  A class of nonlocal Lorentzian quantum field theories is introduced in
arXiv:1502.01655 and arXiv:1411.6513, where the d'Alembertian operator $\Box$
is replaced by a non-analytic function of the d'Alembertian, $f(\Box)$. This is
inspired by the Causal Set program where such an evolution arises as the
continuum limit of a wave equation on causal sets. The spectrum of these
theories contains a continuum of massive excitations. This is perhaps the most
important feature which leads to distinct/interesting phenomenology. In this
paper, we study properties of the continuum massive modes in depth. We derive
the path integral formulation of these theories. Meanwhile, this derivation
introduces a dual picture in terms of local fields which clearly shows how
continuum massive modes of the nonlocal field interact. The dual picture, in
principle, provides a path to extension beyond scalar fields and addressing the
issue of renormalization.
