Table Of ContentAPPLYINGTHEq-ALGEBRASUq′(son)TOQUANTUMGRAVITY
q U′(so )
APPLYING THE -ALGEBRAS q n TO QUANTUM
q
GRAVITY: TOWARDS -DEFORMED ANALOG
SO(n)
1
OF SPIN NETWORKS
A. M. GAVRILIK
UDC538.9;538.915;517.957 Bogolyubov Institute for Theoreti
al Physi
s, Nat. A
ad. S
i. of Ukraine
(cid:13)
2002 (14b, Metrolohi
hna Str., 03143 Kyiv,Ukraine)
4
0
0
2 Nonstandardq-deformedalgebrasUq′(son),proposedade
adeago 2. Simple G = SO(n) Spin Networks
n
for the needs of representation theory, essentially differ from the
a
J sUta(snodna)rdanDdrpinofsesleds(cid:22)sJwimithboreqguaardnttuomthdeefloartmteartaionnuomfbtehreoafligmebproars- GLe=tuSsOfir(nst)brieflydwelluponne
essarysetup
on
erning
15 Utaq′n(tsoand)v,anwtiathgeins.tWweoddqisif
fuesrsenptos
soibnlteexaDtpsploi
faqtiuoannotufmth/eDqq-d-≥aelfgo3erbmreads A gGeneraspliiznednestpwionrknse.twork asso
iated with a Lie
gravity: one
on
erns -deforming of -dimensional ( ) eu- group , a
ording e.g. to [7℄, is defined as a triple
(Γ,ρ,I)
1
lidean gravity, the other applies tog2+1 anti-de Sitter quantum where
v gravity (with spa
e surfa
e ofgenus )intheapproa
h ofNelson Γ
isanorientedgraph,formedbydire
tededgesand
7 andRegge.
6 vertρi
es; e
0 is a labeling of ea
h edge by an irredu
ible rep-
ρ G
1 resentation (irrep) e of ;
I v Γ
0 is a labeling of ea
h vertex of by an intertwin-
4 Iv v
ner mapping tensor produ
t of irreps in
oming at
0 v
1. Introdu
tion to the produ
t of irreps outgoing from .
/
c
Below,weareinterestedinthespinnetworksforthe
q G=SO(n)
Constru
tion of quantum gravity belongs to most fun- parti
ularLiegroup .Moreover,likein[7℄,we
- G = SO(n)
r damental problems of modern quantum theory. During
onsider restri
ted
ase of simple spin net-
g G=SO(n)
: last de
ade and a half, new perspe
tive tools for atta
k- works.Simplespinnetworksasso
iatedwith
v
i ingandsolvingthisproblemhaveappearedamongwhi
h aSrOe(env)a/lSuOat(end−as1F)eynmanintegralsoverStnh−e1
osetspa
e
X we mention, first, the notion of spin networks (see e.g., , i.e. over the sphere . Simpli
i-
SO(n)
r [1℄)
losely
onne
tedwithloopquantumgravityaswell ty means that onSlOy(tnh−e1) representationsof
lass 1
a as with the so-
alled BF-type topologi
al theories, and, (withrespe
tto )labeledbysinglenonnegative
l
se
ond, the powerful methods of quantum groups and integer , are employed.
quantum algebras[2 (cid:22) 5℄. Our goal in this
ontribution Basi
ingredient is the `propatngla(tyo)r,' yex=prceossseθd in
isto
onsiderpotentialappli
abilityoftheso-
allednon- terms of zonal spheri
al fun
tions 00 , or,
q U′(so )
standard -deformed algebras q n introdu
ed in [6℄ in view of the equality [8℄
whi
haredifferentfromthestandard(Drinfeld(cid:22)Jimbo) tNl(cosθ)= Γ(2p)l! Cp(cosθ) , p=(N −2)/2 ,
quantum deformation [2, 3℄ of the Lie algebras of or- 00 Γ(2p+l) l (1)
thogonal groups, while possess a number of rather im-
portant advantages. Here we intend to make a prelimi- dire
tly through the Gegenbauer polynomials:
D
N +2m−2
narystepstowardsextendingthe -dSiOm(eDns)ionalversion G(N)(x,y)= C(N−2)/2(x·y) .
of spin networks (more
on
retely, simple spin m N −2 m (2)
U′(so )
networks) to the
ase of q n related formulation. In Cp(x), l ≥0
the se
ond part of our
ontribution, we briefly dis
uss HeretheGegenbauerpolynomials m ,satisfy
U′(so )
the appearan
e of the q n algebras in the
ontext the defining re
ursion relation
(l+1)Cp (x)=
of anti-de Sitter 2+1 quantum grgavity formulated with l+1
spa
en-p=ar2tgb+ein2g fixed as genus Riemann surfa
e so 2(p+l)xCp(x)−(2p+l−1)Cp (x)
that . l l−1 (3)
1Presented at the XIIIth International HutsulianWorkshop (cid:16)Methods ofTheoreti
al and Mathemati
al Physi
s(cid:17) (September 11 (cid:22)
242000,Uzhgorod (cid:22)Kyiv(cid:22)Ivano-Frankivsk(cid:22)Rakhiv,Ukraine) anddedi
atedtoProf.Dr.W.Kummerontheo
asionofhis65th
birthday.
ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3 213
A.M.GAVRILIK
Cp(x) = 1 g =(m +m +m )/2 g−m ≥0
augmented with the initial value 0 , and obey where 1 2 3 is an integer, i ,
i=1,2,3, p=(N −2)/2
the orthogonality relation and .
N =4
1 For ,
Z (l+1)Clp(x)Cmp(x)(1−x2)p−12dx= θ(4)(m ,m ,m )=(m +1)(m +1)(m +1).
−1 1 2 3 1 2 3 (10)
πΓ(2p+l)
=δ · . There exist a number of other results
on
erning (sim-
lm 22p−1l!(l+p)Γ2(p) (4) SO(n)
ple) spin networks, for generi
situation as well
n=3,4
asfortheparti
ular
asesof ,whi
hwehowever
Anotherimportantpropertyisgivenbythelinearization shall not dis
uss here further.
formulafortheprodu
toftwoGegenbauerpolynomials,
namely q
3. -Deformed Analog of Spin Networks from
l+m q
(n+p)Γ(n+1)Γ(g+2p)
Cp(x)Cp(x)= × -ultraspheri
al Polynomials
l m [Γ(p)]2(g+p)!Γ(n+2p)
n=X|l−m| q
To deal with -deformed
ase we need some fa
ts
on-
q
Γ(g−l+p)Γ(g−m+p)Γ(g−n+p)
erning -ultraspheri
al polynomials. These are defined
× Cp(x)
Γ(g−l+1)Γ(g−m+1)Γ(g−n+1) n (5) through the following re
ursion relations:
(1−qn)C (x;β|q)=2x(1−βqn−1)C (x;β|q)
n n−1
g ≡ 1(l+m+n)
where n2 and the sum ranges olv+ermth+onse −(1−β2qn−2)C (x;β|q), (n≥2),
n−2
valuesof whi
hareofthe sameevennessas . (11)
One of basi
onstru
ts in spin networks is the so-
Θ
alled -graphwhose evaluation is given by the expres- along with spe
ial values
sion C (x;β|q)=1, C (x;β|q)=2(1−β)x/(1−q).
0 1
1 (12)
D(l,m,n;p):=Z Clp(x)Cmp(x)Cnp(x)(1−x2)p−12dx.(6) β =qλ q →1
−1 With , the "
lassi
al"limit yields
q→1
C (x;β|q)−−→Cλ(x).
The result of evaluation is n n (13)
21−2pπ Γ(g+2p)
D(l,m,n;p)= × q
[Γ(p)]4 Γ(g+p+1) The expli
it expression for the -ultraspheri
al polyno-
mials is [9℄ as follows:
Γ(g−m+p)Γ(g−n+p)
× . n (β;q) (β;q)
Γ(g−l+1)Γ(g−m+1)Γ(g−n+1) (7) C (x;β|q)= k n−kei(n−2k)θ
n
(q;q) (q;q)
kX=0 k n−k
Using (6) one easily dedu
es the re
urren
e relation for
D(l,m,n;p) (β;q)
the in the form = neinθ Φ (q−n,β;β−1q1−n;q,qβ−1e−2iθ).
2 1
l+1 2p+l−1 (q;q)n (14)
D(l+1,m,s;p)+ D(l−1,m,s;p)=
p+l p+l
(a;q)
n
In this formula, the notation means:
s+1 2p+s−1
= D(l,m,s+1;p)+ D(l,m,s−1;p) 1, n=0
p+s p+s (8) (a;q)n =(cid:26)(1−a)(1−qa)...(1−qn−1a), n≥1.(15)
(remarkthat in 4 dimensions all the multipliers be
ome
It should be noted that it is also possible to present
equTalhtuos,1)Θ.-graph θ(N)(m1,m2,m3) is nothing but in- Cn(x;β|q) in th4eΦf3orm3wΦh2i
h employs basi
hypergeo-
metri
fun
tion or , see [9, 10℄.
tegral of the produ
t of three normalized propagators q
Orthogonalityrelationsforthe -ultraspheri
alpoly-
defined in (2):
nomials are of prin
ipal importan
e. They are given by
θ(N)(m ,m ,m )=
1 2 3 the relation
π
3
Γ(g+2p)Γ(p+1) (m +p)Γ(g−m +p)
= i i , Cm(cosθ;β|q)Cn(cosθ;β|q)Wβ(cosθ|q)dθ =
Γ(g+p+1)Γ(2p) Γ(p+1)Γ(g−m +1) (9) Z
iY=1 i 0
214 ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3
APPLYINGTHEq-ALGEBRASUq′(son)TOQUANTUMGRAVITY
δmn 1−β2qm−1
= , + D (m−1,n,s,λ)=
hn(β|q) (16) 1−βqm q
where the weight fun
tion and the normalization fa
tor 1−qs+1
= D (m,n,s+1,λ)+
are as follows: 1−βqs q
(e2iθ,e−2iθ;q)
∞
W (cosθ|q)= ,
β (βe2iθ,βe−2iθ;q)∞ (17) 1−β2qs−1
+ D (m,n,s−1,λ).
1−βqs q (22)
(q,β2;q) (q;q) (1−βqn)
∞ n
h (β|q)= ,
n 2π(β,βq;q) (β2;q) (1−β) (18)
∞ n q
Likewise,one
an get evaluationfor other parti
ular( -
deformed analogsof) spin networks.
with
(a ,a ;q) :=(a ;q) (a ;q) ,
1 2 ∞ 1 ∞ 2 ∞
q
4. Covarian
e with Respe
t to -algebras
∞
(a;q) := (1−aqk).
∞
kY=0 Ourmain
on
ernhereisaqpossible relationofthis stuff
to quantum groups and/or -algebraswhi
h
orrespond
SO(n)
to the orthogonal Lie groups and their
orre-
Linearization formula is another important fa
t about
q sponding Lie algebras. As it was shown by Sugitani in
-ultraspheri
alpolynomials.Itisgivenbythefollowing
[11℄,zonalspheri
alfun
tions asso
iatedwith aparti
u-
Rogers' formula [9℄: SN
lar realization q of quantum spheres are proportional
C (x;β|q)C (x;β|q)= q
m n to the -ultrapsheri
al polynomials, that is
min(m,n) (q-)zonal spher. fun
. ↔Ck(N−2)/2(Y;q2) .
= A (β|q)C (x;β|q)
m,n,k m+n−2k (19) Uq(son)
Xk=0 To this eqnd, one starts with sStOan(dna)/rdSO(n−1) and
on-
stru
tsa -analogofthe
oset bymeans
U (so ) SO(n)
q n
with the notation of (
orSrOes(pnon−d1in)g to ) and a
oideal (
or-
(β;q)m−k (β;q)n−k (β;q)k responding to ):
A (β|q)= ×
m,n,k
(q;q)m−k (q;q)n−k (q;q)k e ,...,e ,f ,...,f ,θ ,θ ,qǫ2−1,...,qǫn−1 ,
2 n 2 n 1 2 q−1 q−1
(q;q) (β2;q) 1−βqm+n−2k D E
×(β2;q)mm++nn−−22kk (βq;q)mm++nn−−kk 1−β . (20) Jq := for Bn(n>1) and Dn(n>2) series, (23)
hθ i for N =3
1
q
NaDnoaw(lmoigt,nios,fsnΘ,oλ-tg)hr=aarpdht(o9o),bntaaimnetlhyeresultforthe -deformed hθ1,θ2,qqǫ2−−11i for N =4 .
q
Here
π
= C (x;β|q)C (x;β|q)C (x;β|q)W (cosθ|q)dθ = s·e1+(−1)n−1t·q1/2qǫ1f2···fnfn···f2f1
m n s β
Z
for B (n>1) series,
0 n
= 2π ((βq,,ββ2q;;qq))∞∞((ββq2;;qq))gg((qβ;;qq))gg−−nn((qβ;;qq))gg−−mm((qβ;q;)qg)−g−ss,(21) θ1 :=sf·oer1D+n(−(n1)>n−22)t·sqeǫr1iefs2,···fn−1fnfn−2···f2f1
m+n+s=2g g ≥m g ≥n g ≥s
oRw1bhe−v
eiurqoermussi+o1snymremlateitornyfuonrd,Derqeixs
ohba,tnaginees:dm,in↔then.f↔oNromsti
↔e tmhe. ss··ee11++tt··qq1ǫ1/f2q2ǫ1f(1N =(N4)=; 3),
D (m+1,n,s,λ)+
1−βqm q
(24)
ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3 215
A.M.GAVRILIK
t·q1/2qǫ1f1+(−1)n−1s·e2···enen···e2e1 at n=4 the q-algebraUq′(so4) in addition involves:
for B (n>1) series,
θ2 :=t·qǫ1f1n+(−1)n−2s·e2···en−1enen−2···e2e1 [[[III344+232,,,III443+322]]]qqq ===III434+223,,, [[[III344++131,,,III443++311]]]qqq ===III434++113,,, [[[III244++121,,,III442++211]]]qqq ===III424++112,,,
for D (n>2) series,
n
t·qǫ1f1+s·e2 (N =4), [[II44+32,,II23+11]]==0(q, −[Iq3−2,1I)(4+I12]1=I403−, I32I4+1).
(25)
ThefirstrIel+ationin(27)isviewedasdefinitionforthird
As follows from the results of [11℄, this left
oideal sub- generator 31; with this, the algebrais given in terms of
q q U′(so )
balrgaebUrq′a(s
oonin)
firdoemsw[6qit℄.hNthoetenaolnssotathnadtartdhis-dsaemfoermneodnsatlagne-- er-
aotomrmI3−u1ta=to[rIs2.1,DIqu32→a]lq−
q1o−pw1yhio
fheqnter3sthinevroellvaetsiotnhsesgaemne-
dard (or twisted) -deformed
oideal subalgebra arises as(27),butIw+ithI+ . Similar remarksapply to the
[12, 13℄ when one
onstru
(tns)a/SqOua(nn)tum analogueof the generatoUrs′(s4o2,) 41, as well as (dual
opy of) the whole
symmetri
oset spa
e SU . algebra q 4 .
5. The q-algebra Uq′(son) (Bilinear 6. Deformed Algebras A(n) of Nelson and
Formulation) Regge
(2+1)
Along with the definition inqterms of tUril′i(nseoar)relations For Λ < 0-dimensional gravity with
osmologi
al
on-
originally given in [6℄, the -algebra q n may be stant , the Lagrangian density involves spin
on-
ω ea a,b = 0,1,2
ab
equivalently defined in terms of `bilinear' formulation. ne
tion and dreibein , ,
ombined in
k>l+1, 1≤k,l≤n SO(2,2) ω
AB
Tothis end, thegenerators(set ) the -valued(anti-deSitter)spin
onne
tion
Ik±,l ≡[Il+1,l,Ik±,l+1]q±1 ≡ of the formω 1ea
ω = ab α ,
≡q±1/2I I± −q∓1/2I± I AB (cid:18) −α1eb 0 (cid:19)
l+1,l k,l+1 k,l+1 l+1,l
I ≡ I+ ≡ I− and is given in the Chern(cid:22)Simons (CS) form [17, 18℄
are introdu
ed together with k+1,k k+1,k k+1,k. α 2
Then, the bilinear formulation of the q-algebra Uq′(son) 8(dωAB− 3ωAF ∧ωFB)∧ωCDǫABCD.
reads:
A,B =0,1,2,3 η =(−1,1,1,−1)
[I+ ,I+] =I+ , [I+,I+ ] =I+ , Here ,themetri
is AB ,
lm kl q km kl km q lm Λ
and tΛhe=C−S
1oupling
onstant is
onne
ted wiSthO(2,,2s)o
[I+ ,I+ ] =I+ if k >l >m, that 3α2. The a
tion is invariant under ,
km lm q kl
leadstoPoissonbra
ketsandfieldequations.Theirsolu-
[I+,I+ ]=0 if k>l>m>p or k>m>p>l; tions, i.e. infinitesimal
onne
tions, des
ribe spa
e-time
kl mp (26)
whi
h is lo
ally anti-de Sitter.
[I+,I+ ]=(q−q−1)(I+I+ −I+I+ ) if k>m>l>p. To des
ribe global features of spa
e-time, within
kl mp lp km kp ml
fixed-time formulation, of prin
ipal importan
e are the
S :
I− integrated
onne
tions whi
h provide a mapping
Analogoussetofrelationsexistswhi
hinvolves kl along π1(Σ) → G
q → q−1 ′ Σ of theGh=omSoLtop(2y,Rgr)o⊗upSLfor(a2,sRp)a
e surfa
e
with (qde→not1e this (cid:16)dual(cid:17) set by (2′6)). In the into the group + − (spinorial
SO(2,2)
`
lassi
al' limit , both (26) and (26) redu
e to
so
overingof ) and thoroughlystudied in [19℄. To
n
those of .
n = 3 q U′(so ) generatethealgebraofobservables,onetakesthetra
es
For instan
e, at , the -algebra q 3 is iso- 1
c±(a)=c±(a−1)= tr[S±(a)]
morphi
[14℄ to qthe Fairlie (cid:21) Odesskii algebr[aX[,1Y5,] 1≡6℄ 2
q
(re
all that the -
ommutator is defined as
q1/2XY −q−1/2YX
): where
[I ,I ] =I+, [I ,I+] =I , [I+,I ] =I ; a∈π , S± ∈SL (2,R).
21 32 q 31 32 31 q 21 31 21 q 32 (27) 1 ±
216 ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3
APPLYINGTHEq-ALGEBRASUq′(son)TOQUANTUMGRAVITY
g = 1 Σ A(n)
For (torus) surfa
e , the algebra of (indepen- 7. UIs′o(smoor)phism of the Algebras and
dent) quantumobservableshasbeenderived[19℄,whi
h q n
turned out to be isomorphi
to the
y
li
ally symmet-
A(n)
ri
Fairlie (cid:21) Odesskii algebra [15, 16℄. This latter alge- Toestablishisomorphism[21℄betweenthealgebra
q
biraal,nho=we3ve
ra,seisokfnUoqw′(nsonto).
Sooin,
niadteur[1a4l℄qwueitshtiotnhearsipsees- fUrqo′(mson()30o)neahnads ttohemankoensthtaenfdoallrodwin-gdetfwoormsteedpsa.lgebra
g ≥ 2
whether fqor surfa
esUo′f(shoig)her genera , the non- {K1/2(K−1)−1}aik −→Aik,
standard -algebras q n also play a role. (cid:22) Redefine:
A −→I , K −→q.
Below, the positive answer to this question is given. ik ik
Σ×R Σ (cid:22) Identify:
Forthetopologyofspa
etime ( beinggenus-
g π1(Σ) A(n)
surfa
e), the homotopy group is most effi- Then, the Nelson(cid:22)Regge algebra is seen to trans-
2g + 2 = n q
iently des
ribed in terms of generators late exa
tly into the nonstandard -deformed algebra
t1,t2,...,t2g+2 introdu
ed in [20℄ and su
h that Uq′(son) des
ribed above, see (26). We
on
lude that
2g+2 thesetwodeformedalgebrasareisomorphi
toea
hoth-
K 6= 1 n
t1t3···t2g+1 =1, t2t4,...,t2g+2 =1, ti =1. er (of
goursen, f=or2g+2 ). Re
aKll t=ha(t4α−isilhin)/k(e4dαt+o tihhe)
iY=1 genusα2 =as−1 , while
with Λ.
n(n−1)/2
Classi
al gauge invariant tra
e elements ( in Let us remark that it is the bilinear presentation (2)
q U′(so )
total) defined as ofthe -algebra q n whi
hmakespossibleestablish-
1 ing of this isomorphism. It should be stressed also that
α = Tr(S(t t ···t )), S ∈SL(2,R), A(n)
ij i i+1 j−1
2 (28) the algebra plays the role of (cid:16)intermediate(cid:17) one:
starting with it and redu
ing it appropriately, the al-
generate
on
retealgebrawith Poissonbra
kets,expli
- gebra of quantum observables (gauge invariant global
itly found in [20℄. At the quantum level, to the alge-
hara
teristi
s) is to be finally
onstru
ted. The role of
bra with generators A(2(8n)) there
orresp2o+nd1s quantum Casimir operators in this pro
ess, as seen in [20℄, is of
ommutator algebra Λ spe
ifi
for quantum greatimportan
e.Inthisrespe
tletusmentionqthatthe
g{rja,lv,ikty,mw}it,hj,nle,gka,tmive=1.,.F.o.r,ne,a
h quadruple of indi
es Uqu′a(sdora)ti
anqd higher Casimir ele1ments of the -algebra
su
h that q n ,for beingnotarootof ,areknowninexpli
it
i<j <m<k <i , form[13,22℄ alongwitheigenvaluesoftheir
orrespond-
(29)
ing (representation) operators [22℄.
A(n)
the algebra of quantum observables reads [20℄: As it was shown in detgai=l in1[19℄, the deformed alge-
[a ,a ]=[a ,a ]=0, bra for the
ase of genus surfa
es redu
es to the
mk jl mj kl
desired algebra of three independent quantum observ-
[a ,a ]=(1− 1)(a −a a ), A(3)
jk kl K jl kl jk ables whi
h
oin
ides with , the latteUr′(bseoin)g iso-
[ajk,akm]=(K1 −1)(ajm−ajkakm), (30) morphi
gt=o2the Fairlie (cid:22) Odesskii algebra q 3 . The
aseof issignifi
antlymoreinvolved:hereonehas
[a ,a ]=(K− 1)(a a −a a ). A(6)
jk lm K jl km kl jm to derive, starting with the 15-generator algebra ,
the ne
essary algebra of 6 (independent) quantum ob-
K α
Here the parameter of deformation involves both servables. J.Nelson and T.Regge have su
eeded [23℄ in
and Plan
k's
onstant, namely
onstru
tingsu
hanalgebra.Their
onstru
tionhowev-
K = 4α−ih, α2 =− 1 , Λ<0. er is highly nonunique and, what is moreessential, isn't
4α+ih 3Λ (31) seen to be effi
iently extendable to general situation of
g ≥3
.
SL (2,R)
±
Note that in (28) onlyone
opy of the two is Our goal in this note was to attra
t attention to the
A(n)
indi
ated.In
onjun
tionwiththis,besidesthedeformed isomorphism of the deformed algebras from [20℄
A(n) SL (2,R) q U′(so )
algebra derivedwith, say, + takenin (28) andthenonstandard -deformedalgebras q n intro-
A(n)
and given by (30), another identi
al
opy of (with du
ed in [6℄). The hope is that, taking into a
ount a
K → K−1
the only repla
ement )
an also be obtained signifi
ant amount of the already existing results
on-
SL (2,R) SL(2,R) U′(so )
starting from − taken in pla
e of in
erning diverseaspe
tsof q n (the obtained various
(28).Thisanother
opyisindependentfromtheoriginal
lasses of irredu
ible representations [6, 14, 24(cid:22)28℄ and
one: their generatorsmutually
ommute. others, as well as knowledge of Casimir operators and
ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3 217
A.M.GAVRILIK
theireigenvalues depending on representations,et
.) we 22. Gavrilik A.M. and Iorgov N.Z., // Ibid. (cid:21) P. 310(cid:21)314,
may expe
t for a further progress
on
erning
onstru
- [math.QA/9911201℄.
6g−6
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218 ISSN 0503-1265. Óêð. iç. æóðí. 2002. Ò. 47, N 3
