An analytic nonlinear representation of the Poincaré group. II: The case of helicities \(\pm\)
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.
scientific article; zbMATH DE number 3935353
scientific article