We show that a modification of Wigner’s induced representation for the description of a relativistic particle with spin can be used to construct spinors and tensors of arbitrary rank, with invariant decomposition over angular momentum. In particular, scalar and vector fields, as well as the representations of their transformations, are constructed. The method that is developed here admits the construction of wave packets and states of a many body relativistic system with definite total angular momentum. Furthermore, a Pauli-Lubanski operator is constructed on the orbit of the induced representation which provides a Casimir operator for the Poincaré group and which contains the physical intrinsic angular momentum of the particle covariantly.
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As Kaku17 points out, the usual Pauli-Lubanski operator is only related to the angular momentum of the particle in the rest frame.
Weinberg19 has constructed a somewhat similar mapping for tensor fields using the nonunitary decomposition of the Lorentz algebra in order to achieve the Feynman rules for fields of any spin.
Note that the projection hμν effectively brings the metric into a three dimensional Euclidean space with signature (+++) by the operation hμνgνλhλκ = hμκ.
The second term of (4.5) is Hermitian on integration over the nμ foliation, an intrinsic part of the scalar product on the full Hilbert space.
