AdS/CFT description of D-particle decay

0.47 MB·English

by  Kasper Peeters

AEI-2004-038, hep-th/0405125
AdS/CFT description of D-particle decay
Kasper Peeters and Marija Zamaklar
MPI fu¨r Gravitationsphysik, Am Mu¨hlenberg 1, 14476 Golm, GERMANY∗
(Dated: May 13th, 2004)
UnstableD-particlesintype-IIBstringtheorycorrespondtosphaleronsolutionsinthedualgauge
theory. Weconstructanexplicittime-dependentsolutionforthesphalerondecayonS3×R,aswell
asthecoherentstatecorrespondingtothedecayproduct. Wedevelopamethodtocountthenumber
of bulk particles in the AdS/CFT setup. When applied to our coherent state, the naive number
operatorOˆ†Oˆ isshowntobeinappropriate,eveninthelarge-N limit. Thereasonisthatthefinal
J J
stateconsistsofalargenumberofparticles. Bycomputingallprobabilitiesforfindingmulti-particle
states in the coherent state, we deduce the bulk particle content of the final state of the sphaleron
decay. The qualitative features of this spectrum are compared with the results expected from the
gravity side, and agreement is found.
5
PACSnumbers: 11.25.-w,11.25.Tq
0
0
2
I. INTRODUCTION AND SUMMARY mechanicaldescriptionofthefinalstageofthisdecayus-

n ing a coherent state. We then develop the formalism to
a count the number of bulk particles into which this final
Thespectrumoftype-IIBstringtheorycontains,apart
J
statedecomposes. Inthe lastpart,weapplythis formal-
fromthestableBPSandnon-BPSstates,alsoawideva-
7 ismto aconcretecaseatfinite N andextractqualitative
riety of unstable D-branes. These unstable branes con-

features of the decay process. Although the whole pro-
2 tain a tachyon field on their world-volume, and the con-
cess is highly non-supersymmetric,and thus expected to
v densation of this field corresponds to the decay of the
5 brane. Recently, a lot of progress has been made in un- besubjecttoquantumcorrections,wewillseethatthere
2 derstandingthedynamicalaspectsofthedecayofunsta- are indeed qualitative features which agree with known
1 results derived on the string theory side.1
ble branes. Most of the analysis was performed directly
5
using boundary conformal field theory in flat space, ini-
0
tiated by Sen’s construction of the boundary states for Thedualgaugetheorysystemisstudiedbyconsidering
4
0 decayingD-branes[1],orbyusingthec=1matrixmodel a time dependent solution of the decaying sphaleron on
/ forthedescriptionofthedecayofD-branesin1+1dimen- the three-sphere. We are able to find an analytic, classi-
h
sional string theory [2]. In the present paper we would cal solution for the spherically symmetric decay channel
t
- liketostudytheproblemofdecayingbranesintheset-up of the sphaleron. While the non-abeliancharacterof the
p
of the “standard” AdS/CFT correspondence. gauge theory (i.e. the non-vanishing coupling) is crucial
e
h As was argued by Harvey et al. [3], the unstable D- fortheexistenceofthesphaleronsolutionnearthetopof
: branes in string-theory are equivalents of “sphalerons”: the potential, it turns out that our solution abelianises
v
near the bottom of the potential valley (i.e. it is a solu-
i they are unstable solutions located at a saddle point of
X tion to the free Yang-Mills equations of motion on the
the potential in configurationspace, at the top of a non-
r contractibleloop[4]. InthecontextoftheAdS/CFTcon- sphere). Thisallowsustoconstructacoherentstatecor-
a responding to the final product of the sphaleron decay.2
jecture, this correspondence between unstable D-branes
This coherent state should be dual to the gas of closed
and sphalerons is in fact even more direct. It has
stringparticleswhichisthedecayproductofanunstable
been argued by Drukker et al. [5] that the existence of
D-particle.
sphaleronic saddle points in the potential of the theories
on both sides is a feature which is preserved when go- Inordertomakealinkwithcalculationsonthegravity
ing from weak to strong coupling, despite the fact that side, we then calculate the “number operators” in this
the precise form of the potential receives quantum cor- coherent state c for various single trace operators OˆJ
| i
rections. The unstable D-particles of string theory are
theninprecisecorrespondencewithknownsphaleronso-
lutions of the dual gauge theory. Kinematical aspects of
this correspondence were investigated in detail in [5]. 1 Thisistoacertainextent similartotheattempts tomatchthe
values oftheentropyforAdSblackholes,usingacalculationof
In the present paper, we will analyse dynamical as-
thefreeenergyinfreeYang-Millstheory[6]. Themaindifference
pects of this correspondence. Our analysis consists of with respect to this case, however, is in the dependence on the
three parts. First, we will construct the classical solu- coupling constant. The leading value of the black hole entropy
tion of the decaying sphaleron, and obtain a quantum andthefreeYang-Millsenergyareindependentofthestring/YM
coupling. Aswewillsee,thenumberofparticlesproducedinthe
sphalerondecaydoes dependonthecoupling.
2 SimilardescriptionsoftheStandardModelsphalerondecayusing
a coherent state approach have been discussed by Zadrozny [7]
∗Email:kasper.peeters,[email protected] andHellmundandKripfganz[8].
2
which are dual to closed string particles. There are two This makes the problem of calculating the energy dis-
subtle points in this procedure. One is related to the tribution in the outgoing state very hard to do analyt-
fact that in gravity calculations one uses the “standard” ically. In section IIIC we instead adopt a Monte-Carlo
notion of particles in the bulk as (angular) momentum method in order to compute the state norms,and subse-
eigenstates,and calculatesemissionamplitudes for these quentlyevaluatethesum(1)foralloperatorsintheU(4)
particles. Hence, in order to make a comparison with case. Weshowthat,asexpectedfromstringcalculations,
gravity possible, we cannot directly use the AdS/CFT particles in the final state are suppressed as their mass
correspondence in position space. Instead, we first have increases. We also show that, had one not taken the
to construct boundary operators that are dual to bulk full normsin(1)intoaccount,onewouldincorrectlyfind
angular momentum eigenstates. To construct these op- thatthe energydistributionincreasesformoreandmore
erators one projects the composite operators onto eigen massiveparticles. This is essentially due to the fact that
angularmomentumoperators,bymultiplying them with classical expectation values for all operators grow with
the appropriate tensorial spherical harmonics and inte- their dimension.
grating over the sphere. This construction is explained
in the section IIC and further illustrated on an explicit
example in appendix VA.
The second subtlety in counting particles in the co- Our results agree in a qualitative sense with results
herent state is related to the fact that the operators from previous calculations on the gravity side. The cal-
Oˆ which create elementary bulk particles are, from the culationsonthegravitysidehavealreadybeenperformed
J
point of view of the gauge theory, composite rather than in the literature for decaying D-branes in flat space [9].
elementaryoperators. ThenaivenumberoperatorOˆJ†OˆJ To compare these to the gauge theory calculations, we
turns outtobe inappropriate;we willseethatthis isbe- “embed” these results in the AdS space. A priori, there
cause it only behaves as a counting operator when both is no reason to expect that the flat space results of the
N and the number of particles p in the state satis- decay should be valid for branes in an AdS background.
fies→p∞ N. Therefore, in order to count the number of However, since the D-particles in question are fully lo-
particl≪es corresponding to an operator Oˆ , one needs to calised in the bulk space, one expects that the flat space
J
calculatetheprobabilities forfindingap-particlestate results should carryover,at leastwhen the radius of the
p
individually. Since Oˆ parPticles can appear in combina- AdS is large.3 There are two properties of the spectrum
J
ofthedecayingbranethatwecancomparewiththedual
tion with any arbitrary other (multi-particle) operator
Oˆ , the expression for finding a p-particle Oˆ state is gauge theory calculation. The first property of the spec-
K J trumisconstrainedbythesymmetriesofthesystem,and
(Oˆ )pOˆ c 2 concerns emission amplitudes for the states on the lead-
J K
ing Regge trajectory. By slightly refining the calculation
= . (1)
Pp XOˆK (OˆJ(cid:12)(cid:12)(cid:12))DpOˆK (OˆJ)(cid:12)(cid:12)(cid:12)pOˆEK(cid:12)(cid:12)(cid:12) hc|ci ofofr[9t]hiensesescttaiotensIaIIreBzwereofi.nTdhtehastamalelermesiussltioinsatmhepnlitsuedpeas-
D (cid:12) E
To see why computing (1) is h(cid:12)ard, consider the simplest rately recovered on the gauge theory side by evaluating
(cid:12)
terms inthe sum,whenOˆ isjust the identity operator. thenumberoperatorforthe correspondingdualcompos-
K
This term is ite operators.
2 2
(Oˆ )p c (O )p 0c More important is a second property of the spectrum,
J J
h | i
= , (2) observedin[9],whichreflectsgenuinedynamicalfeatures
(Oˆ(cid:12)(cid:12)(cid:12)JD)p (OˆJ(cid:12)(cid:12)(cid:12))pE(cid:12)(cid:12)(cid:12)hc|ci p! (cid:12)(cid:12)(cid:12)1+ b(Np,2J) +(cid:12)(cid:12)(cid:12)... ofthedecay. Thereisstrongevidence[1,9]thattheopen
D (cid:12) E (cid:16) (cid:17) stringsdecayfully intoclosedstringstates,i.e.thatthere
where OJ w(cid:12)ithout a hat denotes the (positive frequency is no open string remnant left after the decay. This con-
(cid:12)
part of the) classical expectation value of the opera-
clusion is also supported by the matrix model calcula-
torOˆJ. Whenb N,theexpression(1)canbesummed, tions of [2]. As shown in [9], the emission amplitudes
≪
yieldingtheresultonewouldobtainusingthenaivenum- are exponentially suppressed with the level of the emit-
ber operator. However, as indicated, the coefficient b in ted string, at least for high levels (however, due to the
the denominator depends on p and J, and for large p it exponential growth of the available states, most of the
becomes comparable to N2. This invalidates the large- energy of the brane gets transferred into a high-density
N approximation for the sum. Closer analysis of our cloud of very massive closed string states). By studying
coherent state shows that, due to the non-perturbative the dual gauge model we discover the same qualitative
character of the initial gauge configuration, the classical feature: asuppression ofhigher-massstringstatesinthe
expectation values of the operator OˆJ in the coherent decay product.
stateareverylarge. Themaximaltermin(1)isattained
for large p, which grows so fast with N that one cannot
neglect1/N2 andhigherordercorrectionsinthe denom-
inator. Moreover,summingallplanarcontributionsdoes 3 Tobeprecise,theD-particledualtothesphaleronislocalisedin
not yield a good approximation either. theAdSpartofspace-time,whileitissmearedontheS5.
3
II. DECAYING SPHALERONS IN ADS/CFT reduces for our ansatz to the desired form (4) with the
functions g and h given by
A. Classical instability of the sphaleron on S3×R 3
g(t)= R2f˙2 6f2(1 f)2,
−2 − −
(7)
The first step in our analysis is to give a detailed de- 1 g(t)
h(t)= R2f˙2 2f2(1 f)2 = .
scription of the decaying sphaleron on the gauge the-
−2 − − − 3
ory side. Whereas the sphaleron solution on R4 found
To deduce what is the unknown function f(t) we plug
by Klinkhamer andManton[10]is verycomplicated and
the ansatz into the action and derive the action for this
not known analytically, the situation is much simpler on
S3 R. Not only is the solution known in this case, but function. The value of the action for our ansatz is
one×can also find an analytic description of the classical S = 1 dtdΩF Fµν
decay of this metastable state. −4g2 µν
YM Z
Following Drukker et al. [5], one can get a sphaleron (8)
solutiononS3 Rbystartingfromtheinstantonsolution = 24 vol(S3) dt R2f˙2 2f2(1 f)2 ,
on R4. The lat×ter is given by 4g2 R 2 − −
YM Z (cid:18) (cid:19)
xµσ where vol(S3) 2π2 denotes the volume of the unit
Aµ =f(r)(∂µU)U†, U = r µ, r2 =x20+x2i , sphere and R is≡the radius of S3 (also see (73), (77)).
The equation of motion for the function f is
(3)
where f = r2/(r2 +a2). This function interpolates be- R2f¨+4f(1 f)(1 2f)=0. (9)
tween two pure gauge configurations(i.e. the two vacua) − −
f(r = 0) = 0 and f(r = ) = 1. When f(r) = 1/2, This equation can be integrated once, yielding a con-
∞
the system is at the top of the potential barrier. By served quantity, namely the energy (i.e. the component
taking f = 1/2 everywhere one gets a singular solution T00 =48vol(S3)E)
to the equation of motion on R4, which is the so-called
E =R2f˙2+4f2(1 f)2. (10)
“meron”. The f = 1/2 solution is, however, also a solu- −
tion on S3 R, since this manifold can be conformally By introducing a new variable H(t)
mapped to×R4 and Yang-Mills theory in four dimensions
1
is conformally invariant. The solution obtained in this f(t)= (1+H(t)), (11)
way is the Euclidean version of the “sphaleron”, and is 2
non-singular.4 The Lorentzian versionis the same, since the expression for energy becomes
the time component of the potential of the sphaleron is
4E =R2H˙2+(1 H2)2, (12)
zero; the solution is completely time-independent. −
We want to study the decay of the sphaleron, and we which can be further integrated analytically for E = 1.
4
restrict to those modes which preserve the spatial ho- There are two solutions, corresponding to the fact that
mogeneity of the initial sphaleron configuration. This is the sphaleron can roll down on either side of the poten-
because we want to look at the decay of the D-particle tial, to the vacua with Chern-Simons number one and
which sits at the “origin” of the anti-de-Sitter space and zero respectively. The final result reads (see figure 1)
is projected in the same way to all points on the bound-
ary. In other words, we only allow for time dependence, 1 √2
so that energy-momentum tensor should be of the form f (t)= ± +1 . (13)
± 2cosh √2(t t ) 
R − 0
T00 =g(t), Tij =h(t)gij, (4) Onecancheckthattheseso(cid:16)lutionsarei(cid:17)ndeedsolutionsto
thefullequationsofmotion,notjusttotheequationsob-
where g is the metric on S3. So the ansatz we make is
ij tainedfromthereducedaction. Thesesolutionsdescribe
a configuration that starts from the potential maximum
A=f(t)Σiσ , (5)
i at t = (with zero velocity and acceleration), rolls
−∞
down one of the two sides of the hill and up the other
where Σi are the three left-invariant one-forms. The en-
side, where it arrives at t = t . The minimum of the
ergy momentum tensor, 0
potentialenergy(12)is reachedwhenH2 =1 whichcor-
1 respondstot t0 = Rarccosh(√2)/√2 0.62R;the
Tµν =Tr(FµρFνσgρσ)− 4gµνTr(F2) (6) evolution is sy−mmetr±ic around t=t0.5 ≈±
4 The singularity of a meron originates from the region r → 0, 5 After we had derived this solution, we learned that it has been
since the action behaves as S ∼ drr−1. After the conformal obtained before by Gibbons and Steif [11] and Volkov [12, 13],
transformationtheactiondensityreducestoaconstant. albeitinadifferentcontext.
R
4
f(t) B. Coherent state description of the sphaleron
1+√2 decay
2
In order to performan analysis of the spectrum of the
decayinthegaugetheory,asafirststeponeneedstocon-
structaquantumstatedescribingthedecayedsphaleron.
1 Forthatpurpose,itisusefultothinkaboutthesphaleron
2 Epot Ekin
decayinthefollowingway. Nearthetopofthepotential,
mostof the energy of the (perturbed) sphaleronconfigu-
rationisstoredinthepotentialenergy,whicharisesfrom
the non-linear terms in the Yang-Mills action. Precisely
t t t t
− b 0 b →∞ these non-linearitiesinthe actionensurethe existenceof
the sphaleron solution. However, as the sphaleron de-
FIG. 1: The functions f (t) of the decaying sphaleron on
± cays, the potential energy of the sphaleron gets trans-
S3 as given in (13), together with the kinetic and potential
ferred into kinetic energy, and near the bottom of the
energy (with normalisation as given in (12) and R=1).
potential valley all of the energy of the configuration is
stored in the kinetic energy. This can be seen most eas-
ily by performing a finite gauge transformation (78) on
the solution (5) with gauge parameter Λ = U .6 The
†
solution then reduces to
The periodicity of the whole process is natural from
theAdSperspective. SinceAdSeffectivelyactsasabox, Aµ =f˜(t)U†(∂µU), f˜=f 1. (15)
the cloud of outgoing radiationis refocusedto the origin −
of the space, where it arrivesas fine-tuned radiationand Near the bottom of the potential f˜ 0, which means
“re-builds” the D-particle. In this sense the D-particle thatthe derivativepartofthe fieldstr≈ength,ratherthan
never decays, since there is no real dissipation of the en- the non-linear (commutator) part, is dominant. The so-
ergy in the system. However, in the limit of large AdS lution becomes solution of the free Yang-Mills equations
radius, our flat-space intuition should (at least approxi- of motion on S3 R (written in the radiation gauge:
mately) hold. A natural point in time, which should be A = Ai =0) ×
0 i
associated to the decayed brane, is the point where the ∇
sphaleron has rolled down to the the bottom of the po- 1
∂2+ 2 2 Alin. =0. (16)
tential, i.e. when allpotentialenergy has been converted − t R2 ∇S3 − i
to kinetic energy. (cid:16) (cid:0) (cid:1)(cid:17)
Indeed, one can easily see that as t t the solu-
bottom
→
The previous constructioncan easily be generalisedto tion (15) with f given by (13) is very well approximated
describe the decay of a system of coincident D-particles. by the following solution of the linearised equation of
The relevant sphaleron configurations have been given motion (16):
by Drukker et al. [5]. They are obtained by replacing
1 2(t t )
tahcecoPrdauinligmtoatricesin(3)withCliffordalgebragenerators Aliin. =−4sin −Rbottom U†(∂iU). (17)
(cid:18) (cid:19)
Hencenearthebottomofthevalley,onecanthinkabout
the Yang-Mills configuration as dual to a coherent state
σ 0 0
µ ofnon-interactingclosedstringstateswhicharetheprod-
···
0 σ 0
µ uct of the D-particle decay. Our goal will then be to de-
σµ →γµ = ... ... ·.·.·. 0 . (14) termine numbers of various (gravity) “particles” in this
  finalcoherentstate. Whatwepreciselymeanbythiswill
 0 0 σµ
 ···  be explained in the next section. Let us first construct
 
this coherent state.
The fact that our solution abelianises near the bot-
This will make the various sphalerons sit in mutually
tom of the potential valley allows us to write down a
commuting SU(2) factors within U(N). In this case (9)
coherent state for this configuration (see Cahill [14] and
gets replacedby anindependent equationfor eachofthe
Pottinger [15] for related work). In order to do so, we
functions f , and the solutions of those are givenby (13)
i needanexpansionofthe freeYang-Millsgaugepotential
which can differ only by the value of t . In what follows
0
we will restrict to the situations in which all these ini-
tial “phases” are the same, i.e. in which all D-particles
starttodecayatthe sametime. Sincethefieldstrengths
6 Alternatively, this can be seen from the equation (9) in which
will also be block-diagonal,traces of powers of them will
the coupling g is restored; the g → 0 limit leads to the same
decompose as sums of traces of the individual blocks. equation ofmotionaslinearisationoff aroundf =1.
5
in spherical vector harmonics. In the radiation gauge, Aˆfree = 0 holds as an operator equation. Hence the
∇·
anexpansionis givenby (we refer the readerto Hamada coherent state (22) is a legitimate state in the Hilbert
and Horata [16] for more on spherical harmonics) space of the free theory.
Nevertheless,thestate(22)doesnotyetprovideagood
e (2J+1)τ/R description of the system, as it is constructed using the
Aaib = aˆaJbMy − YJiMy(φ,θ,ψ) oscillators of the free theory and does not allow for a
2(2J +1)
JX,y,M(cid:16) smooth deformation to the interacting theory. At finite
p coupling, it does not satisfy the global colour neutrality
e(2J+1)τ/R
+aˆ†JaMby YJ∗Mi y(φ,θ,ψ) , (18) constraint. This constraint arises because the commu-
2(2J+1) tator term in the Gauss law acts as a source, and by
(cid:17)
integratingthe constraintoverthe S3 one finds that this
p
whereM =(m,m)labelstherepresentationsofthesim-
′ total charge has to vanish. One therefore imposes that
ple factors of SO(4)=SU(2) SU(2) and y sums over the
the commutator part of the non-abelian constraint van-
×
physical polarisation states of the gauge field. We have
ishes also at zero coupling [18]. This constraint restricts
alsointroducedmatrixindicesfortheadjointrepresenta-
the coherent state to the colour singlet part,
tionofthe U(N) Lie-algebra. The otherindices runover
the ranges c = c . (24)
singlet singlet
| i P | i
J 1, J y m J +y, In practise, however, we will neither write this projector
≥ 2 − − ≤ ≤ (19)
y = 1, J +y m′ J y. nor construct the projected state explicitly. This is be-
±2 − ≤ ≤ − cause our calculations always involve projections of the
After quantisation, the operators aˆ†lmn and aˆlmn satisfy coherent state onto states which themselves are colour
the canonical commutation relations singlets. Therefore the singlet projection is imposed im-
plicitly throughout. The only thing which we have to
aˆaJbMy,aˆ†Jc′dM′y′ =gY2MδJJ′δMM′δyy′δadδbc. (20) keepinmindisthatwhenthestate|ciisunitnormalised,
the norm of c is much smaller than one; we will
singlet
h i | i
return to this issue in section IIIC when we discuss the
A coherentstate (see Klauder and Skagerstam[17] for
decomposition of the coherent state in a specific exam-
more on coherent states and references to the literature)
ple.7
correspondingtotheclassicalconfigurationgivenby(13)
is constructed by demanding that
aˆab c =Aab c , (21) C. Particles in the AdS/CFT correspondence
JMy| i JMy| i
whereAJMy arethe coefficientsappearingintheFourier In the AdS/CFT correspondence, we have a relation
decompositionoftheclassicalsphaleronconfiguration,as between string states in the bulk and operators in the
in(18). IntheCoulombgaugeAab =0thecoherentstate boundary. These operators are, via the operator–state
0
can be written as mapping, interpreted to create “particles” in the bulk
theory at a particular point on the boundary. That is,
oneneeds to solvefor the waveequationofthe dualfield
|ci=C expgY−M2 Tr AJMyaˆ†JMy |0i, (22) in the bulk in the presence of a delta source inserted at
JX,M,y (cid:0) (cid:1) the boundary. This means that the states created in the
  bulk are noteigen momentum states, anattribute which
The normalisation factor is chosen such that c is of
C | i one usually associates to the notion of a particle in field
unit norm and is given by
C =exp − 21gY−M2 Tr |AJMy|2 . (23)
(cid:16) (cid:0)JXMy (cid:1)(cid:17) 7 Weshouldalsoremarkthatinadditiontotheconfiguration(14)
used in the construction of the coherent state |ci, there exists
A similar construction for the Klinkhamer-Manton a whole familyof inequivalent configurations related to (14) by
sphaleron in Yang-Mills-Higgs theory on R(3,1) has been largegaugerotations. Sincetheparametersofthisfamilydonot
discussedbyZadrozny[7]. Thestate(22)ismostnatural have a counterpart on the gravity side, one needs to integrate
over these configurations when calculating observables on both
from the point of view of the gauge theory; we will dis-
sides. Asimilarsituationoccurswhenonecalculatescorrelation
cussthepossibilityofusingalternativecoherentstatesat
functions inanSU(2) instanton background: whilethesizeand
the end of section IIC. position of the instanton do have an AdS interpretation, the
It is important to note that, due to the properties parameters describing the embedding of the SU(2) instanton in
of the vector spherical harmonics in (18), the coherent SU(N) do not, and thus have tobe integrated over. For allthe
observables we will be calculating from |ci, these integrations
state(22)hasbeenbuiltfromcreationoperatorsthatex-
leadto additional overall groupfactors, whichareirrelevantfor
citeonlyphysicalexcitations: theGausslawconstraintis
ourpurposes. Henceinwhatfollows,wewillignorethistechnical
automatically implemented using these operators, since subtlety.
6
theories. However, since the AdS/CFT correspondence Itisonlyaftertheintegration(27)thatoneobtainsaset
is formulated in position space rather than momentum of orthogonal states. See appendix VA for an explicit
space, these definitions are natural in this context. Our example on S2.
string calculation,onthe otherhand, will be a flatspace Multi-particlestatesareobtainedbyactingrepeatedly
calculation, and for us it will be more natural to use with the Oˆ operators on the vacuum, in analogy with
w
the standard notion of particles in the bulk as angular normal creation operators for elementary particles. In
momentum eigenstates. For that however, we will first contrast to elementary operators, however, there is no
have to construct boundary operators that are dual to simple number operator which can be used to count the
the bulk angular momentum eigenstates. number of composite excitations in a given state. It is
Theoperator–statecorrespondenceisusuallydiscussed true that
in the context of radial quantisation of conformal field
theories (see e.g. Fubini et al. [19] for a discussion in a [Oˆ,Oˆ†]=1+ (N−2), (28)
O
four-dimensionalcontext). OnefirstWickrotatesR R3
× and one might expect that this leads to a well-defined
to the Euclidean regime and then performs a conformal number operator Oˆ Oˆ. However, the coefficients that
transformationsuchthatthe originofR4 correspondsto †
multiply the 1/N2 corrections in (28) are operators, not
t= in the originalframe. Operators inserted at the
−∞ c-numbers. As a consequence, the strength of the 1/N2
origin are then in one-to-one correspondence with states
corrections depends on the state in which the number
in the Hilbert space. The entire procedure can, how-
operator is evaluated,
ever, be formulated without doing the conformal rescal-
ing, which is more naturalin our setup since, as we have c (n)
discussedbefore, the gaugefield configurationonR S3 hn|Oˆ†Oˆ|ni=n+ Ni 2i . (29)
is non-singular while the one on R4 is singular. × Xi
Thestatecorrespondingtoanoperatorwithconformal The numbers c (n) can become arbitrarily large
i
weightwisobtainedbymultiplyingwiththeappropriate when n . Since the coherent state contains such
exponential of Euclidean time and taking the limit τ highly ex→cite∞d states, the operator Oˆ Oˆ cannot be used
→ †
(keeping only the regular part): as a number operator, not even in the N limit.8
−∞ → ∞
We will encounter an explicit manifestation of this fact
|Oˆw(mei)ght-wi=τlim e−wτOˆw(mei)ght-w(τ) 0 in section IIIC, see in particular figure 3.
→−∞ n o(cid:12) (cid:11) (25) We will therefore follow a different route. Instead of
Oˆ(m) 0 . (cid:12) trying to use a number counting operator applied to the
≡ weight-w| i coherent state, we will simply project the coherent state
Thelastexpressionshowstheshorthandnotationthatwe oneachstateintheHilbertspaceofmulti-particlestates.
will use in ordernot to clutter expressionsunnecessarily. Subsequently, using these probabilities, we will calculate
The hermitian conjugate of an operator is given by theaverageenergiesandparticlenumbers. Detailsofthis
procedure will be discussed in the section IIIC.
Letusendthissectionwithacommentonalternatives
Oˆ(τ) † =Oˆ ( τ). (26)
† − to the coherentstate (22). Fromthe point ofview of the
(cid:16) (cid:17) dual string theory, it might seem more natural to con-
bTuhtisapvrooidcesdtuercehnmicimalicpsrtohbeleompserraetloart–edstatotesmolauptipoinnsgwonhiRch4 structacoherentstateusingthecompositeoperatorsOˆJ†
inthe exponent,ratherthanthe elementaryonesaˆ . Af-
†
become singular after the conformal transformation. ter all, the Oˆ correspond to elementary string excita-
Theoperatorswhichweusein(25)areindependentof J
tions. However, a state of the form
the angular coordinates on the sphere, i.e. they are ob-
tained from the position dependent operators as follows |c˜i=C˜exp Oiclass.Oˆi† |0i (30)
(cid:16)Xi (cid:17)
Oˆ(m)(τ)=K(m) dΩ Oˆµ1...µs(τ,φ )Y(m) (φ ). (27) is notacoherentstatein the sense of(21). The expecta-
w w w i µ1...µs i
ZS3 tion value of an operator in this coherent state does not
equal the classical value of that operator,
Here Y(m) denote the lowest lying tensor spherical har-
monics for a given spin s. The index m labels the de- c˜ Oˆ c˜ =Oclass., (31)
i 6 i
generacy of such harmonics. The normalisation con-
stants Kw(m) are chosen such that the states constructed (cid:10) (cid:12)(cid:12) (cid:12)(cid:12) (cid:11)
using (25) are of unit norm. Note that the multiplica-
tion with the time dependent exponent in (25) selects 8 An proper number operator for composite particles, which pro-
outcompositeoperatorsoftherequiredconformaldimen- duces the exact occupation number rather than an expression
whichisonlycorrectuptoN−2corrections,hasbeenconstructed
sion, but when one expresses these operators in terms of
byBrittinandSakakura[20,21]. However,theiroperatorisvery
elementary creation and annihilation operators, one ex-
complicatedanddifficulttohandleinpractise. Weprefertofol-
plicitlyseesthatdifferentoperatorsOˆarenotorthogonal. lowadifferentroutehere.
7
not even up to 1/N corrections. The reason for this is type Oˆ present in the coherent state is now given by
Ji
essentially given in equation (29), with n now being
tgoivuenseb(y22|)nias=theOˆsi†phna|0leir.onThcoisheisreonutrstpartiem|.ei motivation N(Ji):= ∞ ∞ pi (p1;p2;...;pM). (34)
(cid:0) (cid:1) ··· P
pX1=0 pXM=0
The energy stored in these particles, as measured with
III. THE DECAY SPECTRUM
respecttotheglobaltimeinthebulk,isgivenbythecon-
formaldimensionofthecorrespondingoperators. There-
A. Counting procedure and symmetry fore, the total energy is given by the expression
considerations
∞ ∞
Having constructed a coherent state in the gauge the- E(Ji):= ··· ∆JipiP(p1;p2;...;pM), (35)
ory which is dual to the final state of the D-particle de- pX1=0 pXM=0
cay(seesectionIIB),wenowwanttoextractinformation
fromitaboutparticlenumbersinthedecayproduct. By where ∆ is the conformal dimension of the operator
Ji
particle counting, we mean counting of the states con- Oˆ . This is actually why we interpret Oˆ Oˆ 0 as
structed in the previous section. The main subtlety for aJmiulti-particle state: the supergravity ener†g·y··is†s|imiply
this calculationlies,aswe havediscussedinthe previous the sumofthe constituentenergies,despite the factthat
section, in the fact that we want to count states created the norm does not factorise as the product of individual
by composite rather than elementary operators. In this particle norms.
section we will outline the general procedure which we
Due to the general properties of the coherent state,
will use to calculate these numbers, and then apply it to
an evaluation of the numerators in (32) is straightfor-
a special class of operators whose behaviour seems to be
ward. It amounts to evaluating the classical expressions
fully determined by the symmetries of the problem.
for the (multi-)particle operators using the positive fre-
The basic ingredient in our particle counting proce- quency part A+ of the rolling sphaleron solution, near
dure is the probability to find a particular multi-particle the bottom of the potential. When doing this calcula-
state of composite particles in the coherent state. The tiononealsoneeds to use formula(27); i.e. foreachpar-
probability of finding a multi-particle state consisting of ticles in the state separately,one needs to removethe eτ
p1 particles of type OJ1, p2 particles of type OJ2 etc., is factors and then project onto the corresponding lowest-
given by lying harmonics. Hence, even though we know the full
timedependentsphaleronsolution(5)–(13),weneedonly
the part of the solution at the end of the decay for the
(p ;p ;...;p ):=
1 2 M
P evaluation of (32).
(Oˆ )p1...(Oˆ )pM c 2 Sincewearelookingataverysimple,sphericallysym-
J1 JM . metric decay, the final phase of the decay is very much
OˆJ1 p1..(cid:12)(cid:12)(cid:12).DOˆJM pM OˆJ1 p1...(cid:12)(cid:12)(cid:12) OEˆJ(cid:12)(cid:12)(cid:12)M pM c c constrained and is basically independent of the shape of
the potential: it is given by an S-wave on the three-
D(cid:0) (cid:1) (cid:0) (cid:1) (cid:12)(cid:12)(cid:0) (cid:1) (cid:0) (cid:1) E(cid:10) (cid:12) (cid:11)(32) sphere, with an amplitude determined by the height of
(cid:12) (cid:12)
thepotential. Wehavegiventhissolutionin(17),andthe
For this to work it is of course crucial that the basis Euclidean version of its positive-frequency part is given
of multi-particle states is constructed to be orthogonal. by
9 By definition, the average number of particles of the
i 2
A+ = exp t t U (∂ U) f+(t)U (∂ U).
i 8 R − bottom † i ≡ † i
(cid:16) (cid:0) (cid:1)(cid:17) (36)
9 Even ifone has two orthogonal states O1†|0iandO2†|0icreated As we will see in the next section, the real problem
usingcompositecreationoperators,itisgenericallynottruethat
in calculating the numbers of different states is related
h0|(O1)n(O2†)n|0i=0. The notation used in (32) is meant to to the calculationof the denominators in (32). However,
indicate that proper subtraction terms are included, such as to
make all states appearing in the sum orthogonal. Using these for the class of operators which are absent from the de-
orthogonalstates,theprojectionoperatorappearingin(24)takes cay spectrum, i.e. for which numerators in (32) vanish,
theform one does not need to worry about this issue. The first
operatorinthisclassistheenergymomentumtensor. Its
Psinglet=1+ 1/Np1,p2,···pn
vanishing implies thatthere is no gravitationalradiation
{p1,pX2,...pn}
inthebulk,afeaturewhichisexpectedfromthesymme-
× (O1)p1(O2)p2...(On)pn (O1)p1(O2)p2...(On)pn , (33) tryoftheproblem. Namely,sincethedecayisspherically
whe(cid:12)(cid:12)retheNp1,p2,...pn areth(cid:11)e(cid:10)normsofthestates. (cid:12)(cid:12) symmetric,there areno quadrupolemoments turnedon,
8
and hence no gravitationalradiation can be produced.10 calculation,ratherthanoneinAdS,whenmakingacom-
The lowest-massSO(6) singlet that arises from the S5 parison to the gauge theory results.
reduction of the NS-NS two form is given by [22, 23] In order to analyse the decay products of an unstable
D-brane in string theory, one has to solve for the closed-
Tr 1F FαβF + 1F FαβF . (37) string field Ψc in the presence of a time-dependent
µν να βµ αβ µν | i
O ∼ 2 8 brane source,
(cid:18) (cid:19)
The associated state also has a vanishing overlap with (Q+Q¯)Ψ = B . (40)
c
thesphaleroncoherentstate. Thisisduetothefactthat | i | i
thisoperatoriscubicinthefieldstrength,andourgauge
Here B is the boundary state for the unstable D-brane
potentials are “abelianised” SU(N) fields, as explained while|Qiand Q¯ are the BRS operators. The solution
in section IIB.
for Ψ as well as the late-time behaviour was analysed
Inthemassivestringsector,wealsofindthattheradi- in [9|, i24]. For the final state of the decaying D-particle
ation associatedto all twist-two fields vanishes. Namely,
it takes the simple form
N Oˆµ1···µs =0, for all s, (38) |Ψ∞c i:=tl→im∞|Ψci∝
∞ 1
where the opera(cid:0)tors are g(cid:1)iven by d25k exp α0 α¯0 +αi αi
⊥ −n −n −n −n −n
Z LX≥0 hnX=1
Oˆµ1···µs = ghosts (41)
− levelL
vol(2S(3g)2−1NR2)+2s :Tr Fν(µ1∇µ2...∇µs−1Fµs)ν : f(L,k ) k0 =ωk,k ,k =0 i(cid:12)(cid:12)(cid:12)
YM × ⊥ ⊥ k
p (cid:0) (traces)(cid:1). (39) (cid:16) +f∗((cid:12)(cid:12)L,k ) k0 = ωk,k(cid:11),k =0 .
− ⊥ − ⊥ k
(cid:12) (cid:11)(cid:17)
Here the s = 2 operator corresponds to the graviton. HereLdenotestheoscilla(cid:12)torlevelandf(L,k )isafunc-
We will see in section IIIB that all these results can be tion dependent on the level and the transve⊥rse momen-
matched with the string theory prediction. tum.11 This final state can now be projected onto on-
We believe that the vanishing of the amplitudes of
shell closed string states.
the twist-two operators is related to the symmetries of
Thetwist-twostateswhichweareinterestedinareas-
the system, rather than to genuine dynamical proper-
sociated with vertex operators which, for a given level,
ties. Hence, in order to gain insight in real dynamics of
carry maximal spin. This is achieved by using the max-
the problem, we need to consider number operators for
imal number of creation operators for fixed level, i.e. by
generic states, which we will do in section IIIC. building the state using only αµ operators. More pre-
1
cisely, the particle numbers in t−he final state will be de-
termined from the following overlaps
B. Expectations from the string side
Sµ1...µnν1...νn = kµ αµ1...αµnα¯ν1...α¯νn Ψ , (42)
1 1 1 1 ∞c
Thelackofknowledgeaboutstringquantisationonthe
AdS5×S5 backgroundmakesadirectstudyofD-particle where kµ is the m(cid:10)om(cid:12)(cid:12)entum of the centre of(cid:12)(cid:12)mass(cid:11)of the
decay in this background impossible. However, because closed string, related to the level n via the mass shell
theD-particleisafullylocalisedobject,oneexpectsthat conditionk2 =2(1 n). The twist-twostates are associ-
its static and dynamic properties are, at least for large ated with the part−of these vertex operators which have
radii, similar to those of the D-particle in flat space [5]. maximalspin, that is, they are built out from the vertex
We will employ this argument to use a flat-space string operators by contracting them with polarisation tensors
satisfying
ǫ =ǫ , ηµ1ν1ǫ =0.
10 Note that the expression which vanishes is the energy momen- µ1...µnν1···νn (µ1...µnν1···νn) µ1...µnν1···νn (43)
tumtensorevaluatedonthepositivefrequencypartofthesolu-
tion: |h0|Tˆµν|ci|2 = |Tµν(A+coherent)|2 = 0. On the other hand,
the classical expression for the energy momentum tensor of the
fullconfigurationisnon-zero:Tµν(A++A−)6=0. Notealsothat
the Tˆµν which is used here is the abelian expression for the en- 11 Thereisasubtletyconcerningtermsintheoutgoingstatewhich
ergymomentumtensor,sinceallourcalculationsaredoneinthe growexponentiallyintime. Theirinterpretationisatpresentnot
freetheory. Itwouldbeinterestingtoextendtheaboveanalysis entirely clear [24]. Moreover, there exists an alternative deriva-
toincludeinteractions. Inthatcase,however,theremaybenon- tion in which such terms are not present [25]. We do not want
trivialcorrections tothe coherent state, andboth gauge bosons to go into a discussion of this issue here, and consider only the
andscalarswillcontribute tothenumeratorsin(32). termswhicharefiniteatlatetimes.
9
For these polarisation tensors, it is easy to see that the tors which create particles in the out vacuum are
projections (42) vanish. Since only the n = 1 oscillators
1
atbipevcpeoleymarereipndrutohcpeeostrwttoiiosnηt-aµtlνwtαooµ−st1trα¯aaν−tcee1ss.,otAfhtellheepxrppooojlneacertniistoanotisfo(n(441t2e))nestffoherecsn-, OˆJ† = J(gY2MN)J Tr(cid:0)(aˆ†)J(cid:1). (45)
q
whichvanishby(43). Thereis thereforenotwist-twora- These operators are coordinate independent operators,
diation; in particular, there is no gravitational radiation obtained using a procedure similar to (27).
(which is expected because there is no quadrupole mo- With the above normalisationof the operator,the nu-
ment). The radiation into NS-NS two-form states also merators and hence probabilities in (32) depend on the
vanishes, because the polarisation tensor is in this case Yang-Mills coupling in a non-perturbative fashion,
anti-symmetric.
2p
sseolrAec(ltli.eot.fhwethhheeisngehocenosnet-ssdipdoieenrsasnttioaottnesimfcrhpoaomnsegte(h4we3h)p)eo.nlaIonrnispeaatdirootniecsutnleanort-, (cid:12)(cid:12)h0|(cid:0)OˆJ(cid:1)p|ci(cid:12)(cid:12)2 =C2(cid:12)(cid:12)(cid:12) TJr((cid:0)g(Y2aM+)NJ(cid:1))J(cid:12)(cid:12)(cid:12) ≡ JC2p (cid:18)ληJ2J(cid:19)p ,
the dilatonradiationwillnotvanish. These observations (cid:12) (cid:12) (cid:12)q (cid:12) (46)
(cid:12) (cid:12)
matchthe calculationsonthe Yang-Mills side performed (where the last equal(cid:12)ity defines η ; n(cid:12)ote that it is of the
J
in the previous section.12 order N for the configuration(14) and generically scales
asthe number of D-particles). This reflects the fact that
ouroriginalsphaleronconfigurationisanon-perturbative
solution of the equations of motion. Note also that the
C. Decay products in U(4) – dynamical onlywayinwhichthecouplingλappearsin(34)and(35)
considerations is through the combination η2/λJ.
J
The complicatedpartof the calculationofthe average
For a generic operator, the calculation of the numera- particle numbers and energies is the computation of the
tors in (32) is the same as in section IIIA, and amounts norms for the states with an arbitrary number of parti-
toevaluatingthe classicalexpressionofthe (abelianised) cles. The normofthe state withp identicalparticlescan
operatorusingthepositivefrequencypartofthedecayed be written as (see also figure 2)
solution. The main technical problem arises when eval-
uating the denominators of (32). To illustrate this, let (OˆJ)p(OˆJ†)p =p! (OˆJ)(OˆJ†) p
uscsaclaornsfiiedledr.aT“hsiismmploidfieeld”exmhiobditesl,abllasoefdthoentaecnhonni-caablesluiabn- (cid:10)+ p2 2 (OˆJ(cid:11))2(OˆJ†(cid:10))2 conn.(p−(cid:11)2)! (OˆJ)(OˆJ†) (p−2)
tletiesassociatedtothedeterminationofthedecayprod- (cid:18) (cid:19)
ucts. Thecrucialingredientsofthevectorcoherentstate, p 2(cid:10) (cid:11) (cid:10) (cid:11)
namelythatitisconstructedfromthelowest-lyingspher- + 3 hOˆJ3OˆJ†3iconn.(p−3)!hOˆJOˆJ†i(p−3)
icalharmonicsandthatitdependsnon-perturbativelyon (cid:18) (cid:19)
the coupling constant, are preserved by this toy model. p 2 p 2 2 (p 4)!
However,itavoidstheinessentialtechnicalcomplications + 2 −2 (OˆJ)2(OˆJ†)2 2conn. −2! hOˆJOˆJ†i(p−4)
associatedtotheevaluationoftensorsphericalharmonics (cid:18) (cid:19) (cid:18) (cid:19)
(cid:10) (cid:11)
in the numerators of (32). +...
The coherent state for a given classical configuration (47)
in this non-abelian scalar theory is given by The first term is at a leading order independent of 1/N,
thesecondissuppressedas1/N2,thelasttwotermsboth
scaleas1/N4,andsoon. Asimilarbutmorecomplicated
1
c = exp Tr aˆa 0 , expansion can be written for states involving more than
| i C g2 † | i
(cid:18) YM (cid:19) one type of particle.
(cid:0) (cid:1) (44)
1 Naively,onemightexpectthatinthelarge-N limit,all
C =exp −g2 Tr a†a . but the leading term p! in this expansion can be omit-
(cid:18) YM (cid:19) ted. In formula (34), this would produce an exponential
(cid:0) (cid:1)
dependence on the expectation values for the operators
This mimics the construction (22). The unit normalised Oˆ . Since the arguments of the exponent (46) increase
J
(at leading order in 1/N expansion), single-trace opera-
with conformal dimension J, one would conclude that
the number of particles produced during the decay in-
creases withthemassoftheparticle. However,thiskind
of truncation of (47) does not make sense in the case of
12 The conclusions also rely crucially on the fact that we are re- the non-perturbative coherent state (44), as it would ac-
strictingheretotheD-particlecase. Thetwist-tworadiationis,
tually produce probabilities (32) which are larger than
for other boundary states, generically no longer zero. The final
one. The point is that since the numerator (46) is very
state (41) will be more complicated, and the n = 1 part of the
exponentwillnotreducetoηµναµ−1α¯ν−1. large, the maximal probabilities are attained for large
10
valuespmax ofp. Moreover,pmax growswithN,hencein The measureused here is simply a separateintegralover
the large-N limit the sub-leading terms in (47) become the real and imaginary parts of the complex matrix A,
more and more relevant, and are actually comparable to normalised to give unit result when all p in the expres-
i
the leading term. sion above are zero,
In trying to estimate how fast the norms (47) have
to grow with p, one can see that even an exponential N
growth of the norms, say as p!γp (γ = const.), does not dAdA¯=π−N d(ReAab) d(ImAab). (50)
lead to reasonable results. Namely, if we consider the Z aY,b=1
expression (J,p),whichhastobesmallerthanone,
pP ThisapproachhasbeenusedbyKristjansenetal.[26,28]
and assume exponential growth of norms, we would find
P in order to compute several special cases of (49) analyt-
that this sum behaves as
ically. It is still an open problem to extend those exact
∞ ∞ 1 η2 p results to the entire class of correlators, in particular to
(J,p)= 2 J generalsituationsforwhichp >2. Becausewewillneed
P C p! λJγ i
p=0 p=0 (cid:18) (cid:19) these very general correlators, we have decided to use
X X (48)
an alternative approach, in which the integral is evalu-
η2 N
=exp J exp Tr(a a) . ated using Monte-Carlo methods. This provides us with
†
λJγ −λ a technically straightforward way to extract the norms
(cid:18) (cid:19) (cid:18) (cid:19)
for arbitrary operator insertions, even for very large p .
i
Hence we see that even when N (while keeping λ
Ourresultswill,forthisreason,ofcourseberestrictedto
→ ∞
arbitrarybut smaller than one) the result will always be
a fixed value for N, and computer resources put a prac-
larger than 1 for some value of J. Since the calculation
tical limit on the maximum value that can be handled.
of the averagenumber of particles requires a summation
Nevertheless, we will see that interesting results can be
over all J, we conclude that we cannot assume this be-
obtained this way.
haviour of the norms.13
Before we discuss the results, let us present numerical
The situation which we face here is similar in spirit evidence which illustrates the necessity of taking the full
to the double-scaling BMN limit. As observed by Krist- norms in(32)intoaccount,i.e.allplanarandnon-planar
jansen et al. [26] and Constable et al. [27], in the limit contributions. We comparethe results obtained by sum-
N J2 correlatorsingeneralreceivecontributions mingupalargeclassofplanardiagramsin(47)withthe
∼ →∞
fromnon-planargraphsofallgenera. Inthis case,anew numerical results obtained using the Monte-Carlo inte-
expansionparameterJ2/N appears. Inourcase,N gration. For the U(4) case, the Monte-Carlo results are
→∞
aswell,butnowtheadditionalparameterwhichbecomes depictedinfigure3andcomparedtothe analyticanswer
largeis the value ofthe pi for whichthe sum (35) hasits obtainedbyrestrictingtothefirsttwocolumnsofgraphs
maximum term. It would be interesting to understand in figure 2; these columns contain graphs with an arbi-
whether our system also exhibits a double-scaling limit trary number of connected four-blob elements. Clearly,
in which some ratio of powers of p and N is kept fixed. the full norms deviate substantially from this estimate.
In order to determine the correct values of the norms Hence, in the remainder we will only employ the norms
of the states, it is useful to write the norms in terms of obtained by Monte-Carlo integration of (49). Note also
correlators of a complex matrix model, from figure 2 that for largep, the deviation from the ex-
actnormgrowswithincreasingJ (anditalsogrowswith
0 OˆJ1 p1... OˆJn pn OˆJ†n pn... OˆJ†1 p1 0 = ienxcpreecats.ingN),anddoesnotimproveasonemightnaively
(cid:10) (cid:12)h(cid:0) (cid:1) (cid:0) (cid:1) ih(cid:0) (cid:1) (cid:0) (cid:1) i(cid:12) (cid:11) With the correct norms of the multi-particle states at
d(cid:12)AdA¯ OJ1 p1... OJn pn OJ†n pn... OJ†(cid:12)1 p1 hand, one now obtains sensible results for the sums (34)
Z h(cid:0) (cid:1) (cid:0) (cid:1) ih(cid:0) (cid:1) (cid:0) (cid:1) i and (35). An example in U(2) (which is rather trivial
exp Tr(A†A) , (49) becausethereisinthiscaseonlyoneindependentsingle-
× − trace operator) is plotted in figure 4. Note once more
(cid:16) (cid:17)
thatthenumbersplottedherearemuchsmallerthanone.
This is because the norm which multiplies all of these
C
resultsis the normofthe non-singletcoherentstate,and
13 Note that if we would have had a perturbative coherent state thenumberofsingletsinitismuchsmallerthanthetotal
instead of a non-perturbative one, the classical expectation val-
number of states. Note, however, that since the norm
uesain(44)wouldbeofthe forma=gYMη,withη anumber of the coherent state appears in all probabilities as an
independentofthecouplingconstant. Henceformula(48)would
bereplacedwith overall (identical) factor, its absolute value is irrelevant
when considering the ratios of emitted energies or ratios
∞ ∞ 1 η2 p η2
P(J,p)=C2 J =exp J exp −Tr(a†a) . of numbers of particles. See appendix VB for a more
pX=0 pX=0p! NJγ! NJγ! (cid:16) (cid:17) explicit explanation.
Having resolved the computation of the exact norms
Wenowseethatatruncationtothefirsttermin(47)(i.e.setting
γ=1)produces reasonableresultsfortheprobabilities(32). of states, we can now finally compute the energy distri-