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

# Quantization of Gravity in Spherical Harmonic Basis

Renata Kallosh<EMAIL_ADDRESS>
###### Abstract

We perform canonical quantization of gravity in the background of a
Schwarzschild black hole in the generalized Regge-Wheeler gauge proposed in
Kallosh:2021ors . We find that the Hamiltonian at the quadratic level is
unitary and ghost-free. Two canonical degrees of freedom are associated with
Zerilli-Moncrief and Cunningham-Price-Moncrief functions of the metric
perturbations. The ${\ell}<2$ part of the Hamiltonian vanishes. This
quantization with the unitary Hamiltonian for gravity is valid also in
Minkowski space in spherical coordinates.

###### Contents

  1. 1 Introduction
  2. 2  Counting Gravity Physical Degrees of Freedom in the Spherical Harmonic Basis
  3. 3 Quadratic Lagrangian/Hamiltonian for ${\ell}\geq 2$ Modes
    1. 3.1 ${\ell}\geq 2$ even
    2. 3.2 ${\ell}\geq 2$ odd
  4. 4 Quadratic Lagrangian/Hamiltonian for ${\ell}<2$ Modes
    1. 4.1 ${\ell}=1$ even
    2. 4.2 ${\ell}=1$ odd
    3. 4.3 ${\ell}=0$ even
  5. 5 A special role of ${\ell}=0,1$ in quantization of gravity
  6. 6 Quantization of Gravity in Spherical Harmonics Basis in the Flat Background
  7. 7 A comment on Regge-Wheeler and Teukolsky formalism and gravity waves
  8. 8 Summary

##  1 Introduction

This note is a development based on a recent paper Kallosh:2021ors  where we
have performed a covariant (Lagrangian) quantization of gravity in a black
hole background in the Regge-Wheeler set up Regge:1957td ; Zerilli:1971wd ;
Martel:2005ir . The gauge-fixing condition in Kallosh:2021ors  includes the
Regge-Wheeler gauge for ${\ell}\geq 2$ modes, and a certain background
covariant gauge for ${\ell}<2$ modes, where Regge-Wheeler gauge is not valid.
We will refer to the gauge in Kallosh:2021ors  covering all ${\ell}$ modes, as
a ‘generalized Regge-Wheeler gauge’.

The Feynman path integral for gravity, viewed as quantum field theory (QFT),
is defined by De Witt-Faddeev-Popov DeWitt:1967ub ; Faddeev:1967fc  and takes
a form, in absence of sources

$\int
Dh\,J_{\chi}(g,h)\delta\big{(}\chi_{\alpha}(g,h)\big{)}e^{\mathrm{i}S(g+h)}\,.$
(1)

Here we integrate over the perturbations $h$ in the background metric $g$. The
gauge-fixing conditions are $\chi_{\alpha}(g,h)=0$. The Jacobian designed to
make this path integral independent on the choice of the gauge-fixing
conditions can be presented with the help of the Faddeev-Popov (FP) ghosts
Faddeev:1967fc

$J_{\chi}=\int
D\bar{C}^{\alpha}DC_{\beta}\,e^{\mathrm{i}\int\mathrm{d}^{4}x\,\bar{C}^{\alpha}(x)Q_{\alpha}{}^{\beta}(g,h)C_{\beta}(x)}\,.$
(2)

The differential operator in the ghost action is defined by the gauge
variation of the gauge-fixing finctions
$\delta\chi_{\alpha}=Q_{\alpha}{}^{\beta}\xi_{\beta}$. For the choice of the
gauge-fixing functions made in Kallosh:2021ors , which in addition to Regge-
Wheeler gauge for $l\geq 2$ modes includes gauges for $l<2$ modes, a
generalized Regge-Wheeler gauge, we have found that the ghost actions do not
have time derivatives in Schwarzschild coordinates. We therefore predicted
that in the a generalized Regge-Wheeler gauge Kallosh:2021ors  the canonical
Hamiltonian according to the rules for gauge theories Faddeev:1969su ;
Fradkin:1970pn ; Faddeev:1973zb  is expected to be unitary.

In this note we will present the quadratic in gravitational perturbations $h$
part of the gravity Hamiltonian in the spherical harmonic basis. Before doing
this we will perform the standard counting of physical degrees of freedom in
this case. The structure of the Hamiltonian will confirm this counting.

The standard counting of physical degrees of freedom in gauge theories in the
QFT context of the Feynman path integral is the same in either Lagrangian or
Hamiltonian quantization, and it is also gauge independent, if performed
correctly. The general counting formula is formulated for the number of gauge
field components equal to $n+k$ in case of $k$ gauge symmetries. The total
number of physical degrees of freedom is

$n-k$ (3)

This final counting formula in QFT is valid for any choice of gauge-fixing,
but the procedure is different for unitary and pseudo-unitary gauges. For
example in 4D the metric has $n+k=10$ components and there are $k=4$ gauge
symmetries, the counting is $n-k=(10-4)-4=2$.

In QFT in the class of unitary gauges Hamiltonians have manifestly ghost-free
underlying Hilbert spaces. There are $(p^{*},q^{*})$ variables in Faddeev’s
theorem Faddeev:1969su  as described in Kallosh:2021ors . This means that all
$n-k$ physical states have positive definite metric. The S-matrix is unitary.

$\\#\,{\rm degrees\,of\,freedom}_{\rm unitary\,H}\,=n-k$ (4)

Meanwhile, in other gauges, for example, 4D Lorentz covariant gauges in
gravity, the Hamiltonians are “pseudo-unitary” with underlying state spaces
with negative-norm ghost degrees of freedom Fradkin:1977hw ; Batalin:1977pb .
In such case the counting goes as follows: there are $n+k$ states with
positive norm and $2k$ states with negative norm presented by FP anti-
commuting ghosts, so the total counting, with account of negative norm states,
is the same as in unitary gauges

$\\#\,{\rm degrees\,of\,freedom}_{\rm pseudo-unitary\,H}=n+k-2k\Rightarrow
n-k$ (5)

The S-matrix is pseudo-unitary in a space of states with the indefinite
metric.

We will see that the quadratic in $h$ part of the gravity Hamiltonian in
spherical harmonic basis does support this counting. In the class of gauges
used in Kallosh:2021ors  the canonical Hamiltonian is unitary, as predicted
there.

The complete form of the Hamiltonian to all orders of $h$ is beyond the scope
of this paper. However, in Kallosh:2021ors  we have argued that the non-linear
couplings of ghosts to all orders in $h$ are free of time derivatives on
ghosts. Therefore one would expect that the non-linear in $h$ terms in $H$
will be consistent with the unitarity of the Hamiltonian which will be deduced
in this note at the level quadratic in $h$.

The corresponding part of the action $S(g+h)$, quadratic in $h$, is of the
form

$S={1\over 2}\int h_{\mu\nu}S^{\mu\nu\lambda\delta}(g)h_{\lambda\delta}$ (6)

Here $S^{\mu\nu\lambda\delta}(g)$ is a differential operator depending on the
background metric $g$. The left hand side of equations of motion ${\delta
S\over\delta h_{\mu\nu}}=0$ linear in $h$ takes the form

$Q^{\mu\nu}\equiv{\delta S\over\delta
h_{\mu\nu}}=S^{\mu\nu\lambda\delta}(g)h_{\lambda\delta}$ (7)

One can restore the action in eq. (6) from the information available in eq.
(7).

In the spherical harmonic basis the 4D spacetime is split into
$\mathcal{M}=\mathcal{M}_{2}\times\mathbb{S}^{2}$ with coordinates
$(x^{a},\theta^{A})$, $a=1,2$. The 4D perturbations $h_{\mu\nu}$ are
represented by 2D fields for each $({\ell},m)$ Regge:1957td ; Zerilli:1971wd ;
Martel:2005ir . The corresponding equations are known and we will use them as
derived in Martel:2005ir  in Schwarzschild coordinates. Once the quadratic
Lagrangian is known, it is possible to derive the relevant quadratic in $h$
Hamiltonian. For ${\ell}\geq 2$ modes the corresponding quadratic Hamiltonian
was constructed by Moncrief in Moncrief:1974am  where also the relevant Regge-
Wheeler Regge:1957td  and Zerilli Zerilli:1971wd  equations were re-derived in
the form of Hamiltonian equations of motion. The Hamiltonian was derived in
Moncrief:1974am  in absence of source terms. For ${\ell}<2$ the Hamiltonian
was not studied, to the best of our knowledge. In Moncrief:1974am  it was
explained that the attention was restricted to modes with $l\geq 2$ since the
modes with $l<2$ are nonradiative and require a special treatment.

Here we will use the known field equations (7) in the form given in
Martel:2005ir  in Schwarzschild coordinates, which allow to derive the
Lagrangian in (6). From the quadratic Lagrangian we derive a canonical
quadratic part of the Hamiltonian, with account of the algebraic constraints
in our gauges. We will conclude that there are no physical degrees of freedom
suitable for quantization at ${\ell}<2$. Our definition of quantized degrees
of freedom involves the QFT quantization conditions in 2D space of the form

$[q(r,t),p(r^{\prime},t)]=i\delta(r-r^{\prime})$ (8)

The classical field equations for low multipoles in presence of sources are
known to have non-trivial solutions. For example for the monopoles
${\ell}=m=0$ there are solutions like $h^{00}_{tt}\sim{\delta M\over r}$, they
are known to affect the the black hole mass. However, there are no solutions
of the constraint equations compatible with the quantization condition (8) for
${\ell}<2$.

All our results are valid for any mass $M$ of the Schwarzschild black hole,
and the limit to $M=0$ is continuous. This means that they apply not only to
the quantization in the black hole background, but also to the unitary
quantization of the gravitational field in the Minkowski space background in
spherical coordinates.

##  2  Counting Gravity Physical Degrees of Freedom in the Spherical Harmonic
Basis

The ansatz of Regge-Wheeler for the metric perturbations $h_{\mu\nu}$ with
spherical harmonics of definite parity is given in Regge:1957td ;
Zerilli:1971wd ; Martel:2005ir . In our recent paper Kallosh:2021ors  it was
adapted for the purpose of quantization following the formalism and notations
in Martel:2005ir . In particular, we have presented the gauge symmetry
transformations to all orders in $h$. The background metric in Schwarzschild
coordinates is

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-f(r)\,\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}(x)\,\mathrm{d}\Omega^{2}_{2}\,,\qquad
f(r)=1-\frac{2GM}{r}$ (9)

The 2D fields representing all components of $h_{\mu\nu}$ in 4D include the
following

$\displaystyle h_{ab}^{\ell m(+)},\quad j_{a}^{\ell m(+)},\quad K^{\ell
m(+)},\quad G^{\ell m(+)}\,\hskip 85.35826pt{\ell}>1,\quad{\rm even}$ (10)
$\displaystyle h_{a}^{\ell m(-)},\quad h_{2}^{\ell m(-)}\,\hskip
184.9429pt{\ell}>1,\quad{\rm odd}$ (11) $\displaystyle h_{ab}^{1m(+)},\quad
j_{a}^{1m(+)},\quad K^{1m(+)}\qquad\hskip 113.81102pt{\ell}=1,\quad{\rm even}$
(12) $\displaystyle h_{a}^{1m(-)}\hskip 233.3125pt{\ell}=1,\quad{\rm odd}$
(13) $\displaystyle h_{ab}^{00(+)},\quad K^{00(+)}\hskip
187.78836pt{\ell}=0,\quad{\rm even}$ (14)

The gauge symmetries are also expanded in spherical harmonics. In the form
given in our recent paper Kallosh:2021ors  these are

$\displaystyle\xi^{{\ell}>1}\quad{\rm even}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{\ell m(+)},\xi^{\ell m(+)}\\}$ (15)
$\displaystyle\xi^{{\ell}>1}\quad{\rm odd}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi^{\ell m(-)}\\}$ (16)
$\displaystyle\xi^{{\ell}=1}\quad{\rm even}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{1m(+)},\xi^{1m(+)}\\}$ (17)
$\displaystyle\xi^{{\ell}=1}\quad{\rm odd}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi^{1m(-)}\\}$ (18)
$\displaystyle\xi^{{\ell}=0}\quad{\rm even}\qquad$
$\displaystyle\Rightarrow\qquad\\{\xi_{a}^{00(+)}\\}$ (19)

The gauge symmetry parameters $\xi_{a}^{\ell m(+)}$, $\xi^{\ell m(+)}$,
$\xi^{\ell m(-)}$ can be regarded as scalar and vector fields on
$\mathcal{M}_{2}$.

The counting of physical degrees of freedom in these 5 sectors is

  1. 1.

${\ell}>1\quad{\rm even}:\quad\,n+k=7,\quad k=3\,\quad\Rightarrow\quad
n+k-2k=7-2\cdot 3=1$

  2. 2.

${\ell}>1\,\,\quad{\rm odd}:\quad\,n+k=3,\quad k=1\,\quad\Rightarrow\quad
n+k-2k=3-2\cdot 1=1$

  3. 3.

${\ell}=1\quad{\rm even}:\quad\,n+k=6,\quad k=3\,\quad\Rightarrow\quad
n+k-2k=6-2\cdot 3=0$

  4. 4.

${\ell}=1\,\,\quad{\rm odd}:\quad\,n+k=2,\quad k=1\,\quad\Rightarrow\quad
n+k-2k=2-2\cdot 1=0$

  5. 5.

${\ell}=0\quad{\rm even}:\quad\,n+k=4,\quad k=2\,\quad\Rightarrow\quad
n+k-2k=4-2\cdot 2=0$

Thus we find that in $l\geq 2$ sector there is one even and one odd physical
degree of freedom for each $({\ell},m)$. There are no degrees of freedom for
any of ${\ell}<2$.

##  3 Quadratic Lagrangian/Hamiltonian for ${\ell}\geq 2$ Modes

###  3.1 ${\ell}\geq 2$ even

There are 7 fields here, $h_{ab}^{\ell m(+)},\quad j_{a}^{\ell m(+)},\quad
K^{\ell m(+)},\quad G^{\ell m(+)}$. There are 7 equations of motion for these
fields. Now we can add the 3 Regge-Wheeler gauge-fixing conditions

$G=j_{a}=0$ (20)

The remaining 4 fields are $h_{ab}^{\ell m(+)},K^{\ell m(+)}$. We expect to
identify 3 constraints which will leave us with just one canonical degree of
freedom. These equations are according to Martel:2005ir

$\displaystyle Q^{tt}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial
r^{2}}{K}-\frac{3r-5M}{r^{2}f}\frac{\partial}{\partial
r}{K}+\frac{f}{r}\frac{\partial}{\partial
r}{h}_{rr}+\frac{(\lambda+2)r+4M}{2r^{3}}{h}_{rr}+\frac{\mu}{2r^{2}f}{K},$
(21) $\displaystyle Q^{tr}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial t\partial
r}{K}+\frac{r-3M}{r^{2}f}\frac{\partial}{\partial
t}{K}-\frac{f}{r}\frac{\partial}{\partial
t}{h}_{rr}-\frac{\lambda}{2r^{2}}{h}_{tr},$ (22) $\displaystyle Q^{rr}$
$\displaystyle=$ $\displaystyle-\frac{\partial^{2}}{\partial
t^{2}}{K}+\frac{(r-M)f}{r^{2}}\frac{\partial}{\partial
r}{K}+\frac{2f}{r}\frac{\partial}{\partial
t}{h}_{tr}-\frac{f}{r}\frac{\partial}{\partial r}{h}_{tt}+\frac{\lambda
r+4M}{2r^{3}}{h}_{tt}-\frac{f^{2}}{r^{2}}{h}_{rr}-\frac{\mu f}{2r^{2}}{K},$
(23) $\displaystyle Q^{\flat}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial
t^{2}}{h}_{rr}+2\frac{\partial^{2}}{\partial t\partial
r}{h}_{tr}-\frac{\partial^{2}}{\partial
r^{2}}\tilde{h}_{tt}-\frac{1}{f}\frac{\partial^{2}}{\partial
t^{2}}{K}+f\frac{\partial^{2}}{\partial
r^{2}}\tilde{K}+\frac{2(r-M)}{r^{2}f}\frac{\partial}{\partial
t}{h}_{tr}-\frac{r-3M}{r^{2}f}\frac{\partial}{\partial r}{h}_{tt}$
$\displaystyle\mbox{}-\frac{(r-M)f}{r^{2}}\frac{\partial}{\partial
r}{h}_{rr}+\frac{2(r-M)}{r^{2}}\frac{\partial}{\partial r}{K}+\frac{\lambda
r^{2}-2(2+\lambda)Mr+4M^{2}}{2r^{4}f^{2}}{h}_{tt}-\frac{\lambda r^{2}-2\mu
Mr-4M^{2}}{2r^{4}}{h}_{rr}$

Here

$\lambda={\ell}({\ell}+1)\qquad\mu=({\ell}-1)({\ell}+2)$ (25)

The quadratic in $h$ Lagrangian can be restored from these equations as
explained in eqs. (6), (7). One can proceed by defining for each of the 4
fields their canonical momenta. For example, there is no time derivative on
$h_{tt}$ in the action, therefore $p_{tt}=0$, the other 3 coordinates in the
form of ${\cal L}(q,\dot{q})$ do have time derivatives, however, two more
combinations of $q$’s and $p$’s are constrained. Only one independent
canonical degree of freedom out of 4 is left.

The Hamiltonian of the related system starting with the Arnowitt, Deser,
Misner construction was derived in Moncrief:1974am . We skip the details of
the derivation here starting with the field equations (21) since the answer
for the corresponding Lagrangian can be also reconstructed from the Zerilli-
Moncrief function Zerilli:1971wd ; Moncrief:1974am  which in Regge-Wheeler
gauge is

$\Psi_{\rm
even}=\frac{2r}{{\ell}({\ell}+1)}\biggl{[}{K}+\frac{2f}{\Lambda}\biggl{(}f{h}_{rr}-r{K}_{,r}\biggr{)}\biggr{]},\qquad
l\geq 2$

where $\Lambda=({\ell}-1)({\ell}+2)+6M/r$. The equation of motion in the form
of the Zerilli-Moncrief function $\Psi^{lm}_{even}$ as given in Martel:2005ir
is

$(\Box-V_{\rm even})\Psi_{\rm even}=S_{\rm even}$ (26)

where $\Box=g^{ab}{\cal D}_{a}{\cal D}_{b}$ is the Laplacian operator on
${\cal M}_{2}$, $V_{\rm even}$ depends on $r$ as well as on $M$ and on $l$,
and $S_{\rm even}$ is the contribution from sources. We refer to details given
in Martel:2005ir , where also the relation between Zerilli-Moncrief function
and the original Regge-Wheeler function is explained. Equation (26) can be
derived from the Lagrangian of the form (6)

${\cal L}=\sum_{{\ell}\geq 2,m}\Big{[}{1\over 2}\Psi_{\rm even}(\Box-V_{\rm
even})\Psi_{\rm even}-\Psi_{\rm even}S_{\rm even}\Big{]}$ (27)

This can be rewritten in the form producing a quadratic part of the
Hamiltonian. With $\Psi_{\rm even}\equiv Q_{\rm even}$ and its canonically
conjugate $P_{\rm even}$ and, in absence of sources

$H_{{\ell}\geq 2,\rm even}={1\over 2}\sum_{{\ell}\geq
2,m}\int\Big{[}drf(P^{{\ell},m})^{2}_{\rm even}+f(Q_{,r}^{{\ell},m})_{\rm
even}^{2}+V_{\rm even}(Q^{{\ell},m})_{\rm even}^{2}\Big{]}$ (28)

where

$V_{\rm
even}=\frac{1}{\Lambda^{2}}\biggl{[}\mu^{2}\biggl{(}\frac{\mu+2}{r^{2}}+\frac{6M}{r^{3}}\biggr{)}+\frac{36M^{2}}{r^{4}}\biggl{(}\mu+\frac{2M}{r}\biggr{)}\biggr{]}$
(29)

This is an example of the Faddeev’s theorem Faddeev:1969su , which we
described in Kallosh:2021ors , where starting from the original constrained
variables $(p_{i},q^{i})$ with constraints $\phi^{\alpha}(p,q)$ one can
perform a canonical transformation with
$p^{\prime}_{\alpha}=\chi_{\alpha}(p,q)=0$ and
$q^{{}^{\prime}\alpha}=q^{{}^{\prime}\alpha}(p^{*},q^{*})$ so that the
independent set of canonical variables is $(p^{*},q^{*})$. In this particular
case we find just one set of $(p^{*},q^{*})$, which are the Zerilli-Moncrief
function $\Psi$ of the original variables, and its canonical conjugate.

###  3.2 ${\ell}\geq 2$ odd

There are 3 fields in this sector: $h_{a}^{\ell m(-)},\quad h_{2}^{\ell
m(-)}$. In the RW gauge

$h_{2}^{\ell m(-)}=0\,.$ (30)

Equations of motion for the remaining two fields are

$\displaystyle P^{t}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial t\partial
r}{h}_{r}+\frac{\partial^{2}}{\partial
r^{2}}{h}_{t}-\frac{2}{r}\frac{\partial}{\partial t}{h}_{r}\ -\frac{\lambda
r-4M}{r^{3}f}{h}_{t},$ $\displaystyle P^{r}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}{h}_{r}-\frac{\partial^{2}}{\partial t\partial
r}{h}_{t}+\frac{2}{r}\frac{\partial}{\partial t}{h}_{t}+\frac{\mu
f}{r^{2}}{h}_{r},$

Restoring the quadratic Lagrangian and using partial integration one can
identify one field which enters into Lagrangian without a time derivative,
this is $h_{t}$.

${\cal L}=h_{t}\Big{(}-\frac{\partial^{2}}{\partial t\partial
r}{h}_{r}+{1\over 2}\frac{\partial^{2}}{\partial
r^{2}}{h}_{t}-\frac{2}{r}\frac{\partial}{\partial t}{h}_{r}\ -{1\over
2}\frac{\lambda r-4M}{r^{3}f}{h}_{t}\Big{)}+{1\over
2}h_{r}\Big{(}\frac{\partial^{2}}{\partial t^{2}}{h}_{r}+\frac{\mu
f}{r^{2}}{h}_{r}\Big{)}$ (31)

Thus we find

$\displaystyle p_{t}=0$ (32) $\displaystyle p_{r}=h_{t,r}-h_{r,t}-{2\over
r}h_{t}$ (33)

and there is a constraint for $h_{t}$ algebraically related to $p_{r}$

$\Big{(}\partial_{r}+\frac{2}{r}\Big{)}\Big{(}p_{r}+h_{t,r}-{2\over
r}h_{t}\Big{)}-{h}_{t,rr}+\frac{\lambda r-4M}{r^{3}f}{h}_{t}=0$ (34)

Therefore there is one independent degree of freedom $(h_{r},p_{r})$. These
are Faddeev’s $(p^{*},q^{*})$ variables, exactly one set in agreement with the
counting give above. One can write the corresponding Hamiltonian
$H(h_{r},p_{r})$ and the field equations.

On the other hand, the Hamiltonian for this system was already derived in
Moncrief:1974am  in the framework of the Arnowitt, Deser, Misner construction.
The field equations were derived in Cunningham:1978zfa , where the
corresponding Cunningham-Price-Moncrief function was introduced. In notation
of Martel:2005ir , this function is

$\Psi_{\rm
odd}^{lm}=\frac{2r}{({\ell}-1)({\ell}+2)}\biggl{(}{h}_{t,r}^{{\ell}m}-{h}_{r,t}^{{\ell}m}-\frac{2}{r}{h}_{t}^{{\ell}m}\biggr{)}.$
(35)

This function in terms of canonical variables above depends on
$(p_{r},h_{r})$. As in the even case discussed above we are lead to a single
field equation for the Cunningham-Price-Moncrief function

$(\Box-V_{\rm odd})\Psi_{\rm odd}=S_{\rm odd}$ (36)

Here the expressions for $V_{\rm odd}$ and $S_{\rm odd}$ are given in
Martel:2005ir , where also the relation between Cunningham-Price-Moncrief
function and the original Regge-Wheeler function is explained. With $\Psi_{\rm
odd}\equiv Q_{\rm odd}$ the Hamiltonian is

$H_{{\ell}\geq 2,\rm odd}={1\over 2}\sum_{{\ell}\geq
2,m}\int\Big{[}drf(P^{{\ell},m})^{2}_{\rm odd}+f(Q_{,r}^{{\ell},m})_{\rm
odd}^{2}+\Big{(}\frac{{\ell}({\ell}+1)}{r^{2}}-{6M\over
r^{3}}\Big{)}(Q^{{\ell},m})_{\rm odd}^{2}\Big{]}$ (37)

##  4 Quadratic Lagrangian/Hamiltonian for ${\ell}<2$ Modes

###  4.1 ${\ell}=1$ even

Our 6 fields are $h_{ab}^{1m(+)},j_{a}^{1m(+)},K^{1m(+)}$. We take a gauge-
fixing condition Kallosh:2021ors

$j_{a}^{1m(+)}=K^{1m(+)}=0$ (38)

The remaining filelds $h_{ab}^{1m(+)}$ in this gauge satisfy the field
equations

$\displaystyle Q^{tt}$ $\displaystyle=$
$\displaystyle\frac{f}{r}\frac{\partial}{\partial
r}{h}_{rr}+\frac{2(r+M)}{r^{3}}{h}_{rr},$ $\displaystyle Q^{tr}$
$\displaystyle=$ $\displaystyle-\frac{f}{r}\frac{\partial}{\partial
t}{h}_{rr}-\frac{1}{r^{2}}{h}_{tr},$ $\displaystyle Q^{rr}$ $\displaystyle=$
$\displaystyle\frac{2f}{r}\frac{\partial}{\partial
t}{h}_{tr}-\frac{f}{r}\frac{\partial}{\partial
r}{h}_{tt}+\frac{r+2M}{r^{3}}{h}_{tt}-\frac{f^{2}}{r^{2}}{h}_{rr}$

We can therefore reconstruct the Lagrangian of the form (6) which will produce
these equations.

${\cal L}=h_{tt}Q^{tt}+\Big{(}\frac{\partial}{\partial
t}h_{tr}\Big{)}\frac{2f}{r}{h}_{rr}-h_{tr}\frac{1}{2r^{2}}{h}_{tr}-h_{rr}\frac{f^{2}}{2r^{2}}{h}_{rr}$
(39)

We now define $q\equiv h_{tr},\,p\equiv\frac{2f}{r}{h}_{rr}$ and
$h_{tt}\equiv\lambda$

${\cal L}=\dot{q}p+\lambda
Q^{tt}(p,\partial_{r}p)-q^{2}\frac{1}{2r^{2}}-{1\over 8}p^{2}$ (40)

We integrate out the Lagrange multiplier and find

${\cal L}=\dot{q}p-q^{2}\frac{1}{2r^{2}}-{1\over 8}p^{2}$ (41)

where

$\frac{f}{r}\frac{\partial}{\partial r}{rp\over
2f}+\frac{2(r+M)}{r^{3}}{rp\over 2f}=0\qquad\Rightarrow\qquad p_{,r}+F(r)p=0$
(42)

The algebraic constraint which $p$ has to satisfy contradicts the commutation
relation which have to be imposed for quantization, as shown in eq. (8). There
is no solution of the algebraic constraint (42) for the canonical momentum
$p(t,r)$ which would be consistent with the quantization condition, only $p=0$
is a consistent one. We conclude there that there are no physical degrees of
freedom left in this sector,

$H_{{\ell}=1,\rm even}=0$ (43)

This is in agreement with the counting we presented above.

###  4.2 ${\ell}=1$ odd

There are 2 fields: $h_{a}^{1m(-)}$. We take a gauge-fixing condition
$h_{r}^{1m(-)}=0$ Kallosh:2021ors . In this gauge the remaining field equation
is

$P^{t}=\frac{\partial^{2}}{\partial r^{2}}{h}_{t}-\frac{2}{r^{2}}{h}_{t}$ (44)

The Lagrangian which will generate this equation is

${\cal L}={1\over 2}h_{t}\Big{(}\frac{\partial^{2}}{\partial
r^{2}}{h}_{t}-\frac{2}{r^{2}}{h}_{t}\Big{)}$ (45)

There is one field here where the Lagrangian ${\cal L}(q)$ does not have time
derivative of this field, therefore $p={\delta{\cal L}\over\dot{h}_{t}}=0$.
There are no canonical variables here and the Hamiltonian vanishes

$H_{{\ell}=1,\rm odd}=0$ (46)

This is in agreement with the counting we presented above.

###  4.3 ${\ell}=0$ even

There are 4 fields here: $h_{ab}^{00(+)},\quad K^{00(+)}$. We take a gauge-
fixing conditions $K=h_{tr}=0$ Kallosh:2021ors . The remaining field equations
are

$\displaystyle Q^{tt}$ $\displaystyle=$
$\displaystyle\frac{f}{r}\frac{\partial}{\partial
r}{h}_{rr}+\frac{r+2M}{r^{3}}{h}_{rr},$ $\displaystyle Q^{rr}$
$\displaystyle=$ $\displaystyle-\frac{f}{r}\frac{\partial}{\partial
r}{h}_{tt}+\frac{2M}{r^{3}}{h}_{tt}-\frac{f^{2}}{r^{2}}{h}_{rr}$

The Lagrangian which will generate these equations is

${\cal L}=h_{tt}\Big{(}\frac{f}{r}\frac{\partial}{\partial
r}{h}_{rr}+\frac{(r+2M)}{r^{3}}{h}_{rr}\Big{)}-{f^{2}\over 2r^{2}}h_{rr}^{2}$
(47)

There are 2 fields, $q^{1},q^{2}$, but there are no time derivatives in the
Lagrangian, $p_{1}=p_{2}=0$, no canonical variables and the Hamiltonian
vanishes

$H_{{\ell}=0}=0$ (48)

This is again in agreement with the counting we presented above.

##  5 A special role of ${\ell}=0,1$ in quantization of gravity

Is there any relation between the well known fact about the absence of
radiation from monopoles and dipoles in gravity and the fact we observed here,
that there are no quantum physical degrees of freedom in monopoles and dipoles
when gravity is quantized in spherical harmonics basis? The answer is yes, and
it has to do with the tensor nature of gravity, so that radiation starts with
quadrupoles ${\ell}\geq 2$.

Regge-Wheeler ansatz for ${\ell}\geq 2$ has 10 functions depending on
coordinates of ${\cal M}_{2}$ listed in eqs. (10)-(14). Here we show them in
the matrix form contracted with spherical functions.

$h_{\mu\nu}^{{\ell}>1}=\begin{pmatrix}h^{{\ell}m}_{ab}Y^{{\ell}m}&&&&{\color[rgb]{0,0,1}j^{{\ell}m}_{a}Y_{B}^{{\ell}m}}\\\
\cr{\color[rgb]{0,0,1}j^{{\ell}m}_{a}Y_{B}^{{\ell}m}}&&&&r^{2}K^{{\ell}m}\Omega_{AB}Y^{{\ell}m}+{\color[rgb]{1,0,0}G^{{\ell}m}Y_{AB}^{{\ell}m}}\end{pmatrix}^{(+)}+\begin{pmatrix}0&&&&{\color[rgb]{0,0,1}h^{{\ell}m}_{a}X_{B}^{{\ell}m}}\\\
\cr{\color[rgb]{0,0,1}h^{{\ell}m}_{a}X_{B}^{{\ell}m}}&&&&{\color[rgb]{1,0,0}h^{{\ell}m}_{2}X_{AB}^{{\ell}m}}\end{pmatrix}^{(-)}\,.$
(49)

The number gauge symmetries in all cases with ${\ell}>0$ is the same since
$\xi_{\mu}$ is a vector

$\xi_{\mu}^{{\ell}>0}=\begin{pmatrix}\xi^{{\ell}m}_{a}Y^{{\ell}m}\\\
\cr{\color[rgb]{0,0,1}\xi^{{\ell}m}Y_{A}^{{\ell}m}}\end{pmatrix}^{(+)}\,+\begin{pmatrix}0\\\
\cr{\color[rgb]{0,0,1}\xi^{{\ell}m}_{a}X_{B}^{{\ell}m}}\end{pmatrix}^{(-)}\,.$
(50)

Therefore we find that instead of 10 fields (even and odd) as for ${\ell}\geq
2$ we have 8 fields (even and odd) for ${\ell}=1$, no fields in red

$h_{\mu\nu}^{{\ell}=1}=\begin{pmatrix}h^{{\ell}m}_{ab}Y^{{\ell}m}&&&&j^{{\ell}m}_{a}Y_{B}^{{\ell}m}\\\
\cr
j^{{\ell}m}_{a}Y_{B}^{{\ell}m}&&&&r^{2}K^{{\ell}m}\Omega_{AB}Y^{{\ell}m}\end{pmatrix}^{(+)}+\begin{pmatrix}0&&&&h^{{\ell}m}_{a}X_{B}^{{\ell}m}\\\
\cr h^{{\ell}m}_{a}X_{B}^{{\ell}m}&&&&0\end{pmatrix}^{(-)}\,.$ (51)

Therefore from 10-2 =8 states we subtract a double set of 4 symmetries, and
find no degrees of freedom for ${\ell}=1$ since 8-8=0.

At ${\ell}=0$ $Y_{AB}^{00}=X_{AB}^{00}=0$, the terms in red are absent, but
also $Y_{A}^{00}=X_{A}^{00}=0$, all blue terms are absent.

$h_{\mu\nu}^{{\ell}=0}=\begin{pmatrix}h^{{\ell}m}_{ab}Y^{{\ell}m}&&&&0\\\ \cr
0&&&&r^{2}K^{{\ell}m}\Omega_{AB}Y^{{\ell}m}\end{pmatrix}^{(+)}+\begin{pmatrix}0&&&&0\\\
\cr 0&&&&0\end{pmatrix}^{(-)}\,.$ (52)
$\xi_{\mu}^{{\ell}=0}=\begin{pmatrix}\xi^{{\ell}m}_{a}Y^{{\ell}m}\\\ \cr
0\end{pmatrix}^{(+)}\,+\begin{pmatrix}0\\\ \cr 0\end{pmatrix}^{(-)}\,.$ (53)

We are left with 4 fields and 2 gauge symmetries, there are no degrees of
freedom for ${\ell}=0$: 4-4=0.

##  6 Quantization of Gravity in Spherical Harmonics Basis in the Flat
Background

The procedure of Lagrangian quantization performed in Kallosh:2021ors  as well
as the values of the unitary quadratic Hamiltonians presented in this paper,
have a smooth limit from the Schwarzschild background to a flat one. In
Schwarzschild coordinates this means that the limit $M\rightarrow 0$ is
regular.

In particular, Zerilli-Moncrief function for ${\ell}\geq 2$ in Regge-Wheeler
gauge in the limit $M\rightarrow 0$ is

$\Psi_{\rm
even}^{{\ell}m}=\frac{2r}{{\ell}({\ell}+1)}\biggl{[}{K}+\frac{2}{({\ell}-1)({\ell}+2)}\biggl{(}{h}_{rr}-r{K}_{,r}\biggr{)}\biggr{]},\qquad{\ell}\geq
2$ (54)

The Cunningham-Price-Moncrief function is

$\Psi_{\rm
odd}^{{\ell}m}=\frac{2r}{({\ell}-1)({\ell}+2)}\biggl{(}{h}_{t,r}^{{\ell}m}-{h}_{r,t}^{{\ell}m}-\frac{2}{r}{h}_{t}^{{\ell}m}\biggr{)},\qquad{\ell}\geq
2$ (55)

The quadrartic part of the Hamiltonian in both cases is

$H_{\rm even/odd}={1\over 2}\sum_{{\ell}\geq
2,m}\int\Big{[}dr(P^{{\ell},m})^{2}_{\rm even/odd}+(Q_{,r}^{{\ell},m})_{\rm
even/odd}^{2}+\frac{{\ell}({\ell}+1)}{r^{2}}(Q^{{\ell},m})_{\rm
even/odd}^{2}\Big{]}$ (56)

Here $Q_{\rm even/odd}=\Psi_{\rm even/odd}$ and $P_{\rm even/odd}$ is the
corresponding canonical conjugate. At the quadratic level these are the only 2
physical states which appear in the unitary Hamiltonian.

The higher order terms in the each of the quantized actions, at the black hole
background and in the flat background still have to be constructed.

##  7 A comment on Regge-Wheeler and Teukolsky formalism and gravity waves

The Cunningham-Price-Moncrief (CPM) master function and the Zerilli-Moncrief
(ZM) master function, which were identified here as canonical variables in the
gravity, appear to play some role also in a more interesting case of the Kerr
black holes. Namely, as pointed out in a review Pound:2021qin , there is a
relation via Chandrasekhar transformation between these functions and
Teukolsky radial function. Note that Teukolsky equations for the Weyl tensor
components use the expansion in terms of the spin-weighted spheroidal
harmonics. Such and expansion for the metric starts with ${\ell}=2$.

There is also an interesting relation between the metric perturbation far from
the source and our canonical variables in the generalized Regge-Wheeler gauge.
Namely, according to Pound:2021qin  the gravitational wave strain can be
determined directly from CPM and ZM functions of the metric. Using the
Chandrasekhar transformation between these functions and Teukolsky radial
function, and some properties of $\psi_{4}=C_{n\bar{m}n\bar{m}}$ the
gravitational strain was given as

$r(h_{+}-ih_{x})=\sum_{{\ell}\geq 2}\sum_{|m|\leq{\ell}}{D\over
2}\Big{(}\Psi_{\rm even}^{{\ell}m}-i\Psi_{\rm even}^{{\ell}m}\Big{)}\,\hskip
2.84526pt{}_{-2}Y_{{\ell},m}(\theta,\phi)$ (57)

where ${}_{-2}Y_{{\ell},m}(\theta,\phi)$ is the the spin-weighted spheroidal
harmonic. That equality holds in the limit $r\rightarrow\infty$ (at fixed
$u=t-r_{*}$). Here the constant

$D=\sqrt{({\ell}-1)({\ell}+1)({\ell}+1)}$ (58)

is the Schwarzschild limit of the constant that appears in the Teukolsky-
Starobinsky identities. Clearly, the cases ${\ell}=0,1$ drop from the formula
for the gravitational waves. This is in agreement with the fact established in
this paper that these modes have no physical degrees of freedom.

##  8 Summary

In this note we have counted the number of physical quantized degrees of
freedom of Einstein gravity in spherical harmonic basis using the standard
formula: this number is given by $n-k$, where $n+k$ is the number of
components of gauge fields and the gauge theory has $k$ gauge symmetries. For
example, in 4D the graviton has $n+k=6+4=10$ components and there are $k=4$
gauge symmetries. The number of physical degrees of freedom is
$n-k=(n+k)-2k=10-8=2$.

In spherical harmonic basis we have found that for each ${\ell},m$ in
${\ell}\geq 2$ sector there is one degree of freedom for even parity states
and one degree of freedom for odd parity states. In ${\ell}<2$ sector of
gravity we have found that there are no physical degrees of freedom.

To construct the Hamiltonian we start with the Regge-Wheeler formulation
Regge:1957td ; Zerilli:1971wd ; Martel:2005ir  of Einstein gravity in
spherical harmonic basis in the background of a Schwarzschild black hole. The
part of the action $S(g+h)$ quadratic in perturbations $h_{\mu\nu}$ in eq. (6)
can be presented in spherical harmonic basis using the explicit form of
equations of motion linear in perturbations, as shown in eq. (7). We take
these explicit expressions $Q^{\mu\nu}={\delta S(g,h)\over\delta h_{\mu\nu}}$,
which are linear in $h_{\mu\nu}$, from Martel:2005ir , and reconstruct the
part of the action $S(g+h)$ quadratic in perturbations $h_{\mu\nu}$. We impose
the generalized Regge-Wheeler gauge Kallosh:2021ors . The action quadratic in
fields we take in Schwarzschild coordinates and proceed with canonical
quantization, defining canonical momenta and constraints.

For ${\ell}\geq 2$ fields the procedure leads to one independent degree of
freedom for even and one for odd modes in each case with ${\ell},m$, in
agreement with the counting of physical degrees of freedom. We conclude that
up to a canonical transformation such a Hamiltonian is equivalent to the one
presented in Moncrief:1974am  where the corresponding canonical variables are
Zerilli-Moncrief function Zerilli:1971wd ; Moncrief:1974am  for even modes and
a Cunningham-Price-Moncrief function Cunningham:1978zfa  for odd modes. In
Moncrief:1974am  the modes with ${\ell}<2$ were not studied.

We apply our method also for ${\ell}<2$ modes. In each sector for ${\ell}=1$,
even and odd case and for ${\ell}=0$ we first reproduce the action from the
explicit expressions $Q^{\mu\nu}={\delta S(g,h)\over\delta h_{\mu\nu}}$ linear
in $h_{\mu\nu}$. We use the gauge-fixing condition for low multipoles in
Kallosh:2021ors  and identify the canonical variables and constraints. In each
case the conclusion is that there are no independent unconstrained canonical
variables suitable for the quantized Hamiltonian. This is again in agreement
with the counting of degrees of freedom performed earlier.

The original goal of this investigation was to develop a consistent method of
quantization of gravitational field in the background of a Schwarzschild black
hole Kallosh:2021ors . However, we found that in Schwarzschild coordinates the
limit $M\rightarrow 0$ is regular, and therefore the quantization procedure is
valid in the Minkowski background as well. In this paper we found the
Hamiltonian describing unitary evolution of gravitational perturbations in
spherical coordinates, which equally well applies to quantization of gravity
in Minkowski background as well as in the Schwarzschild black hole background.
The choice of the generalized Regge-Wheeler gauge in Kallosh:2021ors  where
the gravity Hamiltonian is unitary requires to use the spherical harmonic
basis for the metric perturbations. This unitary gauge is a Regge-Wheeler
gauge $G^{\ell m(+)}=j_{a}^{\ell m(+)}=h_{2}^{\ell m(-)}=0$ for ${\ell}\geq
2$. For ${\ell}=1$ it is $j_{a}^{1m(+)}=K^{1m(+)}=h_{r}^{1m(-)}=0$ and for
${\ell}=0$ it is $K^{00}=h_{tr}^{00}=0$.

In this generalized Regge-Wheeler gauge, the quadratic part of the Hamiltonian
for ${\ell}<2$ modes is vanishing, whereas for ${\ell}\geq 2$ it is given in
eqs. (28), (37) in the black hole background and in eq. (56) in Minkowski
background.

## Acknowledgement

I am grateful to A. Barvinsky, E. Coleman, A. Linde, E. Poisson, A. Rahman, P.
Stamp, A. Starobinsky, A. Vainshtein, A. Van Proeyen and I. Volovich for
stimulating and helpful discussions. I am supported by the SITP, by the US
National Science Foundation Grant PHY-2014215 and by the Simons Foundation
Origins of the Universe program (Modern Inflationary Cosmology collaboration).

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