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How to quantize gravity and how not to quantize gravity

Taking the quantization of electromagnetism as the paradigm, we show how this procedure cannot work for Einstein gravity. However, it does work for conformal gravity, a fourth-order derivative, renormalizable theory of gravity that Bender and Mannheim have shown to be ghost free. We show that in any gravity theory gravity cannot be quantized canonically. Rather, because of an interplay between the zero-point energy of gravity and that of its matter source, gravity is quantized purely by its coupling to a quantized matter source, with gravity being intrinsically quantum mechanical. Treating the zero-point energy this way provides a solution to the cosmological constant problem. With the gravitational zero-point energy issue having been ignored in standard Einstein gravity, it is not possible to solve the cosmological constant problem in standard gravity, since without the zero-point contribution gravity does not know where the zero of energy is.

they can be written in the special relativistic form as is. With general relativity they are rewritten as And they do not get modified in form following quantization. What happens instead is that the fields are reinterpreted as q-number operators with nontrivial, -dependent, commutators. The fact that the Maxwell equations do not get modified in form is because quantum electrodynamics is a renormalizable field theory, with matrix elements of the quantum operators in states with an indefinite number of photons obeying the classical Maxwell field equations, albeit with renormalized parameters. This limit is not the limit in which we set to zero, as never is zero in the real world. It instead is a many-particle macroscopic limit in which interference effects cause -dependent terms to mutually cancel each other, with the center of a wave packet for instance following the classical trajectory (Ehrenfest's theorem). Similarly, because of interference, a stationary phase approximation to the Feynman path integral is just one path, the stationary classical path solution to the classical equations. While this stationary solution itself does not depend on , the value of the path integral itself in the stationary solution, viz. e iI ST AT / , does, where I ST AT is the stationary classical action. As we will see below, this quantum-mechanical phase can be measured by gravity, the Colella-Overhauser-Werner experiment. There are no separate classical electrodynamics and quantum electrodynamics. Rather, classical electrodynamics is output to an underlying quantum electrodynamic theory. For instance, in the double slit experiment (Fig. 1) the observed classical pattern on a screen is formed by an underlying quantummechanical behavior at the level of each individual photon, with only their cumulative statistical distribution being observed because large numbers of photons arrive at the screen simultaneously. Each individual photon lands at one point on a screen, with a sequence of many photons then forming a pattern of maxima and minima on the screen, with the positions of the maxima and minima on the screen being given by the classical optical path differences.

Classical electrodynamics follows from quantum electrodynamics and not vice versa
Historically, quantum theory was developed by replacing classical Poisson brackets by quantum commutators (canonical quantization). However, this is doing things backwards, classical follows from quantum not the other way round. Thus the task of quantizing gravity is to find a sensible microscopic quantum theory, with classical gravity being its many-graviton macroscopic limit. If that limit is to have the same form as the quantum gravitational equations, then the theory would need to be renormalizable (cf. conformal gravity), and as has been observed by the present author and by John Donoghue, that would have been the natural choice had quantum field theory and SU (3) × SU (2) × U (1) been developed prior to general relativity.
We thus propose to try to quantize gravity by replicating the treatment given for quantum electrodynamics. While this will work there will be a surprise: gravity cannot be quantized on its own. Rather, it is quantized by its coupling to quantized matter, with there being no 0 term, and we should expand as a power series in Planck's constant and not as a power series in the gravitational coupling constant. This will depend heavily on making sense of the zero-point energy, and is central to solving the cosmological constant problem.
In terms of path integrals, for a gravitational field g µν and generic matter field ψ we look for a gravitational path integral of the form with classical gravitational and classical matter actions I GRAV and I M . We require that, for some appropriate choice of action and some appropriate domain for the path integral measure, the path integral be well defined to all orders of perturbation theory. And if we can find an appropriate action and an appropriate domain then we have a consistent theory of quantum gravity. This procedure does not work for Einstein gravity, and so one has to apply "radical surgery" to quantum field theory and pursue a different approach such as string theory or loop quantum gravity. However, it does work for conformal gravity, as it is renormalizable (it has a dimensionless coupling constant), and its ghost/unitarity problem is resolved [1,2] by continuing the theory into the complex plane, just as is needed to make the conformal gravity path integral exist.
2 Does gravity actually know about quantum mechanics?

Experimental considerations
Before discussing how one might quantize gravity we need to discuss whether we need to. Macroscopically, there are two established sources of gravity that are intrinsically quantum-mechanical: (i) the Pauli degeneracy pressure of white dwarf stars with Chandrasekhar mass M CH ∼ ( c/G) 3/2 /m 2 p , and (ii) the energy density ρ = π 2 k 4 B T 4 /15c 3 3 and pressure p = ρ/3 of the cosmic microwave background black-body radiation in cosmology.

Fig. 2 Colella-Overhauser-Werner experiment
Microscopically, the Colella-Overhauser-Werner experiment [3] shows that the quantum-mechanical phase of the wave function of a neutron of mass m and velocity v is modified as it traverses the gravitational field g of the earth. In the vertical ABCD interferometer shown in Fig. 2 CD lies vertically above AB. The incoming neutron beam splits at A with one component traveling horizontally to B and the other component traveling vertically to C. The components at B and C are then reflected so that they interfere at D. With CD being at higher gravitational potential than AB, interference fringes are seen at D. With a change in the action of the neutron being of the form ∆I = −mgH 2 /v (i.e., change compared to the ABCD interferometer lying in the horizontal), the phase shift is given by ∆φ COW = ∆I/ = −mgH 2 /v (see e.g. [4]), where AB = BD = DC = CA = H, yielding an observable fringe shift at D even though H is only of the order of centimeters and m is the minuscule mass of a neutron -it is just that ∆I is not small on the scale of Planck's constant. Thus gravity can measure the actual value of the stationary action and thus can measure the mass m, even though it drops out of the classical geodesic. The quantum-mechanical version of the equivalence principle is thus that the inertial and gravitational de Broglie wavelengths /m i v and /m g v are equal (i.e., interference in the horizontal and vertical).
On measuring a nonvanishing fringe shift, Colella, Overhauser and Werner provided the first laboratory evidence of its kind that shows that gravity knows about quantum mechanics.

Theoretical considerations and concerns
In equations such as the Einstein equations: we note that if the Einstein tensor is to be equal to the matter field energymomentum tensor, then either both sides are classical c-numbers or both sides are quantum-mechanical q-numbers. Otherwise, if the gravity side were to be classical while the matter side were to be quantum mechanical, then the quantum T µν M would have to be equal to a c-number in every single field configuration imaginable. This is actually not impossible for a specific field configuration since a quantum-mechanical quantity such as the equal time field commutator [φ(t,x), π(t,ȳ)] = i Zδ 3 (x −ȳ) is a c-number -it is just impossible if it has to be true for every field configuration. However, from gravitational experiments described above we know that the source of gravity is quantummechanical, and not only that, we know that gravity knows it. Hence gravity must be quantized.
However, since these gravitational experiments do not involve quantum gravity effects themselves (graviton loops), for phenomenological purposes it is conventional to use a hybrid in which we keep gravity classical but take c-number matrix elements of its source, to give involves products of fields at the same point, so it is not finite. Thus we additionally subtract off the infinite zero-point part and take the source to be the normal-ordered T µν M FIN = T µν M − T µν M DIV instead, to thereby yield: It is in this subtracted form that the standard applications of gravity are made. Thus in Σ(a † a + 1/2) ω we keep the Σa † a ω term but ignore the Σ(1/2) ω zero-point energy density term in T 00 M , precisely as is done in determining the Chandrasekhar mass or the black body contribution to cosmology. Also we ignore the zero-point pressure in the spatial T ij M . As of today there is no known justification for using (2.3), with this subtracted hybrid not having been derived from a fundamental theory or having been shown to be able to survive quantum gravitational corrections. Moreover, it is this very hybrid that is used in cosmology, and the cosmological constant problem is then the phenomenological need to make T 00 M FIN be small. Absent a first-principles derivation of this is not the right starting point for attacking the cosmological constant problem.
There is no apparent reason why the zero point energy density of the matter sector should not gravitate. Moreover, while one only needs to consider energy differences in flat space physics, in gravity one has to consider energy itself, with the hallmark of Einstein gravity being that gravity couples to everything. Hence for gravity one cannot ignore zero-point contributions. And if we throw them away then gravity does not know where the zero of energy is -and that is what creates the cosmological constant problem.
Moreover, even if one does start with the subtracted hybrid, then the order G contribution to gravity is given by the flat spacetime Ω|T µν M |Ω FIN , with Lorentz invariance allowing a finite flat spacetime Ω|T µν M |Ω FIN to be of the generic form −Λg µν . Thus even if one ignores the matter sector zero-point energy contributions one still has a vacuum energy problem; with the standard strong, electromagnetic, and weak interactions typically then generating a huge such Λ. We thus recognize two types of vacuum problem, zero-point and −Λg µν problems. Moreover, if we take gravity to be quantum-mechanical (as we must), it too will have divergent zero-point contributions.
As has been shown in [5] and as we discuss below, rather than the divergent zero-point energy density in the gravity sector being yet another vacuum energy problem, instead it is its interplay with the matter field zero-point contribution that actually leads to a solution to the cosmological constant problem. Thus the cosmological constant problem arises entirely due to ignoring how T µν M FIN got to be finite in the first place, and then using R µν − (1/2)g µν R α α = −8πG T µν M FIN as the starting point. However, in order to be able to discuss gravitational zero-point contributions, we need to quantize gravity. As we now show, such quantization cannot be done canonically, with gravity actually being quantized by the source that it is coupled to, rather than by being quantized on its own.

Canonical quantization of fermion fields
For massless flat space fermion fields with matter action I M = d 4 xi ψ γ µ ∂ µ ψ, stationary fields obey δI M /δψ = i γ µ ∂ µ ψ = 0. We introduce creation and annihilation operators according to Quantizing the fermion field according to {ψ α (x, t), ψ † β (x , t)} = δ 3 (x − x )δ α,β then requires that its creation and annihilation operators obey We construct the fermion field energy-momentum tensor by varying the same fermion action I M (as momentarily written in curved space) with respect to a background metric g µν . In the Fock space vacuum the vacuum matrix element of the energy-momentum tensor is given by We can identify zero point energy density and pressure terms of the form Thus we have two zero-point infinities to deal with, viz. ρ M and p M , and neither behaves like a cosmological constant (for which p = −ρ), since p M and ρ M have same sign. Now the fact that Ω|T M µν |Ω is nonzero is not of concern in flat space field theory, and one can ignore it. Hence canonical quantization is based on stationarity with respect to the matter fields, and it is not of concern if it forces products of fields in the energy-momentum tensor to lead to a nonzero Ω|T M µν |Ω since it can be normal-ordered away. But, what happens if gravity is present, and does this same canonical prescription work for gravity? 4 Quantization of the gravitational field 4.1 Quantization of the gravitational field cannot be canonical For gravity let us introduce some general coordinate scalar action function of the metric I GRAV , and let us identify the energy-momentum tensor of gravity itself according to (Such an identification has been suggested in e.g. [6] and [7].) While the prescription is generic, for the Einstein-Hilbert action for instance we have For the general I GRAV the gravitational equations of motion take the form: and in a typical linearization around flat spacetime of the form g µν = η µν +h µν would lead to a first-order wave equation of the form For the Einstein-Hilbert action for instance we would obtain ∇ α ∇ α h µν = 0 (in the harmonic gauge).
If we now were to quantize h µν canonically by identifying the coefficients of its negative and positive frequency modes as creation and annihilation operators, we would find that the term that is second order in h µν , viz. T µν GRAV (2), would not only not be zero, its vacuum expectation value would actually possess zero-point infinities. However, since the stationarity condition T µν GRAV = 0 is to hold to all orders in h µν , stationarity with respect to the metric requires that Now for gravity we cannot normal order away its zero-point contributions since gravity couples to gravity, and thus we have a contradiction since canonical quantization yields We anyway should not get rid of T µν GRAV (2) since the one-graviton matrix element of T 00 GRAV (2) is the energy carried by a gravity wave. Hence unlike all other fields for which the zero-point contribution does not conflict with canonical quantization, for gravity there is a conflict, since its stationarity condition is a statement about its zero point contribution, and this zero-point contribution involves an infinite product of fields at the same point. Hence, gravity cannot be quantized canonically, and in any quantization prescription for it one would have to take into account its zero-point infinities.

Quantization of gravity via coupling to a quantized matter source
Consider a generic action for the universe consisting of gravitational plus matter actions of the form: Stationary variation with respect to the metric yields If both T µν GRAV and T µν M are well-defined quantum-mechanically, we can treat the condition T µν UNIV = 0 as a quantum-mechanical operator identity, with its finite and divergent parts separately having to obey Moreover, even if T µν GRAV is not well-defined quantum-mechanically, we can still apply T µν GRAV + T µν M = 0 if we can (and which we do) make it be well defined quantum mechanically. Now for massless fermions Ω|T µν M |Ω is given by the quartically divergent whee K is a momentum cutoff. Thus for four-component fermions we get a zero-point energy of −2 ω for each momentum statek, i.e. − ω for each of two two-component fermions. In order to cancel the −2 ω fermion zero-point contribution we need to generate +2 ω in the gravity sector Ω|T µν GRAV |Ω . Since the graviton zero-point contribution is quadratic in h µν , we need h µν to be of order 1/2 .
Moreover, we would have to set h µν = 0 if we tried to quantize gravity on its own, since we would then need to also satisfy the second-order Ω|T µν GRAV (2)|Ω = 0. However, if gravity is coupled to a matter source, then the nonvanishing of the matter Ω|T µν M (2)|Ω requires that the gravitational Ω|T µν GRAV (2)|Ω be nonvanishing too. Hence coupling to a matter source forces gravity to have to choose the nontrivial solution to the first-order Ω|T µν GRAV (1)|Ω = 0, with h µν then being forced to nontrivially be of order 1/2 . Gravity is not quantized on its own. It is quantized by being coupled to a quantum-mechanical source.
To see how the quantization works in detail, for Einstein gravity for instance, we have one massless transverse traceless graviton with two helicity components, but a zero point of only + ω/2 for each momentum state, to give a total of + ω. Thus we need to quantize the graviton field according to the generic and thus need to set The normalization of canonical commutators or anti-commutators in the matter sector and the matter content in the matter sector thus fixes the normalization in the gravity sector, with the gravity sector adjusting to whatever its source might be. It is this interplay that makes T µν GRAV +T µν M be well defined. Still to be addressed is what is to happen when we take higher-order gravitational corrections into consideration, and what is to happen when we generate masses dynamically via a change in vacuum from a normal vacuum |N to a spontaneously broken vacuum |S . To address these issues we need a theory of gravity in which radiative corrections are under control, and so we turn now to conformal gravity, as it is a renormalizable theory of gravity, one for which mass generation would have to be spontaneous since unbroken conformal symmetry requires that all masses be zero. Such an unbroken symmetry also requires that any fundamental cosmological constant be zero, to thus provide a very good starting point to attack the cosmological constant problem.
Moreover, in regard to conformal invariance in general, the fermion-gauge boson sector of the standard SU (3) × SU (2) × U (1) is conformal invariant at the level of the Lagrangian. In fact, this is the reason that they are secondorder gauge theories in the first place, with conformal invariance then requiring dimensionless coupling constants and thus renormalizability. However, the double-well potential with its −µ 2 φ 2 term would violate the conformal symmetry, and thus it must be excluded, with the symmetry breaking then being through dynamical fermion bilinears, to then (a la [8]) yield dynamical fermion masses and a dynamical Higgs boson as a fermion-antifermion bound state (a recent calculation may be found in [5]). Now the standard Einstein gravitational theory is not conformal invariant (which is why it is not renormalizable), and so we turn now to a gravity theory that is, viz. conformal gravity. Thus in the same way that G Fermi is induced by dynamics we look to generate G Newton dynamically too.

Conformal gravity
Conformal gravity is based on the requirement of local conformal invariance of the action under g µν → e 2α(x) g µν . The Weyl conformal tensor possesses a remarkable geometric property, namely that under a local conformal transformation on the metric it transforms as C λµνκ → e 2α(x) C λµνκ with all derivatives of α(x) dropping out identically. Given this fact, a locally conformal invariant gravitational action of polynomial form can then uniquely be prescribed to be of the form: Then, since the coupling constant α g is dimensionless, conformal gravity is power-counting renormalizable. Stationary variation of the conformal gravity action with respect to the metric leads to gravitational equations of motion of the form where (see e.g. [9,10]) with both W µν and a conformal T µν M being both covariantly conserved and traceless.
A remarkable feature of (5.3) is that when T µν M is zero, (5.3) admits of a solution in which R µν is zero. The conformal theory thus recovers the Ricci flat Schwarzschild solution, and thus meets the classical solar system tests of General Relativity without the use of the Einstein equations. However, the vanishing of the fourth-order derivative W µν allows for other solutions, with the general one having four integration constants and being of the form −g 00 = 1/g rr = w − u/r + vr − kr 2 [9,10]. The integration constants are not dimensionless, and must be generated by spontaneous breakdown since the scale invariant W µν itself possesses no scale. In this way G Newton is induced dynamically. The linear and quadratic terms are important at large distance and enable the conformal theory to fit the galactic rotation curves of spiral galaxies with universal parameters and without either dark matter or its two free parameter per galaxy dark matter halos. In [11][12][13] very good fits to 138 galaxies were reported without the need for any of the 276 free parameters needed in dark matter fits to the same galaxy sample 6 Quantizing conformal gravity 6

.1 Linearizing around flat
If we linearize around flat spacetime with a transverse traceless K µν we can set Setting T µν GRAV (1) = 0 has generic solution K µν = A µν e ik·x + (n · x)B µν e ik·x , n µ = (1, 0, 0, 0), (6.2) and full solution [14] K µν (x) = where the fields are quantized according to The fact that some of these commutators are zero is indicative of the Hamiltonian being of nondiagonalizable Jordan-block form with an incomplete set of eigenvectors, a point we return to below. Evaluating now the second-order term, we find that is associated with an action and yields Unlike Einstein gravity this time there are two graviton-type modes to thus give a zero-point of +2 ω. As previously With one four-component massless fermion matter source, through orderh we obtain As we see, Z(k) is only nonzero because the matter zero point Ω|T µν M |Ω is nonzero.

Some phenomenology
If we have M massless gauge bosons and N massless two-component spinors, we obtain  (20) if all fermions are to be in the same multiplet [15]. So quantum conformal gravity implies that the only possible grand-unifying groups with all fermions in the same multiplet are in the quite narrow window consisting of SO(16), SO (18) and SO (20). In general we have Thus the zero-point fluctuations of gravity and matter take care of each other. Moreover, the divergent piece comes from the vacuum while the finite piece comes from excitations out of the vacuum. Thus the measured cosmological constant in cosmology is just the finite piece, and not the entire finite plus divergent pieces combined. Moreover, because of (6.9), i.e., because conformal gravity adjusts to whatever its matter source is, trace anomalies take care of each other since the condition g µν T µν UNIV = 0 is not a conformal Ward identity, but is instead an equation of motion, with the conformal gravity and matter sector trace anomalies having to cancel each other identically. Our ability to solve the traceanomaly problem is because unlike in the standard case we do we not start with (−1/8πG)G µν = T µν M and try to then show that T µν M is anomaly-free all on its own.
Since T µν UNIV = T µν GRAV + T µν M = 0 is an identity, it holds in any vacuum. Thus even after we spontaneously break the conformal symmetry, (T µν GRAV ) FIN will remain traceless since W µν is traceless, and thus so will (T µν M ) FIN . This then constrains the cosmological constant that is induced by the change in vacuum, so that it is neither bigger nor smaller than any other contribution. Now if all particle physics mass and length scales are dynamical, then all such scales are intrinsically quantum-mechanical. However, it is mass and length scales that characterize spacetime curvature. Hence in the absence of quantum mechanics there can be no curvature, and gravity must thus be intrinsically quantum-mechanical.
We thus need to expand the gravitational equations not as power series in the gravitational coupling constant but as a power series in Planck's constant, with no 0 term, just as we found with the effect of the zero point on quantization of the gravitational field. On solving the cosmology associated with conformal gravity we obtain [10] a luminosity distance (d L ) versus redshift (z) relation of the form with the current era deceleration parameter being q 0 , and with a deceleration parameter q that the conformal theory constrains to obey −1 ≤ q ≤ 0 (6.13) in all epochs, no matter what values the parameters in (T µν M ) FIN take. I.e., the theory is naturally accelerating in all epochs without any fine-tuning of parameters. Equation (6.12) is found [10] to give a very good fit (see Fig. 3) to the accelerating universe data [16,17] without any fine tuning; with the fitted value obtained for q 0 being found to be q 0 = −0.37, viz. right in the −1 ≤ q 0 ≤ 0 range required by (6.13). In the figure we have also included the standard Ω M (t 0 ) = 0.3, Ω Λ (t 0 ) = 0.7 fit for comparison. As we see, at high enough redshift the two fits behave markedly differently. For further comparison purposes we have also included the so-called coasting or empty universe with Ω M (t 0 ) = 0 and Ω Λ (t 0 ) = 0, so that q 0 = 0.

Ghosts and unitarity
Since the conformal gravity coupling constant α g is dimensionless, conformal gravity is power-counting renormalizable. However, since the action contains four derivatives of the metric, the equations of motion are fourth-order derivative equations, and for a long time were thought to lead to states of negative norm (ghost states) and violations of unitarity. However, Bender and Mannheim [1,2] have reanalyzed fourth-order theories and shown that they have to be reinterpreted as P T theories in which the inner product is given as the overlap of a ket with its P T conjugate rather than with its Hermitian conjugate; with the Hilbert space then being free of any states with either negative norm or negative energy. Conformal gravity is thus recognized as a consistent quantum theory of gravity, one expressly formulated in four spacetime dimensions, the only spacetime dimensions for which we have any evidence.

Why there cannot be a ghost problem
Consider the Dirac action for a massless fermion coupled to a background geometry of the form where the V µ a are vierbeins and is the spin connection that enables I D to be locally Lorentz invariant.
As constructed, Γ µ also enables I D to be locally conformal invariant under We thus get local conformal invariance for free, with the massless fermion Dirac action naturally having an underlying local conformal symmetry. 't Hooft [18] has also argued that there should be an underlying local conformal symmetry in nature. We now introduce the path integral D[ψ]D[ψ] exp iI D = exp(iI EFF ), and on performing the path integration on ψ andψ obtain an effective action with leading term [19] where C is a log divergent constant. Thus in the standard model itself we generate the fourth-order derivative conformal gravity action. But the standard model is ghost free. Since the fermion path integral is equivalent to a one loop Feynman diagram and since one cannot change the signature of a Hilbert space in perturbation theory, conformal gravity must be ghost free too. And if it were not the standard SU (3)×SU (2)×U (1) model would not remain unitary when coupled to gravity. Radiative corrections to SU (3) × SU (2) × U (1) in an external gravitational field generate fourth-order conformal gravity whether we like it or not. So we have to deal with it one way or another. 9 Why do people think that there is a ghost issue?
We can write the conformal gravity 1/k 4 propagator as the limit with some of the poles having residues that are negative, to thus be associated with ghost states with negative norm. That there might be something wrong with this conventional reasoning can be seen by writing the 1/k 4 propagator as a different limit and now there are no negative residue concerns. The first case has two sets of poles (i.e., two gravitons), while the second one only has one set of poles (i.e., one graviton). So which is right?
10 So what is wrong with the conventional reasoning?
A propagator is a c-number and we cannot determine the structure of the underlying quantum theory from it. One might think that the propagator represents a matrix element such Ω|T [φ(x)φ(0)]|Ω where φ represents g µν . And if it does then there would be states of negative norm, since one can recover the 1/(k 2 − M 2 ) − 1/k 2 propagator by inserting the completeness relation However, when Bender and Mannheim constructed the underlying quantum theory they found that (10.1) did not hold. Specifically, they quantized the fourth-order theory and found that the states of the theory were not normalizable. Since the completeness relation given in (10.1) assumes normalizable states it is not valid. With the states not being normalizable the Hamiltonian is not Hermitian (we cannot integrate by parts and throw away surface terms). However, all the poles of the propagator are on the real axis. Thus despite the lack of Hermiticity all energy eigenvalues are real. Now Hermiticity is only sufficient for real eigenvalues, with the necessary condition being that the Hamiltonian have an antilinear symmetry such as P T [20,21]. The theory thus falls into the class of P T theories developed by Carl Bender [22,23]. In these theories the bra state is not the Hermitian conjugate of the ket state but the P T conjugate of it. Thus the propagator should be identified as Similarly, the completeness relation should be given by with all n P T |m norms being positive definite [1] even as the insertion of (10.3) into (10.2) recovers the 1/(k 2 − M 2 ) − 1/k 2 propagator [2]. Since the states are not normalizable we have to continue into the complex plane in order to make them normalizable. Then the theory is well-defined, with the domain of the measure needed for the path integral also having to be continued into the complex plane in order to make it be well-defined too.

One more surprise
The partial fraction decomposition of the propagator into becomes invalid when we set M 2 = 0. This singular behavior translates into there not actually being two sets of gravitons after all. The Hamiltonian becomes of nondiagonalizable Jordan-block form and no longer possesses a complete set of eigenstates. Thus we finish up with only one set of gravitons, and the correct representation of the propagator is as the limit: with the conformal gravity theory being ghost free and unitary. To see what happens to the eigenstates consider ω 1 = ω + , ω 2 = ω − . In the limit → 0 we obtain e iω1t → e iωt , e iω2t → e iωt . (11.3) Thus both wave functions collapse onto the same wave function, and so we lose an eigenstate. Now consider e iω1t − e iω2t 2 → ite iωt . (11.4) This second combination has a well-defined, nonsingular limit, and in the limit becomes nonstationary. So it is no longer an energy eigenstate. Since the two one-particle states were orthogonal to each other before we took the limit, and since they become the same eigenvector in the limit, in the limit the surviving eigenvector is both parallel to and orthogonal to itself. It is thus has zero norm. This is just as we obtained from quantizing conformal gravity, viz.
[ with the existence of commutators that are zero indicating the presence of zero norm states.

Summary
The Einstein equations were first derived for solar system physics. If we continue the Einstein equations to galaxies we get the dark matter problem. If we continue the Einstein equations to cosmology we get the cosmological constant problem. If we continue the Einstein equations to strong gravity we get the singularity problem. If we quantize the Einstein equations we get renormalizability and zero-point problems. Conclusion: the standard extrapolation of solar system wisdom is not reliable. If all these problems thus have a common origin, then maybe they have a common solution. So we seek an alternate extrapolation. Hence conformal gravity, a theory that contains the Schwarzschild solution and controllable departures from it, that is renormalizable and unitary, and that forbids any fundamental cosmological constant at the level of the Lagrangian. In a renormalizable theory of gravity the equations of motion can be treated as well-defined operator identities of the form: They couple the gravity sector infinities to those in the matter sector, with the matter sector quantization forcing the gravity sector to be quantized so as to cause all these infinities to mutually cancel each other while fixing the gravitational renormalization constant Z in the gravity sector. These equations hold in no matter what states one takes matrix elements, but in a mass generating spontaneously-broken vacuum the gravitational Z will readjust [5] so as to then control the cosmological constant that is induced by the breaking. If all mass and length scales come from symmetry breaking, then all scales come from quantum mechanics, i.e., without length scales there cannot be any curvature. Thus spacetime curvature is intrinsically quantum-mechanical and gravity is quantized by its matter source.
To conclude, we note that at the beginning of the 20th century studies of black-body radiation on microscopic scales led to a paradigm shift in physics. Thus it could that at the beginning of the 21st century studies of black-body radiation, this time on macroscopic cosmological scales, might be presaging a paradigm shift all over again.