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In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the symmetry group of Minkowski space. Namely, it is the group of transformations which preserve the invariant interval between any pair of points (also called events). It was first used in Minkowski's 1908 lecture "Space and Time".

Basic explanation

Each pair of points of the Minkowski space defines a vector which can be either time-like, null, or space-like. An automorphism of the Minkowski space is its mapping to itself such that each vector v would retain its magnitude ${\displaystyle \eta _{\mu \nu }{\mathbf {v} }^{\mu }{\mathbf {v} }^{\nu }}$. For a time-like vector, its time interval would be preserved. For a space-like vector, its length would be preserved. And any null (light-like) vector would remain null. It is necessary and sufficient that such transform would not change the proper time along any possible world line (although its direction can be reversed). From physical point of view it means that all lengths and time intervals does not change, although coordinates change, and both time direction and orientation of the space are not necessarily preserved.

This group of transformations can be understood as ten-dimensional space generated by:

• translation through time (1),
• translation through any of the three dimensions of space (3),
• rotations around the three spatial axes (3), and
• boosts in any of the three spatial directions (3).

Technical explanation

The Poincaré group is the group of automorphisms of Minkowski space. 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 the origin. The Poincaré group itself is the minimal subgroup of the affine group which includes all translations and Lorentz transformations, and, more precise, it is a semidirect product of the translations and the Lorentz group:

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

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 invariant 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:

• ${\displaystyle =0\,}$
• ${\displaystyle {\frac {1}{i}}=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho }P_{\mu }\,}$
• ${\displaystyle {\frac {1}{i}}=\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho }\,}$

where P is the generator of translations, M is the generator of Lorentz transformations and η is the Minkowski metric (see sign convention #Metric signature).

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, 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.

See also

• Euclidean group
• Representation theory of the Poincaré group
• Wigner's classification

Footnotes

1. Inappropriately called an isometry in most physical texts, although Minkowski space is pseudo-Euclidean, not a metric space.

References

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

• Lie groups
• Particle physics
• Quantum field theory
• Special relativity
• Symmetry