Ghost busting:PT-symmetric interpretation of the Lee model

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 2 0 , 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 in the V /N θ 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 .

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

“…The importance of the work of Källén and Pauli was emphasised by Salam in his review of their paper [57] and the appearance of the ghost was assumed to be a fundamental defect of the Lee model. However, in 2005 it was shown that the non-Hermitian Lee-model Hamiltonian is P T symmetric and when the norms of the states of this model are determined using the C operator, which can be calculated exactly and in closed form, the ghost state is seen to be an ordinary physical state having positive norm [9]. Thus, the following assertion by Barton [3] is not correct: "A non-Hermitian Hamiltonian is unacceptable partly because it may lead to complex energy eigenvalues, but chiefly because it implies a non-unitary S matrix, which fails to conserve probability and makes a hash of the physical interpretation.…”

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

“…For the past half century, there have been many attempts to make sense of the Lee model as a valid quantum theory (starting as early as [4]), but it was not until 2005 that it was shown that it is possible to formulate the theory without a ghost [5]. The solution to the Lee-model-ghost problem is that when the coupling constant exceeds its critical value, the Hamiltonian transits from being Dirac Hermitian to being PT symmetric, i.e., symmetric under combined parity reflection and time reversal.…”

Exact isospectral pairs of {\cal P}{\cal T} symmetric Hamiltonians

“…The figure shows a closed periodic classical orbit that begins at x = 2i. The period T of this orbit is exactly that given in (19). classical orbit makes a 270…”

“…[17,18]). A few other exactly equivalent pairs of Hamiltonians have been found [19,20,21,22,23,24,25].…”