Jackiw-Teitelboim and Kantowski-Sachs quantum cosmology

The action of the JT model is Jackiw:1984je ; Teitelboim:1983ux

$$S=\int d^{2}x\sqrt{-g}\phi(R-2H^{2})-2\int_{B}\phi_{b}K,$$

(2.1)

where $R$ is the $2D$ spacetime curvature, $H={\rm const}$, $\phi$ is the dilaton field, $\phi_{b}$ is its value at the boundary, and $K$ is the extrinsic curvature of the boundary curve $B$. Throughout the paper we shall assume that $H>0$. Variation with respect to $\phi$ yields $R=2H^{2}$, telling us that the $2D$ spacetime is a de Sitter space with expansion rate $H$.

With the metric represented as

$$ds^{2}=-N^{2}dt^{2}+a^{2}dx^{2},$$

(2.2)

where $0<x<2\pi$ and $N$ is the lapse function, the state vector is a functional

$$\Psi[a(x),\phi(x)].$$

(2.3)

We can choose the gauge so that $\phi_{b}={\rm const}$ at the boundary. Furthermore, we are going to adopt a minisuperspace picture, where $a=a(t)$, independent of $x$, and the boundary is a circle, $t={\rm const}$. Then $\Psi$ is an ordinary function $\Psi(a,\phi)$. It has been shown in Iliesiu that due to the simplicity of the model, the wave functional (2.3) can be recovered from the minisuperspace wave function $\Psi(a,\phi)$. Here, we shall restrict our analysis to the minisuperspace model with $a=a(t),~{}\phi=\phi(t)$.

After integration by parts the action (2.1) can be represented as

$$S=-4\pi\int dt\left(\frac{{\dot{a}}{\dot{\phi}}}{N}+NH^{2}a\phi\right),$$

(2.4)

where dots stand for derivatives with respect to $t$. We are going to use the gauge $N={\rm const}$. The momenta conjugate to $a$ and $\phi$ are

$$\Pi_{a}=-\frac{4\pi}{N}{\dot{\phi}},~{}~{}~{}~{}\Pi_{\phi}=-\frac{4\pi}{N}{\dot{a}}.$$

(2.5)

The equations of motion obtained by varying the action with respect to $a$ and $\phi$ are

$${\ddot{a}}-H^{2}a=0,$$

(2.6)

$${\ddot{\phi}}-H^{2}\phi=0,$$

(2.7)

where we have set $N=1$. The Hamiltonian constraint is obtained by varying with respect to $N$:

$${\dot{a}}{\dot{\phi}}=H^{2}a\phi,$$

(2.8)

or

$$\Pi_{a}\Pi_{\phi}=16\pi^{2}H^{2}a\phi.$$

(2.9)

The classical solution of these equations is

$$a=a_{0}\cosh(Ht),~{}~{}~{}~{}\phi=\phi_{0}\sinh(Ht)$$

(2.10)

with $a_{0},\phi_{0}={\rm const}$. We shall set $a_{0}=H^{-1}$, so that the metric covers the full de Sitter space in a nonsingular way.

To lowest order in the WKB approximation, the wave function is given by

$$\Psi\sim e^{iS_{cl}},$$

(2.11)

where $S_{cl}$ is the classical action,

$$S_{cl}=-2\int_{0}^{2\pi}dx~{}a\phi_{b}K=-4\pi a\phi_{b}K.$$

(2.12)

Here, we used the fact that $R=2H^{2}$ in the classical solution, so only the surface terms make a contribution, and that $\phi$ and $K$ are constant on the boundary. Following the no-boundary philosophy, we assume that the (Euclideanized) geometry closes off smoothly, so that there is no boundary contribution at $a=0$.

In the classically allowed region $(a>H^{-1})$, the extrinsic curvature $K$ is given by

$$K=\frac{\dot{a}}{a}=H\tanh(Ht)=a^{-1}\sqrt{H^{2}a^{2}-1},$$

(2.13)

where we have used Eq. (2.10) with $a_{0}=H^{-1}$. Substituting this in Eqs. (2.12) and (2.11), we obtain

$$\Psi\propto\exp\left(-4\pi i\phi_{b}\sqrt{H^{2}a^{2}-1}\right)$$

(2.14)

A linearly independent WKB wave function is a complex conjugate of (2.14). A general WKB solution is a linear combination of the two. The semiclassical approximation applies when the action is large, $\phi_{b}\sqrt{H^{2}a^{2}-1}\gg 1$.

The momentum operator $\Pi_{\phi}$ acting on $\Psi$ gives

$$\Pi_{\phi}\Psi=-i\partial_{\phi}\Psi=-4\pi\sqrt{H^{2}a^{2}-1}\Psi.$$

(2.15)

The classical momentum is given by Eq. (2.5), so we get

$${\dot{a}}=\sqrt{H^{2}a^{2}-1}.$$

(2.16)

This agrees with the expanding branch of the classical solution (2.10). The complex conjugate wave function describes a contracting universe.

The WDW equation corresponding to the Hamiltonian constraint (2.8) is

$$(\partial_{a}\partial_{\phi}+16\pi^{2}H^{2}a\phi){\tilde{\Psi}}=0.$$

(2.17)

Here,

$${\tilde{\Psi}}=\Psi/a$$

(2.18)

and the factor $1/a$ comes from the factor ordering indicated by the exact quantization of the JT model by Henneaux Henneaux (see IKTV Iliesiu for a detailed explanation). Following MTY Maldacena , we introduce new variables

$$u=\phi^{2},~{}~{}~{}v=(2\pi)^{2}(H^{2}a^{2}-1).$$

(2.19)

Then the WDW equation becomes

$$(\partial_{u}\partial_{v}+1){\tilde{\Psi}}=0.$$

(2.20)

Yet another change of variables

$$T=\sqrt{uv},~{}~{}~{}\xi=\frac{1}{2}\ln\frac{v}{u}$$

(2.21)

brings the equation to a separable form

$$-\frac{1}{T}\partial_{T}(T\partial_{T}{\tilde{\Psi}})+\frac{1}{T^{2}}\partial_{\xi}^{2}{\tilde{\Psi}}-4{\tilde{\Psi}}=0.$$

(2.22)

With the ansatz

$${\tilde{\Psi}}_{m}=e^{m\xi}f_{m}(T)$$

(2.23)

we obtain an equation for $f_{m}(T)$:

$${f_{m}}^{\prime\prime}+\frac{1}{T}{f_{m}}^{\prime}-\frac{m^{2}}{T^{2}}f_{m}+4f_{m}=0.$$

(2.24)

The solution is

$$f_{m}(T)=Z_{m}(2T),$$

(2.25)

where $Z_{m}$ is a Bessel function.

Following MTY, IKTV required that the wave function should describe an expanding universe in the limit of large $a$. Then the appropriate choice of Bessel functions is $H_{m}^{(2)}(2T)$. The general solution of the WDW equation is then a linear combination of functions of the form (2.23):

$${\tilde{\Psi}}_{m}=\left(\frac{v}{u}\right)^{m/2}H_{m}^{(2)}(2T).$$

(2.26)

In terms of the variables $a$ and $\phi$, the argument of the Bessel functions is

$$2T=4\pi\phi\sqrt{H^{2}a^{2}-1}.$$

(2.27)

The asymptotic form of the Bessel functions at large $T$ is $H_{m}^{(2)}(2T)\propto T^{-1/2}e^{-2iT}$; hence

$${\Psi}_{m}(Ha\phi\gg 1)\propto\left(\frac{a}{\phi}\right)^{m+1/2}\exp\left(-4\pi i\phi\sqrt{H^{2}a^{2}-1}\right),$$

(2.28)

where we have accounted for the factor $1/a$ relating $\Psi_{m}$ and ${\tilde{\Psi}}_{m}$. All these functions have the same asymptotic exponential factor as the WKB wave function (2.14). So in order to choose between them one has to determine the pre-exponential factor.

In the path integral formulation, the semiclassical pre-factor is determined by quantum fluctuations about the classical solution. In the JT model these are fluctuations in the shape of the boundary curve, which are described by the Schwarzian theory and yield a one-loop pre-factor $(\phi/a)^{3/2}$ at large $a$ StanfordWitten . It is shown in StanfordWitten that this result is one-loop exact, so there are no further corrections. This pre-factor is obtained by setting $m=-2$ in (2.28). Then the exact wave function takes the form

$$\Psi(a,\phi)=\frac{a\phi^{2}}{H^{2}a^{2}-1}H_{2}^{(2)}\left(4\pi\phi\sqrt{H^{2}a^{2}-1}\right)~{}~{}~{}~{}(Ha>1).$$

(2.29)

Analytic continuation of this wave function to $Ha<1$ is not unique because of the singularity at $Ha=1$. IKTV specify the wave function in the entire range of $a$ by replacing the Hankel function in (2.29) with $K_{2}\left(i\sqrt{\phi^{2}(H^{2}a^{2}-1-i\epsilon)}\right)$, which gives³³3The inclusion of the term $i\epsilon$ with $\epsilon\to+0$ is needed to make the solution well-defined on the branch cut.

$$\Psi(a,\phi)=\frac{2i}{\pi}\frac{a\phi^{2}}{H^{2}a^{2}-1}K_{2}\left(4\pi\phi\sqrt{H^{2}a^{2}-1}\right)~{}~{}~{}~{}(Ha<1).$$

(2.30)

IKTV identify the wave function (2.29),(2.30) with the HH wave function for JT gravity. There are however some problems with this identification. We first note that one of the defining properties of the HH wave function is that it is real. This can be interpreted as reflecting the CPT invariance of the HH state HHH . On the other hand, the tunneling wave function is specified by the outgoing wave boundary condition, which in the present context requires that the large $a$ asymptotic of $\Psi$ should only include terms corresponding to expanding universes. This seems to suggest that the wave function described by Eqs.(2.29),(2.30) is more appropriately interpreted as the tunneling wave function.

More importantly, the wave function (2.29) has a singularity at $a=H^{-1}$. It is actually not a solution of the WDW equation. We will show in Sec.III.C that it satisfies

$${\cal H}\Psi\propto\delta(\phi)\delta^{\prime\prime}(a-H^{-1}).$$

(2.31)

Hence it is not suitable for the role of HH or tunneling wave function.

IKTV have also proposed another candidate for the HH wave function:

$$\Psi(a,\phi)=\frac{a\phi^{2}}{H^{2}a^{2}-1}J_{2}\left(4\pi\phi\sqrt{H^{2}a^{2}-1}\right).$$

(2.32)

This wave function is real and non-singular. However, its behavior in the classically forbidden range $a<H^{-1}$ is very different from what is expected for the semiclassical HH wave function. We have

$$\Psi(a<H^{-1})=\frac{a\phi^{2}}{1-H^{2}a^{2}}I_{2}\left(4\pi\sqrt{\phi^{2}(1-H^{2}a^{2})}\right)\sim\frac{a\phi^{3/2}}{\sqrt{4\pi}(1-H^{2}a^{2})^{5/4}}\exp\left(4\pi|\phi|\sqrt{1-H^{2}a^{2}}\right),$$

(2.33)

where the last expression is the asymptotic form of $\Psi$ assuming that the argument of $I_{2}$ is large. As $a$ varies from $a=0$ to $a\sim H^{-1}$, the exponential factor in $\Psi$ decreases, which is opposite to the expected behavior of the HH wave function.

MTY discussed the relation between JT and Einstein $4D$ gravity using dimensional reduction from a nearly extremal Schwarzschild-de Sitter solution. Since our emphasis is on the cosmological aspects of the theory, we find it more useful to consider a cosmological $4D$ model describing a universe with spatial sections having $S^{1}\times S^{2}$ topology and the metric

$$ds^{2}=-dt^{2}+a^{2}(x,t)dx^{2}+b^{2}(x,t)d\Omega^{2}.$$

(2.34)

Here, $0<x<2\pi$ and $d\Omega^{2}$ is the metric on a unit sphere. Substituting this in the Einstein-Hilbert action (in Planck units)

$$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g^{(4)}}\left(R^{(4)}-2\Lambda\right),$$

(2.35)

where $\Lambda$ is the $4D$ cosmological constant, and integrating over the angular variables we obtain

$$S=\int d^{2}x\sqrt{-g}\left[\frac{b^{2}}{4}R+\frac{1}{2}(\nabla b)^{2}+\frac{1}{2}-\frac{1}{2}\Lambda b^{2}\right].$$

(2.36)

Here, $R$ and $g$ are respectively the $2D$ curvature scalar and the metric determinant and we have omitted the boundary term.

Following Louis-Martinez , we can remove the gradient term in the action by a conformal rescaling

$${\bar{g}}_{\mu\nu}=\Omega^{2}(b)g_{\mu\nu}$$

(2.37)

with

$$\frac{d\ln\Omega}{d\ln b}=\frac{1}{2}.$$

(2.38)

This has the solution

$$\Omega=(b/2)^{1/2},$$

(2.39)

where we have chosen the normalization factor for future convenience. The action then reduces to

$$S=\int d^{2}x\sqrt{-{\bar{g}}}\left[{\bar{\phi}}{\bar{R}}-V({\bar{\phi}})\right],$$

(2.40)

where ${\bar{\phi}}=b^{2}/4$ and

$$V({\bar{\phi}})=2\Lambda\sqrt{\bar{\phi}}-\frac{1}{2\sqrt{\bar{\phi}}}.$$

(2.41)

We define

$${\bar{\phi}}=\phi_{0}+{\phi},$$

(2.42)

where $\phi_{0}=1/4\Lambda$, so that $V(\phi_{0})=0$. We shall assume that $\Lambda\ll 1$, so $\phi_{0}\gg 1$. Then, for $|\phi|\ll\phi_{0}$ we can expand the potential (2.41) around $\phi=0$. Neglecting quadratic and higher order terms in the expansion, we obtain an approximate JT action

$$S\approx\phi_{0}\int d^{2}x\sqrt{-{\bar{g}}}{\bar{R}}+\int d^{2}x\sqrt{-{\bar{g}}}{\phi}\left({\bar{R}}-2{\bar{\Lambda}}\right),$$

(2.43)

where ${\bar{\Lambda}}=2\Lambda^{3/2}$.

Since the second term in (2.43) already includes a factor of $\phi$, we can use

$${\bar{g}}_{\mu\nu}\approx\frac{1}{2\sqrt{\Lambda}}g_{\mu\nu},~{}~{}~{}{\bar{R}}\approx 2\sqrt{\Lambda}R.$$

(2.44)

Hence, in the same approximation we can rewrite the action in terms of the original metric $g_{\mu\nu}$ and the cosmological constant $\Lambda$ as

$$S\approx\phi_{0}\int d^{2}x\sqrt{-{g}}{R}+\int d^{2}x\sqrt{-{g}}{\phi}\left({R}-2{\Lambda}\right),$$

(2.45)

The above analysis suggests that in the appropriate limit the dynamics of the $4D$ cosmological model (2.34) is well approximated by that of the JT gravity (2.1) with $\Lambda=H^{2}$. The radius of the sphere $S^{2}$ in this limit is $b\approx H^{-1}$. We will focus on this regime in most of the paper, but in the next section we will see that in the minisuperspace setting the two models are even more closely related and can be mapped onto one another for arbitrary values of the scale factors $a$ and $b$.

As already mentioned, our focus in this paper will be on homogeneous minisuperspace models. Hence we will use a homogeneous version of the model (2.34), with the scale factors $a$ and $b$ independent of $x$, for dimensional reduction. This is the Kantowski-Sachs (KS) model KS describing a homogeneous universe with spatial sections of $S^{1}\times S^{2}$ topology.

Following Halliwell and Louko HL , we represent the metric of the KS model as

$$ds^{2}=-\frac{N^{2}}{a^{2}}d\tau^{2}+a^{2}dx^{2}+b^{2}d\Omega^{2},$$

(3.1)

where $N$, $a$ and $b$ are functions of time $\tau$, which we can choose to vary in the range $0<\tau<1$. After substituting this in the Lorentzian Einstein-Hilbert action with a cosmological constant $\Lambda\equiv H^{2}$ and integrating over $x$ and over the angular variables, the action reduces to

$$S=-\pi\int_{0}^{1}d\tau\left[\frac{{\dot{b}}{\dot{c}}}{N}+N(H^{2}b^{2}-1)\right],$$

(3.2)

where we have introduced a new variable $c=a^{2}b$.

The factor $1/a^{2}$ is added in the first term of (3.1) in order to simplify the equations of motion, which take the form

$${\ddot{b}}=0,$$

(3.3)

$$\frac{\ddot{c}}{N^{2}}-2H^{2}b=0,$$

(3.4)

where overdots stand for derivatives with respect to $\tau$. The constraint equation is obtained by varying the action with respect to $N$:

$$\frac{{\dot{b}}{\dot{c}}}{N}-N(H^{2}b^{2}-1)=0.$$

(3.5)

An important solution of these equations is obtained by setting ${\dot{b}}=0$. Then Eq.(3.5) tells us that $b=H^{-1}$ and Eq.(3.4), expressed in terms of the proper time variable $t=\int d\tau/a(\tau)$, becomes

$$\frac{d^{2}a}{dt^{2}}=H^{2}a,$$

(3.6)

which has a solution

$$a(t)=H^{-1}\cosh(Ht),$$

(3.7)

where we have set $N=1$. This is the Nariai solution, which is a product of a $2D$ de Sitter space and a 2-sphere of radius $H^{-1}$ Nariai .

It follows from Eq.(3.3) that ${\dot{b}}$ cannot change sign, indicating that the Nariai solution is unstable. If we perturb it by giving the radius of the sphere $b$ a slight velocity, the sphere will collapse if ${\dot{b}}<0$ and will expand to infinite size if ${\dot{b}}>0$.

The Euclidean continuation of the Nariai solution is a product of two spheres of radius $H^{-1}$. It is often referred to as the Nariai instanton and describes nucleation of extremal black holes in de Sitter space Ginsparg:1982rs ; Bousso:1996au .

The quantum cosmology of the KS model has been studied by a number of authors Laflamme ; HL ; Conradi ; Anninos:2012ft ; Hertog and Conti . Some exact solutions of the WDW equation have been found and semiclassical methods have been used to study more general solutions. Here we will follow the method of Halliwell and Louko (HL) HL which allows one not only to find the saddle points of the action, but also helps to determine which saddle points contribute to the semiclassical wave function. This method will also be useful for interpreting the solution (2.29) found by IKTV.

The momenta conjugate to the variables $b$ and $c$ are

$$p_{b}=-\pi{\dot{c}}/N,~{}~{}~{}~{}p_{c}=-\pi{\dot{b}}/N.$$

(3.8)

Using this in the constraint equation (3.5) and replacing $p_{b}\to-i\partial/\partial b$, $p_{c}\to-i\partial/\partial c$, we obtain the WDW equation

$$\pi{\cal H}\Psi=\left[\partial_{b}\partial_{c}+\pi^{2}(H^{2}b^{2}-1)\right]\Psi=0.$$

(3.9)

This equation can be simplified by introducing a new variable $\xi$, which is related to $b$ as $d\xi=\pi^{2}(H^{2}b^{2}-1)db$. Choosing the integration constant so that $\xi(Hb=1)=0$, we have

$$\xi=\frac{\pi^{2}}{3H}(H^{3}b^{3}-3Hb+2)=\frac{\pi^{2}}{3H}(Hb-1)^{2}(Hb+2).$$

(3.10)

We also introduce the variable $\rho=c-H^{-3}$; then the WDW equation becomes

$$(\partial_{\xi}\partial_{\rho}+1)\Psi=0.$$

(3.11)

We note that this equation has the same form as Eq.(2.20) for the JT model. The difference is that Eq.(2.20) is for the function ${\tilde{\Psi}}=\Psi/a$, where the factor $1/a$ appeared due to a particular choice of the factor ordering. Following the same steps as in Sec.2.3, we find that Eq.(3.11) has exact solutions

$$\Psi_{m}=\left(\frac{\rho}{\xi}\right)^{m/2}H_{m}^{(2)}(2\sqrt{\xi\rho}).$$

(3.12)

The solution with $m=-2$ corresponds to the IKTV solution (2.29). We expect this solution to agree with (2.29) when $b\approx H^{-1}$. The argument of the Hankel function in (3.12) is

$$2\sqrt{\xi\rho}=\frac{2\pi}{H^{2}}\sqrt{\frac{Hb+2}{3}}(Hb-1)\sqrt{H^{3}a^{2}b-1}\approx\frac{2\pi}{H^{2}}(Hb-1)\sqrt{H^{2}a^{2}-1}.$$

(3.13)

Comparing this with the argument of the Hankel function in (2.29), we can identify

$$\frac{Hb-1}{H^{2}}\approx 2\phi.$$

(3.14)

It is interesting to note that the correspondence between the two wave function extends beyond this approximation. If we define

$${\tilde{a}}=(Hb)^{1/2}a,~{}~{}~{}~{}{\phi}=\frac{(Hb-1)}{2H^{2}}\sqrt{\frac{Hb+2}{3}},$$

(3.15)

then ${\tilde{a}}\Psi({\tilde{a}},{\phi})$ exactly reproduces the wave function (2.29). More generally, the transformation (3.15) can be used to obtain a solution to the WDW equation of the JT model from that of the KS model and vice versa. Note also that ${\tilde{a}}$ and ${\phi}$ are simply related to the variables $\xi$ and $\rho$:

$$\xi=4\pi^{2}H^{3}{\phi}^{2},~{}~{}~{}~{}\rho=H^{-3}(H^{2}{\tilde{a}}^{2}-1).$$

(3.16)

We thus see that JT and KS minisuperspace models are formally equivalent to one another. This equivalence, however, does not extend beyond minisuperspace. In the JT case, the minisuperspace wave function can be extended to a wave function in full superspace, but in the KS model the number of variables in the wave function and the pre-exponential factor depend on which perturbation modes are included in the minisuperspace truncation. Equivalence between the two models at the minisuperspace level will nevertheless be sufficient for our purposes here.

We now consider the transition amplitude from the initial state $\{b^{\prime},c^{\prime}\}$ to the final state $\{b,c\}$. We will be particularly interested in the initial state

$$\{b^{\prime},c^{\prime}\}=\{H^{-1},H^{-3}\}$$

(3.17)

corresponding to the bounce point of the Nariai solution. We shall refer to it as ‘Nariai initial conditions’.

The general framework for calculating transition amplitudes has been discussed by HL HL . For a general minisuperspace model described by the Hamiltonian

$${\cal H}=\frac{1}{2}f^{\alpha\beta}(q)p_{\alpha}p_{\beta}+V(q),$$

(3.18)

where $q^{\alpha}$ are the generalized coordinates and $p_{\alpha}$ their conjugate momenta, the transition amplitude between $q^{\prime}$ and $q$ is

$$G(q|q^{\prime})=\int_{0}^{\infty}dN\int\mathcal{D}p\mathcal{D}qe^{iS}=\int_{0}^{\infty}dN\langle q,N|q^{\prime},0\rangle.$$

(3.19)

Here $N$ is the lapse parameter, the action is

$$S=\int_{0}^{1}d\tau\left(p_{\alpha}{\dot{q}}^{\alpha}-N{\cal H}\right)$$

(3.20)

and the path integral is over histories interpolating between $q^{\prime}$ and $q$.

HL show that the amplitude (3.19) satisfies

$${\cal H}G(q|q^{\prime})=-i\langle q,0|q^{\prime},0\rangle=-i\delta(q,q^{\prime}),$$

(3.21)

where

$${\cal H}=-\frac{1}{2}\nabla^{2}+\zeta R+V$$

(3.22)

is the Hamiltonian operator, $\nabla^{2}$ and $R$ are the Laplacian and the curvature scalar in the metric $f_{\alpha\beta}$, and $\zeta$ is the conformal coupling. The magnitude of $\zeta$ depends on the dimension of superspace and vanishes in the case of $2D$, which is of interest to us here. We see from Eq.(3.21) that $G$ is a Green’s function of the WDW equation.

For the KS model the Hamiltonian is quadratic and the path integral in (3.19) can be performed exactly. Then, up to an overall multiplicative constant, HL found that the amplitude reduces to

$$G(b,c|b^{\prime},c^{\prime})=\int_{0}^{\infty}\frac{dN}{N}\exp\left[\frac{i}{2}\left(\alpha N-\frac{\beta}{N}\right)\right],$$

(3.23)

where

$$\alpha=1-\frac{H^{2}}{3}(b^{2}+bb^{\prime}+{b^{\prime}}^{2}),~{}~{}~{}\beta=(2\pi)^{2}(c-c^{\prime})(b-b^{\prime}).$$

(3.24)

This amplitude should satisfy

$${\cal H}G(b,c|b^{\prime},c^{\prime})\propto-i\delta(b-b^{\prime})\delta(c-c^{\prime}).$$

(3.25)

The integral over $N$ in (3.23) can be expressed in terms of Bessel functions HL . With Nariai initial conditions (3.17) we have

$$\alpha=-\frac{1}{3}(Hb-1)(Hb+2),~{}~{}~{}\beta=\left(\frac{2\pi}{H^{2}}\right)^{2}(H^{3}c-1)(Hb-1)$$

(3.26)

and

$$G=-i\pi H_{0}^{(2)}\left(\sqrt{-\alpha\beta}\right)~{}~{}~{}~{}~{}(H^{3}c>1)$$

(3.27)

$$G=2K_{0}\left(\sqrt{\alpha\beta}\right)~{}~{}~{}~{}~{}(H^{3}c<1)$$

(3.28)

The amplitude $G$ in Eqs.(3.27),(3.28) is similar to the IKTV wave function (2.29),(2.30). The difference is in the prefactor and in the index of the Bessel functions. The two objects are closely related, as we will now show.

The Bessel functions appearing in Eqs.(3.27),(3.28) can be expressed as $Z_{0}(X)$, where

$$X=\sqrt{H^{3}c-1}f(b)=4\pi{\phi}\sqrt{H^{2}{\tilde{a}}^{2}-1}$$

(3.29)

with

$$f(b)=\frac{2\pi}{H^{2}}(Hb-1)\sqrt{\frac{Hb+2}{3}}=4\pi{\phi},$$

(3.30)

where we have used the notation $Z_{\nu}$ for a Bessel function (of any kind) with index $\nu$ and Eqs.(3.15) relating $a$ and $b$ to ${\tilde{a}}$ and ${\phi}$. Differentiating twice with respect to $c$, we obtain

$$\frac{\partial^{2}Z_{0}}{\partial c^{2}}=\frac{H^{6}f^{2}(b)}{4(H^{3}c-1)}\left[-Z_{1}^{\prime}+\frac{1}{f(b)\sqrt{H^{3}c-1}}Z_{1}\right]=\frac{H^{6}f^{2}(b)}{4(H^{3}c-1)}Z_{2}=(2\pi H^{3})^{2}\frac{{\phi}^{2}}{{\tilde{a}}^{2}-1}Z_{2}.$$

(3.31)

Here prime stands for a derivative with respect to the argument and we used an iteration formula for Bessel functions in the second step.

Now, the expression on the right-hand side of Eq.(3.31) has the same form as the wave function (2.29) of IKTV. We conclude that

$$\Psi_{IKTV}=C\frac{\partial^{2}}{\partial c^{2}}G(b,c|H^{-1},H^{-3})$$

(3.32)

with $C={\rm const}$. Furthermore, it follows from Eq.(3.25) that⁴⁴4Note that $\partial/\partial c$ commutes with ${\cal H}$.

$${\cal H}\Psi_{IKTV}\propto-i\delta(b-H^{-1})\delta^{\prime\prime}(c-H^{-3}).$$

(3.33)

Hence $\Psi_{IKTV}$ is not a solution of the WDW equation. It has a distributional source at $a=b=H^{-1}$, which is more singular than that of a Green’s function.

To discuss the boundary conditions for the no-boundary path integral, it is more convenient to represent the Euclideanized KS metric as

$$ds_{E}^{2}=N^{2}dt^{2}+a^{2}(t)dx^{2}+b^{2}(t)d\Omega^{2}.$$

(4.2)

The Euclidean action is then HL

$$S_{E}=\pi\int_{0}^{1}dt\left[-\frac{1}{N}\left(a{\dot{b}}^{2}+2b{\dot{a}}{\dot{b}}\right)+Na(H^{2}b^{2}-1)\right]+S_{b},$$

(4.3)

where $S_{b}$ is the boundary term⁵⁵5 The Gibbons-Hawking boundary terms in the gravitational action cancel out after integration by parts if the geometry has two boundaries – at $t=0$ and $t=1$. But for a compact geometry with a single boundary at $t=1$, the boundary term at $t=0$ has to be kept HL .

$$S_{b}=-\left[\frac{\pi}{N}\frac{d}{dt}(ab^{2})\right]_{t=0}.$$

(4.4)

The time variable $t$ is defined so that $0<t<1$ with $t=1$ corresponding to the boundary ${\cal B}$ and $t=0$ corresponding to the ”bottom” ${\cal B}_{0}$ of the 4-geometry $g$.

The boundary conditions at $t=1$ fix the values of $\{a,b\}$, while the boundary conditions at $t=0$ should be chosen so that the geometry closes smoothly at ${\cal B}_{0}$. HL show that for a classical 4-geometry(4.2) there are two choices:

$$a(0)=0,~{}~{}\frac{1}{N}{\dot{a}}(0)=\pm 1,~{}~{}\frac{1}{N}{\dot{b}}(0)=0$$

(4.5)

and

$$b(0)=0,~{}~{}\frac{1}{N}{\dot{b}}(0)=\pm 1,~{}~{}\frac{1}{N}{\dot{a}}(0)=0,$$

(4.6)

where overdots now stand for derivatives with respect to $t$.

The time derivatives ${\dot{a}}$ and ${\dot{b}}$ are related to the (Euclidean) momenta conjugate to $a$ and $b$:

$$p_{a}=-\frac{2\pi}{N}b{\dot{b}},~{}~{}p_{b}=-\frac{2\pi}{N}(a{\dot{b}}+b{\dot{a}}),$$

(4.7)

so these boundary conditions correspond to fixing $\{a,p_{a},p_{b}\}$ or $\{b,p_{a},p_{b}\}$ at ${\cal B}_{0}$. It is however inconsistent in quantum theory to fix a coordinate and its conjugate momentum. Hence the best one can do is to impose two out of the three conditions, for example

$$a(0)=0,~{}~{}p_{b}(0)=\mp 2\pi b(0)$$

(4.8)

or

$$b(0)=0,~{}~{}p_{a}(0)=0.$$

(4.9)

HL note that if classical field equations hold, then with either of these choices all three conditions in (4.5) or (4.6) are satisfied. One can therefore expect that in the semiclassical regime the path integral will be dominated by regular geometries. Since we are interested in dimensional reduction of KS model with the $S^{2}$ part integrated out, we will focus on the boundary conditions (4.8), which correspond to fixing $a$ and ${\dot{a}}$ on ${\cal B}_{0}$.

HL suggest that a better choice of variables, suitable for the boundary conditions (4.8), would be

$$A=2\pi b^{2},~{}~{}B=2\pi ab$$

(4.10)

with the conjugate momenta

$$P_{A}=-\frac{\dot{a}}{2N},~{}~{}P_{B}=-\frac{\dot{b}}{N}.$$

(4.11)

The boundary conditions (4.8) then take the form

$$B^{\prime}=0,~{}~{}{P_{A}}^{\prime}=\mp\frac{1}{2},$$

(4.12)

where primes indicate the values at $t=0$.

The HH wave function can now be expressed as HL

$$\Psi_{NB}(A,B)=G(A,B|{P_{A}}^{\prime},B^{\prime})=\int dN\int{\cal D}Q^{\alpha}{\cal D}P_{\alpha}\exp(-S_{E}),$$

(4.13)

where the path integral is taken over histories with fixed $\{A,B\}$ and $\{{P_{A}}^{\prime},B^{\prime}\}$. Unlike the case of fixed initial values $a^{\prime}$ and $b^{\prime}$, this path integral cannot be evaluated exactly. We therefore use the WKB method to express $\Psi_{HH}$ approximately as

$$\Psi_{HH}(Q^{\alpha})=\int_{C}\mu\left(Q^{\alpha},Z^{\beta},N\right)\exp\left[-S_{E}(Q^{\alpha};N|Z^{\beta})\right]dN,$$

(4.14)

where $Q^{\alpha}=\{A,B\},~{}Z^{\beta}=\{{P_{A}}^{\prime},B^{\prime}\}$, $\mu$ is the semiclassical prefactor of the propagator and

$$S_{E}=\pi\left[\frac{H^{2}}{3}N(b^{2}+bb^{\prime}+{b^{\prime}}^{2})-N-\frac{1}{N}(b-b^{\prime})\left(a^{2}b-\frac{{B^{\prime}}^{2}}{b^{\prime}}\right)+2b^{\prime 2}{P_{A}}^{\prime}+2B^{\prime}{P_{B}}^{\prime}\right]$$

(4.15)

is the Euclidean action evaluated on a history satisfying the boundary conditions and the second order equations (3.3),(3.4) for $a$ and $b$, but not the constraint equation (3.5). Note that the last term in (4.15) can be neglected since it does not contribute to the path integral and the semiclassical prefactor. The integration contour $C$ in (4.14) is generally complex; we shall discuss the choice of this contour in Sec.4.4. The calculation of the prefactor is discussed in Sec. 4.3.

HL discussed the calculation of the HH wave functions for KS model only for the case of a vanishing cosmological constant, $H=0$. Eqs.(4.14),(4.15) apply for arbitrary $H$, but from this point on we cannot directly use the results of HL and will have to extend their analysis to $H>0$.

The initial value $b^{\prime}$ in Eq.(4.15) has to be expressed in terms of the boundary values $A,B$ (or $a,b$), $B^{\prime},{P_{A}}^{\prime}$, and the lapse $N$. This can be done using the solutions ${\bar{a}}(\tau),{\bar{b}}(\tau)$ of the second order field equations (3.3),(3.4) (but not of the constraint equation). HL give these solutions in terms of the time variable ${\tau}$, which is related to $t$ as $d{\tau}=a(t)dt$:

$${\bar{b}}({\tau})=(b-b^{\prime}){\tau}+b^{\prime}$$

(4.16)

$${\bar{a}}^{2}({\tau}){\bar{b}}({\tau})=-\frac{H^{2}N^{2}}{3}(b-b^{\prime}){\tau}^{3}-H^{2}N^{2}b^{\prime}{\tau}^{2}+\left[a^{2}b-{a^{\prime}}^{2}b^{\prime}+\frac{H^{2}N^{2}}{3}(b+2b^{\prime})\right]{\tau}+{a^{\prime}}^{2}b^{\prime}.$$

(4.17)

Expressed in terms of $\tau$, the boundary condition $(1/N)(da/dt)=\pm 1$ takes the form

$$\frac{\bar{a}}{N}\frac{d{\bar{a}}}{d{\tau}}({\tau}=0)=\mp 2{P_{A}}^{\prime}=\pm 1.$$

(4.18)

To implement this boundary condition, we differentiate Eq.(4.17) with respect to ${\tau}$ and take the limit ${\tau}\to 0$. This gives (after dividing by ${b^{\prime}}^{2}$)

$$\frac{4N{P_{A}}^{\prime}}{b}{b^{\prime}}=-a^{2}+\frac{{B^{\prime}}^{2}}{{b^{\prime}}^{2}}-\frac{H^{2}N^{2}}{3b}(b+2b^{\prime}).$$

(4.19)

where we have used $a^{\prime}=B^{\prime}/b^{\prime}$.