Massless Lifshitz Field Theory for Arbitrary $z$

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.