Batalin-Tyutin quantization of dynamical boundary of AdS₂

Some decades ago, Jackiw and Teitelboim (JT) proposed a two-dimensional model for anti-de Sitter (AdS) space Jackiw:1984je ; Teitelboim:1983ux . Recently, Almheiri and Polchinski (AP) also studied a modified model allowing one to set up more meaningful holographic dictionary through analysis of the boundary dynamics Almheiri:2014cka . Since the holographic AdS/CFT dictionary is still incomplete, such lower dimensional examples amenable to our needs would provide an essential feature of AdS/CFT correspondence. In this regard, Engelsöy, Mertens, and Verlinde studied the black hole evaporation process from AdS₂ holography Engelsoy:2016xyb . One of the interesting conclusions is that the time coordinate becomes dynamical on the boundary, and the one-dimensional boundary action is given by the Schwarzian derivative Maldacena:2016upp . They also obtained the relevant quantum-mechanical Hamiltonian by taking into account a coupling between the bulk and the boundary.

The above Hamiltonian for the dynamical boundary of AdS₂ can be translated into a corresponding Lagrangian through the Legendre transformation. Then, we immediately find two primary constraints from the definition of momenta, but they are unexpectedly second-class without secondary constraints when classified by the Dirac method dirac2001lectures . What second-class constraints imply is that a local symmetry implemented by constraints as symmetry generators would be broken. If the second-class constraint system is converted into a first-class one, then the remaining quantization process will follow ordinary methods in Refs. Fradkin:1975cq ; Henneaux:1985kr ; Becchi:1975nq ; Kugo:1979gm ; Batalin:1986fm ; Batalin:1991jm .

In fact, there are largely two ways to realize first-class constraint systems. The first one is to use the action. Faddeev and Shatashivili Faddeev:1986pc introduced the Wess-Zumino action Wess:1971yu in order to cancel out the gauge anomaly responsible for second-class constraint algebra and they eventually obtained the first-class constraint algebra. Subsequently, the Wess-Zumino action to cancel the gauge anomaly was also studied in Refs. Babelon:1986sv ; Harada:1986wb ; Miyake:1987dk . The second one is to use the Hamiltonian formalism such as Batalin-Tyutin Hamiltonian method Batalin:1991jm . Interestingly, Banerjee Banerjee:1993pm applied the method to the Chern-Simons field theory of second-class constraint system and obtained a strongly involutive constraint algebra in an extended phase space, which yields a new Wess-Zumino type action which cannot be derived from the action level. And its non-Abelian extension was also done in Ref. Kim:1994np . In addition, the method was applied to wide variety of cases of interest: anomalous gauge theory Fujiwara:1989ia ; Kim:1992ey ; Banerjee:1993pj ; Fujiwara:1990rx , non-gauge theories Hong:1999gx ; Hong:2000bp ; Hong:2000ex , and chiral bosons Amorim:1994ft ; Amorim:1994np ; Kim:2006za .

In this paper, we will consider a Lagrangian describing the dynamical boundary of AdS₂ compatible with the Hamiltonian in Ref. Engelsoy:2016xyb . Then, we obtain two primary constraints from the definition of momenta; however, they turn out to be second-class constraints. Thus, we would like to study how to get the first-class constraint system by the use of the Batalin-Tyutin Hamiltonian method. The organization of the paper is as follows. In Sec. II, we will recapitulate the derivation of the Hamiltonian for the dynamical boundary of AdS₂ from the AP model. In Sec. III, we obtain the corresponding Lagrangian to the Hamiltonian studied in Sec. II. The two primary constraints are shown to be second-class so that the Lagrange multipliers are fully determined without any further secondary constraints. In Sec. IV, using the Batalin-Tyutin Hamiltonian method, we realize the first-class constraint system and obtain the involutive Hamiltonian. Finally, a conclusion will be given in Sec. V.

In the AP model Almheiri:2014cka , we encapsulate the derivation of the Hamiltonian describing the dynamical boundary of AdS₂ from a holographic renormalization process Almheiri:2014cka ; Engelsoy:2016xyb . The action is given as

where $S_{\rm AP}$ is the AP action Almheiri:2014cka , $S_{\rm GH}$ is the Gibbons-Hawking term Gibbons:1976ue , $S_{\rm matt}$ is some arbitrary matter system coupled to the two dimensional metric $\differential s^{2}=g_{\mu\nu}\differential x^{\mu}\differential x^{\nu}$, and $\Phi$ is a dilaton. In the conformal gauge, the length element is written as $\differential s^{2}=-e^{2\omega(X^{+},X^{-})}\differential X^{+}\differential X^{-}$ where $X^{\pm}=X^{0}\pm X^{1}=T\pm Z$. From the action (1), equations of motion are derived as

where the stress tensor for matter is $T_{\mu\nu}=-(2/\sqrt{-g})\delta{S_{\rm matt}}/\delta{g^{\mu\nu}}$. The general solution to Eq. (4) is obtained as the AdS₂ geometry

For $N$ conformal fields of $T_{+-}=0$, the dilaton solution takes the following form:

where integrated source terms are given by $I_{+}=\int_{X^{+}}^{\infty}\differential s(s-X^{+})(s-X^{-})T_{++}(s)$ and $I_{-}=\int_{-\infty}^{X^{-}}\differential s(s-X^{+})(s-X^{-})T_{--}(s)$.

If the conformal matter fields received quantum corrections, then the stress tensor for matter would be no longer traceless. In our work, we will focus on the classically conformal matter fields without trace anomaly. The integration constant $a$ in Eq. (8) is assumed to be positive, which prevents strong coupling singularity from reaching the boundary Almheiri:2014cka . In particular, the infalling source is assumed to be $T_{++}=0$ and $T_{--}=E\delta(s)$, then Eq. (8) can be explicitly expressed by

where $\kappa=8\pi G/a$ and $\Theta$ is a step function. Note that the Poincaré vacuum and the massive black hole are characterized by dilaton profile for $X^{-}<0$ and for $X^{-}>0$, respectively. For the latter case, the black hole can be expressed in terms of a static form by the use of the coordinate transformations $X^{\pm}(\sigma^{\pm})=({1/\sqrt{\kappa E}})\tanh(\sqrt{\kappa E}\sigma^{\pm})$, where $\sigma^{\pm}=t\pm\sigma$. Then, the future and the past horizon are found at $X^{+}(\infty)\to 1/\sqrt{\kappa E}$ and $X^{-}(-\infty)\to-1/\sqrt{\kappa E}$, respectively. In addition, the unperturbed boundary is located at $X^{+}(\sigma^{+})=X^{-}(\sigma^{-})$ which is assumed to be coincident with $\sigma^{+}=\sigma^{-}$, i.e., $\sigma^{1}=0$. Then, a dynamical boundary time is naturally defined as $X^{+}(t)=X^{-}(t)=T(t)$.

Using an infinitesimally small cut-off $\epsilon$ from the unperturbed boundary, one can define two quantities such as $X^{+}(t+\epsilon)+X^{-}(t-\epsilon)=2T(t)$ and $X^{+}(t+\epsilon)-X^{-}(t-\epsilon)=2\epsilon\dot{T}(t)$. In fact, the essential requirement in Ref. Engelsoy:2016xyb is that the asymptotic behaviour of the dilaton is the same as that of the Poincaré patch at the boundary so that

which dictates $\dot{T}(t)=1-\kappa(I_{+}(t)+I_{-}(t))$. Hence, the equation of motion for the dynamical boundary can be obtained as

where $P_{+}=\int_{T}^{\infty}\differential sT_{++}(s)$ and $P_{-}=-\int_{-\infty}^{T}\differential sT_{--}(s)$. Using Eq. (11), near the boundary, one can get the metric (7) and the dilaton (8) as

where $\{T,t\}=\dddot{T}/\dot{T}-(3/2)(\ddot{T}/\dot{T})^{2}$ is the Schwarzian derivative.

Let us now get a boundary stress tensor of the dual CFT through the holographic renormalization procedure Almheiri:2014cka ; Engelsoy:2016xyb . Varying the renormalized on-shell action of $S_{\rm ren}=S_{\rm AP}+S_{\rm ct}$ including a counter term $S_{\rm ct}=1/(8\pi G)\int\differential t\sqrt{-\gamma}(1-\Phi^{2})$ with respect to the boundary metric $\hat{\gamma}_{tt}$, one can obtain

where $\hat{\gamma}^{tt}=\lim_{\epsilon\to 0}\gamma^{tt}/\epsilon^{2}$ is the metric of the boundary. Thus, the boundary stress tensor is obtained as

by plugging Eq. (12) into Eq. (14). The boundary stress tensor $\langle\hat{T}_{tt}\rangle$ reflects excitations of the boundary in terms of the static time coordinate $t$ and it must be a Hamiltonian for the dynamical boundary. In order to describe the coupling between the matter sector and the dynamical boundary theory, the authors in Ref. Engelsoy:2016xyb introduced a new variable $\varphi=\log\dot{T}$ and then arrived at the Hamiltonian

where $\pi_{\varphi}$ and $\pi_{T}$ are conjugate momenta corresponding to $\varphi$ and $T$, respectively, and the Hamiltonian reduces to $H_{\rm EMV}=\kappa\pi_{\varphi}^{2}+e^{\varphi}(P_{+}-P_{-})$ upon setting $\pi_{T}=0$.

We derive a Lagrangian corresponding to the Hamiltonian (16) by means of the canonical path-integral. Thus, we consider the partition function as

In Eq. (17), the integration with respect to $\pi_{T}$ gives the delta functional, which is rewritten in terms of a new variable $\lambda$ in the last line. Thus, we obtain the action describing the dynamical boundary of AdS₂ as

where $\lambda$ plays the role of Lagrange multiplier to implement $\dot{T}-e^{\varphi}=0$ and $P_{\pm}$ are assumed to be background sources.

The canonical momenta conjugate to variables $(\varphi,T,\lambda)$ are defined as

Then, the canonical Hamiltonian is obtained through the Legendre transformation of the Lagrangian (18) as

which is the same as Eq. (16) as it must be when $\lambda$ is replaced by $\pi_{T}$. The standard Poisson brackets are imposed as follows,

In Eq. (19), we now identify two primary constraints as dirac2001lectures

and then construct the primary Hamiltonian by adding the two primary constraints to the canonical Hamiltonian as

where $u_{1}$ and $u_{2}$ are arbitrary Lagrange multipliers. The stability of primary constraints with respect to time evolution is

Note that the two Lagrange multipliers can be chosen as $u_{1}=e^{\varphi}$ and $u_{2}=0$, and thus, they are fully fixed since the primary constraints are second-class. Accordingly, the Dirac bracket between canonical variables can be defined as

where the Dirac matrix is $C_{ij}=\{\Omega_{i},\Omega_{j}\}_{\rm PB}=-\epsilon_{ij}$ with $\epsilon_{12}=1$. Hence, the non-vanishing Dirac brackets are

The equations of motion are given as

In Eq. (28), the combined first-order differential equations result in

In Eq. (29), $\pi_{T}$ turns out to be constant so that $\lambda$ must also be constant through the constraint $\Omega_{1}$. For simplicity, if we set $\lambda=0$, then Eq. (30) is identical with the dynamical equation of motion (11) and the reduced Hamiltonian is $H_{r}=\kappa\pi^{2}_{\varphi}+e^{\varphi}(P_{+}-P_{-})$.

Consequently, the boundary system dual to the bulk AdS₂ is found to be the second-class constraint system. Thus, we will make the second-class system to the first-class one in virtue of the Batalin-Tyutin Hamiltonian embedding.

Following the Batalin-Tyutin quantization method Batalin:1991jm , we introduce new auxiliary one-dimensional fields $\theta^{1}$ and $\theta^{2}$ in order to convert second-class constraints into first-class ones in the extended phase space. Let us assume the Poisson algebra satisfying

In the extended phase space, the modified constraints $\tilde{\Omega}_{i}$ are assumed to be Batalin:1991jm

with the boundary condition $\tilde{\Omega}_{i}=\Omega_{i}$ for $\theta^{i}=0$. The first order correction is

for some matrix $X_{ij}$, and thus, we require

As was emphasized in Ref. Banerjee:1993pm , there exists an arbitrariness in choosing $\omega_{ij}$ and $X_{ij}$, which corresponds to canonical transformation in the extended phase space Batalin:1986fm ; Batalin:1991jm . Now, the simplest choice for $\omega^{ij}$ and $X_{ij}$ is

where $\epsilon^{12}=-1$. Thus, the modified constraints in the extended phase space are given as

without any higher order corrections.

On the other hand, in the extended phase space, the involutive Hamiltonian is defined by Batalin:1991jm

with the $n$th order correction $H^{(n)}$ being

where $\omega_{ij}$ and $X^{ij}$ are inverse matrices of $\omega^{ij}$ and $X_{ij}$, respectively. The generating functions $G^{(n)}_{a}$ are given as

where $\mathcal{O}$ implies that the given Poisson brackets are calculated with respect to the original variables. Explicitly, they are obtained as

Hence, there only exists a linear order correction so that

The final expression for the involutive Hamiltonian (37) is given as

and the time evolution of the modified constraints does not generate any more constraints since

In the above analysis for the constraint system, we see that the original second-class constraint system can be converted into the first-class system by introducing two auxiliary fields in the extended phase space.

Next, let us consider the phase space partition function,

where $S^{\prime}=\int\differential t(\pi_{\varphi}\dot{\varphi}+\pi_{T}\dot{T}+\pi_{\lambda}\dot{\lambda}+\theta^{2}\dot{\theta}^{1}-\tilde{H})$ and $\Gamma_{i}$ are gauge conditions. Integrating out the momenta of $\pi_{\varphi},\pi_{T},\pi_{\lambda}$ in Eq. (45), we finally get

Expectedly, if we choose the unitary gauge condition as $\Gamma_{i}=\theta^{i}=0$, the extended theory reduces to the original action (18). The action (47) is the new type of Wess-Zumino action derived from the Batalin-Tyutin Hamiltonian formalism. Note that this formalism, embedding of familiar second-class constraint systems  Fujiwara:1989ia ; Kim:1992ey ; Banerjee:1993pj ; Fujiwara:1990rx reproduces the Wess-Zumino action derived from usual Lagrangian procedures Babelon:1986sv ; Harada:1986wb ; Miyake:1987dk . The present Batalin-Tyutin formulation is one of the different applications to non-gauge theories studied in Refs. Hong:1999gx ; Hong:2000bp ; Hong:2000ex ; Amorim:1994ft ; Amorim:1994np ; Kim:2006za , so the origin of the second-class nature is not the result of a genuine gauge symmetry breaking.

Now, let us elaborate gauge conditions and discuss the role of the auxiliary fields $\theta^{1}$ and $\theta^{2}$. The total action (46) which consists of the original action and the Wess-Zumino action can be neatly rewritten as

Note that the modified constraints (36) as symmetry generators indicate that the action (48) is invariant under the following local transformations

implemented by a local symmetry generator

where $\epsilon_{1}(t)$ and $\epsilon_{2}(t)$ are arbitrary local parameters. The auxiliary fields can be eliminated by choosing gauge conditions. Choosing special local parameters, one can take an unitary gauge condition such as $\Gamma_{1}=\theta^{1}=0$ and $\Gamma_{2}=\theta^{2}=0$, which results in the original action (18). In that sense, the auxiliary degrees of freedom are gauge artefacts in this special gauge. However, thanks to the local symmetry, a different kind of gauge condition, for example, $\Gamma_{1}=\lambda=0$ and $\Gamma_{2}=T=0$ can be chosen. Then, the reduced action becomes

where physical contents are the same as those of the original action (18) because $\theta_{1}$ and $\theta_{2}$ play the role of $\lambda$ and $T$. Consequently, the original theory turns out to be the gauge fixed version of the extended theory.

On the other hand, the AdS/CFT correspondence tells us that the classical action on the gravity side is the quantum effective action for the dual conformal theory on the boundary. In the present JT model coupled to matter, the boundary theory could be described by the quantum-mechanical Hamiltonian (16). However, the Hamiltonian system on the boundary was found to belong to the second-class constraint system which would indicate that a certain local symmetry for the boundary theory was broken. In order to retrieve the broken local symmetry, the auxiliary degrees of freedom were added without changing net physical degrees of freedom in the Hamiltonian. In the spirit of the AdS/CFT correspondence, one might ask what the extended bulk theory corresponding to the extended boundary theory (46) is. The total bulk system may have a richer symmetry-structure compared to that of the original AdS₂. This issue was unsolved.

In conclusion, the dynamical boundary of the two-dimensional AdS space, described by the one-dimensional Hamiltonian having a coupling between the bulk and boundary system, we obtained the Lagrangian corresponding to the Hamiltonian. In Dirac’s constraint analysis, there were two primary constraints which are fully second-class. In order to convert the second-class constraint system into the first-class constraint system, we employed the Batalin-Tyutin Hamiltonian method and obtained the closed constraint algebra in the extended space. The extended system, of course, reduces to the original one for the unitary gauge condition. From the viewpoint of the AdS/CFT correspondence, it raises a question regarding the existence of the extended bulk gravity corresponding to the extended boundary theory, which deserves further study.

As a comment, the Hamiltonian (16) reproduces the equation of motion for the dynamical boundary of AdS₂ (11) upon setting $\pi_{T}=0$ Engelsoy:2016xyb . In the Dirac method also, Eq. (11) was obtained by setting $\lambda=0$ in Eq. (20), which is actually the same condition as $\pi_{T}=0$ because they are related to each other through the constraint (22). Thus, one might wonder how about introducing additional constraint to enforce $\lambda=0$ by means of new auxiliary $\lambda_{1}$. Now, we can add $\lambda\lambda_{1}$ to the starting action (18). After some tedious calculations, we find that $\lambda_{1}$ still exists in the final equation of motion, so we need to introduce additional term $\lambda_{1}\lambda_{2}$ to enforce $\lambda_{1}=0$. Unfortunately, the repeated infinite process would not warrant the condition $\lambda=0$. We hope this issue will be addressed elsewhere.