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 re. flection (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! Does a PT-symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. In terms of 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 provides an excellent 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. nd a physical interpretation for the ghost, but all such attempts failed. The correct 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-. eld-theoretic models.