on dimensional arguments. We remind the reader that the tensor $X$ is not
assumed to be symmetric in the group of all covariant indices, here $\rho$,
$\nu$, $\mu$ since this is the order of conformal covariant derivatives, that
also do not commute. But as an accident we see that this tensor is symmetric
in $\alpha$ and $\beta$ indices. This is the result originating from the
definition in (83), symmetry of two GR-covariant derivatives on a scalar and
explicit symmetry of Schouten tensor in its indices. Of course, as denoted by
horizontal spacing of group of these indices on the tensor $X$, here on
$X^{\gamma}{}_{\rho\nu\mu}{}^{\alpha\beta}$, the last group of two
contravariant indices plays a completely different role than the first
contravariant index $\gamma$, which is contracted with first derivative on the
parameter $\Omega$, while the former ones are contracted with conformal
derivatives acting on scalar $\phi$. Hence there cannot be any general
symmetry between all three contravariant indices here. Explicitly, here we can
define formally the tensor $X$ as the result of the following linear operation
of double variational derivative:

$X^{\gamma}{}_{\rho\nu\mu}{}^{\alpha\beta}=\frac{\delta^{2}\left(\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)\delta\left(\overset{c}{\nabla}_{\alpha}\overset{c}{\nabla}_{\beta}\phi\right)}$
(95)

and its explicit expression will be given below in a case with general number
$N$ of conformal covariant derivatives acting on a scalar $\phi$.

Equipped with the knowledge of the case of three conformal covariant
derivatives acting on the scalar field $\phi$ we can now present the general
construction of a sequence of $N$ consecutive derivatives acting there. This
construction is based inductively on the knowledge of $N-1$ conformal
derivatives and the fact that these $N-1$ conformal derivatives transform
under conformal transformations in a decent way as explained above (that is
that only the terms with maximum up to the first derivative of the parameter
$\Omega$ are present). We define that

$\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi=\nabla_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi+P_{\rho_{1}\gamma}X^{\gamma}{}_{\rho_{2}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-2}}\overset{c}{\nabla}_{\beta_{1}}\cdots\overset{c}{\nabla}_{\beta_{N-2}}\phi$
(96)

in a way very similarly to the previous definitions. This definition of a
sequence of $N$ conformally covariant derivatives works for $N\geqslant 2$.
The general expression for the $X$ tensor, here with total of $2(N-1)$ indices
with half of them covariant and other half contravariant, reads

$X^{\gamma}{}_{\rho_{2}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-2}}=\sum_{k=1}^{N-1}\left(w_{0}-N+k+1\right)\delta_{\rho_{k+1}}^{\gamma}\delta_{\rho_{2}}^{\beta_{1}}\cdots\delta_{\rho_{k}}^{\beta_{k-1}}\delta_{\rho_{k+2}}^{\beta_{k}}\cdots\delta_{\rho_{N}}^{\beta_{N-2}}+$
$+\sum_{k=2}^{N-1}\sum_{l=1}^{k-1}\left(g_{\rho_{k+1}\rho_{l+1}}g^{\gamma\beta_{k-1}}-\delta_{\rho_{k+1}}^{\gamma}\delta_{\rho_{l+1}}^{\beta_{k-1}}\right)\delta_{\rho_{2}}^{\beta_{1}}\cdots\delta_{\rho_{l}}^{\beta_{l-1}}\delta_{\rho_{l+2}}^{\beta_{l}}\cdots\delta_{\rho_{k}}^{\beta_{k-2}}\delta_{\rho_{k+2}}^{\beta_{k}}\cdots\delta_{\rho_{N}}^{\beta_{N-2}}\,.$
(97)

This expression corectly reproduces cases for $N=2$ and $N=3$ as reported
above in text and in (90). In the expression (96), one finds the skeleton of
the same construction as originally in (83), but this time the tensor $X$
possesses $2(N-1)$ indices grouped into three sets. And it is contracted with
$(N-2)$-nd order conformal derivative on the same scalar $\phi$. The first
special index on the tensor $X$ is $\gamma$ to be contracted with precisely
the first derivative on $\Omega$ in the conformal transformation law. The
other group of indices from $\rho_{2}$ to $\rho_{N}$ are remaining free
$(N-1)$ indices on the tensor of $N$-th order conformal derivative. In the
final group we find $(N-1)$ indices $\beta_{1}$ to $\beta_{N-1}$ which are all
contracted with indices on the $(N-1)$-st order conformal derivative standing
to the most right in (96). The equivalent form of the formula (96) can be
written slightly shorter as

$\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi=\nabla_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi+P_{\rho_{1}\gamma}\frac{\delta\left(\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)}\,,$
(98)

where we use in recursion the transformation properties of a sequence of
precisely $(N-1)$ conformal derivatives. In this formula we can call the
sequence of the derivatives
$\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi$ as an
object $Y$ (and forget its indices). It is important that the object $Y$
transforms up to terms with the first derivatives of the $\Omega$ parameter.
And then we write a general formula for adding new conformal derivative
$\overset{c}{\nabla}_{\rho_{1}}$ on an arbitrary object $Y$

$\overset{c}{\nabla}_{\rho_{1}}Y=\nabla_{\rho_{1}}Y+P_{\rho_{1}\gamma}\frac{\delta\left(\left.\left(\delta_{c}Y\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)}\,.$
(99)

In general, we have for the infinitesimal conformal variation of the arbitrary
long sequence of conformal derivatives

$\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\nabla\Omega}$
(100)

and we find as a consequence of global conformal symmetry that

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\Omega}=w_{0}\Omega\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi.$
(101)

For the part linear in derivatives of $\Omega$, we have a general definition
that

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\rho_{1}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-1}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\cdots\overset{c}{\nabla}_{\beta_{N-1}}\phi,$
(102)

where here we refer to the tensor $X$ of the one level higher, that is it must
have above precisely $2N$ indices. The general conclusion is that the $X$
tensor with $2N$ indices appears in the conformal transformation law of a
sequence of exactly $N$ conformal derivatives, but is heavily used in the
definition of conformal derivative for the sequence of $N+1$ derivatives, like
this is done in (96) for the case of $N$ derivatives. In a formal way we
obtain a general expression for the tensor $X$ as

$X^{\gamma}{}_{\rho_{1}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-1}}=\frac{\delta^{2}\left(\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)\delta\left(\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\cdots\overset{c}{\nabla}_{\beta_{N-1}}\phi\right)}\,.$
(103)

With the explicit formula for the tensor $X$ in (97) valid for any $N\geqslant
1$, one can check that indeed the sequence of conformal derivatives
constructed in a way as in (96) at the level with $(N+1)$ derivatives changes
under conformal transformation only up to maximal first derivatives of the
parameter $\Omega$. For this one needs to establish a certain quadratic (so
non-linear) relation for the $X$ tensors that can be verified tediously and
explicitly knowing the expression in (97). It reads explicitly

$X^{[\gamma}{}_{\rho_{1}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-1}}X^{\gamma^{\prime}]}{}_{\beta_{1}\ldots\beta_{N-1}}{}^{\alpha_{1}\ldots\alpha_{N-2}}=0,$
(104)

where the antisymmetrization is only between two indices $\gamma$ and
$\gamma^{\prime}$, while all indices $\beta_{1}$ to $\beta_{N-1}$ are in
contraction. Moreover, in the above condition the indices which are
additionally free are $\rho_{1}$ to $\rho_{N}$ and in a different set
$\alpha_{1}$ to $\alpha_{N-2}$. This condition can be solved having the
initial condition for $N=1$ that says
$X_{\mu}^{\gamma}=w_{0}\delta_{\mu}^{\gamma}$ and an useful recurrence
relation satisfied by these $X$ tensors relating the $X$ tensor with $2N$
indices to the one with $2N-2$. The recurrence relation satisfied by the $X$
tensor is

$X^{\gamma}{}_{\rho_{1}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-1}}=\left(w_{0}-N\right)\delta_{\rho_{1}}^{\gamma}\delta_{\rho_{2}}^{\beta_{1}}\cdots\delta_{\rho_{N}}^{\beta_{N-1}}-\sum_{k=2}^{N}\delta_{\rho_{k}}^{\gamma}\delta_{\rho_{2}}^{\beta_{1}}\cdots\delta_{\rho_{k-1}}^{\beta_{k-2}}\delta_{\rho_{1}}^{\beta_{k-1}}\delta_{\rho_{k+1}}^{\beta_{k}}\cdots\delta_{\rho_{N}}^{\beta_{N-1}}+$
$+\sum_{k=2}^{N}g_{\rho_{1}\rho_{k}}g^{\gamma\beta_{k-1}}\delta_{\rho_{2}}^{\beta_{1}}\cdots\delta_{\rho_{k-1}}^{\beta_{k-2}}\delta_{\rho_{k+1}}^{\beta_{k}}\cdots\delta_{\rho_{N}}^{\beta_{N-1}}+\delta_{\rho_{1}}^{\beta_{1}}X^{\gamma}{}_{\rho_{2}\ldots\rho_{N}}{}^{\beta_{2}\ldots\beta_{N-1}}.$
(105)

The general tensor $X$ possesses the same properties as discussed above in the
case $N=3$. It is completely covariantly conserved and it does not change
under conformal transformations. There is no any assumed symmetry between all
indices covariant $\rho_{1}$ to $\rho_{N}$, neither between contravariant ones
since $\gamma$ is a special type of index, neither between the group of
$(N-1)$ indices $\beta_{1}$ to $\beta_{N-1}$. With its explicit form one can
write in full detail the result for the infinitesimal conformal variation of
the sequence of $N$ conformal derivatives on the scalar $\phi$. This formula
(part linear in the derivative of the $\Omega$ parameter) is given by

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi\right)\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\rho_{1}\ldots\rho_{N}}{}^{\beta_{1}\ldots\beta_{N-1}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\cdots\overset{c}{\nabla}_{\beta_{N-1}}\phi=$
$=\sum_{k=1}^{N}\left(w_{0}+k-N\right)\nabla_{\rho_{k}}\Omega\overset{c}{\nabla}_{\rho_{1}}\cdots\overset{c}{\nabla}_{\rho_{k-1}}\overset{c}{\nabla}_{\rho_{k+1}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi-$
$-\sum_{k=2}^{N}\sum_{l=1}^{k-1}\nabla_{\rho_{k}}\Omega\overset{c}{\nabla}_{\rho_{1}}\cdots\overset{c}{\nabla}_{\rho_{l-1}}\overset{c}{\nabla}_{\rho_{l+1}}\cdots\overset{c}{\nabla}_{\rho_{k-1}}\overset{c}{\nabla}_{\rho_{l}}\overset{c}{\nabla}_{\rho_{k+1}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi+$
$+\sum_{k=2}^{N}\sum_{l=1}^{k-1}g_{\rho_{k}\rho_{l}}\nabla_{\gamma}\Omega\overset{c}{\nabla}_{\rho_{1}}\cdots\overset{c}{\nabla}_{\rho_{l-1}}\overset{c}{\nabla}_{\rho_{l+1}}\cdots\overset{c}{\nabla}_{\rho_{k-1}}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\rho_{k+1}}\cdots\overset{c}{\nabla}_{\rho_{N}}\phi.$
(106)

With the usage of conformally covariant derivatives
$\overset{c}{\nabla}_{\mu}$ and the tensor of conformal curvature (Weyl tensor
$C_{\mu\nu\rho\sigma}$), metric tensor of spacetime $g_{\mu\nu}$ and basic
conformally covariant matter fields, like the field $\phi$, we can in
principle construct any invariant and conformally covariant expressions (that
is not transforming or transforming only with the $\Omega$ parameter not
differentiated respectively). The theorem here is that any conformal invariant
or conformally covariant expression (or a differential operator) can be
written in a form that uses only these basic building blocks as mentioned
above, and nothing else. The algebra of conformally covariant and invariant
terms is spanned by metric tensors, matter fields $\phi$ and any completely
symmetrized sequences of conformal derivatives acting on basic fields or on
Weyl tensors in various combinations of products. This is the same like in GR
where all invariant and GR-covariant expressions and differential operators
are generated by metric tensors, matter fields $\phi$ and any completely
symmetrized sequences of covariant derivatives acting on basic fields or on
curvature tensors (here Riemann tensors) and all this in various combinations
in tensor products. Here, for conformal symmetry, we must be precise since the
above building elements generate algebra of all expressions and differential
operators that may transform up to the first derivative of the $\Omega$
parameter. Eventually, to assess whether the expression, tensor or the
operator is conformally covariant or invariant we must check whether the first
derivative of $\Omega$ cancel out in the final conformal transformation law.

We also remark on the range of applicability of our just given definition of
arbitrary Wunsch conformal covariant derivatives. Above, we have defined it as
acting on a sequence of previously defined and constructed conformal
derivatives (with one less derivative in the sequence). In this way of
proceeding we see a clear path from basic conformally covariant expressions,
like the scalar field $\phi$ leading to arbitrary long expressions with a
sequence of conformal derivatives. Finally such different expressions can be
multiplied tensorially to form a product expression or a differential operator
with the desired conformal transformation law. But one can ask for a
definition of conformal derivative on a general object $Y$ (all Lorentz
indices hidden for writing here), be it a tensorial or operatorial expression
and abstracting from its previous form as a product of expressions with a
sequence of conformal derivatives. Here there is an important restriction for
the possible form of the expression–object $Y$ on which our definition of
$\overset{c}{\nabla}_{\mu}$ is valid. The conformal transformation of the
object $Y$ must be up to terms with first derivatives of the parameter
$\Omega$. For example, when the object $Y$ transforms with $\square\Omega$ or
$\nabla_{\mu}\nabla_{\nu}\Omega$ terms, then we cannot define the result of
acting on $Y$ with the conformal derivative $\overset{c}{\nabla}_{\mu}$.
Because in a sense we cannot rectify the fact there are already higher than
the first derivative of $\Omega$ present in the transformation law of $Y$ and
then they would also be present in a tentative expression for
$\overset{c}{\nabla}_{\mu}Y$ and we cannot make it transforming only up to the
first derivatives of $\Omega$. Then we have two cases: or the infinitesimal
transformation law is

$\delta_{c}Y=w(Y)\Omega Y$ (107)

or

$\delta_{c}Y=w(Y)\Omega
Y+\left.\left(\delta_{c}Y\right)\right|_{\nabla\Omega},$ (108)

where the last term in the second formula is written very schematically.
Obviously, in the first case $Y$ is a conformally covariant tensor expression
or differential operator. In the second case, it is a an expression
transforming with up to the first derivative of the $\Omega$ parameter. In
these two cases we can very easily define the action of conformal covariant
derivative. We have, with the conformal object $Y$ (first case)
$\overset{c}{\nabla}_{\mu}Y=\nabla_{\mu}Y$ and its conformal transformation
law is simply

$\delta_{c}\left(\overset{c}{\nabla}_{\mu}Y\right)=w(Y)\Omega\overset{c}{\nabla}_{\mu}Y+w(Y)\nabla_{\mu}\Omega
Y.$ (109)

In that case, $Y$ is simply a basic conformally covariant tensor or matter
field (e.g. it could be the metric $g_{\mu\nu}$, Weyl tensor
$C_{\mu\nu\rho\sigma}$ or a scalar field $\phi$ with assigned conformal
weight). In the second case, $Y$ is a more complicated object and it arises
only as a result of conformal derivatives acting in a sequence on some basic
conformally covariant object or a tensorial product of such expressions. But
we can forget about this detail of the structure of the $Y$ object in this
case and consider general and abstract case here. Then the first conformal
derivative of the object $Y$ is defined as in formula (99) and that is why in
the transformation of the second case in (108) we explicitly singled out the
term with first derivatives of the $\Omega$ parameter. Then the conformal
transformation law is simply

$\delta_{c}\left(\overset{c}{\nabla}_{\mu}Y\right)=w(Y)\Omega\overset{c}{\nabla}_{\mu}Y+\left.\left(\delta_{c}\overset{c}{\nabla}_{\mu}Y\right)\right|_{\nabla\Omega},$
(110)

where the last term in the above equation is written schematically. However,
using the recurrence relations (105) as satisfied by the $X$ tensors, this
$\left.\left(\delta_{c}\overset{c}{\nabla}_{\mu}Y\right)\right|_{\nabla\Omega}$
term can be related to $w(Y)$ and
$\left.\left(\delta_{c}Y\right)\right|_{\nabla\Omega}$. Actually, for this
last operation, as one can see from solving (97) of the recurrence relations
in (105), one needs to count the number of conformal covariant derivatives
acting on basic conformal tensors/scalars as hidden in the general object $Y$.
In the other case, when $Y$ is a product of terms, each one possibly
transforming up to the first derivatives of the $\Omega$ parameter, we first
should apply the Leibniz rule for the conformal derivative and then the new to
be defined conformal derivative $\overset{c}{\nabla}_{\mu}$ in each term in
the resulting sum is adding only to the one existing sequence of conformal
derivatives acting on a basic conformal object. So then the above
argumentation with counting the number of these already existing conformal
derivatives and using recurrence for $X$ tensors applies again.

Here we also mention some important properties of the conformal covariant
derivative mimicking the discussion of them as in the standard GR case. The
conformal covariant derivative is a differential operator mapping general
objects transforming with up to the first derivative of $\Omega$ to another
class of objects conformally transforming with up to the first derivative of
$\Omega$. As such differential operator it is a linear operator satisfying
also the standard Leibniz rule for the derivative on the product. This
conformal covariant derivatives preserves the metric tensor, hence we can
define in an unambiguous way the conformal covariant box operator as it is
done in the next subsection. In general, the conformal derivatives do not
commute, similarly as GR-covariant derivatives. However, the law for conformal
commutation of derivatives is slightly more complicated as we also show below
since it is important for clarifying any ambiguities regarding the order of
conformal derivatives in some cases. Here from the simple commutation of two
covariant derivatives on a vector field with some assigned conformal weight,
we define the Weyl tensor as the tensor of conformal curvature, the result of
such a commutator.

Finally, some properties do not hold contrary to the case of standard
differential calculus with covariant derivatives in GR. Firstly, the conformal
covariant derivative as an operation on some terms in general does not commute
with the infinitesimal conformal transformations of the same object. (In GR we
have that the covariant derivative obviously commutes with the infinitesimal
diffeomorphism symmetry transformations since the covariant derivative is
constructed in such a way to be symmetric there.) For example, we do not get a
simplified expression for any conformal analogue version of the Bianchi
identity, with conformally covariant derivatives and on Weyl tensor. This was
the case even with standard GR-covariant derivatives and Weyl tensor, where
the differential Bianchi identity hold only when derivatives are applied on
Riemann tensor. We remind the reader that here on Weyl tensors the conformal
covariant derivative is identical to the GR-covariant one. As the last
property which does not hold (or holds in a restricted and modified sense) we
mention integration by parts. One has to be very careful with integration by
parts of some conformal derivatives under the action integral. Due to the
presence of the Schouten tensor in the definition in (99), in general the
result of integration by parts does not produce this conformal derivative
acting on the rest of tensorial expression together with a minus sign. A term
with Schouten tensor constitutes here a correction term. For this last
property under the spacetime volume integral we do not have to require that
the integral is conformally invariant, only properties under conformal
transformation of each of the terms matter and not of the full spacetime
integral with the proper measure. For example, when all the terms in the
integral are conformally covariant objects, then one use of conformal
derivative is identical to the standard GR one and this conformal derivative
can be integrated by parts without any correction terms. But in general the
Schouten tensor appears after the process of integration by parts under volume
spacetime integral of conformal covariant derivatives.

The fact that the first conformal derivative on any conformally covariant
tensor is equal from the definition to the ordinary GR-covariant derivative
has important consequences as we wrote also earlier. In simple words we start
compensating for the conformal transformations in the definition of a new
conformally adjusted conformal derivative only from the level of second
conformal derivative. Moreover, we do not use a special independent connection
to compensate for transformation in the definition of these conformal
derivatives. From the second level up we use only the Schouten tensor to
compensate since its infinitesimal conformal transformation law is in a simple
form which contains precisely two derivatives of the $\Omega$ parameter.
Because only the second and higher conformal derivatives are properly adjusted
to compensate (but only partially and not completely) for conformal
transformations, we have that all the objects defined with conformal
derivatives transform up to the first derivative of $\Omega$ parameter. There
was no a modification for the first conformal derivative and this is the mere
reason why the objects with proper conformal derivatives are not fully
conformally covariant. They in most cases do have also the terms in the
transformation law, which are linear in the first derivative of the $\Omega$
parameter. This is precisely what we mean by only partial compensation for the
conformal transformations. Another implication of this fact is that as
explained above we must use the complicated construction of arbitrary order of
conformal derivatives employing the definition of $X$ tensors, where the
special care is given to terms linear in the first derivative of $\Omega$. We
remind the reader that the same procedure is exploited for standard
compensation of gauge transformations in the definition of gauge covariant
derivatives, where the gauge connection, which is added on the level of the
first (and also all higher) covariant derivative uses the infinitesimal
transformations with the level of linearity in the $\Omega$ parameter (but not
in its derivatives). In this simple case, one could also speak about the
expression for the corresponding $X$ tensor and its infinitesimal gauge
transformation law, but obviously they are much simpler. The need for special
care on terms linear in $\nabla\Omega$ also gives a small drawback of the
whole formalism of such constructed conformal derivatives, where the terms
with first derivative of $\Omega$ must be properly accounted for and finally
the cancellation of them must be sought for the fully conformal operators,
tensors and expressions.

We comment that the procedure of constructing covariant derivatives as
presented by Wunsch, can be generalized to any other symmetry group that acts
non-linearly on the curvature tensors constructed out of the metric, while on
the metric as we know the conformal group in the infinitesimal form does act
linearly. We think that there was nothing special here about conformal
symmetry, although of course, this is a paradigmatic case. If we do not want
to use additional connection related to this non-linear symmetry, and then if
we are able to find some curvature tensor that can play the role of the
compensator for some number of derivatives, then we could employ the same
procedure as proposed by Wunsch. But then generally within this procedure we
must remember that the constructed general symmetry covariant new derivatives
are not exactly “covariant” with respect to this symmetry, but transform with
some terms containing up to fixed number of derivatives on the parameter of
the symmetry transformation. (This number is by one smaller than the number of
derivatives on the level, when we first time used the compensator tensor for
the symmetry. For example, for conformal symmetry as resolved by Wunsch, we
used Schouten tensor, hence this level was with two derivatives.) Therefore,
by using only conformally covariant derivatives as pioneered by Wunsch, we do
not have a certainty that the objects constructed with them are always totally
conformally covariant (transforming infinitesimally only linearly in $\Omega$
and without derivatives of it) and we must always eventually verify that the
terms with first derivatives of $\Omega$ in the conformal transformation law
do cancel out. On the other hand, in a search for conformally invariant and
covariant expressions, we might take an approach in which all objects are
built using only Wunsch conformal covariant derivatives and other conformally
covariant tensors (like Weyl or Bach tensor and generalization thereof). In a
sense, if an invariant or conformally covariant expression (or operator)
exists, then it is always possible to write it using the conformal derivatives
and other conformal tensors, but as we saw before this way of presentation of
it is not mandatory.

Above, we discussed in detail the construction of conformal covariant
derivative, where the “good” covariant objects transform up to first
derivatives of the parameter of the transformation. For a general symmetry and
along the lines presented above we can still construct a sequence of $N$
covariant derivatives with respect to some symmetry. If we add a compensating
tensor on the level with $N_{0}$ derivatives, then it means that objects in
general transform up to $N_{0}-1$ derivatives of the parameter. Then we think
that the formulas like (96), (100) and (103) can still work, just that we need
more $\gamma$ type of indices (precisely $N_{0}-1$ of them) and more indices
on the tensor $X$, and more indices on the compensating tensor $P$. The
decomposition of the general infinitesimal transformation law can be still as
in (100), but we need to write there all the terms up to $(N_{0}-1)$ GR-
covariant derivatives of the parameter $\Omega$ of symmetry. All previous
terms in such decomposition are defined and determined by previous order.
However, only the last term (with $(N_{0}-1)$ derivatives of $\Omega$ could be
expressed via the newly defined and more general $X$ tensor, similarly to what
was written above in (102) for the case of $N_{0}=2$, now being contracted
with precisely $(N_{0}-1)$ derivatives on $\Omega$. In reverse, the general
definition of the $X$ tensor via variational derivative as in (103) can be
always applied. Here we only need to derive with respect to $(N_{0}-1)$-st
order of derivatives on the $\Omega$ parameter. We believe that the analog
version of the recurrence relation and its solutions also exist in that more
general case. As it is clearly obvious, above we discussed explicitly the case
of conformal transformations as they act on the metric in GR and we choose
$N_{0}=2$ with the help of Schouten tensor and this is why we were able to
write all formulas explicitly, but the procedure is more general for any non-
linearly acting symmetries on curvatures and for any $N_{0}$.

###  VI.2 Conformal box operator and its powers

Naturally, with the conformal covariant derivative at hand we can construct a
scalar differential operator, which would play the role of the analogue of the
GR-covariant d’Alembertian operator. We use formula that

$\overset{c}{\square}\phi=g^{\nu\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi.$
(111)

However, due to the remark made earlier about the character of all operators
and tensors made out of $\overset{c}{\nabla}_{\mu}$ under conformal
transformation law, we must say that the result of the action of this operator
is in general not a conformally covariant object even if the scalar field was
such object. From the point of view of GR the operator
$\overset{c}{\square}\phi=\phi^{\prime}$ is clearly a new scalar. However,
this scalar with the conformal weight $w_{0}-2$ (because of the contraction
with the contravariant metric $g^{\nu\mu}$ in (111)), does not only transform
conformally covariantly. In general, here there is also a part with the first
derivative of the parameter $\Omega$ of the transformation. We have explicitly
that

$\delta_{c}\left(\overset{c}{\square}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\nabla\Omega},$
(112)

with

$\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\Omega}=\left(w_{0}-2\right)\Omega\overset{c}{\square}\phi$
(113)

and

$\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
g^{\nu\mu}X^{\gamma}{}_{\nu\mu}{}^{\alpha}\overset{c}{\nabla}_{\alpha}\phi=\nabla_{\gamma}\Omega\left\\{2\left(w_{0}-1\right)+d\right\\}g^{\gamma\alpha}\overset{c}{\nabla}_{\alpha}\phi=$
$=\left\\{2\left(w_{0}-1\right)+d\right\\}\nabla^{\alpha}\Omega\overset{c}{\nabla}_{\alpha}\phi\neq
0$ (114)

based on (88) and (90). Still, as with all other Wunsch conformal derivatives,
there are no terms with two (or more) derivatives on $\Omega$ in the conformal
transformation law for $\overset{c}{\square}\phi$. This means that one has to
be careful in defining powers of the $\overset{c}{\square}$ conformal
differential operator as acting on some conformally covariant scalars or
tensors, since strictly speaking they are not powers since the result of
action, that is $\phi^{\prime}=\overset{c}{\square}\phi$ has different
transformation properties than the original field $\phi$ which is conformally
covariant with the weight $w_{0}$. (This is different than in the case of
normal GR-covariant d’Alembertian operator, where
$\phi^{\prime}=\square\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi$ is also a
scalar and we can simply take powers of the same operator $\square$, acting in
the same representation of diffeomorphism group.) Here not only the weight
changes, namely $w(\phi^{\prime})=w_{0}-2\neq w_{0}$, but also we have these
first derivatives $\nabla\Omega$ in the transformation law for
$\phi^{\prime}$. Still formally we can define the powers of the
$\overset{c}{\square}$ operator acting on the scalar field $\phi$ as

$\overset{c}{\square}{}^{N}\underset{{\rm df}}{=}g^{\rho_{1}\rho_{2}}\cdots
g^{\rho_{2N-1}\rho_{2N}}\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\cdots\overset{c}{\nabla}_{\rho_{2N}}\phi,$
(115)

where the order of contractions here matters a lot since in general the
conformal covariant derivatives do not commute (the same like ordinary GR-
covariant derivatives). One here can immediately notice that the
transformation law of the $\overset{c}{\square}{}^{N}$ operator is determined
from some (partial) trace of the corresponding $X$ tensor. Namely, we have

$\delta_{c}\left(\overset{c}{\square}{}^{N}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\square}{}^{N}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\square}{}^{N}\phi\right)\right)\right|_{\nabla\Omega}=$
$=\left(w_{0}-2N\right)\Omega\overset{c}{\square}{}^{N}\phi+\nabla_{\gamma}\Omega
X^{\gamma\beta_{1}\ldots\beta_{2N-1}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\cdots\overset{c}{\nabla}_{\beta_{2N-1}}\phi,$
(116)

where we used the new definition of the partially contracted $X$ tensor as

$X^{\gamma\beta_{1}\ldots\beta_{2N-1}}=g^{\rho_{1}\rho_{2}}\cdots
g^{\rho_{2N-1}\rho_{2N}}X^{\gamma}{}_{\rho_{1}\ldots\rho_{2N}}{}^{\beta_{1}\ldots\beta_{2N-1}}=$
$=\sum_{k=1}^{N}\left(2w_{0}+4k-2-4N+d\right)g^{\beta_{1}\beta_{2}}\cdots
g^{\beta_{2k-3}\beta_{2k-2}}g^{\gamma\beta_{2k-1}}g^{\beta_{2k}\beta_{2k+1}}\cdots
g^{\beta_{2N-2}\beta_{2N-1}},$ (117)

so we find that

$\nabla_{\gamma}\Omega
X^{\gamma\beta_{1}\ldots\beta_{2N-1}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\cdots\overset{c}{\nabla}_{\beta_{2N-1}}\phi=$
$=\sum_{k=1}^{N}\left(2w_{0}+4k-2-4N+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}{}^{k-1}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{N-k}\phi.$
(118)

Hence finally we can write

$\delta_{c}\left(\overset{c}{\square}{}^{N}\phi\right)=\left(w_{0}-2N\right)\Omega\overset{c}{\square}{}^{N}\phi+\sum_{k=1}^{N}\left(2w_{0}+4k-2-4N+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}{}^{k-1}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{N-k}\phi.$
(119)

However, in the above formula we cannot simplify further since in general the
conformal covariant derivative does not commute with the conformal box
operator. This issue here is even more complicated than in the case of
standard GR-covariant box operator and GR-covariant derivatives, because the
conformal weight of the result is different than the original of the scalar
field $\phi$. (In GR we have that the result of $\square^{N}$ on any tensorial
representation of the diffeomorphism group stays in the same representation.)
Moreover, we have also other various terms in the commutation rule for two
Wunsch conformal derivatives, which are related to the form of $X$ tensor. For
definiteness and exact formulas about this issue, we can refer the interested
reader to the original literature [105]. In general, the law of commutation of
conformal derivatives $\overset{c}{\nabla}_{\mu}$ (conformal Ricci identity)
states that

$\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]=\left.\left(\vphantom{\overset{c}{\nabla}_{\nu}}\left[\nabla_{\mu},\nabla_{\nu}\right]\right)\right|_{R_{\alpha\beta\rho\sigma}\to
C_{\alpha\beta\rho\sigma}}+\frac{1}{d-3}\nabla_{\alpha}C_{\mu\nu\gamma}{}^{\alpha}\frac{\delta\left(\left.\left(\delta_{c}\cdot\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)},$
(120)

where it is understood that it is applied on any conformal tensorial object
standing to the right. About this object it is assumed that under
infinitesimal conformal transformations it changes only with terms up to the
first derivative of the $\Omega$ parameter. The first term in the above
formula describes the same commutation as between ordinary GR-covariant
derivatives, but only with a substitution of Riemann tensor by the proper
tensor of conformal curvature, that is by Weyl tensor. All terms with any
contractions of Riemann tensor here are substituted by zero since Weyl tensor
is completely traceless. In the second term, we find a modification
proportional to the first contracted derivative of Weyl tensor and to the
conformal variation of the original tensorial object from which the part with
exactly the first derivatives of $\Omega$ is considered and stripped of. By an
explicit derivation, one can convince oneself that a coefficient of this term
is singular in $d=3$ spacetime dimensions. However, this does not lead to any
problem in $d=3$, since there the Weyl tensor explicitly vanishes and in the
formula (120) we are temporarily left only with the first term on the right
giving standard commutation as in GR, which finally in turn also vanishes
since the subsequent substitution of Riemann by Weyl tensor. Hence our
conclusion that in $d=3$ all Wunsch conformally covariant derivatives
completely commute. They also commute when the spacetime background is
conformally flat, that is when Weyl tensor vanishes, $C_{\mu\nu\rho\sigma}=0$.
This proves that on conformal background the sufficient conformal operator is
just $\overset{c}{\square}{}^{N}$.

Another feature of the formula (120) is that it shows that it is more
difficult to define the generalized tensor of curvature for conformal symmetry
as acting on some tensor representation on the right. For the first term on
the right hand side of (120) we see this without a problem since this is in
general a matrix which mixes Lorentz indices of the tensor representation, but
this is always the same tensor representation (scalars, vectors, tensors,
etc.) with the same number of indices before the commutator on the right of
$\left[\nabla_{\mu},\nabla_{\nu}\right]$ and in the result. However, the last
term in (120) is different since it mixes indices on the tensor representation
with one Lorentz index less (in a strict sense with one less conformal
covariant derivative) than the original tensor representation on which action
of the commutator
$\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]$ should be
expressed. This is because of the presence of one derivative on the Weyl
tensor in the last term in (120) and also consistently due to the fact that in
the infinitesimal transformation law we extract the part with precisely one
derivative acting on the $\Omega$ parameter. Then this means that here the
matrix in Lorentz indices operates on a smaller tensorial representation than
the original (one less conformal derivative on the main tensor, while at the
same time one more derivative on the parameter $\Omega$ of the
transformation). This difference between the two representations (original and
the smaller ones used in the last term in (120)) is the main problem in an
attempt of describing the uniform matrix of the tensor of generalized
conformal curvature here.

From the analysis of the formula (119), we see that the
$\overset{c}{\square}{}^{N}\phi$ operator does not in general behaves like a
conformally covariant object (except the case of $d=3$ spacetime dimensions).
Although it is a fact that this is a scalar operator with precisely $2N$
derivatives as considered on the flat spacetime background. Because of its
naturalness and simplicity of the transformation law in (119), we could use
it, following Wunsch, as a starting point for a successful construction of
completely conformally covariant differential operators on scalars with many
derivatives. For this purpose, we first can look at the case of small $N$ of
its power exponents.

It is easily seen that for $N=1$, the operator $\overset{c}{\square}\phi$
reproduces exactly the Ørsted-Penrose operator as discussed earlier. From the
technical side, one can just consider the equation for the vanishing of the
second coefficient in (119), namely

$2w_{0}+4k-2-4N+d=0$ (121)

for $N,\,k=1$ and find as a solution for the weigth $w_{0}=\frac{2-d}{2}$, the
same result as previously. The expansion of $\overset{c}{\square}\phi$ using
(75), (83) and (111) produces exactly the operator in a GR-explicit form

$\overset{c}{\square}\phi=\left(\square-\frac{d-2}{4(d-1)}R\right)\phi$ (122)

and this also proves that this is the only conformally covariant operator
possible here, with two derivatives, that is
$\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\nabla\Omega}=0$
(and also
$\left.\left(\delta_{c}\left(\overset{c}{\square}\phi\right)\right)\right|_{\Omega}=-\frac{d+2}{2}\Omega\overset{c}{\square}\phi$).
Moreover, as it is well known in $d=2$ the conformal operator coincides with
the standard GR-covariant box operator, namely we have
$\left.\overset{c}{\square}\phi\right|_{d=2}=\square\phi$. This is also
explained by the fact that in $d=2$ we do not see any singularity in (122) and
the trace of the Schouten tensor in (75) is not singular, because the interior
of the parenthesis in (75) vanishes after contracting with the contravariant
metric $g^{\mu\nu}$ and hence the Schouten tensor and also its trace overall
are not singular in $d=2$. This is in agreement with the previous discussion
of the form of the Schouten tensor in the case of two dimensions. In an
explicit expansion of the formula (122) after (83), we see that the trace of
the Schouten tensor, which is non-vanishing in $d=2$, is multiplied by the
factor $(d-2)$ and this is why in this dimension the Ricci scalar completely
does not participate in the construction of the conformally covariant box
operator. In $d=2$, the Schouten tensor and its trace are regular, but there
is no such contribution multiplying $R$, because the conformal weight $w_{0}$
must vanish in this dimension since it is a critical dimension for a two-
derivative scalar theory.

For the case $N=2$ we start seeing the advantage of using the formula with
conformal covariant derivatives. We derive that the square of the conformal
box operator $\overset{c}{\square}{}^{2}\phi$ behaves correctly under
conformal transformations. For this case, the law of transformation reads

$\delta_{c}\left(\overset{c}{\square}{}^{2}\phi\right)=\left(w_{0}-4\right)\Omega\overset{c}{\square}{}^{N}\phi+\sum_{k=1}^{2}\left(2w_{0}+4k-2-8+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}{}^{k-1}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2-k}\phi=$
$=\left(w_{0}-4\right)\Omega\overset{c}{\square}{}^{N}\phi+\left(2w_{0}+d-6\right)\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi+\left(2w_{0}+d-2\right)\nabla_{\gamma}\Omega\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\phi.$
(123)

However, one can see by explicit computation that for three conformal
consecutive derivatives $\overset{c}{\nabla}_{\mu}$ contracted in a form of
conformal box operator $\overset{c}{\square}$ and one derivative free, the
order does not matter and we have as a consequence that
$\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\phi=\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi$.
(This result about the commutation can be understood since in GR-covariant
case the result is proportional to the Ricci tensor only and does not involve
at all usage of the Riemann tensor. With conformal derivatives Ricci tensor
cannot appear, since it is a trace of Riemann, but we know that the tensor of
conformal curvature – commutator of two conformal covariant derivatives – Weyl
tensor is completely traceless. Moreover, from dimensional reasons and by
counting indices we cannot have any derivative of the Weyl tensor and we
cannot contract indices on Weyl tensor, so we must have that
$\left[\overset{c}{\square},\overset{c}{\nabla}{}^{\gamma}\right]\phi=0$.)
This means that the law of transformation we simplify to the case

$\delta_{c}\left(\overset{c}{\square}{}^{2}\phi\right)=\left(w_{0}-4\right)\Omega\overset{c}{\square}{}^{N}\phi+\left(4w_{0}+2d-8\right)\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi.$
(124)

Then the condition for conformal covariance again determines the weight in
this case. We find that $4w_{0}+2d-8=0$ implies that $w_{0}=\frac{4-d}{2}$. If
the conformal weight is assigned, then we have obviously that the operator has

$\delta_{c}\left(\overset{c}{\square}{}^{2}\phi\right)=-\frac{d+4}{2}\Omega\overset{c}{\square}{}^{2}\phi,$
(125)

so this is a conformally covariantly transforming expression. A bit longer and
tedious exercise shows that the operator $\overset{c}{\square}{}^{2}$ on
scalar fields with the weight $w_{0}=\frac{4-d}{2}$ reproduces completely the
Paneitz operator. In this case, the operator is not completely unique since we
have also an ambiguity in adding a conformally covariant scalar endomorphism.
As explained earlier we can add here
$C^{2}\phi=C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}\phi$ term with arbitrary
front coefficient. This is all consistent because the weight of the expression
$C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}=C^{\mu\nu\rho}{}_{\sigma}C_{\mu\nu\rho}{}^{\sigma}$
equals $-4$ the same as in the overall expression for
$\overset{c}{\square}{}^{2}$ (because of two contravariant metric tensors
$g^{\mu\nu}$ used in its construction). Therefore, the most general linear
operator acting on a scalar field $\phi$ possessing the conformal weight
$w_{0}=\frac{4-d}{2}$, containing up to 4 derivatives in the leading term
takes the form

$\overset{c}{\square}{}^{2}+\alpha C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma},$
(126)

where $\alpha$ is arbitrary real dimensionless and numerical constant. In that
case we acknowledge the fact that there is a one-parameter freedom in defining
the conformal operator. One can see that the singularity $(d-2)$ in the
denominator in the definition of the Schouten tensor in (75) is here
responsible for the fact that the Paneitz operator
$\overset{c}{\square}{}^{2}$ is not possible to be defined in the limit $d\to
2$, as noticed also earlier. Moreover, here we cannot construct any
differential operator with four derivatives, where all indices on these
conformal derivatives are not contracted with themselves, so we must consider
only scalar operators acting on scalars. Due to the commutation rules of
conformal covariant derivatives mentioned above, we have that when acting on a
scalar field

$\overset{c}{\nabla_{\gamma}}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla_{\beta}}\overset{c}{\nabla}{}^{\beta}\phi=\overset{c}{\nabla_{\gamma}}\overset{c}{\nabla_{\beta}}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\beta}\phi=\overset{c}{\nabla_{\gamma}}\overset{c}{\nabla_{\beta}}\overset{c}{\nabla}{}^{\beta}\overset{c}{\nabla}{}^{\gamma}\phi=\overset{c}{\square}{}^{2}\phi,$
(127)

hence there is only one unique order of conformal derivatives here. For the
cases of $N=1$ and $N=2$, we see that the situation is quite simple and
obvious. Due to the algebraic properties we do not have any other possibility
of constructing the conformal operator and in the transformation law (119) the
simplest operators constructed must already have good conformal transformation
laws (without derivatives of the parameter $\Omega$). This is caused by the
small size of the algebraic space of independent terms that we could write
here, like in (123) (where there is only one independent term
$\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\phi$ within the last two
terms) and in (127). As we shall see for higher $N$ the spaces are bigger,
there is more freedom and therefore the construction of conformally covariant
operators is also more complicated.

Starting with $N=3$ the construction becomes more intricate and the size of
the algebraic spaces of terms grows. We find that the operator
$\overset{c}{\square}{}^{3}$ in action on the scalar field $\phi$ with the
proper conformal weight $w_{0}$ is not conformally covariant. It transforms
with non-zero terms up to the first derivative of the $\Omega$ parameter.
According to the general formula in (119), for the case of $N=3$, we find

$\delta_{c}\left(\overset{c}{\square}{}^{3}\phi\right)=\left(w_{0}-6\right)\Omega\overset{c}{\square}{}^{3}\phi+\sum_{k=1}^{3}\left(2w_{0}+4k-14+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}{}^{k-1}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{3-k}\phi=$
$=\left(w_{0}-6\right)\Omega\overset{c}{\square}{}^{3}\phi+\left(2w_{0}-10+d\right)\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2}\phi+$
$+\left(2w_{0}-6+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi+\left(2w_{0}-2+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}{}^{2}\overset{c}{\nabla}{}^{\gamma}\phi.$
(128)

In the formula above we could use the commutation of three last derivatives
acting on $\phi$ using the previous laws which also apply here, namely
$\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\phi=\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi$
independently on the weight of the scalar field $\phi$, but this $\phi$ must
be with a definite weight (and with no derivatives of $\Omega$ parameter in
the infinitesimal conformal transformation law, so it cannot be for example
$\phi^{\prime}$ as used earlier). Then we would get that

$\delta_{c}\left(\overset{c}{\square}{}^{3}\phi\right)=\left(w_{0}-6\right)\Omega\overset{c}{\square}{}^{3}\phi+\left(2w_{0}-10+d\right)\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2}\phi+$
$+2\left(2w_{0}-4+d\right)\nabla_{\gamma}\Omega\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\phi$
(129)

and even using the a priori known value of the conformal weight (which is
determined from considerations of global conformal symmetry) and which is
$w_{0}=\frac{6-d}{2}$ we remain with the formula

$\delta_{c}\left(\overset{c}{\square}{}^{3}\phi\right)=\left(w_{0}-6\right)\Omega\overset{c}{\square}{}^{3}\phi-4\nabla_{\gamma}\Omega\left[\overset{c}{\nabla}{}^{\gamma},\overset{c}{\square}\right]\overset{c}{\square}\phi.$
(130)

Here we cannot commute derivatives and the conformal box operator (i.e. change
the order of $\overset{c}{\nabla}{}^{\gamma}$ and $\overset{c}{\square}$) for
free, because they all act on $\phi^{\prime}=\overset{c}{\square}\phi$, which
does not have a transformation law without derivative of the parameter of the
conformal transformations for value of $w_{0}=\frac{6-d}{2}$, which does not
coincide with the value $\frac{2-d}{4}$ required for the invariance of
$\overset{c}{\square}\phi$ for the case of $N=1$ only. This means that the
operator $\overset{c}{\square}{}^{3}$ is not conformally covariant and we need
to add modifications to it, if we think of a six-derivative conformal
differential operator. On different vein, here we see that the space of terms
that could appear in
$\left.\left(\delta_{c}\left(\overset{c}{\square}{}^{3}\phi\right)\right)\right|_{\nabla\Omega}$
is bigger, we cannot make all the independent terms there to cancel out by
solely choosing the weight $w_{0}$ and therefore the operator
$\overset{c}{\square}{}^{3}\phi$ is not conformally covariant one.

Following Wunsch, we can stick to the $\overset{c}{\square}{}^{3}$ as the
leading part of the operator, containing the highest number of conformal
derivatives, here 6, and just add some subleading terms in number of them.
These other terms due to dimensional reasons must contain powers of the
conformal gravitational curvatures or of its derivatives. For the sake of the
conformal formalism here we can use Weyl tensor (or Bach tensor – just to be
defined) and not other GR tensors of curvatures (like Riemann tensor, Ricci
tensor or Ricci scalar) and on these we should act with conformal derivatives
only. This is a necessary condition for writing the modification in the
conformal formalism, but it is not sufficient since we must also check that in
the special modification of the operator $\overset{c}{\square}{}^{3}$, being
the combination of various terms written in the formalism, the terms with
(only first) derivatives of $\Omega$ do cancel out. The similar form for this
minimal operator appeared also in works by Branson [66, 107, 108, 109, 110,
115, 111]. When the coefficient in front of $\overset{c}{\square}{}^{3}$ term
in this combination is fixed to unity (just for normalization of the total
operator), Wunsch found the following in a sense “minimal” six-derivative
conformal operator

$D_{(6)}^{\min}\phi=\left(\overset{c}{\square}{}^{3}+\frac{16}{(d-3)(d-4)}B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\right)\phi,$
(131)

where the Bach tensor $B^{\mu\nu}$ is defined as

$B_{\mu\nu}=\overset{c}{\nabla}{}^{\rho}\overset{c}{\nabla}{}^{\sigma}C_{\rho\mu\nu\sigma}.$
(132)

The Bach tensor is symmetric in its indices. However, it is conformally
covariant tensor only in $d=4$ spacetime dimensions (with the conformal weight
$w\left(B_{\mu\nu}\right)=-2$) and there it has other interesting properties
(like covariant conservation and it is an Euler-Lagrange tensor, so it is a
variational derivative with respect to the metric of some gravitational action
functional). Here, we use it just as a building block for construction of the
operator $D_{(6)}^{\min}$ and the most important is only its infinitesimal
conformal transformation law, which reads

$\delta_{c}\left(B_{\mu\nu}\right)=-2\Omega
B_{\mu\nu}+2(d-4)\nabla^{\gamma}\Omega\nabla^{\rho}C_{\rho\mu\nu\gamma}.$
(133)

Already here we notice that in $d=3$ special odd dimension, we have that
simply $D_{(6)}^{\min}=\overset{c}{\square}{}^{3}$, because there the Weyl and
Bach tensor explicitly vanish. Hence for construction of conformal operators
we can just use conformal derivatives acting on any object well transforming
under conformal transformations. These conformal derivatives still have to be
contracted since our conclusion relies here on complete commutativity of
conformal derivatives and the formula (119) for the conformal variation of the
powers of the conformal box operator. These operators
$\overset{c}{\square}{}^{N}$ in $d=3$ for any $N$ are still different than
standard GR $\square^{N}$ differential operators, because we need to use
Wunsch conformal covariant derivatives $\overset{c}{\nabla}_{\mu}$, instead of
GR-covariant $\nabla_{\mu}$ ones, with the exploitation of the construction
with the Schouten tensor (75), which in $d=3$ takes some non-zero finite
values. Explicit computation in $d=3$ based on (119) and with the weight
$w_{0}=\frac{2N-3}{2}$ shows that indeed general terms of the form
$\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{N-1}\phi$
in (119) do consistently cancel out giving us the result that
$\overset{c}{\square}{}^{N}$ is the conformal operator with the total weight
$w=-\frac{2N+3}{2}$ of the expression $\overset{c}{\square}{}^{N}\phi$.

In general, the operator $D_{(6)}^{\min}$ has the leading term in derivatives,
in expansion around flat spacetime, when we can neglect all gravitational
curvatures, as $\square^{3}$, so with precisely six derivatives. In a general
framework, we have the following transformations of this operator under
infinitesimal conformal changes:

$\delta_{c}D_{(6)}^{\min}\phi=\left(w_{0}-6\right)\Omega
D_{(6)}^{\min}\phi=-\frac{d+6}{2}\Omega D_{(6)}^{\min}\phi$ (134)

with $w_{0}=\frac{6-d}{2}$ – the conformal weight of the basic scalar field
$\phi$ here.

One notices that the expression for the minimal six-derivative conformal
operator according to Wunsch in (131), is singular in the case of dimensions
$d=3$ and $d=4$. We are more concerned with the last case here. This was also
confirmed by our studies in the previous sections (IV), where for the
coefficients of expansion, in front of some terms, we have found the
singularity $\frac{1}{d-4}$ near the dimension $d=4$. Therefore, we have
another confirmation of the theorem by Graham [112, 113] that in $d=4$ we
cannot construct a conformally covariant differential operator, with the
leading term $\square^{3}=\partial^{6}$ on flat spacetime in Cartesian
coordinates. Moreover, the same applies to the dimensions $d=2$ where again
the operator with six derivatives and the leading term $\square^{3}$ on flat
spacetime background cannot be constructed; this time due to problems with the
limit $d\to 2$ in the definition of the Schouten tensor. One can here put a
conjecture (following Branson and Wunsch [107, 105]) that the operator with
$2N$ derivatives cannot be constructed in all even spacetime dimensions
smaller than $2N$, that is for $d<2N$, while the dimension $d=2N$ is the
critical.

We may comment here on the general structure of singularities in the
definition of the operators, like the generalized Hamada operator. From the
arguments of the previous section, and also from the results derived in the
tractor formalism [124, 125], we know that the operators are undefined in the
first even subcritical dimension, that is for $d=2N-2$. For $N\geqslant 3$ we
see that these dimensions are greater than 2. In the general construction of
the conformally covariant operators we can use $\overset{c}{\square}{}^{N}$
supplemented with some other terms built out with Weyl tensors, Bach tensors
and conformally covariant derivatives acting on them or on the scalar field
$\phi$, standing the most to the right of each expressions. In the definition
of these objects and also of Schouten tensor, we see that there is never a
place where we could generate singularities, like $\frac{1}{d-4}$ for Hamada,
or $\frac{1}{d-6}$ for the operator with 8 derivatives, etc., since we only
contract metrics which produces positive powers of dimensions $d$ in
numerators. (In the definition of Schouten tensor we see singularities only in
$d=1$ and $d=2$; the definition of conformal covariant derivatives as in (96)
is apparently free from any singularities; the $X$ tensor does not develop any
singularity, and in the conformal weights $w_{0}$ we see the dimensions
appearing only in the numerator. The law for commutation of conformal
covariant derivatives as in (120) contains only $\frac{1}{d-3}$ explicit
singularity. Moreover, the Weyl and Bach tensors can be defined without any
problems in any dimensions $d\geqslant 4$.) Still, in the construction of
conformal operators we must see the singularity in these first even
subcritical dimensions, because the operators are not constructible in such
situations. This singularity must show up in some rational functions of the
dimension $d$ of spacetime. Because of the nature of algebraic operations
described below, these can be only rational functions. The only reason why
singularity can appear here is due to a solution of some algebraic linear
system of equations for coefficients in front of additional terms that have to
be added to $\overset{c}{\square}{}^{N}$ to construct a minimal conformally
covariant operator with $2N\geqslant 6$ derivatives in a form
$D_{(2N)}^{\min}=\overset{c}{\square}{}^{N}+\ldots$. In other words, for
operators with $2N\geqslant 6$ derivatives, we need terms besides the
$\overset{c}{\square}{}^{N}$ and all these terms are constructed in Wunsch
conformal way (having transformation law only up to the first derivatives of
$\Omega$ parameter) and in their construction we do not see any singularity
for dimension $d=2N-2$. However, all these additional terms must come with
some definite coefficients in order to construct a minimal operator
$D_{(2N)}^{\min}$. These coefficients are determined from the condition that
$\left.\left(\delta_{c}\left(D_{(2N)}^{\min}\right)\right)\right|_{\nabla\Omega}=0$.
This last requirement as a tensor equation can be expanded in some basis and
therefore we are led to the linear system of equations for apriori unknown
coefficients. By solving such an algebraic system we determine the front
coefficients and in this procedure we may exploit inverse functions, where the
dimension $d$ appeared only in numerators, hence we may end up with
coefficients being rational functions of the dimension $d$. These rational
functions finally may contain the singularities like $\frac{1}{d-4}$ or
$\frac{1}{d-6}$, etc.. This explains the reason why the singularities like
$\frac{1}{d-4}$ must appear in the subcritical dimensions and what is the
algebraic origin of them. Such singular coefficients for the definition of
minimal operators $D_{(2N)}^{\min}$ are inevitable for $N\geqslant 3$.
Moreover, since the solution of some system of equations is a necessary
ingredient here, it is very probable that there does not exist a procedure or
a new definition of conformal derivative or $\overset{c}{\square^{\prime}}$
that would produce $D_{(2N)}^{\min}$ directly, without algebraic solving for
the coefficients in the basis of all terms that could be potentially added
here.

Following Wunsch, we can now discuss the addition of other operators to the
minimal form of the operator $D_{(6)}^{\min}$ in arbitrary dimension. These
must be terms with subleading number of derivatives, but still the total
energy dimension of the term must agree with 6 as it is dimension of
$\square^{3}$. These terms may possibly contain still 4, 3, 2 or one conformal
covariant derivative acting on the scalar field $\phi$ and we also allow for
endomorphic terms that do not contain any derivatives at all acting on the
scalar field. Due to the fixed energy dimension of terms in such combination
we can use terms with one, two or three powers of curvatures and possibly also
with the derivatives acting on it. Actually, we see that because of the
dimensionality of curvature tensor, we cannot have 5 derivatives on the scalar
and further insights into the construction of conformally covariant terms show
that we can have only 2, 1 or no derivatives acting on scalar in such terms.
For the sake of the conformally covariant formalism (up to first derivatives
in $\Omega$) we shall use only conformally covariant derivatives and only Weyl
tensor as curvatures. Since in the formula (132) we defined Bach tensor, we
can use it although it is a fully conformal tensor only in $d=4$ due to the
transformation law in (133). So we could treat Bach tensor as another
expression which transforms in general up to first derivatives of $\Omega$. We
consider the following terms:

$D_{1}\phi=\overset{c}{\nabla}_{\mu}\left[\left(C^{\mu}{}_{\rho\sigma\kappa}C^{\nu\rho\sigma\kappa}-\frac{1}{4}g^{\mu\nu}C_{\rho\sigma\kappa\lambda}C^{\rho\sigma\kappa\lambda}\right)\overset{c}{\nabla}_{\nu}\phi\right]+\frac{(d-4)(d-6)}{4(d-3)^{2}}\overset{c}{\nabla}_{\mu}C^{\mu\rho\sigma\kappa}\overset{c}{\nabla}_{\nu}C^{\nu}{}_{\rho\sigma\kappa}\phi$
(135)

and

$D_{2}\phi=2(d-10)\overset{c}{\nabla}_{\mu}\left[C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}\overset{c}{\nabla}{}^{\mu}\phi\right]+(d-6)\overset{c}{\square}\left(C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}\right)\phi.$
(136)

In the formula above, we explicitly showed the position and presence of the
scalar field $\phi$ not to avoid some ambiguities in order and the range of
derivatives, where they act only on the first tensor they find on their right
or otherwise their range of action is marked by parentheses. We use above
compact formulas with unexpanded form of the action of differential operators,
although, of course, here we could use the Leibniz rule to write all of them
in the fully expanded form without parentheses marking the action of
derivatives. Each of these terms (namely $D_{1}$ and $D_{2}$) is separately
conformally covariant provided that $w_{0}=\frac{6-d}{2}$ and $d\geqslant 4$
and moreover the total conformal weight of these two expressions is the same
and equal to $w=-\frac{6+d}{2}$. Therefore, we can consider a following
“nonminimal” combination as a proper more generalized differential operator
with up to six derivatives

$D=D_{(6)}^{\min}+\alpha D_{1}+\beta D_{2}$ (137)

with arbitrary real coefficients $\alpha$ and $\beta$. Still as a result of
tedious computation we find that

$\delta_{c}\left(D\phi\right)=-\frac{6+d}{2}\Omega D\phi,$ (138)

where it is important that all terms linear in the first derivative of
$\Omega$ cancel out separately in $D_{(6)}^{\min}$, $D_{1}$ and $D_{2}$. One
notices that in both formulas (135) and (136) the last terms on the right hand
side vanish in dimension $d=6$. By looking at the formulas (135) and (136) one
can easily understand that we have in each combinations after Leibniz
expansion terms with two, one or no conformal derivatives on the scalar field
$\phi$. It is easy to understand why terms with four and three derivatives on
the scalar field $\phi$ cannot be constructed and added here. In such a case
the curvature part of the term had to be Weyl tensor or the first derivative
of the Weyl tensor. But due to antisymmetry of the indices on the Weyl tensor,
we have that the presence of four or three derivatives acting in a sequence on
the scalar field is superfluous. This is because the indices on these
derivatives must be contracted with Weyl tensor to form a scalar from the
point of view of GR and antisymmetry (in at least three indices found on
derivatives acting on scalar we can find a pair of them coinciding with a pair
of antisymmetric indices on Weyl) implies that we should form a commutator of
these derivatives. Finally this leads to a generation of one or more Weyl
curvature tensor and hence the number of derivatives is consequently reduced
to 2, 1 or 0 in action on the scalar field. Additionally, we remind the reader
that only in the very special combinations (135) and (136) with very specific
numerical frontal coefficients (dependent on the dimension $d$ in a rational
way as a result of solving some system of algebraic equations), we find that
the whole expressions in (135) and (136) transform covariantly under conformal
transformations. Single terms there separately do not possess such properties.

Another observation pertaining to terms found in two combinations above for
$D_{1}$ and $D_{2}$ is that the last terms found in both, which are as
mentioned proportional to $(d-6)$, are purely endomorphic since the
derivatives there act only on Weyl tensors. These are absent in the critical
dimension $d=6$ for six-derivative operators as conjectured earlier for any
number $N$ of derivatives. We also verified in $d=4$ for the Paneitz operator
that endomorphic terms appear only separately as some non-minimal fully
endomorphic additions and in this critical dimension do not enter in the
construction of the “minimal” conformal operator $\overset{c}{\square}{}^{2}$
with $2N=4$ derivatives in the leading term. We see that also in the
$D_{(6)}^{\min}$ operator as chosen by Wunsch the $\overset{c}{\square}{}^{3}$
operator must be supplemented by the non-endomorphic term proportional to
$B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}$ and its frontal
coefficient does not vanish in $d=6$ spacetime dimensions. This is obviously
consistent with our conjecture since these terms do contain derivatives on the
scalar field $\phi$ and their addition is crucial even in the critical
dimension $d=6$ for a six-derivative operator.

Finally, we can add here some purely endomorphism terms. They do not contain
any derivatives on the scalar field $\phi$. However, the derivatives can be in
action on Weyl tensors. We find in general dimension $d$ three such terms,
which we can write in the following combination

$D_{{\rm
end}}=\gamma_{1}C_{\mu\nu\rho\sigma}C^{\rho\sigma}{}_{\kappa\lambda}C^{\kappa\lambda\mu\nu}+\gamma_{2}C^{\mu\nu\rho\sigma}C_{\mu\kappa\lambda\sigma}C_{\nu}{}^{\kappa\lambda}{}_{\rho}+\gamma_{3}T.$
(139)

They of course have very stringent relation to terms defining conformal
gravitation with six-derivatives, so in the six-dimensional spacetime. The
scalar $T$ is defined as

$T=\frac{10-d}{2}\left(\overset{c}{\nabla}_{\mu}C_{\nu\rho\sigma\kappa}\overset{c}{\nabla}{}^{\mu}C^{\nu\rho\sigma\kappa}-\frac{4(d-2)}{(d-3)^{2}}\overset{c}{\nabla}_{\mu}C^{\mu\rho\sigma\kappa}\overset{c}{\nabla}_{\nu}C^{\nu}{}_{\rho\sigma\kappa}\right)-2\overset{c}{\square}\left(C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\right).$
(140)

While the two first terms in (139) have obviously very good conformal
transformation laws with the weight given by $w=-6$, to prove that the last
scalar $T$ also transforms covariantly (without first derivatives of $\Omega)$
is a bit more involved. But at the end of the day, one finds again that $w=-6$
and for the scalar $T$ the infinitesimal conformal transformation law is
$\delta_{c}T=-6\Omega T$ with no terms with derivatives of the $\Omega$
parameter. This means that for the expression $D_{{\rm end}}\phi$, we find the
transformation law: $\delta_{c}\left(D_{{\rm
end}}\phi\right)=-\frac{6+d}{2}\Omega D_{{\rm end}}\phi$ for arbitrary real
values of the parameters $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$.
Therefore, the most general combination which will give an expression linear
in the scalar field $\phi$ is

$D_{{\rm gen}}\phi=D\phi+D_{{\rm end}}\phi$ (141)

and from this the most general conformal operator with leading terms of six
derivatives takes the form

$D_{{\rm gen}}=D+D_{{\rm end}}=D_{(6)}^{\min}+\alpha D_{1}+\beta
D_{2}+\gamma_{1}C_{\mu\nu\rho\sigma}C^{\rho\sigma}{}_{\kappa\lambda}C^{\kappa\lambda\mu\nu}+\gamma_{2}C^{\mu\nu\rho\sigma}C_{\mu\kappa\lambda\sigma}C_{\nu}{}^{\kappa\lambda}{}_{\rho}+\gamma_{3}T$
(142)

with five arbitrary real constants $\alpha$, $\beta$, $\gamma_{1}$,
$\gamma_{2}$ and $\gamma_{3}$.

One could ask whether in the list of endomorphic terms in (139) we could have
also some with four derivatives on one Weyl tensor. Obviously three
derivatives on Weyl tensor do not match the total energy dimensionality of the
term like $\square^{3}$. The reason for the absence of four derivatives on one
Weyl tensor is the same antisymmetry argument as put above after the formula
(138). The term is reduced to ones having two less derivatives since all
indices on derivatives are again contracted with Weyland antisymmetry is in
action here. Eventually, here we can have terms with two Weyl tensors and two
conformal derivatives (they could act on one Weyl tensor, then they could be
contracted to form $\overset{c}{\square}$; or they act on two different Weyl
tensor), present in the scalar $T$ or terms with only three Weyl tensors with
all indices properly contracted. These last ones give rise to two
possibilities in the general dimension of construction of Weyl cube terms,
which of course have obvious conformal transformation laws. All terms present
in the definition of the scalar $T$ in (140) are again written in the
unexpanded form where we could use Leibinz rule for conformal derivatives
(working the same as in the standard GR case). The terms that are present
there are the most general constructible all consistent with the symmetries of
the Weyl tensor and Bianchi identities.

When the scalar and conformally covariant term $T$ is properly densitized and
integrated over the whole spacetime volume integral, one gets a prototypical
integral for the action of conformal gravitation with the leading term of six
derivatives. Since $w(T)=-6$, then obviously the action is conformally
invariant only when $d=-w(T)=6$. Only then we can speak of a conformal
gravitation with six-derivative terms. We have explicitly

$S_{{\rm grav},6{\rm d}}=\int
d^{6}x\sqrt{|g|}\alpha_{C}\left(2\left(\overset{c}{\nabla}_{\mu}C_{\nu\rho\sigma\kappa}\overset{c}{\nabla}{}^{\mu}C^{\nu\rho\sigma\kappa}-\frac{16}{9}\overset{c}{\nabla}_{\mu}C^{\mu\rho\sigma\kappa}\overset{c}{\nabla}_{\nu}C^{\nu}{}_{\rho\sigma\kappa}\right)-2\overset{c}{\square}\left(C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\right)\right),$
(143)

where $\alpha_{C}$ is a positive coupling constant (for reflection-positivity
of the theory in Euclidean setup). By using the formula for Ørsted-Penrose
operator (122) and integrating by parts ordinary GR-covariant box operator we
find an equivalent but a simpler form

$S_{{\rm grav},6{\rm d}}=\int
d^{6}x\sqrt{|g|}\alpha_{C}\left(2\left(\overset{c}{\nabla}_{\mu}C_{\nu\rho\sigma\kappa}\overset{c}{\nabla}{}^{\mu}C^{\nu\rho\sigma\kappa}-\frac{16}{9}\overset{c}{\nabla}_{\mu}C^{\mu\rho\sigma\kappa}\overset{c}{\nabla}_{\nu}C^{\nu}{}_{\rho\sigma\kappa}\right)+\frac{4}{5}RC_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\right).$
(144)

We here remind that we could neglect total derivatives and total box
operators, when they are made with GR-covariant derivatives. But we cannot
proceed with the same about total conformal covariant derivatives and total
conformal box operators $\overset{c}{\square}$. However, in the last formula
not all the terms look conformally covariant (even with the proviso that they
are conformal only up to the first derivatives of $\Omega$ parameter of
conformal transformation). The presence of the Ricci scalar is here essential,
however under the action integral it transforms up to total derivatives of the
parameter $\Omega$ hence can be used, although, of course, the expression in
(143) looks better from the point of view of conformal formalism.

The action (143) describes conformal gravitation in $d=6$ spacetime dimension.
It can also be used in Euclidean signature where it would define a six-
dimensional conformal shape dynamics. As obvious from the term in (143) this
is a six-derivative theory, where the graviton kinetic term has naturally six
derivatives in its action. And hence the propagator is even more suppressed in
comparison to the four-derivative Stelle gravitation considered as a flag
model of higher-derivative gravitational theories in $d=4$ dimensions. The
last term in (144) of the type $RCC$ obviously describes only the non-trivial
graviton interactions on the flat spacetime backgrounds. Moreover, two terms
with $\gamma_{1}$ and $\gamma_{2}$ in (139) can be as well added to the
6-dimensional conformal gravitation action with arbitrary real coefficients.
They do not influence the flat spacetime graviton propagator, however give
rise to new graviton interaction vertices. Altogether the theory defined by
volume integral of (139) in $d=6$ is conformally invariant pure gravitational
theory that is also fully renormalizable on the quantum level. The interested
reader can peruse following references (and citations therein) to the works of
Maldacena and others on 6-dimensional conformal gravitation [32, 103, 104].

Here, we understand that the sense in which the operator $D_{(6)}^{\min}$ is
minimal is largely arbitrary since we could add the part of operators $D_{1}$
or $D_{2}$ or any parts from endomorphisms with whatever real coefficients and
still the resulting operator is conformally covariant. Probably the motivation
behind this choice of $D_{(6)}^{\min}$ as in (131) was that its formula in
this shape is more compact. As clearly visible from formulas for $D_{1}$,
$D_{2}$ and $D_{{\rm end}}$ they contain terms which are quadratic in
conformal curvatures (Weyl tensors) and finally in $D_{{\rm end}}$ there are
terms cubic in curvature in terms multiplied by $\gamma_{1}$ and $\gamma_{2}$.
In the scalar $T$ we find instead only terms quadratic in curvatures. Finally,
the Bach tensor is understood as an expression linear in Weyl curvature;
actually two conformal derivatives act here on Weyl tensor according to the
definition in (132). Hence Bach tensor is also an expression linear in
(conformal) curvature. This distinguishes the term of addition to
$\overset{c}{\square}{}^{3}$ in (131) from other terms in $D_{1}$, $D_{2}$ and
$D_{{\rm end}}$ since only in the first case (with Bach tensor) the necessary
addition is linear in curvatures, while in last cases it is quadratic or cubic
in conformal Weyl tensor. Therefore the differential operator as proposed by
Wunsch $D_{(6)}^{\min}$ in (131) can be viewed as minimal in a sense, because
it contains only terms with no conformal curvature
$\overset{c}{\square}{}^{3}$ and other compensating terms are only linear in
conformal curvature (but with the possibility that conformal derivatives act
on this curvature). In this sense the addition of the Bach tensor in (131)
with the prescribed coefficient, dependent on dimension $d$, is a unique
choice for the minimal operator $D_{(6)}^{\min}$ with only linear in curvature
correction terms.

Therefore, the choice of the minimal operator is never unique and invariantly
and for comparisons we can only consider the most general operator as in (142)
and count its free parameters. We see that in the sector of proper
differential operators we have two-parametric freedom exemplified by two
constants $\alpha$ and $\beta$. Finally, in the sector of endomorphisms we
have a bigger three-parameter freedom with three constants $\gamma_{1}$,
$\gamma_{2}$ and $\gamma_{3}$. We compare this with our findings from the
previous sections and found a complete agreement with the number of free real
parameters needed to define the most general operator $D_{{\rm gen}}$ in
arbitrary dimensions $d>4$, which contains up to 6 derivatives acting on the
scalar field $\phi$, possibly with other terms giving some curvature
corrections to this formula.

Before we emphasized the singularities in the construction for generalized
conformal operators on a general gravitational backgrounds in some special
even dimensions. However, here we remark that in $d=4$ the operators with six
derivatives in the kinetic terms can still be well defined and considered on
flat or conformally flat spacetimes, the problem is only with their
generalization to properly conformally curved manifold setting, where the Weyl
curvature does not vanish. In that case for the Hamada or Wunsch operators, we
find the simple form, namely $\overset{c}{\square}{}^{3}$, without any
conformal curvature correction as in (131). The expansion of this
$\overset{c}{\square}{}^{3}$ differential operator in terms of standard GR-
covariant derivatives, standard GR-curvatures is without any problem in
dimension $d=4$ and hence it perfectly exists there. As seen in (131) the
problem in $d=4$ was really with conformal curvature corrections needed on the
general background. But when the background is conformally flat then these
conformal objects do not arise and their singular frontal coefficients are
here immaterial. (For example, one can take the smooth limit $d\to 4$, staying
all the time on conformal backgrounds $C_{\mu\nu\rho\sigma}=0$, and the
expression for the operator does not develop any singularity.) Moreover, the
condition of always staying on conformally flat backgrounds is preserved by
conformal transformations and hence these operators like
$\overset{c}{\square}{}^{3}$ do transform well (i.e. conformally covariantly)
under any conformal transformation in $d=4$ as well. The background is still
conformally flat when we change the metric and transform the operator, but of
course the standard curvature tensors as in GR, they are not left invariant.

##  VII New construction of generalized Hamada operator

In the previous section we discussed the construction of the minimal six-
derivative conformal operator acting on the scalar field along the lines of
the developments done by Wunsch and Graham. Here in this last main section of
the paper, we plan to present a new different derivation of an equivalent form
for the Hamada operator. We accept the fact that just the operator
$\overset{c}{\square}{}^{3}$ is not conformally covariant. In previous
discussion we had decided to add correcting curvature terms. However, there
exists also another formal solution to this problem. Since generally
conformally covariant derivatives do not commute, we may try to construct
operators with six derivatives but with a different order of them. There is a
hope that in a specific combination of such terms we will be able to cancel
terms with the first derivatives of the $\Omega$ parameter. In this way we
expect to derive all terms which are non-endomorphic in the general form of a
six-derivative operator. Of course, endomorphic terms can be added easily just
from $D_{{\rm end}}$. It is well known that curvature terms arise as the
result of doing commutation of covariant derivatives. Here we want to somehow
invert this process and see whether the six-derivative Hamada operator can be
written without curvature terms but as a combination of terms with different
orders of conformally covariant derivatives. We also want to understand when
this process is possible.

To start the explanation of this way of deriving the terms we must remind the
reader that the cube of the conformally covariant scalar box operator
$\overset{c}{\square}{}^{3}$ we view as a particular order of six conformal
covariant derivatives. Namely, we have

$\overset{c}{\square}{}^{3}=g^{\mu\nu}g^{\rho\sigma}g^{\kappa\lambda}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\sigma}\overset{c}{\nabla}_{\kappa}\overset{c}{\nabla}_{\lambda}=\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\rho}.$
(145)

For definiteness of our solutions, we can keep the coefficient of this leading
term $\overset{c}{\square}{}^{3}$ with the default order of conformal
derivatives $\overset{c}{\nabla}_{\mu}$ set to one in our procedure. Due to
the fact that the metric tensor is also covariantly conserved by conformal
derivatives $\overset{c}{\nabla}_{\mu}g_{\nu\rho}=0$, the position of
contracted indices and their order is not important in (145).

Since we have in general six indices on six conformal derivatives, then the
number of completely arbitrary metric contractions initially is
$(6-1)!!=5!!=15$. This number of combinations is too big since it does not
take into account the symmetries that exist between various terms, when we do
not have to commute derivatives since for example the indices on them are
symmetrized by the usage of contravariant metric tensor as in the formula
above. For this one has to carefully look at the commutation law for conformal
derivatives as this was stipulated in the formula (120). Based on this formula
one sees that two (the most internal) conformal derivatives acting on the
scalar field $\phi$ must commute, that is we have

$\overset{c}{\nabla}_{\kappa}\overset{c}{\nabla}_{\lambda}\phi=\overset{c}{\nabla}_{\lambda}\overset{c}{\nabla}_{\kappa}\phi.$
(146)

This is because standard GR-covariant derivative on a scalar commute and here
we cannot construct any contraction of the Weyl tensor due to counting of the
number of indices and due to energy dimensionality reasons. However, we warn
the reader that it is not true for example that higher order conformal
derivatives commute, and we have

$\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\sigma}\overset{c}{\square}\phi\neq\overset{c}{\nabla}_{\sigma}\overset{c}{\nabla}_{\rho}\overset{c}{\square}\phi,$
(147)

since now some contraction of the Weyl tensor can be done with the indices on
derivatives resulting from splitting of the box operator
($\overset{c}{\square}=\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\rho}$).
Moreover, since the last two conformally covariant derivatives on the scalar
do commute, then by considerations of various orders of all 6 derivatives and
by doing the commutation of them we never produce endomorphic terms in which
all derivatives would be substituted by Weyl curvature tensors (or their
derivatives). In a sense, we can never replace the last two derivatives by
Weyl tensor, since their commutator vanishes. Therefore, the endomorphic
terms, like the scalar $T$ and two contractions of Weyl cube have to appended
by hand for the most general operator. Here we construct only the part of the
operator which is non-endomorphic, i.e. contains derivatives acting on the
scalar field $\phi$.

Another corollary from the commutation law (120) is that the following
commutator vanishes, when acting directly on the scalar $\phi$:

$\left[\overset{c}{\square},\overset{c}{\nabla}_{\sigma}\right]=0.$ (148)

The first part of the commutation law does not apply here, since in standard
GR this commutator is
$\left[\square,\overset{c}{\nabla}_{\mu}\right]=R_{\mu\nu}\nabla^{\nu}$ and
for Weyl tensor all traces, like Ricci tensor to Riemann tensor, vanish. The
absence of the second part is again due to the reason that we cannot contract
indices in the second part of the law in (120) when acting with derivatives on
Weyl tensor, when only index $\Omega$ has to be free, but we have also
constraints due to dimensionality of possible terms. Again, in the more
general situation, like for the commutator

$\left[\overset{c}{\square},\overset{c}{\nabla}_{\sigma}\right]\overset{c}{\square}\phi$
(149)

we cannot simplify since the second term on the right hand side of (120) can
be now constructed in which additional indices are contracted with these on
conformal derivatives resulting after the splitting of the most internal
$\overset{c}{\square}$ operator.

Furthermore, we have other following commutators between single conformal
derivatives that vanish here:

$\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]\phi=\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]\overset{c}{\nabla}{}^{\mu}\phi=\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]\overset{c}{\nabla}{}^{\nu}\phi=\left[\overset{c}{\nabla}_{\mu},\overset{c}{\nabla}_{\nu}\right]\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\phi=0$
(150)

and the reasons for their vanishing are all of the same type as discussed
above. Due to the commutation law and the fact expressed in (147) we see that
the numer of independent terms can be bigger than in the corresponding case
for standard GR-covariant derivatives, where we have significantly more
reductions. Since two conformal derivatives do not in general commute when
acting on a GR-scalar like $\overset{c}{\square}{}^{k}\phi$ for $k>0$, then
such terms are independent and cannot be simplified as it would be the case in
GR. This is because in the transformation law (120) we can pull out some
terms, derivatives and indices from the general expression
$\overset{c}{\square}{}^{k}\phi$ after splitting.

Using various symmetries and looking carefully at the commutation law in
(120), one arrives at an irreducible basis of terms with six derivatives,
which has precisely 6 terms. Therefore all the terms in the combinations, we
are looking for, can be presented as

${\cal
O}=C^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}}\overset{c}{\nabla}_{\rho_{1}}\overset{c}{\nabla}_{\rho_{2}}\overset{c}{\nabla}_{\rho_{3}}\overset{c}{\nabla}_{\rho_{4}}\overset{c}{\nabla}_{\rho_{5}}\overset{c}{\nabla}_{\rho_{6}},$
(151)

where the $C$ tensor is

$C^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}}=\alpha_{1}g^{\rho_{1}\rho_{2}}g^{\rho_{3}\rho_{4}}g^{\rho_{5}\rho_{6}}+\alpha_{2}g^{\rho_{1}\rho_{3}}g^{\rho_{2}\rho_{4}}g^{\rho_{5}\rho_{6}}+\alpha_{3}g^{\rho_{1}\rho_{4}}g^{\rho_{2}\rho_{3}}g^{\rho_{5}\rho_{6}}+\alpha_{4}g^{\rho_{1}\rho_{4}}g^{\rho_{2}\rho_{5}}g^{\rho_{3}\rho_{6}}+$
$+\alpha_{5}g^{\rho_{1}\rho_{5}}g^{\rho_{2}\rho_{4}}g^{\rho_{3}\rho_{6}}+\alpha_{6}g^{\rho_{1}\rho_{5}}g^{\rho_{2}\rho_{6}}g^{\rho_{3}\rho_{4}}$
(152)

and it defines all 6 basis elements. In other words, our combination for the
conformal differential operator with six derivatives is

${\cal
O}=\alpha_{1}\overset{c}{\square}{}^{3}+\alpha_{2}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\square}+\alpha_{3}\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square}+\alpha_{4}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\rho}+$
$+\alpha_{5}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\rho}+\alpha_{6}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}.$
(153)

As mentioned above for definiteness we can take $\alpha_{1}=1$ since this only
changes the overall normalization of the operator.

In line with what was mentioned extensively in the previous section, in the
construction of the conformally covariant operators using formalism of Wunsch
one must finally check whether the terms with precisely the first derivative
of the $\Omega$ parameter cancel out in the combination. This will signify
that the operator is conformally covariant and transforms only linearly with
$\Omega$ undifferentiated, so the operator ${\cal O}$ receives only the
conformal weight. To check this we need to know $\left.\left(\delta_{c}{\cal
O}\right)\right|_{\nabla\Omega}$. Based on the consideration of the previous
section this is simply given by

$\left.\left(\delta_{c}{\cal
O}\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}}{}^{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\beta_{5}}C^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}},$
(154)

hence for cancellation we only require that the contraction of the $C$ tensor
with the $X$ tensor explicitly vanish, when this expression with six free
indices $(\gamma,\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5})$ is
expanded within some basis of terms. For the explicit computation we can use
the formula for the $X$ tensor (97) with $N=7$ (in our notation from the
previous section, where we also have to shift indexing of indices $\rho$: from
the sequence $\rho_{2}$ to $\rho_{7}$ into a different one from $\rho_{1}$ to
$\rho_{6}$). Then we are left with the expression with free indices
$\gamma,\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5}$. However, since it
is contracted with a sequence of $N-2=5$ conformal derivatives on the right of
it, this tensor also inherits the symmetry of exchange of indices between
$\beta$. Here again due to the properties of conformal covariant derivatives
and their commutation law (when they act on the scalar field $\phi$ on the
most right), we have some identities between various terms, when we expand the
result in the basis of all orders of five conformal derivatives. We remark
that the tensor contraction $X$ with $C$ (apart from some numerical
coefficients) is built out only with the contravariant metric tensor of
spacetime $g^{\mu\nu}$. Hence, again the number of possible orders and the
combinations here for $6$ indices is $(6-1)!!=5!!=15$. This is before the
symmetries of five conformal derivatives are taken into account. Similarly to
the previous case one finds exactly $9$ identities and the number of elements
in the independent basis here for writing such terms is $6$. This is not a
mere coincidence since our system for six unknown coefficients $\alpha_{i}$
for $i=1,\ldots,6$ has to be determined by six equations to remove the
possible situation of algebraic degeneracy or inconsistency. These identities
are summarized below:

$1)\quad
g^{\gamma\beta_{1}}g^{\beta_{2}\beta_{4}}g^{\beta_{3}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{1}}g^{\beta_{2}\beta_{3}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\to\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2}$
$2)\quad
g^{\gamma\beta_{1}}g^{\beta_{2}\beta_{5}}g^{\beta_{3}\beta_{4}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{1}}g^{\beta_{2}\beta_{3}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}=\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\to\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2}$

$3)\quad
g^{\gamma\beta_{2}}g^{\beta_{1}\beta_{4}}g^{\beta_{3}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{2}}g^{\beta_{1}\beta_{3}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\to\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square}$
$4)\quad
g^{\gamma\beta_{2}}g^{\beta_{1}\beta_{5}}g^{\beta_{3}\beta_{4}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{2}}g^{\beta_{1}\beta_{3}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}=\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\to\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square}$
$5)\quad
g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{5}}g^{\beta_{2}\beta_{4}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{4}}g^{\beta_{2}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\to\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}$
$6)\quad
g^{\gamma\beta_{4}}g^{\beta_{1}\beta_{2}}g^{\beta_{3}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{2}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\square}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\nu}=\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\nu}\to\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}$
$7)\quad
g^{\gamma\beta_{5}}g^{\beta_{1}\beta_{2}}g^{\beta_{3}\beta_{4}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{2}}g^{\beta_{4}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\square}{}^{2}\overset{c}{\nabla}{}^{\gamma}=\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\gamma}\to\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}$
$8)\quad
g^{\gamma\beta_{5}}g^{\beta_{1}\beta_{3}}g^{\beta_{2}\beta_{4}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{4}}g^{\beta_{1}\beta_{3}}g^{\beta_{2}\beta_{5}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\gamma}\to\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\nu}$
$9)\quad
g^{\gamma\beta_{5}}g^{\beta_{1}\beta_{4}}g^{\beta_{2}\beta_{3}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}\to
g^{\gamma\beta_{4}}g^{\beta_{1}\beta_{5}}g^{\beta_{2}\beta_{3}}\overset{c}{\nabla}_{\beta_{1}}\overset{c}{\nabla}_{\beta_{2}}\overset{c}{\nabla}_{\beta_{3}}\overset{c}{\nabla}_{\beta_{4}}\overset{c}{\nabla}_{\beta_{5}}$

equivalent to

$\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\gamma}=\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\gamma}\to\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}.$
(155)

With this system we end up with the basis of terms for contraction of the $X$
and $C$ tensors contracted with a sequence of five conformal derivatives

$\left\\{\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}{}^{2},\,\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square},\,\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu},\,\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square},\,\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\nu},\,\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\nabla}{}^{\mu}\right\\}$
(156)

or equivalently for terms built with metric only (without contractions with
derivatives)

$\left\\{g^{\gamma\beta_{1}}g^{\beta_{2}\beta_{3}}g^{\beta_{4}\beta_{5}},\,g^{\gamma\beta_{2}}g^{\beta_{1}\beta_{3}}g^{\beta_{4}\beta_{5}},\,g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{4}}g^{\beta_{2}\beta_{5}},\,g^{\gamma\beta_{3}}g^{\beta_{1}\beta_{2}}g^{\beta_{4}\beta_{5}},\,g^{\gamma\beta_{4}}g^{\beta_{1}\beta_{3}}g^{\beta_{2}\beta_{5}},\,g^{\gamma\beta_{4}}g^{\beta_{1}\beta_{5}}g^{\beta_{2}\beta_{3}}\right\\}.$
(157)

The choice of bases in (153) as well as in (156) is quite arbitrary. We made
one because we had to use an irreducible set of basis elemenets to solve the
system of algebraic equations for six coefficients $\alpha_{1}$ to
$\alpha_{6}$ unambiguosly. The basis defined in (153) is not special, and of
course, others may be used here as well. We do not find why this basis could
be thought of as a preferred one; it is just a basis. We will soon present
some results and interesting features independent of the choice of the basis.

Requiring that the contraction of the $X$ with $C$ tensor vanish expanded in
the basis (157), we get a system of 6 equations for 6 unknown coefficients.
The system of six linear equations reads explicitly, where we also used the
fact that here $w_{0}=\frac{6-d}{2}$,

$\displaystyle\frac{1}{2}\left(-8\alpha_{1}-(d+2)\left(\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (158)
$\displaystyle\frac{1}{2}\left(-4\alpha_{2}-4\alpha_{3}-d\left(\alpha_{4}+\alpha_{5}+\alpha_{6}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (159) $\displaystyle
4\alpha_{1}+\frac{d-2}{2}\alpha_{2}+\frac{d-2}{2}\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}$
$\displaystyle=$ $\displaystyle 0$ (160) $\displaystyle 0$ $\displaystyle=$
$\displaystyle 0$ (161) $\displaystyle
4\alpha_{2}+(d-1)\alpha_{4}+\frac{d-2}{2}\alpha_{5}+2\alpha_{6}$
$\displaystyle=$ $\displaystyle 0$ (162) $\displaystyle
4\alpha_{3}+\alpha_{4}+\frac{d+2}{2}\alpha_{5}+(d-2)\alpha_{6}$
$\displaystyle=$ $\displaystyle 0,$ (163)

where we also notice that in the basis (157) the contraction $XC$ does not at
all produce a term proportional to
$\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}$ (the
fourth equation in the system above). This is obviously not a problem since we
want a null solution to all equations here. Already this means that our system
of homogenous equation will have some freedom and not all $\alpha_{1}$ to
$\alpha_{6}$ coefficients will be determined uniquely. This fact is consistent
with the discussion from the previous section, where we indeed found some
freedom in defining the six-derivative Hamada operator without any endomorphic
parts.

System of equations (158) to (163) can be solved for various set of
coefficients. Before this answer we should put a question how big is the
freedom in the system, which is a statement independent of the basis and it
defines the dimension of the non-vanishing kernel of the system. We already
know that the system is not maximal, with a kernel 6-dimensional because of
the absence of terms proportional to
$\overset{c}{\square}\overset{c}{\nabla}{}^{\gamma}\overset{c}{\square}$.
Actually, the matrix rank of the system (158) to (163) is equal to three, so
we have only 3 independent equations and 3-parametric freedom of the
solutions. As we remember we can always take as for the normalization
$\alpha_{1}=1$ and then the freedom is reduced to a 2-parametric one.

We decided to solve the system for three unknown coefficients $\alpha_{3}$,
$\alpha_{5}$ and $\alpha_{6}$ (leaving $\alpha_{1}=1$ and undetermined
$\alpha_{2}$ and $\alpha_{4}$ playing here the role of real parameters). The
explicit solutions are

$\alpha_{3}=\frac{8d}{(d-4)(d+2)}+\alpha_{2},$ (164)
$\alpha_{5}=-\frac{128}{(d-6)(d-4)(d+2)}-\frac{8}{d-6}\alpha_{2}-\frac{2(d-3)}{d-6}\alpha_{4},$
(165)
$\alpha_{6}=\frac{32(d-2)}{(d-6)(d-4)(d+2)}+\frac{8}{d-6}\alpha_{2}+\frac{d}{d-6}\alpha_{4}.$
(166)

We see that these solutions are not defined in dimensions $d=4$ and $d=6$ as
well. This first singularity in all determined coefficients $\alpha_{3}$,
$\alpha_{5}$ and $\alpha_{6}$ for $d=4$ is here obvious and consistent with
the previously mentioned theorem due to Graham that such an operator with six-
derivative leading term does not exist in dimensions $d=4$. However, we will
also comment on this in more detailed later and discuss whether such result is
basis independent and whether it depends on the choice of the set of three
coefficients for which to solve here.

Actually, the situation in dimensions $d=6$ is more interesting since this
singularity is only apparent, but still there could be some problem or the
defect of defined operator that we will comment on below. To see that the
singularity is only an artefact of the choice of coefficients to solve for,
one can choose another set of them, for example, $\alpha_{3}$, $\alpha_{4}$
and $\alpha_{5}$. Then the system in $d=6$ of the explicit form

$\displaystyle
4\left(-\alpha_{1}-\left(\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}\right)\right)$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle-2\alpha_{2}-2\alpha_{3}-3\left(\alpha_{4}+\alpha_{5}+\alpha_{6}\right)$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
4\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle
4\alpha_{2}+5\alpha_{4}+2\alpha_{5}+2\alpha_{6}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle
4\alpha_{3}+\alpha_{4}+4\alpha_{5}+4\alpha_{6}$ $\displaystyle=$
$\displaystyle 0$ (167)

has solutions for $\alpha_{3}$, $\alpha_{4}$ and $\alpha_{5}$ which are

$\alpha_{3}=-3-\alpha_{2},$ (168)
$\alpha_{4}=-\frac{4}{3}-\frac{4}{3}\alpha_{2},$ (169)
$\alpha_{5}=\frac{10}{3}+\frac{4}{3}\alpha_{2}-\alpha_{6},$ (170)

where we also used the normalization condition that $\alpha_{1}=1$. So we see
in the system of solutions (168) to (170) that it is not singular in $d=6$,
just that we have to choose a good set to solve for. The singularity or its
disappearance depends on the set of final parameters here. However, there is
another problem, when one computes the following combination of all
coefficients:
$\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}$ for the
just constructed non-singular operator in $d=6$ dimensions. This vanishes and
this implies that the operator ${\cal O}$ just constructed in $d=6$ does not
have a leading term $\square^{3}$ on flat spacetime. On flat spacetime order
of conformal covariant derivatives does not matter (actually it does not
matter on any conformally flat spacetime, where Weyl tensor explicitly
vanishes $C_{\mu\nu\rho\sigma}=0$) and then the operator reduces to GR-
covariant $\square^{3}$ with all GR-covariant derivative with the frontal
coefficient which is the sum of $\alpha_{i}$ for $i=1$ to $i=6$. On curved
spacetime (it has to be not conformally flat) the operator has other terms, so
this is not equal to zero, but in all of these terms the conformal derivatives
can be ordered and terms higher in conformal curvature arises, and nothing out
of them survives on conformally flat spacetimes. This is a defect in $d=6$ of
the proposals for the operator ${\cal O}$ that does not depend on the choice
of three-element set of coefficients for which we solve for (or in
complementary way on the set of two parameters). It also does not depend on
the choice of writing our operator as in (153) in particular basis of terms
with order of six covariant derivatives. This can be seen already from the
first of the equation in system (167) in $d=6$ which tells us that
$\sum_{i=1}^{6}\alpha_{i}=0$. This is a defect of the operator ${\cal O}$ only
in critical dimension for six derivatives. In higher dimensions nothing like
this happens since it can be proven that independently on the choice of the
three-element set of coefficients to solve for, from the system of equations
(158) to (163), we derive a general expression for the sum
$\sum_{i=1}^{6}\alpha_{i}$ which is independent on the choice of terms in the
basis. By using just the first equation (158) we derive

$\frac{d-6}{2}\alpha_{1}=\frac{d+2}{2}\left(\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}\right),$
(171)

so

$\sum_{i=1}^{6}\alpha_{i}=\frac{d-6}{d+2}\alpha_{1}$ (172)

and for normalization $\alpha_{1}\neq 0$ we have that the sum is non-zero in
every dimension $d\neq 6$. This confirms that the problem only persists in
critical dimension $d=6$ and that it is independent on the basis of how we
write the operator in (153).

One can also analyze the system of equations (158) to (163) in dimensions
$d=4$ in a basis-independent way. The system is explicitly in the form

$\displaystyle-4\alpha_{1}-3\left(\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}\right)$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle-2\left(\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}\right)$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
4\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle
4\alpha_{2}+3\alpha_{4}+\alpha_{5}+2\alpha_{6}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle
4\alpha_{3}+\alpha_{4}+3\alpha_{5}+2\alpha_{6}$ $\displaystyle=$
$\displaystyle 0.$ (173)

One can clearly see, for example, by analyzing the first and second equation
in the above system that the inevitable conclusion here is that
$\alpha_{1}=0$, which means that we cannot normalize our operator ${\cal O}$
by setting the coefficient in front of $\overset{c}{\square}{}^{3}$ to one. It
must vanish. Since this happens, again we stay with the same problem that
$\sum_{i=1}^{6}\alpha_{i}=0$, because in the formula (172) we must use
unfortunately that $\alpha_{1}=0$. When one uses a different set of
coefficients to solve for (different than $\alpha_{3}$, $\alpha_{5}$ and
$\alpha_{6}$), a singularity in $d=4$ is not seen any more. Instead we have as
solutions

$\alpha_{1}=0,\quad\alpha_{2}=\frac{1}{4}\left(-3\alpha_{4}-\alpha_{5}-2\alpha_{6}\right),\quad\alpha_{3}=\frac{1}{4}\left(-\alpha_{4}-3\alpha_{5}-2\alpha_{6}\right)$
(174)

with the values of $\alpha_{4}$, $\alpha_{5}$ and $\alpha_{6}$ real and
completely arbitrary. We see that in this case we still have a three-parameter
freedom. Following [105] we can expand comments on this case. It was noticed
that in $d=4$ six-derivative operator with leading term $\square^{3}$ on flat
background is not constructible; however there exists an operator which is a
part of the Wunsch construction for $D_{(6)}^{\min}$ which is conformally
covariant in this dimension and whose expansion starts with terms linear in
curvature (and with conformally covariant derivatives acting upon it). This
operator is unique and it equals to $D_{{\rm
Bach}}\phi=B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\phi$
when in action on the scalar field. Here we can forget about the general
coefficient dimension-dependent which was present in (131). In special
dimension $d=4$ the Bach tensor is itself conformally covariant with the
weight $w\left(B_{\mu\nu}\right)=-2$ according to (133). However, the
operatorial action of
$B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}$ on a scalar
field with the weight $w_{0}=1$, brings another conformally covariant
expression $D_{{\rm Bach}}\phi$, when the coefficient in front can be here
completely arbitrary. The resulting conformal weight of this expression is
$w\left(B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\phi\right)=-5$.
In general, in this dimension we can construct a general expression with the
same resulting weight $w=-5$, which is an arbitrary linear combination of
$D_{{\rm Bach}}$, $D_{1}$ and $D_{2}$ operators acting on a scalar field
$\phi$. According to the definitions in (135) and (136), in $d=4$ the form of
these operators simplify to

$D_{1}=\overset{c}{\nabla}_{\mu}\left[\left(C^{\mu}{}_{\rho\sigma\kappa}C^{\nu\rho\sigma\kappa}-\frac{1}{4}g^{\mu\nu}C_{\rho\sigma\kappa\lambda}C^{\rho\sigma\kappa\lambda}\right)\overset{c}{\nabla}_{\nu}\right]$
(175)

and

$D_{2}=-12\overset{c}{\nabla}_{\mu}\left[C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}\overset{c}{\nabla}{}^{\mu}\right]-2\overset{c}{\square}\left(C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}\right).$
(176)

This is exactly the same three-parameter freedom that we find with arbitrary
real values of $\alpha_{4}$, $\alpha_{5}$ and $\alpha_{6}$ for the combination
of different orders of derivatives in the operator ${\cal O}$ in (153). The
explicit form of this operator ${\cal O}$ (with the three real parameters
$\alpha_{4}$, $\alpha_{5}$ and $\alpha_{6}$) can be presented as

${\cal
O}=\frac{1}{4}\left(-3\alpha_{4}-\alpha_{5}-2\alpha_{6}\right)\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\square}+\frac{1}{4}\left(-\alpha_{4}-3\alpha_{5}-2\alpha_{6}\right)\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square}+\alpha_{4}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\rho}+$
$+\alpha_{5}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\rho}+\alpha_{6}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}.$
(177)

Here we just can express it or as different order of conformal derivatives or
by an expression explicitly with conformal curvatures (Weyl tensors or Bach
tensors) and conformal derivatives on them. We just have to remember that in
$d=4$ it is impossible to have the operator with the leading term
$\overset{c}{\square}{}^{3}$ and all our operators in whatever form of writing
them start with terms linear in conformal curvatures.

Finally, we comment that in dimension $d=2$ the operator ${\cal O}$ cannot be
defined as the results of the known and previously discussed problems with the
definition of Schouten tensor in this dimension. So in all even dimensions
$d=2$ and $d=4$ smaller than the critical we have the impossibility to
construct such operators with six derivatives according to a conjecture that
such constructions are impossible in all even dimension. In the critical
dimension in turn we see the problem with the conformally flat spacetime limit
of the operator constructed only as a combination of different orders of
conformal derivatives. For all dimensions – all odd and even ones above the
critical one, we do not see any problem with such a construction of the
generalization of the Hamada operator.

We come back now to the discussion of the situation within the critical
dimension $d=6$ for a generalized Hamada operator. From our previous
discussion we understand now that the minimal operator $D_{(6)}^{\min}$ as
proposed by Wunsch cannot be reproduced as results of commutation of different
orders of conformal derivatives, from a general form of the operator ${\cal
O}$ in (153). Its explicit form in $d=6$ reads

$D_{(6)}^{\min}=\overset{c}{\square}{}^{3}+\frac{8}{3}B^{\mu\nu}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}.$
(178)

We see that also in this case we can only reproduce a linear combination of
$D_{1}$ and $D_{2}$ as acting on an conformally inert scalar field $\phi$
(with the vanishing weight since this is a situation in critical dimension).
Therefore the operator ${\cal O}$ given explicitly by (where
$\alpha_{2},\alpha_{6}\in\mathbb{R}$)

${\cal
O}=\overset{c}{\square}{}^{3}+\alpha_{2}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\square}+\left(-3-\alpha_{2}\right)\overset{c}{\nabla}_{\mu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\square}+\left(-\frac{4}{3}-\frac{4}{3}\alpha_{2}\right)\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\rho}+$
$+\left(\frac{10}{3}+\frac{4}{3}\alpha_{2}-\alpha_{6}\right)\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}{}^{\nu}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\rho}+\alpha_{6}\overset{c}{\nabla}_{\mu}\overset{c}{\nabla}_{\nu}\overset{c}{\square}\overset{c}{\nabla}{}^{\mu}\overset{c}{\nabla}{}^{\nu}$
(179)

corresponds to a two-parameter freedom of defining the operator as a linear
combination of $D_{1}$ and $D_{2}$. Again we find for consistency the same
degree of freedom. In $d=6,$ we have the following simplification for the
explicit form of the operators $D_{1}$ and $D_{2}$, according to (135) and
(136):

$D_{1}=\overset{c}{\nabla}_{\mu}\left[\left(C^{\mu}{}_{\rho\sigma\kappa}C^{\nu\rho\sigma\kappa}-\frac{1}{4}g^{\mu\nu}C_{\rho\sigma\kappa\lambda}C^{\rho\sigma\kappa\lambda}\right)\overset{c}{\nabla}_{\nu}\right]$
(180)

and

$D_{2}=-8\overset{c}{\nabla}_{\mu}\left[C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}\overset{c}{\nabla}{}^{\mu}\right].$
(181)

One can understand that in a special dimension $d=6$ we cannot produce the
operator (178), specifically its part with Bach tensor, as the result of
commutation of conformal derivatives. This is a particular realization of the
fact that not all expressions with curvatures can be reproduced as commutators
of covariant derivatives. Another example is the square of Weyl tensor
$C^{2}=C^{\rho\sigma\kappa\lambda}C_{\rho\sigma\kappa\lambda}$ acting by
scalar multiplication on some sequence of conformal derivatives, like for
example in the expression $C^{2}\overset{c}{\square}\phi$. Such expressions we
indeed find in the definitions of the operators $D_{1}$ and $D_{2}$ in (135)
and (136). However, there we see that covariant derivatives in some terms
after Leibniz expansion act on these invariants like $C^{2}$ and hence lead to
a splitting of the contraction of all indices and some curvature tensors there
can be really “uncommmuted”. In the sole expression like
$C^{2}\overset{c}{\square}\phi$ again, curvatures cannot be ever written as a
square of the commutator since such operation of commutation of conformal
derivatives never leads to a full contraction of indices between two
commutators. Some indices have to be internally contracted between the
commutator and the tensor on which it acts. (For example in GR we write
$\left[\nabla_{\mu},\nabla_{\nu}\right]v_{\rho}=R_{\mu\nu\rho}{}^{\sigma}v_{\sigma}$
and here $\sigma$ is this internally contracted index and $v_{\rho}$ is
arbitrary covariant vector field.) Therefore, it is not a surprise that some
terms in the expansion of the Hamada operator cannot be expressed as the
commutator of conformal derivatives, so in this case the “uncommutation”
procedure is unsuccessful, like for example for the full $D_{(6)}^{\min}$
operator. One must remember that in general curvature tensors are objects in
differential geometry that live their own life, independently how they were
defined. For example, they can be introduced as the result of some variational
derivative, when we effectively differentiate with respect to a tensor on
which they act upon. Then the curvature is properly defined and we can
abstract and forget on which particular tensor in differential geometry this
commutator was evaluated. The idea of one expression with a different order of
derivatives is that in such expression the operator acts on precisely on one
tensor on the right. However, when we take a square of the curvature, for
example like $C^{2}$ term, then if we would like to write them back all these
curvatures as resulting from commutations of derivatives, then we would have
to take variational derivatives two times and with respect to different
arguments. But in the original expression there was precisely one tensor as
the argument.

In general dimension, it is however important that we find the same degree of
freedom in number of free parameters needed to define the non-endomorphic
parts of the six-derivative operator. No matter, whether we use explicitly
Wunsch construction with $D_{(6)}^{\min}$, $D_{1}$ and $D_{2}$ operators or
the operator ${\cal O}$ with arbitrary real parameters ($\alpha_{2}$ and
$\alpha_{4}$ for example). We have all the time two-parameter freedoms in
construction of the derivative part of the conformal operators. By analysis of
various solutions of the system (158) to (163) in arbitrary dimension $d$ for
different three dependent coefficients, we find that even if we completely
simplify the choice of the independent parameters such that $\alpha_{1}=1$ and
two free real parameters are set to zero, therefore in this sense looking for
some minimal operator, the combination in the operator ${\cal O}$ always
contains four terms. And this number of different orders for conformal
derivatives cannot be reduced further. In general, the non-endomorphic parts
of the generalized Hamada operator, can have $4,$ $5$ or $6$ terms with
different orders of conformal derivatives. Therefore the result of
“uncommutation” technique produces generally more terms and does not lead to a
direct one simple linear combination of orders of six derivatives that would
give a tentative answer for “minimal” conformal operator. It is generally
believed that commuting the derivatives reduces the number of terms and this
is true, for example, for the construction of $D_{(6)}^{\min}$ operator in
(131), which is built only with two terms, with only term of curvature
corrections. With different orders of all six conformal derivatives we in
principle, need more terms to write and construct the conformal operator even
in its shortest form.

With the procedure of studying of various order of conformal derivatives
outlined above, the search for such combinations also for higher number of
derivatives, higher than $2N=6$, is possible, since this is now revealed as
only an algebraic task of solving a big system of linear equations. However,
we envisage that the number of terms and of possible orders of conformal
derivatives grows quite fast with the number $2N$ of derivatives (rough
estimate is that it grows as $(2N-1)!!$, when we do not take into account any
type of special identity discussed at the beginning of this section). Hence,
the size and the complexity of the matrix of linear equations that has to be
diagonalized here is also big for even relatively small $N$ and this is left
as the task only for symbolic and numerical algorithms on computers. We
believe that even “minimal” form of conformal operators for higher $N$ is not
so short since they may contain many necessary curvature correction terms, or
many necessary different orders of a sequence of $2N$ conformal derivatives.

##  VIII Discussion

One could also perform a comparison of three methods for finding the explicit
expression for the generalized Hamada operator and their algebraic complicacy.
In all these methods we need to solve some big system of algebraic linear
equations, hence the important signal of the effectiveness of the methods (and
new methods that we have proposed in this paper) would be the size of the
algebraic system involved. As we remarked earlier there does not exist any
signigicant hope that the expression for the generalized Hamada operator can
be found without solving such system of equations for linear coefficients of
terms and as a result of using of some new powerful conformal covariant
derivative or the conformally covariant box operator.

In the first method, when we use the basis of all terms that can be used in a
construction of generally covariant scalar invariants for the expression
$D_{(6)}\phi$, we use all three types of possible curvature tensors ($R$,
$R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$) and normal GR-covariant derivatives
acting on them (so we use operators $\nabla_{\mu}$ or $\square$). This basis
of writing all possible terms in $D_{(6)}\phi$ has many elements as we have
seen in previous chapters. The reason is that all these terms are not designed
for having good conformal transformation properties. They are just all
possible scalar GR-invariants. Hence they are not specially adjusted for the
sake of finding some conformal invariant operator like $D_{(6)}$. The Riemann
tensor, Ricci tensor and Ricci scalar do not have simple conformal
transformation laws (even in the infinitesimal form it contains up to second
derivatives of the $\Omega$ parameter). Had we used here in the basis Weyl
tensor, this task of curvature choice would be simplified, since the last is
conformal curvature tensor and it is covariantly transforming under conformal
transformations. Moreover, the covariant derivatives $\nabla_{\mu}$ and
$\square$ are not selected in any way to facilitate the conformal
transformation laws of the object on which they act. Again, these differential
operators do not transform in any way that would be in line with conformal
transformations.

In conclusion, this basis of writing terms is not well prepared from the point
of view of looking for conformally invariant expressions and conformally
covariant operators like $D_{(6)}$. This is also why it has many terms. This
implies that we have to solve the system of algebraic equations for many
coefficients of terms in a general combination giving at the end conformally
invariant expression. Moreover, since we use the basis of writing terms which
is only covariant from the point of view of GR, the requirement for good
conformal transformation law is equivalent here to checking that the terms
with first and all higher level of derivatives on the $\Omega$ parameter must
cancel out in the combination for generalized conformal operator
$D_{(2N)}\phi$. In the case of Hamada operator we need to check up to the
sixth derivative of the $\Omega$ parameter and this is still on the
infinitesimal level, so all terms are linear in $\Omega$ or its derivative.
Luckily this also guarantees that the finite transformations will not generate
problem and the combination for the operator $D_{(6)}\phi$ will be conformally
covariant, if it is shown that on the infinitesimal level the expression
$D_{(6)}\phi$ transforms only up to the terms linear in $\Omega$ and
completely without derivatives of it. This again implies that we have many
equations: cancellation of all terms at the order $\nabla\Omega$, cancellation
of all terms at the order $\nabla^{2}\Omega$, etc., so we have to have here
many linear equations with many unknowns and the system can be quite
complicated.

When we used the second method with basis constructed only with conformal
curvature tensors (Weyl and Bach tensors and conformal derivatives acting on
them) and we exploited the conformal derivatives there
$\overset{c}{\nabla}_{\mu}$ and $\overset{c}{\square}$, we generically have a
much smaller basis of all possible terms that we have to use to construct the
conformal operator $D_{(6)}$ when acting on the scalar field $\phi$. This is
because we use the adjusted conformal curvature tensors specially built to
facilitate conformal transformation laws and they are also conformal tensors,
meaning that they transform covariantly – without derivatives of the $\Omega$
parameter. The same arguments also applies to usage of conformally covariant
derivatives like $\overset{c}{\nabla}_{\mu}$ and $\overset{c}{\square}$ since
with them the law of transformation of any object is only up to the first
derivatives of the $\Omega$ parameter. Therefore, here we expect to have less
number of unknown coefficients in the combinations giving conformal operators.
(For example, in the case of the minimal operator in (131) we needed to find
only one relative coefficient in front of the Bach tensor.) Finally, since the
usage of conformal curvature tensors and other basic conformal objects (like
the scalar field with the assigned conformal weight $w_{0}$) and of
conformally covariant Wunsch derivatives creates other tensorial expressions
that under conformal transformations change only up to the first derivatives
of the $\Omega$ parameter, we at the end of the computation need to check only
if terms linear in $\nabla\Omega$ do cancel out in our proposed combination
for conformal operators. Other higher derivatives like $\nabla^{2}\Omega$
never arise in this formalism of conformal derivatives and other conformal
tensors. Hence we conclude that here generically we expect to have less number
of linear equations and also smaller number of unknown coefficients, so
algebraically speaking this system is more effective. Of course, this is like
that here since our building blocks are already conformally prepared. This
reduction of algebraic complication is clearly visible when one compares the
computations of sections IV and VI and VII of our paper.

With the advantages of the second method (with conformal curvatures and
conformal derivatives), there come also a few restrictions or shortcomings.
They are the most visible when instead of creating operators like $D_{(6)}$
one considers conformally covariant Lagrangians of the form $\phi
D_{(2N)}\phi$, which after the proper densitization give rise to conformally
invariant Lagrangian densities $\sqrt{|g|}\phi D_{(2N)}\phi$, which are at the
basis for conformally invariant scalar actions. In the case of the basis of
general GR-covariant curvatures and GR-covariant derivatives, on the level of