Batalin-Tyutin quantization of dynamical boundary of AdS$_2$

In a two-dimensional AdS space, a dynamical boundary of AdS space was described by a one-dimensional quantum-mechanical Hamiltonian with a coupling between the bulk and boundary system. In this paper, we present a Lagrangian corresponding to the Hamiltonian through the Legendre transformation with a constraint. In Dirac's constraint analysis, we find two primary constraints without secondary constraints; however, they are fully second-class. In order to make the second-class constraint system into a first-class constraint system, we employ the Batalin-Tyutin Hamiltonian method, where the extended system reduces to the original one for the unitary gauge condition. In the spirit of the AdS/CFT correspondence, it raises a question whether a well-defined extended bulk theory corresponding to the extended boundary theory can exist or not.


I. INTRODUCTION
Some decades ago, Jackiw and Teitelboim (JT) proposed a two-dimensional model for anti-de Sitter (AdS) space [1,2]. 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 [3]. 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 AdS 2 holography [4]. 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 [5]. 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 AdS 2 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 secondclass without secondary constraints when classified by the Dirac method [6]. What secondclass 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 firstclass one, then the remaining quantization process will follow ordinary methods in Refs. [7][8][9][10][11][12].
In fact, there are largely two ways to realize first-class constraint systems. The first one is to use the action. Faddeev and Shatashivili [13] introduced the Wess-Zumino action [14] 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. [15][16][17]. The second one is to use the Hamiltonian formalism such as Batalin-Tyutin Hamiltonian method [12].
Interestingly, Banerjee [18] applied the method to the Chern-Simons field theory of secondclass 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. [19]. In addition, the method was applied to wide variety of cases of interest: anomalous gauge theory [20][21][22][23], non-gauge theories [24][25][26], and chiral bosons [27][28][29].
In this paper, we will consider a Lagrangian describing the dynamical boundary of AdS 2 compatible with the Hamiltonian in Ref. [4]. 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 AdS 2 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.

II. HOLOGRAPHIC RENORMALIZATION OF THE AP MODEL
In the AP model [3], we encapsulate the derivation of the Hamiltonian describing the dynamical boundary of AdS 2 from a holographic renormalization process [3,4]. The action is given as where S AP is the AP action [3], S GH is the Gibbons-Hawking term [30], S matt is some arbitrary matter system coupled to the two dimensional metric ds 2 = g µν dx µ dx ν , and Φ is a dilaton.
In the conformal gauge, the length element is written as From the action (1), equations of motion are derived as where the stress tensor for matter is T µν = −(2/ √ −g)δS matt /δg µν . The general solution to Eq. (4) is obtained as the AdS 2 geometry For N conformal fields of T +− = 0, the dilaton solution takes the following form: where integrated source terms are given by 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 [3]. In particular, the infalling source is assumed to be T ++ = 0 and T −− = Eδ(s), then Eq. (8) can be explicitly expressed by where κ = 8πG/a and Θ 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 Then, the future and the past horizon are found at respectively. In addition, the unperturbed boundary is located at is assumed to be coincident with σ + = σ − , i.e., σ 1 = 0. Then, a dynamical boundary time Using an infinitesimally small cut-off ǫ from the unperturbed boundary, one can define two quantities such as X In fact, the essential requirement in Ref. [4] is that the asymptotic behaviour of the dilaton is the same as that of the Poincaré patch at the boundary so that which dictatesṪ (t) = 1 − κ(I + (t) + I − (t)). Hence, the equation of motion for the dynamical boundary can be obtained as 1 2κ where (11), near the boundary, one can get the metric (7) and the dilaton (8) as where {T, t} = ...
Let us now get a boundary stress tensor of the dual CFT through the holographic renormalization procedure [3,4]. Varying the renormalized on-shell action of S ren = S AP + S ct including a counter term S ct = 1/(8πG) dt √ −γ(1 − Φ 2 ) with respect to the boundary metricγ tt , one can obtain whereγ tt = lim ǫ→0 γ tt /ǫ 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 T tt 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. [4] introduced a new variable ϕ = logṪ and then arrived at the Hamiltonian where π ϕ and π T are conjugate momenta corresponding to ϕ and T , respectively, and the Hamiltonian reduces to H EMV = κπ 2 ϕ + e ϕ (P + − P − ) upon setting π T = 0.

III. HAMILTONIAN FORMULATION OF THE DYNAMICAL BOUNDARY
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 π T gives the delta functional, which is rewritten in terms of a new variable λ in the last line. Thus, we obtain the action describing the dynamical boundary of AdS 2 as where λ plays the role of Lagrange multiplier to implementṪ − e ϕ = 0 and P ± are assumed to be background sources.
The canonical momenta conjugate to variables (ϕ, T, λ) 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 λ is replaced by π T . The standard Poisson brackets are imposed as follows, In Eq. (19), we now identify two primary constraints as [6] 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 ϕ 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 = {Ω i , Ω j } PB = −ǫ ij with ǫ 12 = 1. Hence, the non-vanishing Dirac brackets are The equations of motion are given aṡ In Eq. (28), the combined first-order differential equations result in 1 2κφ + (λ + P + − P − )e ϕ = 0.
In Eq. (29), π T turns out to be constant so that λ must also be constant through the constraint Ω 1 . For simplicity, if we set λ = 0, then Eq. (30) is identical with the dynamical equation of motion (11) and the reduced Hamiltonian is H r = κπ 2 ϕ + e ϕ (P + − P − ). Consequently, the boundary system dual to the bulk AdS 2 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.

IV. BATALIN-TYUTIN HAMILTONIAN QUANTIZATION
Following the Batalin-Tyutin quantization method [12], we introduce new auxiliary onedimensional fields θ 1 and θ 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 constraintsΩ i are assumed to be [12] with the boundary conditionΩ i = Ω i for θ i = 0. The first order correction is for some matrix X ij , and thus, we require As was emphasized in Ref. [18], there exists an arbitrariness in choosing ω ij and X ij , which corresponds to canonical transformation in the extended phase space [11,12]. Now, the simplest choice for ω ij and X ij is where ǫ 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 [12] with the nth order correction H (n) being where ω ij and X ij are inverse matrices of ω ij and X ij , respectively. The generating functions a are given as where 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 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 ′ = dt(π ϕφ + π TṪ + π λλ + θ 2θ1 −H) and Γ i are gauge conditions. Integrating out the momenta of π ϕ , π T , π λ in Eq. (45), we finally get Expectedly, if we choose the unitary gauge condition as Γ i = θ 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 [20][21][22][23] reproduces the Wess-Zumino action derived from usual Lagrangian procedures [15][16][17]. The present Batalin-Tyutin formulation is one of the different applications to non-gauge theories studied in Refs. [24][25][26][27][28][29], 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 θ 1 and θ 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 ǫ 1 (t) and ǫ 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 Γ 1 = θ 1 = 0 and Γ 2 = θ 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, Γ 1 = λ = 0 and Γ 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 θ 1 and θ 2 play the role of λ 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 AdS 2 . This issue was unsolved.

V. CONCLUSION
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 AdS 2 (11) upon setting π T = 0 [4]. In the Dirac method also, Eq. (11) was obtained by setting λ = 0 in Eq. (20), which is actually the same condition as π T = 0 because they are related to each other through the constraint (22). Thus, one might wonder how about introducing additional constraint to enforce λ = 0 by means of new auxiliary λ 1 . Now, we can add λλ 1 to the starting action (18). After some tedious calculations, we find that λ 1 still exists in the final equation of motion, so we need to introduce additional term λ 1 λ 2 to enforce λ 1 = 0. Unfortunately, the repeated infinite process would not warrant the condition λ = 0. We hope this issue will be addressed elsewhere.