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Quantum massive conformal gravity

We first find the linear approximation of the second plus fourth order derivative massive conformal gravity action. Then we reduce the linearized action to separated second order derivative terms, which allows us to quantize the theory by using the standard first order canonical quantization method. It is shown that quantum massive conformal gravity is renormalizable but has ghost states. A possible decoupling of these ghost states at high energies is discussed.


Introduction
Massive conformal gravity [1] is a recently developed conformal theory of gravity in which the gravitational action is the sum of the fourth order derivative Weyl action [2] with the second order derivative Einstein-Hilbert action conformally coupled to a scalar field [3].
The gravitational potential of the theory, which is composed by an attractive Newtonian potential and a repulsive Yukawa potential, reproduces the rotation curves of the major number of galaxies. In addition, the momentum space propagators of massive conformal gravity have a good high-energy behavior, which makes the theory power-counting renormalizable. However, one of these propagators has a negative sign between its terms, which is a common feature of fourth order derivative theories of gravity. In such theories this negative sign imply that either the energy eigenvalue spectrum is unbounded from below or the Hilbert space norms are negative [4]. Several attempts to solve the negative energy (or negative norm) problem in higher derivative gravity have been carried out in the literature (see, e.g., [5][6][7][8][9][10]).
In this paper, we analyze the consequences of the negative sign term in massive conformal gravity. In Sect. 2 we derive a second order derivative linearized massive conformal gravity action by introducing auxiliary variables. In Sect. 3 we canonically quantize the massive conformal gravity fields and show a e-mails: felfrafar@hotmail.com; fff@uespi.br that the theory has ghosts, states with negative norm, which do not necessarily spoil the unitarity of the S-matrix. Finally, in Sect. 4 we present our conclusions.

Linearized action
Let us consider the gravitational action of massive conformal gravity, which is given by 1 where α is a dimensionless constants, λ =h/mc (h is the Planck constant and m is the graviton mass), k = 8π G/c 4 (G is the gravitational constant and c is the speed of light in vacuum), is the Weyl tensor, ϕ is a scalar field, R α μβν is the Riemann tensor, R μν = R α μαν is the Ricci tensor, and R = g μν R μν is the scalar curvature. It is worth noting that (1) is invariant under the conformal transformations where θ(x) is an arbitrary function of the spacetime coordinates.
With the help of the Lanczos identity, we can write (1) in the form Then, using the weak-field approximations and keeping only the terms of second order in h μν and σ , we find that (5) reduces to wherē is the linearized Ricci tensor, is the linearized scalar curvature, and is the linearized Einstein-Hilbert Lagrangian density, with h = η μν h μν . The linearized action (8) is invariant under the coordinate gauge transformation where ξ μ is an arbitrary spacetime dependent vector field, and under the conformal gauge transformations where is an arbitrary spacetime dependent scalar field. We can fix these gauge freedoms by imposing the coordinate gauge condition and the conformal gauge condition to (8). However, this procedure is not suitable for the quantum analysis of the theory, since it introduces complication in the definition of the canonical commutation relations. Another procedure to eliminate the gauge freedoms of the theory consists on adding gauge fixing terms to the action such that the field equations obtained from the action plus the gauge fixing terms are the same as the gauge fixed field equations obtained from the action alone. Thus, by adding the gauge fixing terms 2 to (8), and integrating by parts, we obtain the diagonalized action where = ∂ ρ ∂ ρ and In order to obtain a first order canonical form, we choose the method of the decomposition into oscillator variables [11] and write the action (19) as Varying this action with respect to μν gives and with this the field equations obtained from action (21) are equivalent to the field equations obtained from action (19). Finally, with the change of variables we find the action which is dynamically equivalent to action (8).
The action (25) contains a positive energy massless spin-2 field A μν , a negative energy massive spin-2 field B μν , and a negative energy massive spin-0 field σ . Classicaly, since the theory is not interacting, there is no problem with the negative energy fields. However, when interactions are introduced instabilities can appear. Thus a careful analysis is necessary on the interaction of massive conformal gravity with matter fields, which is beyond the scope of this paper. We will deal with the negative energy problem at the quantum level in the next section.

Canonical quantization
Varying the action (25) with respect to A μν , B μν , and σ , we obtain the field equations 3 The most general real solutions of these equations are given by where ω A p = |p|, ω B p = |p| 2 + m 2 , ω σ p = |p| 2 + m 2 , the creation and annihilation operators obey the commutation relations with all the other commutators equal to zero, and the polarization tensors satisfy the orthonormality and completeness relations 3 In this section we use "absolute units" in which c =h = 16π G = 1.
In order to find the energy spectrum of massive conformal

Final remarks
We have shown in this paper that massive conformal gravity is a renormalizable quantum theory of gravity which has two massive ghost states. A careful analysis is needed to check if these ghost states are decoupled from the theory at high energies, which would ensure the unitarity of the theory. While this analysis is not done we cannot rule out massive conformal gravity as a viable quantum theory of gravity. We are investigating this issue right now and we hope that this investigation help to show that quantum massive conformal gravity is not only renormalizable but also unitary.
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