††institutetext: Stanford Institute for Theoretical Physics and Department of
Physics,
Stanford University, Stanford, CA 94305, USA

# Quantization of Gravity in the Black Hole Background

Renata Kallosh and Adel A. Rahman<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a covariant (Lagrangian) quantization of perturbative gravity in
the background of a Schwarzschild black hole. The key tool is a decomposition
of the field into spherical harmonics. We fix Regge-Wheeler gauge for modes
with angular momentum quantum number $l\geq 2$, while for low multipole modes
with $l$ $=$ $0$ or $1$—for which Regge-Wheeler gauge is inapplicable—we
propose a set of gauge fixing conditions which are 2D background covariant and
perturbatively well-defined. We find that the corresponding Faddeev-Popov
ghosts are non-propagating for the $l\geq 2$ modes, but are in general
nontrivial for the low multipole modes with $l=0,1$. However, in Schwarzschild
coordinates, all time derivatives acting on the ghosts drop from the action
and the low multipole ghosts have instantaneous propagators. Up to possible
subtleties related to quantizing gravity in a space with a horizon, Faddeev’s
theorem suggests the possibility of an underlying canonical (Hamiltonian)
quantization with a manifestly ghost-free Hilbert space.

###### Contents

  1. 1 Introduction
  2. 2 Gravity in the Schwarzschild Black Hole Background
  3. 3 Gravitational Perturbations in the Spherical Harmonic Basis
    1. 3.1 Decomposition into Spherical Harmonics
    2. 3.2 Gauge Freedom in Spherical Harmonics
    3. 3.3 Boundary Conditions
    4. 3.4 Regge-Wheeler-Zerilli-Martel-Poisson Gauge
    5. 3.5 Comments on Monopoles $l=0$ and Dipoles $l=1$
  4. 4 Covariant Quantization of Gravity in the Schwarzschild Background
    1. 4.1 Gauge-Fixed Action
    2. 4.2 Covariant Quantization of the $l\geq 2$ Modes: Decoupling of Ghosts
    3. 4.3 Covariant Quantization of the $l=1$ Even (Electric Dipole) Modes
    4. 4.4 Covariant Quantization of the $l=1$ Odd (Magnetic Dipole) Modes
    5. 4.5 Covariant Quantization of the $l=0$ (Monopole) Modes: Results to All Orders
  5. 5 Faddeev’s Theorem and a Possible Hamiltonian Origin
    1. 5.1 Gauges with Unitary and Pseudo-Unitary Hamiltonians
    2. 5.2 $l\geq 2$
    3. 5.3 $l=1$ even
    4. 5.4 $l=1$ odd
    5. 5.5 $l=0$
  6. 6 Discussion
  7. A De Witt-Faddeev-Popov Covariant Quantization of Gravity
  8. B Faddeev-Fradkin-Vilkovisky Canonical Quantization of Gravity
  9. C Non-Linear Part of the Ghost Action in Schwarschild Coordinates
    1. C.1 $l\geq 2$ Modes
    2. C.2 $l=1$ Even Modes
  10. D Comments on Gravitational Propagator in the Black Hole Background

##  1 Introduction

We would like to perform a perturbative quantization of gravity in the
background of a Schwarzschild black hole, in a setting originally studied by
Regge and Wheeler Regge:1957td , and later studied by Zerilli Zerilli:1971wd
and by Martel and Poisson Martel:2005ir , which was recently discussed in the
context of quantization by Gaddam:2020rxb ; Gaddam:2020mwe .

As is well known, a key obstacle to a straightforward perturbative
quantization of gravity is the presence of a gauge symmetry, diffeomorphism
invariance, which leaves the naïve Feynman path integral ill-defined. A
procedure for defining the covariant (Lagrangian) path integral for quantum
field theories with gauge symmetries was proposed in Faddeev:1967fc ;
DeWitt:1967ub  and involves breaking the gauge symmetry using a set of gauge-
fixing conditions and introducing compensating Faddeev-Popov (FP) ghost
fields, whose action is computed according to the rules introduced in
Faddeev:1967fc ; DeWitt:1967ub . See appendix A for a review.

In the past such a quantization of gravity was performed either in a general
type of curved background satisfying the Einstein field equations
DeWitt:1967ub ; tHooft:1974toh ; Kallosh:1974yh ; Grisaru:1975ei , with a
background-covariant and Lorentz-covariant gauge choice such as background
covariant harmonic gauge, or in a flat Minkowski background Faddeev:1969su ;
Fradkin:1970pn ; Faddeev:1973zb  with either a Lorentz-covariant gauge choice
such as harmonic gauge or a non-Lorentz-covariant gauge choice such as Dirac
gauge Dirac:1958jc .

In the background-covariant and Lorentz-covariant harmonic gauge and its
generalizations, the 1- and 2-loop computations in gravity were performed in
tHooft:1974toh ; Grisaru:1975ei ; Kallosh:1978wt ; Barvinsky:1985an ;
Goroff:1985th ; vandeVen:1991gw . In these gauges the Fadeev-Popov (FP) ghosts
are propagating fields which give an important contribution to the Feynman
rules. In tHooft:1974toh ; Grisaru:1975ei ; Goroff:1985th  the standard
background-covariant and Lorentz-covariant de Donder gauge fixing condition
$F_{\mu}=\nabla_{\nu}h\indices{{}^{\nu}_{\mu}}-\frac{1}{2}\nabla_{\mu}h\indices{{}^{\nu}_{\nu}}$,
linear in the metric perturbation $h_{\mu\nu}$, was used. In vandeVen:1991gw
the choice of the gauge-fixing condition involved in addition to the standard
de Donder term a set of extra terms nonlinear in $h_{\mu\nu}$ (for example
terms like $h^{\nu\lambda}\nabla_{\lambda}h_{\mu\nu}$), with arbitrary
numerical or depending on scalars of the theory, coefficients. The result, in
this case of the 2-loop UV divergence in gravity, is independent of the 7
additional parameters defining the generalized gauge fixing condition. The FP
ghost contributions were computed in these generalized classes of gauges
according to standard rules, and the final results were gauge-independent,
since the FP ghosts were taken into account.

The correct Feynman path integral for computing loop diagrams has been studied
both in covariant (Lagrangian) quantization and canonical (Hamiltonian)
quantization, see for example DeWitt:1967ub ; Faddeev:1969su ; Fradkin:1970pn
; Faddeev:1973zb ; Fradkin:1976xa ; Fradkin:1977hw ; Batalin:1977pb . We
describe the main features of both ways of defining the perturbative Feynman
path integral in Appendices A and B, respectively.

Here we discuss briefly the relation between these two types of quantization
and their equivalence. In the covariant case, the gauge symmetry of the action
$S[\phi]=\int\mathcal{L}(\phi)$ with infinitesimal gauge parameters
$\xi_{\alpha}(x)$, $\alpha=1,\dots,m$ has to be broken in order to lift the
degeneracy of the kinetic terms and define the propagators. This is
accomplished by adding a gauge-fixing condition to the classical theory, for
example in the form $\chi_{\alpha}(\phi)=0$, where the functions
$\chi_{\alpha}(\phi)$ depend on the fields $\phi$ of the classical action and
possibly their derivatives. The gauge fixing functions $\chi_{\alpha}$
transform under infinitesimal gauge transformations as

$\delta\chi_{\alpha}=Q_{\alpha}{}^{\beta}\xi_{\beta}$ (1)

where $Q_{\alpha}{}^{\beta}$ is in general some differential operator which
depends on the fields $\phi$ of the classical action and their derivatives.
This then defines the Lagrangian for the Fadeev-Popov (FP) ghosts
corresponding to the given gauge-fixing condition via:

${\cal
L}_{\mathrm{ghost}}[C,\bar{C}]=\bar{C}^{\alpha}Q_{\alpha}{}^{\beta}C_{\beta}$
(2)

where $\bar{C}^{\alpha},C_{\beta}$ are known as FP anti-ghosts and ghosts,
respectively. The gauge-fixing condition $\chi_{\alpha}$ is viewed as
admissible provided $\det||Q_{\alpha}{}^{\beta}||\neq 0$ and provided that
arbitrary field configurations $\phi$ can always be transformed to
configurations satisfying $\chi_{\alpha}(\phi)=0$ via gauge transformations
respecting the boundary conditions (see section 3.3). Covariant quantization
has relatively simple rules which guarantee the gauge-independence of the
physical observables in the theory.

One important feature of the covariant quantization is that the local measure
of integration is not strictly defined. We refer to Fradkin:1977hw  where this
issue is discussed in detail. In particular, in the case that
$Q_{\alpha}{}^{\beta}(\phi)$ is a local function of fields without
differential operators acting on $C_{\beta}(x)$, the relevant ghost action
simply becomes $\bar{C}^{\alpha}(x)\tilde{C}_{\alpha}(x)$ where
$\tilde{C}_{\alpha}(x)\equiv Q_{\alpha}{}^{\beta}(\phi(x))C_{\beta}(x)$. The
corresponding ghosts are non-propagating and drop from Feynman rules. The
functional determinant resulting from the transformation from $C$ to
$\tilde{C}$ may contribute to the local measure of integration some field
dependent divergent term proportional to $\delta^{D}(0)$ where $D$ is the
dimension of the space-time on which $x$ is a coordinate (in our case we will
have $D=2$ since our fields will become effectively two-dimensional after
decomposition into spherical harmonics). Terms of this nature can be neglected
in a regularized theory or can be shown to cancel, as explained in
Fradkin:1977hw .

The canonical quantization procedure is more fundamental, being based on a
canonical Hamiltonian, but it is also more involved. The issue of unitarity of
the S-matrix can only be addressed in the canonical quantization method.
However, according to Faddeev:1969su ; Faddeev:1973zb ; Fradkin:1976xa ;
Fradkin:1977hw ; Batalin:1977pb  the result of covariant quantization is
equivalent to the canonical one. In the canonical formulation of theories with
gauge symmetries, one encounters first class constraints
$\phi^{\alpha}(p_{i},q^{i})=0$ where $i=1,\dots,n$ runs over the naïve
configuration space and again $\alpha=1,\dots,m$. An additional set of
conditions $\chi_{\alpha}(p,q)=0$ has to be added to perform the canonical
quantization, and it is required that the Poisson brackets of constraints with
the additional conditions have a non-vanishing determinant,
$\det||\\{\chi_{\alpha},\phi^{\beta}\\}||\neq 0$.

There are two different classes of conditions $\chi_{\alpha}$. One corresponds
to the case of “unitary” Hamiltonians in gauge theories, which have manifestly
ghost-free underlying Hilbert spaces; the other corresponds to the case of
“pseudo-unitary” Hamiltonians in gauge theories, which have underlying state
spaces with negative-norm ghost degrees of freedom which must be quotiented
out to yield the physical Hilbert space. In the second case the S-matrix is
pseudo-unitary in a space of states with the indefinite metric. More details
on this are in Sec. 5.

We will show here that in the Regge-Wheeler gauge Regge:1957td ;
Zerilli:1971wd ; Martel:2005ir  for gravitational perturbations with angular
momentum quantum number $l\geq 2$, the corresponding FP ghosts are non-
propagating and decoupled in the covariant quantization method. In the low
multipole $l=1,0$ sector in the two-dimensionally background covariant gauges
we propose here, the FP ghosts are present and in general propagating in
covariant quantization. However, in Schwarzschild coordinates in covariant
quantization the FP ghosts have “instantaneous” propagators containing only
space derivatives, which suggests that our covariant quantization may have a
corresponding canonical quantization which belongs to the first class of
theories mentioned above, with a unitary Hamiltonian and manifestly ghost-free
Hilbert space.

We will provide some evidence that this covariant result is in agreement with
one originating from a canonical quantization, as would be expected according
to Fadeev’s theorem Faddeev:1969su . There may be subtleties in directly
applying the logic of Faddeev:1969su  to the situation here, since
Schwarzschild coordinates are singular on the event horizon, which was absent
in quantization theorems of Faddeev:1969su ; Faddeev:1973zb ; Fradkin:1976xa ;
Fradkin:1977hw . We leave the study and resolution of such subtleties for
future work.

##  2 Gravity in the Schwarzschild Black Hole Background

Consider a 4D asymptotically flat spacetime $(\mathcal{M},\bar{g}_{\mu\nu})$
whose spacetime manifold $\mathcal{M}=\mathcal{M}_{2}\times\mathbb{S}^{2}$ can
be endowed with coordinates $(x^{a},\theta^{A})$, $a=1,2$ in which the metric
takes the form

$\bar{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$ (3)

with $g_{\mu\nu}$ the metric of the Schwarzschild black hole written in a
“spherically symmetric” form

$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=g_{ab}\,\mathrm{d}x^{a}\mathrm{d}x^{b}+r^{2}(x)\,\mathrm{d}\Omega^{2}_{2}$
(4)

Here $r(x)$ is defined globally and invariantly by

$4\pi r(x)^{2}\equiv\mathrm{Area}\Big{(}\text{$\mathrm{SO}(3)$ orbit of $x$ in
$\mathcal{M}$}\Big{)}$ (5)

As noted in appendix A, we work in the context of 4D asymptotically flat
general relativity, for which local diffeomorphisms (i.e. diffeomorphisms
obeying certain fall-off conditions, see section 3.3 and eqs. 30 and 31) are
gauge symmetries which act infinitesimally as

$\delta\bar{g}_{\mu\nu}=\overline{\nabla}^{(\bar{g})}_{\mu}\xi_{\nu}+\overline{\nabla}^{(\bar{g})}_{\nu}\xi_{\mu}$
(6)

where $\overline{\nabla}^{(\bar{g})}_{\mu}$ denotes the (torsion-free)
covariant derivative of $\bar{g}_{\mu\nu}$ and where we take the gauge
parameter to be the covector111The corresponding finite gauge transformation
is the diffeomorphism generated by $\bar{g}^{\mu\nu}\xi_{\nu}$. $\xi_{\mu}$.
In the context of the background field method within which we will work, the
diffeomorphism eq. 6 acts on the background $g_{\mu\nu}$ and perturbation
$h_{\mu\nu}$ as

$\delta g_{\mu\nu}=0,\qquad\delta
h_{\mu\nu}=\overline{\nabla}^{(\bar{g})}_{\mu}\xi_{\nu}+\overline{\nabla}^{(\bar{g})}_{\nu}\xi_{\mu}$
(7)

respectively. Note that in this context the diffeomorphisms eq. 6 are kept
distinct from diffeomorphisms/coordinate transformations of the background
$(\mathcal{M},g_{\mu\nu})$.

Our formalism (based on that of Martel:2005ir ) will be covariant with respect
to two-dimensional background diffeomorphisms/coordinate transformations of
$(\mathcal{M}_{2},g_{ab})$, i.e. diffeomorphisms/coordinate transformations of
the background $(\mathcal{M},g_{\mu\nu})$ which are constant on the two-
sphere. Letting $\mathcal{D}_{a}$ and $\epsilon_{ab}$ denote the covariant
derivative and volume form of $(\mathcal{M}_{2},g_{ab})$ respectively, it is
helpful to define

$r_{a}(x)=\mathcal{D}_{a}r(x)\quad\text{and}\quad
t^{a}(x)=-\epsilon^{ab}r_{b}(x)$ (8)

which are orthogonal (but not orthonormal) by construction. Here and below all
indices will be raised and lowered with the 2D background metric $g_{ab}$.
Here $t^{a}$ is a pseudo-vector which agrees with the stationary Killing
vector of the Schwarzschild background up to its transformation properties
under parity $\epsilon_{ab}\to-\epsilon_{ab}$. Defining

$f(r)\coloneqq g_{ab}r^{a}r^{b}=-g_{ab}t^{a}t^{b}=1-\frac{2GM}{r}$ (9)

we have that

$g^{ab}=\frac{1}{f(r)}\left(-t^{a}t^{b}+r^{a}r^{b}\right)$ (10)

Due to the warp factor $r^{2}(x)$, there are “cross-term” Christoffel symbols
for the coordinate systems eq. 4 which prevent the simple factorization

$\mathrm{d}x^{\mu}\,\nabla_{\mu}\neq\mathrm{d}x^{a}\mathcal{D}_{a}+\mathrm{d}\theta^{A}D_{A}$
(11)

where we have let $\nabla_{\mu}$ denote the covariant derivative of
$(\mathcal{M},g_{\mu\nu})$ and $D_{A}$ denote the covariant derivative of
$(\mathbb{S}^{2},\Omega_{AB})$. These “cross-term” Christoffel symbols are
given by

$\Gamma\indices{{}^{a}_{A}{}_{B}}=-r\,g^{ab}r_{b}\,\Omega_{AB},\qquad\Gamma\indices{{}^{\mu}_{a}{}_{B}}=\frac{1}{r}\,r_{a}\,\delta^{\mu}_{B}$
(12)

The remaining “cross-term” Christoffel symbols vanish, i.e.
$\Gamma\indices{{}^{A}_{a}{}_{b}}=0$.

“Schwarzschild coordinates”

$g_{ab}\,\mathrm{d}x^{a}\mathrm{d}x^{b}=-f(r)\,\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}$
(13)

which will serve an important role in the latter part of this paper, cover the
right static patch (right outer domain of communications) of the Schwarzschild
black hole and are adapted to its $\mathbb{Z}_{2}\rtimes\mathbb{R}$ “static
$\rtimes$ stationary” isometry. Here the “radial coordinate” $r$ is the
function defined by eq. 8 and $t$ is an affine parameter along the timelike
orbits of $t^{a}$ which obeys $t^{a}\mathcal{D}_{a}t=1$ in the given patch. It
is useful to note that, in these coordinates,

$r_{a}\mathrm{d}x^{a}=\mathrm{d}r,\qquad
r^{a}\partial_{a}=f(r)\partial_{r},\qquad t^{a}\partial_{a}=\partial_{t}$ (14)

Note that $t^{a}\partial_{a}=\partial_{t}$ is timelike throughout this patch,
asymptoting to a null (pseudo-)vector along the event horizon and a unit $t$
translation at infinity. Note that the slices $\\{\Sigma_{t}\\}$ of constant
$t$ are everywhere spacelike and constitute a family of Cauchy surfaces for
this patch.

##  3 Gravitational Perturbations in the Spherical Harmonic Basis

###  3.1 Decomposition into Spherical Harmonics

It is helpful to take advantage of the spherical isometries and orientability
of the background—and, in particular, of the explicit spherical symmmetry of
our background gauge choice eq. 4—to expand the metric perturbation
$h_{\mu\nu}$ in spherical harmonics of definite parity. In the “spherical”
coordinates eq. 4, the components of $h_{\mu\nu}$ transform under the action
of the background $\mathrm{SO}(3)$ spherical symmetry associated to each two-
sphere of fixed $x^{a}$ (and under parity $\epsilon\to-\epsilon$) as three
scalars $h_{ab}$, two vectors $h_{aA}$, and one (symmetric) second-order
tensor $h_{AB}$. We can thus resolve $h_{\mu\nu}$ into spherical harmonics of
definite parity via Regge:1957td ; Zerilli:1971wd ; Martel:2005ir

$h_{\mu\nu}=\underbrace{\begin{pmatrix}p_{ab}&p^{(+)}_{aB}\vspace{0.5em}\\\
p^{(+)}_{bA}&p^{(+)}_{AB}\end{pmatrix}}_{\text{parity even}}\ +\
\underbrace{\begin{pmatrix}0&p^{(-)}_{aB}\vspace{0.5em}\\\
p^{(-)}_{Ab}&p^{(-)}_{AB}\end{pmatrix}}_{\text{parity odd}}$ (15)

with

$\displaystyle p_{ab}$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{|m|\leq\ell}h_{ab}^{\ell m}Y^{\ell
m}$ (16) $\displaystyle p_{aA}^{(+)}$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{|m|\leq\ell}j_{a}^{\ell m}\,Y^{\ell
m}_{A}$ $\displaystyle p_{aA}^{(-)}$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{|m|\leq\ell}h_{a}^{\ell m}\,X^{\ell
m}_{A}$ (17) $\displaystyle p^{(+)}_{AB}$
$\displaystyle=r^{2}\left(\sum_{\ell=0}^{\infty}\sum_{|m|\leq\ell}K^{\ell
m}\Omega_{AB}Y^{\ell m}+\sum_{\ell=2}^{\infty}\sum_{|m|\leq\ell}G^{\ell
m}Y^{\ell m}_{AB}\right)$ $\displaystyle p^{(-)}_{AB}$
$\displaystyle=\sum_{\ell=2}^{\infty}\sum_{|m|\leq\ell}h_{2}^{\ell m}X^{\ell
m}_{AB}$ (18)

Here $Y^{\ell m}(\theta^{A})$ are the usual unit-normalized spherical
harmonics on $\mathbb{S}^{2}$. The even-parity vector and tensor spherical
harmonics are defined by ${Y^{\ell m}_{A}\equiv D_{A}Y^{\ell m}}$ and
${Y^{\ell
m}_{AB}\equiv\left[D_{A}D_{B}+\frac{1}{2}\ell\left(\ell+1\right)\Omega_{AB}\right]Y^{\ell
m}}$ respectively. The odd-parity vector and tensor spherical harmonics are
defined by ${X^{\ell m}_{A}\equiv-\epsilon\indices{{}_{A}^{B}}D_{B}Y^{\ell
m}}$ and ${X^{\ell
m}_{AB}\equiv-\frac{1}{2}\left[\epsilon\indices{{}_{A}^{C}}D_{C}D_{B}+\epsilon\indices{{}_{B}^{C}}D_{C}D_{A}\right]Y^{\ell
m}}$ respectively, where $\epsilon_{AB}$ is the volume form of the round unit
sphere and all indices have been raised and lowered with $\Omega_{AB}$.

Note that $p_{ab}$ can be regarded as an $\mathcal{M}_{2}$ tensor and an
$\mathbb{S}^{2}$ scalar; $p_{aA}$ as both an $\mathcal{M}_{2}$ vector and an
$\mathbb{S}^{2}$ vector; and $p_{AB}$ as an $\mathcal{M}_{2}$ scalar and an
$\mathbb{S}^{2}$ tensor. The decomposition eqs. 15, 16, 17, and 18 completely
specifies the angular dependence of the perturbation $h_{\mu\nu}$. In
particular, the coefficient functions $h_{ab}^{\ell m}$, $j_{a}^{\ell m}$,
$K^{\ell m}$, $G^{\ell m}$, $h_{a}^{\ell m}$, $h_{2}^{\ell m}$ are “angle
independent” (constant on the two-sphere and hence functions only of the two-
coordinate $x^{a}$) and can be regarded as scalar, vector and tensor fields on
$\mathcal{M}_{2}$.

###  3.2 Gauge Freedom in Spherical Harmonics

Note that, since the different spherical harmonics decouple from one another
in the kinetic term of the action, we can make independent gauge choices for
each choice of $(\pm)\ell m$. In order to understand how the gauge freedom eq.
7

$h_{\mu\nu}\to
h_{\mu\nu}+\overline{\nabla}^{(\bar{g})}_{\mu}\xi_{\nu}+\overline{\nabla}^{(\bar{g})}_{\nu}\xi_{\mu}$
(19)

acts on a given harmonic, we resolve the gauge parameter $\xi_{\mu}$ into
spherical harmonics as

$\xi_{\mu}=\underbrace{(\Xi_{a},\Xi_{A}^{(+)})}_{\text{parity even}}\ +\
\underbrace{(0,\Xi_{A}^{(-)})}_{\text{parity odd}}$ (20)

with

$\Xi_{a}=\sum_{\ell=0}^{\infty}\sum_{|m|\leq\ell}\xi_{a}^{\ell m}Y^{\ell m}$
(21) $\Xi_{A}^{(+)}=\sum_{\ell=1}^{\infty}\sum_{|m|\leq\ell}\xi^{(+)\ell
m}Y^{\ell
m}_{A},\qquad\Xi_{A}^{(-)}=\sum_{\ell=1}^{\infty}\sum_{|m|\leq\ell}\xi^{(-)\ell
m}X^{\ell m}_{A}$ (22)

The $\xi_{a}^{\ell m}$, $\xi^{(+)\ell m}$, $\xi^{(-)\ell m}$ are “angle
independent” (constant on the two-sphere and hence functions only of the two-
coordinate $x^{a}$) and can be regarded as scalar and vector fields on
$\mathcal{M}_{2}$.

In terms of these variables the gauge transformation eq. 7 reads

$\displaystyle\delta p_{ab}$
$\displaystyle=\mathcal{D}_{a}\Xi_{b}+\mathcal{D}_{b}\Xi_{a}-2\widehat{\Gamma}^{\mu}_{ab}\xi_{\mu}$
(23) $\displaystyle\delta p_{aB}$
$\displaystyle=\mathcal{D}_{a}\Xi_{B}+D_{B}\Xi_{a}-\frac{2}{r}\,r_{a}\Xi_{b}-2\widehat{\Gamma}^{\mu}_{aB}\xi_{\mu}$
(24) $\displaystyle\delta p_{AB}$
$\displaystyle=D_{A}\Xi_{B}+D_{B}\Xi_{A}+2rg^{ab}r_{a}\Xi_{b}\Omega_{AB}-2\widehat{\Gamma}^{\mu}_{AB}\xi_{\mu}$
(25)

The last term involves the linear tensor $\widehat{\Gamma}^{\rho}_{\mu\nu}$
which relates the full $\overline{\nabla}_{\mu}^{(\bar{g})}$ and background
$\nabla^{(g)}_{\mu}$ covariant derivatives:

$\widehat{\Gamma}^{\rho}_{\mu\nu}\xi_{\rho}\equiv\overline{\nabla}_{\mu}^{(\bar{g})}\xi_{\nu}-\nabla^{(g)}_{\mu}\xi_{\nu}$
(26)
$\widehat{\Gamma}^{\rho}_{\mu\nu}\equiv\frac{1}{2}\,\bar{g}^{\rho\sigma}\Big{(}{\nabla^{(g)}_{\mu}h_{\sigma\nu}+\nabla^{(g)}_{\nu}h_{\mu\sigma}-\nabla^{(g)}_{\sigma}h_{\mu\nu}}\Big{)}$
(27)

The key fact that will be useful to us is that such terms do not involve
derivatives acting on the gauge parameters $\xi_{\mu}=(\Xi_{a},\Xi^{(\pm)})$,
and the contributions of such terms to any FP ghost actions will therefore not
involve derivatives acting on ghosts. Since any such terms are at least linear
in $h_{\mu\nu}$, they do not contribute to the ghost propagators, but rather
provide the couplings for interaction vertices of the form $\bar{C}h^{n}C$
with $n\geq 1$. It is important to stress here that such terms only involve
objects which, after decomposing into spherical harmonics, have coefficients
which are proper scalars, vectors, or tensors on $\mathcal{M}_{2}$, which can
be understood after close inspection of eq. 27.

In general, the terms eq. 26 are complicated power series in $h_{\mu\nu}$
which couple different harmonics of the expansions eqs. 16, 17, 18, 21, and
22, and outside of the monopole ($l=0$) sector we do not have explicit closed-
form expressions for the sums of these series. The general properties of these
terms that we will use are that 1) they do not contribute to ghost propagators
and 2) their decomposition into harmonics/2D fields only involves proper
representations of the Lorentz group (in particular, the coefficients of all
terms in the decomposition and expansion will be proper Lorentz scalars). This
latter fact will be used in Appendix C where we study the beyond leading order
in $h_{\mu\nu}$ part of the $l\geq 1$ ghost actions. These general properties
will be sufficient to establish our results. In the monopole ($l=0$) sector,
we have been able to find an explicit closed-form expression eq. 85, valid to
all orders in perturbation theory which explicitly displays these two
properties.

###  3.3 Boundary Conditions

In gravity, the issue of boundary conditions is especially important, since
this will affect the asymptotics of the spacetime on top of which our fields
propagate, as well as important physical quantities such as the ADM mass and
the flux of gravitational radiation measured at infinity. In the context of
asymptotically flat gravity that we consider here, this additional
consideration places a restriction on the asymptotic falloff of the gauge
parameters $\xi_{\mu}$ which relate physically equivalent field configurations
of $h_{\mu\nu}$. All other choices of $\xi_{\mu}$ correspond to “large”
diffeomorphisms which act nontrivially on the physical space of states222An
example of a family of such “large” diffeomorphisms which act nontrivially on
the physical space of states of asymptotically flat gravity are the BMS
Supertranslations BBM ; Sachs . See e.g. Strominger  for a recent review..

For 4D general relativity with asymptotically flat boundary conditions333As
noted in SatishchandranWald , these asymptotics are slightly weaker than
requiring smoothness at future null infinity $\mathscr{I}^{+}$. ADM ;
Faddeev:1973zb ; SatishchandranWald

$h_{\mu\nu}\ \underset{r\to\infty}{\sim}\
O\bigg{(}\frac{1}{r}\bigg{)},\qquad\partial_{\rho}h_{\mu\nu}\
\underset{r\to\infty}{\sim}\ O\bigg{(}\frac{1}{r^{2}}\bigg{)}$ (28)

the gauge parameters which relate physically equivalent field configurations
of $h_{\mu\nu}$ are those which fall off at least as fast as

$\bar{g}^{\mu\nu}\xi_{\nu}\ \underset{r\to\infty}{\sim}\
O\bigg{(}\frac{1}{r}\bigg{)},\qquad\bar{g}^{\mu\nu}\partial_{\rho}\xi_{\nu}\
\underset{r\to\infty}{\sim}\ O\bigg{(}\frac{1}{r^{2}}\bigg{)}$ (29)

or, in terms of the decomposition eqs. 21 and 22,

$\xi_{a}^{\ell m}\ \underset{r\to\infty}{\sim}\
O\bigg{(}\frac{1}{r}\bigg{)},\qquad\partial_{r}\xi_{a}^{\ell m}\
\underset{r\to\infty}{\sim}\ O\bigg{(}\frac{1}{r^{2}}\bigg{)}$ (30)

and

$\frac{1}{r^{2}}\,\xi^{(\pm)\ell m}\ \underset{r\to\infty}{\sim}\
O\bigg{(}\frac{1}{r}\bigg{)},\qquad\frac{1}{r^{2}}\,\partial_{r}\xi^{(\pm)\ell
m}\ \underset{r\to\infty}{\sim}\ O\bigg{(}\frac{1}{r^{2}}\bigg{)}$ (31)

In eqs. 28, 29, 30, and 31 we work with the coordinates and radial function
eq. 8 of the background eq. 4, but these statements hold more generally for
any asymptotically Minkowskian coordinate system with
$r=\sqrt{\|\vec{x}\|^{2}}$ its spatial radial function. eq. 29 can be easily
proven using the covariant phase space methods of WaldLee ; WaldIyer . The
essential point is that an infinitesimal diffeomorphism of an asymptotically
flat metric represents a zero mode of the (appropriately restricted)
presymplectic form associated to the 4D Einstein-Hilbert action (this is the
precise sense in which $\xi_{\mu}$ relates physically equivalent field
configurations) if and only if certain surface integrals at spacelike infinity
$i_{0}$ vanish. With the asymptotically flat boundary conditions eq. 28, this
requires the falloff conditions eq. 29.

It is important that our gauge-fixing conditions $\chi_{\alpha}$ be chosen
such that any field configuration of $h_{\mu\nu}$ can always be put into a
form satisfying $\chi_{\alpha}=0$ via a gauge transformation whose generator
satisfies eqs. 30 and 31. This will ensure that the corresponding gauge slice
includes (at least) one representative from each gauge equivalence class of
field configurations. We have been careful to check that this is the case for
each of the gauge-fixing conditions chosen below.

The falloff conditions eqs. 30 and 31 exclude transformations of the form (19)
for which the gauge parameters $\xi_{\mu}$ do not vanish at infinity. This
clarifies an issue raised in Martel:2005ir  concerning the gauge-fixing of the
low multipole $l=0,1$ modes. It was observed in Martel:2005ir  that their
choice of gauge-fixing conditions did not seem to fully determine the gauge,
since their conditions were preserved by particular families of additional
gauge transformations, with particular generators $\xi_{\mu}(v,r)$ presented
in ingoing Edington-Finkelstein (EF) coordinates. However, these generators
all violate the conditions eqs. 30 and 31: rather than falling off at large
$r$, they are either constant or grow with $r$.

After imposing the falloff conditions eqs. 30 and 31, the gauge-fixing
conditions for low multipoles suggested in Martel:2005ir  in EF coordinates,
or the coordinate independent ones suggested here, fully determine the
gauge444They also already fully determine the gauge beyond linear order in
$h_{\mu\nu}$ . Note that such falloff conditions are also required in quantum
field theories with gauge symmetries in order to derive the Ward identities
which govern the physical observables.

###  3.4 Regge-Wheeler-Zerilli-Martel-Poisson Gauge

Regge-Wheeler gauge Regge:1957td  (see also Zerilli Zerilli:1971wd  and
Martel-Poisson Martel:2005ir ) was often used in studies of linearized
perturbations of the classical Einstein equations near the Schwarzschild black
hole background. We will now review this gauge-fixing condition and its
inapplicability for low-multipole modes with $l=0,1$, in anticipation of its
use in covariant quantization in section 4 below.

As noted above (see eqs. 15 and 18), after decomposition into spherical
harmonics, the gravitational field $h_{\mu\nu}(x^{a},\theta^{A})$ becomes
encoded in a set of functions, which, as noted above, may be regarded as 2D
fields on $\mathcal{M}_{2}$. In the notation of Martel:2005ir , these
functions/2D fields are

$h_{ab}^{\ell m(+)},\quad j_{a}^{\ell m(+)},\quad K^{\ell m(+)},\quad G^{\ell
m(+)},\quad h_{a}^{\ell m(-)},\quad h_{2}^{\ell m(-)}\,\qquad l\geq 2$ (32)

At $l\geq 2$ all functions/2D fields in eq. (32) are available, while for low
multipoles only a restricted set of fields is available due to a lack of
vector and/or tensor harmonics for those modes. One finds for electric
dipoles, magnetic dipoles, and monopoles respectively, the following fields

$\displaystyle h_{ab}^{1m(+)},\quad j_{a}^{1m(+)},\quad K^{1m(+)},\qquad
l=1,\quad{\rm even}$ (33) (34) $\displaystyle h_{a}^{1m(-)}\hskip
125.19194ptl=1,\quad{\rm odd}$ (35) (36) $\displaystyle h_{ab}^{00(+)},\quad
K^{00(+)},\hskip 73.97733ptl=0$ (37)

All functions in (32)-(37) can be regarded as fields on $\mathcal{M}_{2}$:
$h_{ab}$ may be regarded as a tensor on $\mathcal{M}_{2}$, $j_{a},h_{a}$ as
vectors on $\mathcal{M}_{2}$, and $K,G,h_{2}$ as scalars on $\mathcal{M}_{2}$.
The angular dependence of the perturbation $h_{\mu\nu}$ is encoded in the
discrete dependence of these fields on the labels $(\pm)lm$.

As also noted above (see eqs. 21 and 22), after decomposition into spherical
harmonics, the gauge parameters $\xi_{\mu}(x^{a},\theta^{A})$ also become
encoded in a set of functions which may be regarded as 2D fields on
$\mathcal{M}_{2}$. At $l\geq 2$ for each $l,m$ there are 4 gauge symmetries.
At $l=1(+)$ there are 3 gauge symmetries for each $m=-1,0,1$, at $l=1(-)$
there is one gauge symmetry for each $m=-1,0,1$ and at $l=0$ there are two
gauge symmetries.

$\displaystyle\xi^{l\geq 2}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{\ell m(+)},\xi^{\ell
m(+)},\xi^{\ell m(-)}\\}$ (38) $\displaystyle\xi^{l=1(+)}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{1m(+)},\xi^{1m(+)}\\}$ (40)
$\displaystyle\xi^{l=1(-)}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi^{1m(-)}\\}$ (42)
$\displaystyle\xi^{l=0}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{00(+)}\\}$ (44)

All functions in (38)-(44) can similarly be regarded as fields on
$\mathcal{M}_{2}$: $\xi_{a}$ may be regarded as a vector on $\mathcal{M}_{2}$
and $\xi^{(\pm)}$ may be regarded as scalars on $\mathcal{M}_{2}$. The angular
dependence of the gauge parameter $\xi_{\mu}$ is similarly encoded in the
discrete dependence of these fields on the labels $(\pm)lm$.

The Regge-Wheeler gauge condition for modes with $l\geq 2$ is

$j_{a}^{\ell m(+)}=G^{\ell m(+)}=h_{2}^{\ell m(-)}=0$ (45)

It involves one vector and two scalars, leading to a total of four gauge-
fixing conditions. One can check that an arbitrary configuration of the fields
involved in eq. 45 can always be brought to Regge-Wheeler gauge by means of a
gauge transformation whose generator satisfies the fall-off conditions eqs. 30
and 31, so these facts together tell us that there is always one and only one
field configuration of (the $l\geq 2$ part of) $h_{\mu\nu}$ in each gauge
equivalence class which satisfies the Regge-Wheeler gauge condition eq. 45.
This tells us that Regge-Wheeler gauge is a good gauge-fixing condition for
covariant quantization. Note that the Regge-Wheeler gauge condition eq. 45 is
independent of the choice of coordinates $x^{a}$ on $\mathcal{M}_{2}$.

Some of the fields involved in eq. 45 are absent at $l<2$ as one can see in
eqs. (33)-(37). Therefore Regge-Wheeler gauge is not a valid gauge choice for
low multipole modes with $l<2$, and we will present an alternative choice in
section 4 below.

###  3.5 Comments on Monopoles $l=0$ and Dipoles $l=1$

An important feature of the low multipole modes established in Zerilli:1971wd
; Martel:2005ir  is the following. First off, the Regge-Wheeler gauge is not
valid and one has to impose a different set of gauges at $l=0,1$ since for
these modes some of the functions in eq. (45) are absent. Examples of such
gauge conditions were proposed in Zerilli:1971wd ; Martel:2005ir , where it
was also observed that the classical equations of motion for linearized
perturbations in this sector have simple local in time solutions.

This is associated with the feature of the low multipoles $l=0$ and $l=1$ that
they do not contain radiative degrees of freedom. The gravitational
perturbations near future null infinity were studied in Martel:2005ir  in the
retarded coordinate system $(u,r,\theta,\phi)$ where $u=t-r-2M\ln(r/2M-1)$. It
was shown there that the energy carried away by the gravitational radiation
near future infinity at $u,r\rightarrow\infty$ is proportional to $l(l-1)$. At
the event horizon the radiation was studied in advanced coordinates
$(v,r,\theta,\phi)$ where $v=t+r-2M\ln(r/2M-1)$ and again the result is
proportional to $l(l-1)$. In both cases the radiation involves quadrupoles and
higher modes, monopoles and dipoles drop from radiation in agreement with the
standard expectation that $l=0$ and $l=1$ perturbations do not contain
radiative degrees of freedom.

In absence of additional sources all solutions of Einstein equations for
perturbations with $l=0,1$ can be gauged away according to Zerilli:1971wd ;
Martel:2005ir  by an appropriate choice of coordinate transformation. In the
presence of additional sources, like a point particle of mass $m_{0}$ moving
towards the black hole, or a particle orbiting a black hole with a fixed
angular momentum $a$, solutions for linearized perturbations take a specific
form. In the $l=0$ monopole case one finds $h_{tt}\sim{m_{0}\over r}$,
Zerilli:1971wd . This solution provides a linear correction to the black hole
mass proportional to $m_{0}$. In the $l=1$ odd parity (magnetic dipole) case,
the perturbed metric is shown to describe a slightly rotating black hole: One
finds according to Zerilli:1971wd  that the $l=1$ solution for odd-parity
perturbations is of the form $\sim{m_{0}a\over r^{2}}$ which represents the
linearization of the Kerr metric with respect to its angular-momentum
parameter, determined here by the angular momentum of the source.

In Martel:2005ir  where advanced time coordinates were used, the monopole
solution is $h_{vv}\sim{2\delta M\over r}$ and the magnetic dipole solution is
$h_{v}\sim{2\delta J\over r}$. This again confirms that classical solutions
for small perturbations lead to small changes in the black holes mass
(position of the horizon) and add a small rotation. Finally for the $l=1$
even-parity (electric dipole) case, solutions for small perturbations are
vanishing, even in presence of sources, and are interpreted as simply encoding
a switch to a non-inertial coordinate system with respect to the original
Schwarzschild space-time Martel:2005ir .

##  4 Covariant Quantization of Gravity in the Schwarzschild Background

###  4.1 Gauge-Fixed Action

The general form of the BRST-invariant gauge-fixed action Becchi:1975nq ;
Tyutin:1975qk  consists of 3 terms: the classical action, the gauge-fixing
part of the action and the FP ghost action. In case of gravity it takes the
form

$S_{\mathrm{gf}}(g,B,\bar{C},C;h)=S_{\mathrm{cl}}(g+h)+\int
B^{\alpha}\chi_{\alpha}(g;h)+\int\bar{C}^{\alpha}Q_{\alpha}{}^{\beta}(g;h)C_{\beta}$
(46)

where the gauge-fixing conditions $\chi_{\alpha}=0$ result from integrating
out the auxiliary fields $B^{\alpha}$. The Feynman path integral acquires the
form shown in eq. (98).

We now propose the following (two-dimensionally) background covariant and
perturbatively well-defined gauge-fixing conditions, including the Regge-
Wheeler case (45) as well as a gauge-fixing of the low multipole modes:

$\displaystyle j_{a}=G=h_{2}=0\hskip 28.45274ptl\geq 2,\quad{\rm
even},\quad{\rm odd}$ (47) (48) $\displaystyle K=j_{a}=0\hskip
51.21504ptl=1,\quad{\rm even}$ (49) (50) $\displaystyle r^{a}h_{a}=0\hskip
65.44142ptl=1,\quad{\rm odd}$ (51) (52) $\displaystyle
K=t^{a}r^{b}h_{ab}=0\hskip 28.45274ptl=0,\quad{\rm even}$ (53)

All gauge fixing functions here as well as in (45) are 2D scalars or vectors,
with exception of the case $t^{a}r^{b}h_{ab}=0$ which is a pseudoscalar. This
simply means that all auxiliary fields $B^{\alpha}$ for our choice of gauge-
fixing functions $\chi_{\alpha}$ are 2D scalars or vectors, with the exception
of the case $Bt^{a}r^{b}h_{ab}$ where $B$ is a pseudoscalar, so that the total
$\int B^{\alpha}\chi_{\alpha}(g;h)$ contribution to the action is a 2D Lorentz
scalar, including the monopole term $Bt^{a}r^{b}h_{ab}$. One can check that
arbitrary field configurations of $h_{\mu\nu}$ can be made to satisfy eq. (53)
by acting with gauge transformations whose generators satisfy eqs. 30 and 31.
Since (53) also exhausts the gauge freedom (19), we see that there is always
one and only one field configuration of $h_{\mu\nu}$ in each gauge equivalence
class which satisfies the gauge condition (53). This tells us that (53) is a
good gauge-fixing condition for covariant quantization.

###  4.2 Covariant Quantization of the $l\geq 2$ Modes: Decoupling of Ghosts

We apply to our theory the De Witt-Faddeev-Popov procedure DeWitt:1967ub ;
Faddeev:1967fc  for the covariant quantization of quantum field theories with
gauge symmetries, see Appendix A for a short review. We choose the four Regge-
Wheeler gauge-fixing conditions used in Regge:1957td ; Zerilli:1971wd ;
Martel:2005ir  for all modes starting with quadrupoles and above ($l\geq 2$),
is given, for each $(l,m)$, in eq. (45). After writing the gauge symmetry (19)
in terms of the spherical harmonic decomposition eqs. 15, 16, 17, and 18 for
the gravitational field, we will show here that all FP ghosts corresponding to
the gauge-fixing conditions (45) for even and odd $l\geq 2$ modes decouple.

From eq. 25, we see that the gauge transformation eq. 1 of each of the gauge
fixing functions in (45) is:

$\displaystyle\delta j_{a}^{(+)}$ $\displaystyle=$
$\displaystyle\xi_{a}^{(+)}+\mathcal{D}_{a}\xi^{(+)}-\frac{2}{r}\,r_{a}\xi^{(+)}+f^{(+)}_{a}[h,\xi],$
(54) $\displaystyle\delta G^{(+)}$ $\displaystyle=$
$\displaystyle\frac{2}{r^{2}}\,\xi^{(+)}+f_{(G)}^{(+)}[h,\xi],$ (55)
$\displaystyle\delta h_{2}^{(-)}$ $\displaystyle=$ $\displaystyle
2\xi^{(-)}+f_{2}^{(-)}[h,\xi],$ (56)

Here the last terms in these equations $f[h,\xi]$ are functionals of the
background metric, the gauge parameters $\xi_{\mu}$, and the metric
perturbation $h_{\mu\nu}$ or spacetime derivatives of $h_{\mu\nu}$ which obey
$f[0,\xi]=0$ and tensorial linearity in $\xi_{\mu}$ (and hence linearity in
$\xi_{a}$ and $\xi^{(\pm)}$, with no dependence on derivatives of $\xi_{a}$
and $\xi^{(\pm)}$). These terms originate from the tensor eq. 26 which relates
the covariant derivative of the background to the full covariant derivative
appearing in eq. 25. As explained above, the contribution to the ghost action
due to these terms does not involve derivatives acting on ghosts, and since
these terms are at least linear or higher power in $h$, they do not contribute
to the ghosts propagators, but rather define couplings for the interaction
terms $\bar{C}h^{n}C$ with $n\geq 1$.

The total ghost Lagrangian is given by

$\displaystyle\bar{C}^{a}\Big{(}C_{a}^{(+)}+(\mathcal{D}_{a}-\frac{2}{r}\,r_{a})C^{(+)}+f^{(+)}_{a}[h,C]\Big{)}+\bar{C}^{(+)}\Big{(}\frac{2}{r^{2}}\,C^{(+)}+f_{(G)}^{(+)}[h,C]\Big{)}$
(57) $\displaystyle+\bar{C}^{-}\Big{(}2C^{(-)}+f_{2}^{(-)}[h,C]\Big{)}$ (58)

where we have not specified the details of the interaction terms. The odd
sector ghost action is algebraic and decoupled from the even sector. We can
integrate out $\bar{C}^{(-)}$ which leads to the constraint
$2C^{(-)}+f_{2}^{(-)}[h,C]=0$, so that the odd ghost action vanishes.

We now integrate over $\bar{C}^{(+)}$ to find the constraint

$\frac{2}{r^{2}}\,C^{(+)}+f_{(G)}^{(+)}[h,C]=0$ (59)

The remaining ghost Lagrangian is

$\bar{C}^{a}\Big{(}C_{a}^{(+)}-{r^{2}\over
2}(\mathcal{D}_{a}-\frac{2}{r}\,r_{a})f_{(G)}^{(+)}[h,C]+f^{(+)}_{a}[h,C]\Big{)}$
(60)

where we have used eq. (59). To establish the perturbative Feynman rules we
will now look only at the terms quadratic in ghosts and anti-ghosts, without
couplings to $h_{\mu\nu}$ (i. e. neglecting terms with $f[h,C]$) to find the
ghosts propagators in the Regge-Wheeler gauge in the background of the
Schwarzschild black hole. We find that the terms quadratic in quantum fields
are just

$\bar{C}^{a(+)}C_{a}^{(+)}$ (61)

Thus, the quadratic term for the remaining ghosts is algebraic. We conclude
that all ghosts for the Regge-Wheeler gauge are non-propagating and can
therefore be neglected for all even and odd $l\geq 2$ modes. We study the non-
linear part of the ghost action in the Appendix C where we argue that in
Schwarzschild coordinates there are no time derivatives acting on the ghosts
at all in the $l\geq 2$ sector.

###  4.3 Covariant Quantization of the $l=1$ Even (Electric Dipole) Modes

From eq. 25, we see that the gauge transformation eq. 1 of each of the gauge
fixing functions in the $l=1$ even (electric dipole) part of (53) is:

$\displaystyle\delta j_{a}^{(+)}$ $\displaystyle=$
$\displaystyle\xi_{a}^{(+)}+\mathcal{D}_{a}\xi^{(+)}-\frac{2}{r}\,r_{a}\xi^{(+)}+f^{(+)}_{a}[h,\xi],$
(62) $\displaystyle\delta K^{(+)}$ $\displaystyle=$
$\displaystyle-\frac{2}{r^{2}}\,\xi^{(+)}+\frac{2}{r}\,r^{a}\xi^{(+)}_{a}+f_{(K)}^{(+)}[h,\xi]$
(63)

The total ghost Lagrangian is therefore given by

$\displaystyle\bar{C}^{a}\Big{(}C_{a}^{(+)}+(\mathcal{D}_{a}-\frac{2}{r}\,r_{a})C^{(+)}+f^{(+)}_{a}[h,C]\Big{)}$
(64)
$\displaystyle+\bar{C}^{(+)}\Big{(}-\frac{2}{r^{2}}\,C^{(+)}+\frac{2}{r}\,r^{a}C^{(+)}_{a}+f_{(K)}^{(+)}[h,C]\Big{)}$
(65)

We integrate over $\bar{C}^{(+)}$ to yield the constraint

$C^{(+)}=rr^{a}C^{(+)}_{a}+\frac{r^{2}}{2}\,f_{(K)}^{(+)}[h,C]$ (66)

The remaining ghost Lagrangian is

$\displaystyle\bar{C}^{a}\Big{(}C_{a}^{(+)}+(\mathcal{D}_{a}-2\,r_{a})(r^{a}C^{(+)}_{a}+\frac{r}{2}\,f_{(K)}^{(+)}[h,C])+f^{(+)}_{a}[h,C]\Big{)}$
(67)

The part of this Lagrangian quadratic in ghosts (which defines their
propagator) is now

$\bar{C}^{a}\Big{(}C_{a}^{(+)}+(\mathcal{D}_{a}-2\,r_{a})r^{b}C^{(+)}_{b}\Big{)}$
(68)

Thus, there is in general a propagating ghost field $C^{(+)}_{a}$.

In Schwarzschild coordinates ($t,r$) where $r_{a}=(0,1),r^{a}=(0,f)$, eq. 68
becomes

$\bar{C}^{t}C_{t}^{(+)}+\bar{C}^{r}C_{r}^{(+)}+\bar{C}^{t}\mathcal{D}_{t}fC_{r}+\bar{C}^{r}(\mathcal{D}_{r}-2\,)fC^{(+)}_{r}$
(69)

This action is linear in $C_{t}^{(+)}$, which we can therefore integrate out
to impose the constraint $\bar{C}^{t}=0$. The remaining part of the Lagrangian
quadratic in the ghosts is

$\bar{C}^{r}[(\mathcal{D}_{r}-2\,)f+1]C^{(+)}_{r}$ (70)

The non-linear part of this Lagrangian is studied in Appendix C where we argue
that in Schwarzschild coordinates there are no time derivatives acting on the
ghosts in $l=1$ even sector.

###  4.4 Covariant Quantization of the $l=1$ Odd (Magnetic Dipole) Modes

In the $l=1$ odd (magnetic dipole) sector, there is a single gauge symmetry
$\xi^{1m(-)}$ for each $m=-1,0,1$, which we fix using the 2D background
covariant gauge-fixing condition

$r^{a}h_{a}^{(-)}=0$ (71)

From eq. 25, we see that the gauge transformation eq. 1 of the gauge-fixing
condition is

$r^{a}\delta
h_{a}^{(-)}=r^{a}\mathcal{D}_{a}\xi^{(-)}+r^{a}f^{(-)}_{a}[h,\xi]$ (72)

This variation involves, at the quadratic level defining the ghost kinetic
term, a derivative of $\xi^{(-)}$. This means that in general, these ghosts
are propagating and coupled to gravitational modes. The corresponding ghost
action is

$\int\bar{C}^{(-)}\Big{(}r^{a}(\partial_{a}+V_{a}(h))\Big{)}C^{(-)}$ (73)

where the coupling $r^{a}V_{a}(h)$ represents vertices where the ghosts for
the $l=1$ odd modes interact with $h_{\mu\nu}$.

In Schwarzschild coordinates where $r^{a}=(0,f)$ the ghosts action simplifies
to

$\bar{C}^{(-)}f(\partial_{r}+V_{r}(h))C^{(-)}$ (74)

Here the propagator is instantaneous since the kinetic terms has only
$r$-derivatives. There are no derivatives acting on this ghost field at higher
order in $h_{\mu\nu}$, so in Schwarzschild coordinates there are no time
derivatives acting on the ghosts at all in the $l=1$ odd sector.

###  4.5 Covariant Quantization of the $l=0$ (Monopole) Modes: Results to All
Orders

In the $l=0$ (monopole) sector there is a single 2D vector $\xi_{a}^{00(+)}$,
a parameter of a gauge symmetry, which we fix using the 2D background
covariant gauge-fixing conditions

$K=t^{a}r^{b}h_{ab}=0$ (75)

The first one $K=0$ is a 2D scalar while the second one $t^{a}r^{b}h_{ab}=0$
is a 2D pseudoscalar since $t^{a}=-\epsilon^{ab}r_{b}$ is a pseudovector.

The gauge transformation eq. 1 of the gauge-fixing conditions is now given by

$\displaystyle t^{a}r^{b}\delta h_{ab}$
$\displaystyle=t^{a}r^{b}(\mathcal{D}_{a}\xi_{b}+\mathcal{D}_{b}\xi_{a}+f_{ab}[h,K,\xi])$
(76) $\displaystyle\delta K$
$\displaystyle=\frac{2}{r}\,r^{a}\xi_{a}+f_{(K)}[h,K,\xi]$ (77)

which leads to a ghost Lagrangian

$\bar{C}^{(K)}\Big{(}\frac{2}{r}\,r^{a}C_{a}+f_{(K)}[h,C]\Big{)}+\bar{C}^{(h)}\Big{(}t^{a}r^{b}(\mathcal{D}_{a}C_{b}+\mathcal{D}_{b}C_{a}+f_{ab}[h,C])\Big{)}$
(78)

We can integrate over $\bar{C}^{(K)}$ which imposes the constraint

$\frac{2}{r}\,r^{a}C_{a}+f_{(K)}[h,C]=0$ (79)

The first term in the ghost action now vanishes. We now note that, since
$f_{(K)}[h,C]$ is a Lorentz scalar, it can only depend on $r^{a}C_{a}$ (not
the pseudo-scalar $t^{a}C_{a}$), so that

$f_{(K)}[h,C]=\hat{f}_{(K)}[h]\,r^{a}C_{a}$ (80)

for some functional $\hat{f}_{(K)}[h]$ of $h_{ab}^{00}$. This means that the
constraint eq. 80 really reads

$\quad\Big{(}\frac{2}{r}+\hat{f}_{(K)}[h]\Big{)}r^{a}C_{a}=0$ (81)

and so sets $r^{a}C_{a}=0$ up to a local functional determinant. Such changes
of variables were discussed in the Introduction below eq. (2) and simply
contribute a term proportional to $\delta^{2}(0)$ to the local measure of
integration. We neglect these following the arguments in Fradkin:1977hw .

The second term in the Lagrangian is

$\bar{C}^{(h)}\Big{(}t^{a}r^{b}(\mathcal{D}_{a}C_{b}+\mathcal{D}_{b}C_{a}+f_{ab}[h,C])\Big{)}$
(82)

In general the kinetic term, coming from the part of the action quadratic in
ghosts, has both space and time derivatives, and a coupling is present between
the ghosts and the 2D monopole sector metric $h^{00}_{ab}$, encoded in the
term $\bar{C}^{(h)}t^{a}r^{b}f_{ab}[h,C]$.

However, in the Schwarzschild coordinates, we find from (81) that $C_{r}=0$
and eq. 82 reduces to

$\bar{C}^{(h)}f(r)\Big{[}\Big{(}\mathcal{D}_{r}-{f^{\prime}\over
f}\Big{)}C_{t}+f_{tr}[h,C_{t}]\Big{]}$ (83)

The propagator is again instantaneous since the kinetic terms has only
$r$-derivatives, and the non-linear term $f_{tr}[h,C_{t}]$ has no derivatives
acting on ghosts. Therefore there are again no time derivatives acting on the
ghost fields in Schwarzschild coordinates.

Explicit All Orders in $h_{ab}^{00}$ Ghost Action:

The monopole $l=0$ sector has the advantage that the relevant parts decouple
from other modes even at the non-linear level. Therefore we are able to find
an explicit expression for the ghost action, including all non-linear terms.
With our gauge choices $K=t^{a}r^{b}h_{ab}^{00}=0$ a convenient
parameterization for the remaining two components of $h_{ab}$ is

$h_{ab}^{00}\equiv\frac{1}{f(r)^{2}}\,At_{a}t_{b}+Br_{a}r_{b}$ (84)

The Lagrangian for the monopole ghosts is given exactly (to all orders in
$h_{\mu\nu}$) by

$\bar{C}^{(K)}\,\left(\frac{2}{r}\frac{1}{1+f(r)B}\right)r^{a}C_{a}\\\
+\bar{C}^{(h)}\left[\left(t^{a}\partial_{a}-\frac{f(r)(t^{b}\partial_{b}B)}{1+f(r)B}\right)r^{c}C_{c}+\left(r^{a}\partial_{a}-\frac{f(r)f^{\prime}(r)-(r^{d}\partial_{d}A)}{f(r)-A}\right)t^{c}C_{c}\right]$
(85)

Note that here the anti-ghost field $\bar{C}^{(h)}$ is a 2D pseudo-scalar,
same as the corresponding auxiliary field $B$ in $Bt^{a}r^{b}h_{ab}$ in Sec.
4.1 so the the total ghost action is a scalar since $\bar{C}^{(h)}$ multiplies
a piece of the Lagrangian which is linear in $t^{a}$.

Now we take into account equation (81) and we find that the remaining ghost
action is given by

$\bar{C}^{(h)}\Big{(}r^{a}\partial_{a}-\frac{f(r)f^{\prime}(r)-(r^{d}\partial_{d}A)}{f(r)-A}\Big{)}t^{c}C_{c}$
(86)

In general this describes propagating ghosts coupled to monopole gravitational
perturbations. However, in Schwarzschild coordinates we find only space
derivatives acting on ghosts.

$\bar{C}^{(h)}f(r)\left(\partial_{r}-\frac{f^{\prime}(r)-\partial_{r}h^{00}_{tt}}{f(r)-h^{00}_{tt}}\right)C_{t}$
(87)

Therefore the propagator is instantaneous as in the cases above.

##  5 Faddeev’s Theorem and a Possible Hamiltonian Origin

###  5.1 Gauges with Unitary and Pseudo-Unitary Hamiltonians

In the Hamiltonian formalism in gauge theories where we have 1st class
constraints $\phi^{\alpha}(t,\vec{x})$ and additional conditions
$\chi_{\alpha}(t,\vec{x})$, the Poisson bracket

$\\{\chi_{\alpha}(t,\vec{x}),\phi^{\beta}(t,\vec{y})\\}=M_{\alpha}{}^{\beta}\delta^{3}(\vec{x}-\vec{y})$
(88)

defines a differential operator $M_{\alpha}{}^{\beta}$. As we suggested in the
Introduction, there are two different classes of gauge-fixing functions
functions $\chi_{\alpha}$ which either lead to a theory with a unitary
Hamiltonian, or to a theory with the pseudo-unitary Hamiltonian.

Case of Unitary Hamiltonian $H(p^{*},q^{*})$ in Gauge Theories

The first class of conditions $\chi_{\alpha}(p,q)$ is usually associated with
Lorentz non-covariant gauges, like Coulomb gauge in Yang-Mills theory or Dirac
gauge in gravity. This class includes all instances where the differential
operator $M_{\alpha}{}^{\beta}$ does not involve time derivatives. In such a
case there exists a unitary Hamiltonian in a space with $(n-m)$ physical
degrees of freedom $(p^{*},q^{*})$, where $*=1,\dots,(n-m)$ and which is
manifestly ghost-free.

In covariant quantization as defined in Faddeev:1969su  one can arrange that

$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}$ (89)

at $\chi_{\alpha}=\phi^{\alpha}=0$ Therefore the differential operator
$M_{\alpha}{}^{\beta}$ defined by the Poisson bracket eq. (88) of canonical
quantization actually defines the FP ghost action
$\bar{C}^{\alpha}Q_{\alpha}{}^{\beta}C_{\beta}$ in covariant quantization.
When the operator $M_{\alpha}{}^{\beta}$ has only space derivatives one finds
that, though a nontrivial FP ghost action is present in the covariant
quantization, it has the particular feature that the ghosts propagators are
instantaneous.

It is known555We are grateful to A. Weinstein and M. Shifman for the
clarifying discussion of this issue, namely the absence of ghosts in the
canonical Feynman rules in the Coulomb gauge in Yang-Mills theory
Khriplovich:1969aa , Faddeev:1969su ; Fradkin:1970pn , and the presence of
instantaneous ghosts as well as instantaneous gluons in the covariant Feynman
rules.  in the example of Coulomb gauge in Yang-Mills theory that the ghosts
loops with instantaneous propagators are cancelled by the loops of the
instantaneous part of the gluon part of the propagator, to all orders in
perturbation theory Zwanziger:1998yf . It is also known that there are no
closed instantaneous loops when the Feynman rules are deduced from the
Hamiltonian path integral and the $S$ matrix is computed as time ordered
product of the unitary Hamiltonian. In this case the equivalence of the
Hamiltonian perturbative Feynman rules in QCD Khriplovich:1969aa ,
Faddeev:1969su ; Fradkin:1970pn  and the Lagrangian De Witt-Faddeev-Popov
rules in QCD is clearly established. In gravity in the Dirac gauge
Dirac:1958jc  the ghosts in the covariant quantization also have instantaneous
propagators whereas the underlying Hamiltonian is unitary and the Hilbert
space of physical degrees of freedom is ghost-free Faddeev:1969su ;
Fradkin:1970pn .

In Appendix B we explain the general relation between the original constrained
canonical variables $q$ and $p$ in eq. (104) and the independent physical
canonical variables $(q^{*},p^{*})$ eq. (110).

Case of Pseudo-Unitary Hamiltonian in Gauge Theories,
$H(q^{A},p_{A},\eta^{a},{\cal P}_{a})$

The other class involves situations where the condition
$\chi_{\alpha}=\chi_{\alpha}(q,p,\lambda,\dot{\lambda})$ depends not only on
the naïve canonical variables $(q,p)$, but also on Lagrange multipliers
$\lambda$ and their time derivative $\dot{\lambda}$, where the Lagrange
multipliers originate from the imposition of the constraints in the
Hamiltonian form of the action:

$S(q,p,\lambda)=\int\mathrm{d}t\left(p_{i}\dot{q}^{i}-H(q,p)-\lambda_{\alpha}\phi^{\alpha}(q,p)\right)$
(90)

For example, in QCD, the field $A_{0}$ is a Lagrange multiplier since the
classical action does not depend on $\partial_{0}A^{0}$. In the Lorentz
covariant gauge $\partial_{\mu}A^{\mu}=0$, there is a time derivative of the
Lagrange multiplier $\lambda=A^{0}$, since $\partial_{0}A^{0}=\dot{\lambda}$

The set of canonical coordinates $(q,p)$ is now enlarged by the Lagrange
multipliers and their canonical momenta so that the total naïve phase space is
spanned by $(q^{A},p_{A})$ with $A=1,\dots,n+m$. This allows for the
accommodation of Lorentz covariant gauges, like Feynman gauge in Yang-Mills
theory and de Donder type gauges in gravity. In this case, one must add to the
system $2m$ additional degrees of freedom $(\eta^{a},{\cal P}_{a})$,
$a=1,\dots,2m$, with opposite statistics, which correspond to propagating FP
ghosts and anti-ghosts. The net number of commuting minus anti-commuting
canonical variables is $n+m-2m=n-m$, in agreement with the counting of
physical states $(q^{*},p^{*})$ in the unitary gauges described above, where
neither Lagrange multipliers nor FP ghosts/antighosts are propagating degrees
of freedom. The Hamiltonian in such an extended space
$(q^{A},p_{A},\eta^{a},{\cal P}_{a})$ defines an S-matrix which is pseudo-
unitary in a state space with indefinite metric, as explained in
Fradkin:1977hw .

Examples of gauges with pseudo-unitary Hamiltonian in gravity include
harmonic/de Donder gauge, which has propagating (non-instantaneous) FP ghosts
in the covariant quantization method. The underlying Hamiltonian in this class
of gauges is pseudo-unitary. The proof of unitarity of the S-matrix in this
class of gauges, where ghost degrees of freedom appear even in the
canonical/Hamiltonian construction, follows only because of the equivalence of
this S-matrix to the one in the class of gauges with the ghost-free unitary
Hamiltonian.

###  5.2 $l\geq 2$

Our 4 gauge fixing functions, for each $(l,m)$ with $l\geq 2$, are
$\chi_{\alpha}=\\{j_{a},G,h_{2}\\}$. To leading order in $h_{\mu\nu}$ they
transform under the 4 gauge parameters
$\xi_{\alpha}=\\{\xi_{a},\xi^{(+)},\xi^{(-)}\\}$ with the same $l$ and $m$ via

$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}+O(h_{\mu\nu})$
(91)

where the first term on the right hand side of eq. 91 is given by the leading
order terms in eqs. (54)-(56). The corresponding leading order ghost action—in
particular the ghost kinetic term—is associated with the determinant of the
matrix $\det||\\{\chi_{\alpha},\phi^{\beta}\\}||\neq 0$ as explained in (106),
which is given by

$\det\begin{pmatrix}\mathds{1}&\mathcal{D}_{a}-\frac{2}{r}r_{a}&0\\\
0&\frac{2}{r^{2}}&0\\\ 0&0&2\end{pmatrix}=\det\begin{pmatrix}\mathds{1}&0&0\\\
0&\frac{2}{r^{2}}&0\\\ 0&0&2\end{pmatrix}$ (92)

Such a determinant is algebraic and will contribute to the action as
$\delta^{2}(0)$. We can see therefore that in both covariant as well as in
canonical quantization there are not expected to be propagating ghosts for the
modes with $l\geq 2$. One can further show that in Schwarzschild coordinates,
time derivatives of ghosts are absent even in the higher order terms. Thus,
the absence of a nontrivial ghost action for the $l\geq 2$ modes provides
evidence that a unitary Hamiltonian is available for these modes.

###  5.3 $l=1$ even

Our 3 gauge fixing functions, for each $m=-1,0,1$, are
$\chi_{\alpha}=\\{j_{a},K\\}$. To leading order they transform under the 3
gauge parameters $\xi_{\alpha}=\\{\xi_{a},\xi^{(+)}\\}$ with the same $m$ via
the Poisson bracket
$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}+O(h_{\mu\nu})$
given by the leading order terms in eqs. (62) and (63). The corresponding
determinant $\det||\\{\chi_{\alpha},\phi^{\beta}\\}||$ is given by

$\det\begin{pmatrix}1&\mathcal{D}_{a}-\frac{2}{r}\,r_{a}\\\
\frac{2}{r}\,r^{a}&-\frac{2}{r^{2}}\end{pmatrix}=\det\left(-\frac{2}{r^{2}}\right)-\det\left(\frac{2}{r}\,r^{a}\left(\mathcal{D}_{a}-\frac{2}{r}\,r_{a}\right)\right)$
(93)

In Schwarzschild coordinates, with $r_{a}=(0,1)$ and $r^{a}=(0,f)$, this
becomes

$\left.\det\begin{pmatrix}1&\mathcal{D}_{a}-\frac{2}{r}\,r_{a}\\\
\frac{2}{r}\,r^{a}&-\frac{2}{r^{2}}\end{pmatrix}\right|_{\mathrm{Schld}}=\det\left(-\frac{2}{r^{2}}\right)-\det\left(\frac{2f(r)}{r}\left(\mathcal{D}_{r}-\frac{2}{r}\right)\right)$
(94)

Note that this determinant has only space derivatives. We can deduce therefore
from the canonical quantization in Schwarzschild coordinates, that the FP
ghosts in a covariant quantization will have instantaneous propagator. One can
further show that in Schwarzschild coordinates, time derivatives of ghosts are
absent even in the higher order terms. Again, the absence of the time
derivatives in the ghosts action for $l=1$ even modes provides an evidence
that the unitary Hamiltonian is available for these modes.

###  5.4 $l=1$ odd

Our 1 gauge fixing function, for each $m=-1,0,1$, is
$\chi_{\alpha}=\\{r^{a}h_{a}\\}$ which to leading order transforms under a the
gauge parameter $\xi_{\alpha}=\\{\xi^{(-)}\\}$ with the same $m$ via the
Poisson bracket
$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}+O(h_{\mu\nu})$
given by the leading order terms in eq. (72). In Schwarschild coordinates we
see the corresponding determinant is given by
$\det||\\{\chi_{\alpha},\phi^{\beta}\\}||=\det(\mathcal{D}_{r})$, which
involves a derivative operator in the $r$ direction only. The corresponding FP
ghost action shown in eq. (73) is not vanishing but leads to ghost loop
diagrams with an instantaneous propagator.

Thus, the ghost action for magnetic dipoles, which has only space derivatives
in the action in Schwarschild coordinates, provides evidence that the unitary
Hamiltonian is available for these modes.

###  5.5 $l=0$

In Schwarschild coordinates we have the two gauge-fixing conditions
$\chi_{\alpha}=\\{K,h_{tr}\\}$ and the two gauge parameters $\xi_{r},\xi_{t}$.
The determinant of the corresponding Poisson bracket
$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}$ is given to
all orders in $h_{ab}^{00}$ by

$\det\left(\begin{array}[]{cc}{2\over r}{f(r)\over 1+f(r)h_{rr}}&\,\,\,0\\\
\cr f(r)\left(\partial_{t}-{f(r)\partial_{t}h_{rr}\over
1+f(r)h_{rr}}\right)&\hskip
14.22636pt\,\,f(r)\left(\partial_{r}-{f^{\prime}(r)-\partial_{r}h_{tt}\over
f(r)-h_{tt}}\right)\end{array}\right)\\\ =\det\left(\begin{array}[]{cc}{2\over
r}{f(r)\over 1+f(r)h_{rr}}&\,\,\,0\\\ \cr 0&\hskip
14.22636pt\,\,f(r)\left(\partial_{r}-{f^{\prime}(r)-\partial_{r}h_{tt}\over
f(r)-h_{tt}}\right)\end{array}\right)$ (95)

Here again we can see that the determinant of the Poisson bracket is in
agreement with the expression for the action of the FP ghosts in the covariant
method. There are only space derivatives on monopole FP ghosts. Therefore in
the covariant quantization they have instantaneous propagators and it is
expected that their loops will cancel with the instantaneous part of the
gravitational fields.

Time derivatives in Schwarzschild coordinates explicitly vanish to all order
here. Thus, the ghosts action for monopoles, which has only space derivatives
in the action, provides evidence that the unitary Hamiltonian is available
also in the monopole sector.

##  6 Discussion

A rather surprising feature of the covariant quantization of perturbative
gravity in the Schwarzschild black hole background in Regge-Wheeler gauge is
that FP ghosts are absent (i.e. decoupled) for all modes with the exception of
monopoles and dipoles. These latter modes are known to be related to the ADM
mass and angular momentum of the perturbed black hole; for example, on-shell
and to linear order, they are known to encode the linearized perturbation of
the ADM mass and the linearized rotation of the perturbed black hole solution
in presence of sources Regge:1957td ; Zerilli:1971wd ; Martel:2005ir .

In the well known background-covariant and Lorentz-covariant gauges in
tHooft:1974toh ; Goroff:1985th ; vandeVen:1991gw  the gauge fixing
functions—e.g. for de Donder gauge
$F_{\mu}=\nabla_{\nu}^{(g)}h^{\nu}{}_{\mu}-{1\over
2}\nabla_{\mu}^{(g)}h^{\nu}{}_{\nu}$—depend on background covariant
derivatives of the gravitational perturbation $h_{\mu\nu}$.The ghosts are
propagating due to the tensorial nature of the gravitational perturbations
$\delta h_{\mu\nu}=2\overline{\nabla}^{(\bar{g})}_{(\mu}\xi_{\nu)}$ which
involves both space and time derivatives acting on $\xi_{\nu}$. In gravity,
Dirac gauge Dirac:1958jc  is the only one known to have FP ghosts with
instantaneous propagators in a covariant quantization, as shown in
Faddeev:1969su ; Fradkin:1970pn ; Faddeev:1973zb , and to have a unitary
Hamiltonian (in the sense described in section 5.1 above).

Here we have studied covariant quantization of perturbative gravity in a
Schwarzschild back hole background, using the Regge-Wheeler Regge:1957td ;
Zerilli:1971wd ; Martel:2005ir  framework where the quantum fields are
expanded in spherical harmonics. In the Schwarzschild black hole background
there is a natural split of the space into a warped product
$\mathcal{M}=\mathcal{M}_{2}\times\mathbb{S}^{2}$ of two 2D submanifolds. All
dependence on the $\mathbb{S}^{2}$ is encoded in the the discrete indices of
the harmonics, $l,m$ and $\pm$. For each $(l,m,\pm)$, the quantization is
reduced to a quantization of a quasi-two dimensional theory on
$\mathcal{M}_{2}$ (with the additional information of a radial function
$r(x)$): The gravity perturbations with fixed values of $(l,m,\pm)$ depend
only on $x^{a},a=1,2$, the two coordinates of $\mathcal{M}_{2}$. For low
multipoles for which the Regge-Wheeler gauge condition, is not valid, we have
proposed a two-dimensionally background covariant set of gauges in eq. (53).
They can can written out using any choice of coordinates, including, for
example, Schwarzschild, Eddington-Finkelstein or Kruskal-Szekeres coordinates.
We found the following:

1\. For all even and all odd modes with $l\geq 2$, the corresponding FP ghosts
are not propagating: their kinetic terms do not contain derivatives.

2\. All even and all odd dipole modes with $l=1$, and monopole modes with
$l=0$, have nontrivial FP ghosts in covariant quantization, since their
kinetic terms involve derivatives.

In the special case of Schwarzschild coordinates our results are:

1\. In Schwarzschild coordinates, the low multipole ghosts have no time
derivatives, only space derivatives. Their propagators are therefore
instantaneous, as in Coulomb gauge in QCD Khriplovich:1969aa ; Faddeev:1969su
; Fradkin:1970pn  and Dirac gauge in gravity Faddeev:1969su ; Fradkin:1970pn ;
Faddeev:1973zb .

2\. We have provided evidence that with our choice of gauge-fixing functions
our covariant quantization rules when viewed in Schwarzschild coordinates are
consistent with the existence of an underlying unitary Hamiltonian in a
manifestly ghost-free Hilbert space with a positive-definite metric, as
suggested by Faddeev’s theorem Faddeev:1969su .

An open issue which needs to be addressed with regard to a potential canonical
quantization of gravity in the black hole background concerns the fact that
the existing constructions of the canonical (Hamiltonian) path integral in
Faddeev:1969su ; Fradkin:1970pn ; Faddeev:1973zb ; Fradkin:1976xa ;
Fradkin:1977hw  were only performed in a flat Minkowski background. Meanwhile
in the black hole background in Schwarzschild coordinates there is an event
horizon along which the relevant coordinates break down. The concept of the
Hamiltonian and of the Hilbert space of states might be more subtle, if well-
defined at all.

Nevertheless, Faddeev’s theorem Faddeev:1969su , valid at least in the flat
Minkowski background, suggests that a ghost-free unitary Hamiltonian ought to
exist for the gauge-fixing conditions studied here in the black hole
background in Schwarzschild coordinates. Namely, we have found that in the
covariant quantization in Schwarzschild coordinates, there are no time
derivatives acting on the ghost fields. This suggests that such a Hamiltonian,
if explicitly constructed, might belong to the class we described in section
5.1, the case of the unitary ghost-free Hamiltonian $H(p^{*},q^{*})$ depending
on $(n-m)$ degrees of freedom described by $2(n-m)$ canonical variables
$(p^{*},q^{*})$.

The reason this is a likely outcome of the canonical quantization is that the
case of pseudo-unitary Hamiltonian in a Hilbert space of states with the
indefinite metric, also described in section 5.1,
$H(q^{A},p_{A},\eta^{a},{\cal P}_{a})$, would be inconsistent with the absence
of time derivatives acting on the ghosts, which we found in this paper. Here
$A=1,\dots,n+m$ involves commuting fields, and $a=1,\dots,2m$ involves anti-
commuting fields. The total counting of degrees of freedom is therefore
$n+m-2m=n-m$. But we have just shown that all of our anti-commuting fields (FP
ghosts and anti-ghosts) have no time derivatives, so they are not expected to
contribute to a space of states with indefinite metric in a process of
canonical quantization.

Note that in Eddington-Finkelstein and Kruskal-Szekeres coordinates the
situation is different and has to be studied separately. Although we have
performed a covariant quantization which is valid in any of these coordinate
systems, the canonical quantization is still to be explored.

To summarize, it would be very interesting to perform a canonical quantization
of gravity in the Schwarzschild black hole background in the class of gauges
presented here. We leave this for future work.

## Acknowledgement

We are grateful to A. Barvinsky, R. Bond, V. Chandrasekaran, R. Flauger, A.
Linde, R. Mahajan, M. Mirbabayi, E. Poisson, G. Satishchandran, M. Shifman, V.
Shyam, E. Silverstein, P. Stamp, D. Stanford, A. Starobinsky, A. Vainshtein,
A. Van Proeyen, I. Volovich, and R. Wald for stimulating and helpful
discussions. RK and AR are supported by the Stanford Institute of Theoretical
Physics and by the Grant PHY-2014215 of National Science Foundation (United
States/US). RK is additionally supported by the Simons Foundation Origins of
the Universe program (Modern Inflationary Cosmology collaboration (United
States/US). AR is additionally supported by the National Science Foundation
GRF Program under Grant No. DGE-1656518 and by a Fletcher Jones Foundation
National Science Foundation Graduate Fellowship in the Stanford School of
Humanities & Sciences (United States/US).

##  Appendix A De Witt-Faddeev-Popov Covariant Quantization of Gravity

We consider 4D asymptotically flat Einstein gravity in the background of a
Schwarzschild black hole $(\mathcal{M},g_{\mu\nu})$. The total classical
action depends on $\bar{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$, where $h_{\mu\nu}$
is the quantized perturbative gravitational field (graviton field) and the
background field is the Schwarzschild black hole metric $g_{\mu\nu}$. The
action $S(g+h)$ is invariant under the gauge symmetries

$\delta
h_{\mu\nu}=\overline{\nabla}^{(g+h)}_{\mu}\xi_{\nu}+\overline{\nabla}^{(g+h)}_{\nu}\xi_{\mu},\qquad\delta
g_{\mu\nu}=0$ (96)

where $\overline{\nabla}^{(g+h)}_{\mu}$ is the covariant derivative operator
of the full metric $g+h$.

Due to the gauge symmetries the naive path integral

$\int Dh\,e^{\mathrm{i}S(g+h)}$ (97)

has to be defined using De Witt-Faddeev-Popov procedure DeWitt:1967ub ;
Faddeev:1967fc . This procedure in the simple case suitable for our purpose
involves a set of gauge-fixing conditions $\chi_{\alpha}(g;h)=0$ which
constrain the gravitational fields. The path integral becomes

$\int
Dh\,J_{\chi}(g;h)\,\delta\big{(}\chi_{\alpha}(g;h)\big{)}\,e^{\mathrm{i}S(g+h)}$
(98)

Here the Jacobian $J_{\chi}(g;h)$ is defined by the variation of the gauge-
fixing function $\chi_{\alpha}(g;h)$ under the gauge symmetry with the
parameters $\xi_{\beta}$

$J_{\chi}(g,h)=\exp{{\rm Tr}\ln}\,Q_{\alpha}{}^{\beta}(g,h)$ (99)

where

$\delta\chi_{\alpha}=Q_{\alpha}{}^{\beta}(g,h)\,\xi_{\beta}$ (100)

This Jacobian can be also presented with the help of the FP ghosts
Faddeev:1967fc  as follows

$J_{\chi}=\int
D\bar{C}^{\alpha}DC_{\beta}\,e^{\mathrm{i}\int\mathrm{d}^{4}x\,\bar{C}^{\alpha}(x)Q_{\alpha}{}^{\beta}(g,h)C_{\beta}(x)}$
(101)

When $Q_{\alpha}{}^{\beta}(g;h)$ is a local function of $(g;h)$ without
differential operators acting on $C_{\beta}(x)$, the relevant ghost action
becomes $\bar{C}^{\alpha}(x)\tilde{C}_{\alpha}(x)$ where
$\tilde{C}_{\alpha}(x)=Q_{\alpha}{}^{\beta}(g,h)C_{\beta}(x)$. The
corresponding ghosts are non-propagating and drop from Feynman rules. When
$Q_{\alpha}{}^{\beta}(g;h)$ involves differential operators with time and
space derivatives, the corresponding ghosts are propagating and generically
give important contribution to the Feynman diagrams. When
$Q_{\alpha}{}^{\beta}(g;h)$ involves a differential operator with only space
derivatives, the corresponding ghosts are said to have “instantaneous”
propagators.

##  Appendix B Faddeev-Fradkin-Vilkovisky Canonical Quantization of Gravity

In this appendix, we would like to present a brief summary of the most
relevant results obtained in Faddeev:1969su ; Faddeev:1973zb ; Fradkin:1976xa
; Fradkin:1977hw  concerning the canonical quantization of gravity. We would
like to stress that the standard Faddeev-Fradkin-Vilkovisky quantization of
gravity was formulated in the context of a flat Minkowski background, whereas
in this paper we perform the quantization in the Schwarzschild black hole
background using the covariant quantization method developed by De Witt
DeWitt:1967ub ; tHooft:1974toh ; Kallosh:1974yh ; Grisaru:1975ei .

It might be useful to clarify the relevant observation made by Weinberg in Ch.
15.8 on p. 41 in Weinberg:1996kr . He noticed that in “theories like general
relativity there is no way of choosing a coordinate system in which the ghosts
decouple. Such theories may be dealt with by the BRST-quantization method
described at the end of the previous section, using BRST invariance to prove
that the S-matrix in a physical ghost-free Hilbert space is unitary.”

To clarify this statement we need to explain the precise meaning of the words
“the ghosts decouple”: one has to make a clear distinction between ghosts
decoupling in a covariant BRST quantization procedure and in canonical
quantization. For example, one may wonder what exactly the statement above
means for e.g. Dirac gauge Dirac:1958jc  in view of the fact that in
Faddeev:1969su ; Fradkin:1970pn ; Faddeev:1973zb  gravity was canonically and
covariantly quantized in the Dirac gauge, with no FP ghosts appearing in the
canonical quantization. The clarified statement is: in canonical quantization
in Dirac gauge in gravity there is a unitary Hamiltonian and there are no FP
ghosts. In covariant quantization in the Dirac gauge, using the BRST method,
there are FP ghosts with instantaneous propagators. To explain this we proceed
with a review of the results in Faddeev:1969su ; Fradkin:1970pn ;
Faddeev:1973zb .

The Faddeev-Fradkin-Vilkovisky construction involves defining the canonical
(Hamiltonian) Feynman integral for systems with singular Lagrangians, which
have the property that the equation

$p_{i}={\partial L(q,\dot{q})\over\partial\dot{q}^{i}}$ (102)

cannot be solved for $\dot{q}^{i}$ as a function of the $q^{i}$’s and
$p_{i}$’s, which is a constrained system studied first by Dirac. The
Hamiltonian $H(q^{i},p_{i}),i=1,\dots,n$ depends on 2n variables. For such
singular Lagrangians with $m$ gauge symmetries, the naïve canonical variables
$(q^{i},p_{i})$ of the naïve phase space $\Gamma$ satisfy a set of first class
constraints

$\phi^{\alpha}(q,p)=0,\,\qquad\alpha=1,\dots m$ (103)

These are in involution with each other as well as with the Hamiltonian. The
constraints define a surface $M$ of dimension $(2n-m)$ in $\Gamma$.

For such Lagrangians $L(q,\dot{q})$ all equations of motion, including the
constraints, can be obtained from the constrained action

$S(q,p,\lambda)=\int\mathrm{d}t\left(p_{i}\dot{q}^{i}-H(q,p)-\lambda_{\alpha}\phi^{\alpha}(q,p)\right)$
(104)

For such systems the Feynman path integral is not well-defined unless an
additional set of $m$ conditions $\chi_{\alpha}(q,p)$ on the canonical
variables is introduced. These conditions666A more general choice of
conditions is $\chi_{a}(q,p,\lambda,\dot{\lambda})$ where there is a
dependence on the Lagrange multiplier and its time derivatives. In these cases
the underlying Hamiltonian was also constructed, and depends on $(n+m)$
degrees of freedom. The S-matrix in this case is pseudo-unitary, since the
Hilbert space has an indefinite metric Fradkin:1977hw . This case includes
Lorentz covariant gauges like a harmonic gauge.

$\chi_{\alpha}(q,p)=0$ (105)

define a submanifold $\Gamma^{*}$ in $M$. These functions must satisfy the
condition that

$\det||\\{\chi_{\alpha},\phi^{\beta}\\}||\neq 0$ (106)

since only in this case can the surface $\Gamma^{*}$ be defined. Here
$\\{f,g\\}$ are Poisson brackets in $\Gamma$. Here we remind the reader that,
as explained near eq. (89), in covariant quantization one can arrange that at
$\chi_{\alpha}=\phi^{\alpha}=0$ the variation of the function $\chi_{\alpha}$
is

$\delta\chi_{\alpha}=-\\{\chi_{\alpha},\phi^{\beta}\\}\xi_{\beta}$ (107)

Therefore the Poisson bracket

$\\{\chi_{\alpha}(t,\vec{x}),\phi^{\beta}(t,\vec{y})\\}=M_{\alpha}{}^{\beta}\delta^{3}(\vec{x}-\vec{y})$
(108)

of the canonical quantization defines (via $M_{\alpha}{}^{\beta}$) the FP
ghost action $\bar{C}^{\alpha}Q_{\alpha}{}^{\beta}C_{\beta}$ in covariant
quantization.

A convenient choice is when $\\{\chi_{\alpha},\chi_{\beta}\\}=0$, in which
case we can perform a canonical transformation in $\Gamma$ to obtain the new
canonical variables

$p^{\prime}_{\alpha}=\chi_{\alpha}(p,q)=0\,,\qquad
q^{{}^{\prime}\alpha}=q^{{}^{\prime}\alpha}(p^{*},q^{*})$ (109)

where $q^{{}^{\prime}\alpha}$ and $p^{\prime}_{\alpha}$ are canonically
conjugate, and where the $q^{{}^{\prime}\alpha}$ are parameterized by the
physical canonical variables $q^{*},p^{*}$, which are independent coordinates
on $\Gamma^{*}$. Therefore on $\Gamma^{*}$ we are left with $2(n-m)$
independent canonical variables $(p^{*},q^{*})$. In such case the correct
Feynman integral can be given in a canonical/Hamiltonian form as

$\int\prod_{t}{\mathrm{d}p^{*}\mathrm{d}q^{*}\over
2\pi}\,\exp\left\\{\mathrm{i}\left[p^{*}\dot{q}^{*}-{\cal
H}(p^{*},q^{*})\right]\mathrm{d}t\right\\}$ (110)

This Hamiltonian defines the unitary S-matrix and depends on the $2(n-m)$
physical canonical variables, i. e. on $(n-m)$ physical degrees of freedom.
The Hilbert space of physical states has a positive-definite metric and is
manifestly ghost-free.

The basic result of Faddeev-Fradkin-Vilkovisky for the class of constraints in
eq. (105) is that the path integral in (110) based on a unitary Hamiltonian
can be given in the form corresponding to a covariant quantization as defined
in DeWitt:1967ub ; Faddeev:1967fc , where the constraint function
$\chi_{\alpha}(q,p)$ defines a gauge-fixing function $\chi_{\alpha}(\phi)$,
depending on the classical fields of the theory. In addition, observables
computed using this path integral will be independent on the specific choice
of the additional condition/gauge-fixing function.

A simple example of this situation is the Coulomb gauge in QCD. In this case
there is a unitary Hamiltonian with only physical degrees of freedom. In
notation of Faddeev:1969su  there is a constraint with the Lagrange multiplier
$A^{0}$. The configuration space coordinate is $q^{i}=A^{i}$ with canonical
conjugate $p_{k}=E_{k}\equiv F_{0k}$. The constraint and additional condition
are

$\phi(q,p)=\partial_{k}E_{k}+[A_{k},E_{k}]\,,\qquad\chi(q,p)=\partial_{i}A_{i}$
(111)

The Poisson bracket $\\{\chi_{\alpha},\phi^{\beta}\\}$ is computed in this
case and it gives

$\\{\chi_{\alpha},\phi^{\beta}\\}=-[\delta^{\alpha\beta}\partial_{i}\partial_{i}+\epsilon^{\alpha\beta\gamma}A_{k}^{\gamma}\partial_{k}]\delta^{3}(\vec{x}-\vec{y})$
(112)

It is clear from this example why the determinant of the Poisson bracket at
equal times $\\{\chi_{\alpha}(q,p),\phi^{\beta}(q,p)\\}$ can only produce
space derivatives, and not time derivatives.

The quantization of gravity in Dirac gauge performed in Faddeev:1969su ;
Fradkin:1970pn ; Faddeev:1973zb  has the same property: There is a unitary
Hamiltonian with only physical degrees of freedom. In the covariant Lagrangian
quantization there are ghosts, but only with instantaneous propagators.

##  Appendix C Non-Linear Part of the Ghost Action in Schwarschild Coordinates

Here we will argue that the higher order in $h_{\mu\nu}$ part of the ghost
action corresponding to the gauge-fixing conditions (53) does not involve time
derivatives acting on ghosts when written in Scwarzschild coordinates.

###  C.1 $l\geq 2$ Modes

We expect that the scalar $f_{(G)}^{(+)}[h,C]$ in eq. (55) can depend on the
scalars $r^{a}C_{a}$ and $C^{(+)}$ (but not on the pseudoscalar $t^{a}C_{a}$)
so that the constraint eq. 59 can be expanded as

$\frac{2}{r^{2}}\,C^{(+)}+\tilde{f}_{(G)}^{(+)}[h]\,C^{(+)}+\hat{f}_{(G)}^{(+)}[h]\,r^{b}C_{b}=0$
(113)

which allows us to express $C^{(+)}$ as a multiple (with an
$h_{\mu\nu}$-dependent coefficient) of $r^{a}C_{a}$. It follows that, after
imposing this constraint, $f_{(G)}^{(+)}[h,C]$ depends only on $r^{b}C_{b}$

$f_{(G)}^{(+)}[h,C]\to-\hat{F}[h]\,r^{b}C_{b}$ (114)

We similarly expect that the vector $f^{(+)}_{a}[h,C]$ appearing in eq. (54)
depends on $C_{a}$ and on $r_{a}r^{b}C_{b}$ (since $C^{(+)}$ is now a function
of $r^{b}C_{b}$), i.e.
$f^{(+)}_{a}[h,C]=\alpha[h]\,C_{a}+\beta[h]\,r_{a}r^{b}C_{b}$, so that the
residual ghost action (60) reads

$\bar{C}^{a}\big{(}1+\alpha[h]\big{)}C_{a}+\bar{C}^{a}\Big{(}{r^{2}\over
2}(\mathcal{D}_{a}-\frac{2}{r}\,r_{a})\hat{F}[h]r^{b}C_{b}+\beta[h]\,r_{a}r^{b}C_{b}\Big{)}$
(115)

In Schwarzschild coordinates ($t,r$) where $r_{a}=(0,1),r^{a}=(0,f)$ there is
a dependence on $C_{t}$ only in the first term

$\bar{C}^{t}\big{(}1+\alpha[h]\big{)}C_{t}$ (116)

so we can integrate over $C_{t}$ to impose the constraint $\bar{C}^{t}=0$,
which reduces the ghost action to

$\bar{C}^{r}{r^{2}\over
2}\Big{(}\mathcal{D}_{r}+\left(\beta[h]-\tfrac{2}{r}\right)\Big{)}f(r)\,C_{r}$
(117)

We thus find that in Schwarzschild coordinates, the ghost action does not
contain any time derivatives acting on the ghosts fields even at the higher
order in $h_{\mu\nu}$ level. This indicates that the ghost free unitarizing
Hamiltonian may be expected in this sector of the theory.

###  C.2 $l=1$ Even Modes

The argument is about the same as in $l\geq 2$ case. As argued above, we
expect that the higher order in $h_{\mu\nu}$ terms $f[h,C]$ may depend on
$\bar{C}^{a}$ contracted with $r_{a}r^{b}C_{b}$ or $C_{a}$. In Schwarzschild
coordinates they therefore depend on $\bar{C}^{r}Q[h]C_{r}$,
$\bar{C}^{a}\mathscr{D}_{a}Q[h]C_{r}$, or $\bar{C}^{t}Q[h]C_{t}$ where $Q[h]$
is some functional of $h_{\mu\nu}$ which does not contain differential
operators acting on $C_{t}$ or $C_{r}$. We therefore expect that the higher
order in $h_{\mu\nu}$ terms will not change the condition $\bar{C}^{t}=0$,
which we derived earlier at leading order in $h_{\mu\nu}$. We therefore expect
that terms with time derivatives acting on ghosts will not appear at higher
orders in $h_{\mu\nu}$, since the only terms with derivatives will be
$\bar{C}^{a}\mathcal{D}_{a}C_{r}\to\bar{C}^{r}\mathcal{D}_{r}C_{r}$. We
therefore expect that the higher order in $h_{\mu\nu}$ ghost action for the
$l=1$ even modes will be given by an expression analogous to the one in eq.
(117), above, with all time derivatives acting on the ghosts dropping from
this expression in Schwarzschild coordinates.

In case of $l=1$ odd modes and $l=0$ modes the absence of time derivatives in
Schwarzschild coordinates to all orders was already demonstrated in Secs. 4.4
and 4.5, respectively.

##  Appendix D Comments on Gravitational Propagator in the Black Hole
Background

In Gaddam:2020rxb  an expression for the tensorial part of the gravitational
propagator in Regge-Wheeler gauge for the even modes was proposed as

$\mathcal{P}^{abcd}~{}=~{}\dfrac{1}{4}\,f_{\ell}\left(k^{2}\right)\left(\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}\right)\,.$
(118)

where

$f_{\ell}\left(k^{2}\right)=-\dfrac{4R^{2}_{S}}{\left(\lambda+1\right)}-\dfrac{2R^{4}_{S}k^{4}}{\left(\lambda+1\right)\left(\lambda-3\right)\left(k^{2}+R^{-2}_{S}\lambda\right)}\,,$
(119)

and $\lambda\equiv\ell^{2}+\ell+1$. The relevant part of the effective two-
dimensional theory was taken in the form

$S_{2d}~{}=~{}\sum_{l,m}\dfrac{1}{4}\int\mathrm{d}^{2}x\left(\mathfrak{h}^{ab}_{lm}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}_{lm}+\mathfrak{h}^{ab}_{lm}\Delta^{-1}_{L,ab}\mathcal{K}_{lm}+\mathcal{K}_{lm}\Delta^{-1}_{R,ab}\mathfrak{h}^{ab}_{lm}+\mathcal{K}_{lm}\Delta^{-1}\mathcal{K}_{lm}\right)\,,$
(120)

for all $l$. It was important here that $\mathcal{K}_{lm}\not=0$.

Meanwhile our results show that the action in cases $l=0,1$ has to be
considered separately from $l\geq 2$ cases since in these cases Regge-Wheeler
gauge fixing with $\mathcal{K}_{lm}\not=0$ is not valid. In particular, as we
see from equations (53), for low multipoles $l=0,1$

$\mathcal{K}_{lm}=0\ ,\qquad l=0,1$ (121)

It is therefore not surprising that the propagator in (118), (119) has strange
properties, noticed in Gaddam:2020rxb  : the $l=0$ mode causes a change in
sign in the second term, the $l=1$ mode has a pole.

Thus, the derivation of the propagator (118), (119) from the action (120) is
not valid for low multipoles. Instead of that, for $l=0,1$ one should consider
the action (120) with $\mathcal{K}_{lm}=0$

$S_{2d}~{}=~{}\sum_{l=0,1;m}\dfrac{1}{4}\int\mathrm{d}^{2}x\
\mathfrak{h}^{ab}_{lm}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}_{lm}\,.$ (122)

Even this action is still a bit dangerous, since the gauge-fixing condition
$t^{a}r^{b}h_{ab}=0$ still has to be added into account for the monopole
ghosts.

We leave the derivation of the full graviton propagator in the Schwarzschild
black hole background for future work.

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