<p>We consider Einstein gravity with positive cosmological constant coupled to matter in an asymptotically de Sitter universe with sphere boundary at timelike infinity. In this setting we show that, to one-loop order and at late time, the norm of the no-boundary state vanishes, going as <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msup> <mi>e</mi> <msub> <mi>S</mi> <mn>0</mn> </msub> </msup> <mfrac> <mrow> <mi>Z</mi> <msubsup> <mi>S</mi> <mn>0</mn> <mrow> <mo>−</mo> <mi>d</mi> <mfenced close=")" open="("> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mo>/</mo> <mn>4</mn> </mrow> </msubsup> </mrow> <mrow> <mi>vol</mi> <mfenced close=")" open="("> <mrow> <mi>SO</mi> <mfenced close=")" open="(" separators=","> <mi>d</mi> <mn>1</mn> </mfenced> </mrow> </mfenced> </mrow> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( {e}^{S_0}\frac{Z{S}_0^{-d\left(d+1\right)/4}}{\textrm{vol}\left(\textrm{SO}\left(d,1\right)\right)} \)</EquationSource> </InlineEquation> with <i>S</i><sub>0</sub> the tree-level entropy of the static patch, <i>d</i> the spacetime dimension, and <i>Z</i> non-negative. We show that the presence of an observer stabilizes the norm to a large, positive value.</p>

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Norm of the no-boundary state

  • Jordan Cotler,
  • Kristan Jensen

摘要

We consider Einstein gravity with positive cosmological constant coupled to matter in an asymptotically de Sitter universe with sphere boundary at timelike infinity. In this setting we show that, to one-loop order and at late time, the norm of the no-boundary state vanishes, going as e S 0 Z S 0 d d + 1 / 4 vol SO d 1 \( {e}^{S_0}\frac{Z{S}_0^{-d\left(d+1\right)/4}}{\textrm{vol}\left(\textrm{SO}\left(d,1\right)\right)} \) with S0 the tree-level entropy of the static patch, d the spacetime dimension, and Z non-negative. We show that the presence of an observer stabilizes the norm to a large, positive value.