No-Boundary State for Klein Space

Analytic continuation from $(3,1)$ signature Minkowski to $(2,2)$ signature Klein space has emerged as a useful tool for the understanding of scattering amplitudes and flat space holography. Under this continuation, past and future null infinity merge into a single boundary ($\mathcal{J}$) which is the product of a null line with a $(1,1)$ signature torus. The Minkowskian $\mathcal{S}$-matrix continues to a Kleinian $\mathcal{S}$-vector which in turn may be represented by a Poincar\'e-invariant vacuum state $|\mathcal{C}\rangle$ in the Hilbert space built on $\mathcal{J}$. $|\mathcal{C} \rangle$ contains all information about $\mathcal{S}$ in a novel, repackaged form. We give an explicit construction of $|\mathcal{C}\rangle$ in a Lorentz/conformal basis for a free massless scalar. $\mathcal{J}$ separates into two halves $\mathcal{J}_\pm $ which are the asymptotic null boundaries of the regions timelike and spacelike separated from the origin. $|\mathcal{C}\rangle$ is shown to be a maximally entangled state in the product of the $\mathcal{J}_\pm $ Hilbert spaces.