Sci..
R{:p.Kanazft:
waUniv~
VoL2
1，No.1，pp.各1‑116June1
975，

A NoteonYang;‑Mi
HsField
I日asatoshi
Kan
αzawa
(ReceivedApri
I30，1976)

Js
tr
ad Aresultofa
nattempttoobtai
nasolutio
nofYang‑MiIIsfield己quatlOn
Ali
spr
芭sentedunderbothagaugeconditionpA~ =qA~ ， wher日 ρ
andq arεnmnericalndacondition(戸 θ~+qåc') )(a=o，whereXαisar
bitr
ar
yphys
icalq
u
ant
it
y.
const
ant
s，a~

1 Intr:odu
ctiollar
enows
日ar
chin
ga
fte
re
xactc
lassicalsolu
tionso
fthe
for
Inthi
snote we presentpreliminaryresultsIorthisnonabelian gauge field!
)川 市
i
mi
te
dre
su
lt
su
nde
rsev
er
econstraints. The equations we s
hallatte
mpt，namely linv
estigateare婦DνG~νニ/自(1)ら νAρDν=0，(2)whereDνdenotescovariantderivative，G~νfield str
engthoffieidA~ ，andJ，'ffer
mioncurrem.
Throughoutthisnoteweshal
lfixt
hegaugew
ith
jう0~

~

(A~iAg)，(3)whereρandq arenumer
icalconstants. A
lsothroughoutt
hi
snot
ewes
hal
lassumeaconditiona ，̲
.
1 aa(p一一+q‑ーν at )X = 0，(4)

where Xa is any arbitraryquantit)
人Eq.
(4)i
sconsitstentwiththεgauge condition3).Eq.
(4)showsanyq
uantityisIunctionof(qz‑jうct). Wei
ntendtohaveasolutionnde
rtwoconditionsthata)J
ι J f andJ~ (j~ willbedefinedinofeqs.
(1)and(2)ueq.
(8)alwaysv
anishes，respectively，forQ = むo/q)
2‑1=0case，andJ~ andJ~ alwaysespectiv
ely，for Q弓と ocase，andb)fiel
dA，
'fa
ndf
ieldstrengthG，'fνv
anishesv
anishes，rr)b
ecomes infinitely remote，respectively. Here x，y， andz arewhenx andy
Z arecy
1indricalcoordinates(x=rcose，y=rsine). 羽1ecartesiancoordinatesand15) and (16)of which onlyas
pecialsituationwillbesha
11obtainasetofeqs.
(14)，(打。

‑ 41‑

MasatoshiYAMAZAKI

42considered. Togetasolutionofeqs.
(14)，(15) and(16)underlighterconstraintisafutureproblem.

~

2GeneralEquations

Undertwoconditionseqs.
(3)and(4)weobtainfollow
ingeqs.(5)~ (1 3).
ρIH~-qE~ ，E~0"(5)

ρIH~ ='qE~ ，whereG~3 =H~ (cyclic)andG4
f=‑iE1 ・J~=DyH~ +QDzH';

J~ =-DxH~ -QDzH~(6)
J~

DxH'; -DyH~Cρα=(ρtjq)J~，

whereQ=(ρtjq)
2‑1 andJZ=(j~， Jf ， J~， icρα).

Jf'= _gfabC(AtH~+QA~H~)J
f= gfα bC(A~H~+ QA~H;)

h = gfabc(AßH;-A~H~)(7)c
予α = ゆI/q)Jff，

whereJaρ=‑gf
abcAB
G%
J.
/.
J~

ayH~ +QazH~J;
r= ‑axHff‑QåzH~J~

åxH~ ‑ ayH~c
声a = (Þl q)J~ ，
whereJ
Z= J
Z+JZ;

(8)

43A1¥!
oteonYan~Mills FieldJ.~A = 0=0(9)JgA= (t/q)CρaACρaA-gf'α bC(A~Hf: 十 A~H長十 A%H~) ，

whereJsA=一(1/2)白 山，gfab
CAiGfp(withE1234= 1).
=QH~(DxI#-DyHf)- QH~DzH~-H'1 DxH'1
Ky

QHf(DyH'}‑DxH3)‑QH~DzH~-H'1 D)，H'1

Kz

QH~Dz l-I3 十 QH'}DzH~ 十 H3DyH'1 +H~DxHg(10)

Ko = (t/q)K，
z
whereKμ=GZνJe=(Kx，Ky，Kz，iKo).
AA一2mM1AYDA沼山岡////ytL

q h 旧日刊
ゆQQUZK K K

xyzo

一一UH‑‑一一一一一一

左一一一一一一一一一一一一AXAYAZAo

kkkk

whereKμ(11)GZvld'andK合=一 (l!2)ε川ρGfpf~A.

θEc=針。'/q)H~H'!iθf"‑i(pjq)H3Hl'(12)θiLzj(ρ，/q) {(H~)2 +(H3)2Iand(‑Q/2) {(Hf)
2‑(H3)
2I一(1/2)(Hl')
2，‑QH?
:H3，H'}H~-θ ~=Tij = 1 ‑QH<;H'}，(Q/2){(Hf)2ー (H'j)
2I一(1/2)(H1')
2，H'
jHi!
'Hi!
'Hf，Hi!
'H'j，(‑1/2) {印/1q)
2十 1f {(Hf)2+(H'j)2f 十 (1/2)(Hi!
')
2j，(13)where θgvdenotessymmetrizedenergy‑momentumtensoroffieldAs
羽Te intendt
ohaveasolutionwhenJf，J3，JfandJ~ v
anishes，respectively，forQ=Ocase，andJ~ andJ~ v
anishes，respectively，for Qヰ ocase.AxialcurrentsJ~A andJ'jAespectively，asshowni
neq.
(9).Thenwehave followingthr
eesetsofalreadyv
anishes，r44I
VlasatoshiYAMAZAKI
whichthefieId。 oåeH~and aj‑nトH，;
JandHi'0forQ = 0，andH'i}must

IorQヰ Oojab
CAßH~ = 0 and0 forQ = びjabC(A$H'i+QAgHg)= 0 forQヰ θaxHfe++= -gfα bC(AUJ~+AßH3-ト()xH~- åyH~α 吋 A~H3-1nthefirstplacewetrea
tQ=0caseandi
nthenextQ宇 oc
ase，~

3 Q=OCa自信

FromIie
ldstrengt
hisindependentofc
oor
dinatesx andy Weimposetheboundaryc
onditiont
hatfiei
dandfieldmustv
anishwhenc
oor
dinat
ex andybecomes infinitely remote，
Therefore H~ rnustvanish0Eqs.
(15)aresatisfiedwhenQ=0andHi'=O.
+

become-gfabc(A~H~+AßH~)(17)
吋A~H~-VVhent
hei
nter
na
lsymmetrygroupi
sSUI臼~ anda
fterwetransformeqs.
(17)，
weobtaina
identicali
nformwithequationofmotionfor
whereA
andJ~ isidentified，respectively，withmomentum，andmomento
fexternalforceininternalisospace. Fromweobtain
(1/2}
axl

-(H~)2 f(‑1/2)()yl(Hi')
2一ト êJy(H~H~)‑H~J~

i 十 ()AH~H~)

H';:J~，(18)

wheresummationsabouti
ndicesa areunderstood Eqs.
(18)correspondwiththespacepar
toffo
renergy‑momentumconserv
ation.I
fwerestr
ictou
rgaugegroup
tobeS[~白人 we obtainfromH~ = 03)，4~=(‑2/g)E;abcrf}
axr
tc

Ae ( ‑ ε二二ゆ，Lff
iculttoobtainagener
alsolut
io
nofeqs.
functiun. So we considera
vanishJ
becomerespecttoeforgivenandassumedformo
fJ~casewithaxial symmetry (al!deriv
ativ
eswithANote0河 1切zg‑
MillsField45H8He(1/r)
d，‑(rl=H8.l~.
Butwehave(1/r)
iJr(rHe)ヰ月。 AlsoweobtainAgニ oandH~ = O.

(21)

Allcomponentso
fLorent
zfor
ceKμvanishwhen Q=0 and H~
A~ satisfyarelationHe

òzA~0 FieldsA~ 3.nd固(22)θγA~+gfabCA~A~.

The Q=0 casecorrespondstothesituationwhere0.1 quantitie
smovetoward z‑axisw
ithlightvelocity.

~

4 QニギoCa
問Inthissectio
nwe considerthecase Q=(p!q)
2‑1ヰ O. Eqs.
(14) and (15) becomemore complicatedones when Qヰ O. So we solelytreatspecialsituation where a1lderiv
ativ
eswithrespecttocoor
dinatez v
anish. Thisisthεsituationwhere allder‑iv
ativ
eswithrespectt
otimetvani
sh，too. Thenweobtainfromeqs.
(14)
(23)

dBH~= 0明Tehave onlyoneeq.
(23)ingeneralas compared wit
h two eqs.
(14)for Q=0 case，becauseo
fconserv
ationoftotalv
ectorcurrent. So dγ H~ 宇 o ingenera.
l Wecannotngenerali
ncontrastwit
h Q=θcase. Butwearenowtreatinga
havevanishingH~ isituation where alld. Inthissituationonly we can assumeθrH~=O withou
tz=Oconflic
twit
hconserv
ationoftotalv
ectorcur
r
ent.T
hi
si
sanadhoca
ssu
mption. Then，
asint
hep
r
ecedingsect
ion，H~ mustv
anisheverywhere. Soeq.
(15)giv
esjabcAgH3O.

(24)18)andNowwec
onfronteqs.
(18)withInthepresentsectionwehavea
lsoeqs.
(17)，(additionalconstr
ainteqs.
(24)，ascomparedw
ithQ=0case. Herea
1sowetreataspecial21)，(22) and furthereq.
(24).
casewithaxial symmetry. Then we have eqs.
(20)，(fLorentzforceactin
gonfermionv
anishes，respectiv
ely，ComponentKe，Kz，andKooandsolenon‑v
anishingcomponentKγbecomes(25)= QH$ (1/r)ar(rHe ) ニ QHe.l~

Lorentzforceisnegativ
ewhenQ<α Hg>αandJ~>OI
nsummarywehavet
hefollowingnonv
anishingquantitieswhenQ 0，d，Ot=O，zニ 0de=0 andarH~= 0:*A~ ， A~ ， Ag.l~=(q/ρ)
cρα

戸Hg，andE~ = ρ(yq)H3(1/r)Dr(rHg)

M
asatoshiYAMAZAKJ46)
一

CnσαQJ'O77

HH(﹀nC GP)
Y

AM‑一();

ZU

忽

子

==HGB

(日イペベd︐何ρ
ρa z ‑戸山‑ ︑qMY
CCGQJ//

h u f a 刊r u t ‑//︐ //flhq
diq(i︑huJ ︑S F Y γ ザ 町 ム u7G/︹↑一一一一一一一

月 mH・ a k o(26)= ‑Te
e= (Q/2)(Hg)
2andTZ
Z= ( ̲ 園InadditionwehavearelationjabcA
onbosonv
anish=0. Axialcurren
tandLorentzIorces，

ニoa n d μ = 0~羽TesupposeourQヰ

5 R肥marksocaserepresentamodelofhadronwhichi
8infinitelylonginz‑direction. I
norderto havεa closedstr
ingi
nsteadof an infini
telyone，wemU8thaveasolutionofeqs(14)，(15)andwithou
tthec
onstr
aintaa=0andaz=0atニ
Inou
rQヰ oc
as
eLor
ent
zfor
ce(Kァ<0)balance
sthep
r
essu
r
eP du
eto
manyofandwhiche
xisti
nsideofandoutsideo
fthesur
faceo
fourstring.Y
Vehavefollowingrelations:圃K =マlD，Ha• マ P = 0，andJa・マ p =016)under.lesssev
er
econst
子Wea
r
enow
t
ohaveasolutionofand(aints:
than those done int
hisnote，‑a solut
ion which may also have，vv
e hope，thepropertyshownineqs.
(27)
Theauthorwouldliketoexpresshisthan
kstothememberso
fDepartment
forherortheirusefu
ldiscussions and toMrs. NLof Kanazawa Universi匂T ftypewntmg.
ー @Refer
el1ces.N.Yanga
ndR L.!
Vlills，Phys.R
ev96(1954)1911)C.Ishidaa
ndA Hosoya，Lett.NuovoCim.13(1975)2372)J3)M.Hirayama，].IshidaandT.Kawabe，Prepri
日ltToyamョ‑20March，1
976，，