No-boundary state for Klein space

Abstract 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">J</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> ) which is the product of a null line with a (1, 1) signature torus. The Minkowskian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">S</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -matrix continues to a Kleinian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">S</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -vector which in turn may be represented by a Poincaré-invariant vacuum state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> in the Hilbert space built on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">J</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> contains all information about <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">S</mml:mi> </mml:mrow> </mml:mrow> </mml:math> in a novel, repackaged form. We give an explicit construction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> in a Lorentz/conformal basis for a free massless scalar. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">J</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> separates into two halves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">J</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>±</mml:mo> </mml:msub> </mml:mrow> </mml:math> which are the asymptotic null boundaries of the regions timelike and spacelike separated from the origin. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> is shown to be a maximally entangled state in the product of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">J</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>±</mml:mo> </mml:msub> </mml:mrow> </mml:math> Hilbert spaces.