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Abstract
Higher derivative theories are frequently avoided because of undesirable properties, yet they occur naturally as corrections to general relativity and cosmic strings. We discuss some of their more interesting and disturbing problems, with examples. A natural method of removing all the problems of higher derivatives is reviewed. This method of "perturbative constraints" is required for at least one class of higher derivative theories, those which are associated with non-locality. Non-locality often appears in low energy theories described by effective actions. The method may also be applied to a wide class of other higher derivative theories. An example system is solved, exactly and perturbatively, for which the perturbative solutions approximate the exact solutions only when the method of "perturbative constraints" is employed. Ramifications for corrections to general relativity, cosmic strings with rigidity terms, and other higher derivative theories are explored.
Next, flat space is shown to be perturbatively stable, to first order in $h$, against quantum fluctuations produced in semiclassical (or $1/N$ expansion) approximations to quantum gravity, despite past indications to the contrary. It is pointed out that most of the new "solutions" allowed by the semiclassical corrections do not fall within the perturbative framework, unlike the effective action and field equations which generate them. It is shown that excluding these non-perturbative "pseudo-solutions" is the only self-consistent approach. The remaining physical solutions do fall within the perturbative formalism, do not require the introduction of new degrees of freedom, and suffer none of the pathologies of unconstrained higher derivative systems. The presence of the higher derivative terms in the semiclassical corrections may be related to non-locality.
