Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions

Conformal field theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the noncompact maximal Abelian subgroup of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>S</a:mi><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>d</a:mi><a:mo>,</a:mo><a:mn>2</a:mn><a:mo stretchy="false">)</a:mo></a:mrow></a:math>. Reduction of a conformal field theory four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>H</e:mi></e:math> bi-invariance, i.e., <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>F</g:mi><g:mo stretchy="false">(</g:mo><g:mi>h</g:mi><g:mi>g</g:mi><g:msup><g:mi>h</g:mi><g:mo>′</g:mo></g:msup><g:mo stretchy="false">)</g:mo><g:mo>=</g:mo><g:mi>F</g:mi><g:mo stretchy="false">(</g:mo><g:mi>g</g:mi><g:mo stretchy="false">)</g:mo></g:math>, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>g</m:mi><m:mo>∈</m:mo><m:mi>S</m:mi><m:mi>O</m:mi><m:mo stretchy="false">(</m:mo><m:mi>d</m:mi><m:mo>,</m:mo><m:mn>2</m:mn><m:mo stretchy="false">)</m:mo></m:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>H</q:mi></q:math> an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously. Published by the American Physical Society 2024