Fourth Order Theories without Ghosts *

Using the Dirac constraint method we show that the pure fourth-order PaisUhlenbeck oscillator model is free of observable negative norm states. Even though such ghosts do appear when the fourth order theory is coupled to a second order one, the limit in which the second order action is switched off is found to be a highly singular one in which these states move off shell. Given this result, construction of a fully unitary, renormalizable, gravitational theory based on a purely fourth order action in 4 dimensions now appears feasible. As a theory conformal gravity is somewhat enigmatic. With it being based on a local invariance (viz. gμν(x) → e gμν(x)) and with it possessing a gravitational action (viz. IW = −αg ∫ dx(−g)CλμνκC λμνκ where C is the conformal Weyl tensor) whose coupling constant αg is dimensionless, it has a structure which is remarkably similar to that which underlies the other fundamental interactions. Moreover, not only is the conformal gravitational theory power counting renormalizable in four spacetime dimensions, as a quantum theory it turns out to even be asymptotically free [1], while as a classical theory it has [2,3] the confining V (r) = −β/r+γr/2 as its exact classical potential, to thus reproduce some of the most desirable properties of non-Abelian gauge theories. And further, through use of this linear potential the theory has been found capable [4] of accounting for the perplexing galactic rotation curves without the need to introduce any galactic dark matter, with the cosmology associated with the theory being found [5] to be free of any flatness, horizon, universe age or cosmological dark matter problem, with it being a cosmology which naturally (i.e. without fine tuning) explains the recently detected cosmic acceleration, and which, through its underlying scale invariance, naturally solves [6] the cosmological constant problem. Despite all of these attractive features, standing on the debit side is a potential ghost problem which could render the theory non-unitary. Specifically, being a fourth order theory it has a D(k) = 1/k propagator, a propagator which by being rewritten as the M → 0 limit of D(k,M) = [1/k − 1/(k + M)]/M can immediately be seen as hep-th/0001115, January 19, 2000