Coupling of Quantum Fields to Classical Gravity

Abstract

We consider ensembles on configuration space that consist of quantum fields which interact with and are the source of a classical gravitational field. These are hybrid systems where gravity remains classical while matter is described by quantized fields. There are some well known arguments in the literature which claim that such models are not possible. However, an examination of the most prominent ones, that are detailed enough to allow scrutiny, indicates that the hybrid models considered here are not excluded by any of the consistency arguments. We illustrate the approach with two examples. Our first example is a cosmological model. We consider the case of a closed Robertson–Walker universe with a massive quantum scalar field and solve the equations using a particular ansatz which selects a highly non-classical solution, one in which the scale factor of the Robertson–Walker universe is restricted to discrete values as a consequence of the interaction of the classical gravitational field with the quantized scalar field. We discuss this cosmological model in two approximations, that of a minisuperspace model and that of a midisuperspace model. Our second example concerns black holes. We consider CGHS black holes and show that we recover Hawking radiation from the equations that describe a hybrid system consisting of a classical CGHS black hole in a collapsing geometry interacting with a quantized scalar field. We also show that the hybrid model provides a natural resolution to the well known problem of time in quantum gravity.

Appendix: Ground State Gaussian Functional Solution

We follow the presentation given in App. C of Ref. [22]. To solve Eq. (11.15) for the case $S=0$,

$$\begin{aligned}&\int \mathrm{d}r N \left[ -\frac{1}{2\varLambda R^2} \left( \frac{1}{A}\frac{\delta ^{2}A}{\delta \phi ^{2}}\right) \right. + \lambda \frac{\varLambda R^2}{2} + V \nonumber \\&\quad + \left. \frac{R^2 }{2 \varLambda }\phi ^{\prime 2} + \frac{\varLambda R^{2} m^2}{2 } \phi ^{ 2} \right] = 0, \end{aligned}$$

(11.102)

consider the ansatz

$$\begin{aligned} A \sim \exp \left\{ -\frac{1}{2} \int \int \mathrm{d}y \, \mathrm{d}z \, \varLambda _y \, \varLambda _z \, R_y^2 \, R_z^2 \; \phi _y \, K_{yz} \, \phi _z \right\} . \end{aligned}$$

(11.103)

Evaluate $\frac{\delta ^{2}A}{\delta \phi ^{2}}$ and collect terms that have the same power of $\phi$. This leads to the following pair of equations for the kernel $K_{xy}$,

$$\begin{aligned} \int \mathrm{d}r\, N \left[ \lambda \frac{\varLambda R^2}{2} + V(R,\varLambda ) + \frac{\varLambda _r R_r^2}{2} K_{rr} \right] =0 \end{aligned}$$

(11.104)

and

$$\begin{aligned} \int \mathrm{d}r \, \varLambda _r R_r^2 \, N \left[ \frac{\phi _r^{\prime 2}}{2 \varLambda _r^2} + \frac{m^2}{2}\phi _r^{ 2} - \frac{1}{2} \int \int \mathrm{d}y \mathrm{d}z \,\varLambda _y R_y^2 \varLambda _z R_z^2 \,\phi _{y}K_{yr}K_{rz}\phi _{z} \right] =0 . \end{aligned}$$

(11.105)

After an integration by parts in Eq. (11.105), the equations for $K_{xy}$ take a form which is standard in the context of the Schrödinger functional representation of quantum field theory in curved spacetimes [19, 23–25].

Assume a foliation of spaces of constant positive curvature and a constant lapse function N and look for a solution valid under these conditions. Note that the $\varLambda$ and R that appear in the line element of Eq. (10.24) satisfy $\sqrt{h}=\varLambda R^2$ and $h^{rr}=\varLambda ^{-2}$, where $h^{kl}$ is the inverse metric tensor on the three-dimensional spatial hypersurface of constant curvature. Then, Eq. (11.105) can be written in the form

$$\begin{aligned}&\int \mathrm{d}r \sqrt{h_r} \left[ \frac{1}{2}h^{rr}\frac{\partial \phi _r}{\partial r}\frac{\partial \phi _r}{\partial r} \right. + \frac{m^2}{2}\phi _r^{ 2} \nonumber \\&\quad -\left. \frac{1}{2} \int \int \mathrm{d}y \mathrm{d}z \sqrt{h_y} \sqrt{h_z}\,\phi _{y}K_{yr}K_{rz}\phi _{z} \right] = 0, \end{aligned}$$

(11.106)

and the kernel $K_{xy}$ satisfies

$$\begin{aligned} \int \mathrm{d}r \sqrt{h_r} K_{yr}K_{rz} = \left[ -\frac{1}{\sqrt{h_y}}\frac{\partial }{\partial y}\left( h^{yy} \sqrt{h_y} \frac{\partial }{\partial y} \right) + m^2 \right] \delta (y,z), \end{aligned}$$

(11.107)

where $\delta (y,z)=\frac{1}{\sqrt{h_y}}\delta (y-z)$ is the delta function on the hypersurface.

To get an explicit expression for $K_{xy}$ that solves Eq. (11.107), introduce a fixed, particular set of coordinates for the line element of Eq. (10.24). Let

$$\begin{aligned} N = 1, \quad N_r=0, \quad \varLambda =a_0, \quad R=a_0 \, \sin \, r, \end{aligned}$$

(11.108)

where $r\in [0,2\pi )$ and $a_0$ can be interpreted as the scale factor of a closed Robertson–Walker universe. Then the solution of Eq. (11.107) is given by

$$\begin{aligned} K_{xy} = \frac{1}{2 a_0^4} \sum _n \sqrt{\gamma _n } \, \psi ^{(n)}_x \psi ^{(n)}_y, \end{aligned}$$

(11.109)

where the basis functions $\psi ^{(n)}_r$ are solutions of a Schrödinger-type equation in a space of constant curvature,

$$\begin{aligned} -\frac{1}{\sin ^2r}\,\frac{\partial }{\partial r} \left( \sin ^2 r \, \frac{\partial \psi _r^{(n)}}{\partial r} \right) + m^2 a_0^2 \, \psi _r^{(n)} = \gamma _n \psi _r^{(n)}. \end{aligned}$$

(11.110)

The $\psi _n(r)$ satisfy orthonormality and completeness relations. The eigenvalues $\gamma _n$ are given by

$$\begin{aligned} \gamma _n = n^2 - 1 + m^2 a_0^2, \quad n=1,2,3\ldots . \end{aligned}$$

(11.111)

Given the solution of Eq. (11.105), $a_0$ can be expressed in terms of the cosmological constant and the energy E of the quantized scalar field using Eq. (11.104), since [25]

$$\begin{aligned} E \sim a_0^3 \int \mathrm{d}r\, \sin ^2r \, K_{rr}. \end{aligned}$$

(11.112)

However, $\int \mathrm{d}r \, \sin ^2r \, K_{rr} \sim \sum _{n} \gamma _n$, which diverges. This is a consequence of the infinite zero-point energy of the quantum field. Therefore, to extract a finite result for $a_0$ it becomes necessary to introduce renormalization procedures.