On the Super-Renormalizablity of Quantum Gravity in the Linear Approximation

TL;DR

This paper demonstrates that one-loop quantum gravity contributions in the linear approximation are trivial on physical states, suggesting potential simplifications for the broader problem of quantum gravity's renormalizability.

Contribution

It proves that one-loop contributions in massless gravity are coboundaries, indicating a possible path to super-renormalizability in quantum gravity.

Findings

One-loop contributions are coboundaries and vanish on physical states.

The result may extend to higher orders, simplifying quantum gravity.

Supports the conjecture of super-renormalizability in linear quantum gravity.

Abstract

We compute the one-loop contributions of the chronological products for massless gravity in the second order of the perturbation theory. We prove that the loop contributions are coboundaries i.e. expressions which give zero when averaged on physical states. We conjecture that such a result should be true in higher orders of the perturbation theory also. This result should make easier the problem of constructive quantum field theory.

Equations

$$d_{Q}A=[Q,A]$$

$$Q^2=0⟺d_Q^2=0$$

$$[Q,T]=i{\partial }_{\mu }{T}^{\mu }$$

$$T(f){\mathcal{H}}_{\mathrm{phys}}\subset {\mathcal{H}}_{\mathrm{phys}}$$

$$[Q,T]=i{\partial }_{\mu }{T}^{\mu },[Q,T^{\mu }]=i{\partial }_{\nu }{T}^{\mu \nu },[Q,T^{\mu \nu }]=i{\partial }_{\rho }{T}^{\mu \nu \rho }[Q,T^{\mu \nu \rho }]=0$$

$$\omega (T^I)={\omega }_0=5,\forall I$$

$$gh(T^I)=|I|.$$

$$d_Q{T}^I=i{\partial }_{\mu }{T}^{I\mu }.$$

$$T^{I_1,\dots ,I_n}(x_1,\dots ,x_n)\equiv T(T^{I_1}(x_1),\dots ,T^{I_n}(x_n))$$

$$d_Q{T}^{I_1,\dots ,I_n}=i\sum _{l=1}^n(-1{)}^{s_l}\frac{\partial }{\partial x_l^{\mu }}{T}^{I_1,\dots ,I_l\mu ,\dots ,I_n}$$

$$s_l\equiv \sum _{j=1}^{l-1}|I{|}_j.$$

$$(\delta C{)}^{I_1,\dots ,I_n}=\sum _{l=1}^n(-1{)}^{s_l}\frac{\partial }{\partial x_l^{\mu }}{C}^{I_1,\dots ,I_l\mu ,\dots ,I_n}.$$

$${\delta }^2=0.$$

$$s=d_Q-i\delta ,\bar{s}=d_Q+i\delta$$

$$s\bar{s}=\bar{s}s=0.$$

$$sC=0$$

$$C=\bar{s}B.$$

$$sT=0.$$

$$<\Psi ,\bar{s}B{\Psi }^{\prime }>=<Q\Psi ,B{\Psi }^{\prime }>-<\Psi ,BQ{\Psi }^{\prime }>+i<\Psi ,\delta B{\Psi }^{\prime }>$$

$$=i<\Psi ,\delta B{\Psi }^{\prime }>\to 0$$

$$<\Omega ,h_{\mu \nu }(x_1)h_{\rho \sigma }(x_2)\Omega >=-\frac{i}{2}({\eta }_{\mu \rho }{\eta }_{\nu \sigma }+{\eta }_{\nu \rho }{\eta }_{\mu \sigma }-{\eta }_{\mu \nu }{\eta }_{\rho \sigma })D_0^{(+)}(x_1-x_2),$$

$$<\Omega ,u_{\mu }(x_1){\tilde{u}}_{\nu }(x_2)\Omega >=i{\eta }_{\mu \nu }{D}_0^{(+)}(x_1-x_2),$$

$$<\Omega ,{\tilde{u}}_{\mu }(x_1)u_{\nu }(x_2)\Omega >=-i{\eta }_{\mu \nu }{D}_0^{(+)}(x_1-x_2)$$

$$h_{\mu \nu }^{†}=h_{\mu \nu },u_{\rho }^{†}=u_{\rho },{\tilde{u}}_{\sigma }^{†}=-{\tilde{u}}_{\sigma }.$$

$$[Q,h_{\mu \nu }]=-\frac{i}{2}({\partial }_{\mu }{u}_{\nu }+{\partial }_{\nu }{u}_{\mu }-{\eta }_{\mu \nu }{\partial }_{\rho }{u}^{\rho }),[Q,u_{\mu }]=0,[Q,{\tilde{u}}_{\mu }]=i{\partial }^{\nu }{h}_{\mu \nu }$$

$$Q\Omega =0$$

$$[h_{\mu \nu }(x_1),h_{\rho \sigma }(x_2)]=-\frac{i}{2}({\eta }_{\mu \rho }{\eta }_{\nu \sigma }+{\eta }_{\nu \rho }{\eta }_{\mu \sigma }-{\eta }_{\mu \nu }{\eta }_{\rho \sigma })D_0(x_1-x_2)\cdot I,$$

$$[u(x_1),\tilde{u}(x_2)]=i{\eta }_{\mu \nu }{D}_0(x_1-x_2)\cdot I$$

$$[Q,[h_{\mu \nu }(x_1),{\tilde{u}}_{\sigma }(x_2)]]+\mathrm{cyclic}\mathrm{permutations}=0$$

$$h\equiv {\eta }_{\mu \nu }{h}^{\mu \nu }$$

$$[h_{\mu \nu }(x_1),h(x_2)]=[h(x_1),h_{\mu \nu }(x_2)]=i{\eta }_{\mu \nu }\cdot I,$$

$$[h(x_1),h(x_2)]=4i{D}_0(x_1-x_2)\cdot I$$

$$d_{Q}h=i{\partial }_{\mu }{u}^{\mu }.$$

Full text

**On the Super-Renormalizablity of Quantum Gravity in the Linear Approximation

**

D. R. Grigore, 111e-mail: grigore@theory.nipne.ro

Department of Theoretical Physics

Institute for Physics and Nuclear Engineering “Horia Hulubei”

Bucharest-Măgurele, P. O. Box MG 6, ROMÂNIA

We compute the one-loop contributions of the chronological products for massless gravity in the second order of the perturbation theory. We prove that the loop contributions are coboundaries i.e. expressions which give zero when averaged on physical states. We conjecture that such a result should be true in higher orders of the perturbation theory also. This result should make easier the problem of constructive quantum field theory.

1 Introduction

We remind a few facts about perturbative quantum field theory following essentially [5]. The general framework of perturbation theory consists in the construction of the chronological products: for every set of Wick monomials $W_{1}(x_{1}),\dots,W_{n}(x_{n})$ acting in some Fock space ${\cal H}$ one associates the operator $T^{W_{1},\dots,W_{n}}(x_{1},\dots,x_{n});$ all these expressions are in fact distribution-valued operators called chronological products. It will be convenient to use another notation: $T(W_{1}(x_{1}),\dots,W_{n}(x_{n})).$ These operators are constrained by Bogoliubov axioms [1], [4], [2] and their construction can be done recursively according to Epstein-Glaser prescription [4] (which reduces the induction procedure to a distribution splitting of some distributions with causal support). These products are not uniquely defined but there are some natural limitation on the arbitrariness. This limitation is a bound on the degree of the singularity of the vacuum averages of the chronological products. One imposes that this singularity degree should be as low as possible. If this arbitrariness does not grow with the order $n$ of the perturbation theory then we say that the theory is renormalizable; the most popular point of view is that only such theories are physically meaningful. Apparently, this power counting argument excludes quantum gravity. We will argue that there is a way out of this no-go result. The basic idea is that quantum gravity must be considered, at least in the perturbation theory, as a theory of particles of zero mass and helicity $2$. Such theories are best described using physical and non-physical fields (the ghost fields) as well. This means that the Fock space will contain physical and non-physical states. Therefore one should investigate the ultraviolet behavior of the chronological products resticted to the subspace of physical states. If we do that we can prove by direct computation that the one-loop contributions are in fact null if we restrict ourselves to the subspace of physical states. We have given in [7] a pure cohomological argument of this assertion. Here we use a new approach based on explicit computations of the loop contributions and explicit proof of their triviality.

From the technical point of view, one must first construct a Fock space ${\cal H}$ with indefinite metric, generated by physical and un-physical fields (called ghost fields). One selects the physical states assuming the existence of an operator $Q$ called gauge charge which verifies $Q^{2}=0$ and such that the physical Hilbert space is by definition ${\cal H}_{\rm phys}\equiv Ker(Q)/Im(Q).$

One has a natural grading in the Hilbert space ${\cal H}$ and the graded commutator $d_{Q}$ of the gauge charge with any operator $A$ is defined by

[TABLE]

where in the right hand side we understand that $[\cdot,\cdot]$ is the graded commutator. Because

[TABLE]

it means that $d_{Q}$ is a co-chain operator.

A gauge theory assumes also that there exists a Wick polynomial of null ghost number $T(x)$ called the interaction Lagrangian such that

[TABLE]

for some other Wick polynomials $T^{\mu}.$ This relation means that the expression $T$ leaves invariant the physical states, at least in the adiabatic limit. Indeed, if this is true we have:

[TABLE]

up to terms which can be made as small as desired (making the test function $f$ flatter and flatter). In the case of quantum gravity we also have the Wick polynomials $T^{\mu},~{}T^{\mu\nu},~{}T^{\mu\nu\rho}$ such that:

[TABLE]

and the expressions $T^{\mu\nu},~{}T^{\mu\nu\rho}$ are completely antisymmetric in all indexes so we can also use a compact notation $T^{I}$ where $I$ is a collection of indexes $I=[\nu_{1},\dots,\nu_{p}]~{}(p=0,1,\dots,)$ and the brackets emphasize the complete antisymmetry in these indexes. All these polynomials have the same canonical dimension

[TABLE]

and because the ghost number of $T\equiv T^{\emptyset}$ is supposed null, then we also have:

[TABLE]

One can write compactly the relations (1.5) as follows:

[TABLE]

If the interaction Lagrangian $T$ is Lorentz invariant, then one can prove that the expressions $T^{I},~{}|I|>0$ can be taken Lorentz covariant as well.

Now we can construct the chronological products

[TABLE]

according to the recursive procedure of Epstein and Glaser. We say that the theory is gauge invariant in all orders of the perturbation theory if the following set of identities generalizing (1.8):

[TABLE]

are true for all $n\in\mathbb{N}$ and all $I_{1},\dots,I_{n}.$ Here we have defined

[TABLE]

We introduce some cohomology terminology. We consider a cochains to be an ensemble of distribution-valued operators of the form $C^{I_{1},\dots,I_{n}}(x_{1},\dots,x_{n}),~{}n=1,2,\dots$ (usually we impose some supplementary symmetry properties) and define the derivative operator $\delta$ according to

[TABLE]

We can prove that

[TABLE]

Next we define

[TABLE]

and note that

[TABLE]

We call relative cocycles the expressions $C$ verifying

[TABLE]

and a relative coboundary an expression $C$ of the form

[TABLE]

The relation (1.9) is simply the cocycle condition

[TABLE]

The purpose of this paper is to investigate if this condition implies that, at least some contributions of $T$, are in fact coboundaries. Coboundaries are trivial from the physical point of view: if we consider two physical states $\Psi,\Psi^{\prime}$ then

[TABLE]

(in the adiabatic limit).

We will consider here only the second order of the perturbation theory and prove that for massless gravity the loop contributions is a coboundary i.e. the theory is essentially classical. This follows from the fact that in the loop expansion the [math]-loop (or tree) contribution corresponds to the classical theory [3].

In the next Section we present the description of the free fields we use, mainly to fix the notations. In Section 3 we remind Bogoliubov axioms for the second order of the perturbation theory and we give the basic distributions with causal support appearing for loop contributions in the second order of the perturbation theory. In Section 4 we prove the cohomology result for gravity.

2 Massless Particles of Spin $2$ (Gravitons)

We refer to more details to [5]. We consider the vector space ${\cal H}$ of Fock type generated (in the sense of Borchers theorem) by the symmetric tensor field $h_{\mu\nu}$ (with Bose statistics) and the vector fields $u^{\rho},\tilde{u}^{\sigma}$ (with Fermi statistics). We suppose that all these (quantum) fields are of null mass. In this vector space we can define a sesquilinear form $<\cdot,\cdot>$ in the following way: the (non-zero) $2$-point functions are by definition:

[TABLE]

and the $n$-point functions are generated according to Wick theorem, or equivalently assuming that the truncated $n$-point functions are null for $n\geq 3.$ Here $\eta_{\mu\nu}$ is the Minkowski metrics (with diagonal $1,-1,-1,-1$) and $D_{0}^{(+)}$ is the positive frequency part of the Pauli-Jordan distribution $D_{0}$ of null mass. To extend the sesquilinear form to ${\cal H}$ we define the conjugation by

[TABLE]

Now we can define in ${\cal H}$ the operator $Q$ according to the following formulas:

[TABLE]

where by $[\cdot,\cdot]$ we mean the graded commutator. One can prove that $Q$ is well defined. Indeed, we have the causal commutation relations

[TABLE]

and the other commutators are null. The operator $Q$ should leave invariant these relations, in particular

[TABLE]

which is true according to the preceding relations. Then we have:

Theorem 2.1

The operator $Q$ verifies $Q^{2}=0.$ The factor space $Ker(Q)/Im(Q)$ is isomorphic to the Fock space of particles of zero mass and helicity $2$ (gravitons).

If we define

[TABLE]

we also have

[TABLE]

and

[TABLE]

3 General Gauge Theories

We give here the essential ingredients of perturbation theory for the order $n=2$ of the perturbation theory. We asignate to $h_{\mu\nu},u_{\mu},\tilde{u}_{\mu}$ the canonical dimension $1$. A derivative applied to a field raises the canonical dimension by $1$. The ghost number of $h_{\mu\nu}$ is [math] and for the ghost fields $u_{\mu},\tilde{u}_{\mu}$ is $1$. The Fermi parity of a Fermi (Bose) field is $1$ (resp. [math]). The canonical dimension of a Wick monomial is additive with respect to the factors and the same is true for the ghost number and the Fermi parity.

3.1 Bogoliubov Axioms

Suppose that the Wick monomials $A,B$ are self-adjoint: $A^{\dagger}=A,~{}B^{\dagger}=B$ and of fixed Fermi parity $|A|,|B|$ and canonical dimension $\omega(A),\omega(B)$. For gravity we must take $\omega(A),\omega(B)=5$. The chronological products $T(A(x),B(y))$ are verifying the following set of axioms:

• •

Skew-symmetry:

[TABLE]

• •

Poincaré invariance: we have a natural action of the Poincaré group in the space of Wick monomials and we impose that for all elements $g$ of the universal covering group $inSL(2,\mathbb{C})$ of the Poincaré group:

[TABLE]

where $x\mapsto g\cdot x$ is the action of $inSL(2,\mathbb{C})$ on the Minkowski space.

• •

Causality: if $x\succeq y$ i.e. $y\cap(x+\bar{V}^{+})=\emptyset$ then we have:

[TABLE]

• •

Unitarity: If we define the anti-chronological products according to

[TABLE]

then the unitarity axiom is:

[TABLE]

It can be proved that this system of axioms can be supplemented with

[TABLE]

where $A=A_{1}A_{2},B=B_{1}B_{2}$ is an arbitrary decomposition of $A$ and resp. $B$ in Wick submonomials and we have supposed for simplicity that no Fermi fields are present; if Fermi fields are present, then some apropriate signs do appear. This is called the Wick expansion property.

We can also include in the induction hypothesis a limitation on the order of singularity of the vacuum averages of the chronological products:

[TABLE]

where by $\omega(d)$ we mean the order of singularity of the (numerical) distribution $d$ and by $\omega(W)$ we mean the canonical dimension of the Wick monomial $W$.

The contributions verifying

[TABLE]

will be called super-renormalizable.

The operator-valued distributions $D,A,R,T$ admit a decomposition into loop contributions $D=\sum_{l}D_{(l)}$, etc. Indeed every contribution is associated with a certain Feynman graph and the integer $l$ counts the number of the loops. Alternatively, if we consider the loop decomposition of the advanced (or retarded) products we have in fact series in $\hslash$ so the contribution corresponding to $l=0$ (the tree contribution) is the classical part and the loop contributions $l>0$ are the quantum corrections [3].

3.2 Second Order Chronological Products

We go to the second order of perturbation theory using the causal commutator

[TABLE]

where $A(x),B(y)$ are arbitrary Wick monomials and, as always we mean by $[\cdot,\cdot]$ the graded commutator. These type of distributions are translation invariant i.e. they depend only on $x-y$ and the support is inside the light cones:

[TABLE]

A theorem from distribution theory guarantees that one can causally split this distribution:

[TABLE]

where:

[TABLE]

The expressions $A(A(x),B(y)),R(A(x),B(y))$ are called advanced resp. retarded products. They are not uniquely defined: one can modify them with quasi-local terms i.e. terms proportional with $\delta(x-y)$ and derivatives of it.

There are some limitations on these redefinitions coming from Lorentz invariance, and power counting: this means that we should not make the various distributions appearing in the advanced and retarded products too singular.

Then we define the chronological product by:

[TABLE]

In the particular case of a gauge theory (as it is quantum gravity) we need that causal commutators

[TABLE]

with the symmetry property

[TABLE]

and the limitations

[TABLE]

and power counting limitations coming from (3.7). To get some explicit intuition, we need some details about (numerical) distributions with causal support appearing in the second order of the perturbation theory.

3.3 Second Order Causal Distributions

We remind the fact that the Pauli-Villars distribution is defined by

[TABLE]

where

[TABLE]

such that

[TABLE]

This distribution has causal support. In fact, it can be causally split (uniquely) into an advanced and a retarded part:

[TABLE]

and then we can define the Feynman propagator and antipropagator

[TABLE]

All these distributions have singularity order $\omega(D)=-2$. We will consider from now on only the case $m=0.$

It is easy to see that the tree contribution in the second order of perturbation theory of the causal commutator is of the form

[TABLE]

where $p_{j}$ are monomials in the partial derivatives and $W_{j}(x,y)$ are some Wick monomials. Then we can obtain the advanced, retarded and chronological products by simply replacing $D_{0}$ by $D_{0}^{\rm adv}$, $D_{0}^{\rm ret}$ and $D_{0}^{F}$ respectively.

It is not hard to see that a formula similar to (3.22) is valid for the loop contributions

[TABLE]

where we will need the basic distribution

[TABLE]

which is also with causal support and it can be causally split as above in

[TABLE]

and the corresponding Feynman propagators can be defined. These distributions have the singularity order $\omega(D)=0$ so the causal splitting is not unique: we can add an arbitrary contribution of the form $\sim\delta(x).$

In the explicit computations some associated distributions with causal support do appear. We can have two derivatives distributed in two ways on the two factors $D_{0}^{(+)}$:

[TABLE]

It is not hard to prove that we have

[TABLE]

We also have:

[TABLE]

Now we introduce distributions with three derivatives:

[TABLE]

and we can prove that

[TABLE]

[TABLE]

We also have:

[TABLE]

Finally we have distributions with four derivatives:

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

We also have:

[TABLE]

[TABLE]

[TABLE]

For two-loop contributions we have

[TABLE]

where we will need the basic distribution

[TABLE]

which is also with causal support and it can be causally split as above in

[TABLE]

As in the one-loop case we encounter the associated causal distributions

[TABLE]

so in the end we have

[TABLE]

4 The Lagrangian for Massless Gravity

We have the following result [6]:

Theorem 4.1

Let $T$ a a relative cocycle, i.e. a Wick polynomial verifying (1.3) which is tri-linear in the fields and is of canonical dimension $\omega(T)=5$ and ghost number $gh(T)=0.$ Then: (i) $T$ is (relatively) cohomologous to a non-trivial cocycle of the form:

[TABLE]

[TABLE]

where $\kappa\in\mathbb{R}.$

(ii) The relation $d_{Q}T=i~{}\partial_{\mu}T^{\mu}$ is verified by:

[TABLE]

[TABLE]

(iii) The relation $d_{Q}T^{\mu}=i~{}\partial_{\nu}T^{\mu\nu}$ is verified by:

[TABLE]

[TABLE]

(iv) The relation $d_{Q}T^{\mu\nu}=i~{}\partial_{\rho}T^{\mu\nu\rho}$ is verified by:

[TABLE]

[TABLE]

and we have $d_{Q}T^{\mu\nu\rho}=0.$

(v) The cocycles $T,T^{\mu},T^{\mu\nu}$ and $T^{\mu\nu\rho}$ are non-trivial and invariant with respect to parity.

In the preceding expressions we have, for simplicity, omitted the Wick ordering signs.

5 The Generic Expressions for the One-Loop Cochains

We compute the one-loop contribution $D_{(1)}(T(x_{1}),T(x_{2}))=[T(x_{1}),T(x_{2})]_{(1)}$ using Wick theorem: we get

[TABLE]

where the last three terms correspond to tree, one-loop and two-loop contributions . We obtain the following formula for the one-loop contribution:

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using the formulas from the preceding Section one can rewrite everything more compactly:

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The expressions $D_{(1)}(T^{I}(x_{1}),T^{J}(x_{2})),|I|+|J|=1,2$ can be computed in the same way. As far as we know, the expressions (5.11) $+$ (5.12) - (5.18) are new in the literature.

6 Cohomology

In this Section we prove the main result:

Theorem 6.1

The expression (5.11) is in fact a relative coboundary, i.e. of the form:

[TABLE]

where the generic form of the cochains is:

[TABLE]

with $W_{j}$ Wick monomials. The same assertion stays true for one-loop the chronological product $T_{(1)}(T(x_{1}),T(x_{2}))$.

Proof: The proof is very computational: we must make a generic ansatz of the form (6.2) for the expressions $B(x_{1},x_{2})$ and $B^{\mu}(x_{1},x_{2})$. The expression $B$ should be antisymmetric with respect to $1\leftrightarrow 2$ as it is $D_{(1)}(T(x_{1}),T(x_{2}))$.

After some hard computations we can find a non-trivial solution of this problem. It is more interesting to emphasize that every individual term from (5.11) is in fact a coboundary. We have $38$ terms in (5.11):

$D=\partial_{\mu}\partial_{\nu}\square d_{2}(x_{1}-x_{2})~{}[h^{\mu\nu}(x_{1})h(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}\square d_{2}(x_{1}-x_{2})~{}h^{\mu\rho}(x_{1})~{}{h^{\nu}}_{\rho}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\square^{2}d_{2}(x_{1}-x_{2})~{}h^{\mu\nu}(x_{1})h_{\mu\nu}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\square^{2}d_{2}(x_{1}-x_{2})~{}h(x_{1})h(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}d_{2}(x_{1}-x_{2})~{}h^{\mu\nu}(x_{1})h^{\rho\sigma}(x_{2})$

We can take

[TABLE]

and

[TABLE]

$D=\partial_{\mu}\partial_{\rho}\partial_{\sigma}d_{2}(x_{1}-x_{2})~{}[h^{\rho\sigma}(x_{1})\partial^{\mu}h(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\rho}\partial_{\sigma}d_{2}(x_{1}-x_{2})~{}[h^{\rho\sigma}(x_{1})\partial_{\nu}h^{\mu\nu}(x_{2})-(1\leftrightarrow 2)]$

We can take

[TABLE]

and

[TABLE]

$D=\partial_{\mu}\partial_{\rho}\partial_{\sigma}d_{2}(x_{1}-x_{2})~{}[h^{\mu\nu}(x_{1})\partial_{\nu}h^{\rho\sigma}(x_{2})-(1\leftrightarrow 2)]$

We can take

[TABLE]

and

[TABLE]

$D=\partial_{\mu}\partial_{\rho}\partial_{\sigma}d_{2}(x_{1}-x_{2})~{}[u^{\mu}(x_{1})\partial^{\rho}\tilde{u}^{\sigma}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h^{\mu\nu}(x_{1})\partial^{\rho}h_{\nu\rho}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h(x_{1})\partial_{\nu}h^{\mu\nu}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h_{\rho\sigma}(x_{1})\partial^{\mu}h^{\rho\sigma}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h(x_{1})\partial^{\mu}h(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h^{\mu\nu}(x_{1})\partial_{\nu}h(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[h_{\rho\sigma}(x_{1})\partial^{\rho}h^{\mu\sigma}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[u_{\nu}(x_{1})\partial^{\mu}\tilde{u}^{\nu}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[u^{\mu}(x_{1})\partial_{\nu}\tilde{u}^{\nu}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\square d_{2}(x_{1}-x_{2})~{}[u_{\nu}(x_{1})\partial^{\nu}\tilde{u}^{\mu}(x_{2})-(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\mu}h^{\nu\rho}(x_{1})\partial^{\sigma}h_{\rho\sigma}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\rho}h^{\mu\nu}(x_{1})\partial^{\sigma}h_{\rho\sigma}(x_{2})+(1\leftrightarrow 2)]$

We can take

[TABLE]

and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}\partial_{\rho}h^{\mu\rho}(x_{1})\partial_{\sigma}h^{\nu\sigma}(x_{2})$

We can take

[TABLE]

and $B^{\mu}=0.$

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\mu}h(x_{1})\partial_{\rho}h^{\nu\sigma}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}\partial^{\mu}h^{\rho\sigma}(x_{1})\partial^{\nu}h_{\rho\sigma}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}\partial^{\mu}h(x_{1})\partial^{\nu}h(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\mu}h^{\nu\rho}(x_{1})\partial_{\rho}h(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\mu}h_{\rho\sigma}(x_{1})\partial^{\rho}h^{\nu\sigma}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial^{\nu}d_{2}(x_{1}-x_{2})~{}\partial^{\rho}h^{\mu\sigma}(x_{1})\partial_{\rho}h_{\nu\sigma}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}\partial_{\rho}h^{\mu\sigma}(x_{1})\partial_{\sigma}h^{\nu\rho}(x_{2})$

We can take

[TABLE]

and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\rho}u^{\mu}(x_{1})\partial^{\nu}\tilde{u}_{\rho}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial_{\rho}u^{\rho}(x_{1})\partial^{\mu}\tilde{u}^{\nu}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\mu}u^{\nu}(x_{1})\partial_{\rho}\tilde{u}^{\rho}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\partial_{\mu}\partial_{\nu}d_{2}(x_{1}-x_{2})~{}[\partial^{\rho}u^{\mu}(x_{1})\partial_{\rho}\tilde{u}^{\nu}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

$D=\square d_{2}(x_{1}-x_{2})~{}\partial_{\rho}h^{\mu\rho}(x_{1})\partial^{\sigma}h_{\mu\rho}(x_{2})$

We can take

[TABLE]

and $B^{\mu}=0$

$D=\square d_{2}(x_{1}-x_{2})~{}[\partial_{\mu}h(x_{1})\partial_{\nu}h^{\mu\nu}(x_{2})+(1\leftrightarrow 2)]$

We can take

[TABLE]

and

[TABLE]

$D=\square d_{2}(x_{1}-x_{2})~{}\partial_{\mu}h_{\rho\sigma}(x_{1})\partial^{\mu}h^{\rho\sigma}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\square d_{2}(x_{1}-x_{2})~{}\partial_{\mu}h(x_{1})\partial^{\mu}h(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\square d_{2}(x_{1}-x_{2})~{}\partial_{\mu}h_{\nu\rho}(x_{1})\partial^{\rho}h^{\mu\nu}(x_{2})$

We can take $B=0$ and

[TABLE]

$D=\square d_{2}(x_{1}-x_{2})~{}[\partial_{\mu}u^{\mu}(x_{1})\partial_{\nu}\tilde{u}^{\nu}(x_{2})+(1\leftrightarrow 2)]$

We can take $B=0$ and

[TABLE]

We have proved the formula from the statement. If we make $d_{2}\mapsto d_{2}^{F}$ in the expressions $B$ and $B^{\mu}$ we obtain the expressions $B^{F}$ and $B^{F,\mu}$ so the chronological product $T_{(1)}(T(x_{1}),T(x_{2}))$ is also a relative coboundary. $\blacksquare$

Remark 6.2

We emphasize that in the preceding proof we did not need the explicit expressions for the coefficients of the $38$ monomials from (5.11). Moreover, it can be proved that a generic expression of the type (5.11) has $126$ generic terms and all of them are coboundaries, as above.

Let us try to establish cohomological formulas as in the preceding theorem for the loop contributions of the other chronological products. We easily get from (1.3) that

[TABLE]

If we use the formula obtained in the preceding theorem we obtain:

[TABLE]

We cannot apply Poincaré lemma because the cochains space has the particular form (6.2) and the usual homotopy formula used to prove Poincaré lemma does not preserve this form. In the absence of a better idea we can proceed by a brute force analysis. We consider cochains $D^{\mu}(x_{1},x_{2})$ of the form (6.2) of ghost number $1$, canonical dimension $6$, Lorentz covariant and verifying the cocycle condition

[TABLE]

we try to see if such an expression is a coboundary i.e. of the fom

[TABLE]

with the expressions $B^{[\mu\nu]\emptyset}$ and $B^{[\mu][\nu]}$ also cochains of the form (6.2). A hard computation shows that the cohomology is non-trivial. Every cocycle is cohomologous with a cochain of the form

[TABLE]

which is a non-trivial cocycle. So, without explicit computations, we can say that

[TABLE]

If we want to prove that $\lambda=0$ such that $D_{(1))}(T^{\mu}(x_{1}),T(x_{2}))$ is a relative coboundary, then one must use the explicit expressions. Indeed, one can prove that neither $D_{(1))}(T^{\mu}(x_{1}),T(x_{2}))$ or $d_{Q}B^{\mu}(x_{1},x_{2})$ appearing in (6.48) have contributions proportional to $D_{0}^{\mu}(x_{1},x_{2}).$

We iterate the preceding argument in the sector of ghost $2$ and we obtain in the same way a formula of the type (6.49) i.e. a cocycle condition of the type

[TABLE]

where the expressions $D$ are cochains of the form (6.2). We want to establish if these expressions are coboundaries i.e. of the form:

[TABLE]

with $B$ cochains of the form (6.2) and of ghost number $2$. By a long computation one can establish that this is true. As a consequence, the expression $D^{IJ}_{(1)}(x_{1},x_{2}),~{}|I|+|J|=2$ is a relative coboundary.

Finally, because the expression $D_{(2)}(x_{1},x_{2})$ is proportional to $\square^{2}d_{3}$ it is easy to put it in the form of a coboundary of the form $\delta B.$

7 Conclusions

We have proved that the loop contributions to the causal commutator $D^{IJ}_{(1)}$ are of the form $\bar{s}B$ in the pure gravity case. Because the expressions $B$ have also causal support this property stays true after causal splitting. This means that, in the second order of the perturbation theory, the physical contributions is the tree contribution which correspond to the classical theory. We conjecture that this result stays true in all orders of the perturbation theory. This conjecture might be true in the non-perturbative case also.

We remark that our approach differs from some other recent approaches to quantum gravity without ghosts. In these aproaches one cannot establish a super-renormalizability result as above, so at best, one can treat quantum gravity as an effective theory.