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 (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 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.