Astrophysical evidence for the non-Hermitian but PT-symmetric Hamiltonian of conformal gravity

作者：Mannheim Philip D^*

来源：Progress of Physics, 2013, 61(2-3): 140-154.

DOI：10.1002/prop.201200100

• PT symmetry
• non-diagonalizable Hamiltonians
• conformal gravity
• galactic rotation curves

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.

• 出版日期2013-2