Making the Case for Conformal Gravity
We review some recent developments in the conformal gravity theory that has been advanced as a candidate alternative to standard Einstein gravity. As a quantum theory the conformal theory is both renormalizable and unitary, with unitarity being obtained because the theory is a symmetric rather than a Hermitian theory. We show that in the theory there can be no a priori classical curvature, with all curvature having to result from quantization. In the conformal theory gravity requires no independent quantization of its own, with it being quantized solely by virtue of its being coupled to a quantized matter source. Moreover, because it is this very coupling that fixes the strength of the gravitational field commutators, the gravity sector zero-point energy density and pressure fluctuations are then able to identically cancel the zero-point fluctuations associated with the matter sector. In addition, we show that when the conformal symmetry is spontaneously broken, the zero-point structure automatically readjusts so as to identically cancel the cosmological constant term that dynamical mass generation induces. We show that the macroscopic classical theory that results from the quantum conformal theory incorporates global physics effects that provide for a detailed accounting of a comprehensive set of 138 galactic rotation curves with no adjustable parameters other than the galactic mass to light ratios, and with the need for no dark matter whatsoever. With these global effects eliminating the need for dark matter, we see that invoking dark matter in galaxies could potentially be nothing more than an attempt to describe global physics effects in purely local galactic terms. Finally, we review some recent work by ’t Hooft in which a connection between conformal gravity and Einstein gravity has been found.
Keywords:conformal gravity, quantum gravity, cosmological constant problem
pacs:04.60.-m, 04.50.Kd, 04.90.+e
1 Einstein gravity: what must be kept
Following his development of special relativity, Einstein was faced with two immediate problems. The first was to make Newtonian gravity compatible with relativity, and the second was to develop a formalism in which not only uniformly moving observers but also accelerating ones would all be able to agree on the same physics. While these two issues are logically independent (even in the absence of gravity one has to able to write Newton’s second law of motion in an observer-independent way), by imposing general coordinate invariance and by identifying the spacetime metric as the gravitational field, Einstein was able to provide a solution to both problems simultaneously. In this formalism a central role is played by the Christoffel symbols
since in terms of them one can show that the path that minimizes the distance
between two points in any geometry (curved or flat) is the one that obeys the geodesic equation
The key feature of this equation is that even though neither of the two terms that appears in (3) is itself a true coordinate vector, the linear combination of them is, with the vanishing of their sum in any given coordinate system ensuring its vanishing in any other.
Even though the Christoffel symbols are not themselves true coordinate tensors, from them one can construct a quantity that is, viz. the Riemann tensor as defined by
The utility of this tensor is that a given spacetime will be flat if and only if every component of is zero. When all components of are zero, (3) describes Newton’s second law of motion for a free particle in the absence of gravity as viewed in an accelerating coordinate system. And when is non-zero, there is a choice of values for the Christoffel symbols (viz. the Schwarzschild metric ones that are associated with the vanishing of the Ricci tensor ) that enables (3) to describe Newton’s law of gravity, again in an arbitrary accelerating coordinate system. Then, with the Schwarzschild metric also giving relativistic corrections to Newtonian gravity, the observation of the predicted gravitational bending of light by the Sun established the validity of the above description of nature.
One can thus say with confidence that gravity is a covariant metric theory in which the metric describes the gravitational field, and that the geometry in the vicinity of the Sun is given by the Schwarzschild metric
up to the perturbative order to which the metric has so far actually been tested. Thus any viable theory of gravity must embody all of the above, and we note that in the above we have not specified the equation of motion that is to be obeyed by the gravitational field. Rather, we have indicated only what its solution on solar system distance scales needs to look like.
2 Einstein gravity: what could be changed
To complete the theory one thus needs to specify the gravitational field equations themselves. To this end Einstein postulated that the needed equations are to be of the form
a set of equations that can be obtained via functional variation with respect to the metric of an action for the Universe of the form
In (7) and (8) denotes the matter field sector and is the matter field energy-momentum tensor. Given (7), we immediately see that in the source-free region where the Ricci tensor has to vanish, with the solution given in (5) and (6) then following in the region exterior to a static, spherically symmetric source.
is the Weyl conformal tensor. Functional variation of this action with respect to the metric leads to the equation of motion (see e.g. Mannheim2006 ())
to thus yield a gravitational theory that also has as a vacuum solution.111Since is constructed as the functional variation with respect to the metric of a gravitational action that is both general coordinate invariant and locally conformal invariant, it kinematically obeys , . Likewise a matter energy momentum tensor constructed as the functional variation with respect to the metric of a matter action that is general coordinate and locally conformal invariant will obey , . While the tracelessness of forbids the matter fields from having kinematical or mechanical masses, it is important to note Mannheim2006 () that it it does not prevent them from acquiring dynamical masses via spontaneous symmetry breaking. Specifically, as for instance seen explicitly in Sec. (7) below, the scalar order parameter that gives a fermionic matter field a mass also carries energy density and momentum that serve to maintain . Moreover, since in flat space it does this by adding a cosmological constant term on to the standard kinematic perfect fluid (see e.g. (45) below and Mannheim2006 ()), it does not affect energy differences or the conservation of . In flat space is not observable, and one can use the non-traceless to describe macroscopic systems. It is only in its coupling to gravity that presence of can be felt. This analysis thus shows that the Einstein equations given in (7) are only sufficient to give the Schwarzschild solution and its non-relativistic Newtonian limit but not necessary.
The Einstein equations are thus not uniquely selected. Moreover, in and of itself, the requirement that the gravitational action be a general coordinate scalar does not at all restrict the gravitational sector of the action to be of the Einstein-Hilbert form given in (8), with the number of possible general coordinate invariant gravitational actions that one could write down actually being infinite, since one could use arbitrarily high powers of the Riemann tensor and its contractions. This lack of uniqueness is familiarly reflected in the fact that one is free to augment (8) with a term of the form where is the cosmological constant.
Beyond this we note that if one takes (7) as a given and extrapolates it beyond its weak classical gravity solar system origins, one runs into difficulties in essentially every type of extrapolation that one could consider. Thus if one extrapolates the classical theory to galaxies and clusters of galaxies one runs into the dark matter problem, if one extrapolates the classical theory to cosmology one runs into the cosmological constant or dark energy problem, if one extrapolates to strong classical gravity one runs into the singularity problem, and if one quantizes the theory and extrapolates far off the mass shell one runs into the renormalizability and zero-point problems. Now even though no dark matter has yet been detected and dark energy is not yet at all understood, if one nonetheless takes dark matter and dark energy as a given, one then encounters many successes as well (such as big bang nucleosynthesis, anisotropy of the cosmic microwave background, strong lensing). However, to achieve these successes one has to take the energy budget of the Universe to be of order 70% dark energy and 25% dark matter, with only 5% or so being regular luminous baryonic matter. Not only is there as yet no detection of the needed dark matter particles, the required amount of dark energy is 60 orders of magnitude or so less than the amount expected from fundamental elementary particle physics – and if one were to use the large particle physics value the fits would be disastrous. Moreover, with all applications to date of gravity to astrophysics and cosmology having been made with gravity itself being treated classically, there is no guarantee that the current successes of standard Einstein gravity would not be modified by its non-renormalizable quantum corrections. Given these concerns we shall thus look for a completely different extrapolation of solar system wisdom, in a theory that is unambiguously specified. As we shall see, via the imposition of a particular invariance principle, namely local conformal invariance, none of the above extrapolation problems or ambiguities will any longer be encountered.
3 Conformal gravity: an ab initio approach
To see how things work, we note that if we start with the kinetic energy of a free massless fermion in flat spacetime, then in order to obtain an action that is locally gauge invariant under (we suppress internal symmetry group indices), we introduce a gauge field that transforms as , and minimally couple according to
Similarly, if we start with the kinetic energy of a free massless fermion in flat spacetime, in order to obtain an action that is locally coordinate invariant, we introduce the fermion spin connection , with the action taking the form
with and , and with being a vierbein and the four being the special-relativistic Dirac gamma matrices.
Having now obtained (14) this way, we find that without having required it explicitly, this action actually has an additional symmetry, as it is locally conformal invariant under , , . Consequently, we can regard the spin connection as being introduced not to maintain local coordinate invariance but rather to maintain local conformal invariance instead, in exactly the same minimally-coupled way that maintains local gauge invariance. Thus if we require that the kinetic energy of a massless fermion be invariant under complex phase transformations of the form , we will be led to an action
that is locally gauge invariant and locally conformal invariant combined. (Under a local gauge transformation the metric transforms as , while under a local conformal transformation the gauge field transforms as .)
The reason why such a local conformal structure emerges in (14) is that massless particles move on the light cone, and the light cone is not just Poincare invariant, it is invariant under the full 15-parameter conformal group . (If the line element is zero, then so is .) Moreover, the covering group of is . Since is generated by the 15 Dirac matrices (, , , ), its fundamental representation is a fermionic field, and the full conformal structure of the light cone is thus built into a massless fermionic field, with a 4-component Dirac spinor being irreducible under the conformal group even as it is reducible under Poincare.
As we see, it is thus natural to take fermions to be the most basic elements in physics, with internal symmetry gauge fields and a gravitational spin connection being induced once one gives the fermion kinetic energy a local complex phase invariance. However, such a starting point does not generate any kinetic energy terms for the gauge and gravitational fields. To actually generate them rather than just postulate them we take note of a calculation by ’t Hooft 'tHooft2010 (). Specifically, ’t Hooft evaluated the logarithmically divergent part of the path integral associated with (15), and found that after dimensional regularization it took the form of an effective action:
where in spacetime dimension . Comparing with (9) we see that we have generated none other than the conformal gravity action (as then rewritten using the Gauss-Bonnet theorem) together with the Maxwell action, and take note of the fact that this procedure did not generate the Einstein-Hilbert action. With the Maxwell action being invariant under and with the conformal gravity action being invariant under (see e.g. Mannheim2006 ()), the conformal gravity action thus serves as the gravitational analog of the Maxwell action, and in the following we shall thus use local conformal invariance as the principle with which to fix the structure of the gravitational action. In so doing we see that gravity can be generated by gauging the full conformal symmetry of the light cone.
Given the assumption of local conformal invariance, we find that the conformal gravity action is the unique gravitational action that is invariant under the local transformation , with the gravitational coupling constant being dimensionless. Because the coupling constant is dimensionless, the conformal theory is power-counting renormalizable, and thus if we did not include an initial term in the action, we would anyway generate one as a renormalization counter-term. Because the variation of the conformal action leads to fourth-order equations of motion, it had long been thought that the theory would not be unitary. However, as we describe in Sec. (5), Bender and Mannheim Bender2008a (); Bender2008b () have recently shown that one can find a realization of the theory that is unitary. Consequently, conformal gravity is to be regarded as a bona fide quantum gravitational theory. Moreover, the similarity of the theory to Maxwell theory also carries over to the generation of a macroscopic classical limit starting from a microscopic quantum field theory. Specifically, in just the same way as the classical Maxwell equations emerge from the quantum Maxwell equations as matrix elements of the quantum fields in states containing an indefinite number of photons, because of its renormalizability the same will happen for conformal gravity in states containing an indefinite number of gravitational quanta, with both the quantum theory and its macroscopic classical limit obeying the equation of motion given in (11). Consequently, in the following we will be able to use (11) to study both the microscopic zero-point fluctuation problem and the macroscopic behavior of the theory on astrophysical distance scales.
The non-renormalizable Einstein-Hilbert action is expressly forbidden by the conformal symmetry because Newton’s constant carries an intrinsic dimension. However, as noted above, this does not prevent the theory from possessing the Schwarzschild solution and Newton’s law of gravity. In addition, the same conformal symmetry forbids the presence of any intrinsic cosmological constant term as it carries an intrinsic dimension too; with conformal invariance thus providing a very good starting point for tackling the cosmological constant problem.
Now we recall that the fermion and gauge boson sector of the standard model of strong, electromagnetic, and weak interactions is also locally conformal invariant since all the associated coupling constants are dimensionless, and gauge bosons and fermions get masses dynamically via spontaneous symmetry breaking. Other than the Higgs sector (which we shall shortly dispense with), the standard model Lagrangian is devoid of any intrinsic mass or length scales. And with its associated energy-momentum tensor serving as the source of gravity, it is thus quite natural that gravity should be devoid of any intrinsic mass or length scales too. Our use of conformal gravity thus nicely dovetails with the standard model. To tighten the connection, we note that while the standard model is based on second-order equations of motion, an electrodynamics Lagrangian of the form would be just as gauge and Lorentz invariant as the Maxwell action, and there is no immediate reason to leave any such type of term out. Now while an theory would not be renormalizable, in and of itself renormalizability is not a law of nature (witness Einstein gravity). However, such a theory would not be conformal invariant. Thus if we impose local conformal invariance as a principle, we would then force the fundamental gauge theories to be second order, and thus be renormalizable after all. However, imposing the same symmetry on gravity expressly forces it to be fourth order instead, with gravity then also being renormalizable. As we see, renormalizability is thus a consequence of conformal invariance.
Now if the underlying theory is to be locally conformal invariant, there would be no place for a fundamental Higgs field with its tachyonic double-well potential. Instead, mass scales would have to be generated dynamically in the vacuum via dynamical fermion bilinear condensates. The elimination of a fundamental Higgs field has an immediate benefit – one no longer has to deal with the uncontrollable cosmological constant contribution it produces when it acquires a non-vanishing expectation value. However, one still has to make contact with the standard model, and one would thus want to obtain a standard model Lagrangian with some effective scalar field. Such an effective scalar field would have to emerge as a Ginzburg-Landau c-number order parameter, i.e. as the matrix element of a fermion bilinear operator in some possibly spacetime-dependent coherent state. Such an effective c-number scalar field would not be observable in an accelerator.
To see what such an effective theory might look like, we note that if the fermion acquires a mass parameter by some dynamical symmetry breaking mechanism, the associated Hartree-Fock mean-field action would take the same form as given in (15), only with a mass term added, viz.
Evaluating the logarithmically divergent part of the same path integral as before only with (17) this time generates the same as in (16), while adding on the mean-field action 'tHooft2010 () (as written here using the sign convention employed in this paper for ):
Here is the same logarithmically divergent constant as before. Finally, if we give a group index, the same procedure would cause the terms to be replaced by covariant gauge derivatives (see Eguchi1974 (); Mannheim1976 ()), and would yield
When (19) is taken in conjunction with (16), a conformally coupled standard model emerges, but with a c-number scalar parameter that is not a fundamental field. In Secs. (7) and (8) we will show how such a mass parameter could be generated as a fermion bilinear condensate in a conformal invariant theory. And in Mannheim2009a () we explore the degree to which the existence of such a c-number order parameter necessitates the existence of an accompanying dynamical bound state scalar particle.
4 Quantization of gravity through coupling
In trying to find solutions to the conformal theory, we note that given the lack of any intrinsic mass or length scales in the conformal action, without dynamical generation of such scales there could be no non-trivial solutions to the theory. Thus if all mass and length scales are to come from dynamics, and all such mass-generating dynamics is to be quantum-mechanical (c.f. no fundamental Higgs fields), the only allowed geometry in purely classical conformal gravity would be one with no curvature at all, viz. a geometry that is Minkowski. (For there to be any curvature one needs some length scale to characterize it.) Thus if one takes to be a classical fourth-order derivative function, then even though one could readily find an exterior vacuum solution to such as the Ricci-flat one given in (5) and (6), there would be no basis for taking the dimensionful parameter to be non-zero since the theory is as yet scale free. For the solution to a differential equation to have lower symmetry than the equation itself it (i.e. for a solution to contain symmetry-breaking integration constants that are not present in the equation itself), there has to be a spontaneous breakdown of the symmetry. If all symmetry breaking is to be quantum-mechanical, then in the absence of quantum mechanics, geometry would have to be flat. Thus, if the mass that appears in the Schwarzschild radius of a source is quantum-mechanically generated, then the curvature produced by that mass is ipso facto due to quantum mechanics too.
The above remarks require some clarification since in classical electrodynamics one can construct plane wave solutions with even though the classical Maxwell equations are conformal invariant. Specifically, if we have a free classical Maxwell action in flat spacetime, the associated homogeneous wave equation takes the form (in the convenient Lorentz gauge). However, while this equation has as a solution, this solution involves a dimensionful four-vector momentum that is not present in the equation of motion itself. Hence some mechanism is required to generate such a four-momentum. In classical Maxwell theory the mechanism for doing this is not from the sector of the theory at all, but rather via the introduction of a localized source . In the presence of the source the solution to the inhomogeneous is given as where is the massless retarded propagator. Now itself is written entirely in terms of the spacetime coordinates and contains no fundamental scales. Rather, the scales reside in the localized , and if it for instance oscillates with a specific frequency, then the resulting will oscillate with that same frequency too. However, for to be localized and possess such an oscillation frequency, it would not be scale invariant. Hence the classical Maxwell field only possesses frequency scales because the sources to which it couples are taken to possess them, and the sources themselves can only possess such scales if they are not scale invariant. However, if all the particles contained in electromagnetic sources are to acquire length scales quantum-mechanically, then there could be no fundamental classical electromagnetic sources that could possess such length scales in the first place. Classical electromagnetism with localized oscillating sources is thus a macroscopic manifestation of an underlying microscopic quantum Maxwell theory in which scales are generated dynamically. In a truly scale-free classical electrodynamics there could not be any electromagnetic waves. Thus just like gravity, the same conformal invariance will not permit electromagnetic sources to have any nontrivial intrinsically classical component either, with such sources being intrinsically quantum-mechanical.
If momentum modes are not to arise in classical physics, one needs to ask how it is that they then do arise. To this end we note that as well as generating mass scales via dynamics, there is another way in which quantum mechanics produces scales, namely via the quantization procedure itself. Specifically, scales are introduced via canonical commutation relations, with the generic equal-time commutation relation for instance being a non-linear relation that introduces a scale everywhere on a spacelike hypersurface. Since we can set , this is equivalent to introducing a complete basis of momentum modes, with momentum modes thus being intrinsically quantum mechanical. And indeed it is the very existence of this set of modes that gives rise to the zero-point energy density and pressure of a quantized field that we discuss in the following. Moreover, a quantized field will have a zero-point energy density and pressure even when it is massless, i.e. even in the absence of mass generation. Then, when there is mass generation, the momentum modes will obey the mass condition and cause the massless theory zero-point energy density and pressure to readjust. And as we show in Secs. (7) and (8), this readjustment will cancel the cosmological constant that is induced by the same mass generation mechanism.
Given the above remarks, we see that in conformal gravity we should expand the metric as a power series in Planck’s constant rather than as a power series in the gravitational coupling constant, with the zeroth-order term in the expansion being flat. Consequently, in the theory there is no intrinsic classical gravity, with the equations that are to be used for macroscopic systems being associated with matrix elements of the quantum fields in states with an indefinite number of gravitational quanta, Since there is to be no intrinsic classical gravity, there could not be any classical black holes. While conformal gravity thus eliminates the classical gravity singularity problem, and thus simultaneously eliminates the need to have to make such classical singularities compatible with quantum mechanics, it remains to be seen whether the theory might still generate geometric singularities through quantum-mechanical effects, though one might anticipate that the uncertainty principle might spread sources out enough to prevent this from happening.
Even with the requirement that the metric be expanded as a power series in Planck’s constant, quantization of gravity can still not follow the standard canonical quantization prescription that is used for other fields. Specifically, for a matter field one obtains its equation of motion by varying the matter action with respect to the matter field, but one obtains its energy-momentum tensor by instead varying the matter action with respect to the metric. Since involves products of matter fields at the same point, a canonical quantization of the matter field then gives the matter energy-momentum tensor a non-vanishing zero-point contribution. However, in a standard quantization procedure for a given matter field, the non-vanishing of violates no constraint since one does not simultaneously impose the equation of motion of any other field. Thus for a given matter field one does not require stationarity with respect to the metric, with thus not being constrained to vanish.
In contrast however, for gravity the relevant field is the metric itself. If we define the variation of the gravitational action with respect to the metric to be a quantity , the gravitational equation of motion is then given by .222In S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972) and also in P. D. Mannheim, Phys. Rev. D 74, 024019 (2006) it is suggested that one treat the functional variation of the gravitational action with respect to the metric as the energy-momentum tensor of gravity. Specifically it was noted that if in Einstein gravity one perturbs the Einstein equations around some background according to , the first-order term in gives the wave equation obeyed by , while the term that is quadratic in is both covariantly conserved with respect to the background metric and gives the energy density carried by a gravity wave. In Sec. (6) below we provide an equivalent analysis in the conformal case. Then, with containing products of fields at the same point, a canonical quantization of the gravitational field would give a zero-point contribution to , and thus violate the stationarity condition that has to obey. Hence, unlike the matter fields for which there is no constraint on in the absence of any coupling of matter to gravity, gravity itself is always coupled to gravity, with its stationarity condition not permitting it to consistently be quantized on its own.
Despite this, we note that if we impose a stationarity condition with respect to the metric not on the gravity piece or the matter piece alone, but on their sum as given by the total of the universe introduced in (9), we then obtain
In this case it now is possible to quantize gravity consistently, with now being able to be non-zero provided gravity is coupled to some quantized matter field source for which is non-zero. Thus gravity can only be quantized consistently if it is coupled to a quantized matter field. However, in order for the cancellation required of the total to actually take place, the quantization condition imposed on the gravitational sector commutation relations would have to be fixed by the quantization condition in the matter sector in order to enforce , with each term being intrinsically quantum-mechanical. Consequently, gravity is not only quantized though its coupling to quantized matter, its commutation relations are explicitly determined by that coupling, with gravity needing no independent quantization of its own. Finally, we note that not only do the matter fields quantize gravity, the vanishing of entails that the gravity field and the matter field zero-point fluctuations must cancel each other identically. In Secs. (6), (7) and (8) we explore this point in detail.
5 Unitarity via symmetry
When the tensor given in (LABEL:M12) is linearized around a flat spacetime background with metric according to , it is found Mannheim2009a () to be a function of the traceless quantity . In the convenient transverse gauge , the first order term in is found to take the simple form
while the second order term in the conformal action given in (9) takes the form
Since there is no mixing of components of in either (21) or (22), one can explore the unitarity structure of the theory by working with an analog one-component scalar field theory. As such, the condition is one of a broad class of fourth-order equations of motion that have been encountered in the literature, and all of them can be associated with the generic scalar action
Given this action one obtains an equation of motion
and an energy-momentum tensor with component
These equations immediately possess two well-known realizations that exhibit the problems that higher-derivative theories are thought to possess. If one takes the contour for the integration in the propagator to be the standard Feynman one in which all positive energy modes propagate forward in time and all negative energy modes propagate backwards in time, because of the relative minus sign in (25), one finds that some of the poles have the negative residues that occur in an indefinite metric Hilbert space. To avoid such negative residues, one can find Bender2008b () an alternate contour in which the residues of all poles are positive but in which some of the negative energy modes propagate forward in time. While one can quantize this realization with a standard Dirac norm, as the presence of the term in indicates, in this case the energy eigenvalue spectrum is unbounded from below.
With neither of these two possibilities being palatable, higher-derivative theories have long been regarded as being unphysical. However, recently Bender and Mannheim revisited the issue Bender2008a (); Bender2008b () and found a third realization of the theory in which the energy spectrum is bounded from below and there are no negative Hilbert space norms at all. With the appropriate scalar product for a Hilbert space being determined by boundary conditions, to determine the relevant scalar product one needs some asymptotic information. To this end Bender and Mannheim studied the eigenvalue problem for the Hamiltonian in the sector of the theory where the energy eigenvalue spectrum is bounded from below, and found that the associated wave functions were not normalizable on the real axis. In consequence of this, the Hamiltonian of the system could not be Hermitian. However, the wave functions were found to be normalizable on the imaginary axis, and thus the field would have to be anti-Hermitian rather than Hermitian, with the term in then being bounded from below. In addition they noted that if they constructed a path integral for the system, it would not exist with real but would be well-defined if were pure imaginary. So again, one needs to take to be an anti-Hermitian operator.
Now if a Hamiltonian is not Hermitian, one is immediately concerned that its eigenvalues might not all be real. However, while Hermiticity implies reality of eigenvalues, there is no converse theorem that says that a non-Hermitian Hamiltonian must have complex eigenvalues. Consequently, Hermiticity is only sufficient for reality but not necessary. Recently, as part of the general symmetry program that has been developed by Bender and collaborators Bender2007 () a necessary condition for reality has been found, namely that a Hamiltonian have a symmetry where is a linear operator and is an antilinear one. Specifically, it was shown in Bender2002 () that if a Hamiltonian is invariant the secular equation that determines the eigenvalues is real. Then in Bender2010 () the converse was shown, namely that if the secular equation is real, the Hamiltonian must have a symmetry. Consequently, the energy eigenspectrum of a Hamiltonian that is not symmetric must contain some complex eigenvalues.
Noting now that all the poles in the propagator given in (25) lie on the real axis, we see that the Hamiltonian for the fourth-order theory while not Hermitian must instead be symmetric. For such Hamiltonians one can construct a norm, the so-called norm of theories Bender2007 (), that obeys unitary time evolution. For our purposes here we note that for a non-Hermitian Hamiltonian that has a completely real energy eigenspectrum, and must be related by a similarity transform of the form
Thus if has a right-eigenvector according to with real , its conjugate will obey and will not be a left-eigenvector of . Rather, the state defined as will be a left-eigenvector of since it obeys
In this case it will be the norm that will obey unitary time evolution since it evolves as
Thus we see that the needed scalar product is the overlap of a left-eigenvector with a right-eigenvector and not the overlap of a right-eigenvector with its own conjugate. Moreover, in Mannheim2009b () it was shown that the existence of a that can connect and according to is a necessary and sufficient condition for both the existence of a symmetry and for the existence of a unitary scalar product, with the scalar product then being of the form . The existence of a invariance for a Hamiltonian is thus a necessary and sufficient condition for unitary time evolution.
When these ideas are applied to the fourth-order propagator, it is found Bender2008b (); Mannheim2009a () that the relative minus in it is no longer associated with an indefinite metric at all. Rather, it is associated with operator, with the completeness relation being given by and not by the negative norm . Finally, the propagator itself is found to be given by the Green’s function rather than by the familiar . As we see, the unitarity problem for fourth-order propagators only arose because one wanted to represent them as . Once one recognizes that the left vacuum need not be the conjugate of the right vacuum unitarity can then readily be achieved.
Now while the above discussion was developed for the general second- plus fourth-order action given in (23), for the conformal case given in (21) we are interested in a pure fourth-order theory alone where the propagator is given by (a propagator whose poles again are all on the real axis). Since the reduction to a pure fourth-order theory would require setting both and equal to zero in (23), we see that because of the prefactor in (25), the limit is singular. In consequence, the limit is a quite unusual one in which the Hamiltonian becomes a non-diagonalizable, Jordan-block Hamiltonian Bender2008b (); Mannheim2009a (), with some of the states that had been eigenstates being replaced by non-stationary ones. The very fact that the Hamiltonian is not diagonalizable immediately confirms that it could not be Hermitian, just as we had noted above. In this case even though the set of energy eigenstates is not complete, the set of stationary plus non-stationary states combined is complete Bender2008b (), with time evolution of packets built out of the two classes of states combined being unitary Bender2008b (). The unitarity of the pure fourth-order conformal gravity theory is thus established.
As we thus see, in order to establish unitarity for fourth-order theories we need the field , and thus itself, to be anti-Hermitian rather than Hermitian. Now this is not how one ordinarily thinks about the gravitational field, since one would presuppose that it, above all fields, should have a real classical limit. Nonetheless, having an anti-Hermitian gravitational field is is not in conflict with anything that is actually known about gravity. Specifically, in Bender2008b () it was noted that if one replaces by (and thus by ) neither the Christoffel symbols that appear in geodesics nor would be affected at all. In four space-time dimensions would not be affected either. Even though Riemann tensor contractions could generate factors of i, all such factors could be absorbed by redefining the overall multiplicative coefficients in the action (and likewise for the line element). Hence, current gravitational measurements cannot distinguish between a purely real or a purely imaginary gravitational field. And as we have seen, once one takes the gravitational field to be anti-Hermitian, one can construct a consistent, renormalizable and unitary theory of quantum gravity. And perhaps the problems that beset quantum gravity have arisen because one wanted the gravitational field to be Hermitian.
6 The zero-point problem
In current applications of standard gravity to macroscopic astrophysical and cosmological systems, one treats gravity itself as being purely classical. However, one cannot treat its matter source that way too since there are some intrinsically quantum-mechanical sources that are significant macroscopically. Thus, white dwarf stars are stabilized by the Pauli degeneracy pressure of the electrons in the star, and black-body radiation contributes to cosmic expansion.333The Chandrasekhar mass limit for white dwarfs and the Stefan-Boltzmann constant for black-body radiation both intrinsically depend on . Both of these parameters would expressly have to appear in the macroscopic gravitational equations of motion, and for neither of them could one set to zero. To couple these particular effects to classical gravity, both of them are taken to be described as ensemble averages over an appropriate set of positive-energy Fock space states. However, while the Fock space states would be eigenstates of a Hamiltonian of the generic form , the infinite zero-point contribution is ignored, i.e. one takes the Hamiltonian to be of the truncated form instead. Now in flat space one is free to discard the zero-point term (say by a normal ordering prescription) since in flat space one can only measure energy differences. However the hallmark of gravity is that it couple to energy density itself and not to energy density difference, and so discarding anything to which gravity couples would require justification.
Since one would have to cancel infinities in the matter field energy-momentum tensor if the gravitational effects that occur in standard gravity are to be finite, some mechanism needs to be identified that would effect the cancellation. An immediate mechanism that might achieve this would be a cancellation between appropriately chosen matter fields, since bosons and fermions contribute to with opposite signs. In fact such a cancellation will occur if there is an exact supersymmetry between fermions and bosons. However, once the fermion-boson mass degeneracy is broken, the cancellation is lost. With the non-observation to date of any of the requisite superpartners, we know that the supersymmetry breaking scale must be at least in the TeV region, with the uncanceled zero-point energy then being huge.
Nonetheless, the generic idea of a boson-fermion cancellation as enforced by some symmetry still makes sense since fermions and bosons generate vacuum energies with opposite signs no matter what the theory. To take advantage of this, we note that the treatment of standard gravity described above is deficient in one rather serious regard, namely it ignores the effect of quantum mechanics on gravity itself. And as soon as one quantizes gravity, gravity itself will acquire a zero-point contribution. Since gravitational quanta are bosonic, under certain circumstances they may then be able to provide the needed cancellation. Specifically, for such a cancellation to occur one needs three things: a quantum gravity theory that makes sense, a symmetry, and the presence of fermions in the matter field sector. With conformal gravity meeting all of these criteria (as noted above fermions are its building blocks), in the following we will explore its implications for the vacuum energy problem. We will see that when the conformal symmetry is unbroken the needed cancellation does in fact occur. And then, unlike the supersymmetry situation, the cancellation will be maintained even after the conformal symmetry is spontaneously broken and a cosmological constant term is induced.
One of the challenges that gravity theory faces is that zero-point and cosmological constant contributions already occur for matter fields in flat spacetime, i.e. they occur in the absence of gravity. Since gravity is not involved in flat space physics, it is very difficult for gravity to then resolve any problem that it is not responsible for. To enable gravity to resolve such problems we need to put the gravitational field an equal footing with the matter fields. This we can do if there are no intrinsic classical contributions in either the gravitational or the matter sectors and all physics is quantum mechanical, i.e. precisely as conformal symmetry requires. Since the lowest order quantum-mechanical contribution to is a zero-point contribution of order , to cancel it through the vanishing of the total given in (20), we will need the lowest non-trivial gravitational term to be of order too. Since the zero-point contribution is due to products of fields at the same point, the order gravitational zero-point must involve a product of two gravitational fields and thus be given by the second-order () tensor that is obtained by varying the term in (22). To this order in we only need to evaluate in a flat background. It will then generate an order curvature, with the order term in then being curvature dependent. Moreover, with vanishing not only in lowest order but in all orders if both the gravitational and matter field sectors are renormalizable (i.e. in a renormalizable theory (20) is an all-order identity), the zero-point cancellation will occur to all orders. Thus if we decompose and into finite and divergent parts according to , , (20) will decompose into
With (31) we see that all gravitational and matter field infinities cancel each other identically, with and regulating each other. Given this regulation, there is no need to renormalize either of the two terms as their sum is finite, and thus no renormalization anomaly such as the conformal anomaly is generated.444If one did first renormalize each term separately, the associated conformal anomalies would then have to cancel each other identically. Specifically, with the vanishing of being due to stationarity with respect to the metric, such stationarity equally guarantees the vanishing of the trace without any need to impose conformal invariance. Thus even though the vanishing of the individual gravity sector and matter sector traces and do require conformal invariance, and even though conformal symmetry Ward identities might be violated by renormalization anomalies, the vanishing of cannot be affected by the lack of scale invariance of regulator masses. Any anomalies in must thus all mutually cancel each other identically. And with all infinities having been removed, (32) provides us with a completely finite framework for calculating gravitational effects. Thus in (31) we take care of the type terms, and in (32) we are free to use the type terms alone.
With being of order , the gravitational fluctuation must itself be of order , and since the lowest non-trivial term in is of order , it must be the case that in (21) vanish identically. While the vanishing of provides us with a wave equation, the situation is not quite the same as the one that occurs when one expands in a power series in the gravitational coupling constant. Specifically, in that case the first-order fluctuation term on the gravitational side is produced by a first-order fluctuation term on the matter side, so that the gravitational fluctuation would obey an inhomogeneous wave equation with a source. In contrast, in the conformal case the first-order gravitational wave equation is strictly homogeneous on all scales. Then, since this equation is homogeneous, in and of itself it does not force to be non-zero. However, since the order contribution to is non-zero, cannot vanish, and thus cannot vanish either. It is thus quantization of the matter field that forces the gravitational field to be quantized, with the condition fixing the strength of the commutator terms present in the second order . With the matter field fixing the strength of the gravitational sector, the cancelation of both zero-point contributions and conformal anomalies is secured.
To see how things work in detail we consider conformal gravity coupled to a Dirac fermion. To the order of interest to us we can take the fermion to be a free massless fermion in flat spacetime. In this case the matter field energy-momentum tensor is given by , with its vacuum expectation value being given by
where is a lightlike 4-vector with . In (33) we recognize two separate infinite terms, one associated with and the other with . Since the fermion is massless, is traceless and thus these two infinities obey . Such an energy-momentum tensor could not be associated with a cosmological constant term of the form since its trace is given by the non-zero , with the cosmological constant and zero-point fluctuation problems in principle thus being different. Given its structure, the zero-point quantity can be written in the form of a perfect fluid with a timelike fluid velocity vector , viz.
with the fluid thus possessing both a zero-point energy density and a zero-point pressure.555 Even though (33) involves terms that are infinite and thus not well-defined, we note the perfect fluid form given in (34) can be established by integrating over the direction of the 3-momentum vector alone, an integration that is completely finite. A perfect fluid form for (33) can thus be established prior to the subsequent divergent integration over the magnitude of the momentum, with this latter integration not bringing to the form of a cosmological constant. Even though (33) is not well-defined, for our purposes here the perfect fluid form given in (34) is a very convenient way of summarizing the infinities in (33) that need to be cancelled. Since gravity couples to the full and not just to its component, it is not sufficient to only address the vacuum energy density problem, one has to deal with the vacuum pressure as well. There are thus two vacuum problems that need to be addressed, and not just one. The gravitational sector will thus need to cancel both the vacuum energy density and the vacuum pressure of the matter field, and in a conformal theory will readily be able to do so since has an identical traceless vacuum perfect fluid structure. (In a conformal invariant theory the variation with respect to the metric of the pure gravitational sector of the action is automatically traceless.)
For the explicit structure of the gravity sector we follow the discussion given in Mannheim2009a (). On using some residual gauge symmetry the general solution to is given as
as expressed in terms of quantum operators , , and and two transverse traceless polarization tensors (, both of which are normalized to . Since is to not be Hermitian, the creation operators are not the Hermitian conjugates of the annihilation operators. However, in the following all that will matter is that and annihilate the right vacuum while and annihilate the left vacuum. With being equal to , in (35) we recognize the presence of non-stationary modes, which, as noted above, is characteristic of theories with non-diagonalizable Hamiltonians.
As discussed in detail in Mannheim2009a (), the quantization procedure is also characteristic of non-diagonalizable Hamiltonians, with the commutators taking the form
In (36) everything except the possibly dependent renormalization constant is fixed by kinematics (the vanishing of the commutator for instance is needed to cancel all terms contained in ). The constant , however, will be fixed by the dynamics, with the dynamics preventing from being zero and the above commutator algebra from being trivial. Specifically, given (36) we obtain
with the factor appearing in (37) since we have to sum the standard bosonic zero-point energy density per mode over two polarization states of two separate families of massless spin 2 modes (the and sectors). Then with the fermion sector generating a factor of in (33) (the standard fermionic zero-point energy density per mode as summed over negative energy states with spin up and spin down) the cancellation of the fermionic and gravitational contributions in will enforce .
Establishing that is fixed by the coupling of gravity to the matter sector is our key result as it shows that gravity requires no independent quantization of its own, with its quantization strength being fixed by the consistency condition that all zero-point infinities cancel identically. To appreciate the point, it is of interest to take a more general matter source. Thus if we take the source to contain massless gauge bosons and massless two-component fermions (viz. four-component fermion modes), together they will generate units of for each . (For gauge bosons one gets for each of two helicity states.) In this case consistency requires that be given by . This condition shows that cannot be assigned in isolation. Rather it is determined by the dynamics each time. Moreover since must be positive (c.f. no negative norm states) it also provides an interesting constraint on model building, namely that must be greater than . For the standard model for instance, we have gauge bosons and two-component spinors per generation, with then being positive. Intriguingly, for the grand-unified gauge group one has