An algebraic approach to laying a ghost to rest

In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck (1950 Phys. Rev. 79 145-65) which has not had a good reception over the past half a century due to the existence of ghosts in the properties of the quantum mechanical solution. Bender and Mannheim (2008 J. Phys. A: Math. Theor. 41 304018) were successful in persuading the corresponding quantum operator to 'give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physics community and in the process added further credit to the cause of advancement of the use of PT symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Dirac's theory by providing an Hamiltonian that is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order (Nucci M C 1996 J. Math. Phys. 37 1772-5) that it is possible to construct a Hamiltonian that gives rise to a satisfactory quantal description without having to abandon Dirac.