Making Sense of Non-Hermitian Hamiltonians
| dc.creator | Bender, Carl M. | |
| dc.date | 2007-03-09 | |
| dc.date.accessioned | 2026-07-07T11:44:06Z | |
| dc.date.available | 2026-07-07T11:44:06Z | |
| dc.description | The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models. | |
| dc.description | 78 pages, 26 figures, invited review article to be published in Rep. Prog. Phys | |
| dc.identifier | https://arxiv.org/abs/hep-th/0703096 | |
| dc.identifier | http://arxiv.org/abs/hep-th/0703096 | |
| dc.identifier | Rept.Prog.Phys.70:947,2007 | |
| dc.identifier | doi:10.1088/0034-4885/70/6/R03 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/201481 | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | Making Sense of Non-Hermitian Hamiltonians | |
| dc.type | text |