Astrophysical Evidence for the Non-Hermitian but $PT$-symmetric Hamiltonian of Conformal Gravity

Abstract

In this review we discuss the connection between two seemingly disparate topics, macroscopic gravity on astrophysical scales and Hamiltonians that are not Hermitian but $PT$ symmetric on microscopic ones. In particular we show that the quantum-mechanical unitarity problem of the fourth-order derivative conformal gravity theory is resolved by recognizing that the scalar product appropriate to the theory is not the Dirac norm associated with a Hermitian Hamiltonian but is instead the norm associated with a non-Hermitian but $PT$-symmetric Hamiltonian. Moreover, the fourth-order theory Hamiltonian is not only not Hermitian, it is not even diagonalizable, being of Jordan-block form. With $PT$ symmetry we establish that conformal gravity is consistent at the quantum-mechanical level. In consequence, we can apply the theory to data, to find that the theory is capable of naturally accounting for the systematics of the rotation curves of a large and varied sample of 138 spiral galaxies without any need for dark matter. The success of the fits provides evidence for the relevance of non-diagonalizable but $PT$-symmetric Hamiltonians to physics.

LaTex, 15 pages, 21 figures. Expanded version of talks given at the International Seminar and Workshop "Quantum Physics with Non-Hermitian Operators", Dresden, June 2011 and the Symposium "PT Quantum Mechanics 2011", Heidelberg, September 2011. Prepared for a Special Issue of Fortschritte der Physik - Progress of Physics on "Quantum Physics with Non-Hermitian Operators: Theory and Experiment"

References in corpus (6)

• Making Sense of Non-Hermitian Hamiltonians
• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Impact of a global quadratic potential on galactic rotation curves
• Solution to the ghost problem in fourth order derivative theories
• Giving up the ghost

Cited by in corpus (9)

• Stable Self-Interacting Pais-Uhlenbeck Oscillator
• $\mathcal{P}\mathcal{T}$-symmetric wave guide system with evidence of a third-order exceptional point
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Coupling approach for the realization of a $\mathcal{P}\mathcal{T}$-symmetric potential for a Bose-Einstein condensate in a double well
• Vortices in Bose-Einstein condensates with $\mathcal{P}\mathcal{T}$-symmetric gain and loss
• Simple models of three coupled $\mathcal{P}\mathcal{T}$-symmetric wave guides allowing for third-order exceptional points
• Purity oscillations in coupled Bose-Einstein condensates
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method