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 ( \mathcal{P}\mathcal{T} ) symmetry. If H has an unbroken \mathcal{P}\mathcal{T} symmetry, then the spectrum is real. Examples of \mathcal{P}\mathcal{T} -symmetric non-Hermitian quantum-mechanical Hamiltonians are H=\hat{p}^2+\rmi\hat{x}^3 and H=\hat{p}^2-\hat{x}^4 . Amazingly, the energy levels of these Hamiltonians are all real and positive!

Does a \mathcal{P}\mathcal{T} -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 \mathcal{P}\mathcal{T} symmetry, then it has another symmetry represented by a linear operator \mathcal{C} . In terms of \mathcal{C} , one can construct a time-independent inner product with a positive-definite norm. Thus, \mathcal{P}\mathcal{T} -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 \mathcal{P}\mathcal{T} -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. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is \mathcal{P}\mathcal{T} -symmetric. The \mathcal{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 \mathcal{P}\mathcal{T} symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.