Time dependent field correlators from holographic EPR pairs

We consider the $AdS_{d+1}$ metric in the Poincare coordinates given by

where $R$ is the curvature radius. The string in this background is described by the Nambu-Goto action

where $T_{0}$ is the string tension and $h=\text{det}\,h_{ab}$ is the determinant of the induced metric. We choose a static gauge $(\tau,\sigma)=(t,z)$ and the embedding of the string as $X^{\mu}(t,z)=(t,z,x(t,z),0,\cdots,0)$ with $\mu=1,2,3,\cdots,d$. Then, $\sqrt{-h}=\frac{R^{2}}{z^{2}}\sqrt{1-\dot{x}^{2}+x^{\prime 2}}$. The classical equation of motion is given by

where ${}^{\prime}=\frac{\partial}{\partial z}$ and $\cdot=\frac{\partial}{\partial t}$. This equation has an exact solution Xiao_08 ,

where $b$ is a real constant. In the solution, the trajectory of the end-points at $z=0$ describes the motion of two particles along the $x$-direction with uniform deceleration $\frac{1}{b}$. They head toward each other from $x=\pm\infty$ when $t=-\infty$, subsequently stop at $x=\pm b$ when $t=0$ and then turn around in the opposite directions, moving away from each other. We label the position of the particle moving in the positive (negative) $x$ region by $q_{R}(q_{L})$, respectively. The induced metric on the worldsheet with the embedding coordinates is

As noticed in Jensen_13 , this metric has the same casual structure as that of the eternal two-sided $AdS$ back hole (see Fig. 1) and has the bifurcating horizons located at $t=\pm x$, the time-like boundaries at the particle trajectories $x^{2}-t^{2}=b^{2}$, and the space-like “singularity” at $x^{2}-t^{2}\rightarrow-\infty$. However, this space-like singularity is not a physical singularity and just reflects the coordinate singularity in the background AdS as $z\rightarrow\infty$ to approach the Poincare Killing horizon. Due to the analog between the EPR pair and the wormhole geometry ER=EPR , it was proposed in Jensen_13 ; Sonner_13 that this metric (5) gives the gravity description of the entangled quark and anti-quark pair.

We now consider the fluctuations of a string in the direction transverse to the $x$ coordinate, say $y_{1}$, with the embedding in (1) as $X^{\mu}=(t,z,x_{b}(t,z),q(t,z),0,\cdots,0)$ where $q$ is small as compared to $x_{b}$. The quadratic action for $q$ is obtained as

As seen in the action, in this coordinate system, there is no separable solution in the variables $t$ and $z$. However, in the comoving frame of one of the accelerating particles, say the $R$ branch in (4), defined as,

with $0<r<b$, the metric of (1) becomes

As in Xiao_08 , this metric has the event horizon at $r=b$ with the Hawking temperature $T=\frac{1}{2\pi b}$, which shows no causal connection between the two particles. However, on the worldsheet (5), it can be seen that a wormhole connects them. We also expect that in the particle rest frame, the particles will experience quantum fluctuations at finite temperature $T$ to be verified later. Note that this temperature $T$ can also be interpreted as the Unruh temperature that an accelerating particle observes. In this coordinate system, the background worldsheet solution in (4) is

And the quadratic action for the perturbations in the $y_{1}$ direction on the worldsheet becomes

with ${}^{\prime}=\frac{\partial}{\partial r}$ and $\cdot=\frac{\partial}{\partial\tau}$. In terms of the Fourier modes

the equation of motion reads

On the worldsheet, the comoving coordinate $r$ is equal to the static coordinate $z$, and now ${}^{\prime}=\frac{\partial}{\partial z}$. To make the action finite, we introduce a cutoff scale by putting the boundary brane at the finite $z=z_{m}$. Then, with the normalization condition $y_{\omega}(z_{m})=1$, the equation (14) has two independent solutions

where $Y_{\omega}(z)$ ($Y^{*}_{\omega}(z)$) is the incoming (outgoing) wave near the horizon $z=b$. We would like to stress that these are the exact solutions, whereas in Xiao_08 only the approximate solutions in the small-$\omega$ expansion are given.

Using the holographic prescription, the retarded Green’s function of the fields coupled to one end of the string is

which is analytic in the upper half complex $\omega$-plane. Its small $\omega$ limit agrees with the results in Xiao_08 and also our previous papers Lee_13 , giving the effects to the particle of the temperature-dependent damping term $\gamma_{T}=R^{2}T_{0}/b^{2}$ and the effective mass term $M_{T}=M_{0}+\Delta M_{T}$. $M_{T}$ consists of the large temperature-independent mass $M_{0}=R^{2}T_{0}/z_{m}$ and the small temperature-dependent mass correction $\Delta M_{T}=-R^{2}T_{0}z_{m}/b^{2}$ in the limit of $z_{m}/b\rightarrow 0$. This picture of the particle dynamics is captured in the Langevin equation Son_09 ; Lee_19 ,

where $q(\tau)$ is the position of the particle and $\eta(\tau)$ is the noise force with $\langle\eta(\tau)\eta(\tau^{\prime})\rangle$ given by the Hadamard function of the field, $G_{H}(\tau-\tau^{\prime})$. This can also be used to obtain the particle correlator, which in frequency space is given by

This equation incorporates not only the dissipation effect given by the retarded Green’s function $G_{R}$ of the field in (16) but also the force-force correlation function in terms of the Hadamard function that satisfies the fluctuation-dissipation theorem, namely $G_{H}(\omega)=\coth(\omega/T)\,{\rm{Im}}G_{R}(\omega)$ (this relation was shown to be true in very general holographic setups Dimitrio_18 ). To estimate the late time behavior of the particle correlator, we consider the retarded Green’s function in the small $\omega$ expansion in (16). Through the fluctuation-dissipation theorem, the Hadamard function $G_{H}\simeq T\gamma_{T}$. So we have $\langle\tilde{q}(\omega)\tilde{q}(-\omega)\rangle\propto\omega^{-2}$. Thus, to obtain the finite particle correlator, we need to introduce an infrared cutoff, for example, by adding a potential trap $\Omega^{2}q^{2}(\tau)$ to the end of the string¹¹1Another way to introduce the cutoff is by deforming the integral to the complex $\omega$-plane, where the double pole contribution at $\omega=0$ also gives the correlator linear in $\tau$. This prescription can be naturally incorporated in defining different type of Green’s functions by the contour integrals. Here, the potential trap cut-off is more natural as we want to make the saturated entanglement entropy finite as in Lee_19 ., and this gives in the small $\omega$ limit,

When taking away the cutoff, $\Omega\rightarrow 0$, we have the cutoff independent term that grows linearly in $\tau$. This signals the late-time growth of uncertainty in the particle position for observers moving together with the particle. Notice that here $\omega$ is the frequency corresponding to the proper time for the particle. For the particle correlator in the inertial time, $t$, we can see from (38) and the fact that in the late time $\tau=b\ln(t/b)$,

In Lee_19 , we have used this particle correlator, in the setup of strings with only one end on the boundary, to calculate the entanglement entropy between particles and environment fields in the late times, and in that case the divergent piece in $\langle q(t)q(0)\rangle$ is absorbed into the renormalized saturated entropy. Here, we are more interested in the dynamics of two entangled particles in the current paper, but we will leave the detailed study for future works. For the moment, we can qualitatively see the rate for information exchange between a single particle and the field to which it couples.

Furthermore, as far as we know, via a bilinear coupling between a particle and environmental fields, the temperature-dependent damping term can only be achieved from the holographic approach, whereas in the free field theory, the same type of the coupling leads to the state-independent back reaction from the field to the particle JT_1 . It is also worth mentioning that when two ends of the string undergo the acceleration, they are never in causal contact during their journeys. In this case, the energy of the particle transferred to the field does not have a chance to come back, which gives the dissipative dynamics of the particles. Moreover, due to the causality consideration, the retarded Green’s function of the particle has no dependence on the distance between two particles. In contrast, in the static string case, two ends are initially causally disconnected and then become connected at the time scale of the distance between two ends. It thus induces a somewhat different dispersion relation of the particle after two ends reach the causal contact.

To obtain the force-force correlations between two ends of the string, which encode the information of the wormhole, we need to find the generating functional for the field that couples to them. For this purpose, we transform the solution in (15) back to the coordinates in (1). The $R(L)$ branch corresponds to the $+(-)$ background solution in (4). Also, notice that the $R$ and $L$ branches have different relations between the proper time $\tau$ and the coordinate time $t$. Thus, from (9) $\tau_{R/L}=\mp\tanh^{-1}\frac{t}{\sqrt{b^{2}+t^{2}-z^{2}}}$. Then we can write the general solutions as

The boundary values of $q^{R/L}(t,z)$ are interpreted holographically as the sources for the quark and anti-quark, $q^{R/L}(t,z_{m})\equiv q_{0}^{R/L}(t)$. And their “Fourier transform” is defined as

where $u(t)\equiv b\tanh^{-1}\frac{t}{\sqrt{b^{2}+t^{2}-z_{m}^{2}}}$. Thus we have two boundary conditions at $z=z_{m}$ as

The other two boundary conditions for $q^{R/L}(t,z)$ are imposed at the horizon $z=b$. As in Son_03 , the analyticity properties of the boundary conditions for the mode functions cross the horizon of a maximally extended black hole gives the Schwinger-Keldysh correlators for a single particle. In our previous work Lee_15 , we also obtained the same set of correlators by imposing the constraints from the unitarity and periodicity of the finite temperature boundary correlators. In the setting of this paper, the maximally extended black hole does not describe the double copy of a single particle action giving the Schwinger-Keldysh correlators, but the action of the two coupled particles. To achieve it, we will modify the definition of the Kruskal coordinates in Son_03 . We will also give the prescription from the field theory constraints similar to the ones in Lee_15 .

Near the horizon in (10), the mode functions can be simplified when the Kruskal coordinates, say in the $R$ branch, are introduced as

where $r^{*}=b\tanh^{-1}\frac{r}{b}$. Then the mode function can be written as

Notice that $UV=\frac{r-b}{r+b}$. Therefore, we can even define the mode functions inside the horizon when $UV>0$. A similar definition of the mode functions for the $L$ branch is adopted, where now $r^{*}=-b\tanh^{-1}\frac{r}{b}$. So, the value of $r^{*}$ is negative in the $L$ branch and becomes positive in the $R$ branch. In the $L$ branch,

Note that $\tau$ in the $L$ branch has an opposite sign relative to the $R$ branch. Then the mode function in the $L$ branch can be written down as

To match $q^{R}$ with $q^{L}$, we require them to be analytic on the real $U$ and $V$ axes and the lower half of complex $U$ and $V$ planes as in Son_03 . Thus, the matching conditions are given by

and together with (23), they uniquely fix the bulk mode functions $q^{R/L}(t,z)$ with

The bulk on-shell action for this solution, which contains only the boundary terms, is identified as a generating functional for the boundary field correlators. Leaving out the contact terms that should be related to the renormalization of the particle self-energy, we have

with $G_{R}(\omega)$ given in (16). Then we can read off the correlators $G^{ij}(\omega)=\frac{\delta}{\delta\tilde{q}^{i}_{0}}\frac{\delta}{\delta\tilde{q}^{j}_{0}}S[\tilde{q}^{R}_{0},\tilde{q}^{L}_{0}]$ with $i,j=R,L$, and the real-time correlators:

Here, $\tau^{R}(t)=u(t)$ and $(-1)^{R}=1$, while $\tau^{L}(t)=-u(t)$ and $(-1)^{L}=-1$.

From the field theory point of view, we consider the particles $\tilde{q}_{0}^{R/L}$ coupled to the common quantum field at finite temperature through a bilinear coupling with the same coupling strength. Integrating out the degrees of freedom of the field leads to the effective action of the particles. Thus, the generating function of the field requires to be symmetric under the exchange of $R$ and $L$, leading to $G^{RL}=G^{LR}$ and $G^{RR}=G^{LL}$. Also, the Green’s functions $G^{RR}$ and $G^{LL}$ should have the form of the finite temperature time-ordered Green’s functions at temperature $T=\frac{1}{2\pi b}$. The above requirements turn out to be equivalent to (28) from the bulk analyticity condition.

Notice that $G^{RR}(\omega)=G^{LL}(\omega)$ has the same form as the finite-temperature time-order correlators. Even though we have not studied the coupled Langevin equations, the correlation between two entangled particles is, in principle, encoded in the cross correlators, $G^{LR}(\omega)$ and $G^{RL}(\omega)$, just like in the case of the single-particle dynamics (18). $G^{LR}(t,0)=G^{RL}(0,t)$ is plotted numerically in Fig. 2.

We turn to the early-time and late-time analytic behaviors of the $G^{LR}(t,0)=G^{RL}(0,t)$ correlators. Note that the retarded Green’s function, $G_{R}(\omega)$ in (16), has the pole at $\omega=-\frac{i}{z_{m}}$, and the Boltzmann factor gives the poles at $\omega=\frac{in}{b}$ with non-zero integers. We take $b\gg z_{m}$, where $b$ plays a role of an IR cut-off and $z_{m}$ a UV cut-off. Thus, in the limit $t\ll z_{m}$ with the proper time $u(t)\simeq t$, the time evolution of $G^{LR}(t,0)=G^{RL}(t,0)$ is dominated by the pole at $\omega=-\frac{i}{z_{m}}$ giving

As for $t\gg b$ with $u(t)\simeq b\ln(t/b)$, the pole at $\omega=\frac{-i}{b}$ dominates the correlators $G^{ij}(t,0)$ resulting in the power-law decay

This can be compared with the correlators of the free scalar fields on the spacetime points of two observers following the trajectories, say along the $x$-direction with the uniform acceleration $1/b$. The worldlines of two observers are specified by $x_{L}^{\mu}=(b\sinh\frac{\tau_{L}}{b},-b\cosh\frac{\tau_{L}}{b},0,0)$ and $x_{R}^{\mu}=(b\sinh\frac{\tau_{R}}{b},b\cosh\frac{\tau_{R}}{b},0,0)$. In particular, the correlator of the free scalar field at the spacetime point $x_{L}^{\mu}$ and $x_{R}^{\mu}$ is given by

Thus, when $t\ll b$ giving the proper time $\tau\simeq t$,

whereas for $t\gg b$ giving $\tau\simeq b\ln(t/b)$,

instead. By comparing with (40), we see that for strongly coupled fields the correlation between fields at two spacelike-separate points dies out more quickly than the one for free fields, and this potentially induces the interaction between two spacelike-separate particles and gives a shorter time scale for the particles to sustain quantum entanglement between them.