Celestial decomposition of Wigner's particles

TL;DR

This paper offers a detailed, explicit decomposition of Wigner's particles into Lorentz group representations, revealing connections to celestial holography and providing a unique, unitary framework applicable across dimensions.

Contribution

It presents a novel, explicit decomposition of Wigner's particles into Lorentz representations, emphasizing its uniqueness and implications for celestial holography.

Findings

Decomposition involves only principal continuous series Lorentz representations.

Provides explicit formulas for particles of arbitrary mass and spin.

Confirms unitary translations preserve Lorentz representation structure.

Abstract

We provide a detailed decomposition of Wigner's particles, defined as unitary irreducible representations of the Poincar\'e group, in terms of unitary representations of its Lorentz subgroup. As pointed out before us, this decomposition only involves Lorentz representations belonging to the principal continuous series, and further underpins the connection between scattering amplitudes and conformal correlation functions discussed in the context of celestial holography. We provide very explicit formulae for the decomposition of particles of arbitrary mass and (half-)integer spin and for any spacetime dimension. We emphasise that this decomposition is unique and comes with a specific inner product on the corresponding Hilbert space. We also confirm that unitary translations mix Lorentz representations within the principal continuous series only, as required. Implications to the celestial holography program are indicated.