  We examine the Casimir effect for free statistical field theories which have
Hamiltonians with second order derivative terms. Examples of such Hamiltonians
arise from models of non-local electrostatics, membranes with non-zero bending
rigidities and field theories of the Brazovskii type that arise for polymer
systems. The presence of a second derivative term means that new types of
boundary conditions can be imposed, leading to a richer phenomenology of
interaction phenomena. In addition zero modes can be generated that are not
present in standard first derivative models, and it is these zero modes which
give rise to long range Casimir forces. Two physically distinct cases are
considered: (i) unconfined fields, usually considered for finite size embedded
inclusions in an infinite fluctuating medium, here in a two plate geometry the
fluctuating field exists both inside and outside the plates, (ii) confined
fields, where the field is absent outside the slab confined between the two
plates. We show how these two physically distinct cases are mathematically
related and discuss a wide range of commonly applied boundary conditions. We
concentrate our analysis to the critical region where the underlying bulk
Hamiltonian has zero modes and show that very exotic Casimir forces can arise,
characterised by very long range effects and oscillatory behavior that can lead
to strong metastability in the system.
