An analytic nonlinear representation of the Poincaré group. II: The case of helicities \(\pm\)
type: Item
instance of: scholarly article
title: An analytic nonlinear representation of the Poincaré group. II: The case of helicities \(\pm\)
zbMATH Open document ID: 0584.22009
DOI: 10.1007/BF00420574
author: Jacqueline Bertrand
author: Guy Rideau
published in: Letters in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: The existence of analytic nonlinear representations of the Poincaré group in a Banach space is proven for mass zero and helicities \(\pm 1\) (in part I; see the preceding review) and \(\pm\) (in part II). An auxiliary Fock space is constructed in which the linear unitary representations of mass zero and helicities \(\pm 1\) are extended in the usual way. The principal property used to obtain the results is the existence of a cocycle of the extension Z(a,\(\Lambda)\) of the corresponding representation by itself such that [Z(a,\(\Lambda)\), \(Z(a',\Lambda ')]=0\). The same property is proven for helicities \(\pm\), so that the results obtained for \(\pm 1\) are extended to this case. The true nonlinearity is on the Lorentz boosts. It is shown that the nonlinear representations constructed for the Poincaré group are not equivalent to the unitary linear representations of this group. It is suggested that nonlinear representations of the Poincaré group are important in the case of relativistic nonlinear wave equations. Some results are known for systems of nonlinear Klein-Gordon and Schrödinger equations.
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 3935353
zbMATH Keywords: extension cocycle
zbMATH Keywords: nonlinear representations
zbMATH Keywords: Poincaré group
zbMATH Keywords: helicities
zbMATH Keywords: wave equations
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
cites work: An analytic nonlinear representation of the Poincaré group
full work available at URL: https://doi.org/10.1007/bf00420574
OpenAlex ID: W2019518104
Recommended article: Formal linearization of nonlinear massive representations of the connected Poincaré group
Recommended article: Linear covariance for nonlinear formal representations of the Poincaré group in \(2+1\) dimensions
Recommended article: Q4524264
Recommended article: The Cauchy problem for nonlinear Klein-Gordon equations
Recommended article: Nonlinear Representations of the Euclidean Group of the Plane
Recommended article: Q3751793
Recommended article: Nontrivial extensions of a massless representation with definite helicity of the Poincaré group in \(3+1\) dimensions by the tensor product of two other massless representations with definite helicity
Recommended article: Q3840225
Recommended article: Uniqueness problems in formal nonlinear representations of the Poincaré group in 2+1 dimensions
Recommended article: Intrinsic nonlinear spinor wave equations associated with nonlinear spinor representations
description: scientific article; zbMATH DE number 3935353