Poincaré group

In physics and mathematics, the Poincaré group, named after Henri Poincaré,^[1] is the group of isometries of Minkowski spacetime, introduced by Hermann Minkowski.^[2]^[3] It is a Non-abelian Lie group with 10 generators, of fundamental importance in physics.

Basic 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 one ignores 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, altogether 1 + 3 + 3 + 3 = 10.

If one combines such isometries together (implementing one and then the other), the result is also such an isometry (although, in general, a linear combination of the ten basic ones detailed). 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 specific 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 also 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. More precisely, it is a semidirect product of the translations and the Lorentz group,

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

Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the "inhomogeneous Lorentz group". In turn, it can also be obtained as a group contraction of the de Sitter group SO(4,1) ~ Sp(2,2), as the de Sitter radius goes to infinity.

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 −−see Wigner's classification.

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. More specifically, the proper (detΛ=1), orthochronous ( $Λ 00 \geq1$ ) part of the Lorentz subgroup (its identity component), SO⁺(1, 3), is connected to the identity and is thus provided by the exponentiation $exp(ia μ P μ) exp(iω μν M μν /2)$ of this Lie algebra. In component form, the Poincaré algebra is given by the commutation relations,^[4]^[5]

$~[P_{\mu },P_{\nu }]=0\,$

$~{\frac {1}{i}}~[M_{{\mu \nu }},P_{\rho }]=\eta _{{\mu \rho }}P_{\nu }-\eta _{{\nu \rho }}P_{\mu }\,$

$~{\frac {1}{i}}~[M_{{\mu \nu }},M_{{\rho \sigma }}]=\eta _{{\mu \rho }}M_{{\nu \sigma }}-\eta _{{\mu \sigma }}M_{{\nu \rho }}-\eta _{{\nu \rho }}M_{{\mu \sigma }}+\eta _{{\nu \sigma }}M_{{\mu \rho }}\,,$

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

The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, $J i = - ϵ imn M mn /2$ , and boosts, $K i = M i0$ . In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as

$[J_{m},P_{n}]=i\epsilon _{{mnk}}P_{k}~,$

$[J_{i},P_{0}]=0~,$

$[K_{i},P_{k}]=i\eta _{{ik}}P_{0}~,$

$[K_{i},P_{0}]=-iP_{i}~,$

$[J_{m},J_{n}]=i\epsilon _{{mnk}}J_{k}~,$

$[J_{m},K_{n}]=i\epsilon _{{mnk}}K_{k}~,$

$[K_{m},K_{n}]=-i\epsilon _{{mnk}}J_{k}~,$

where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". Note the important simplification $[J m +i K m, J n -i K n] = 0$ , which permits reduction of the Lorentz subalgebra to su(2)⊕su(2) and efficient treatment of its associated representations.

The Casimir invariants of this algebra are $P μ P μ$ and $W μ W μ$ where $W μ$ is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group.

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 squared of each particle (i.e. its mass squared) 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 in practice do violate parity and charge conjugation. In those cases, the $P$ and the $C$ are forfeited. 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, P. 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, with generators J)
boosts, i.e., transformations connecting two uniformly moving bodies, with generators K.

The last two symmetries, J and K, 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. Objects which are invariant under this group are said to possess Poincaré invariance or relativistic invariance.

Notes

↑
- Poincaré, Henri (1905/6), "Sur la dynamique de l’électron", Rendiconti del Circolo matematico di Palermo 21: 129–176 Check date values in: |date= (help)
- Wikisource translation: On the Dynamics of the Electron
↑ Minkowski, Hermann (1907/8), "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111 Check date values in: |date= (help)
- Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies.
↑ Minkowski, Hermann (1908/9), "Raum und Zeit", Physikalische Zeitschrift 10: 75–88 Check date values in: |date= (help)
↑ N.N. Bogolubov (1989). General Principles of Quantum Field Theory (2nd ed.). Springer. p. 272. ISBN 0-7923-0540-X.
↑ T. Ohlsson (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. p. 10. ISBN 1-13950-4320.

References

Wu-Ki Tung (1985). Group Theory in Physics. World Scientific Publishing. ISBN 9971-966-57-3.
Weinberg, Steven (1995). The Quantum Theory of Fields 1. Cambridge: Cambridge University press. ISBN 978-0-521-55001-7.
L.H. Ryder (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. p. 62. ISBN 0-52147-8146.

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