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

TL;DR

This paper links astrophysical observations with quantum mechanics by showing that conformal gravity's non-Hermitian, $PT$-symmetric Hamiltonian can explain galaxy rotation curves without dark matter, supporting the physical relevance of such Hamiltonians.

Contribution

It demonstrates that conformal gravity's non-Hermitian, $PT$-symmetric Hamiltonian is consistent and can successfully fit galaxy rotation data without dark matter, revealing the importance of non-diagonalizable Hamiltonians in physics.

Findings

Conformal gravity's Hamiltonian is non-Hermitian and $PT$-symmetric.

The theory fits galaxy rotation curves without dark matter.

Non-diagonalizable $PT$-symmetric Hamiltonians are physically relevant.

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.