  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.
