Unitarity in the presence of closed timelike curves

We conjecture that, in certain cases, quantum dynamics is consistent in the presence of closed timelike curves. We consider time dependent orbifolds of three-dimensional Minkowski space describing, in the limit of large anti-de Sitter (AdS) radius, Bañados, Teitelboim, Zanelli (BTZ) black holes inside the horizon. Although perturbative unitarity fails, we show that, for discrete values of the gravitational coupling, particle propagation is consistent with unitarity. This quantization corresponds to the quantization of the black hole angular momentum, as expected from the dual conformal field theory (CFT) description. Note, however, that we recover this result by analyzing the physics inside the horizon and near the singularity. The space-time under consideration has no AdS boundary, and we are therefore not using any assumption regarding a possible dual formulation. We perform the computation at very low energies, where string effects are irrelevant and interactions are dominated by graviton exchange in the eikonal regime. We probe the noncausal structure of space-time to leading order, but work to all orders in the gravitational coupling.

Figure 1

Leading correction to the free propagation of a scalar field due to gravitational interactions with virtual particles winding closed causal curves.

Figure 2

Penrose diagram of the extremal BTZ black hole (a). The shaded area represents the region behind the chronological singularity, where closed causal curves are present. In the limit $\ell \to \infty$, $J$ fixed, one focuses on the region inside the black hole horizon and obtains a flat space orbifold with Penrose diagram (b).

Figure 3

A representation of the basic features of the orbifold geometry. The null Killing direction $y^{+}$ and the transverse direction $y$ are drawn on the plane. The compact $y^{-}$ circle is spacelike for positive $y$ and timelike in the shaded region after the chronological singularity. Represented are also the first polarization surfaces at $y_w=w^2/(24E)$. Finally we show a closed timelike curve and a closed null curve. The latter starts and ends on the first polarization surface.

Figure 4

Scalar propagator for a particle winding $w$ times the compact $y^{-}$ direction. The incoming and outgoing momenta are related by the action of the orbifold generator and the usual propagator is multiplied by a momentum dependent phase.

Figure 5

A two-loop graph, with loop winding numbers ${\omega }_1$, ${\omega }_2$. The winding numbers $w_i$ for the propagators can be chosen arbitrarily, as long as ${\omega }_1=\sum _{i=1}^4{w}_i$ and ${\omega }_2=w_5-w_3$.

Figure 6

Scattering process in the orbifold geometry. The dynamics is strongly coupled in the shaded region due to gravitational interactions in the presence of CCC’s, which naively violate unitarity. We expect, on the other hand, to be able to define particle states and a consistent S–matrix for discrete values of the gravitational coupling constant $M^{-1}$.

Figure 7

The correction to the free scalar propagator due to interactions. The conserved momenta $p_{\pm }$ flow through the diagram, whereas $\lambda$, ${\lambda }^{\prime }$ are the off–shell mass squared of the external legs. The shaded blob is computed using the Feynman rules of Sec. 3 and includes internal propagators winding the compact $y^{-}$ direction.

Figure 8

Complete scalar propagator ${\Gamma }_{p_{+},p_{-}}({\lambda }^{\prime },\lambda )$ and tadpole expanded in increasing number of internal propagators with nonvanishing winding number, represented by dashed lines. The remaining effective vertices, shown with light gray blobs, are computed in the parent theory to all orders in the coupling constant, with internal propagators with vanishing winding number.

Figure 9

Leading nontrivial contribution to the two–point function ${\Gamma }_{p_{+},p_{-}}(\lambda ,{\lambda }^{\prime })$, as shown in Fig. 8. The loop momentum has nonvanishing winding number $w$, whereas the blob $\mathcal{A}$ represents the four-point amplitude in the parent theory to all orders in the coupling constants.

Figure 10

The one-loop scalar tadpole. The on-shell particle winding the compact $y^{-}$ direction contributes an imaginary part to the tadpole, which is nonvanishing only for $p<0$. In position space the tadpole diverges at the polarization surfaces.

Figure 11

Generalized ladder diagram contributing to the scattering amplitude $\mathcal{A}$ of scalar particles in the eikonal regime $s\gg |t|$. The exchanged particle can have in general spin $j$ but, in the kinematical regime of interest, the gravitational interaction $j=2$ will dominate.

Figure 12

Phase shift for a scattering process in the gravitational background created by the target. In three dimensions, the background geometry is a conical space with deficit angle proportional to $\sqrt{s}/M$. The phase shift $\delta$ is proportional to the impact parameter $x$ and is negative.

Figure 13

Poles of the integrand in Eq. (46) in the $s$-plane. The pole denoted with a dot comes from the winding propagator, whereas the poles marked with a cross come from the eikonal amplitude $\mathcal{A}$. Recall that $s=u+\sigma$, with $\sigma =8p{p}_{-}/|w|$.