Ghost busting: PT -symmetric interpretation of the Lee model

1550-7998=20 The Lee model was introduced in the 1950s as an elementary quantum field theory in which mass, wave function, and charge renormalization could be carried out exactly. In early studies of this model it was found that there is a critical value of g2, the square of the renormalized coupling constant, above which g0, the square of the unrenormalized coupling constant, is negative. Thus, for g2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. It was also discovered that in this nonHermitian regime a new state appears whose norm is negative. This state is called a ghost state. It has always been assumed that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess PT symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having PT symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a new time-independent operator called C. In terms of C one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this paper the C operator for the Lee model in the ghost regime is constructed in the V=N sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g2.