  In this paper we will study Lorentz-invariant, infinite derivative quantum
field theories, where infinite derivatives give rise to non-local interactions
at the energy scale $M_s$, beyond the Standard Model. We will study a specific
class, where there are no it new dynamical degrees of freedom other than the
original ones of the corresponding local theory. We will show that the Green
functions are modified by a non-local extra term that is responsible for
acausal effects, which are confined in the region of non-locality, i.e.
$M_s^{-1}.$ The standard time-ordered structure of the causal Feynman
propagator is not preserved and the non-local analog of the retarded Green
function turns out to be non-vanishing for space-like separations. As a
consequence the local commutativity is violated. Formulating such theories in
the non-local region with Minkowski signature is not sensible, but they have
Euclidean interpretation. We will show how such non-local construction
ameliorates ultraviolet/short-distance singularities suffered typically in the
local quantum field theory. We will show that non-locality and acausality are
inherently off-shell in nature, and only quantum amplitudes are physically
meaningful, so that all the perturbative quantum corrections have to be
consistently taken into account.
