Hilbert space of quantum field theory in de Sitter spacetime

We study the decomposition of the Hilbert space of quantum field theory in (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>d</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>)-dimensional de Sitter spacetime into unitary irreducible representations (UIRs) of its isometry group <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>SO</c:mi><c:mo stretchy="false">(</c:mo><c:mn>1</c:mn><c:mo>,</c:mo><c:mi>d</c:mi><c:mo>+</c:mo><c:mn>1</c:mn><c:mo stretchy="false">)</c:mo></c:mrow></c:math>. First, we consider multiparticle states in free theories starting from the tensor product of single-particle UIRs. Second, we study conformal multiplets of a bulk conformal field theory with symmetry group <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>SO</g:mi><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:mo>,</g:mo><g:mi>d</g:mi><g:mo>+</g:mo><g:mn>1</g:mn><g:mo stretchy="false">)</g:mo></g:mrow></g:math>. Our main tools are the Harish-Chandra characters and the numerical diagonalization of the (truncated) quadratic Casimir of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi>SO</k:mi><k:mo stretchy="false">(</k:mo><k:mn>1</k:mn><k:mo>,</k:mo><k:mi>d</k:mi><k:mo>+</k:mo><k:mn>1</k:mn><k:mo stretchy="false">)</k:mo></k:mrow></k:math>. We introduce a continuous density that encodes the spectrum of irreducible representations contained in a reducible one of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>SO</o:mi><o:mo stretchy="false">(</o:mo><o:mn>1</o:mn><o:mo>,</o:mo><o:mi>d</o:mi><o:mo>+</o:mo><o:mn>1</o:mn><o:mo stretchy="false">)</o:mo></o:math>. Our results are complete for <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>d</s:mi><s:mo>=</s:mo><s:mn>1</s:mn></s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>d</u:mi><u:mo>=</u:mo><u:mn>2</u:mn></u:math>. In higher dimensions, we rederive and extend several results previously known in the literature. Our work provides the foundation for future nonperturbative bootstrap studies of quantum field theory in de Sitter spacetime.