Spinor-helicity representations of particles of any mass in dS4 and AdS4 spacetimes

The spinor-helicity representations of massive and (partially) massless particles in four-dimensional (anti–)de Sitter (A)dS spacetime are studied within the framework of the dual pair correspondence. We show that the dual groups (also known as “little groups”) of the anti–de Sitter and de Sitter groups are, respectively, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msup><e:mrow><e:mi>O</e:mi></e:mrow><e:mrow><e:mo>*</e:mo></e:mrow></e:msup><e:mo stretchy="false">(</e:mo><e:mn>2</e:mn><e:mi>N</e:mi><e:mo stretchy="false">)</e:mo></e:mrow></e:math>. For <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>N</i:mi><i:mo>=</i:mo><i:mn>1</i:mn></i:math>, the generator of the dual algebra <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi mathvariant="fraktur">so</k:mi><k:mo stretchy="false">(</k:mo><k:mn>2</k:mn><k:mo stretchy="false">)</k:mo><k:mo>≅</k:mo><k:msup><k:mrow><k:mi mathvariant="fraktur">so</k:mi></k:mrow><k:mrow><k:mo>*</k:mo></k:mrow></k:msup><k:mo stretchy="false">(</k:mo><k:mn>2</k:mn><k:mo stretchy="false">)</k:mo><k:mo>≅</k:mo><k:mi mathvariant="fraktur">u</k:mi><k:mo stretchy="false">(</k:mo><k:mn>1</k:mn><k:mo stretchy="false">)</k:mo></k:mrow></k:math> corresponds to the helicity operator, and the spinor-helicity representation describes massless particles in (A)<v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mtext>dS</v:mtext><v:mn>4</v:mn></v:msub></v:math>. For <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>N</x:mi><x:mo>=</x:mo><x:mn>2</x:mn></x:math>, the dual algebra is composed of two ideals, <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="fraktur">s</z:mi></z:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi mathvariant="fraktur">m</cb:mi><cb:mi mathvariant="normal">Λ</cb:mi></cb:msub></cb:math>. The former ideal <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi mathvariant="fraktur">s</gb:mi><gb:mo>≅</gb:mo><gb:mi mathvariant="fraktur">so</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:mn>3</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math> fixes the spin of the particle, while the mass is determined by the latter ideal <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi mathvariant="fraktur">m</mb:mi><mb:mi mathvariant="normal">Λ</mb:mi></mb:msub></mb:math>, which is isomorphic to <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi mathvariant="fraktur">so</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>2</qb:mn><qb:mo>,</qb:mo><qb:mn>1</qb:mn><qb:mo stretchy="false">)</qb:mo></qb:math>, <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:mi mathvariant="fraktur">iso</vb:mi><vb:mo stretchy="false">(</vb:mo><vb:mn>2</vb:mn><vb:mo stretchy="false">)</vb:mo></vb:math>, or <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mi mathvariant="fraktur">so</ac:mi><ac:mo stretchy="false">(</ac:mo><ac:mn>3</ac:mn><ac:mo stretchy="false">)</ac:mo></ac:math> depending on the cosmological constant being positive, zero, or negative. In the case of a positive cosmological constant, namely <fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:mrow><fc:msub><fc:mrow><fc:mi>dS</fc:mi></fc:mrow><fc:mrow><fc:mn>4</fc:mn></fc:mrow></fc:msub></fc:mrow></fc:math>, the spinor-helicity representation contains all massive particles corresponding to the principal series representations and the partially massless particles corresponding to the discrete series representations leaving out only the light massive particles corresponding to the complementary series representations. The zero and negative cosmological constant cases, which had been addressed in earlier references, are also discussed briefly. Finally, we consider the multilinear form of helicity spinors invariant under (A)dS group, which can serve as the (A)dS counterpart of the scattering amplitude, and discuss technical differences and difficulties of the (A)dS cases compared to the flat spacetime case.