AdS/CFT—Non-equilibrium

Abstract

The GKP-Witten relation is the most important equation to apply AdS/CFT to nonequilibrium physics. We explain the relation using simple examples.

Appendix: More About Massive Scalars \({}^\blacklozenge \)

Holographic renormalization   A holographic renormalization is necessary to derive the field/operator correspondence for a massive scalar field. As an example, we consider the case $p=2$ and $m^2=-2$, or $(\varDelta _-, \varDelta _+)=(1,2)$.

The counterterm action is given by

$$\begin{aligned} \mathsf {S}_\text {CT} = - \frac{\varDelta _-}{2} \int d^{p+1}x\, \sqrt{-\gamma }\, \phi ^2. \end{aligned}$$

(10.70)

The on-shell action becomes

$$\begin{aligned} \underline{\mathsf {S}}= \underline{\mathsf {S}_\text {bulk}} + \underline{\mathsf {S}_\text {CT}} = \left. \int d^3x\, \left( \frac{1}{2u^2}\phi \phi ' - \frac{1}{2u^3}\phi ^2\right) \right| _{u=0}. \end{aligned}$$

(10.71)

The first term is the bulk contribution. Substituting the asymptotic form of the scalar

$$\begin{aligned} \phi \sim \phi ^{(0)}\left( u + \phi ^{(1)}u^2 \right) , \quad (u\rightarrow 0), \end{aligned}$$

(10.72)

one gets a finite result:

$$\begin{aligned} \underline{\mathsf {S}}= \int d^3x\, \frac{1}{2} \phi ^{(0)\,2}\phi ^{(1)}. \end{aligned}$$

(10.73)

Then, the one-point function is given by

$$\begin{aligned} \langle O\rangle _{s}= \frac{\delta \underline{\mathsf {S}}\big [\phi ^{(0)}\big ]}{\delta \phi ^{(0)}} = \phi ^{(1)}\phi ^{(0)}. \end{aligned}$$

(10.74)

The general formula is

$$\begin{aligned} \langle O\rangle _{s}&= (\varDelta _+ - \varDelta _-) \phi ^{(1)}\phi ^{(0)}\end{aligned}$$

(10.75)

$$\begin{aligned}&= \{ 2\varDelta _+ - (p+1) \} \phi ^{(1)}\phi ^{(0)}. \end{aligned}$$

(10.76)

When $p=3$ and $m=0$, one reproduces Eq. (10.20), $\langle O\rangle _{s}= 4\phi ^{(1)}\phi ^{(0)}$.

“Alternative quantization”   From the scaling dimension (10.42), the above procedure gives an operator with dimension $\varDelta _+ \ge (p+1)/2$. However, it is known that unitarity allows a local scalar operator with dimension $\varDelta \ge (p-1)/2$. The missing operators $(p-1)/2 \le \varDelta < (p+1)/2$ are provided by the slow falloff. When

$$\begin{aligned} -\frac{(p+1)^2}{4} \le m^2 \le - \frac{(p+1)^2}{4}+1, \end{aligned}$$

(10.77)

the slow falloff is also normalizable so that one can exchange the role of the external source and the operator [7]. The procedure we described so far is known as the “standard quantization” whereas this procedure is known as the “alternative quantization” (Fig. 10.2).

Fig. 10.2

Scaling dimension $\varDelta$ and $m^2$ from Eq. (10.42). On the thick curve, the falloffs can be interpreted as operators

For the alternative quantization, the counterterm action is given by

$$\begin{aligned} \mathsf {S}_\text {CT} = \int d^{p+1}x\, \sqrt{-\gamma }\, \left( \phi n^M \nabla _M \phi + \frac{\varDelta _-}{2} \phi ^2 \right) , \end{aligned}$$

(10.78)

where $n^M$ is the unit normal to the boundary, so $g_{\textit{MN}} n^{M}n^{N}=1$ and $n^u = -1/\sqrt{g_{uu}}$.

Substituting the asymptotic form of the scalar

$$\begin{aligned} \phi \sim \phi ^{(1)}\left( \phi ^{(0)}u + u^2 \right) , \quad (u\rightarrow 0), \end{aligned}$$

(10.79)

the on-shell action becomes

$$\begin{aligned} \underline{\mathsf {S}}= \underline{\mathsf {S}_\text {bulk}} + \underline{\mathsf {S}_\text {CT}}&= \left. \int d^3x\, \left( \frac{1}{2u^2} \phi \phi '- \frac{1}{u^2} \phi \phi ' +\frac{1}{2u^3} \phi ^2 \right) \right| _{u=0} \end{aligned}$$

(10.80)

$$\begin{aligned}&= -\int d^3x\, \frac{1}{2} \phi ^{(1)\,2}\phi ^{(0)}. \end{aligned}$$

(10.81)

The one-point function is given by

$$\begin{aligned} \langle O\rangle _{s}&= \frac{\delta \underline{\mathsf {S}}[\phi ^{(1)}]}{\delta \phi ^{(1)}} \end{aligned}$$

(10.82)

$$\begin{aligned}&= - \phi ^{(0)}\phi ^{(1)}= -(\varDelta _+ - \varDelta _-) \phi ^{(0)}\phi ^{(1)}. \end{aligned}$$

(10.83)

The result is just the exchange of the external source and the operator ($\phi ^{(0)}\leftrightarrow \phi ^{(1)}$ and $\varDelta _+\leftrightarrow \varDelta _-$).

It is often important to generalize the boundary conditions at the AdS boundary, and this provides one example.