The Poincare group is the semidirect product of spacetime translations by the Lorentz group. It is the isometry group of Minkowski spacetime.
The Poincare algebra is generated by translations and Lorentz transformations . Its mixed commutator is .
The Pauli-Lubanski pseudovector is , up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.
The little group of a momentum is its stabilizer in the Lorentz group. Its unitary irreducible representations label the internal degrees of freedom of one-particle states on the orbit of .
Wigner's classification obtains irreducible positive-energy representations of the Poincare group by inducing from unitary irreducible representations of the little group of a standard momentum.
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The Poincaré group is a fundamental algebraic structure in the field of theoretical physics, particularly in the context of special relativity and quantum field theory. It describes the symmetries of spacetime in four dimensions and serves as the group of isometries for Minkowski spacetime. The group includes the following transformations: 1. **Translations**: These are shifts in space and time.
Full set of all possible special relativity symmetries:
In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.
There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.
- space and time shifts, because their space origin and time origin (time they consider 0, i.e. when they started their timers) are not synchronized. This can be modelled with a 4-vector addition.
- their space axes are rotated relative to one another. This can be modelled with a 4x4 matrix multiplication.
- and they are moving relative to each other, which leads to the usual spacetime interactions of special relativity. Also modelled with a 4x4 matrix multiplication.
The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.