A theorem concerning twisted and untwisted partition functions in U(N) and SU(N) lattice gauge theories

0.16 MB·English

by  Takuya Kanazawa

A theorem concerning twisted and untwisted partition
functions in U(N) and SU(N) lattice gauge theories
9
0
0
2 TakuyaKanazawa∗

n
a DepartmentofPhysics,Universityof Tokyo, Tokyo 113-0033,Japan
J

8
January 9,2009

]
t
a
l
-
p Abstract
e
h
[

3 Inordertogetacluetounderstandingthevolume-dependenceofvortexfreeenergy(whichisdefinedas
v
2 the ratio of the twisted against the untwisted partition function), we investigate the relation between vortex
4
7 freeenergiesdefinedonlatticesofdifferentsizes. Anequalityisderivedthroughasimplecalculationwhich
2
.
5 equatesagenerallinearcombinationofvortexfreeenergiesdefinedonalatticetothatonasmallerlattice.The
0
8
couplingsinthedenominatorandinthenumeratorhowevershowsadiscrepancy,andwearguethatitvanishes
0
:
v inthethermodynamiclimit. ComparisonbetweenourresultandtheworkofTomboulisisalsopresented. In
i
X
theappendixwecarefullyexaminetheproofofquarkconfinementbyTomboulisandsummarizeitsloopholes.
r
a
PACSnumbers:11.15.Ha,12.38.Aw
keywords:latticegaugetheory,vortexfreeenergy,Migdal-Kadanofftransformation,quarkconfinement
∗Email:[email protected]
1
§1 Introduction
Quarkconfinement,or(moregenerally)colorconfinementisoneofthemostlong-standingproblemsintheo-
reticalphysics[1]. SofarmanyproposalshavebeenmadeconcerningthenonperturbativedynamicsofQCD
whichyieldconfinement,includingthedualsuperconductivityscenario[2]andcentervortexscenario[3](see
[4]forareview),butatrulysatisfyingpictureseemstobestillmissingandpreciserelationshipbetweendif-
ferentscenariosiselusive. InthelatticegaugetheoryitisformulatedasthearealawoftheWilsonloopinthe
absenceofdynamicalfermions,whichindicatesalinearstaticpotentialV(r)(cid:181) rbetweeninfinitelyheavyquark
and anti-quark. So far the area law has been rigorouslyproved(in the physicists’ sense) for quite restricted
models[5,6,7]althoughithasbeennumericallycheckedbyMonteCarlosimulationsforyears[8].
The concept of center vortices in non-abelian gauge theories was introduced long time ago [3, 6, 7, 9].
Thispicturesuccessfullyexplainsmanyaspectsofinfra-redpropertiesofYang-Millstheoryatthequalitative
level, and also at the quantitative level it is reported that Monte Carlo simulations show that the value of
thestringtensioncanbemostlyrecoveredbytheeffectivevortexdegreesoffreedom(‘P-vortex’)whichone
extractsviaa procedurecalled ‘centerprojection’[10]. Itisalso reportedthatquenchingP-vorticesleadsto
thedisappearanceofarealawandtherestorationofchiralsymmetryatthesametime[11],whichsuggeststhat
thevorticesrepresentinfra-redpropertiesofthetheoryinacomprehensivemanner.
Inaddition,thepicturethatthepercolationofcentervorticesleadstothearealawhasafirmgroundbased
ontheTomboulis-Yaffeinequality[12]:
hW(C)i≤2 1(1−e−Fv) AC/Lm Ln (1)
2
n o
forSU(2),whichcanbeprovedrigorouslyonthelattice. HereA denotestheareaenclosedbyarectangleC
C
whichliesina[m ,n ]-plane,W(C)istheWilsonloopassociatedtoCinthefundamentalrepresentationandLm
isthelengthofthelatticeinthem -thdirection.Accordingtothisinequalitythearealawofthel.h.s. followsif
thevortexfreeenergyF vanishesinthethermodynamiclimitinsuchawaythatF ≈e−r Lm Ln . Thequantityr
v v
givesalowerboundofthestringtension,andiscalledthe’tHooftstringtension.ThisbehaviorofF isverified
v
withinthestrongcouplingclusterexpansion[13]andisalsosupportedbyMonteCarlosimulations[14]. Soit
isworthwhiletostudythevolumedependenceofF particularlyatintermediateandweakcouplings.
v
Inthisworkweattempttomakeaconnectionbetweenvortexfreeenergiesdefinedonlatticesofdifferent
2
sizes. Thesetupofthisnoteisasfollows.In§2weestablishanequalitywhichrelatestheratiooftheordinary
andthetwistedpartitionfunctiononalatticetothatonasmallerlattice,withoutanyrestrictiononthecoupling
strength. The cost is that there is a slight discrepancy between the couplingsin the denominatorand in the
numerator, but it can be shown to tend to zero in the thermodynamic limit. In §3 we examine the relation
between our result and the work of Tomboulis [15].1 Final section is devoted to summary and concluding
remarks.Intheappendixwecarefullyexaminetheproofofquarkconfinementinref.[15],listingitsloopholes.
Inviewoftheseriousscarcityofrigorousresultsinthisareaofresearch,ouranalyticalworkwhichinvolves
noapproximationseemstobeofbasicimportance,andwehopethatthisresultwillserveasabuildingblock
oftheproofofquarkconfinementinthefuture.
§2 The main result
Letusbeginbydescribingthebasic set-upoflatticegaugetheory. LetL a d-dimensionalhypercubiclattice
of length Lm (m =1,...,d) in each direction with the periodic boundary condition imposed. Let L (n) a d-
dimensionalhypercubiclattice of lengthLm /bn(m =1...,d) in each directionwith a parameterb∈N. The
numberofplaquettesinL isdenotedby|L |. ThepartitionfunctiononthelatticeL isdefinedas
ZL ({cr})≡ (cid:213) dUb (cid:213) fp(Up)≡ (cid:213) dUb (cid:213) 1+(cid:229) drcr(b )c r(Up) , (2)
Z b∈L p⊂L Z b∈L p⊂L r6=1
h i
wheredUisthenormalizedHaarmeasureofgaugegroupGandUpisaplaquettevariable;Up≡Ux,m Ux+m ,n Ux†+n ,m Ux†,n .
InwhatfollowsweonlyconsiderG=U(N)andSU(N). Thesubscriptrlabelsirreduciblerepresentationsof
G and d is the dimension, c is the characterof the r-th representation(1 is the trivialrepresentation). The
r r
coefficients{c (b )}canbedeterminedthroughthecharacterexpansionof(forinstance)theWilsonaction:
r
eb ReTrU dU′eb ReTrU′ ≡ f (U)=1+(cid:229) d c (b )c (U). (3)
p r r r
Z r6=1
.
Itcanbecheckedthatc ≥0holdsforeveryr, whichguaranteesthereflectionpositivityofthemeasureand
r
unitarityofthecorrespondingquantum-mechanicalsystem.
MultiplyingaplaquettevariableU byanontrivialelementofthecenterofthegaugegroupiscalledatwist
p
which, inphysicalterms, generatesa magneticfluxpiercingthe plaquette. LetC(G)denotethecenterofG;
1SomeaspectsofTomboulis’paperwhicharenottoucheduponinthisnoteareexaminedinref.[16].
3
C(SU(N))=Z andC(U(N))=U(1). Thetwistedpartitionfunctionreads
N
ZLg({cr})≡ (cid:213) dUb (cid:213) fp(gUp) (cid:213) fp(Up′), g∈C(G). (4)
Z b∈L p⊂V p′⊂L \V
HereV isasetofstackedplaquetteswhichwindsaround(d−2)ofthed periodicdirectionsofL forminga
(d−2)-dimensionaltorusontheduallattice. (UsingreflectionpositivityonecanshowZLg ≤ZL [17]andthe
vortexfreeenergyFvg associatedtothetwistbygisdefinedbye−Fvg =ZLg/ZL .) Ourmainresultisasfollows:
Theorem 1. Let H an arbitrary discrete subgroupofU(1) for G=U(N) and H =Z for G=SU(N). Fix
N
a setofpositivecoefficients{c′} andn∈N, andchooseaconstantAg>0foreachg∈H arbitrarily. Then
r
there exists l ({c′r})>0 such that for any 0<a <l ({c′r}) and for sufficiently large |L | there exists a L ≡
1
a L (a ,{cr},{c′r})suchthat|a −a L |≤O |L | and
(cid:16) (cid:17)
ZL ({cr}) = ZL (n)({a c′r}) (5)
ZL ({cr}) ZL (n)({a L c′r})
with
ZL ({cr})≡ (cid:229) AgZLg({cr}). (6)
g∈H
Proof. LetusdefinefunctionsCL (a ,{cr},{c′r}),CL (a ,{cr},{c′r})by
ZL ({cr})=eCL (a ,{cr},{c′r})|L |ZL (n)({a c′r}), (7)
ZL ({cr})=eCL (a ,{cr},{c′r})|L |ZL (n)({a c′r}). (8)
Takingtheratioof(7)and(8)yields
ZL ({cr}) =e{CL (a ,{cr},{c′r})−CL (a ,{cr},{c′r})}|L |ZL (n)({a c′r}). (9)
ZL ({cr}) ZL (n)({a c′r})
From(9)and2Ae< ZL ({cr}), ZL (n)({a c′r}) < (cid:229) Agwehave
ZL ({cr}) ZL (n)({a c′r}) g∈H
1
CL (a ,{cr},{c′r})−CL (a ,{cr},{c′r}) ≤O |L | . (10)
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
Nowletusrewrite(7)intomoreusefulform:
1 |L (n)| 1
|L | logZL ({cr})=CL (a ,{cr},{c′r})+ |L | "|L (n)| logZL (n)({a c′r})#. (11)
2edenotestheunitelementofH.
4
Thel.h.s. hasawell-definedthermodynamiclimitwhichwedenotebyz({c }). Itiswellknown[13,18]that
r
strongcouplingclusterexpansionhasanon-vanishingradiusofconvergencewhichisindependentofthetotal
volume,sothereexistsl ({c′})suchthat[...]inther.h.s.of(11)canbewellapproximatedbythelowestorder
r
strongcouplingclusterexpansionfor0<a <l ({c′}),giving
r
z({cr})=C¥ (a ,{cr},{c′r})+ bandd (cid:229) dr2 a c′r 6+O a c′r 10 , (12)
r6=1
(cid:0) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
1 2
wherea3≡ 3,a4≡ 3 andwedenominatedC¥ (a ,{cr},{c′r})≡|Lli|→m¥ CL (a ,{cr},{c′r}). Thus
dC¥ (a ,{cr},{c′r}) ≃−C a 5 for 0<a <l ({c′}), (13)
da 0 r
dCL
where the factor C is independent of the volume. Therefore has a non-vanishing lower bound for
0 da
(cid:12) (cid:12)
arbitrarilylarge|L |.Fromthisfactand(10)followsthatif|L |iss(cid:12)(cid:12)ufficie(cid:12)(cid:12)ntlylargethereexistsa L (a ,{cr},{c′r})
1
suchthat|a −a L |≤O |L | andCL (a ,{cr},{c′r})=CL (a L ,{cr},{c′r}). Hence(5)isproved.
(cid:16) (cid:17)
Onecanrelaxthecondition{Ag>0|g∈H}withinascopewhichdoesnotaffecttheexistenceofastrictly
ZL ({cr})
positivelowerboundfor .
ZL ({cr})
Note that a and {c′} have no dependence on {c }, for they are parameters we introduced by hand. As
r r
the free energy density z({c }) in the l.h.s. of (12) is a quantity which reflects the phase structure of the
r
model, it is not necessarily analytic in {c } and in b . In the r.h.s. of (12), such a nonanalyticity in {c }
r r
residesinC¥ (a ,{cr},{c′r})simplybecausetherestofther.h.s. of(12)isindependentof{cr}. Notealsothat
C¥ (a ,{cr},{c′r}) is Taylor-expandableand analytic in a ; the analyticity ina and a possible nonanalyticity
in{c }shouldbestrictlydistinguished.3
r
Letuslookat(5)inmoredetail.Sincea L convergestoa inthethermodynamiclimit,onemaybetempted
ZL ({cr})
toarguethat lim isindependentof{c }because
r
|L |→¥ ZL ({cr})
lim ZL ({cr}) = lim ZL (n)({a c′r}) = lim ZL (n)({a c′r}). (14)
|L |→¥ ZL ({cr}) |L |→¥ ZL (n)({a L c′r}) |L |→¥ ZL (n)({a c′r})
Itwouldbevaluabletoclarifywhythisclaimisincorrect.Letusrewrite(5)as
ZL ({cr}) = ZL (n)({a c′r}) ZL (n)({a c′r}) . (15)
ZL ({cr}) ZL (n)({a c′r}) ZL (n)({a L c′r})
3TheauthorisgratefultoK.R.Itoforvaluablecorrespondenceonthispoint.
5
|a −a L |≤O |L1| allows us to write a L =a + gL (a ,|{L c(rn})|,{c′r}) with |gL |≤O(1). For 0<a ,a L <
(cid:16) (cid:17)
l ({c′r})wemayapplytheconvergentclusterexpansiontologZL (n)({a c′r})andlogZL (n)({a L c′r})toobtain
Z ({a c′}) 1 1
ZLL(n(n))({a L cr′r}) ≡exp"|L (n)| |L (n)| logZL (n)({a c′r})− |L (n)| logZL (n)({a L c′r})!# (16)
=exp |L (n)|(cid:229) sr,k (a c′r)k−(a L c′r)k (17)
" r,k #
n o
=exp"−gL (cid:229)r,ksr,k(c′r)kka k−1+O |L 1(n)| # (18)
(cid:16) (cid:17)
−→ exp −g¥ (cid:229) sr,k(c′r)kka k−1 as|L |→¥ . (19)
" r,k #
{sr,k} are coefficientsof clusters. (19) clearly shows how the discrepancybetween a and a L persists in the
thermodynamiclimitandwenowseewhythepreviousclaimisincorrect.Note,furthermore,thatthefunction
g¥ (a ,{cr},{c′r})≡ lim gL (a ,{cr},{c′r})isnotnecessarilyanalyticin{cr};informationinthel.h.s. of(15)
|L |→¥
aboutthephasestructureofthemodelisnowpackagedwithinasingleunknownfunctiong¥ .
We wouldlike tocommentonapossiblepathfromtheorem1toaproofofquarkconfinement. Suppose
G=SU(N).
Theorem2. IfN-alityofther-threpresentationisnon-zero,wehave
g
1 (cid:229) ZL ({cr}) AC/Lm Ln
|hW(C)i|≤2 1− (20)
r
n N g∈ZN ZL ({cr})o
for the normalized Wilson loopW (C)≡ 1 c P(cid:213) U , where P implies path-orderingand h...i in the
r r b
d
r b∈C
(cid:0) (cid:1)
l.h.s. istheexpectationvaluew.r.t. ZL ({cr}).
1
Detailedproofof(20)isgiveninref.[17]andweskipithere. LetAg= for∀g∈Z andassumethat
N
N
CL (a 0,{cr},{c′r})=CL (a 0,{cr},{c′r})holdforsomea 0∈[0,l ({c′r})]. Substitutinga =a 0 into(9)gives
ZL ({cr}) ≡ 1 (cid:229) ZLg({cr}) = ZL (n)({a 0c′r}). (21)
ZL ({cr}) (cid:16) N g∈ZN ZL ({cr})(cid:17) ZL (n)({a 0c′r})
Fromthedefinitionofl ({c′}),ther.h.s. canbeestimatedbytheconvergentclusterexpansion,giving
r
ZL (n)({a 0c′r}) ≈1−O(e−r L(mn)Ln(n))=1−O(e−r Lm Ln /b2n) (22)
Z ({a c′})
L (n) 0 r
whereL(mn)≡Lm /bnisthelengthofL (n) inm -thdirection.Inserting(22)into(20)weobtain
|hW (C)i|.e−r AC/b2n, (23)
r
6
hencethequarkconfinementfollows.Whethertheabovestrategy(tosearchfortheintersectionpointofcurves
CL (·,{cr},{c′r})andCL (·,{cr},{c′r}))isviableornotremainstobeseen.
§3 Comparison with Tomboulis’ approach
Thepurposeofthissectionistoelucidatethesimplificationandgeneralizationachievedintheprevioussection,
throughthecomparisonwiththeapproachinref.[15],whereonlyG=SU(2)istreatedexplicitly. Thewhole
argumentinref.[15]seemstorestontheMigdal-Kadanoff(MK)renormalizationgrouptransformationbelow:
b2r
dU f (U,n) bd−2 1 c (U)
p j
c (n+1)≡Z dj  ∈[0,1], j= 1,1, 3,... (24)
j (cid:2) (cid:3) bd−2 2 2
dU f (U,n)
 p 
 Z 
 (cid:2) (cid:3) 
 
wherer∈(0,1)isanewlyintroducedparameterand
b
exp TrU
fp(U,n)≡1+(cid:229) djcj(n)c j(U), fp(U,0)≡ (cid:16) 2 b (cid:17) , b ≡ g42 . (25)
j6=0 dU exp TrU
2
Z
(cid:16) (cid:17)
Whenr=1,(24)reducestotheoriginalMKtransformation[19,20]. Thereasonwhyr isintroducedwillbe
brieflyexplainedlater. Inadditionthefollowingquantityisdefined:
bd−2 b2
F (n)≡ dU f (U,n) . (26)
0 p
Z
(cid:16) (cid:2) (cid:3) (cid:17)
Startingpointistheinequality
ZL ≤F0(1)|L (1)|ZL (1)({cj(1)}) (27)
whichis provedin appendixA ofref.[15]. A variablea ∈[0,1]andan interpolationfunctionh(a ,t)isthen
introduced,whichissupposedtosatisfy
¶ h ¶ h
>0, <0, h(0,t)=0, h(1,t)=1. (28)
¶a ¶ t
Thedomainoft∈Risarbitrary.
From1≤ZL and(27),weseethatthereexistsavaluea =a L(1,h)(t)∈[0,1]suchthat
ZL =F0(1)h(a L(1,h)(t),t)|L (1)|ZL (1)({a L(1,h)(t)cj(1)}). (29)
7
Similarlyitcanbeshownthatthereexistsavaluea =a +(1)(t)∈[0,1]suchthat
L ,h
ZL+=F0(1)h(a L+,(h1)(t),t)|L (1)|ZL+(1)({a L+,(h1)(t)cj(1)}) (30)
1
where Z+ ≡ (Z+Z(−)) is introduced for a technical reason 4 ; Z(−) is the twisted partition function for
2
SU(2). Notethatboth(29)and(30)areindependentoft. Letustaketheirratio5:
ZL+ = F0(1)h(a L+,(h1)(t),t)|L (1)|ZL+(1)({a L+,(h1)(t)cj(1)}) (31)
ZL F0(1)h(a L(1,h)(t),t)|L (1)|ZL (1)({a L(1,h)(t)cj(1)})
= F0(1)h(a L+,(h1)(t),t)|L (1)| ZL+(1)({a L+,(h1)(t)cj(1)}) ZL+(1)({a L(1,h)(t)cj(1)}). (32)
F0(1)h(a L(1,h)(t),t)|L (1)| ZL+(1)({a L(1,h)(t)cj(1)}) ZL (1)({a L(1,h)(t)cj(1)})
From
1 < ZL+ <1 and 1 < ZL+(1)({a L(1,h)(t)cj(1)}) <1, (33)
2 ZL 2 Z ({a (1)(t)c (1)})
L (1) L ,h j
wehave
1 < F0(1)h(a L+,(h1)(t),t)|L (1)| ZL+(1)({a L+,(h1)(t)cj(1)}) <2. (34)
2 F0(1)h(a L(1,h)(t),t)|L (1)| ZL+(1)({a L(1,h)(t)cj(1)})
ThisandthefactthatZ+ ({a c (1)})isamonotonicallyincreasingfunctionofa yield
L (1) j
1
a (1)(t)−a +(1)(t) ≤O . (35)
L ,h L ,h |L (1)|
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
This implies that h(a (1)(t),t) and h((cid:12)a +(1)(t),t) can be(cid:12)made arbitrarily close to each other if one lets |L |
L ,h L ,h
sufficientlylarge.Thereforeaslightshiftoft→t+d twillenableustoget
h(a (1)(t+d t),t+d t)=h(a +(1)(t),t), (36)
L ,h L ,h
dh(a (1)(t),t)
providedthat L ,h 6→0for|L |→¥ . Thatthenewlyintroducedparameterr∈(0,1)guaranteesthis
dt
canbeprovedthrougharatherinvolvedcalculation;then(31)gives
ZL+ = ZL+(1)({a L+,(h1)(t+)cj(1)}). (37)
ZL Z ({a (1)(t)c (1)})
L (1) L ,h j
Repeatingaboveprocedure,thefollowingisproved[15]:
4ThemeasureofZ+isreflectionpositive,whichisnecessarytoderive(30).ThemeasureofZ(−)isnotreflectionpositive.
5Onecanchoosedifferentvaluesoft’sin(31)fornumeratoranddenominator,althoughwedon’tdosohere.
8
Theorem3. Foranyn∈Nandsufficientlylarge|L |,thereexista (n)(t ),a +(n)(t+)∈[0,1]suchthat
L ,h n L ,h n
1
a (n)(t )−a +(n)(t+) ≤O and
L ,h n L ,h n |L (n)|
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12) ZL+ = ZL+(n)({a L+,(hn)(tn+)cj(n)}). (38)
ZL Z ({a (n)(t )c (n)})
L (n) L ,h n j
1
Itisclearthattheorem3followsfromtheorem1asaspecialcase H=Z ,c′ =c (n)andA1=A−1= .
2 j j 2
(cid:16) (cid:17)
A differenceworthnotingisthattheproofoftheorem1necessitatesneithertheMKtransformation(andthe
related inequality(27)) northe speciallinear combinationZ+ ≡(Z+Z(−))/2. We could entirelyavoid the
complicationcausedbyr,whichseemstobeasignificantsimplification.
§4 Summary and concluding remarks
Inthisnoteweinvestigatedthelatticegaugetheoryforgeneralgaugegroupswithnontrivialcenter,andproved
a formula which relates the ratio of twisted and untwisted partition functions to that on the smaller lattice.
Althoughthe couplingsin the numeratorandin the denominatorcannotbe exactlymatched, we showedthe
discrepancyto bevanishinglysmallin the thermodynamiclimit. We presenteda strategyto provethequark
confinement,andalsocomparedourworkwithTomboulis’approachinref.[15]clarifyingthatgreatsimplifi-
cationhasoccurredinourformulation.
As has already been clear, our theorem is correct both for SU(N) and forU(1). Whether the theory is
confiningor not, or whetheris asymptoticallyfree ornot, has nothingto do with the theorem, and the same
isalsotruefortheorem3. Althoughtheorem3ispresentedinref.[15]asacornerstonefortheproofofquark
confinement, it must be confessed that his and our formalism are not quite successful in incorporating the
dynamicsofthetheory;entirelynewtechniquemightbenecessarytoprovethequarkconfinementfollowing
thestrategydescribedin§2.
Acknowledgment
The author is gratefulto Tamiaki Yoneya, Yoshio Kikukawa, Hiroshi Suzuki, Shoichi Sasaki, Shun Uchino,
KeiichiR.ItoandTerryTomboulisforusefuldiscussions. HealsothanksTetsuoHatsudaandTetsuoMatsui
9
who stimulated him to investigate ref.[15]. This work was supported in part by Global COE Program “the
PhysicalSciencesFrontier”,MEXT,Japan.
A Appendix
Inthefollowingwecommentonthevalidityoftheadvocatedproofofquarkconfinementinfourdimensional
SU(2) lattice gauge theory [15], to help the reader understand the precise relation between our work and
ref.[15]. Thereaderwhoisonlyinterestedinaqualitativeunderstandingontheviabilityoftheproposedproof
maywanttogodirectlytothelastparagraphofthisappendix,inwhichalesstechnical,quiteintuitiveoverview
ofthesituationispresented.
A.1 Correctness
Firstofall,theproofismathematicallyincompleteatleastinfouraspects.
1. In theorem 3 of this paper, a +(n)(t+) numerically differs from a (n)(t ) in general. However, if we
L ,h n L ,h n
couldfindt∗∈R(atleastforn≫1forwhichc (n)≪1;wewilldiscussthispointshortly)suchthat
j
a +(n)(t∗)=a (n)(t∗), then the usualstrongcouplingclusterexpansionmethodcouldbe appliedto the
L ,h L ,h
r.h.s. of(38),thusgivingaproofofquarkconfinement. InthiswayTomboulisreducedtheproblemof
quarkconfinementtotheproblemofshowingtheexistenceofsucht∗∈R. Furthermoreheattemptedto
findsucht∗byinterpolatinga +(n)(t)−a (n)(t)=0(theequationtobesolved)andanotherindependent
L ,h L ,h
equationwhosesolutionwe knowdoesexist, byoneparameter0≤l ≤1. Theresultisa singletwo-
variableequationY (l ,t)=0,whereY (1,t)=0isequivalenttoa +(n)(t)−a (n)(t)=0andY (0,t)=0
L ,h L ,h
is the other equation that can be solved by some t ∈R. Assuming the existence of t(l ) such that
0
Y (l ,t(l ))=0andthendifferentiatingbothsidesbyl ,wehave
dt Y ,l (l ,t)
=− (39)
dl Y (l ,t)
,t
where the subscripts denote partial derivatives. (Y (l ,t)6=0 is assumed in (39), but it is rigorously
,t
provedinref.[15].) Thuswhatweshoulddoistotosolvethedifferentialequation(39)withtheinitial
conditiont(0)=t andfindt(1),whichisnothingbutt∗. Herecomesthecruxoftheproblem:heargues
0
thatY (l ,t)6=0for0≤l ≤1isa sufficientconditionfortheexistenceoft(1)sinceonecanextend
,t
10