DIAS-STP-03-5
A Fuzzy Th ee Sphe e and Fuzzy To i
B ian P. Dolana),b)∗and Denjoe O’Conno b)†
a)Dep . o Ma hema ical Physics, NUI, Maynoo h, I eland
b)School o Theo e ical Physics,
Dublin Ins i u e o Ad anced S udies,
10 Bu ling on Rd., Dublin 8, I eland
Sep embe 23, 2005
Abs ac
A uzzy ci cle and a uzzy 3-sphe e a e cons uc ed as subspaces
o uzzy complex p ojec i e spaces, o complex dimension one and
h ee, by modi ying he Laplacians on he la e so as o gi e un-
wan ed s a es la ge eigen alues. This lea es only s a es co esponding
o uzzy sphe es in he low ene gy spec um ( his allows he commu a-
i e algeb a o unc ions on he con inuous sphe e o be app oxima ed
o any equi ed deg ee o accu acy). The cons uc ion o a uzzy ci cle
opens he way o uzzy o i o any dimension, hus ci cum en ing he
p oblem o powe law co ec ions in possible nume ical simula ions on
hese spaces.
1 In oduc ion
One o he p incipal goals o he s udy o ield heo ies on uzzy spaces is
o de elop an al e na i e non-pe u ba i e echnique o he amilia la ice
one [1]. To da e, his new app oach in he case o ou dimensional ield
heo ies has been limi ed o s udies o Euclidean ield heo y on S2×S2
[2], CP2[3] and S4[4]. All bu S2×S2ha e addi ional complica ions. Fo
∗[email p o ec ed]
†[email p o ec ed]
1
example, CP2is no spin bu spincand S4is eally a squashed CP3and
includes many unwan ed massi e Kaluza-Klein ype modes. E en S2×S2
is no ideal since i has cu a u e e ec s ha d op o as powe co ec ions
a he han exponen ially as in he case o o oidal geome ies.
The uzzy app oach does, howe e , ha e he ad an age o p ese ing con-
inuous symme ies such as he SU(2) symme y o a ound S2and does no
su e om e mion doubling [5]. The ad an ages a e gained a he cos o in-
oducing a non-locali y associa ed wi h he non-commu a i i y o he uzzy
sphe e. The e is he e o e a balance o ad an ages and disad an ages asso-
cia ed wi h he uzzy app oach. The inal decision on whe he he app oach
has eal ad an ages o e he s anda d la ice app oach should be de e mined
by doing genuine simula ions. Fo his eason Mon e Ca lo simula ions o
he uzzy app oach a e now in p og ess. In he la ice app oach non-locali y
is also a p oblem when e mions a e included. So ou expec a ion is ha as
a as Mon e Ca lo simula ions a e conce ned he uzzy app oach will no be
compe i i e wi h he la ice one un il e mions a e included. The app oach
will gain u he ad an ages in si ua ions whe e symme ies a e mo e impo -
an . I also ex ends na u ally o allow o supe symme y. (see [6] whe e
a uzzy supe sphe e was cons uc ed). So we expec he ue powe o he
app oach o eme ge when supe symme y and chi al symme y a e p esen
in a model.
A adically di e en al e na i e o he Euclidean Mon e Ca lo app oach
becomes a ailable once one has a uzzy h ee-dimensional space. Such a
space has he ad an age ha i allows one o de elop e y di e en non-
pe u ba i e me hods, since now one can add ess he non-pe u ba i e ques-
ions om a Hamil onian poin o iew.
The pu pose o his a icle is o in oduce p ecisely such uzzy h ee-
dimensional spaces. We will begin by p esen ing a uzzy e sion o he ci cle
S1
F, om which one can ob ain o i o a bi a y dimension. We will hen
p esen a uzzy app oxima ion o he h ee-sphe e, S3
F. Un o una ely, bo h
o hese spaces a e s ill no ideal in ha hey in ol e many unwan ed addi-
ional deg ees o eedom which we supp ess so ha hey do no con ibu e
o he low ene gy physics. The p esence o addi ional deg ees o eedom is
p obably una oidable as i seems o be he p ice one pays o he classical
space no being a phase space. The h ee-sphe e is also cu ed and hence
he esul s ob ained om s udies o ield heo ies on his space will app oach
hose o a la h ee-dimensional space wi h polynomial co ec ions. I has,
howe e , he ad an ages o p ese ing he ull SO(4) symme y o a ound
S3. F om he cons uc ion i seems clea ha bo h o hese spaces will also
be ee o e mion doubling p oblems.
We will es ic ou ocus he e o scala ield heo ies and demons a e how
2
he unwan ed deg ees o eedom can be supp essed so ha he limi ing la ge
ma ix heo y o a scala ield heo y eco e s ield heo y on he commu a i e
spaces. We will a gue ha he da a speci ying he geome ies can be cleanly
speci ied by gi ing a sui able Laplace- ype ope a o o he scala ield, which
oge he wi h he ma ix algeb a and i s Hilbe space s uc u e gi es a
spec al iple.
Aside om ou pe sonal mo i a ions, non-commu a i e geome y has e-
cen ly become a e y popula a ea o esea ch om bo h he poin o iew o
possible new physics in s ing heo y and D-b ane heo y, [7, 8], and as a new
egula isa ion echnique in o dina y quan um ield heo y, [2]-[4] and [9]-[12].
In bo h hese endea ou s “ uzzy” spaces play an impo an ˆole. Roughly
speaking a uzzy space is a ini e ma ix app oxima ion o he algeb a o
unc ions on a con inuous mani old, he seminal example being he uzzy
wo-sphe e, [13]. I has he impo an p ope y o p ese ing he isome ies
o he space ha i is app oxima ing. As such he idea can se e as a sou ce
o examples ela ed o ma ix models in s ing heo y and as a egula isa ion
echnique o o dina y quan um ield heo y. As a egula isa ion me hod
i p o ides one ha p ese es he unde lying space- ime symme ies and is
amenable o nume ical compu a ion.
Fuzzy sphe es in dimensions o he han wo we e analysed in [14]-[17], bu
he cons uc ion he e was incomple e. They also ad oca e p ojec ing ou
he unwan ed modes and wo king wi h a non-associa i e algeb a which we
conside unsa is ac o y. Also he case o odd sphe es wo ks e y di e en ly
o ha o e en sphe es. An al e na i e app oach o he uzzy ou -sphe e,
S4
F, was gi en in [4], based on he ac ha uzzy CP3and CP1∼
=S2a e well
unde s ood [18], and, in he con inuum limi , CP3is an S2bundle o e S4.
In his pape we show how he odd-dimensional uzzy sphe es S1
Fand S3
F
can be ex ac ed om he ma ix algeb as associa ed wi h he uzzy complex
p ojec i e spaces CP1
Fand CP3
F. An al e na i e app oach o ob aining a
ini e app oxima ion o S3∼
=SU(2), based on con o mal ield heo y, was
p esen ed in [19], howe e , in his app oach i is unclea how he unwan ed
modes a e o be supp essed. Ou me hod uses a simila supp ession mecha-
nism o ha used o S4
Fin [4]. Al hough he e is no closed ini e dimensional
ma ix algeb a o SN
Funless N= 2, he ele an deg ees o eedom when
N= 1 and N= 3 a e con ained in he ma ix algeb as o CP1and CP3
espec i ely. One can he e o e ob ain unc ional in eg als o ield heo ies
o e S1
Fand S3
Fby s a ing wi h unc ional in eg als o e CP1
Fand CP3
F
and hen supp essing he unwan ed modes so ha hey do no con ibu e
o he unc ional in eg al. Because o he high deg ee o symme y inhe en
in he cons uc ion, he unwan ed modes can be supp essed simply by using
app op ia e quad a ic Casimi s in he Laplacian. In his way we by-pass
3
he p oblems associa ed wi h he ac ha he algeb a o ma ices associa ed
wi h unc ions on he sphe e does no close on he sphe e, bu necessa ily
li s in o he en eloping complex p ojec i e space. In a simila ashion we
expec ha when a Hamil onian app oach o ield heo y is de eloped using
hese spaces he unwan ed modes will cause no di icul ies since hey can be
made a bi a ily di icul o exci e.
The pape is o ganized as ollows. In sec ion 2 we summa ize how a
gi en geome y is cap u ed in he uzzy app oach. Sec ion 3 hen gi es ou
cons uc ion o a uzzy ci cle, S1
F. Sec ion 4 summa ises he cons uc ion
o S4
Fp esen ed in [4] and in sec ion 5 we p esen ou uzzy h ee-sphe e,
S3
F. Sec ion 6 gi es an al e na i e cons uc ion o S3
Fwhich lends i sel o a
gene alisa ion o SN
F o any N[20].
2 Encoding he geome y o a uzzy space
F ¨ohlich and Gaw¸edzki [19] ( ollowing Connes, [21]) ha e demons a ed ha
he abs ac iple (H, A,∆γ), whe e His he Hilbe space o squa e in e-
g able unc ions on he mani old M, wi h Laplace-Bel ami ope a o ∆γ,γ
being he me ic, and A=C∞(M) is he algeb a o smoo h bounded unc-
ions on M, cap u es a opological space oge he wi h i s me ical geome y.
In a simila ashion one can speci y a uzzy space, MF, as he sequence
o iples
MF:= (HL,Ma dL,∆L) (1)
pa ame e ized by L, whe e HL=Cd2
Lis he Hilbe space ac ed o he com-
ple e ma ix algeb a Ma dLo dimension d2
Lwi h inne p oduc < M, N >=
1
dLT (M†N) and ∆Lis a sui able Laplacian ac ing on ma ices. One can
eadily ex ac in o ma ion such as he dimension o he space om hese
da a. The Laplacian comes wi h a cu o and so he dimension can be ead
om he g ow h o he numbe o eigen alues.
The da a con ained in he iple (H, A,∆γ) a e p ecisely he da a ha go
in o he Euclidean ac ion o a scala ield heo y on he space Mand hence
speci ying he scala ac ion is a con enien me hod o p esc ibing hese da a.
In he uzzy app oach he algeb a will always be a ma ix algeb a and we
will e ain he Hilbe space inne p oduc speci ied abo e so he only da a
om he iple, (HL,Ma dL,∆L), emaining o be supplied a e he pe mi ed
ma ix dimensions, dLand a ealiza ion o he Laplacian, ∆L. Once his
in o ma ion is gi en he uzzy geome y is speci ied.
Though i may be con enien o gi e a map o unc ions his is no neces-
sa y. Once he Laplacian is gi en i s eigenma ices and spec um can be used
o p o ide such a map i needed. Suppose o example ha he spec um o
4
∆Lis iden ical o ha o ∆γup o some cu o and a comple e se o eigen-
ma ices is gi en by ˆ
Ψλwi h he co esponding commu a i e eigen unc ions
being Ψλ, hen he symme ic symbol-map D gi en by
D =
d2
L
X
λ
Ψλˆ
Ψλ(2)
p o ides a map o unc ions wi h
M=1
dL
T (DM) (3)
he unc ion co esponding o he ma ix M. By cons uc ion he map has
no ke nel and he symbol-map induces a ∗p oduc on unc ions gi en by
M∗D N=1
dL
T (DMN) (4)
which ep esen s ma ix mul iplica ion in e ms o an ope a ion on he image
unc ions. The ∗p oduc depends on D, a di e en bu equi alen one could
be ob ained by gi ing a nonze o weigh ing cλ(L) o he di e en e ms in he
sum (2). In he case o CPNa pa icula choice o he cλ(L) will gi e he
diagonal cohe en s a e p esc ip ion1as discussed in [18].
I he symbol-map (2) has he p ope y ha
∆γ M=1
dL
T (D∆LM) (5)
whe e ∆γis a na u al Laplacian o he space o be app oxima ed, hen he
spec um o he uzzy space will be p ecisely a cu o e sion o ha o he
commu a i e space M. This is p ecisely wha happens in he case o CPN
F,
see [18].
Howe e , i is con enien o ex end he de ini ion o uzzy space o he
case whe e he spec um coincides o low-lying eigen alues, bu de ia es o
a amily o eigen alues ha can be gi en a bi a ily high alue and which
co espond o deg ees o eedom ha ha e no coun e pa in he commu a-
i e space M. This allows us o ob ain uzzy app oxima ions o addi ional
spaces — in pa icula , as we will see, o o i and he h ee sphe e.
1In he case whe e he symbol-map is he p ojec o o cohe en s a es he unc ion M
is e e ed o as he co a ian symbol o he ma ix Mand since he coe icien s cλ(L) a e
no one i will di e om he co esponding con a a ian symbol, see Be ezin [22]. The
symbol-map is e e ed o as symme ic when i s co a ian and con a a ian symbols a e
equal and coincides wi h he case o cλ(L) = 1.
5

I one akes he Euclidean quan um ield heo y poin o iew hen he
desi ed geome y appea s as ha associa ed wi h he accessible con igu a-
ions o he ield heo y and he de ia ions a e supp essed in a p obabilis ic
ashion.
A success ul me hod o supp essing he unwan ed modes would be o add
o he scala ac ion a e m SI[Φ] which is non-nega i e o any Φ, ze o only
o ma ices ha co espond o unc ions on M, and posi i e o hose ha
do no . The modi ied ac ion would he e o e be o he o m S[Φ] + hSI[Φ].
The pa ame e hshould be chosen o be la ge and posi i e. The p obabili y
o any gi en ma ix con igu a ion hen akes he o m
P[Φ] = e−S[Φ]−hSI[Φ]
Z(6)
whe e
Z=Zd[Φ]e−S[Φ]−hSI[Φ] (7)
is he pa i ion unc ion o he model. I he p esc ip ion is o wo k o ee
ield heo ies, hen SI[Φ] should be a mos quad a ic in Φ. This can hen
be hough o as a modi ica ion o he Laplacian in he iple (1).
Fu he mo e he p oblem o UV/IR mixing in scala heo ies can be e-
mo ed by including a highe de i a i e ope a o in he quad a ic e m o he
ield heo y such ha i ende s all diag ams ini e when he ma ix size is
sen o in ini y. Wi h such a p esc ip ion since each diag am has a limi ing
commu a i e alue in he la ge ma ix limi each diag am mus ake his
alue and hence no UV/IR mixing can occu . The p esc ip ion o sending
he ma ix size o in ini y and sending he coe icien o he i ele an highe
de i a i e ope a o o ze o do no commu e. This p esc ip ion o adding an
i ele an ope a o o he ac ion is simple han he no mal o de ing p e-
sc ip ion p oposed in [23] and wo ks o any dimension.
F om he abo e discussion i should be clea ha he en i e p oblem o
cons uc ing a uzzy app oxima ion o a space is he p oblem o gi ing a
sui able p esc ip ion o he ma ix Laplacian.
3 App oxima ing a ci cle om a uzzy sphe e
Conside he ini e ma ix algeb a ep esen a ion o he uzzy sphe e S2
F[13].
The algeb a o (L+ 1) ×(L+ 1) ma ices, which will be deno ed by Ma L+1,
has he same dimension as he numbe o deg ees o eedom in a sphe ical
6
ha monic expansion o a unc ion on S2, unca ed a angula momen um L,
L(θ, φ) =
L
X
l=0
l
X
m=−l
lmYlm(θ, φ).(8)
Tha is L
X
l=0
(2l+ 1) = (L+ 1)2.(9)
The p ecise iden i ica ion be ween a ma ix Φ ∈Ma L+1 and a cu -o unc-
ion L(θ, φ), as discussed in he p eceding sec ion, is no unique, bu he pos-
sible maps can be gi en in e ms o cohe en s a es o he symme ic symbol-
map D o (2), and he esul ing p oduc o unc ions is non-commu a i e o
ini e L. I is c ucial o ou cons uc ion ha only maps o which he p od-
uc o unc ions becomes commu a i e in he limi L→ ∞ be conside ed.
The symbol-map (2) associa es o hono mal (L+ 1) ×(L+ 1) pola isa ion
enso s ˆ
Ylm wi h sphe ical ha monics Ylm(θ, φ). The con en ions used he e
will be ha
ˆ
Ylm =1
√L+ 1
ˆ
Tlm (10)
whe e he pola isa ion enso s ˆ
Tlm a e hose o [24].
The SO(3) symme ic Laplacian, L2, on he uzzy sphe e ac s on ma ices
Φ and is ep esen ed by he second o de Casimi co esponding o he adjoin
ac ion o he angula momen um gene a o s Liin he (L+ 1) ×(L+ 1)
ep esen a ion:
L2Φ = [Li,[Li,Φ]].(11)
Hence he ac ion can be aken o be
S[Φ] = 1
L+ 1T 1
2Φ†L2Φ + V(Φ)(12)
o some scala po en ial V(Φ)†=V(Φ), which is assumed o be bounded
below. This ac ion can hen be used in a pa i ion unc ion which in ol es
o dina y in eg a ion o e (L+ 1)2deg ees o eedom
Z=ZDΦe−S[Φ].(13)
The p obabili y dis ibu ion o ield con igu a ions is hen
P[Φ] = e−S[Φ]
Z(14)
7
whe e S[Φ] gi en is by (12). This p obabili y dis ibu ion is associa ed wi h
he geome y (HL,Ma L+1,L2) which speci ies a ound uzzy sphe e. The
ield heo y wi h quad a ic po en ial, howe e , su e s om UV/IR mixing
p oblems [23, 25]. I we add he e m aL4 o he Laplacian and use he
iple (HL,Ma L+1,L2+aL4) he UV/IR mixing p oblem is emo ed and we
eco e a ield heo y on he commu a i e S2in he in ini e ma ix size limi .
The pa ame e acan inally be sen o ze o wi h he esul ha he c i ical
alue o he mass pa ame e will be sen o in ini y. The p ocess o aking
he la ge ma ix limi and sending a o ze o do no commu e. To ob ain he
commu a i e heo y on he sphe e he ma ix size mus be sen o in ini y
o non-ze o a.
The e is no ini e ma ix app oxima ion o he algeb a o unc ions on
S1. Ne e heless, he deg ees o eedom ele an o a ci cle a e ce ainly
con ained in Ma L+1. Focusing on he op ha monic in (8), wi h l=L, he
YLm con ain all −L≤m≤Land hus ep oduce unc ions on he ci cle as
m→ ∞. This implies ha he pa i ion unc ion and co ela ion unc ions
o a ield heo y on a ci cle can be ex ac ed om ha o he uzzy sphe e
by supp essing all he modes wi h l < L in (13). One way o achie ing his
is o penalise modes wi h l < L by gi ing hem a la ge posi i e weigh in he
ac ion. To his end we modi y he ac ion (12) o
Sh[Φ] = 1
L+ 1T 1
2Φ†[L3,[L3,Φ]] + h
2Φ†L(L+ 1) −L2Φ + V(Φ).
(15)
All modes wi h l < L now ha e he w ong sign o L2and, when his
e y la ge, a e hea ily penalised in he pa i ion unc ion (13), con ibu ing
no hing as h→ ∞. In his limi only he modes wi h l=L emain and
hese ha e he co ec sign o hei kine ic ene gy, because he e m linea
in h anishes on hese and only hese modes. The ‘w ong sign’ o he L2
con ibu ion o he kine ic ene gy he e is analogous o an an i- e omagne ic
coupling in a la ice heo y and jus as in he la ice heo y wi h an an i-
e omagne ic coupling he ac ion he e is also bounded below. Tha he
ac ion emains bounded om below is in ima ely ela ed o he ac ha he e
is an ul a iole cu o in he model and he e o e a maximum eigen alue o
he Laplacian o equi alen ly a sho es wa eleng h.
To see ha he commu a i e algeb a o unc ions on S1is eco e ed in he
l=Lsec o o he uzzy sphe e as L→ ∞, we i s decompose he ma ix
Φ using he basis o pola isa ion enso s:
Φ =
L
X
l=0
l
X
m=−l
Φlm ˆ
Ylm.(16)
8
In ou con en ions (10) he commu a o o he pola isa ion enso s is gi en
by (see e.g. [24] page 191, equa ion (46))
[ˆ
Yl1m1,ˆ
Yl2m2] = (2l1+ 1)(2l2+ 1)
L+ 1
L
X
l=0
(−1)L−l1−(−1)l1+l2+l
×l1l2l
L/2L/2L/2Clm
l1m1,l2m2
ˆ
Ylm,
(17)
whe e l1l2l
L/2L/2L/2a e 6j-symbols and Clm
l1m1,l2m2a e Clebsch-Go don
co-e icien s. Now o la ge L
l1l2l
L/2L/2L/2≈1
√L+ 1Cl0
l10,l20(18)
and Cl0
l10,l20= 0 when l1+l2+lis odd. Thus
[ˆ
Yl1m1,ˆ
Yl2m2]→0 (19)
and he algeb a is commu a i e when L→0 as p omised. In pa icula
[ˆ
YLm1,ˆ
YLm2]→0 (20)
and he op ha monic alone ep oduces he commu a i e algeb a o unc ions
on S1in he con inuum.
To summa ize we can encode he geome y speci ying a uzzy ci cle by
he iple
S1
F:= HL,Ma L+1,L2
3+hL(L+ 1) −L2.(21)
wi h h >> 1. This picks ou he uzzy ci cle om he op angula momen um
pola iza ion enso ˆ
YL,m.
One could equally pick i ou om a lowe one, ˆ
YL0,m by modi ying he
e m p opo ional o h o (L0(L0+ 1) −L2)2. This la e choice may ha e
ad an ages o he supp ession o UV/IR mixing e ec s in he uzzy con ex .
I oughly co esponds o a mix u e o ‘nea es neighbou ’ and nex nea es
neighbou e omagne ic and an i- e omagne ic couplings.
Ha ing cons uc ed a uzzy ci cle i is now clea ha he e is no obs acle
o cons uc ing uzzy o i o a bi a y dimension, simply by aking p oduc s
o uzzy ci cles. This has he ob ious ad an age o nume ical simula ion o
a oiding powe -law cu a u e e ec s.
9
I we can penalise all modes wi h 2l < L and m6= 0 o 2l=Lin
a unc ional in eg al o e SO(5)/SO(3) ×SO(2) hen we will eally be
doing a unc ional in eg al o e S3
F. This is easily achie ed since 2l=Land
m= 0 has he la ges second o de Casimi ,
C(5)
2(L, 0) = L(L+ 3),(51)
o all he SO(5) ep esen a ions in Ma d0
L. In he now amilia manne he
unwan ed modes in he unc ional in eg al o e SO(5)/SO(3)×SO(2)can
be supp essed by using he Laplacian
L2
h0=1
2[Jαβ,[Jαβ,·]] + h0L(L+ 3) −L2
(5),(52)
which ac s on ields Φ ∈Ma d0
Land Le en. The unwan ed modes a e
comple ely elimina ed in he limi h0→ ∞, gi ing S3
F unca ed a le el L.
The cons ain ha Lis e en does no change he ac ha we ge he ull
con inuum S3as L→ ∞.
7 Conclusions
By s a ing wi h he known ini e ma ix algeb as o CP3and CP1, he
uzzy CP3
Fand he uzzy sphe e CP1
F∼
=S2
F, ini e unc ional in eg als o
scala ield heo ies on S3
Fand S1
Fha e been cons uc ed. The geome y o
a uzzy space is speci ied by a iple (HL,Ma dL,∆L) and, al hough he e
is no known closed associa ed algeb a gi ing a uzzy S1as a iple di ec ly,
CP1
Fne e heless con ains he s a es equi ed o a S1
Fplus o he unwan ed
s a es. The unwan ed s a es a e gi en la ge eigen alues by modi ying he
Laplacian on CP1
F, as in equa ion (21), lea ing only he s a es o S1
Fin he
low ene gy spec um o he Laplacian. In a simila way CP3
Fcon ains he
s a es necessa y o a uzzy desc ip ion o S3( ia S4
F) and he Laplacian on
CP3
Fcan be modi ied, as in equa ion (42), so ha s a es no ela ed o S3
Fa e
gi en la ge eigen alues, lea ing only S3
Fs a es in he low ene gy spec um.
An al e na i e cons uc ion o S3
F, based on supp essing modes on a uzzy
e sion o he o hogonal G assmannian SO(5)/SO(3) ×SO(2), has been
p esen ed in sec ion 6. This has he ad an age o ha ing a na u al ex ension
o SN
F o any N, [20]. Thus S3
Fcan be ob ained ei he in wo s eps, ia he
uzzy S4
Fcons uc ed in [4], CP3
F→S4
F→S3
F, o al e na i ely in a single
s ep om he uzzy e sion o SO(5)/[SO(3)×SO(2)] as desc ibed in sec ion
6.
16

The cons uc ion o he uzzy ci cle allows uzzy o i o be de ined in
an ob ious way, by aking p oduc s o uzzy ci cles, hus opening he way
o nume ical simula ions on o i while p ese ing he ull U(1) ×···×U(1)
isome y g oup and a oiding he e mion doubling p oblem [5]. One i s
w i es down a ini e unc ional in eg al o a ield heo y on S2
F× ··· ×
S2
F, which con ain he modes ele an o p opaga ion S1
F× ··· × S1
Fin i s
spec um, and hen damps he unwan ed modes. This can be done, in a
manne ha p ese es he isome ies o he o us, by in oducing app op ia e
combina ions o second o de Casimi s in o he p opaga o s.
I is pleasu e o acknowledge A. P. Balachand an, Pe e P eˇsnajde , Ha -
ald G osse and Daniel Roggenkamp o help ul discussions.
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