Reviving QFT in (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>)-dimensional de Sitter spacetime

We consider a conformally coupled scalar quantum field theory (QFT) on (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>)-dimensional static Einstein universe <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>R</c:mi><c:mo>×</c:mo><c:msup><c:mi>S</c:mi><c:mn>2</c:mn></c:msup></c:math> and write down the free theory Hilbert space. We explain that this theory is secretly a QFT in (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mn>2</e:mn><e:mo>+</e:mo><e:mn>1</e:mn></e:mrow></e:math>)-dimensional de Sitter space because all the quantum observables experience “quantum revivals,” which naturally restricts the timelike <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>R</g:mi></g:math> to the appropriate de Sitter time range. Our construction circumvents the causal obstruction to formulating QFT in de Sitter due to event horizons. There are not any in static Einstein. The “unitary gauge” description of the theory is realized by the zonal harmonics <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>P</i:mi><i:mo>ℓ</i:mo></i:msub><i:mo stretchy="false">(</i:mo><i:mover accent="true"><i:mi>n</i:mi><i:mo stretchy="false">^</i:mo></i:mover><i:mo>·</i:mo><i:msup><i:mover accent="true"><i:mi>n</i:mi><i:mo stretchy="false">^</i:mo></i:mover><i:mo>′</i:mo></i:msup><i:mo stretchy="false">)</i:mo></i:math>. We verify that interactions with conformally invariant external sources are mediated only by these modes. Hence these modes comprise the complete basis of the “bulk” theory. When the theory is cut off in the UV, the basis dimension scales as the Bekenstein-Hawking formula. Published by the American Physical Society 2024