More on half-wormholes and ensemble averages

Jia Tian and Yingyu Yang

Kavli Institute for Theoretical Sciences (KITS),University of Chinese Academy of Science,100190 Beijing,China

Abstract We continue our study Half-Wormholes and Ensemble Averages about the half-wormhole proposal.By generalizing the original proposal of the half-wormhole,we propose a new way to detect half-wormholes.The crucial idea is to decompose the observables into self-averaged sectors and non-self-averaged sectors.We find the contributions from different sectors have interesting statistics in the semi-classical limit.In particular,dominant sectors tend to condense and the condensation explains the emergence of half-wormholes and we expect that the appearance of condensation is a signal of possible bulk description.We also initiate the study of multi-linked half-wormholes using our approach.

Keywords: half-wormholes,factorization problem,quantum gravity,ensemble averages,SYK

1.Introduction

Recent progress in quantum gravity and black hole physics impresses on the fact that wormholes play important roles1For an up-to-date review,see [1]..Many evidences suggest an appealing conjectural duality between a bulk gravitation theory and an ensemble theory on the boundary[2–54].For example,the seminal work[2]shows that Jackiw-Teitelboim(JT)gravity is equivalent to a random matrix theory.On the other hand,this new conjectural duality is not compatible with our general belief about the AdS/CFT correspondence.A sharp tension is the puzzle of factorization[55,56].In [57],this puzzle is studied within a toy model introduced in[37],where they find that (approximate) factorization can be restored if other saddles which are called half-wormholes are included.Motived by this idea,in[58]a half-wormhole saddle is proposed in a 0-dimensional (0d) Sachdev–Ye–Kitaev (SYK)model,followed by further analyses in different models[59–66].

Another scenario where half-wormholes are crucial is in the example of the spectral form factor.Chaotic systems exhibit a linear ramp behavior in their spectral form factor at late times[67].In gravity theory,the smooth portion of the ramp can be explained by the emergence of a ‘wormhole’ saddle [68].The oscillation portion should also be captured by the gravity theory,and the half-wormhole saddle is a potential candidate.This phenomenon has been observed in other SYK-like models[66],even though their half-wormhole concept differs from ours[69].

In our previous works [44,69],we pointed out the connection between the gravity computation in [57] and the field theory computation in [58] and tested the half-wormhole proposal in various models.The main difficulty of this proposal is the construction of the half-wormhole saddles.Furthermore,the ansatz proposed in [58,62] seems to rely on the fact the ensemble is Gaussian with zero mean value.As a result,the 0d SYK model only has non-trivial cylinder wormhole amplitude.However for a generic gravity theory for example the JT gravity,disk and all kinds of wormhole amplitudes should exist.In our previous work[69],we find even turning on disk amplitude in 0d SYK model will change the half-wormhole ansatz dramatically.

In this work,we generalize the idea of [57] and propose a method of searching for half-wormhole saddles.In our proposal,the connection between[57]and[58]will manifest.One notable benefit of our approach is that it does not depend on the trick of introducing a resolution identity used in [58],the collective variables emerge automatically.More importantly,our proposal can be straightforwardly generalized to non-Gaussian ensemble theories.

2.Gaussian distribution or the CGS model

In [57],the main model is the Coleman and Giddings-Strominger (CGS) model.The CGS model is a toy model of describing spacetime wormholes and it is more suggestive to obtain it from the Marolf–Maxfield (MM) model [37] by restricting the sum over topologies to only include the disk and the cylinder [57].

Let the amplitudes of the disk and cylinder be μ and t2,i.e.

The crucial idea of[57]is that the correlation functions of partition function do not factorize in general but they factorize between α-states which are the eigenstates of

The α-state is also created by a generation operator acting on∣H〉H

Noting that 〈ψα〉=P(Zα),where P(Z) is the PDF of Z,we find that (8) coincides with the trick used in [69] and [62] of rewritingas a formal average

2.1.Half-Wormhole in CGS-like model

In the CGS model,because Z satisfies the Gaussian distribution there is a more concrete expression for the half wormhole saddle as shown in [57].The key point is the fact that when Z is Gaussian,it can be thought of as the position operator of a simple harmonic oscillator so there exists a natural orthogonal basis,the number basis{n}which is called the n-baby Universe basis in the context of the gravity model.If we insert the complete basis∑i∣i〉〈i∣into (9) we can get2Note that our convention is Z=μ+t(a+a?).

then (14) coincides with results in [69].So we confirm the result that within the Gaussian approximation (only keep the first two cumulants),can be decomposed as (14) and it suggests that θi’s are the convenient building blocks of possible half-wormhole saddles.Some examples of the decomposition (14) are3θ(i) is simply the (unnormalized) Hermite polynomial.

with

In general we have

Considering the following ensemble average

and expanding both sides into Taylor series of u1and u2one can find

In CGS model,since there is only a single random variable Z so it does not admit any approximation related to large N or small GN.Therefore the wormhole or half-wormhole are not true saddles in the usual sense.To breathe life into them we should consider a model with a large number N of random variables such as random matrix theory or SYK model which can be described by certain semi-classical collective variables like the G,∑in SYK,which potentially have a dual gravity description.However,we find that it is illustrative to first apply the factorization proposal to some simple statistical models as we did in [69].

2.2.Statistical model

Let us consider a function Y(Xi) of a large number N independent random variables Xi.Assuming that Xi’s are drawn from the Gaussian distribution then we have the decomposition

where Γkdenotes different sectors,in particular Γ0=〈Yn〉.This kind of model can be also thought of as the CGS model with species [57].

2.2.1.Simple observables.The simplest operator is

Apparently for n=1 there are only two sectors

and for n=2 there are three sectors

In general the parameters μ and t2are N independent therefore Ynis self-averaged Yn≈〈Yn〉in the large N limit.This is also true even Xiare not Gaussian because of the central limit theorem.But we also know in the literature that in order to have well-defined semi-classical approximation,the parameters μ and t2should depend on N in a certain way like in SYK model.Interestingly in this case if t2～μ2N,the selfaveraged part and non-self-averaged part are comparable and we should keep them both.This is exactly what we have encountered in the 0-SYK model.But a crucial difference is that for this simple choice of observables,all the non-selfaveraged sectors are also comparable so it is not fair to call any of them the half-wormhole saddle and to restorefactorization we have to include all the non-self-averaged sectors.The extremal case is t2?μ2N.In this limit we find that the sector with highest level dominates.For example,

Figure 1.Poisson distribution (44).(a) Poisson distribution with β2t2=100,(b) Poisson distribution with β2t2=0.01.

then it is reasonable to identify Θ1with half-wormhole and identify Φ2with the 2-linked half-wormhole.Similarly we can introduce n-linked half-wormholes.For example,in this extremal case,we can approximate Y3with

where the sector Λ3should describe the 3-linked halfwormhole.We will consider a similar construction in the 0-SYK model.

2.2.2.Exponential observables.In the Random Matrix Theory or quantum mechanics,the most relevant observable is the exponential operator Tr(eβH) since it relates to the partition function.So it may be interesting to consider a similar exponential operator

in the toy statistical model.By a Taylor expansion of the exponential operator we find the following decomposition

Interestingly the ratio rkfollows the Poisson distribution Pois(βt2),some examples are in figure 1.When βt ?1 the dominant sector is Θ0while for βt ? 1 the Poisson distribution approaches Gaussian distribution N(β2t2,β2t2)so we have to include all the sectors in the peak k ∈(β2t2-βt,β2t2+βt) to have a good approximation.We can decompose Y2in a similar way

The behavior is similar.When βt ?1,the dominant sector is the self-averaged sector Φ0.When 2β2t2＞log N(47)approaches the Gaussian N(4β2t2,4β2t2).On the other hand,when 1 ?2β2t2?log N (47) approaches the Gaussian N(2β2t2,2β2t2).In the end when 2β2t2～log N,(46)will have two comparable peaks,see figure 2 as an example.However the half-wormhole ansatz proposed in [62,69] which can be written as

only works for small value of βt.

To summarize our proposal,by introducing the basis{θi}which is the generalization of n-baby Universe basis [57] we can decompose the observables or partition functions into a single self-averaged sector and many non-self-averaged sectors.These sectors are independent in the sense of (28).The contributions from each sector have interesting statistics:in the large N limit leading contributing sectors may condense to peaks.This condensation is a signal that the observable potentially has a bulk description (or semi-classical description) in the large N limit.If the self-averaged sector survives then it means the observable is approximately self-averaging.The surviving non-self-averaged sectors in the large N limit are naturally interpreted as the (n-linked) half-wormholes which are the results of sector condensation.In the extremal case,only one non-self-averaging survives reminiscing the famous Bose–Einstein condensation.

2.3.0-SYK model

In this section we apply our proposal to the 0-SYK model which has the ‘a(chǎn)ction’

where we found in [69] in order to have a semi-classical description u should also have a proper dependence

We sometimes use the collective indies A,B to simplify the notation

Integrating out the Grassmann numbers directly gives4Here we choose the measure of Grassmann integral to be∫ dNψψ1…N=i-N/2.:

where the expression (54) is nothing but the hyperpfaffian Pf(J).According to (30),we can similarly decompose it as

thus by matching the power of θ(1)we get a integral expression of Θk

Next following [69] we can introduce G,∑variables directly as

The expression of 〈z2〉 is derived in [69]

By matching the power of t2we can identify

The coefficient is very involved so let us first consider some simple cases.If p=2,then there are only three sectors

Taking the large N limit,we find

which implies that

so that we have the approximation

Similarly when p=3,we can find

This turns out be general: when p ?N the dominant term is Θp.Therefore,the self-averaged 〈z〉 will not survive.This behavior is the same as we found in the simple statistical model in the regime when the cylinder amplitude is much larger than the disk amplitude.

On the other hand,if q ?N then

the situation is very different.As a simple demonstration let us consider the case of q=2

Fig.2.Plot of (47) when they are two comparable peaks.log N=298,βt=10.

The dominant term is neither 〈z〉nor Θpbut some intermediate term Θkas argued in[69].With this detailed analysis we find that we should also include some‘sub-leading’sectors.The distribution of the surviving sectors in the large N limit has a peak centered at the ‘dominant’ sector with a width roughlyOne possible interpretation of this result is the surviving sectors are only approximate saddles or constrained saddles with some free parameters.Even though each approximate saddle contribution is as tiny asbut after integrating over the free parameters the total contribution is significant.Note that similar approximate saddles are also found for the spectral form factor in the SYK model [68].We plot the ratioas function of k in figure 3.With increasing q or equivalently decreasing p,the peak moves to the left (small k) and becomes sharper and sharper.This is consistent with our analysis of limit of small p where there is only one dominant saddle,Θp.So our result shows that the wormhole(actually disk in this case)does not persist but the half-wormhole appears.As we found in[69]〈z2〉 can be computed by a trick of introducing the collective variables

and doing the path integral.The final expression is

Actually we can derive a different G,∑expression from Θidirectly in a more enlightening way.Because ψiare Grassmann numbers and q is even then the exponential in (57) factorizes

Using Tyler expansion the definition of θ(i)one can derive a useful identity

The integral (92) is not convergent but we can introduce the generating function

which can be computed with a saddle point approximation and theis given by

As a simple test,we know that the exact result of F(v) is just

2.3.1.Half-wormhole in z2.To make the half-wormhole saddle manifest below we will set u=0.In this case‘Bose–Einstein’condensation happens.As found in [58] for the square of partition function z2the wormhole persists and there is only one dominant non-self-averaged sector.Applying (30) directly leads to the decomposition

and then to substitute it into the integral form of z2

By matching the power of t2we can extract the expression of Φi.Note that the expressions of Φihave been derived in[62]based on the proposal of[58].In[62]the non-dominant sectors are derived as fluctuations of the dominant saddle Φ2pwith the help of introducing G,∑variables.Because our derivation here does not rely on G,∑trick so it can be used to derive possible n-linked half-wormholes in zn.First we notice that〈z2〉2=〈〉is in the same order of 〈z4〉≈〈z2〉2as proved in [58] so the wormhole saddle persists.To confirm that Φ2pis the only dominant non-self-averaged saddle we only need to show

2.3.2.Half-wormhole in z3.As we argued in the statistical toy model,there should exist n-linked half-wormholes.For simplicity let us focus on 3-linked half-wormholes and z3.Similar to (104),z3can be rewritten as

Again the expression of Λican be extracted by matching the power of t2.Since〈z3〉=0,so the self-averaged sector does not exist and z3is only dominated by non-self-averaged sectors which we expect are Λ3p:

which is expected to be one of the dominant non-self-averaged sector in the large N limit.

2.4.0+1 SYK model

Now let us apply our proposal to the 1-SYK model.The partition function is defined as

with JA’s satisfy (51).We will assume that (122) is approximately valid at least semi-classically.In other words,the saddle point can be derived from(122).The possible problem of(122)in the one-dimensional SYK model is that the fermions are not Grassmann numbers but Majorana fermions.As a result,ψAdoes not commute with ψBif there are odd number of common indexes in the collective indexes A and B.Therefore (89) is not exact anymore.The reason why we expect such subtlety is negligible in the large N limit is that when we introduce standard G,∑variables in the SYK model we already ignore this fact and it is shown in [68] this approximation is correct in the large N limit.

2.4.1.Half-wormhole in z and complex coupling.First let us consider z(β+iT)

where we have defined the operator

The reason we consider z(β+iT)is that its square〈z(β+iT)z(β-iT)〉is the spectral form factor(SFF)which has universal behaviors for chaotic systems like SYK model and random matrix theories.When T is small,SFF is self-averaged so it is dominated by disconnected piece 〈z(β+iT)z(β -iT)〉≈〈z(β+iT)z(β -iT)〉.Because the one point function decays with respect to time and so is SFF.This decay region of SFF is called the slope.Because of the chaotic behavior SSF should not vanish in the late time.It will be the non-self-averaged sector dominates which are responsible for the ramp of the SFF.Therefore,in the ramp region we expect the approximation

which describes the wormhole saddle considering that we can introduce the GLRas

so the saddle point solution of(130)is the same saddle point solution of〈z2〉with GLL=GRR=0.Such solutions are found in[68].To be more precise,these solutions found in[68]are time-dependent and only in the ramp region we have GLL,GRR→0.This is why we stress that only in the ramp region our approximation is good.Away from this region,we have to include other sectors which can be obtained by the expansion (125) as

2.4.2.Half-wormhole in z2and factorization.Let us consider z(iT)z(-iT) and apply our decomposition proposal (122)

Motivated by the result of 0-SYK model,we expect that there is also a ramp region where the dominant non-self-averaged sector is given by the 2-linked half-wormhole5Note that we have normalized the fermionic integral such that ∫ dψ=0 thus 〈Φ〉=0.

as shown in figure 4.Thus it implies the approximate factorization

2.5.Random matrix theory

whose ensemble average is given by

where t2is usually taken to be 1/N.

2.5.1.Half-wormhole.First let us consider the non-selfaveraged sector in z.It is useful to study a simpler observable TrHnto get some intuitions about the non-self-averaged sector of matrix functions.For the random variable Hijwe can not use the decomposition (30)directly.One possible way of adapting to(30)is to rewrite Hijas a linear combination of the Gaussian random variables.However this rewriting is not very convenient.Alternatively,we can transfer the matrix integral into the integral over eigenvalues

where Δ(λ) is the Vandermonde determinant

Then the simple single-trace observable translates to

However those eigenvalues are not Gaussian random variables.As a result,even though we can still do the sector decomposition but the resulting different sectors are not orthogonal anymore.Although when the level is finite,we can obtain a new orthogonal basis by a direct diagonalization but it is still very cumbersome.We will make some preliminary analysis beyond Gaussian distribution in next section.Here we will take a similar approach as before.Considering the non-vanishing correlator 〈HijHji〉=t2we should define

So the highest level sector can also be understood as the observable in the ‘normal order’.Applying this rule of decomposition to the single-trace observables we get

where the normal ordered terms are explicitly given by

Like the Wick transformation in quantum field theory,the normal order or the highest level sector can be defined as

or we can introduce the formal integral

which is more convenient sometimes.Therefore we can rewrite the decomposition as

where the Conkmeans choosing all possible k pairs of matrix elements from[f(H)] and replacing each pair HabHcdwith its expectation value 〈HabHcd〉.It implies the identification

For these single-trace observables,in the large N limit their correlation functions factorize so the dominant sector is always the self-averaged sector.The more interesting observable is z(iT) whose expectation value is

where Ckis a famous Catalan number andSo in the late time,the non-self-averaged sector becomes important.The lowest sector can be simply obtained by expanding z and picking the term with θ(1)6There is a 1/N in front because one of the summation of indexes gives the trace of θ(1) instead of a factor of N.For example:

Similarly we find that the next sector is7The factor 6×2 comes from the adjacent terms like and factor 3 comes from the pairs like

where we have dropped the terms Trθ(1)Trθ(1)because they are suppressed by 1/N.Comparing with the known results8For example see [65].of the wormhole contribution to 〈z(iT)z(iT2)〉c

we will show in the Appendix that

Fig.4.The illustration of (138).

3.Beyond Gaussian distribution or the generalized CGS model

One of simplest way to go beyond CGS model is again starting from the MM model but including connected spacetimes with other topologies in the Euclidean path integral.So the next simplest case beyond CGS model is the Disk-Cylinder-Pants model.Let the amplitudes of the disk,cylinder and pants to be

Figure 5. Contributions of different sectors(170),α= 1,T = 3.The horizontal axis represents different sectors,and the vertical axis is the proportion of each sector.

The generating function is

such that using the trick (9) we can decomposeinto different sectors which are exactly like (18)–(21).In other words,the number basis or {θ(i)} is still the basis for decomposition.But {θ(i)} should be determined from the recursion relations (18)–(21).For example,in the Disk-Cylinder-Pants model the first few θ(i)are

Because of the inclusion of new wormholes,the pants,the basis is not orthogonal anymore in the sense

It is easy to find that

Naturally θ(i)can be understood as the i-linked half wormhole as shown in figures 7 and 8.

3.1.Toy statistical model

We start from the simplest operator

The modification starts to show up in

In this special case since〈θ(1)θ(3)〉=0,there is no cross terms in 〈Y6〉

which is the analog of (36).

3.2.0-SYK model

Let us reconsider the 0-SYK model but assume the random couplings satisfying

we will determine the scaling of κ3in a moment.Then the averaged quantity is

which can be computed by introducing the collective variables

where mpis defined in (69) and

In general decomposing z3is still very complicated.Let us consider some simple examples.If p=2,then we have

and there are seven different sectors.A simple way to derive the explicit expression of each sector is to first decompose eachas (18)–(21):

then collect the terms in the same sector:

Fig.6.Behaviors of the sectors(170)in z(iT)(141),the horizontal axis is the time T and the vertical axis is the value of sectors.(a) The sum of two sectors.(b) An individual sector.

In large N limit the relevant parameters have the following asymptotic behaviors

then the approximation can be given as

In general we find that when p ?N(or q ?1),z3is not self-averaged,i.e.the wormhole does not persists,but the (threelinked) half-wormhole emerges.This fact can be intuitively understood as the following.In this limit because of the scaling(213),the three-mouth-wormhole amplitude is favored thus the possible dominate sectors are Δ0,Δ3p-3and Δ3p:

4.Discussion

Fig.7.Illustrations of the metric (180).

Fig.8.Illustrations of the metric (180).

In this paper,we have generalized the factorization proposal introduced in[57].The main idea is to decompose the observables into the self-averaging sector and non-self-averaging sectors.We find that the contributions from different sectors have interesting statistics in the semi-classical limit.When the selfaveraging sector survives in this limit,the observable is selfaveraging.An interesting phenomenon is the sector condensation,meaning the surviving non-self-averaging trend to condense,and in the extreme case,only one non-self-averaging sector is left-over,resembling the Bose–Einstein condensation.Then the half-wormhole saddle is naturally understood as the condensed sectors.We apply this proposal to a simple statistical model,a 0-SYK model and a random matrix model.Half-wormhole saddles are identified and they are in agreement with the known results.With our proposal,we also show the equivalence between the results in[57]and[58].We also studied multi-linked-half-wormholes and their relations.There are some future directions.

4.1.Sector condensation

It is interesting to understand the sector condensation better.We expect that it is some criterion for an ensemble theory or a statistical observable to potentially have a bulk description,and so it deserves to be studied in other gravity/ensemble theories.Definitely,the extreme case mimicking the Bose–Einstein condensation is the most interesting one.We have not understood when it will happen and could it be used as some order parameter.We expect by studying the ‘phase diagram’ in the sector space we can obtain more information about the observables and systems.

4.2.Complex coupling and half-wormholes

In [62],it shows that factorization is related to the complex couplings.In our approach,the complex coupling emerges as an auxiliary parameter to obtain the half-wormhole saddle.The trick here is similar to the one used by Coleman,Giddings and Strominger [70–72],where the non-local effect of spacetime wormhole is ‘localized’ with the price of introducing random couplings.But the current analysis shows that this is only possible when ‘Bose–Einstein’ happens such that the dominant sector can be obtained from this trick.So it would be interesting to explore the relation between complex coupling and half-wormhole further using our approach.

4.3.Relations to other factorization proposal

Besides the half-wormhole proposal,there exists other proposals of factorization.For example,in [65] it shows twodimensional gravity can be factorized by including other branes in the gravitational path integral.These new branes correspond to specific operators in the dual matrix model.From the point of view of our approach,inserting operators may be related to adding back the contributions from nonself-averaging sectors.In [73],it is argued that factorization can be restored by adding other kinds of asymptotic boundaries corresponding to the degenerate vacua.It is clear that from our approach,this is equivalent to introducing new random variables.It would be interesting to see how this changes the statistic of contribution form different sectors.

Acknowledgments

We thank Cheng Peng for valuable discussions and comments on an early version of the draft.We thank many of the members of KITS for interesting related discussions.JT is supported by the National Youth Fund No.12105289 and funds from the UCAS program of special research associates.YY is supported by the Fundamental Research Funds for the Central Universities,by funds from the University of Chinese Academy of Science (UCAS),and NSFC NO.12175237.

Notice that in the large N limit[TrHn]is a linear combination of single trace operator so we should expand each 1/(w -λi)into Taylor series and only keep terms with

thus we arrive at the final result

These contour integral can be evaluated exactly by using the expansion

By expanding with respect to u,indeed we get the correct normal-ordered operators

We can also obtain a generating function of Θk

which,unfortunately,does not have a simple closed form but the ensemble average 〈TrHkTreiTH〉 can be computed with the generating function (230).