Making sense of non-Hermitian Hamiltonians

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 = p2 + ix 3 and H = p2 -x4 . Amazingly, the energy levels of…

• UD
  U.S. Department of Energy
• LA
  Los Alamos National Laboratory