Algebras and States in JT Gravity

The action of JT gravity with negative cosmological constant on a spacetime $M$ can be written, in the notation of HJ ; JK , as

$$I_{JT}=\int_{M}{\mathrm{d}}^{2}x\sqrt{-g}\upphi(R+2)+2\int_{\partial M}{\mathrm{d}}t\sqrt{|\gamma|}\upphi(K-1)+\cdots,$$

(1)

where $g$ is the bulk metric with curvature scalar $R$, $\gamma$ is the induced metric on the boundary, $K$ is the extrinsic curvature of the boundary, and we have omitted a topological invariant related to the classical entropy $S_{0}$. Upon integrating first over $\upphi$ to impose the equation of motion $R+2=0$, with a boundary condition that fixes the boundary value of $\upphi$, the action reduces to

$$I_{\partial M}=2\int_{\partial M}{\mathrm{d}}t\sqrt{|\gamma|}\upphi(K-1).$$

(2)

The condition $R+2=0$ implies that $M$ is locally isomorphic to a portion of ${\mathrm{AdS}}_{2}$, a homogeneous manifold of constant curvature $-2$. ${\mathrm{AdS}}_{2}$ is the universal cover of what we will call ${\mathrm{AdS}}_{2}^{(0)}$, namely the quadric $X^{2}+Y^{2}-Z^{2}=1$ with metric ${\mathrm{d}}s^{2}=-{\mathrm{d}}X^{2}-{\mathrm{d}}Y^{2}+{\mathrm{d}}Z^{2}$. ${\mathrm{AdS}}_{2}^{(0)}$ has an action of ${{SL}}(2,{\mathbb{R}})$ generated by vector fields

	$\displaystyle j_{1}$	$\displaystyle=X\partial_{Y}-Y\partial_{X}$		(3)
	$\displaystyle j_{2}$	$\displaystyle=Y\partial_{Z}+Z\partial_{Y}$		(4)
	$\displaystyle j_{3}$	$\displaystyle=-X\partial_{Z}-Z\partial_{X},$		(5)

satisfying $[j_{a},j_{b}]=\epsilon_{ab}^{c}j_{c}$, where the metric on the Lie algebra is $\eta_{ab}=\mathrm{diag}(-1,1,1)$. Coordinates $T,\sigma$ with

	$\displaystyle X$	$\displaystyle=\cos T\cosh\sigma$		(6)
	$\displaystyle Y$	$\displaystyle=\sin T\cosh\sigma$		(7)
	$\displaystyle Z$	$\displaystyle=\sinh\sigma$		(8)

give a useful parametrization of the universal cover ${\mathrm{AdS}}_{2}$. In these coordinates, the metric is

$${\mathrm{d}}s^{2}={\mathrm{d}}\sigma^{2}-\cosh^{2}\sigma\,{\mathrm{d}}T^{2},~{}~{}~{}~{}-\infty<\sigma,T<\infty.$$

(9)

The vector fields (3) on ${\mathrm{AdS}}_{2}^{(0)}$ lift to vector fields on ${\mathrm{AdS}}_{2}$ that generate an action of $\widetilde{{SL}}(2,{\mathbb{R}})$, the universal cover of ${{SL}}(2,{\mathbb{R}})$.

${\mathrm{AdS}}_{2}$ has a “right” conformal boundary at $\sigma\to+\infty$ and a “left” conformal boundary at $\sigma\to-\infty$. In studies of JT gravity, $M$ is usually taken to be “almost all” of ${\mathrm{AdS}}_{2}$ AP ; Malda . This is achieved as follows. First of all, the left and right boundaries could be defined simply by functions $\sigma_{L}(T)$, $\sigma_{R}(T)$. However, in “nearly ${\mathrm{AdS}}_{2}$ holography,” one assumes that the boundary is parameterised by a distinguished parameter $t$, the time of the boundary quantum mechanics, and one parametrizes the right boundary curve by functions $\sigma_{R}(t)$, $T_{R}(t)$, and similarly for the left boundary. To get nearly ${\mathrm{AdS}}_{2}$ spacetime, one further imposes the boundary conditions

$$\gamma_{tt}=-\frac{1}{\epsilon^{2}},~{}~{}~{}\upphi|_{\partial M}=\frac{\text{\textphi}_{b}}{\epsilon},$$

(10)

with constant $\text{\textphi}_{b}$ and with very small $\epsilon$. For small $\epsilon$, the condition $\gamma_{tt}=-1/\epsilon^{2}$ reduces to

$$e^{\sigma_{R}}=\frac{2}{\epsilon}\frac{1}{\dot{T}_{R}},~{}~{}e^{-\sigma_{L}}=\frac{2}{\epsilon}\frac{1}{\dot{T}_{L}},$$

(11)

where dots represent derivatives with respect to $t$. Thus, the left and right boundary curves lie, for small $\epsilon$, at very large negative or positive $\sigma$, and each of them is determined by a single function $T_{L}(t)$ or $T_{R}(t)$.

It is useful to define

$$e^{\sigma_{R}}=\frac{2\text{\textphi}_{b}}{\epsilon}e^{\widetilde{\sigma}_{R}},~{}~{}~{}e^{-\sigma_{L}}=\frac{2\text{\textphi}_{b}}{\epsilon}e^{-\widetilde{\sigma}_{L}},$$

(12)

where (in view of eqn. (11)) $\widetilde{\sigma}_{L}$, $\widetilde{\sigma}_{R}$ remain finite for $\epsilon\to 0$. Here $\widetilde{\sigma}_{L}$, $\widetilde{\sigma}_{R}$ are renormalized length parameters, in the sense that, for $\epsilon\to 0$, the length of a geodesic between the left and right boundaries is

$$\ell=\widetilde{\sigma}_{R}-\widetilde{\sigma}_{L}+{\mathrm{constant}}.$$

(13)

The constant depends only on $T_{L},\,T_{R}$ and not on $\widetilde{\sigma}_{L},\,\widetilde{\sigma}_{R}$.

With a small calculation, one finds that for $\epsilon\to 0$, the boundary action (2) becomes

$$I_{\partial M}=\text{\textphi}_{b}\int{\mathrm{d}}t\left(-{\dot{T}_{R}}^{2}+\left(\frac{\ddot{T}_{R}}{\dot{T}_{R}}\right)^{2}\right)+\text{\textphi}_{b}\int{\mathrm{d}}t\left(-{\dot{T}_{L}}^{2}+\left(\frac{\ddot{T}_{L}}{\dot{T}_{L}}\right)^{2}\right),$$

(14)

which is known as the Schwarzian action because it is a linear combination of the Schwarzian derivatives $\{T_{R},t\}$ and $\{T_{L},t\}$.

However, there is another convenient way to describe the problem KS ; ZY . One term in the boundary action (2) is just $-2\upphi L$, where $L$ is the length of $\partial M$; using the boundary condition on $\upphi$, this is $-\frac{2\text{\textphi}_{b}}{\epsilon}L$. The other term involving the integral of $K$ can be expressed, using the Gauss-Bonnet theorem, in terms of $\int_{M}{\mathrm{d}}^{2}x\sqrt{g}R$ (together with a topological invariant, the Euler characteristic of $M$); since $R=-2$, this is just $-2A$, with $A$ the area of $M$. Thus the boundary action is

$$I_{\partial M}=\frac{2\text{\textphi}_{b}}{\epsilon}\left(A-L\right)+{\mathrm{constant}}.$$

(15)

The area form of ${\mathrm{AdS}}_{2}$ is $\sqrt{-g}{\mathrm{d}}\sigma{\mathrm{d}}T=\cosh\sigma{\mathrm{d}}\sigma{\mathrm{d}}T={\mathrm{d}}(\sinh\sigma{\mathrm{d}}T)$, so

$$A=\int{\mathrm{d}}t\left(\sinh\sigma_{R}\frac{{\mathrm{d}}T_{R}}{{\mathrm{d}}t}-\sinh\sigma_{L}\frac{{\mathrm{d}}T_{L}}{{\mathrm{d}}t}\right).$$

(16)

As for the length term, the sum over all paths of length $L$ is a random walk of that length. A random walk on a manifold describes a process of diffusion which can also be described by the heat kernel $e^{-t\Delta}$, where $\Delta$ is the Laplacian (or its Lorentz signature analog). On a general Riemannian manifold, if we take for the action the usual kinetic energy of a nonrelativistic particle, $I_{\mathrm{kin}}=\frac{1}{2}\int{\mathrm{d}}tg_{ij}\dot{x}^{i}\dot{x}^{j}$, then the corresponding Hamiltonian is $\Delta/2$, appropriate to describe diffusion. The upshot is that the $L$ term in the action can be replaced by an action of the form $I_{\mathrm{kin}}$. The resulting action for the right boundary is then

$$I_{R}=\text{\textphi}_{b}\int{\mathrm{d}}t\left(\dot{\sigma}_{R}^{2}-\cosh^{2}\sigma_{R}\dot{T}_{R}^{2}\right)+\frac{2\text{\textphi}_{b}}{\epsilon}\int{\mathrm{d}}t\sinh\sigma_{R}\dot{T}_{R},$$

(17)

with a similar action for the left boundary. Proofs of the relationship²²2This relationship involves some renormalization, leading to a divergent additive constant in the Hamiltonian that will be dropped in the next paragraph. between (15) and (17) can be found in KS ; ZY , in part following chapter 9 of Polyakov .

For our purposes, we will just verify that³³3This derivation was explained to us by Z. Yang; a similar calculation in a different coordinate system can be found in ZY . (17) is equivalent to (14) in the limit $\epsilon\to 0$ (apart from an additive constant that has to be dropped from the Hamiltonian). The canonical momenta deduced from $I_{R}$ are $p_{\sigma_{R}}=2\text{\textphi}_{b}\dot{\sigma}_{R}$, $p_{T_{R}}=2\text{\textphi}_{b}(-\cosh^{2}\sigma_{R}\dot{T}_{R}+\frac{1}{\epsilon}\sinh\sigma_{R})$. The Hamiltonian is then

$$H_{R}=\frac{p_{\sigma_{R}}^{2}}{4\text{\textphi}_{b}}-\frac{1}{4\text{\textphi}_{b}\cosh^{2}\sigma_{R}}\left(p_{T_{R}}-\frac{2\text{\textphi}_{b}}{\epsilon}\sinh\sigma_{R}\right)^{2}.$$

(18)

Making the change of variables (12), where $p_{\widetilde{\sigma}_{R}}=p_{\sigma_{R}}$, the $\epsilon\to 0$ limit of the Hamiltonian comes out to be (after discarding an additive constant)

$$H_{R}=\frac{1}{2\text{\textphi}_{b}}\left(\frac{1}{2}p_{\widetilde{\sigma}_{R}}^{2}+p_{T_{R}}e^{-\widetilde{\sigma}_{R}}+\frac{1}{2}e^{-2\widetilde{\sigma}_{R}}\right).$$

(19)

The action of any Hamiltonian system has a canonical form $I_{\mathrm{can}}=\int{\mathrm{d}}t(\sum_{i}p_{i}\dot{q}^{i}-H)$. In the present case, this is

$$I_{\mathrm{can}}=\int{\mathrm{d}}t\left(p_{\widetilde{\sigma}_{R}}\dot{\widetilde{\sigma}}_{R}+p_{T_{R}}\dot{T}_{R}-\frac{1}{2\text{\textphi}_{b}}\left(\frac{1}{2}p_{\widetilde{\sigma}_{R}}^{2}+p_{T_{R}}e^{-\widetilde{\sigma}_{R}}+\frac{1}{2}e^{-2\widetilde{\sigma}_{R}}\right)\right).$$

(20)

Here, $I_{\mathrm{can}}$ is linear in $p_{T_{R}}$, so $p_{T_{R}}$ behaves as a Lagrange multiplier setting $e^{-\widetilde{\sigma}_{R}}=2\text{\textphi}_{b}\dot{T}_{R}$. After also integrating out $p_{\widetilde{\sigma}_{R}}$, which appears quadratically, by its equation of motion, we see that the action (20) is equivalent to the Schwarzian action⁴⁴4Introducing and simplifying the Hamiltonian has given an efficient way to do this calculation; however, one can reach the same conclusion by analyzing how the solutions of the equations of motion behave for small $\epsilon$. We will in any case need the formula for the Hamiltonian. (14).

In terms of the variables

$$\chi_{R}=-\widetilde{\sigma}_{R},~{}~{}~{}~{}~{}~{}\chi_{L}=\widetilde{\sigma}_{L}$$

(21)

used in JK , the Hamiltonian on the right boundary is

$$H_{R}=\frac{1}{2\text{\textphi}_{b}}\left(\frac{1}{2}p_{\chi_{R}}^{2}+p_{T_{R}}e^{\chi_{R}}+\frac{1}{2}e^{2\chi_{R}}\right).$$

(22)

By a similar derivation, the Hamiltonian on the left boundary is⁵⁵5In this derivation, a minus sign in the formula (16) for the area is compensated by a relative minus sign in the definitions of $\chi_{R},~{}\chi_{L}$.

$$H_{L}=\frac{1}{2\text{\textphi}_{b}}\left(\frac{1}{2}p_{\chi_{L}}^{2}+p_{T_{L}}e^{\chi_{L}}+\frac{1}{2}e^{2\chi_{L}}\right).$$

(23)

The renormalized geodesic length between the left and right boundaries is

$$\ell=-\chi_{R}-\chi_{L}+\log\left(\frac{1+\cos(T_{L}-T_{R})}{2}\right).$$

(24)

The left and right boundaries of $M$ are thus described by variables $T_{L},\chi_{L},T_{R},\chi_{R}$ and their canonical conjugates. Quantum mechanically, we can describe these boundaries by a Hilbert space ${\mathcal{H}}_{0}$ consisting of $L^{2}$ functions $\Psi(T_{L},\chi_{L},T_{R},\chi_{R})$.

However HJ ; KS ; ZY ; MLZ ; JK , ${\mathcal{H}}_{0}$ is not the appropriate bulk Hilbert space for JT gravity, for two reasons. One reason involves causality, and the second reason involves the gauge constraints. We will discuss causality first. Classically, one can describe a solution of JT gravity by specifying a pair of functions $T_{L}(t)$, $T_{R}(t)$ that satisfy the equations of motion derived from the Schwarzian action (14). Not all pairs of solutions are allowed, however; one wants the two pairs of boundaries to be spacelike separated. For the metric (9), the condition for this is that

$$|T_{L}(t)-T_{R}(t^{\prime})|<\pi$$

(25)

for all real $t,t^{\prime}$. Quantum mechanically, the observables $T_{R}(t)$ at different times are noncommuting operators that cannot be simultaneously specified; the same applies for $T_{L}(t)$. So we cannot directly impose the condition (25) at all times. Fortunately, one can check that in the classical theory it is sufficient to impose the condition (25) at one pair of times $t,t^{\prime}$. As we discuss briefly below, the classical dynamics then ensure that (25) holds at all times so long as the the two trajectories have vanishing total ${\widetilde{{SL}}}(2,{\mathbb{R}})$ charge – i.e. the solution satisfies the gauge constraints. We will define the quantum theory in the same way: we impose the condition $|T_{L}(t)-T_{R}(t^{\prime})|<\pi$ at some chosen times, say $t=t^{\prime}=0$, and then hope that after imposing the gauge constraints the quantum dynamics lead to a causal answer. We impose this initial condition by refining the definition of the Hilbert space ${\mathcal{H}}_{0}$ to say that it consists of $L^{2}$ functions $\Psi(T_{L},\chi_{L},T_{R},\chi_{R})$ whose support is at $|T_{L}-T_{R}|<\pi$.

Having made this definition, we then have to ask whether it leads to quantum dynamics that are consistent with causality. For JT gravity with matter, we will eventually get a fairly satisfactory answer, along the following lines. We will define algebras ${\mathcal{A}}_{L}$, ${\mathcal{A}}_{R}$ of observables on the left and right boundaries, respectively. ${\mathcal{A}}_{L}$ and ${\mathcal{A}}_{R}$ will contain, respectively, all quantum fields inserted on the left or right boundary at arbitrary values of the quantum mechanical time. The two algebras will commute with each other, and this will be a reasonable criterion for saying at the quantum level that the two boundaries are out of causal contact. For JT gravity without matter, an explanation along those lines is unfortunately not available, since there are not enough boundary observables. However the fact we end up with sensible boundary Hamiltonians on a Hilbert space constructed from wavefunctions with $|T_{L}-T_{R}|<\pi$ is itself evidence that the boundaries remain out of causal contact at all times.

Even after imposing the condition $|T_{L}-T_{R}|<\pi$, ${\mathcal{H}}_{0}$ is not the physical Hilbert space of JT gravity, because we have to impose the constraints. Since we are interested in the intrinsic geometry of $M$, not in how it is identified with a portion of ${\mathrm{AdS}}_{2}$, we have to regard two sets of variables $T_{L},\chi_{L},T_{R},\chi_{R}$ that differ by the action on ${\mathrm{AdS}}_{2}$ of $\widetilde{{SL}}(2,{\mathbb{R}})$ to be equivalent. In other words, we have to treat ${\widetilde{{SL}}}(2,{\mathbb{R}})$ as a group of constraints.

The constraint operators are

$$J_{a}=J_{a}^{L}+J_{a}^{R},$$

(26)

where $J_{a}^{L}$ and $J_{a}^{R}$ are the generators of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ acting on the right and left boundaries, namely

	$\displaystyle J_{1}^{R}$	$\displaystyle=p_{T_{R}}$
	$\displaystyle J_{2}^{R}$	$\displaystyle=(\cos T_{R})p_{T_{R}}-(\sin T_{R})p_{\chi_{R}}+e^{\chi_{R}}\cos T_{R}+\frac{{\mathrm{i}}}{2}\sin T_{R}$		(27)
	$\displaystyle J_{3}^{R}$	$\displaystyle=(\sin T_{R})p_{T_{R}}+(\cos T_{R})p_{\chi_{R}}+e^{\chi_{R}}\sin T_{R}-\frac{{\mathrm{i}}}{2}\cos T_{R}.$

and⁶⁶6The formulas for $J_{a}^{L}$ used in JK differ from these by $T_{L}\to T_{L}\pm\pi$, reversing the signs of $J_{2}^{L}$ and $J_{3}^{L}$. We will not make this change of variables as that would make the discussion of causality less transparent.

	$\displaystyle J_{1}^{L}$	$\displaystyle=p_{T_{L}}$		(28)
	$\displaystyle J_{2}^{L}$	$\displaystyle=-(\cos T_{L})p_{T_{L}}+(\sin T_{L})p_{\chi_{L}}-e^{\chi_{L}}\cos T_{L}-\frac{{\mathrm{i}}}{2}\sin T_{L}$		(29)
	$\displaystyle J_{3}^{L}$	$\displaystyle=-(\sin T_{L})p_{T_{L}}-(\cos T_{L})p_{\chi_{L}}-e^{\chi_{L}}\sin T_{L}+\frac{{\mathrm{i}}}{2}\cos T_{L}.$		(30)

These operators are self-adjoint and obey $[J_{a}^{R},J_{b}^{R}]={\mathrm{i}}\epsilon_{ab}{}^{c}J_{c}^{R}$, $[J_{a}^{L},J_{b}^{L}]={\mathrm{i}}\epsilon_{ab}{}^{c}J_{c}^{L}$. Here $\epsilon_{abc}$ is completely antisymmetric with $\epsilon_{123}=1$; Lie algebra indices are raised and lowered with the metric $\eta_{ab}=\mathrm{diag}(-1,1,1)$.

The derivation of the formulas (2.2), (28) can be understood as follows. The terms in $J_{a}^{L},J_{b}^{R}$ that are linear in the momenta give the $\sigma\to\pm\infty$ limit of the group action on the AdS₂ coordinates $(\sigma,T)$ generated by the vector fields (3). The imaginary terms in $J_{a}^{L}$, $J_{b}^{R}$ are there simply to make those operators self-adjoint. Finally, the terms proportional to $e^{\chi_{R}}$ and $e^{\chi_{L}}$ are neccessary to give the correct action on the conjugate momenta $p_{\chi_{R}},p_{T_{R}}$ and $p_{\chi_{L}},p_{T_{L}}$. This action can be computed from the ${\widetilde{{SL}}}(2,{\mathbb{R}})$-invariant action (17) by taking the $\epsilon\to 0$ limit. However, it is somewhat easier to instead derive the ${\widetilde{{SL}}}(2,{\mathbb{R}})$ charges in the Hamiltonian description. In this description, symmetry group generators must commute with $H_{L}$ and $H_{R}$, which we have already determined. This forces the inclusion of the terms proportional to $e^{\chi_{R}}$, $e^{\chi_{L}}$. Actually, $H_{R}$ and $H_{L}$ are essentially the quadratic Casimir operators for the action of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ on the right and left boundary degrees of freedom:

$$2\text{\textphi}_{b}H_{R}=\frac{1}{2}\left(\eta^{ab}J_{a}^{R}J_{b}^{R}-\frac{1}{4}\right),~{}~{}~{}~{}~{}2\text{\textphi}_{b}H_{L}=\frac{1}{2}\left(\eta^{ab}J_{a}^{L}J_{b}^{L}-\frac{1}{4}\right).$$

(31)

Before discussing how to impose these constraints at the quantum level, we first describe how they are implemented in classical JT gravity. The classical phase space procedure for dealing with a gauge symmetry is known as a symplectic quotient, and involves a two-step procedure. The starting point is a $\text{{\teneurm g}}^{*}$-valued function called a “moment map” where g is the Lie algebra of the gauge group and $\text{{\teneurm g}}^{*}$ is its dual. This moment map should generate the gauge group action via Poisson brackets. In our case, the moment map is just the total ${\widetilde{{SL}}}(2,{\mathbb{R}})$ charge $J_{a}=J_{a}^{L}+J_{a}^{R}$, where the conserved charges $J_{a}^{L}$ and $J_{a}^{R}$ are given by the formulas (2.2) and (28) above, except that the imaginary terms can be dropped because we are in the classical limit. To take a symplectic quotient, we first consider the subspace of phase space on which the moment map is zero. To recover a symplectic manifold (i.e. a sensible phase space), we then also identify points on this constrained space that are related by the action of the gauge group. Each of these two steps reduces the phase space dimension by the dimension of the gauge group. In our case, the unconstrained phase space is eight dimensional, and the group ${\widetilde{{SL}}}(2,{\mathbb{R}})$ is three dimensional, so the physical phase space will be two dimensional.

The qualitative properties of a classical orbit depend on whether the Casimir $\eta^{ab}J^{R}_{a}J^{R}_{b}$ is positive, negative, or zero. If $\eta^{ab}J^{R}_{a}J^{R}_{b}<0$, then up to an $SL(2,{\mathbb{R}})$ rotation, we can assume that $J^{R}_{1}\not=0$, $J^{R}_{2}=J^{R}_{3}=0$. The conditions $J^{R}_{2}=J^{R}_{3}=0$ imply via eqn. (2.2) that $p_{\chi_{R}}=0$ and $e^{\chi_{R}}=-p_{T_{R}}$, so that we must have $J_{1}^{R}=p_{T_{R}}<0$. The ${\widetilde{{SL}}}(2,{\mathbb{R}})$ constraint implies that the left boundary particle has $J^{L}_{a}=-J^{R}_{a}$, and now the conditions $J^{L}_{2}=J^{L}_{3}=0$ lead to $J_{1}^{L}=p_{T_{L}}=-e^{\chi_{L}}<0$. But as $J_{1}^{R},J_{1}^{L}$ are then both negative, it is impossible to satisfy the constraint $J_{1}^{R}+J_{1}^{L}=0$. So orbits with $\eta^{ab}J^{R}_{a}J^{R}_{b}<0$ cannot satisfy the constraints. A similar analysis shows that the same is true of orbits with $\eta^{ab}J^{R}_{a}J^{R}_{b}=0$.

Thus, we have to consider orbits with $\eta^{ab}J_{a}^{R}J_{b}^{R}>0$. Any such orbit is related by ${\widetilde{{SL}}}(2,{\mathbb{R}})$ to one with $J_{2}^{R}>0$ and $J_{1}^{R}=J_{3}^{R}=0$; again, the ${\widetilde{{SL}}}(2,{\mathbb{R}})$ constraint requires $J_{a}^{L}=-J_{a}^{R}$. The conditions $J_{1}^{R}=J_{1}^{L}=0$ give $p_{T_{R}}=p_{T_{L}}=0$ and the other conditions can be solved to give

	$\displaystyle e^{\chi_{R}}$	$\displaystyle=J_{2}^{R}\cos T_{R}$
	$\displaystyle e^{\chi_{L}}$	$\displaystyle=-J_{2}^{L}\cos T_{L}=J_{2}^{R}\cos T_{L}.$		(32)

Any orbit of this type therefore has

$$2\pi n_{R}-\pi/2<T_{R}<2\pi n_{R}+\pi/2,~{}~{}~{}~{}2\pi n_{L}-\pi/2<T_{L}<2\pi n_{L}+\pi/2$$

(33)

for some integers $n_{L},n_{R}$. An element of the center of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ will shift $n_{L},n_{R}$ by a common integer, so only the difference $n_{R}-n_{L}$ is invariant. If this difference vanishes, then $|T_{L}(t)-T_{R}(t^{\prime})|<\pi$ for all $t,t^{\prime}$ and two boundaries are spacelike separated at all times. If the difference is nonzero, then $|T_{L}(t)-T_{R}(t^{\prime})|>\pi$ always, and the two boundaries are timelike separated at all times. Thus it is necessary to impose a condition that the two boundaries are spacelike separated, and if this condition is imposed at one time, it remains valid for all times.

At this stage, we have reduced the phase space to a three-dimensional space parameterised by the value of $J_{2}^{R}$, or equivalently of the Hamiltonians $H_{L}=H_{R}=\frac{1}{4\text{\textphi}_{b}}(J_{2}^{R})^{2}-\frac{1}{16\text{\textphi}_{b}}$, along with the locations of the two boundary particles along their trajectories. To complete our analysis, we note that the gauge symmetry generator $J_{2}=J_{2}^{L}+J_{2}^{R}$ preserves the gauge charges and hence preserves the two boundary trajectories. In fact (up to an energy-dependent rescaling), it generates forwards time-translation of the right boundary and backwards time-translation of the left boundary. After quotienting by this action, we obtain the final two-dimensional phase space HJ parameterised by the boundary energy $H_{L}=H_{R}$ along with the “timeshift” between the two boundary trajectories.

Let us now discuss what happens in the quantum theory. Because the constraint group ${\widetilde{{SL}}}(2,{\mathbb{R}})$ is non compact, imposing the constraints on quantum states is somewhat subtle. Suppose that a group $G$ acts on a Hilbert space ${\mathcal{H}}_{0}$, with inner product $(~{},~{})$, and one wishes to impose $G$ as a group of constraints. In our case, $G={\widetilde{{SL}}}(2,{\mathbb{R}})$ and ${\mathcal{H}}_{0}$ was defined earlier. Naively, one imposes the constraints by restricting to the $G$-invariant subspace of ${\mathcal{H}}_{0}$. This is satisfactory if $G$ is compact, but if $G$ is not compact, this procedure can be problematical because $G$-invariant states are typically not normalizable, so there may be few or no $G$-invariant states in ${\mathcal{H}}_{0}$. A procedure that often works better for a noncompact group and that has been extensively discussed in the context of gravity (see for example MarolfReview ; marolf ) is to define a Hilbert space of coinvariants of the $G$ action, rather than invariants. This means that one considers any state $\Psi\in{\mathcal{H}}_{0}$ to be physical, but one imposes an equivalence relation $\Psi\cong g\Psi$ for any $g\in G$. The equivalence classes are called the coinvariants of $G$. $G$ acts trivially on the space of coinvariants, since by definition $\Psi$ and $g\Psi$ are in the same equivalence class for any $\Psi\in{\mathcal{H}}_{0}$, $g\in G$. Thus, the coinvariants are annihilated by $G$, even if they cannot be represented by invariant vectors in the original Hilbert space ${\mathcal{H}}_{0}$. If (as in the case of ${\widetilde{{SL}}}(2,{\mathbb{R}})$) the group $G$ has a left and right invariant measure ${\mathrm{d}}\mu$, one can try to define an inner product on the space of coinvariants by integration over $G$:

$$\langle\Psi^{\prime}|\Psi\rangle=\int_{G}{\mathrm{d}}\mu~{}(\Psi^{\prime},R(g)\Psi).$$

(34)

Here $R(g)$ is the operator by which $g\in G$ acts on ${\mathcal{H}}_{0}$. If the integral in eqn. (34) is convergent (as is the case for the states that will be introduced presently in eqn. (35)), then $\langle\Psi^{\prime}|\Psi\rangle$ depends only on the equivalence classes of $\Psi$ and $\Psi^{\prime}$, so the formula defines an inner product on the space of coinvariants and enables us to define the Hilbert space ${\mathcal{H}}$ of coinvariants.

The general procedure to impose constraints is really BRST quantization, or its BV generalization. Both the space of invariants and the space of coinvariants are special cases of what is natural in BRST-BV quantization. See shvedov or Appendix B of CLPW for background. BRST-BV quantization in general (see Henneaux for an introduction) permits one to define something intermediate between the space of invariants and the space of coinvariants. For example, in perturbative string theory, where one wants to impose the Virasoro generators $L_{n}$ as contraints, one usually imposes a condition $L_{n}\Psi=0$, $n\geq 0$, on physical states, and also an equivalence relation $\Psi\cong\Psi+L_{n}\chi$, $n<0$. This means that one takes invariants of the subalgebra generated by $L_{n}$ for $n\geq 0$ and coinvariants of the subalgebra generated by $L_{n}$, $n<0$. BRST quantization generates this mixture in a natural way. Such a mixture is also natural, in general, in gauge theory and gravity.

In the case of JT gravity, such refinements are not necessary. We can just define the Hilbert space ${\mathcal{H}}$ of JT gravity to be the space of coinvariants of the action of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ on ${\mathcal{H}}_{0}$. We will see that this definition leads to efficient derivations of useful results, some of which have been deduced previously by other methods. In fact, JT gravity is simple enough that it is possible, as shown in the literature, to get equivalent results, sometimes with slightly longer derivations, by working with unnormalizable ${\widetilde{{SL}}}(2,{\mathbb{R}})$ invariant states and correcting the inner product by formally dividing by the infinite volume of ${\widetilde{{SL}}}(2,{\mathbb{R}})$.

To minimize clutter, we henceforth write just $T,T^{\prime},\chi,\chi^{\prime}$ for $T_{R},T_{L},\chi_{R},\chi_{L}$. For any $T,T^{\prime},\chi,\chi^{\prime}$ satisfying the causality constraint $|T-T^{\prime}|<\pi$, there is always a unique element of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ that sets $T=T^{\prime}=0$, $\chi=\chi^{\prime}$. This means that the space of coinvariants is generated by wavefunctions of the form

$$\Psi=\delta(T)\delta(T^{\prime})\delta(\chi-\chi^{\prime})\psi(\chi).$$

(35)

Such wavefunctions are highly unnormalizable in the inner product of ${\mathcal{H}}_{0}$, but in the natural inner product (34) of the space ${\mathcal{H}}$ of coinvariants, we have simply

$$\langle\Psi,\Psi\rangle=\int_{-\infty}^{\infty}{\mathrm{d}}\chi\,\overline{\psi}\psi.$$

(36)

The form (35) of the wavefunction is preserved by the operator $\chi$, acting by multiplication, along with $\widetilde{p}_{\chi}=p_{\chi}+p_{\chi^{\prime}}=-{\mathrm{i}}(\partial_{\chi}+\partial_{\chi^{\prime}})$. Of course, $[\widetilde{p}_{\chi},\chi]=-{\mathrm{i}}$. In short, the physical Hilbert space ${\mathcal{H}}$ can be viewed as the space of square-integrable functions of $\chi$ (or $\chi^{\prime}$), and the algebra of operators acting on ${\mathcal{H}}$ is generated by the conjugate operators $\chi$ and $\widetilde{p}_{\chi}$.

Now we can evaluate the left and right Hamiltonians $H_{L}$ and $H_{R}$ as operators on ${\mathcal{H}}$. In doing so, we note that by definition any $\Psi\in{\mathcal{H}}$ is annihilated by the constraint operators $J_{a}=J_{a}^{L}+J_{a}^{R}$. This statement is just the derivative at $g=1$ of the equivalence relation $\Psi\cong g\Psi$, $g\in{\widetilde{{SL}}}(2,{\mathbb{R}})$. Acting on a state of the form given in eqn. (35), we have

	$\displaystyle J_{1}\Psi$	$\displaystyle=(p_{T}+p_{T^{\prime}})\Psi$
	$\displaystyle J_{2}\Psi$	$\displaystyle=(p_{T}-p_{T^{\prime}})\Psi$
	$\displaystyle J_{3}\Psi$	$\displaystyle=(p_{\chi}-p_{\chi^{\prime}})\Psi.$		(37)

So as operators on ${\mathcal{H}}$, $p_{T}$ is equivalent to $(J_{1}+J_{2})/2$ and hence can be set to zero, and $p_{\chi}$ is equivalent to $\frac{1}{2}\widetilde{p}_{\chi}+\frac{1}{2}J_{3}$ and so can be replaced by $\frac{1}{2}\widetilde{p}_{\chi}$. Likewise $p_{T^{\prime}}$ can be replaced by 0 and $p_{\chi^{\prime}}$ by $-\frac{1}{2}\widetilde{p}_{\chi}$. With these substitutions, we get

$$2\text{\textphi}_{b}H_{L}=2\text{\textphi}_{b}H_{R}=\frac{\widetilde{p}_{\chi}^{2}}{8}+\frac{e^{2\chi}}{2}.$$

(38)

From eqn. (24) (with $\chi_{L}=\chi_{R}=\chi$, and after absorbing a constant shift in $\ell$), the renormalized length $\ell$ of the geodesic between the two boundaries is $\ell=-2\chi$, so alternatively

$$2\text{\textphi}_{b}H_{L}=2\text{\textphi}_{b}H_{R}=\frac{p_{\ell}^{2}}{2}+\frac{1}{2}e^{-\ell}.$$

(39)

As noted in JK , before imposing the ${\widetilde{{SL}}}(2,{\mathbb{R}})$ constraints, the operators $H_{L}$, $H_{R}$ are not positive-definite. On the other hand, after imposing the constraints, we have arrived at manifestly positive formulas for $H_{L}$ and $H_{R}$; the negative energy states have all been removed by the constraints. This is the quantum analogue of our observation that, in classical JT gravity, orbits with $\eta^{ab}J^{R}_{a}J^{R}_{b}<0$ cannot satisfy the constraints.

The fact that $H_{L}=H_{R}$ after imposing constraints is analogous to the fact that in higher dimensions, the ADM mass of an unperturbed Schwarzschild spacetime is the same at either end. It can be deduced directly from the relation (31) between the Hamiltonians and the Casimir operators. We have

$$2\text{\textphi}_{b}(H_{R}-H_{L})=\frac{1}{2}\left(\eta^{ab}J_{a}^{R}J_{b}^{R}-\eta^{ab}J_{a}^{L}J_{a}^{R}\right)=\frac{1}{2}\eta^{ab}(J_{a}^{R}+J_{a}^{L})(J_{b}^{R}-J_{b}^{L}).$$

(40)

The operator on the right hand side annihilates physical states, since any operator of the general form $\sum_{a}J_{a}X^{a}$, where $J_{a}$ are the constraint operators and $X^{a}$ are any operators, annihilates ${\mathcal{H}}$. Hence $H_{R}-H_{L}=0$ as an operator on ${\mathcal{H}}$. Once we know this, it follows easily that $H_{R}$ and $H_{L}$ are positive after imposing the constraints. Since $H_{R}=H_{L}$ as operators on ${\mathcal{H}}$, if one of them is negative, so is the other. From eqns. (22) and (23), we see that for this to happen, $p_{T}$ and $p_{T^{\prime}}$ must be negative, but in this case $J_{1}=p_{T}+p_{T^{\prime}}$ is negative, contradicting the fact that $J_{1}$ annihilates physical states.

It is pleasantly straightforward to include matter fields in this construction. As we will see, $H_{L}$ and $H_{R}$ remain positive.

As in many recent papers, we add to JT gravity a “matter” quantum field theory that does not couple directly to the dilaton field $\upphi$ of JT gravity. Quantized in ${\mathrm{AdS}}_{2}$, such a theory has a Hilbert space ${\mathcal{H}}^{\mathrm{matt}}$. Since $\widetilde{{SL}}(2,{\mathbb{R}})$ acts on ${\mathrm{AdS}}_{2}$ as a group of isometries, any relativistic field theory on ${\mathrm{AdS}}_{2}$, whether conformally invariant or not, is $\widetilde{{SL}}(2,{\mathbb{R}})$-invariant. Hence the group $\widetilde{{SL}}(2,{\mathbb{R}})$ acts naturally on ${\mathcal{H}}^{\mathrm{matt}}$, say with generators $J_{a}^{\mathrm{matt}}$, obeying the ${\widetilde{{SL}}}(2,{\mathbb{R}})$ commutation relations.

In the context of coupling to JT gravity, the matter system should be formulated on a large piece $M$ of ${\mathrm{AdS}}_{2}$, not on all of ${\mathrm{AdS}}_{2}$. However, in the limit $\epsilon\to 0$ that was reviewed in section 2.1, this distinction is unimportant because the boundary of $M$ is, in the relevant sense, near the conformal boundary of ${\mathrm{AdS}}_{2}$. Hence we can think of the matter theory as “living” on all of ${\mathrm{AdS}}_{2}$. Therefore, prior to imposing constraints, we can take the Hilbert space of the combined system to be ${\mathcal{H}}_{0}\otimes{\mathcal{H}}^{\mathrm{matt}}$, where ${\mathcal{H}}_{0}$ is defined as in section 2.2.

On this we have to impose the ${\widetilde{{SL}}}(2,{\mathbb{R}})$ constraints. The relevant constraint operators are now the sum of the constraint operators of the gravitational sector and the matter system:

$$J_{a}=J_{a}^{R}+J_{a}^{L}+J_{a}^{\mathrm{matt}}.$$

(41)

Now it is straightforward to impose the constraints and construct the physical Hilbert space ${\mathcal{H}}$. We define ${\mathcal{H}}$ to be the space of coinvariants of the action of ${\widetilde{{SL}}}(2,{\mathbb{R}})$ on ${\mathcal{H}}_{0}\otimes{\mathcal{H}}^{\mathrm{matt}}$. As before, because ${\widetilde{{SL}}}(2,{\mathbb{R}})$ can be used to fix $T=T^{\prime}=0$, $\chi=\chi^{\prime}$ in a unique fashion, the coinvariants are generated by states of the form

$$\Psi=\delta(T)\delta(T^{\prime})\delta(\chi-\chi^{\prime})\psi(\chi).$$

(42)

The only difference is that $\psi(\chi)$, instead of being complex-valued, is now valued in the matter Hilbert space ${\mathcal{H}}^{\mathrm{matt}}$. Evaluation of the inner product (34) now gives

$$\langle\Psi,\Psi\rangle=\int_{-\infty}^{\infty}{\mathrm{d}}\chi\,\,(\psi(\chi),\psi(\chi)),$$

(43)

where here $(~{},~{})$ is the inner product on ${\mathcal{H}}^{\mathrm{matt}}$. So the Hilbert space of coinvariants is ${\mathcal{H}}=L^{2}({\mathbb{R}})\otimes{\mathcal{H}}^{\mathrm{matt}}$, where $L^{2}({\mathbb{R}})$ is the space of $L^{2}$ functions of $\chi$. The algebra of operators on ${\mathcal{H}}$ is generated by $\chi$, $\widetilde{p}_{\chi},$ and the operators on ${\mathcal{H}}^{\mathrm{matt}}$.

Now we want to identify the boundary Hamiltonians $H_{R}$ and $H_{L}$ as operators on ${\mathcal{H}}$. To do this, we just have to generalize eqn. (2.2) to include $J_{a}^{\mathrm{matt}}$. On a state of the form (42), the constraint operators $J_{a}$ act by

	$\displaystyle J_{1}\Psi$	$\displaystyle=(p_{T}+p_{T^{\prime}}+J_{1}^{\mathrm{matt}})\Psi$
	$\displaystyle J_{2}\Psi$	$\displaystyle=(p_{T}-p_{T^{\prime}}+J_{2}^{\mathrm{matt}})\Psi$
	$\displaystyle J_{3}\Psi$	$\displaystyle=(p_{\chi}-p_{\chi^{\prime}}+J_{3}^{\mathrm{matt}})\Psi.$		(44)

With the aid of these formulas, one finds that as operators on ${\mathcal{H}}$,

	$\displaystyle 2\text{\textphi}_{b}H_{R}$	$\displaystyle=\frac{1}{8}(\widetilde{p}_{\chi}-J_{3}^{\mathrm{matt}})^{2}-\frac{1}{2}(J_{1}^{\mathrm{matt}}+J_{2}^{\mathrm{matt}})e^{\chi}+\frac{1}{2}e^{2\chi}$
	$\displaystyle 2\text{\textphi}_{b}H_{L}$	$\displaystyle=\frac{1}{8}(\widetilde{p}_{\chi}+J_{3}^{\mathrm{matt}})^{2}-\frac{1}{2}(J_{1}^{\mathrm{matt}}-J_{2}^{\mathrm{matt}})e^{\chi}+\frac{1}{2}e^{2\chi}.$		(45)

The operators $(\widetilde{p}_{\chi}\pm J_{3}^{\mathrm{matt}})^{2}$, $e^{2\chi}$, and $e^{\chi}$ are manifestly positive, and in a moment, we will show that the operators $-(J_{1}^{\mathrm{matt}}\pm J_{2}^{\mathrm{matt}})$ are non-negative. So $H_{L}$ and $H_{R}$ are positive as operators on the physical Hilbert space ${\mathcal{H}}$. One can also verify using eqn. (2.3) that $[H_{L},H_{R}]=0$, as expected since this is true even before imposing the constraints.

To understand the statement that the operators $-(J_{1}^{\mathrm{matt}}\pm J_{2}^{\mathrm{matt}})$ are non-negative, we need to discuss in more detail the meaning of the constraints. Let $\Phi$ be one of the matter fields that can be inserted on the boundary of ${\mathrm{AdS}}_{2}$, say on the right side. The constraints are supposed to commute with boundary insertions such as $\Phi(T(t))$, while reparameterising $T$. Since $J_{1}^{R}=p_{T}=-{\mathrm{i}}\partial_{T}$, we have $[J_{1}^{R},T(t)]=-{\mathrm{i}}$. To get $[J_{1}^{R}+J_{1}^{\mathrm{matt}},\Phi(T(t))]=0$, we then need $[J_{1}^{\mathrm{matt}},\Phi(T)]=+{\mathrm{i}}\partial_{T}\Phi(T)$. Comparing to the standard quantum mechanical formula $[H,\Phi(T)]=-{\mathrm{i}}\partial_{T}\Phi$, where $H$ is the Hamiltonian, we conclude that actually $J_{1}^{\mathrm{matt}}=-H$. In quantum field theory in ${\mathrm{AdS}}_{2}$, $H$ is non-negative and annihilates only the ${\widetilde{{SL}}}(2,{\mathbb{R}})$-invariant ground state. So therefore $J_{1}^{\mathrm{matt}}$ is non-positive. For $-1<a<1$, the operator $J_{1}^{\mathrm{matt}}+aJ_{2}^{\mathrm{matt}}$ is conjugate in ${\widetilde{{SL}}}(2,{\mathbb{R}})$ to a positive multiple of $J_{1}^{\mathrm{matt}}$, so it is again non-positive. Taking the limit $|a|\to 1$, the operators $J_{1}^{\mathrm{matt}}\pm J_{2}^{\mathrm{matt}}$ are likewise non-positive, and therefore $-(J_{1}^{\mathrm{matt}}\pm J_{2}^{\mathrm{matt}})$ is non-negative, as claimed in the last paragraph.