2021/09/27 by Philip D. Mannheim, Mannheim, Philip D. · 3 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories
paper · doi:10.48550/arxiv.2109.12743
openalex publication_date 2021/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
With standard Einstein gravity not being renormalizable at the quantum level there is much interest in studying higher-derivative quantum gravity theories. Thus just as a Ricci-scalar-based action produces a propagator that behaves as a non-renormalizable 1/k2 at large k2, an action based on the square of the Ricci scalar behaves as a renormalizable 1/k4 at large k2. An action based on both the Ricci scalar and its square leads to a renormalizable propagator of the generic Pauli-Villars form. However, given the form of the Hamiltonian and the propagator such theories are thought to be plagued by either energies that are unbounded from below or states of negative Dirac norm (the overlap of a ket with its Hermitian conjugate bra). But when one constructs the quantum Hilbert space one finds (Bender and Mannheim) that in fact neither of these problems is actually present. The Hamiltonian turns out to not be Hermitian but to instead have an antilinear PT symmetry, and for this symmetry the needed inner product is the overlap of a ket with its PT conjugate bra. And this inner product is positive definite. Moreover, for the pure 1/k4 propagator the Hamiltonian turns out to not be diagonalizable, and again there are no states of negative energy or of negative norm. Instead there are states of zero norm, non-standard but perfectly acceptable states that serve to maintain probability conservation. With the locally conformal invariant fourth-order derivative conformal gravity theory being in this category, it can be offered as a candidate theory of quantum gravity that is renormalizable and unitary in four spacetime dimensions.