Abstract
We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the S matrix is unitary when the cosmological constant vanishes. The model is the simplest of its class. However, infinitely many similar options are allowed, which raises the issue of uniqueness. To deal with this problem, we propose a new quantization prescription, by doubling the unphysical poles of the higher-derivative propagators and turning them into Lee-Wick poles. The Lagrangian of the simplest theory of quantum gravity based on this idea is the linear combination of R, RμνRμν, R2 and the cosmological term. Only the graviton propagates in the cutting equations and, when the cosmological constant vanishes, the S matrix is unitary. The theory satisfies the locality of counterterms and is renormalizable by power counting. It is unique in the sense that it is the only one with a dimensionless gauge coupling.
Highlights
The complex energy hyperplane is divided into disjoint regions Ai of analyticity, which can be connected to one another by a well defined, but nonanalytic procedure
We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions
We investigate the main options for quantum gravity that are offered by the nonanalytic Wick rotation of Euclidean higher-derivative theories, combined with extra tools that we introduce anew
Summary
The first option that we consider is a superrenormalizable higher-derivative gravity, formulated by nonanalytically Wick rotating its Euclidean version. Its most general Lagrangian LQG is given by. Where λC, ζ, γ, η, ξ, α1, · · · , α6 are dimensionless constants, κ has dimension −1 in units of mass and M is the Lee-Wick mass scale. We expand the metric tensor gμν around the Galilean metric ημν =diag(1, −1, −1, −1) by writing gμν = ημν + 2κhμν , where hμν is the quantum fluctuation. After the expansion around flat space, we raise and lower the indices by means of the Galilean metric. It is convenient to choose a gauge-fixing function that is linear in the fluctuation hμν, such as the De Donder function. We complete the gauge-fixing following the steps of ref. [10], so as to obtain the gauge-fixed. We begin by studying the renormalization of the theory and discuss the conditions under which it is perturbatively unitary
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