIJHOHshy
Hoe rrpoCTPaBC TBO rrpe~CT~ermlI ~OrrOrrHffB ero YTJIOBHMH rrepeMeHHYMH
OlLHaKO eCJIB rrpoTHlKeHHilll od~eKT 06~aJLaeT CBO~CTBOM c~p~eCKOH
CI1MMeTplm TO OKa3UBaeTClI ~OCTaTOiHO nOCTpoHTb ~e IDmapHaHTHllX KOshy
opIJffiaTu BpeMeHHyIO H paJJlfanHyro
Sc()c) = (ctr~)rpr) l~ x)l coo LJlql() (35)
~TO HaxO~TCR B COOTBeTCTBllH C pe3YJIbTaTaMll pa60T9
KaK j~e OTMelaJIOCb Bume paCCMOTpeHH8 dOJIee od~ero CJIYiaJI MOshy
ue1IH cifgtepWleCKI1 HeCHMMe TpHlHoro od~eKTa TpedyeT rrepexoua K dOJIse
C~CraquoUUlM npelLCT~eH1UlM lJJm OrmCaHMH CTaTHleCKoro ~eiJoPMrrpOBaHHOrO
06beKra MH BH6epeM rrpe~CT~8ID8 anredpu lIyaHKape 13 iPJPMe
J C l e lL r I f f11S P = Igt IV) = (i f - ~I ) shyJ ) ) ( (J 1)
)-) K1 -- (C( pd - ~r~ ~ ~ - c - J
1) fj =t ~ )(36) j ~ - J
1 r~ [S ) f ~1 Li ( y) 7 ( - ( r 1 1= (C Coil S~
~ro npSuCT8BJI8HH8 peam3yeTcH B rrpoC Tp aHCTBe IWH1QWlli If yJ ~ r~e X - yrJill rrpoCTpaIlCTBCHHHX BpamellmL TIo ifJH3H1eCKOMY CMliCJIy KO[shy
JIeKTIlBHlle KOOPlUrnaTH r HI ol1IJe~eJIRlOT rrMOlKefDlle ~8txlpMllpOBaHHOshyr o od~eKTa 1lt1 ero opHeraquoTaIJJm B rrpo CTpaHCTB MaTp~ vl lre( YJ B (35 )
3aJLaIDT BHn r eHepaTopoB rpymru Bpame~ S t I l) pyt
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
IJ8CCMO TPJIM Hrl Atlno C()CTOIlHHJI coo rHuTC InYlI1li1 1I0 KOlweMYCJl OOmiddotH KlY
~~ ( bull l )
c )H Ij n i 1 I () ( 6)
~ )l i )1 rT ) 1) 111
jfl1 COOTIOUlelllR I ypan BlIUlR (44 ) Ilol11(CJ11JYJOT jO ~~i(lJ1W1e IlTU ra3ll0JWHlUI
orrcpAToF3 ~
12
M ~ -ll l U 1 (1 c -= t x ( (J ((L ) Mi - gt
AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
~ I ~ )CPl XJ _ 0II -_ ( lB )3t TA gt
JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
~ I ~ III( l (X JJ - J) ) G-I ~ = D I u ) x ( 1 ( u ) I L
ITPU)JCT8BI1M IToJIe ([II C ) R13Jrmomeecp lBWeHveM (4 5 ) H Blfne Pamp3JlomiddotshyteHJUl no nOJllioll CHCTeMe pem8H1f~ YP8BHSHIfl
J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
YCJWBI1H (4 8a) (4 8d) flCIltJ1X)laroT fl p a ilIoJtEilTllil J(OMIIOHGIITLI COOTlef shy
cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
IIIElnCll 18W(IlIWM YjlEllIllt llIlfl (1r) ~I 1I0)I IIIWIIlrj O yoqOllllHM (middotI tmiddot) I lIonyshyIJiM lU~fWlltf l lIll
1 laquo) (I) (4 Ill)I ( ~
HrullleM fII 1(lMMY1lHlIICI II II II
11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
(2 6) rrpli yllCpl8IiID B pa3Jolle lUIR BI1pall81ildi (2 3) CJIara ElMHX He Bhllll8
nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
- [ 6 Ip) R l~) lt t) I -COafeCrHOCTb pa3JOlfeIDIR (410) H COOTHOm8HHA (4 n) rrpImOllwr K rrpashyBwry KBaHTOBalliUl
-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
TaKHM o dpa30M B lCKOMOM rropU)(e T80Pruf B03MymCHHIl nOJIe ctshynpellCTaBHMO B ~e
1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
OJ I i ) JIBJIJIeTCR peWeHItOM KJIaCCIIlel)oro yIaBHeHIUI (45 ) ae BTOPtllUIO lltBaJiTOB8HHoe IlOJIe J1 1 1) ()rrpelleJIReTCR pa3JlOIIIBHl1eM
(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
Hmm 110 KlllUlJ I) ITO 111(IIpLH IJ83TIIIlJW( rtl)onCTalVlllI1H 8JIluOpU ITyaHIC1shy
pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
leCIWM nrllI~-1t 11m nocTllOOHll npouenypa KBaHTOllrumR nOKaJIbUOlO reli-30Ildl11JCIIIC JtOIO nOAA B MOlleJIH CTaTIPieCKoro neq()PltIiij)oBatlHoro II[oOTRshy
lIWHJlOIO o01ewra
C 3TOH ~eJnlO B sect C npOli3BSnSHO rrpeo dpa30Brurne K HOBUM Tlq)eM8Hshy
HLIM n03DOJUlOlltee cTporo yqecTb 3axOll~ COxpaH8lffiR II OTIleJUfTigt KHIJ8THshy
iecKYlO laCT 3anaTffi OT Jl1Ill8MJ~lecKOI1 11 pS3yJllii8T8 rrpo06pa30BalHfR apshyryMellTaMH nOJIR CTaHOBRTCR CneUKanbHHe HllBapHaHTH~e n eVeMeHHU8 ~TH
nepeMeHH~e ~TCR KOMdllll~eR reHepaToI~B rpynTIH IlBIIXeHHR II B
IltJIaCClfleCKOM npellene XapaKTepH3YJlT I(BIDtelrue npOTmteHHoro 06bampKT8 181
u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
OT B~dopa npencTaBJIeflliR aJIre6ptl nyruIKapo 110 B CBOID Olepellb onpeneshy
JmeTCR ~3lfleCKO~ CHTYauHeK B paCOTe paCCMOTpeHU rrpenCTaBJIe1~R 00shy
OTBeTCTBJ1OUU1e nBIDIteHIDO JUfIlffi onHo ro ITfJOTRZeHHoro Odb9YT8 _ 13 sect 1 IlJlH
pemeIHR JIRHaMHeCKOH laCTH 3alIa1Jll I1CnOlTh30DaHO lltBaSHKJlACCfImiddotreCKOO
npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
B03MYUleHIrt CTPOHTCR JOOIlI nO nepBoro IIOPRllKa pS3JlOlKemm nOAA no
5t BKJIlOliHT8JIbHO
ABTop 6JIarDll8PHT 11 II BorOJIlOdoBa K ACBellllliKOBa Ii H E CKaKoua 3a 0 6CYJKlleHHe BOnpOCOB SaTpOHYTUX B CTa fhe 11 p e syJIb TarO B paCOTti
JI H T C I a Y P a
I Cj llmiddot_~t J ~ I l cc T U I _middot~_middot bull _-f 1 -( 1J~~j O I il ~ t t - 1Lff
TOHUO d i ~J middotn middot ~J J~_ ulc IC f 1 I ~ - ~ -J4
Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
3 YM J)l3aB8 X llauyMoTo X TaTuKil - TCtMOIIOJl l Jlftrl Jl)1I1HMHI fJ It 1(1l11shy
neHc HpoBaIlHtle COC1OJIHllH -1gt1 1 HIL) l Jl I X
4 CBEUIHIgtIKOB ILA_ l1 I~L N ~ CI-i ll
i _ 1 ()1fM~J(nII 11 11 JlljlMrIMl 1 -~ I IA Ill 1 U-1 II1I1I11
T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
7 j r LI
_ I shy 11 141 bull I U 1 1 lHlt
I i
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
~ c (l) (( ~) OleBHlUlO YOBJIeTBOPReT ClCr eMe
ypaBHermii (31) Hai1l1eM Te n epb TJlH IlPyrRX HHB8PHaHTHWC peIDeHRH cncshyTeMil yenPaBHeHHil (3 I) IIBJIHI[lllHXC$l JIHHeiiHliMll no -t H RIlHO 3aBffCRlItltX
0 1 yrJIOBbIX nepeMeHHIDC llruJ 3TO IleJIJ MbI paCCMOTpHM CHa~ ypaBHeHHH
orrpe~e~e TpaHC~OH~ HHBapnaHTHOCTb KCROMHX pemeHR~
J l1 () c ~ (T S (37)
0611HM perueHKeM ypaalieHIDl (37) OqeBlmHO HBJIReTCSl ltPYtultmm ~ u I( J I ) y S) == c (I 1 ) J ~ ) C YleTOM IIOryq9HHOIO pemsmm
CMCTeMa ypaTmelllHl rapaHTIIPYfOIllWl mrnafiHaHTHOCTlgt pemeHlUl OT80ClIlTe7IbHO
rrpoCTp aHCTBeHHhIX Bpa1I9HHii BJJIJlR)tllr C19I(ylOlllIDgtl 06pa30M
- (t lt r amp _~ Ij ~ L~ i 1 I c ) SJ~LL ) 1 I 11 ip rJ r ~P p ~ (18)
JlilHeilHhle no _t peweHKc1 yprumelDfH (3l ) HMelOT IDlll
So (J J ( fl f ~) = I (J) j(J( l bull J ~)pe ) U t ) - MaTpoUB TPeXMePHHX 13P8U(OHlol11 3am1CalUlaJT illllTpl1Mep B napaMeTpif31lltIDl ~JIe-pa
I i
U ) - l
( )1 ) ~ ~ll I t)
l~ UIIJr 1 1 dk
( P)middot -
( P) pound y~ p ) ~
j l ( J II 1J I I) ( l II) ) ) j ) ) 1( P) a lljJ) Fr jJ) I r)) 111 11 ) - npol~ 3BOJIbHble PYJiRltlm SIBJIftlOOUI8Cf CIUlshy
nfYpflMH OTIlOCHTeJIhHO npoCTVaIiCTBeHIiliX BpamellJiJi nepeMeHRl1X p
~ 1 t r 11 OTIICllCJl1IM 113 p8Ul8HM OCTaBIDKXCR
ypaBHeHIlli CHCTeMIi (3 I) KOToplie BupalKaKlT c060H c BOkicTBO HmapHaHTshyHOC TH IfCROMliX ltWH~I OTHOCIITeJIbHO [ZCTO JlOpeaneBClltJ1X Bpauterurlt )
t it pJs ~ r tl i f 12 F i- pJY ld f _
( rIt ) PtI ( 11 )l r () rf-f~fjij ~ (39
= LaC 1 T (1 ~L 1 i) L I )1( 1 J( J11 Jp ~
TIOCJIe nOllCTaHOBKPI B ypaBHeHIHl (39) pemeHlUi (38) 1I0JIyltmM ypaB-
HeHHH Ha HeH3BeCTHJle JiYHKl(KJI ~ L ZF U ~ Hal1neM HX peweHH8
iz~ l-I (Ill I ll( P) -I - _ p - =- Ili r f )
T8KHM 06pa30M MH n ORa3aJrJt ~TO B IpellCTarureHID (36) mmapH8BTshylWMH OTHOCHT8J1bllO npeo6pa30Ba~ rpynnh IlyaHK8pe (3 I) P9JIJITlrIBIICTCll1shyMIl ROOpItHHa TaMll JIBJtIllOTCH nepeM9HHlle BIfJJB
raquo
Scmiddot
p o) = (1 rJl c () ) = A) j) X t ) (31u) J
eP) r f n fgt n oS rlXdt) =- [( X ri)k _ (L rq -g 1 ik r lle -- f-r p j
LlJklP) = (a Jk -T p~ rgt( _ ) M Mt ~h r )
HanOMRHM ITO no 111lBalHaHTHOCrM1 n epeMeHHlLX (3 ro) nOllpa3YMoshy
BaeTCR CJI8~ee CBOtlCTnO nycTb T(l = fJl ) ( 1 c1 [raquo 1-1 Lf = c~ (L ~f) eCTb YHHTapHble onepaTopu TpaHCJImm0 H JlOpeHUeBcIGJX Bpamemrn COO 1shy
BeTCTB9HHO TOrna
U T --ltgt) T Umiddot L -= S(I(4~)( fa)) i() tl lJ R LL)
OlemIJrfIO 110
LS degC)J S ( )c ) ]=Ugt (3 1I )
TaR KaK S~ lt
(x) 6eSpa3MepHl1
KOMMYT8UHOHlrue COOTHomellIDl Meany S ~Y) HMeJOT BHn
l c 0) l tI~~-J1 =- c ~fS~_ P Ilaquo l~~5) ( 3 L 7 (fl-r Pj
TrurnM o6pasoM B rrpecTaBJI81ID1 anre6pu r8liepaTopoB rpymru TIyaushyKape 3aAaHHOM COOTHOWeHHRMI1 (3 6) cymecTByeT 4 KHBapHaHTHUe ve~~-
1rnWCTCKlW KOo]1JlHHaTIi ~(( 2 ) (3 IO ) AAHeiltlHl1e no t ll IDIIeJOmHe
MellnY eo60M KOMMYTattHOHHhle COOTHomemm (3 II) ReKoMMYTaIyenBHOCTb
BeJmllJfJ l 4U) (3 TI) orpatlHllffiaeT B031JlOlKHOCTh IIOCT]XlSHo$l pery-
JISJpHoi TeOplrn B03Mymelllfi no J lIlUi nOJIR 0(X) (2 6) TI03TOMY B
CJIenllUleM p a3llOJle Te MlJ zCIloJlbyeM nepCtvleHHll8 IJ1IR lJOCTshyS- (2) poeHHH UOJUI B Moneml CTamqeCKoro ]J9cJxlpmpOBalUlOIO 06b9KTa Bce Beshy
JmtUrnbi eoprul paCC~mTpltBamplTCR TOJIhKO B lByx nepBux fIeI1C~e3amplnnrr noshy
pJInK8X IlX pa3JI01leHIDI B IM no napaM8 -r1Y 17 (ItIm lleROTOpoii CTeneHR
J ) ilepSMeHH18 71 (3 10) JllIJI1IM o6pa30M Bup~alOTCJ tIepea reHepa-
TOphi lpyYlIlH PI L~ fr )
reHcparop paCTRllteHKi1 j) v bull i I I
iC ~ =- L 112 i ] 111
I)
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
TIepei-inUM TeneJb K Rli110JlY TPX oJJ aHl1PHHii K IOTOPHM nplHlOlll1T
lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
aJITlIOCTb nonxona ooucneI1Ba8T rrpaalVTMIJjO ~laBHCJlMOC r h iOHCIaHT JlBIIJlte shy~HHH 01 JIlIHeliHoil 11 yrJlOBO ffOpOc nd1 nO ~lTOMY yCJlOHlIR CnRal1
IJ8CCMO TPJIM Hrl Atlno C()CTOIlHHJI coo rHuTC InYlI1li1 1I0 KOlweMYCJl OOmiddotH KlY
~~ ( bull l )
c )H Ij n i 1 I () ( 6)
~ )l i )1 rT ) 1) 111
jfl1 COOTIOUlelllR I ypan BlIUlR (44 ) Ilol11(CJ11JYJOT jO ~~i(lJ1W1e IlTU ra3ll0JWHlUI
orrcpAToF3 ~
12
M ~ -ll l U 1 (1 c -= t x ( (J ((L ) Mi - gt
AHaJI01l-flJIO 13 CBR3efi Clt) C~1 ( r1 noJIYlKM COOTlle[CT I VL l
BSHHO YCIOH1f
~ I ~ )CPl XJ _ 0II -_ ( lB )3t TA gt
JCD~ Q Cshy~d) d X 1 -~ ) (4 ttj)
~ I ~ III( l (X JJ - J) ) G-I ~ = D I u ) x ( 1 ( u ) I L
ITPU)JCT8BI1M IToJIe ([II C ) R13Jrmomeecp lBWeHveM (4 5 ) H Blfne Pamp3JlomiddotshyteHJUl no nOJllioll CHCTeMe pem8H1f~ YP8BHSHIfl
J f f )- ) oJ U ((j)~ ) I = it) t t )
(4 9)~ C~lt) fJA
(C) ( (() = ~ltr
YCJWBI1H (4 8a) (4 8d) flCIltJ1X)laroT fl p a ilIoJtEilTllil J(OMIIOHGIITLI COOTlef shy
cTRYlOUU-Ie IIYJleBIJM MOnBM
J 11~j J ~(j - ((lt ) r I _( J de middot lt)
111 Cf8Jl110HapHUX KJlaCClfleCKlIX UWulIlll1 C ( YCJJ(lBII ( L IIJ) IUiloIIIlI shy
UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
IIIElnCll 18W(IlIWM YjlEllIllt llIlfl (1r) ~I 1I0)I IIIWIIlrj O yoqOllllHM (middotI tmiddot) I lIonyshyIJiM lU~fWlltf l lIll
1 laquo) (I) (4 Ill)I ( ~
HrullleM fII 1(lMMY1lHlIICI II II II
11
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
(2 6) rrpli yllCpl8IiID B pa3Jolle lUIR BI1pall81ildi (2 3) CJIara ElMHX He Bhllll8
nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
- [ 6 Ip) R l~) lt t) I -COafeCrHOCTb pa3JOlfeIDIR (410) H COOTHOm8HHA (4 n) rrpImOllwr K rrpashyBwry KBaHTOBalliUl
-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
TaKHM o dpa30M B lCKOMOM rropU)(e T80Pruf B03MymCHHIl nOJIe ctshynpellCTaBHMO B ~e
1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
OJ I i ) JIBJIJIeTCR peWeHItOM KJIaCCIIlel)oro yIaBHeHIUI (45 ) ae BTOPtllUIO lltBaJiTOB8HHoe IlOJIe J1 1 1) ()rrpelleJIReTCR pa3JlOIIIBHl1eM
(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
Hmm 110 KlllUlJ I) ITO 111(IIpLH IJ83TIIIlJW( rtl)onCTalVlllI1H 8JIluOpU ITyaHIC1shy
pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
leCIWM nrllI~-1t 11m nocTllOOHll npouenypa KBaHTOllrumR nOKaJIbUOlO reli-30Ildl11JCIIIC JtOIO nOAA B MOlleJIH CTaTIPieCKoro neq()PltIiij)oBatlHoro II[oOTRshy
lIWHJlOIO o01ewra
C 3TOH ~eJnlO B sect C npOli3BSnSHO rrpeo dpa30Brurne K HOBUM Tlq)eM8Hshy
HLIM n03DOJUlOlltee cTporo yqecTb 3axOll~ COxpaH8lffiR II OTIleJUfTigt KHIJ8THshy
iecKYlO laCT 3anaTffi OT Jl1Ill8MJ~lecKOI1 11 pS3yJllii8T8 rrpo06pa30BalHfR apshyryMellTaMH nOJIR CTaHOBRTCR CneUKanbHHe HllBapHaHTH~e n eVeMeHHU8 ~TH
nepeMeHH~e ~TCR KOMdllll~eR reHepaToI~B rpynTIH IlBIIXeHHR II B
IltJIaCClfleCKOM npellene XapaKTepH3YJlT I(BIDtelrue npOTmteHHoro 06bampKT8 181
u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
OT B~dopa npencTaBJIeflliR aJIre6ptl nyruIKapo 110 B CBOID Olepellb onpeneshy
JmeTCR ~3lfleCKO~ CHTYauHeK B paCOTe paCCMOTpeHU rrpenCTaBJIe1~R 00shy
OTBeTCTBJ1OUU1e nBIDIteHIDO JUfIlffi onHo ro ITfJOTRZeHHoro Odb9YT8 _ 13 sect 1 IlJlH
pemeIHR JIRHaMHeCKOH laCTH 3alIa1Jll I1CnOlTh30DaHO lltBaSHKJlACCfImiddotreCKOO
npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
B03MYUleHIrt CTPOHTCR JOOIlI nO nepBoro IIOPRllKa pS3JlOlKemm nOAA no
5t BKJIlOliHT8JIbHO
ABTop 6JIarDll8PHT 11 II BorOJIlOdoBa K ACBellllliKOBa Ii H E CKaKoua 3a 0 6CYJKlleHHe BOnpOCOB SaTpOHYTUX B CTa fhe 11 p e syJIb TarO B paCOTti
JI H T C I a Y P a
I Cj llmiddot_~t J ~ I l cc T U I _middot~_middot bull _-f 1 -( 1J~~j O I il ~ t t - 1Lff
TOHUO d i ~J middotn middot ~J J~_ ulc IC f 1 I ~ - ~ -J4
Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
3 YM J)l3aB8 X llauyMoTo X TaTuKil - TCtMOIIOJl l Jlftrl Jl)1I1HMHI fJ It 1(1l11shy
neHc HpoBaIlHtle COC1OJIHllH -1gt1 1 HIL) l Jl I X
4 CBEUIHIgtIKOB ILA_ l1 I~L N ~ CI-i ll
i _ 1 ()1fM~J(nII 11 11 JlljlMrIMl 1 -~ I IA Ill 1 U-1 II1I1I11
T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
7 j r LI
_ I shy 11 141 bull I U 1 1 lHlt
I i
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
q = _L ~k ii~Jk T Do ~ ~
~ ~
LV t- W-) p M r ue ~L -_ L T CC Ik P I
Vd1 S
3TH ~eTHpeXMepHHe KOOp~TU ~TCH aHanoroM KOOp~T HbDTOHashyBKrHepa B ~eT~eXMepHOM rrpoCTp8HCTBe EnH HKX BUDo~eHH COOTHomeHKH
L l~ rt( J I -0 0 ) L l1 r pJ 1~ - ~~
B 3aRJIJOleHl1e SToro pa3lleJIa MIl paCCMOTpIDJ C~ oCimIDi CJIYiaH MOshylle~ HepaBHOMepHo llBHEyme rocR ~eqopMKpOBaHHoro oCi~eKTa
~ STOro BuCiepeM rrpellCTaBH8HHe rpynnH ITYaHKape lle ttcTB~ee B
MaKCHMarrbHO pacIDHPeHHoM ICOHqiHrypauHOHHOM rrpoCTpaHCTBe napaMeTpoB
rpyrrrru I ( L f 0) ~
f I ( V 4 ltv (3 12 )I ~ i r M = j 1 ( pi Sl )~ -tshy
rlle [ r () LI h LGj rv LC r J 0 r J = L fl t (l~)( r j)
[ S( ) sf J ~ ( fj fS J1 -f IY ))1 VsJJ
B STOM rrpellCTaMBHIDI iPYlnuurn
AgtLl) () ~ ) V (I))r ( ()Pl )middot lt 1 (3 13)
oCipa3yroT TIM M HaCiop mrnapHaHTHRX peweHJtii ypaBHC~ (3 I) B Bupashylte HIDI (3 13) t~ C-) een MaTpHua npco6pa30BaHIiH JIopeI-n18 ~CKOMye peJIRTlmllCTClrne Koopmrn3ThI BIgt Ci epeM B BlUle
)S ( J ) = II I _ fl 1 ~ ) 11 ( 3 14 )
ITepeMeHHHe (3 14) o6Jla)l8JOT cBOilCTB8MH
LJ T C (X) Tt i L1 pound -t
-= S( _1 J ) 80gt
10
rlle ~r
-= it II(()] II
R LSr(~)) SJ(r) J = j [(lj)r1jJ) J_ UIf)(f Y IJ
MTaK HaMH nOKa3aHO ITO C nOMomom BuCiopa pa3~ux rrpellCTaaneshyHHH anreCipu ITyaHKape lepe3 KaHOHHleCKHe KOOpllHHaTU (32 ) (36) (312) MorYT CiUTb nocTpoeHN ouepaTopH03HalHHe fiHBapHaHTHUe nepeMeHshyHUe C oX O- (33) (35) (310) (314) 3TH ltpyHKIlID BXOJfjT KaK
apryMeHTU B rrpaByro laCTb rrpeoCipa30BaHHR (16)
B c~euyromeM naparpaWe MIl Bocno~3yeMcR pa3BHTliM B sect 2 B
qoPM~~noM ~ nocTpoeHMR TeopHH B03MYmeliHH B oKpecTHOcTH ~aCCHshyleCKoro pellle ImR XapaKTepH3YlJl11ero CTaTliJ1lec KIrl lleqopMHpOBaHHlill rrposhyTRlIIeffilm oCi~eKT
sect 4 3aMeHa nepeMeHHux
BepHeMCR K nOCTpoel-Urn B MOll8Jrn (21) nOM cfJ ( t) bull rrpOKBaHshy
TOBaHHOrO C nOMOmhm KaHO~eCKKX KOMMYTauHOHHUX CooTHollleHHM (23) TpYllHOCTH CBR3aHHUe C pa3lle~eImeM onepaTopoB R nOM 0rx) a TaKlKe HeKOMMYTaTRBHOCTbm C - (2) (3 II) lleJIaJOT 3aKOHHllM UOIiaK peshyllieHHR KHllih B KBa3H~aCCHleCKOM rrpH~eHHH
(4 1)ocs) = ~laquo) Vf q L Cs) ~ 0(d)t-
3lleCb 0 (~) COlleplKHT oCTaTolHl1i1 ~eH paMOlKeHMR nOJiR C1JCS) H Heshy
onpeue~eHHOCTb OCiyc~OMeHHy1O npoCiJeMoll ynopRllOleHMR onepa1opoB Clt KaK BI1llHO H3 llaJIbHeilIlJero B ITpHCiJIIDKeHHH (4 I) nepeMeHl-ilie MOlKHO pac-CMaTpHBaTb KaK C -llic~a a BUp1lleIDIR (28) (29) orrpell8JIIDJllU1e CBR3H H KOMMyTauHoHHHe COOTHomeHKH llOCTaTOlHO C~TaTb orrepaTOpHliMR COOTHOmeHRIDm
PaCCMOTpHM 38MeHY nepeMeHHblX B ypaBHeIDIRX lUllIlKeaaa KOTopue B oCiDleM BRlle llMem fllopMy
w llIJgt Cj) - U(Q)) ( f) l X)) 14J gt= 0 (42)
BoCIT0JU30BaBIlIKCh RlIIillIM lJH]lOM nepeM811HllX C ( 3 12 ) H cIoPMY~8MR
II
3
-lt ) APd c j
( [) 1 J)
) A (
HM80M j L J shy
)(4 )I I I 4 --) bullbull L UJ I~ )
~j ~ (4 3)
llIlCJ iL - nepB~l-i I1IeH B pa3JIOllteHlfR
s )~l co ~ ( -0- V I ~ I 1 ( ~1 L ( ( d
~ middot1
( [Ii i = U B clUJ) BlIllOm 8ID1R yCJloJmii CBR 311 CM liIDPie ) TaKHM 0 6pa 30M
nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
I L I (1 L (( L ) ( (44)L
j (4 5 )lJ ((1 l bull ( ur middot )1 t
iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
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J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
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J f f )- ) oJ U ((j)~ ) I = it) t t )
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cTRYlOUU-Ie IIYJleBIJM MOnBM
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11
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1 IJ )
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11
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lIWHJlOIO o01ewra
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JI H T C I a Y P a
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H tJ -JILJ
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J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
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l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
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C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
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llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
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011$111
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IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
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( [) 1 J)
) A (
HM80M j L J shy
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~j ~ (4 3)
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nonyqaeM ITO ypaRHeHI1Jl IIBIDK8Hlltf11 B HYJle1l0M 11 nepOOM nOPMRO oa3JIOllleshy 1
IIlHi no D HOBblX n epSMOH1lUX 11j ilUiHMl1OT Dlfn
1
I
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iteJlJHegtiHOo y paJ3H0JllIe (44 ) IlpencT8BJUIeT codol KJIaCCI1eCKoe ypaBlletme
nB1U8HlIH COOTSOTCTBYOlueo MO)JeJH ( l ) 0110 no npeJIJlOJ[OlKeHIfIO MeeT
HeTpHBlfaJID HOe peOJllrne 1 (t ) xapAKrepI1 3YXlIllec CTallllCClillii Jl(shy
~OPMHllOBaHHWi OOheKT i3 TpeoyeMoM nj ll1(IJIWIWHHI1 Yl)aBHBlllIe (45) onJl8Jleshy
J1ReT BTOPllUlO KBaHTOBaJlHoe n ore n i I )
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lIaJTOlgtlCltI1P YCJJOBlIP CBR311 ( i) Ha nolll q J) OTM P THM ITO KOBAlI11shy
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UTC CCII1 yCJlOUI18 ( 488) YJtOHll lIlOJf Il O Tr ]IJl NUl lion ( HIrI~
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11
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H tJ -JILJ
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J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
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l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
11 YCJIOBI1R KBaHTOOOHHR BMTlJIIfTyn ex C IIOMOUlhlO rrpe odpa30BaHlUl
(2 6) rrpli yllCpl8IiID B pa3Jolle lUIR BI1pall81ildi (2 3) CJIara ElMHX He Bhllll8
nepsoro I10PRIlKa no ~ noJIYUIM
[ (jl(T) (] i l ~I) ( -()1 - LA lJ(t~ ~7) =C )
[G1((rrd () ] 11 t 8(((~)=tfr((()- I1(((~
tr -)T I r- J J P ) jJ 1 I ( - ) I - _ (4II)L 1( J 1 ) I ) r l F ~~ ~ laquo r- ~- 1 -~ It b( t $lt) = a
1 IJ )
(h 0gt( 7) = r)- + ) f7(t~ g(s~F) = L 0 (Jdl l - t l 1 ~
- [ 6 Ip) R l~) lt t) I -COafeCrHOCTb pa3JOlfeIDIR (410) H COOTHOm8HHA (4 n) rrpImOllwr K rrpashyBwry KBaHTOBalliUl
-[ [1 ~ (t 1 ~ ( T
L1i ~
I () -J L( I l ll-- (I T~ I ~ c ~ (4 12 )
ITO JIeIKO npOBepRCTCR Ha aelltTOpe COCTORI-UiR JOKOR ( = 0)
TaKHM o dpa30M B lCKOMOM rropU)(e T80Pruf B03MymCHHIl nOJIe ctshynpellCTaBHMO B ~e
1lh J ) 0 4 1 (~ ) ~ (J l ( ~ S-) -c=
(413)- J ~ J) i - r
[ T t r l L I 11 u ) 1 ~ i ) )
r ue
d (1 ) r t41 1 ~ - bull 1
J (~ lS UT( - ~tiIr) 11 () Il_ j qUl~)
OJ I i ) JIBJIJIeTCR peWeHItOM KJIaCCIIlel)oro yIaBHeHIUI (45 ) ae BTOPtllUIO lltBaJiTOB8HHoe IlOJIe J1 1 1) ()rrpelleJIReTCR pa3JlOIIIBHl1eM
(4 10) bull
sect 5 3 a K Jl 10 Ii e H If e
B llaHHO ~ p a 60 Te C nOMOmhlO rrpeodpa30BBHMR boIO~oBa HaMH H3yqe shy
H8 rrpoCJI8Ma rro c TpO eHnn rrpo ue~yp~ KBaHTOBBHHR rrOJIR B o KpecTHo c TH Heshy
TPRBIII8JIbHOro KJIa CCHi8 CKOro peW8HHR B 18 TllpeXMepHoM npoCTP8HCTB8 bull
11
Hmm 110 KlllUlJ I) ITO 111(IIpLH IJ83TIIIlJW( rtl)onCTalVlllI1H 8JIluOpU ITyaHIC1shy
pe MOllliO OllltOUfl Je nmtlltbllHR IYpOTRlReHHoro 061UKTa n KBOSIOUl8Cll shy
leCIWM nrllI~-1t 11m nocTllOOHll npouenypa KBaHTOllrumR nOKaJIbUOlO reli-30Ildl11JCIIIC JtOIO nOAA B MOlleJIH CTaTIPieCKoro neq()PltIiij)oBatlHoro II[oOTRshy
lIWHJlOIO o01ewra
C 3TOH ~eJnlO B sect C npOli3BSnSHO rrpeo dpa30Brurne K HOBUM Tlq)eM8Hshy
HLIM n03DOJUlOlltee cTporo yqecTb 3axOll~ COxpaH8lffiR II OTIleJUfTigt KHIJ8THshy
iecKYlO laCT 3anaTffi OT Jl1Ill8MJ~lecKOI1 11 pS3yJllii8T8 rrpo06pa30BalHfR apshyryMellTaMH nOJIR CTaHOBRTCR CneUKanbHHe HllBapHaHTH~e n eVeMeHHU8 ~TH
nepeMeHH~e ~TCR KOMdllll~eR reHepaToI~B rpynTIH IlBIIXeHHR II B
IltJIaCClfleCKOM npellene XapaKTepH3YJlT I(BIDtelrue npOTmteHHoro 06bampKT8 181
u eJtoro KaK nOKa3aHO B sect 3 RBHIDi BM IlliBapH81ITH~X nepeMeliJ-illX ~813OCliIT
OT B~dopa npencTaBJIeflliR aJIre6ptl nyruIKapo 110 B CBOID Olepellb onpeneshy
JmeTCR ~3lfleCKO~ CHTYauHeK B paCOTe paCCMOTpeHU rrpenCTaBJIe1~R 00shy
OTBeTCTBJ1OUU1e nBIDIteHIDO JUfIlffi onHo ro ITfJOTRZeHHoro Odb9YT8 _ 13 sect 1 IlJlH
pemeIHR JIRHaMHeCKOH laCTH 3alIa1Jll I1CnOlTh30DaHO lltBaSHKJlACCfImiddotreCKOO
npHd1OOKeHHe 113- 3a H9KOMMYTaTHBHocrn l1H13apIHllITHHIX nepellem llix TeOpl1Jl
B03MYUleHIrt CTPOHTCR JOOIlI nO nepBoro IIOPRllKa pS3JlOlKemm nOAA no
5t BKJIlOliHT8JIbHO
ABTop 6JIarDll8PHT 11 II BorOJIlOdoBa K ACBellllliKOBa Ii H E CKaKoua 3a 0 6CYJKlleHHe BOnpOCOB SaTpOHYTUX B CTa fhe 11 p e syJIb TarO B paCOTti
JI H T C I a Y P a
I Cj llmiddot_~t J ~ I l cc T U I _middot~_middot bull _-f 1 -( 1J~~j O I il ~ t t - 1Lff
TOHUO d i ~J middotn middot ~J J~_ ulc IC f 1 I ~ - ~ -J4
Pa3YMOB AB 1M 1976 JO ji I lc-30
j openl H B ~ tanneeB J1[ Tj 1~75 25 I ~ I47- IG3
2 bull J 1crrLal11 Jucl 1 3 1)( 1 31 0 4 ~ ) - iI J
3 YM J)l3aB8 X llauyMoTo X TaTuKil - TCtMOIIOJl l Jlftrl Jl)1I1HMHI fJ It 1(1l11shy
neHc HpoBaIlHtle COC1OJIHllH -1gt1 1 HIL) l Jl I X
4 CBEUIHIgtIKOB ILA_ l1 I~L N ~ CI-i ll
i _ 1 ()1fM~J(nII 11 11 JlljlMrIMl 1 -~ I IA Ill 1 U-1 II1I1I11
T JYJIII 1 h lll1l1 lIfIYurtrll JIJ1 1 I Ir
W COJlI1Jlf frt1 (Ill I II bull 101 UfI 1M 111 11
H tJ -JILJ
7 j r LI
_ I shy 11 141 bull I U 1 1 lHlt
I i
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object
o bull jte~nmr rJ O uclJhy s l J7b iJ 14 141- 1L gt
idlLBUJUOLO H Umc~~uwa iI l vpestnmutiou N J [J19 flev 1)83 lAu f JtJ lljJ4 - 1~4 u
J Lllwfw l 1 I Sl l( C yw icles a llJ ~i El d s 1)60 L Iw l 1) ) - 1)4
LlUUdlJ ~~ If I l Ui~C Lo II L~fIe~[wlI II P~middotfS tv 11131 ])24 ItL t - - 14
l ~jJft I IJ ~ I oJ 1 bull tJ~ bull u I )4U- 1Jb1 lOJgtOXOR AE lM1gt T9U4 GI bull r 1 64-04
middotu ~l1O I bull I) II j I bull bull l I1~ 1 ~l t J~ ro l ( ) ~(j- 31 0 bull JJT 1 t i lJ U ( L umv II Umezuwu II It middotOHf (J or il i or
tL ) hJ ~) 1(4 -llJtl)
i 1 bullbull r 11 bullJLl I ltJ ~1tl t ~I~ II bull i middotUrr~ r 1 II CI-l Jh ) - 1bull l J
PyKOllllCh ITOCTYTUL1U 3 il3~aT~UbCK~ l OTA~1 ~O aarycTa 19b5 ro~a
Ifl
llll P2-85-632 I- (1 1 I UlIITOR l1HHe B MOlJe nHX rrpOTHlKeHHbIX 06beKTOB
C IIOMOlllblO npe06pa30BaHHH EoroJ1lo60Ba HsyqeHa rrp06neMa
nIHTpoellllH npoue lJYPbl KBaHTOBaHHR rronH B OKpeCTHOCTH HeTpHshy
llHaJlbllOJ O Kn a CCHleCKOrO pemeHHH B qeTblpeXMepHOM rrpOCTpaHCTBe
B p e 3 ynbTaTe rrpe06pa30BaHHR apr)Me HTaf1H nonR CTaHOBRTCR crreshy
UHa nb llblC HHsapHaHTHbie rrepeMeHJbIe OHH HBnRIOTCR KOM6HHaUHeif
rlH Cparo pOB rpyrrrrbi AB HJKeHHR H B KJlaCCHICCKOM rrpeAe n e xapaKshy
TI pll~yIOT lBHlf(eHHI npOTHlKCHHoro 06beKTa KaK uenoro nOKa3aHO
t l n~ 1I11liIIPUl pa~ITHlHble nptACTaBneHHH aITrc6pbl nyaHK a pe lOlKHO
OnlllII l h II Il lllH l lKe ll HR npOTRlKe HHoro 06beK T a B KBa3HKnaccushy
ilC l l ~ IIlJIII1J1I1+l lIll11 nOCT p oeHa rr p OuelJypa KBallTOBaHHR nOKaJlbHOshy
10 I fl I If1 p l UII CKO 10 1T0nR B MOlJeJ1 H eTa THlleCKOrO lJe4l0pMHpollaHshy
HOlll tqllll Hln llllul ( llICKTa
I 1110111 1111 11 11 1 11 JIa60pa TopHH TeopeTH 4eCK Ol1 41H3HKH
011$111
npenpllll) t)(I~ Ulll r tlllIO H IIC THTYT8 RDepH1olX HCCneA08IlHHA IlYCSH8 1985
IJep(lIllJ [I G flIIIl)IIIIJ)I 1111 11
P2-85-632 in MtI l s of Ext e n d e d Objects
The invest i ga L 1011 Iw o hl~O Il lir f middot11l(middotd ~ j l Ihc Laorilt o ry
of T h e o retical Physics J INH Prprint of the Joint Institute InJ NIII I ~u f iI IIH h Dllbna 1985
approximat i object