An algebraic approach to laying a ghost to rest
Abstract
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck Pais-Uhlenbeck 50 a, which has not had a good reception over the last half century due to the existence of ghosts in the properties of the quantum mechanical solution. Bender and Mannheim Bender 08 a 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 physical 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 which 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 that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.