Poincaré group

For the Poincaré group (fundamental group) of a topological space, see Fundamental group.

Group theory

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In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.

1 Simple explanation
2 Technical explanation
3 Poincaré symmetry
4 See also
5 References

Simple explanation

An isometry is a way in which the contents of spacetime could be shifted that would not affect the proper time along a trajectory between events. For example, if everything was postponed by two hours including two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything was shifted five miles to the west, you would also see no change in the interval. It turns out that the length of a rod is also unaffected by such a shift.

If you ignore the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation (by a fixed angle) around any of the three spatial axes, or a boost in any of the three spatial directions. 10=1+3+3+3. If you combine such isometries together (do one and then the other), the result is also such an isometry (although not generally one of the ten basic ones). These isometries form a group. That is, there is an identity (no shift, everything stays where it was), and inverses (move everything back to where it was), and it obeys the associative law. The name of this particular group is the "Poincaré group".

Technical explanation

The Poincaré group is the group of isometries of Minkowski spacetime. It is a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is the affine group of the Lorentz group, i.e. the Poincaré group is a semidirect product of the translations and the Lorentz transformations:

$\mathbf{R}^{1,3} \rtimes O(1,3).\,$

Another way of putting it is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it.

Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer or half integer), and are associated with particles in quantum mechanics.

In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group.

The Poincaré algebra is the Lie algebra of the Poincaré group. In component form, the Poincaré algebra is given by the commutation relations:

$[P_\mu, P_\nu] = 0\,$
$\frac{ 1 }{ i }[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\,$
$\frac{ 1 }{ i }[M_{\mu\nu}, M_{\rho\sigma}] = \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\sigma} %2B \eta_{\nu\sigma} M_{\mu\rho}\,$

where $P$ is the generator of translations, $M$ is the generator of Lorentz transformations and $\eta$ is the Minkowski metric (see sign convention).

The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all elementary particles fall in representations of this group. These are usually specified by the four-momentum of each particle (i.e. its mass) and the intrinsic quantum numbers J^PC, where J is the spin quantum number, P is the parity and C is the charge conjugation quantum number. Many quantum field theories do violate parity and charge conjugation. In those cases, we drop the P and the C. Since CPT is an invariance of every quantum field theory, a time reversal quantum number could easily be constructed out of those given.

As a topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted.

Poincaré symmetry

Poincaré symmetry is the full symmetry of special relativity and includes

translations (i.e., displacements) in time and space (these form the abelian Lie group of translations on space-time)
rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations)
boosts, i.e., transformations connecting two uniformly moving bodies.

The last two symmetries together make up the Lorentz group (see Lorentz invariance). These are generators of a Lie group called the Poincaré group which is a semi-direct product of the group of translations and the Lorentz group. Things which are invariant under this group are said to have Poincaré invariance or relativistic invariance.

References

Weinberg, Steven (1995). The Quantum Theory of Fields. 1. Cambridge: Cambridge University press. ISBN 978-0-521-55001-7.

Poincaré group

Contents

Simple explanation

Technical explanation

Poincaré symmetry

See also

References