Lorentz group, the Poincare group, are the representations that have direct physical relevance.
2.
If the spacetime translations are included, then one obtains the " inhomogeneous Lorentz group " or the Poincare group.
3.
This yields an action of the Poincare group on the space of square-integrable functions defined on the hypersurface in Minkowski space.
4.
Note that the Einstein group approaches but never reaches the Poincare group as the flat spacetime ( special relativity limit ) is approached.
5.
However, if the Poincare group is replaced with a different spacetime symmetry, for example, with the de Sitter group the theorem no longer holds.
6.
In this way, the 0 } } irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.
7.
The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics.
8.
The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the ( generalized ) Lorentz group appears as the little group of the Poincare group of spacelike vectors in higher spacetime dimension.
9.
The first condition for the theorem is that the unified group " G contains a subgroup locally isomorphic to the Poincare group . " Therefore the theorem only makes a statement about the unification of the Poincare group with an internal symmetry group.
10.
The first condition for the theorem is that the unified group " G contains a subgroup locally isomorphic to the Poincare group . " Therefore the theorem only makes a statement about the unification of the Poincare group with an internal symmetry group.
