Ghost Busting: PT-Symmetric Interpretation of the Lee Model

TL;DR

This paper demonstrates that the Lee model, traditionally considered non-Hermitian and non-unitary in certain regimes, can be interpreted as PT-symmetric, allowing for a consistent, unitary quantum theory even with ghost states.

Contribution

The paper constructs the C operator for the Lee model in the ghost regime, showing it is PT-symmetric and thus a valid unitary quantum field theory for all coupling values.

Findings

The ghost state has a positive norm under the new inner product.

The Lee model's Hamiltonian is PT-symmetric in the ghost regime.

The model is unitary for all coupling constants.

Abstract

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 g^2, the square of the renormalized coupling constant, above which g_0^2, the square of the unrenormalized coupling constant, is negative. Thus, for g^2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. It was also discovered that in this non-Hermitian 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 exactly in the V/N-theta 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 g^2.