  By using the notion of fractional derivatives, we introduce a class of
massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary
anisotropy index $z$. The Lifshitz scale invariant ground state of the theory
is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show
that there is a continuous family of ground states with degeneracy
parameterized by the choice of solution to the equation of motion of an
auxiliary classical system. The quantum mechanical path integral establishes a
2d/1d correspondence with the equal time correlation functions of the Lifshitz
scalar field theory. We study the entanglement properties of the Lifshitz
theory for arbitrary $z$ using the path integral representation. The
entanglement measures are expressed in terms of certain cross ratio functions
we specify, and satisfy the $c$-function monotonicity theorems. We also
consider the holographic description of the Lifshitz theory. In order to match
with the field theory result for the entanglement entropy, we propose a
$z$-dependent radius scale for the Lifshitz background. This relation is
consistent with the $z$-dependent scaling symmetry respected by the Lifshitz
vacuum. Furthermore, the time-like entanglement entropy is determined using
holography. Our result suggests that there should exist a fundamental
definition of time-like entanglement other than employing analytic continuation
as performed in relativistic field theory.
