# Ghost busting: Making sense of non-Hermitian Hamiltonians

## Abstract

The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. Early studies of this model in the 1950’s 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*. For *g* ^{2} larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a *ghost*. It has always been thought 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 \(
\mathcal{P}\mathcal{T}
\) symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having \(
\mathcal{P}\mathcal{T}
\) symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called \(
\mathcal{C}
\). In terms of \(
\mathcal{C}
\) 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 talk the \(
\mathcal{C}
\) operator for the Lee model in the ghost regime is constructed in the *V/Nθ* 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}.

## Keywords

Energy Eigenvalue Bare Coupling Symmetric Quantum Versus Particle Charge Renormalization## Preview

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