The virtual black hole in $\boldsymbol{2d}$ quantum gravity

W. Kummer, H. Liebl, and D. V. Vassilevich, ``Integrating geometry in general 2D dilaton gravity with matter,'' Nucl. Phys. B544 (1999) 403, hep-th/9809168.

D. Grumiller, W. Kummer, and D. V. Vassilevich, ``The virtual black hole in 2d quantum gravity,'' Nucl. Phys. B580 (2000) 438, gr-qc/0001038.

P. Fischer, D. Grumiller, W. Kummer, and D. Vassilevich, ``$S$-matrix for $s$-wave gravitational scattering,'' to appear in Phys.Lett. B, gr-qc/0105034.

D. Grumiller, ``Quantum dilaton gravity in two dimensions with matter,''PhD thesis (supervisor: W. Kummer), gr-qc/0105078.

Two questions:

Question 1: What is a virtual black hole (VBH)?

(Something that behaves like a VBH...)

You will get a more satisfying answer during this talk.

Question 2: Why just $d=2$?

(Because we are lazy...)

Because conceptual problems are mostly independent of the dimension.

Why 2d?

cf. e.g. recent review of S. Carlip, gr-qc/0108040: Conceptual questions - dual role of the metric, problem(s) of time, information puzzle - can be answered much more easily in $d=2$ while encountering less technical problems.

Naive $\int \sqrt{-g} R = 8\pi(1-g)$ trivial $\rightarrow$ generalized gravity theories:

Jackiw-Teitelboim 84 ($R+\Lambda$)

Katanaev-Volovich 86 ($R^2+T^2$)

Callan-Giddings-Harvey-Strominger 92 (dilaton black hole)

and many others.

Unifying framework treating all dilaton theories in $d=2$ does exist and is closely related to Poisson-$\sigma$-models (Schaller, Strobl 94).

First order gravity in $d=2$

The first order Lagrangian in terms of Cartan variables $e$ and $w$:

$${\cal{L}}_{\text{car}} = \textcolor{red}{{\cal{L}}^{(g)}}+ \textcolor{magenta}{{\cal{L}}^{(m)}}$$

$$= \textcolor{red}{X^+ D\wedge e^- + X^- D\wedge e^+ + X d\wedge \omega}$$

$$\textcolor{red}{- e^- \wedge e^+ {\cal V}}+ \textcolor{magenta} {{\cal{L}}^{(m)}}$$ (1)

with ${\cal V} = V(X) + X^+X^- U(X)$.

Explanations of the terms present in (1):

• The geometric part ${\cal{L}}^{(g)}$ is just a special case of a Poisson-$\sigma$ model and the matter part will be specified below.

• The ``gauge-covariant'' derivative $D$ acts on the vielbein via

  $$\textcolor{red}{D\wedge e^{\pm} = d \wedge e^{\pm} \pm \omega \wedge e^{\pm}}.$$

  
  
  and $A_i=(\omega ,e^-,e^+), X^i=(X,X^+,X^-)$ .

Constraint analysis

$H_{can} = \alpha _i G_i \approx 0$ as expected.

3 first class primary constraints $\bar{p}_i\approx 0$

3 first class secondary constraints $G_i\approx 0$

structure functions $\left\{G_i,G_j'\right\}=C_{ij}{}^kG_k\delta (x-x')$ with
$\parbox{1cm}{\begin{eqnarray*} && C_{12}{}^2 = -1, \\ && C_{13}{}^3 = 1, \end{eqnarray*}}$ $\parbox{7cm}{\begin{eqnarray*} && C_{23}{}^1 = -\frac{\partial{\cal{... ... \\ && C_{23}{}^3 = -\frac{\partial {\cal{V}}}{\partial X^-}. \end{eqnarray*}}$ $\parbox{1cm}{\begin{equation*}\end{equation*}}$

A certain linear combination of the constraints yields the Virasoro algebra, another useful combination yields an abelian algebra in the matterless limit (``Casimir-Darboux'').

BRST quantization in
``temporal'' gauge

gauge-fixed Hamiltonian:

$$H_{gf} = H_{BRST} + \left\{ \Psi, \Omega \right\}$$

BRST charge: $\Omega = c^iG_i + \frac{1}{2}c^ic^jC_{ij}{}^kp^c_k + b^i\bar{p}_i$gauge-fixing fermion for ``temporal gauge'':

$$\Psi = p^c_2$$

leads (after eliminating all ghosts $b^i,c^i$ and ghost-momenta $p^b_i,p^c_i$) to the path integral

$$W = \int \left({\cal D}q_i\right)\left({\cal D}p^i\right) \textcolor{magenta}{\left({\cal D}S\right)\left({\cal D}P\right)}$$

$$\left( \det \partial_0 \right)^2 \det \left( \partial_0 + X^+U(X) \right)$$

$$\times \exp \left[ i \int \left( \textcolor{magenta}{{\cal L}_{\text{gf}}}+ J_i p^i + j_i q_i + \textcolor{magenta}{Q S}\right) d^2x \right],$$

with

$$\textcolor{magenta}{{\cal L}_{\text{gf}}}= p^i \dot{q}_i + \textcolor{magenta}{P\dot{S} + G_2}.$$

Without matter the result is trivial.

Minimally coupled matter

Main results:

• can reconstruct geometry out of matter fields (albeit perturbatively)
• mass-aspect function jumps $\rightarrow$ ``virtual black hole'' (VBH)
• have to choose boundary values of line element (for $x_0 \to \infty$ $\rightarrow$ take simplest choice (no Schwarzschild- and no Rindler-term)
• with this choice one can use asymptotic Minkowski modes for the evaluation of $S$-matrix elements
• trivial result for massless scalars (either the BH is ``plugged'' or the scattering amplitude diverges)
• nontrivial result for massive scalars for suitable boundary conditions at $r=0$

Integrating out geometry

First ``coordinates'', then momenta $\rightarrow$ exact integration possible, but dependence on homogeneous solutions. Final integration of matter fields is only possible perturbatively $\rightarrow$ obtain to LO propagator contribution (apart from terms of ${\cal O}(\hbar)$) like generalized Polyakov action and to NLO a 4-point vertex.

• 4-point vertex can be extracted formally from the final (non-local and non-polynomial) effective action (complicated)
• 4-point vertex can be obtained by a trick, using classical EOM (easy)

Trick: sufficient to assume second order combinations of scalar localized

$$S_0:=\frac{1}{2}(\partial_0 S)^2=c_0\delta (x-y),$$

$$S_1:=\frac{1}{2}(\partial_0 S)(\partial_1 S)=c_1\delta (x-y).$$

and solve classical EOM to LO in $c_0,c_1$.

The LO solution of classical EOM

$\parbox{2.5cm}{\begin{eqnarray*} && \partial_0 p_1 = p_2, \\ && \pa... ...genta}{S_0}, \\ && \partial_0 p_3 = 2 + \frac{p_2p_3}{2p_1}, \end{eqnarray*}}$ $\parbox{4cm}{\begin{eqnarray*} && \partial_0 q_1 = \frac{q_3p_2p_3}{... ...q_3p_3}{2p_1}, \\ && \partial_0 q_3 = - \frac{q_3p_2}{2p_1}, \end{eqnarray*}}$ $\parbox{1cm}{\begin{equation*}\end{equation*}}$
to linear order in $\textcolor{magenta}{c_0}$ and $\textcolor{magenta}{c_1}$ is found easily:

$$p_1 (x) = x_0 + (x_0 - y_0) \textcolor{magenta}{c_0}y_0 h(x,y),$$

$$p_2 (x) = 1 + \textcolor{magenta}{c_0}y_0 h(x,y),$$

$$q_2 (x) = 4 \sqrt{p_1} + \left(2\textcolor{magenta}{c_0}y_0^{3/2}-\textcolor{magenta}{c_1}y_0\right.$$

$$\quad \left. + (\textcolor{magenta}{c_1}-6\textcolor{magenta}{c_0}y_0^{1/2})p_1 \right) h(x,y),$$

$$q_3 (x) = \frac{1}{\sqrt{p_1}}.$$

with $h(x,y) := \theta (y_0 - x_0) \delta (x_1-y_1)$
(``causal prescription'' for boundary values at $x_0 \to \infty$)

Integrations constants fixed by asymptotic conditions on the effective line element.

Effective line element

Reconstruction of geometry from matter solution possible. To LO the effective line element has outgoing Sachs-Bondi form:

$$(ds)^2=2drdu+\textcolor{magenta}{K(r,u)}(du)^2$$

The Killing-norm equals asymptotically (i.e. $x_0>y_0$) unity by construction:

$$\textcolor{magenta}{K(r,u)}=\left(1-\left(\frac{2\textcolor{magenta}{m}}{r}+ \textcolor{magenta}{a}r\right)\theta(y_0-x_0)\right),$$

$\parbox{5cm}{\epsfig{file=virtualBH.epsi,height=10cm,width=5cm}}$ $\parbox{10cm}{with \textcolor{magenta}{$m = \delta (x_1-y_1)(-c_1 y_... ...\textcolor{red} {$u=2\sqrt{2}x_1$} and \textcolor{red}{$r=\sqrt{p_1(x_0)/2}$}}}$

The virtual black hole

In all $2d$ dilaton theories exists a conserved quantity (trivial in PSM approach: Casimir function ``counting'' the symplectic leaves), cf. Kummer,Schwarz 92, Grosse, Kummer, Presnajder, Schwarz 92, Mann 93, Schaller, Strobl 94, Strobl, Klösch 96, ...

Useful even in the presence of matter (Kummer, Tieber 99, Grumiller, Kummer 00): its geometric part is proportional to the so-called mass aspect function $\rightarrow$ related to BH mass

In SRG ${\cal C}^{(g)} = \frac{p_2p_3}{\sqrt{p_1}} - 4 \sqrt{p_1}$.

$p_1$ and $p_3$ are continuous, but $p_2$ jumps at $x_0=y_0$ $\rightarrow$ ${\cal C}^{(g)}$ is discontinuous $\rightarrow$ this discontinuity phenomenon has been called virtual black hole

VBH enters $S$-matrix $\rightarrow$ idea of 't Hooft 96 to consider BH together with matter fields in scattering processes

The 4-point vertex

$$V^{(4)}_a = \int_x\int_y S_0(x) S_0(y) \left( \frac{d q_2}{d c_0} p_1 + q_2 \frac{d p_1}{d c_0} \right)$$

$$= \int_x \int_y S_0(x) S_0(y) \left\vert \sqrt{y_0}-\sqrt{x_0} \right\vert \sqrt{x_0y_0}$$

$$\quad \left( 3x_0+3y_0+2\sqrt{x_0y_0} \right) \delta (x_1-y_1),$$

and

$$V^{(4)}_b = \int_x\int_y \left(S_0(y) S_1(x) \frac{d p_1}{d c_0} - S_0(x) S_1(y) \frac{d q_2}{d c_1} p_1 \right)$$

$$= \int_x\int_y S_0(x) S_1(y) \left\vert x_0-y_0 \right\vert x_0 \delta (x_1 - y_1),$$

with $\int_x := \int\limits_0^\infty dx^0\int \limits_{-\infty}^\infty dx^1$ .

This non-local vertex for SRG differs qualitatively from the minimally coupled case $\rightarrow$ expect/hope for non-trivial result, even in massless case

Asymptotics

With $t := r + u$ the scalar field satisfies asymptotically the spherical wave equation. For proper $s$-waves only the spherical Bessel function

$$\textcolor{magenta}{R_{k0} (r) = \frac{\sin (kr)}{kr}}$$

survives in the mode decomposition ( $Dk:=4\pi k^2dk$):

$$\textcolor{magenta}{S(r,t) = \frac{1}{(2\pi)^{3/2}}\int\limits_0^{\infty} \frac{Dk}{\sqrt{2k}} R_{k0} \left[a^+_k e^{ikt} + a^-_k e^{-ikt}\right].}$$

With $a^\pm$ obeying the commutation relation $[a_k^-,a_{k'}^+] = \delta (k-k')/(4\pi k^2)$, they are used to define asymptotic states and to build the Fock space. The normalization factor is chosen such that the Hamiltonian reads

$$\textcolor{magenta}{H = \frac{1}{2} \int\limits\limits_0^{\infty}... ...l_t S)^2 + (\partial_r S)^2 \right] = \int\limits_0^{\infty} Dk a^+_k a^-_k k.}$$

The scattering amplitude

After a long and tedious calculation (for details see Grumiller 01 and Fischer 01 or the web-page http://stop.itp.tuwien.ac.at/$\sim$grumil/projects/myself/ thesis/s4.nb) for the $S$-matrix element with ingoing modes $q, q'$ and outgoing ones $k, k'$

$$T(q, q'; k, k') = \frac{1}{2} \left< 0 \left\vert a^-_ka^-_{k'} \left(V^{(4)}_a + V^{(4)}_b \right) a^+_qa^+_{q'}\right\vert 0 \right>$$

having restored the full $\kappa$-dependence we arrive at

$$T(q, q'; k, k') = -\frac{i\kappa \delta \left(k+k'-q-q'\right)}{2(4\pi)^4 \vert kk'qq'\vert^{3/2}} E^3 \tilde{T}$$

with $E=q+q'$,

$$\tilde{T} (q, q'; k, k') := \frac{1}{E^3}{\Bigg [}\Pi \ln{\frac{\Pi^2}{E^6}} + \frac{1} {\Pi} \sum_{p \in \left\{k,k',q,q'\right\}}p^2$$

$$\ln{\frac{p^2}{E^2}} {\Bigg (}3 kk'qq'-\frac{1}{2} \sum_{r\neq p} \sum_{s \neq r,p}\left(r^2s^2\right){\Bigg )} {\Bigg ]},$$

and $\Pi = (k+k')(k-q)(k'-q)$. The interesting part of the scattering amplitude is encoded in the scale independent factor $\tilde{T}$.

Discussion

• simplicity of result surprising, since both individual contributions are vastly more complicated and contain divergencies, which cancel precisely
  
• similar to gauge theories in particle physics, where a particular subset of graphs may be gauge dependent or even divergent, while only the sum must be finite and gauge independent cf. e.g. (Kummer, Lane 73 or Papavassiliou, Sirlin 94)
  
• forward scattering: pole in the amplitude
  
• analogy to cross section for $s$-waves can be defined
  
• decay rate (!) of an ingoing $s$-wave into three outgoing ones can be calculated
  
• caveat: despite of non-minimal coupling not full $4d$ information: on quantum level dimensional reduction anomaly (Frolov, Sutton, Zelnikov 00, Sutton 00), but even on classical level non-spherical modes can contribute to the $s$-wave matrix element

Conclusion & Outlook

• conceptual problems of quantum gravity independent of dimension $\rightarrow$ use $d=2$
  
• $2d$ dilaton quantum gravity treated most easily in first order formalism
  
• geometric parted can be treated exactly
  
• need matter for physical field degrees of freedom
  
• matter part can be treated perturbatively
  
• lowest order 4-point vertex yields interesting $S$-matrix elements for $s$-waves
  
• 1-loop calculation should give insight into information puzzle (in preparation)
  
• non-perturbative calculations could be possible for special cases