NSFKITP08115
Matrix Models for the Black Hole Information Paradox
Norihiro Iizuka^{1}^{1}1
Takuya Okuda^{2}^{2}2
Joseph Polchinski^{3}^{3}3
Kavli Institute for Theoretical Physics
University of California
Santa Barbara, CA 931064030, USA
We study various matrix models with a chargecharge interaction as toy models of the gauge dual of the AdS black hole. These models show a continuous spectrum and powerlaw decay of correlators at late time and infinite , implying information loss in this limit. At finite , the spectrum is discrete and correlators have recurrences, so there is no information loss. We study these models by a variety of techniques, such as Feynman graph expansion, loop equations, and sum over Young tableaux, and we obtain explicitly the leading corrections for the spectrum and correlators. These techniques are suggestive of possible dual bulk descriptions. At fixed order in the spectrum remains continuous and no recurrence occurs, so information loss persists. However, the interchange of the longtime and large limits is subtle and requires further study.
1 Introduction
The black hole information paradox [1] has been a fruitful thought experiment, leading in particular to the discovery of gauge/gravity duality [2]. This duality in turn provides a nonperturbative definition of string theory as quantum gravity in AdS backgrounds in terms of unitary quantum mechanics, and implies that the information escapes with the Hawking radiation, but important questions remain. In particular, how does the argument for information loss, based on the low energy effective field theory of gravity, break down?
Ref. [3], building on ideas of Refs. [4, 5], presented a simple model in which this might be studied further. The quantum mechanical system of a single large matrix oscillator and a single fundamental oscillator displays the key property [6] that information is lost at infinite but not for finite. Since is proportional to in gravity, this result demonstrates that quantum gravity effects are crucial to avoid information loss. In the planar limit the SchwingerDyson equation for this model closes, and moreover can be reduced to a recursion relation with respect to frequency [3]; analytic arguments, and numerical solution, then confirm the desired properties. However, a complete analytic solution seems difficult even in the planar limit, and a systematic study of the corrections is even more difficult.
In the present paper we present more tractable models that have the same degrees of freedom, but where the previous trilinear interaction [3] is replaced by one that is quartic in the oscillators but quadratic in the charges. In Sec. 2 we motivate this by showing that the weakcoupling limit of the trilinear model leads to an effective chargecharge interaction. We then consider more general chargecharge interactions, and present the solution in the planar limit. Like the trilinear model, this displays the essential feature of a continuous spectrum at infinite . The SchwingerDyson equation is algebraic, and the decay of the planar twopoint function is powerlaw rather than exponential at late times. Still, information is lost at large , allowing us to address the paradox in a simpler setting.
Our ultimate goal is to see whether the preservation of information might be reflected in the largetime and/or large order (in ) behaviors of the perturbative expansion. We are also interested in developing the analog of a bulk string/gravity description, to see how information is preserved in this language. To these ends we attempt to solve the model in several different frameworks.
In Sec. 3 we further develop the graphical approach, beyond the planar limit. We find explicitly the first nonplanar amplitude, summing all Feynman graphs that have the topology of a disk with a handle. Our explicit computation of the first correction demonstrates that the spectrum remains continuous. No recurrence occurs and information loss persists.
In Sec. 4 we analyze the theory in terms of loop equations, recursive relations for expectation values of operators. This allows us to recover the planar term and first nonplanar correction, and generalizes efficiently to higher orders.
In Sec. 5 we show that the correlator for this theory can be written for any in terms of a sum over Young tableaux. The planar limit is recovered as a large saddle point.
In Sec. 6 we study various models in which the matrix oscillator is generalized to a rectangular matrix. This allows for several limits of large and small and where a more explicit solution is possible.
In Sec. 7 we discuss the results and future directions. In particular we examine the possibility that one might see signs of the longtime recurrences in the behavior of the expansion.
2 Chargecharge models
2.1 Models
First we recall the trilinear model [3]. In terms of lowering operators and for the adjoint and fundamental, the Hamiltonian is
(2.1) 
where . The final term is needed to stabilize the system, but has been arranged to vanish in the relevant sectors . We take to be large so that is essentially zero in the thermal ensemble.
The free thermal propagators are
(2.3) 
The only singularities in the lowerhalf planes are from the adjoint propagators, at , so we can evaluate these integrals by residues, leaving the fundamental propagators at momenta for various integers . Thus, an individual Feynman graph gives only poles on the real axis, as in the discussion in Ref. [5]. However, it is shown in Ref. [3] that this cannot be a property of the full planar propagator for any nonzero . The point is that perturbation theory is singular, because higher orders of perturbation theory give higher order poles. The most singular graphs as goes to its onshell value 0 are those in which alternates between and 0, since this gives the maximum number of poles.
Thus we can capture the most singular graphs by integrating out the propagators at , leaving an effective quartic interaction with two adjoints and two fundamentals.^{1}^{1}1 This is equivalent to dropping smaller terms in the recursion relations for , as was done in [3]. In other words, we must do degenerate perturbation theory, because states with the same total number of excitations are degenerate in the free theory. Thus
(2.4)  
In the last line we have projected down to the sector (which is annhilated by ), as relevant for .
The final interaction is simply a coupling of the charges of the fundamental and the adjoint,
(2.5) 
with
(2.6) 
Thus, the energy can be expressed in terms of a difference of quadratic Casimirs. This implies a large degeneracy; nevertheless, the model will still have enough mixing to produce a continuous spectrum in the large limit.
We could generalize by giving independent coefficients to the two terms in . In fact, these two terms separately generate commuting ’s, the first acting on the left index of and the second on the right index. We obtain a slightly simpler model (in terms of its SchwingerDyson equation) by keeping only one term,
(2.7) 
2.2 Planar solution
Now let us solve these in the planar approximation. For the model (2.7) the nontrivial planar graphs involve a cycle with vertices, giving the SchwingerDyson equation shown in Fig. 2.
Thus,
(2.8) 
where is the thermal propagator
(2.9) 
At large , has singularities only in the lower half plane, and and both fall as at large frequency, so we can close the integrals in the lower halfplane and pick up residues only from . This gives
(2.10) 
The SchwingerDyson equation becomes
(2.11) 
or
(2.12) 
where is the ’t Hooft coupling. The solution is
(2.13) 
This has a pole of spectral weight at , and a cut from to .
For the original model (2.5) the only modification is the inclusion of an additional copy of each cycle but with the arrows reversed, so that
(2.14) 
with . Then
(2.15) 
This reproduces the cubic equation for , Eq. (32) of [3], which was obtained from the weakcoupling approximation to the recursion relation of the trilinear model. Again, there is a branch cut on a finite segment of the real axis.
2.3 Black hole physics
The continuous spectrum implied by the cut in these models is the signature of a horizon. As noted in Ref. [3], the cut is absent at zero temperature, and also below the HawkingPage transition (which we can simulate in this onematrix model by imposing the singlet constraint). The Fourier transform of the cut gives a behavior at late times. Although this falls off more slowly than the exponential for the real black hole, it is still inconsistent with the properties of a system with finite entropy, and so with the exact correlator. Indeed, the energies in this model are all multiples of , so there are regular recurrences with period . This can be seen by writing the interaction in terms of quadratic Casimirs, as we will do in Sec. 5.1. This time is much shorter than for a fully thermalized system, for which the energy splittings are of order , but still presents us with a version of the information paradox. We should note that the more general model
(2.16) 
cannot be written in terms of commuting Casimirs for generic , and so may give a more realistic model of the black hole.
3 Nonplanar corrections
3.1 SchwingerDyson equation
We now consider the full SchwingerDyson equation, including nonplanar corrections. We focus henceforth on the model (2.7). It is useful first to carry out all of the loop integrations, as in the previous discussion. The number of loops is equal to the number of adjoint propagators, so we can take the adjoint propagator momenta as integration variables. We orient these in the direction of the arrow on the fundamental propagator, as in Figs. 1, 2. The momentum on the fundamental propagator therefore always involves , and so this propagator contributes no poles in the lowerhalf plane. Thus we can close the loop integrals in this halfplane picking up the pole at for each forward propagator (one whose arrow is parallel to that on the fundamental line) and at for each backward propagator (antiparallel to the fundamental line). Further, since each vertex contains one and one , there are always equal numbers of and loop momenta flowing on any internal fundamental line, so this is always at . This is simply a repetition of the point made in Sec. 2.1, that this model isolates the propagators with .
We therefore have the Feynman rules
(3.1) 
The evaluation of the amplitudes is thus reduced to counting graphs,
(3.2) 
where is the number of vertices, is the number of backward propagators, and is the genus of the graph.
We now reduce the sum over graphs by summing over certain classes of subgraph. Start with the sum of the oneparticleirreducible (1PI) fundamental selfenergy graphs , in terms of which the full SchwingerDyson equation reads
(3.3) 
We denote explicitly its dependence on the coupling and the functional form of the propagator for the fundamentals. Consider selfenergy corrections on internal propagators. Because we are assuming that the fundamental oscillator mass is large, there are no loops of the fundamental field, and so there are no corrections to the adjoint propagator, only to the fundamental propagator. By packaging corrections on internal fundamental propagator as , we only have to sum over graphs without them. What we get is precisely the twoparticleirreducible (2PI) fundamental selfenergy ,^{2}^{2}2A 2PI graph is one that cannot be separated into two pieces by cutting two propagators. Such a graph clearly cannot have an internal propagator correction. If a graph in can be separated in this way, at least one fundamental propagator has to be cut, because all vertices lie along a single fundamental line. Since the number of forward minus backward propagators emerging from any subgraph is always zero, the second propagator to be cut must be a fundamental line, isolating the propagator corrections in between. so
(3.4) 
Next, for any vertex, there is a geometric series of graphs obtained by expanding it into vertices, each connected to the next by a forward adjoint propagator and a fundamental propagator. This is illustrated in the SD equation of Fig. 2, where all of the interaction terms can be obtained from the term in this way. Thus, we can sum all such series into effective vertices. We will refer to the propagators to be summed as trivial forward propagators, a term which we will explain in Sec. 3.2. So we can restrict the sum to graphs with no trivial forward propagators (NTF) but with the geometric series incorporated into the vertices:
(3.5) 
Here denotes the sum of the graphs that are fully irreducible (I), i.e. 1PI, 2PI, and NTF.
One form for the SchwingerDyson equation is then
(3.6) 
The total contribution of a given I graph to is
(3.7) 
The first few terms are shown in Fig. 3. The planar contributions have been collapsed into a single term by the summation over graphs containing trivial forward propagators, but even at the first nonplanar order, the number of I graphs is infinite. Summing these is our next exercise.
3.2 The correction
If we take the trace of the fundamental propagator, which is just , then the planar graphs are those that can be drawn on a disk, while the corrections come from graphs that can be drawn on a disk with handles [7]. Thus we wish to enumerate all fully reducible I graphs that can be drawn on a disk with one handle (Fig. 4).
We have marked with an the point where the ends of the fundamental propagator are joined, because this enters into the Feynman rules. We have also marked and cycles; every adjoint propagator is homotopic to for some integers and . For the present discussion we are not distinguishing an orientation on the adjoint propagators.
The possible trivial propagators, , are very limited in a 2PI graph. As is clear from Fig. 2, the forward propagators that have been summed into the NTF vertices are trivial, and in fact these are the only trivial forward propagators as we have already mentioned. To see this, note that a contractible forward propagator divides the Riemann surface into two pieces (that which it crosses when contracting, and that which it does not). It follows that if we cut the fundamental propagators that attach to each end of this forward propagator, the surface is separated into two. Because the graph is 2PI, this is only possible if the two ends are connected by bare fundamental propagator. Thus it is of precisely the type summed by the NTF condition.
If a backward propagator is trivial, a similar argument shows that it also divides the Riemann surface, and so would cutting on the adjacent fundamental lines. This is also excluded by the 2PI condition, the only exception being when the ends of the backward propagator are the first and last adjoint propagators to attach to the fundamental line, since the 1PI condition omits these adjacent propagators. For example, all the planar graphs in Fig. 2 have such a trivial backward propagator. So for any nonplanar graph, either all propagators are homotopically nontrivial, or there is a single trivial backward propagator which separates the from the rest of the graph.
To enumerate the nontrivial propagators, we must be careful not to overcount, because the modular group of the torus allows us to draw the same graph in different ways. Therefore, we specify that as we move along the fundamental line in the direction of the arrow, the first nontrivial propagator that we encounter is homotopic to the cycle.^{3}^{3}3We can exclude a integer multiple of (which would not be modularequivalent to ) because such a propagator would intersect itself. Similarly, we specify that the second nontrivial propagator that we meet (excluding those homotopic to the first) is homotopic to the cycle. This fixes the modular group, and so we can count freely. Note that there is at least one propagator along the cycle and one along the cycle, or else the graph is actually planar.
There are five possible kinds of propagator:

propagators which go under the handle from the left of the marked point ( cycle propagators). As noted above, .

propagators which go directly along the handle (the cycle). Again, if the graph is nonplanar.

propagators which go along the handle in a “twisted” way. These are homotopic to . There are genusone graphs without these twisted propagators, so .

propagators which go under the handle from the right of the marked point. Again, these are homotopic to the cycle, but there need not be propagators of this in this group: .

trivial backward propagators as discussed above, where .
In Fig. 5 we depict in various ways the case , .
The integers are subject to several parity constraints. Because the operators and alternate along the fundamental line, there must be an even number of propagators attaching between the ends of any given propagator. By applying this to given propagators of types 1, and 2, we find that and must be even. Type 3 propagators give the sum of these conditions, type 4 give the same condition as type 1, and type 5 give no conditions. This leaves four cases:
(3.8) 
( even, odd), and in each case or 1.
It remains to determine the number of backward propagators , since each of these brings in a factor of . The number is approximately half the total number of propagators, but a precise count requires us to examine separately each parity case (3.8). For one finds respectively
(3.9) 
where . For ,
(3.10) 
Recall that the weight (3.7) of a given graph is , with . We sum over the integers with the limits , , , , and also over , separating according to parity case:
(3.11)  
Here .
Expand the correlator in ,
(3.12) 
where is identified with (2.13). Also we define . The term in the SD equation is then
(3.13) 
where from the single planar I* graph. The solution is
(3.14) 
The RHS of Eq. (3.14) is a rational function of and of , so its branch cut comes only from that of : it is in the same place as for the planar amplitude . For real , Re is nonzero only if Re is also nonzero. Furthermore the continuous spectrum of Re is not modified by the leading perturbative correction Re. Note that from (3.6), (3.7), each I graph contributing to has the same branch points as the planar amplitude, and the same continuous spectrum. This need not hold after summing an infinite series of I graphs, but we have found that (3.11) is a rational function of and , and so does not introduce new cuts. We expect that this will continue to hold at higher genus, so the continuity of the spectrum and the positions of branch points will be the same at any finite order in .
However, even this first correction does change the nature of the branch point. By inserting the result (2.13) for the planar propagator, one finds that vanishes as at the ends of the cut, so that the denominator vanishes as . The branch point behavior of is then more singular than that of : there is an double pole, as well as a subleading branch cut, as opposed to in the planar term. (This effect, which seems rather accidental in the present approach, will be more evident in the next section.) This implies that grows at long times. In the conclusions we will discuss the possible relevance of this for the information problem.
It would be very useful to extend the graphical solution to all orders in . The main challenge seems to be the treatment of the modular group. It would be interesting to find in particular some string field description, in which the sums over adjoint propagators can be thought of as a string propagator. We hope to return to this in future work.
4 Loop approach
One of our goals is to find a ‘bulk’ description of our model, one analogous to the gravity side of gauge/gravity duality. Bulk quantities are invariant under the gauge symmetries of the CFT, so we must work with invariant objects. Recasting gauge theory in terms of invariants has long been a tantalizing idea for connecting gauge theory with string theory [8, 9], but it has been difficult to implement. In our model, it turns out to be a useful way to calculate.
The invariants that we work with are , and generating functions for these. We have to be careful about ordering because is a matrix whose elements are operators. We specify the natural ordering whenever there is a matrix multiplication, e.g.
(4.1) 
That is, we think of as acting on the tensor product of the index space and the adjoint Hilbert space. We will encounter expressions where other orderings arise, and for these we write the indices explicitly. Also we define . Note that the matrix elements commute with the .
Because the thermal ensemble has no fundamental excitation, we can write the correlator (2.2) as
(4.2) 
(We will write simply as hereafter.) Since we consider the limit that the mass of fundamental field is very large, the number of fundamental field is always one. If we expand the exponential, only adjacent oscillators can be contracted. Thus we can express the correlator in terms of invariants as
(4.3) 
The trace is over the dimensional matrix indices, while the (thermal) expectation value is in the free adjoint Hilbert space.
4.1 Thermal loop equations
In a thermal ensemble, the Heisenberg operators satisfy
(4.4) 
for any . The free field equation then implies
(4.5) 
or
(4.6) 
We now omit the common time argument 0 from all operators. The ‘loop equation’ (4.6) determines all expectation values iteratively, since the right side has two fewer mode operators than the left.
Let us first illustrate this with some simple examples. For we obtain from Eq. (4.6)
(4.7) 
For ,
(4.8) 
For ,
(4.9) 
In particular we obtain the connected contribution, which is down by .
Now let ; this will require a bit more work. First, the loop equation gives
(4.10) 
If were a number matrix, the cyclic property of the trace would give , but here we have commutators,^{4}^{4}4 Alternatively, one can compute , and hence the correlator (4.3), by applying both (4.6) and
(4.12) 
To solve this, we introduce the resolvents
(4.13) 
Multiplying Eq. (4.12) by and summing from to gives
(4.14) 
so that
(4.15) 
To apply this to the recursion (4.10), multiply by and sum, giving
(4.16) 
This can be used immediately to obtain the planar propagator, but we first derive further results that will be useful beyond planar order.
The next case is . The loop equation is
(4.17) 
To simplify the last term we use
(4.18)  
We have used the fact that , and that the matrix elements of and commute. If we could reverse the order of and this would simplify. Thus we proceed
(4.19) 
Now multiply by and sum. Then
(4.20) 
To use this, multiply Eq. (4.18) by and sum on ,
(4.21)  
so
(4.22) 
Now multiply the relation (4.17) by and sum from to obtain
(4.23) 
We can simplify further by using partial fractions,
(4.24) 
Then
(4.25) 
Finally, we consider a very general case . The commutator (4.21) can be written
(4.26) 
Then becomes
(4.27)  
This is our most general form for the loop equation. The trace acts on the matrix indices of and . Note that the matrix elements of commute with and so can all be brought to the right.
By partial fractions the RHS can again be written in terms of expectation vales of , and their derivatives. For example, in the case that ,
(4.28)  
Here we used
(4.29) 
4.2 expansion
We have not yet made any use of large . In the ’t Hooft limit, are of order , is of order , and is of order . We therefore rewrite the loop equation in terms of , and . Note that
(4.30) 
so the second term is nonplanar. Therefore from Eq. (4.27)
(4.31) 
or, using ,
(4.32) 
In the planar limit the expectation value factorizes, and so we have
(4.33) 
where . This reproduces the planar result (2.12) for .
We have assumed large factorization, but we should be able to derive it from the loop equations, since we have argued that these are complete. We give a somewhat formal derivation, as follows. In Eq. (4.32) let the all be equal to . By forming a power series we can conclude for any analytic function that
(4.34) 
The thermal ensemble produces some probability distribution for .^{5}^{5}5Since is the generating function of all Casimirs , this is precisely the probability distribution of Young tableaux discussed in Sec. 5. It is concentrated on a typical tableau determined there. The explicit formulas for in terms of tableaux can be found in [10]. Since Eq. (4.34) holds for arbitrary , it must be that the distribution is concentrated on the zeros of the expression in the bracket. There are two such zeros, but these have different large behaviors, and . Using the known asymptotic behavior we can conclude that the distribution is a delta function on the first solution.
We can now solve iteratively for the higher orders, expanding the general loop equation (4.27) as