Contents

arXiv:0712.4180

On the Quantum Resolution of Cosmological

Singularities using AdS/CFT


Ben Craps, Thomas Hertog and Neil Turok

Theoretische Natuurkunde, Vrije Universiteit Brussel,

Pleinlaan 2, B-1050 Brussels, Belgium International Solvay Institutes, Boulevard du Triomphe,

ULB–C.P.231, B-1050 Brussels, Belgium Institute for Theoretical Physics, KU Leuven, 3001 Leuven, Belgium

DAMTP, CMS, Wilberforce Road, Cambridge, CB3 0WA, UK and

African Institute for Mathematical Sciences, 6-8 Melrose Rd, Muizenberg 7945, RSA

Ben.Craps@vub.ac.be, Thomas.Hertog@fys.kuleuven.be, N.G.Turok@damtp.cam.ac.uk

ABSTRACT

The AdS/CFT correspondence allows us to map a dynamical cosmology to a dual quantum field theory living on the boundary of spacetime. Specifically, we study a five-dimensional model cosmology in type IIB supergravity, where the dual theory is an unstable deformation of supersymmetric gauge theory on . A one-loop computation shows that the coupling governing the instability is asymptotically free, so quantum corrections cannot turn the potential around. The big crunch singularity in the bulk occurs when a boundary scalar field runs to infinity, in finite time. Consistent quantum evolution requires that we impose boundary conditions at infinite scalar field, i.e. a self-adjoint extension of the system. We find that quantum spreading of the homogeneous mode of the boundary scalar leads to a natural UV cutoff in particle production as the wavefunction for the homogeneous mode bounces back from infinity. However a perturbative calculation indicates that despite this, the logarithmic running of the boundary coupling governing the instability generally leads to significant particle production across the bounce. This prevents the wave packet of the homogeneous boundary scalar to return close to its initial form. Translating back to the bulk theory, we conclude that a quantum transition from a big crunch to a big bang is an improbable outcome of cosmological evolution in this class of five-dimensional models.

1 Introduction and Summary

No theoretical framework for cosmology can claim to be complete until it resolves the cosmological singularity, most likely through quantum gravitational effects. We do not know, for example, whether the singularity represents the beginning of the universe and, if not, what happens to the thermodynamic arrow of time there. The no-boundary proposal [1], for example, predicts the arrow of time reverses. Another, perhaps simpler possibility is that the physics responsible for resolving the singularity also describes dynamical evolution across it[2, 3, 4].

Whichever of these options is correct has profound implications for cosmology. If the singularity was essentially the beginning, the horizon and flatness problems, and the problem of the origin of the observed density variations, seem to demand an early epoch of inflation[5]. Although the no-boundary proposal provides some support for this, its formulation in a consistent quantum gravity framework remains a challenge. If, on the other hand, the singularity was not the beginning and there was a preceding epoch of slow cosmological contraction (in Einstein frame) before it, non-inflationary solutions of the classic cosmological problems are possible [3, 6, 7]. Furthermore, in a universe where repeated cycles of evolution occur, technically natural mechanisms whereby the cosmological constant can relax to tiny values become viable [8].

The AdS/CFT correspondence [9] has emerged as an extremely powerful tool for understanding quantum gravity, providing a non-perturbative definition of string theory in asymptotically anti-de Sitter (AdS) spacetimes, in terms of conformal field theories (CFTs) on their conformal boundaries. In this paper, we shall use the AdS/CFT correspondence to study the quantum dynamics near cosmological singularities. We study Maldacena’s original example in type IIB supergravity, but with generalized boundary conditions on some of the negative mass squared scalars (saturating the Breitenlohner-Freedman bound). These generalized boundary conditions allow smooth, asymptotically AdS initial data to evolve into a big crunch singularity, namely a spacelike singularity that reaches the spacetime boundary in finite time [10, 11].

The dual description of these “AdS cosmologies” involves field theories with scalar potentials which are unbounded below and which drive certain boundary scalars to infinity in finite time. The AdS/CFT duality, therefore, relates the problem of resolving cosmological singularities to that of understanding the dynamics of quantum field theories of this type. The present paper describes an attempt to put forward a method that specifies a consistent rule for unitary quantum evolution in the boundary theory.

In our example, the dual field theory is a deformation of Super-Yang-Mills (SYM) theory on by an unbounded double trace potential , with a trace operator quadratic in the adjoint Higgs scalars [12]. The dual field theory has several key properties that allow us to analyze its dynamics quantitatively:

  1. The principal advantage of this five-dimensional setup, compared with similar four-dimensional cosmologies studied previously, is that the undeformed dual boundary theory is well-understood.111 After the present work appeared on the preprint archive, an improved understanding of the M2-brane theory [13] has enabled us to study four-dimensional cosmologies as well [14]. Furthermore, the deformation is renormalizable and a one-loop computation shows that, because of the “wrong” sign of the deformation, the coupling that governs the instability is asymptotically free [15], allowing perturbative field theory computations in the regime near the singularity, i.e., for large , at least at small ’t Hooft coupling. In particular, we can show that quantum corrections do not turn the potential around, so that it is really unbounded below.

  2. As the dual theory evolves towards the singularity, the field evolution becomes ultralocal on any fixed length scale, meaning that spatial gradients become dynamically unimportant. This means that one effectively has an infinite set of decoupled quantum mechanical systems, one at each spatial point, when one approaches the singularity.

  3. Since the conformal boundary has finite spatial volume, the unstable, homogeneous background mode evolves quantum mechanically. The quantum mechanical spread of its wave function will turn out to be crucial for the suppression of particle creation as the wavefunction for the homogeneous mode rolls down the potential and bounces back from infinity.

  4. As the singularity is approached, the semiclassical approximation becomes increasingly accurate, both for the homogeneous background and for the fluctuations. This allows us to study the full quantum dynamics with some analytical precision.

In the unstable dual theory, even if we start with the homogeneous background mode described by a localized wave packet, the wavefunction spreads to infinite scalar field in an arbitrarily short time. Unless suitable boundary conditions are imposed, probability will be lost at infinity. Therefore, in order to study quantum evolution in this theory at all, one has no choice but to impose unitary boundary conditions at large field values, i.e., boundary conditions which restrict the Hilbert space to a subspace on which the Hamiltonian is self-adjoint. In quantum mechanics, such a restriction is known as a “self-adjoint extension” of the original ill-defined theory [16, 17]. In this paper, we will describe an attempt to extend this construction to the full boundary field theory.

The fact that the semiclassical approximation becomes more and more accurate near the singularity allows us to implement the self-adjoint extension using classical solutions and the method of images. The relevant solutions turn out to be generically complex near the singularity as a consequence of the quantum spread in the homogeneous mode. To leading order, therefore, the inhomogeneous modes evolve in a complex classical homogeneous background. The complexity of the background turns out to be essential to the resolution of the singularity, providing an ultraviolet cutoff on quantum particle creation. If the backreaction of particles on the homogeneous background were sufficiently small this procedure would specify a rule for unitary quantum evolution in the boundary theory, in the presence of bulk cosmological singularities.

If the ’t Hooft coupling in the boundary super-Yang-Mills theory is small, we have good control over the field theory. However, for small ’t Hooft coupling the bulk is in a stringy regime. In fact, we shall see that the form of the quantum effective potential remains valid at large ’t Hooft coupling, so at least the key dynamical feature of an unbounded negative potential is shared by both regimes. Further analysis is required, though, to fully treat the boundary theory in the regime where the bulk is well-described by supergravity.

With self-adjoint boundary conditions the homogeneous wave packet rolls down the potential and bounces back. In the bulk this behavior corresponds to a quantum transition from a big crunch to a big bang, as envisioned e.g. in ekpyrotic cosmology. However in order for the expectation value of to return close to its original form the backreaction of the particle production across the bounce must be sufficiently small. We shall find however that in the models we consider here, the logarithmic running of the boundary coupling governing the instability leads, despite the ultraviolet cutoff, to significant particle production. Our estimates indicate that for most of the wave packet the backreaction effect of this appears to prevent the homogeneous mode from rolling back up the potential. Hence we find that a transition from a big crunch to a big bang is rather improbable in this class of models. At the same time, however, our calculations shed light on what might be several key properties of boundary models that do predict a big crunch big bang transition in the bulk.

Let us now briefly review the status of other approaches to cosmological singularity resolution. The simplest string theory models with cosmological singularities are time-dependent orbifolds. Here, however, perturbative string theory tends to break down, with large gravitational backreaction leading to divergences in perturbative string scattering amplitudes [18]. It is natural to hope that a condensation of winding modes can improve the situation, but in general this is still unclear [19]. However, in a specific model it has been argued that the singularity is replaced by a winding tachyon condensate phase, and that the system can be analyzed within perturbative string theory [20].

The ekpyrotic model [3] involves similar time-dependent orbifold solutions to M-theory. Near the singularity, the only light modes in the theory are winding M2-branes [21, 22], and the effective string coupling (in IIA or heterotic frame) vanishes. A study of the classical M2-brane dynamics suggests that the expansion must be replaced by an expansion in  [21], but fully quantum mechanical calculations have not yet been possible.

Other attempts to describe spacelike singularities using a non-perturbative dual description have also been made. Models with two spacetime dimensions have been described using matrix models [23]. Higher-dimensional models with light-like singularities have been studied in the framework of matrix theory [24], which has led to the suggestion that spacetime may be replaced by non-commuting matrices near a singularity [25]. The quantitative study of the dynamics of this regime is still work in progress, however.

In earlier work, the AdS/CFT correspondence has also been used to study the singularity inside black holes, which is closely analogous to a cosmological singularity [26]. Although some progress in this direction has been made, the fact that the singularity is hidden behind an event horizon clearly complicates the problem. The CFT evolution is dual to bulk evolution in Schwarschild time, so the CFT never directly “sees” the singularity. This should be contrasted with the model discussed in the present paper, in which the bulk singularity reaches the boundary in finite time and is thus directly visible in the boundary theory. Other AdS/CFT models of cosmological singularities include [27].

Figure 1: The Penrose diagram of a large black hole in anti-de Sitter space is identical to that of an anti-de Sitter cosmology. In a black hole spacetime, however, time at infinity continues forever whereas in an AdS cosmology the singularity hits the boundary in finite time.

In the setup we consider, the difference between the quantum dynamics of cosmological singularities and the thermalization process that describes the formation of large AdS black holes is apparent in the dual theory. To compare the two situations, we show the Penrose diagram of gravitational collapse to a large black hole in anti-de Sitter space in Figure 1. This is identical to the Penrose diagram of an anti-de Sitter cosmology. Therefore from this point of view, there appears to be little distinction in the classical bulk theory between singularities inside black holes and cosmological singularities extending to the boundary in finite time. However the dual (quantum) description of both types of singularities appears to be qualitatively different: as we discussed above, AdS cosmologies are described in terms of unstable conformal field theories with steep potentials that are unbounded below (Figure 2, left), whereas black holes are interpreted as thermal states requiring a vacuum state in the dual theory (Figure 2, right).

If one “regularizes” the unbounded potential in Figure 2 (left), for example by adding higher order terms to obtain a potential like the one shown in Figure 2 (right), one finds this changes the evolution in the bulk near the big crunch, in the upper corners of the Penrose diagram. This is because the regularization affects the bulk boundary conditions. In particular one finds this causes the cosmological singularity to turn into a large (stable) black hole with scalar hair [11], where the bulk scalar field turned on is dual to the operator in the boundary theory. This is a new type of black hole which does not exist (in a stable form) for the original bulk boundary conditions. It has a natural interpretation in the dual theory as an oscillatory excitation about the global negative minimum of [11] (whereas the usual Schwarschild-AdS black holes correspond to thermal states around the standard vacuum at ). This new black hole with scalar hair is the natural end state of evolution in the bulk corresponding to a wave packet rolling down a regularized potential.

Considering a series of dual theories where the global minimum is taken to be more and more negative, one finds that the horizon size of the black holes with scalar hair – keeping the mass constant – increases. In the limit where the global minimum goes to minus infinity the hairy black holes become infinitely large and we recover the original cosmological solutions. Since for the black hole case (with bounded potentials) one expects that the boundary system will eventually thermalize, it seems plausible that the late time behaviour in the cosmological case (with unbounded potentials) is similar. From a cosmological perspective, the dynamics at intermediate times may be more interesting, however, if one can find models where a wave packet rolling down the potential bounces back a number of times before the system thermalizes and settles down.

We emphasize, however, that although the dual description in terms of a bounded describing the formation of hairy black holes appears to share many qualitative features with the cosmologies considered here, it does not provide us with a precise and consistent model of singularities. This is because the higher order terms needed to regularize the potential generally lead to a non-renormalizable boundary theory. In contrast, in this paper we focus on a class of five-dimensional cosmologies for which the dual field theory is a renormalizable deformation of SYM on by an unbounded double trace potential .

Figure 2: The dual description of AdS cosmologies involves a wave packet rolling down an unstable direction of the field theory potential (left) whereas the formation of large black holes in AdS is described as a thermalization process in a dual theory that has a ground state (right).

The outline of this paper is as follows. In Section 2, we introduce the bulk cosmologies of interest. We review how modifications of the standard AdS boundary conditions allow smooth initial data to evolve into a big crunch singularity, and focus on a specific example for which the dual field theory analysis will be tractable. We also show that the bulk cosmological solution to IIB supergravity is, after a duality taking us to type IIA frame, qualitatively similar to that describing compactified Milne (or colliding orbifold planes) in M-theory. In Section 3, we discuss the dual field theory, an unstable double trace deformation of SYM theory, deriving the effective potential both at weak and strong ’t Hooft coupling. Section 4 discusses self-adjoint extensions in quantum mechanics, as well as an attempt to extend this idea to quantum field theory. In Section 5, we discuss the quantum evolution of the homogeneous background, which exhibits a quantum spread because the field theory lives on a finite volume space. We develop a method for implementing self-adjoint extensions in the semiclassical expansion, using complex classical solutions and the method of images. We also explain how this method extends to potentials with branch points, such as the quantum effective potential of interest in this paper. Section 6 focuses on the inhomogeneous modes, in particular on the question whether abundant particle creation prevents the scalar field from running back up the potential after the bounce. We find that for most of the range of the Schrödinger wavefunction’s argument carrying significant probability, the quantum spread of the homogeneous background provides an ultraviolet cutoff on the wavelength of produced particles. In Section 7, we find nevertheless that the backreaction of the produced particles on the homogeneous mode appears to be strong enough to prevent the wavefunction from rolling back up the potential. Appendix A contains more details on the bulk theory. In Appendix B, we discuss a number of technical details related to the field theory effective potential. Appendix C discusses stress tensor correlators in the boundary theory, which would be an important ingredient in relating a more realistic version of our model to cosmological observables.

2 Anti-de Sitter Cosmology

2.1 Setup

Our starting point is gauged supergravity in five dimensions [28, 29, 30], which is thought to be a consistent truncation of ten-dimensional type IIB supergravity on . The spectrum of this compactification involves 42 scalars parameterizing the coset . We concentrate on the subset of scalars that parameterizes the coset . From the higher-dimensional viewpoint, these arise from different quadrupole distortions of . The relevant part of the action involves five scalars and takes the form [31]

(2.1)

where we have chosen units in which the coefficient of the Ricci scalar is , i.e., the 5d Planck mass is unity. The potential for the scalars is given in terms of a superpotential via

(2.2)

is most simply expressed as

(2.3)

where the sum to zero, and are related to the five ’s with standard kinetic terms as follows [31],

(2.4)

The potential reaches a negative local maximum when all the scalar fields vanish. This is the maximally supersymmetric AdS state, corresponding to the unperturbed in the type IIB theory. At linear order around the AdS solution, the five scalars each obey the free wave equation with a mass that saturates the Breitenlohner-Freedman (BF) bound [32] in five dimensions,

(2.5)

Nonperturbatively, the fields couple to each other and it is generally not consistent to set only some of them to zero. However it is possible to truncate this theory further [29] to gravity coupled to a single -invariant scalar by setting222 There are several inequivalent ways in which this theory can be further truncated to a single scalar as only matter field [31]. An alternative option that was studied in [10, 11] is to take for and , which corresponds to taking only . This choice preserves an symmetry. for and . The action (2.1) then reduces to

(2.6)

with .

2.2 Boundary Conditions

We will work mainly in global coordinates in which the metric takes the form

(2.7)

In all asymptotically AdS solutions, the scalar decays at large radius as

(2.8)

where and generally depend on the other coordinates.

For the dynamics of the theory to be well-defined it is necessary to specify boundary conditions at on the fields. This amounts to specifying a relation between and in (2.8). For example, one can take , leaving totally unspecified. This is the usual boundary condition, which preserves the full AdS symmetry group and which has empty AdS as its stable ground state [33, 34]. Alternatively, one can adopt boundary conditions of the form

(2.9)

where is an essentially arbitrary real smooth function.333 In general, boundary conditions specified by an arbitrary function can be imposed in anti-de Sitter gravity coupled to a tachyonic scalar with mass in the range (see e.g.[35]). Theories of this type have been called designer gravity theories [36], since their dynamical properties depend significantly on the choice of . We will see that under the AdS/CFT duality, this function appears as an additional potential term in the action of the dual field theory.

Boundary conditions of the form (2.9) generically break some of the asymptotic AdS symmetries, but they are invariant under global time translations. The conserved energy associated with this is well-defined and finite [37, 38], but its expression depends on the function .

This can be seen as follows. When is nonzero, the scalar field falls off more slowly than usual. This backreacts on the asymptotic behavior of the metric components, which causes the usual gravitational surface term of the Hamiltonian to diverge. This divergence is exactly canceled, however, by an additional scalar contribution to the surface terms. The total charge can therefore be integrated (provided one has specified a functional relation between and ). Hence one arrives at a finite expression for the conserved mass, which generally contains an explicit finite contribution from the scalar field which depends on . Whether or not the energy admits a positive mass theorem,444 See [39] for recent work on the stability of theories of this type. however, depends on the choice of the function .

Below we will be interested in solutions with the following scalar field boundary conditions

(2.10)

where is an arbitrary constant. The corresponding asymptotic form of the component of the metric is given by

(2.11)

The conserved mass of spherically symmetric configurations with these boundary conditions reads

(2.12)

where is the coefficient of the correction to the component of the metric.

2.3 AdS Cosmologies

We now construct a class of asymptotically AdS big bang/big crunch cosmologies that are solutions of (2.6) with boundary conditions (2.10) on the scalar field, with . This is a straightforward generalization of the four-dimensional cosmologies discussed in [10, 11].

Figure 3: Regular initial data that evolve to a big crunch singularity for boundary conditions with .

A particularly simple example of an open FLRW cosmology can be found from the evolution of an initial scalar field profile obtained from an -invariant Euclidean instanton555See e.g. [10, 11, 40] for a discussion of similar four-dimensional cosmologies.. Indeed, in Appendix A we show that all boundary conditions (2.10), for , admit precisely one such instanton solution.666Instantons also exist for negative provided . However, as we explain below, we do not expect initial data obtained from slicing these instantons across the four sphere to evolve to a big crunch. The slice through the instanton obtained by restricting to the equator of the four sphere defines time symmetric initial data for a Lorentzian solution with mass . The instanton, therefore, specifies negative mass initial data in this theory.777 As mentioned earlier, the mass of initial data obtained from instantons depends on the asymptotic behavior of the fields. For the AdS-invariant boundary conditions discussed in Appendix A (see eq. A.5) one finds the instanton initial data have exactly zero mass, in line with their interpretation as the solution decays into [10]. In Figure 3 we show the initial scalar field profile obtained in this way for .

Analytic continuation of the Euclidean geometry yields a Lorentzian solution that describes the evolution of these initial data under AdS-invariant boundary conditions888For our model these are given by (A.5). [41]. The origin of the Euclidean instanton then becomes the lightcone emanating from the origin of the Lorentzian solution. Outside the lightcone, the scalar field is constant along four-dimensional de Sitter slices of AdS and the scalar remains bounded in this region. On the light cone we have and (since at the origin in the instanton). Inside the lightcone, the symmetry ensures that the solution evolves like an open FLRW universe,

(2.13)

where is the metric on the four-dimensional unit hyperboloid. Under time evolution, rolls down the negative potential. This causes the scale factor to vanish in finite time, producing a singularity that extends to the boundary of AdS in finite global time. A coordinate transformation in the asymptotic region outside the light cone between the usual static coordinates (2.7) for and the invariant coordinates (see Appendix A) shows that . Hence is now time dependent and blows up as , when the singularity hits the boundary.

Figure 4: Anti-de Sitter cosmology. To predict what happens at the singularities one must turn to the dual field theory description.

The boundary conditions (2.10) differ from the AdS-invariant boundary conditions (A.5). This means we cannot obtain the Lorentzian solution simply by analytic continuation from an -invariant instanton. Instead one must evolve the initial data numerically. However, for sufficiently small values of , our boundary conditions are nearly AdS-invariant except when becomes large, and causality then restricts its effect. In particular, since the evolution of the instanton data for AdS-invariant boundary conditions has trapped surfaces, a singularity will still form in the central region and the effect of the modification of the boundary conditions on the evolution will only be appreciable in the corners of the conformal diagram where the singularity hits the boundary at infinity. Hence it is reasonable to expect that a singularity must form under evolution with boundary conditions when is small. This restricts us to positive since for negative we find instantons only for . It would now appear possible, however, for the singularity to be enclosed inside a large black hole instead of extending out to infinity. Since the instanton initial data have slightly negative mass, , these black holes must necessarily have scalar hair. In Appendix A we numerically integrate the field equations to verify whether boundary conditions admit static, spherically symmetric black hole solutions with scalar hair outside the horizon. We find a one-parameter set of spherical hairy black holes, which can be characterized by their conserved mass. However, it turns out that for the hairy black holes are always more massive than a vacuum Schwarzschild-AdS black hole of the same size. Hence the singularity that develops from the spherical negative mass initial data defined by the instanton cannot be hidden behind an event horizon. It is therefore plausible that it extends all the way to the boundary, cutting off all space.

2.4 Ten-dimensional Viewpoint

gauged supergravity is believed to be a consistent truncation of ten-dimensional IIB supergravity on . This means that it should be possible to lift our five-dimensional solution to ten dimensions. At the linearized level, the scalar fields which saturate the BF bound correspond to quadrupole modes on . Since in our example, the scalar field we retain diverges at the classical singularity, one expects that the sphere will become highly squashed.

Even though it is not known how to lift a general solution of supergravity to ten dimensions, solutions that only involve the metric and scalars saturating the BF bound can be lifted to ten dimensions [42]. The ten-dimensional solution involves only the metric and the self-dual five-form. To describe them, we first introduce coordinates on so that the metric on the unit sphere takes the form ()

(2.14)

Letting and , the full ten-dimensional metric is

(2.15)

which preserves an symmetry of the five-sphere, as expected. The five-form is given by

(2.16)

where and are the volume-form and dual in the five-dimensional solution and

(2.17)

In the homogeneous region of the asymptotic space, the metric can be written in Robertson-Walker form (2.13). Near the singularity both the potential and the curvature are unimportant in the Friedmann-Lemaître equations, and we have and . Therefore, over most of the near the singularity, the metric approaches

(2.18)

Introducing a new time coordinate , where , this takes a simple Kasner-like form

(2.19)

The anti de Sitter space and four of the dimensions of the shrink to zero, with Kasner exponents of approximately 0.19 and 0.26 respectively, while the fifth dimension of the , labelled by , blows up with a Kasner exponent of so the becomes spindle-shaped. Following the analysis of [43], we can T-dualize the dimension to obtain a homogeneous spacetime solution of type IIA string theory. The duality-invariant dilaton is , where the scale factor of the ’th spatial dimension is . Under T-duality, and the string-frame Kasner exponent for the dimension becomes . The dilaton was static in the original IIB frame, but in the T-dual theory we have so the string coupling tends to zero at the singularity. The new metric is now contracting in all directions, and hence qualitatively similar to the isotropic cosmological solution to the low energy effective action for string theory in string frame, with for , and . This background solution in type IIA theory corresponds to the solution to 11-dimensional M-theory in which the M-theory dimension and time form compactified Milne spacetime, with the other dimensions are static [4, 22]. It will be very interesting to see whether we can find an unstable mode within the generalized setup which, near the singularity, corresponds precisely to the collapse of the M-theory dimension and which could, once the appropriate boundary deformation is identified, be used to model an end-of-the-world brane collision in 11 dimensions, in the heterotic model.

More generally, it is clear that the cosmological solution we focus on here is only one of many possible cosmologies allowed by generalized boundary conditions on . The AdS/CFT correspondence can thus be used as a “laboratory” for the study of the nonperturbative counterparts (in both and ) of a large class of cosmological solutions to the low-energy effective actions for string theories. Every nontrivial solution possesses a spacelike singularity but, by mapping the theories in each case into an unstable dual quantum field theory, it might be possible to resolve (some of) these singularities and to describe the passage of model universes through them.

3 The Boundary Theory: a Double Trace Deformation of Super-Yang-Mills Theory

In the previous section, we have seen that our bulk theory with boundary conditions (2.10) allows smooth, asymptotically AdS initial data to evolve in finite time into a big crunch singularity that extends all the way to the boundary. Now we discuss the dual CFT counterpart of this phenomenon. First, we review that the boundary conditions (2.10) with correspond to adding an unstable potential to the boundary field theory. Then we argue that, in this particular model, the quantum effective potential shares the property of being unbounded below.

For the usual boundary conditions on the bulk scalars, the dual field theory is super Yang-Mills theory. The bulk scalars that saturate the BF bound in AdS correspond in the gauge theory to the operators , where are the six scalars in super Yang-Mills and is a normalization factor that will be fixed momentarily. The -invariant bulk scalar that we have kept in (2.6) couples to the operator

(3.1)

According to the AdS/CFT correspondence, this means the following. In the asymptotic behavior (2.8), the function of the coordinates along the boundary plays the role of a source for in the field theory: the field theory action has a term . The usual boundary conditions set this source to zero, meaning that the boundary field theory is the undeformed super Yang-Mills theory. On the other hand, the function plays the role of the expectation value of in the field theory; different correspond to different quantum states of the boundary theory.999For a detailed discussion of the relation between the bulk field and the Yang-Mills operator, see for instance [44]. For issues specific to AdS/CFT in Lorentzian signature, see for instance [45].

In general, imposing nontrivial boundary conditions in the bulk corresponds to adding a multi-trace interaction to the CFT action, such that after formally replacing by its expectation value one has [15, 46]

(3.2)

Adding a source term to the action can be considered as a special case, where is a single-trace interaction linear in . The boundary conditions (2.10) that we have adopted correspond to adding a double trace term to the field theory action

(3.3)

The operator has dimension two, so the extra term is marginal and preserves conformal invariance, at least classically (we shall see that conformal invariance is broken quantum mechanically).

In the previous section we have taken the constant to be small and positive in the bulk. The term we have added to the CFT action, therefore, corresponds to a negative potential. Since the energy associated with the asymptotic time translation in the bulk can be negative, the dual field theory should also admit negative energy states and have a spectrum unbounded below. This shows that the usual vacuum must be unstable, and that there are (nongravitational) instantons which describe its decay. After the tunneling, the field rolls down the potential and becomes infinite in finite time. This provides a qualitative dual explanation for the fact that the function of the asymptotic bulk solution (2.8) diverges as , when the big crunch singularity hits the boundary. Since is interpreted as the expectation value of in the dual CFT, this shows that to leading order in , diverges in finite time.101010 In fact, we shall see later that when we consider our theory on , the expectation value of in a state described by a wavepacket diverges well before the center of the wavepacket reaches infinity, so in our deformed theory on a finite volume space it is inappropriate to phrase the dynamics in terms of expectation values. What happens in that case is that the bulk of the wavepacket reaches infinity in finite time.

So the big crunch spacetime in the bulk theory corresponds in the boundary theory to an operator rolling down an unbounded potential in finite time. It is important to know whether quantum corrections preserve the unbounded nature of the potential in the boundary theory. While this was unclear for the AdS model that was the main focus of earlier work [10, 11, 47, 48], we shall now argue that for our model we indeed have an unbounded quantum effective potential.

For this purpose, we first briefly summarize the renormalization properties of a double trace deformation (3.3) of super-Yang-Mills theory; a more detailed discussion can be found in Appendix B. As explained in [15], the computation of amplitudes at order involves matrix elements of

(3.4)

From conformal invariance, on flat (where the constant depends on the normalization factor in (3.1) as well as on ). This leads to a short distance divergence that renormalizes and survives in the large limit:

(3.5)

with an ultraviolet cutoff. This leads to a one-loop beta function for , which does not receive higher loop corrections in the large limit [15]. As in [15], we now fix the normalization constant (and thus ) by demanding that the beta function coefficient should be one. At least for small ’t Hooft coupling, it is easy to see that for large ; therefore with a numerical constant. The coupling can then be kept fixed in the large limit. The existence of a non-vanishing beta function means that the conformal invariance of super-Yang-Mills theory is broken quantum mechanically by the double trace deformation.

We will be interested in an approximation to the quantum effective action that is valid for a large range of field values, in particular for large field values. An appropriate framework is that of [49], where the standard Feynman diagram expansion is resummed and the theory is organized in a derivative expansion and an expansion in the number of loops (see Appendix B). The one-loop effective potential is given by [50]

(3.6)

where is a renormalization scale, a renormalized coupling, and the counterterms have been chosen such that there is no constant and no linear term in , and such that

(3.7)

The renormalization group equation can be obtained by demanding that be independent of :

(3.8)

which shows that the normalization of implicit in (3.6) is indeed such that the beta function coefficient is one. In (3.8), we have ignored a contribution from hitting the in the second term of (3.6), which is justified as long as . Equation (3.8) is solved by

(3.9)

with an arbitrary scale (this implements dimensional transmutation). Choosing , i.e., the renormalization scale is set by the value of the field , the Coleman-Weinberg potential can then be written as

(3.10)

Now suppose that for some value , the coupling is small,

(3.11)

then and (3.10) is trustworthy (i.e., higher order corrections can be ignored) for any such that . As a result, we can conclude that in this case

(3.12)

Other quantum corrections are small in the large regime we will be interested in, as described at the end of Appendix B.

As was shown in [15], these renormalization properties have a precise counterpart in the bulk theory. First, define a new coordinate by . In terms of this coordinate, the conformal boundary of AdS is at . Near a point on the boundary, the metric takes the form

(3.13)

where and replace the coordinates of the three-sphere (which is approximately flat when zooming in on one point) [51]. With the boundary condition , the behavior of the field near the boundary is

(3.14)

It was shown in [52] that the near-boundary (small ) region of the bulk theory corresponds to the UV of the dual field theory. Therefore, to study this UV regime, we introduce a new coordinate

(3.15)

with ; this new coordinate is well-suited to studying the near-boundary (small ) region of AdS. In terms of , the boundary behavior (3.14) reads

(3.16)

with

(3.17)

and . Interpreting as a ratio of renormalization scales, , and interpreting and as the coupling defined at the scales and , respectively, (3.17) implies the following relation between the couplings at different scales:

(3.18)

which is consistent with the renormalization group equation (3.8). The reason that this bulk computation (valid for large ’t Hooft coupling) agrees with the perturbative field theory computation (valid for small ’t Hooft coupling) is the fact that the beta function is one-loop exact for large [15].111111 To relate our version of the above argument to Witten’s, given in Ref. [15], note that our conventions are related to his by . In the original argument, the boundary condition is written as (3.19) where is an arbitrary scale introduced to define the logarithm. One can choose a different mass scale if one “renormalizes” the field and the coupling in such a way that the bulk field is left invariant: (3.20) which implies the relation (3.18) between the couplings at different scales.

In what follows we will mostly concentrate on the steepest negative direction of the effective potential. Fluctuations in orthogonal directions in field space acquire a positive mass and we will see these are suppressed. For the -invariant operator we consider, the most unstable direction comes from the term in (3.10) (see (3.1)). We focus on the dynamics of as it rolls along a fixed direction in :

(3.21)

with a constant Hermitian matrix satisfying , so that is a canonically normalized scalar field. The Coleman-Weinberg potential (3.10) for this scalar is then given by

(3.22)

where

(3.23)

with the numerical constant implicitly defined after (3.5).

4 Unbounded Potentials, Self-Adjoint Extensions and Ultra-Locality

We have seen that our field theory description involves the potential (3.22), which is unbounded below. This implies that the quantum field theory has no ground state. While such unstable theories are usually considered unphysical, we want to explore whether quantum mechanical evolution can be defined for them in a consistent way. A clue is provided by quantum mechanics (as opposed to quantum field theory) with unbounded potentials. As we shall review momentarily, if a potential allows a wavepacket to move off to infinity in finite time, one can nevertheless define unitary quantum evolution by imposing appropriate boundary conditions at infinity. Technically, one restricts the domain of allowed wavefunctions to those on which the Hamiltonian is self-adjoint – this is called a “self-adjoint extension”. In this section, we begin to investigate the possibility that the quantum field theory we are interested in might also possess a self-adjoint extension. In Subsection 4.1, we review unbounded potentials in quantum mechanics. In Subsection 4.2 we show explicitly how the field theory dynamics become “ultralocal” as the singularity is approached, thus making it plausible that the quantum field is described by an independent set of identical quantum mechanical systems, one for each spatial point. When trying to use this to define self-adjoint extensions point by point, it turns out, however, that the ultralocality does not hold beyond the singularity, invalidating the attempt. Therefore, in subsequent sections, we will limit ourselves to considering a self-adjoint extension for the homogeneous field mode only, and treat the inhomogeneous modes perturbatively around the homogeneous background.

4.1 Quantum Mechanics in a Potential for

We are interested in a theory of a scalar field which is classically conformal invariant but unstable. Let us first emphasize the generality of this setup, in a holographic context. We consider a classically conformal-invariant scalar field theory on a dimensional conformal boundary of the form where is time. The action is

(4.1)

where is the Ricci scalar, and the coupling is an arbitrary constant. Conformal invariance requires and . For any , is greater than 2 and provided is positive, will run to infinity in a finite time. When focusing on the behavior near the singularity, where the term dominates in the potential, we shall neglect the term.

The simple fact that the spatial volume of the conformal boundary is finite is very important for the quantum behavior of the boundary theory. Consider the quantum description of the homogeneous mode of the scalar field. Its kinetic term in the action is where the volume of space, , acts as the “mass” of . In the infinite volume limit, this “mass” becomes infinite, and undergoes no quantum spreading: it becomes a classical variable. (This is the essential reason why spontaneous symmetry breaking in possible in quantum field theory but impossible in quantum mechanics). When is finite, as here, the homogeneous mode undergoes quantum spreading. It is convenient to canonically normalize the homogeneous mode, setting and , which is constant (for now, we are ignoring the running of the coupling constant). We then have a unit mass quantum mechanical particle with coordinate and potential

(4.2)

In this subsection, we summarize the operator approach to the quantum mechanics of such potentials [16] as reviewed in [17].121212 For a related recent discussion, see [53]. In Subsection 5.2, we shall implement the self-adjoint extension using complex solutions to the classical equations of motion, which will be useful for describing the evolution of Gaussian wavepackets.

A classical particle rolling down the potential (4.2), with , reaches infinity in finite time. The same is true for a quantum mechanical wavepacket, and this at first sight appears to lead to a loss of probability, i.e., to non-unitary evolution. However, if a self-adjoint Hamiltonian could be defined for this system, unitary quantum mechanical evolution would be guaranteed. As we shall now review, this can be done by carefully specifying an appropriate domain for the Hamiltonian

(4.3)

Since the WKB approximation becomes increasingly accurate at large , we can use it to study the generic behavior of energy eigenfunctions there. The two WKB wavefunctions for fixed energy are proportional to

(4.4)

where the lower limit of the integral may be chosen arbitrarily.

The Hamiltonian is self-adjoint if for any wavefunctions in its domain, . Using integration by parts, one sees that this is equivalent to

(4.5)

This can be arranged if for each energy we select the linear combination of the two WKB wavefunctions which behaves like

(4.6)

at large , where is an arbitrary constant phase. (The angle labels a one-parameter family of inequivalent self-adjoint Hamiltonians.) If is positive, for example, we set

(4.7)

with

(4.8)

which tends to the required form at large . A similar construction can be given for negative [17].

As a consequence of the fact that every energy eigenfunction tends to the same, energy-independent form (4.6), equation (4.5) is satisfied if and with the same value of . The domain of the “self-adjoint extension” of labelled by can now be defined as all wavefunctions

(4.9)

that satisfy , so is square integrable, and , so that is also square integrable, i.e., it is also a normalizable wavefunction. Under these conditions, the inner product makes sense and, as we have just checked, it equals so that is self-adjoint.

One can interpret the parameter as follows [17]. If one placed a “brick wall” at large , it would force the energy eigenfunctions to vanish there. However, since for finite the de Broglie wavelength becomes independent of energy at large , displacing the wall by a half-integral number of de Broglie wavelengths would, in this regime, have no effect. Hence only the location of the brick wall modulo an integer number of half-wavelengths matters physically, and this is the information contained in the phase .

Physically, one can interpret these self-adjoint extensions as follows. A right-moving wavepacket that moves to infinity is always accompanied by a left-moving “reflected” wavepacket that runs back up the hill. The time it takes for a right-moving wavepacket to run to infinity and for the left-moving wavepacket to run back up the potential hill can be shown to be the same as a classical particle would take to fall to infinity and climb up the potential again after being reflected at infinity. If the potential is bounded for , the Hamiltonian allows a continuum of scattering states. It also has an infinite number of bound states with quantized (negative) energies,131313 We note in passing that there is another approach to quantum mechanics in potentials which are unbounded below, based on a symmetry [54]. This approach is motivated by analytic continuation from the harmonic oscillator and it results in a positive energy spectrum. In contrast, for all self-adjoint extensions described above, there is an infinite set of negative energies. Since we know the bulk theory has negative energy solutions (the instanton discussed in Section 2 provides one example), the approach using self-adjoint extensions seems more appropriate. the values of which depend on . However, for the scattering problem – the bounce off the singularity – which we study, to a very good approximation the phase enters only as an overall phase in the final wavefunction and hence has no physical consequence.

In Section 6, we shall compute the Schrödinger wavefunctional for the full quantum field, decomposed into its homogeneous and inhomogeneous parts, . The homogeneous part , discussed in Section 5, behaves like the coordinate considered in this section. The inhomogeneous part is well-described at early times in terms of its Fourier modes, describing a set of harmonic oscillators which we shall take to be in their incoming ground state. We will then look for a range of final values of where it is consistent to treat the inhomogeneous modes to quadratic order while ignoring their backreaction on .

4.2 Ultralocality and Self-Adjoint Extensions in Quantum Field Theory

To address the question whether these methods can be extended to quantum field theory, we shall first consider a simplified model that shares the same finite time singularity, namely the quantum field theory of a single scalar field with a negative quartic potential. In the context of the actual dual quantum field theory, this approximation amounts to concentrating on the scalar that parameterizes the steepest negative direction: the gauge-invariant magnitude of . We shall attempt to extend the method of self-adjoint extensions to this quantum field theory. First, we show that when approaching the singularity, the scalar field evolution becomes ultra-local: spatial gradients become unimportant and the quantum field theory can be thought of as a collection of independent, identical quantum mechanical systems, one for each point in space. As a consequence, one could try to implement the self-adjoint extension method point by point. Unfortunately, though, it turns out that the ultralocality underlying this construction breaks down as soon as the field reaches infinity at any spatial point. Therefore, what we shall do in subsequent sections is apply the method of self-adjoint extensions to the homogeneous mode of the field, and treat inhomogeneous fluctuations around this background perturbatively. For consistency, it will then be needed that the backreaction of the inhomogeneous modes on the homogeneous mode is small, which we investigate in Section 7.

To begin with, we shall study the classical, conformally-invariant field theory on ,

(4.10)

where is the Ricci scalar. We are interested in studying generic solutions, i.e., those specified by the appropriate number of arbitrary functions of space, in the vicinity of the singularity, i.e., the locus where reaches infinity, which we shall assume to be a space-like surface .141414The form of will obviously depend on the specific solution one considers, i.e., on the initial conditions for the field . One may wonder whether will be a smooth, spacelike surface for reasonable initial conditions, and in particular for those assumed in the remainder of the paper, where the inhomogeneous field modes start out in the adiabatic vacuum. These quantum mechanical initial conditions generate a classical solution by the phenomenon of “mode freezing,” essentially in the same way that classical perturbations are generated from quantum fluctuations in inflationary cosmology. For those solutions, we shall verify in Subsection 6.6 that the surface is indeed smooth and space-like, justifying the assumption we are presently making. Note that this conclusion is not in contradiction with the conclusion we shall reach in the present section, namely that the field will evolve in an ultra-local way towards the singularity. While ultralocality implies that field fluctuations grow as the singularity is approached, it does not cause wild fluctuations in the surface , as we shall show in Subsection 6.6. The field equation is

(4.11)

The curvature of the three-sphere plays an important role in determining the initial conditions for the scalar field, but as the field runs down towards the singularity the spatial curvature rapidly becomes irrelevant. Since we are concerned with the behavior of the field near the singularity, in this section we shall ignore the last term in (4.11).

It is also convenient to change variables to so that the equation of motion assumes the form

(4.12)

and the zero-energy, homogeneous background solution is just , for , with an arbitrary constant.

Figure 5: The singular hypersurface , assumed spacelike, upon which is infinite and is zero.

We would like to construct the generic, spatially inhomogeneous solution to (4.12) as a Taylor series in the proper time away from the singularity, which, as mentioned above, we assume to be a spacelike surface . The coordinates of the embedding Minkowski spacetime (recall, we are neglecting the space curvature of the ) are ; the locus of the singular hypersurface is , . As long as is not too strongly curved, for every spacetime point there is a unique corresponding point on , connected to by a geodesic normal to . The unit normal to is

(4.13)

where indicates the gradient with respect to . It is then natural to change coordinates from to , the proper time from , and , the corresponding point on , given by , or

(4.14)

where is (the absolute value of) the proper time from .

To construct the scalar field equation we need the metric in the new coordinates. From the Minkowski line element in , and (4.14), we find the line element in coordinates is , where

(4.15)
(4.16)

Here, is the locus of the singular hypersurface, whose metric is and inverse metric is . All space derivatives are taken with respect to , at fixed . The matrix is the extrinsic curvature (or second fundamental form) of the singular hypersurface . (Since the embedding space is flat, the intrinsic curvature of can also be expressed in terms of , by the Gauss relation , and is thus not an independent quantity.)

In these coordinates, the field equation (4.12) reads

(4.17)

where is the determinant and the inverse of . By substituting (4.15) into (4.17), we obtain an asymptotic expansion for ,

(4.20)

where , , and so on, and traces are taken using the metric . The higher coefficients , , are all determined in terms of , and the higher coeffients , , are determined in terms of the second arbitrary function . The reason for the entrance of the logarithm, and the second arbitrary function, at order is seen by examining the first two terms of (4.17), at order . At this order, it is necessary for consistency to introduce a term proportional to into the series expansion, and the coefficient of is then left undetermined.

Let us comment on the general form of the full nonlinear solution (4.20). First, if we re-express in terms of the global time coordinate,

(4.21)

(4.20) can be viewed as an expansion in powers of as well as in . However, if we linearize in the two arbitrary functions and , we obtain the solution

(4.22)

which is actually completely regular at . Within this linearized approximation, the two arbitrary functions are easily interpreted. We decompose the scalar field into a homogeneous component , treated nonlinearly, and an inhomogeneous component which we treat only at the linearized level. Then is the time delay to or from the singularity, given in linear theory by the limit of as tends to zero. And is the linearized perturbation in the Hamiltonian density, evaluated at the singularity: setting

(4.23)

we find that as , tends to . Note that spatial gradients do not appear in at linearized order because the background solution is spatially homogeneous.

Second, (4.20) is clearly an expansion in , so that gradients become less and less significant in the dynamical evolution as the singularity is approached: the evolution of becomes ultralocal in this limit. To see this, consider the “velocity-dominated” version of the equation of motion (4.12), i.e., the equation of motion with the gradient terms omitted. The solution of the velocity-dominated equations of motion is

(4.24)

Therefore, we see that the full solutions (4.20) differ from the velocity-dominated ones only by terms that are unimportant near the singularity, so that the gradient terms are indeed dynamically unimportant there.

Finally, note that the non-analytic terms in (4.20), involving , enter only at quadratic and higher order in the perturbations. If these nonlinear terms are constructed perturbatively by expanding in the linearized perturbation, then no new functions enter: the nonlinear terms are completely dictated by the equations of motion. Hence, a matching condition within linear theory completely determines the matching of the full nonlinear solution, as long as the latter is well-described by linear perturbation theory.

To summarise, we have exhibited a generic class of classical solutions which are singular on an arbitrary spacelike hypersurface . The solutions are constructed as an expansion in proper time times spatial gradients: if we follow a solution with spatial structure on some wavenumber , gradients become unimportant in the evolution when falls below unity. At this point, the evolution becomes ultralocal so that the field values at different spatial points decouple. This behavior is relevant to the full quantum theory, because the semiclassical expansion becomes exact as the singularity is approached, so that the quantum wavefunction can be constructed from classical solutions.

In contructing the quantum theory, it would then seem natural to use the same self-adjoint extension (or boundary condition at large ) at every spatial point, to evolve the quantum field across the singularity. However, a problem with this attempt becomes clear by studying a simple toy model with discretized space, where the field is represented by particles in a potential at each point of space and the gradient interaction by springs connecting neighboring particles. As soon as one of the particles hits infinity, the gradient terms dominate the dynamics, showing that ultralocality only holds before the singularity is reached. Therefore, we will follow a different approach, where we use a self-adjoint extension for the homogeneous field mode and treat the inhomogeneous modes perturbatively around the corresponding background.

Specifically, we shall implement the self-adjoint extension using the method of images, so the full wavefunction will be constructed from two classical solutions with one, roughly speaking, starting out “behind” the singularity. A key question is whether the UV regime of the quantum field theory can be consistently treated, or whether the presence of structure on smaller and smaller scales will lead to divergences that invalidate perturbation theory. The remainder of this paper will largely be devoted to analysing this question.

5 Quantum Evolution of the Homogeneous Component

We want to compute the Schrödinger wavefunctional for the full quantum field in the semiclassical expansion. We write the field as the sum of its homogeneous and inhomogeneous parts, , and assume that each Fourier mode of the inhomogeneous component starts out in its quantum ground state. As we shall see later, the inhomogeneous quantum fluctuations are suppressed by powers of , so it makes sense to first calculate the wavefunction for homogeneous component , with the inhomogeneous modes ignored.

As emphasized in Subsection 4.1,