  Typically higher-derivative theories are unstable. Instabilities manifest
themselves from extra propagating degrees of freedom, which are unphysical. In
this paper, we will investigate an infinite derivative field theory and study
its true dynamical degrees of freedom via Hamiltonian analysis. In particular,
we will show that if the infinite derivatives can be captured by a Gaussian
kinetic term, i.e. exponential of entire function, then it is possible to prove
that there are only finite number of dynamical degrees of freedom. We will
further extend our investigation into infinite derivative theory of gravity,
and in particular concentrate on ghost free and singularity free theory of
gravity, which has been studied extensively in the Lagrangian approach. Here we
will show from the Hamiltonian perspective that there are only finite number of
degrees of freedom.
