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Poincaré group

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Group of flat spacetime symmetries

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

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ThePoincaré group, named afterHenri Poincaré (1905),^[1] was first defined byHermann Minkowski (1908) as theisometry group ofMinkowski spacetime.^[2]^[3] It is a ten-dimensionalnon-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals ofphysics.

Overview

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The Poincaré group consists of allcoordinate transformations ofMinkowski space that do not change thespacetime interval betweenevents. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that theproper length of an object is also unaffected by such a shift.

In total, there are tendegrees of freedom for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, withrotations being produced as the composition of an even number of reflections.

Inclassical physics, theGalilean group is a comparable ten-parameter group that acts onabsolute time and space. Instead of boosts, it featuresshear mappings to relate co-moving frames of reference.

Ingeneral relativity, i.e. under the effects ofgravity, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article.

Poincaré symmetry

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Poincaré symmetry is the full symmetry ofspecial relativity. It includes:

translations (displacements) in time and space, forming theabelian Lie group of spacetime translations (P);
rotations in space, forming the non-abelian Lie group ofthree-dimensional rotations (J);
boosts, transformations connecting two uniformly moving bodies (K).

The last two symmetries,J andK, together make theLorentz group (see alsoLorentz invariance); thesemi-direct product of the spacetime translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possessPoincaré invariance orrelativistic invariance.

10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, byNoether's theorem, imply 10 conservation laws:^[4]^[5]

1 for the energy – associated with translations through time
3 for the momentum – associated with translations through spatial dimensions
3 for the angular momentum – associated with rotations between spatial dimensions
3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time

Poincaré group

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The Poincaré group is the group of Minkowski spacetimeisometries. It is a ten-dimensionalnoncompact Lie group. The four-dimensionalabelian group ofspacetime translations is anormal subgroup, while the six-dimensionalLorentz group is also a subgroup, thestabilizer of the origin. The Poincaré group itself is the minimal subgroup of theaffine group which includes all translations andLorentz transformations. More precisely, it is asemidirect product of the spacetime translations group and the Lorentz group,

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

with group multiplication

(\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)

.^[6]

Another way of putting this is that the Poincaré group is agroup extension of theLorentz group by a vectorrepresentation of it; it is sometimes dubbed, informally, as theinhomogeneous Lorentz group. In turn, it can also be obtained as agroup contraction of the de Sitter groupSO(4, 1) ~ Sp(2, 2), as thede Sitter radius goes to infinity.

Its positive energy unitary irreduciblerepresentations are indexed bymass (nonnegative number) andspin (integer or half integer) and are associated with particles inquantum mechanics (seeWigner's classification).

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

Inquantum field theory, the universal cover of the Poincaré group

\mathbf {R} ^{1,3}\rtimes \operatorname {SL} (2,\mathbf {C} ),

which may be identified with the double cover

\mathbf {R} ^{1,3}\rtimes \operatorname {Spin} (1,3),

is more important, because representations of $\operatorname {SO} (1,3)$ are not able to describe fields with spin 1/2; i.e.fermions. Here $\operatorname {SL} (2,\mathbf {C} )$ is the group of complex $2\times 2$ matrices with unit determinant, isomorphic to theLorentz-signature spin group $\operatorname {Spin} (1,3)$ .

Poincaré algebra

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ThePoincaré algebra is theLie algebra of the Poincaré group. It is aLie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper ( ${\textstyle \det \Lambda =1}$ ),orthochronous ( ${\textstyle {\Lambda ^{0}}_{0}\geq 1}$ ) part of the Lorentz subgroup (itsidentity component), ${\textstyle \mathrm {SO} (1,3)_{+}^{\uparrow }}$ , is connected to the identity and is thus provided by theexponentiation ${\textstyle \exp \left(ia_{\mu }P^{\mu }\right)\exp \left({\frac {i}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}$ of thisLie algebra. In component form, the Poincaré algebra is given by the commutation relations:^[7]^[8]

${\begin{aligned}[][P_{\mu },P_{\nu }]&=0\,\\{\frac {1}{i}}~[M_{\mu \nu },P_{\rho }]&=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho }P_{\mu }\,\\{\frac {1}{i}}~[M_{\mu \nu },M_{\rho \sigma }]&=\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho }\,,\end{aligned}}$

where $P {\displaystyle P}$ is thegenerator of translations, $M {\displaystyle M}$ is the generator of Lorentz transformations, and $\eta$ is the $(+,-,-,-)$ Minkowski metric (seeSign convention).

A diagram of the commutation structure of the Poincaré algebra. The edges of the diagram connect generators with nonzero commutators.

The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, ${\textstyle J_{i}={\frac {1}{2}}\epsilon _{imn}M^{mn}}$ , and boosts, ${\textstyle K_{i}=M_{i0}}$ . In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as

{\begin{aligned}[][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}~,\end{aligned}}

where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification ${\textstyle [J_{m}+iK_{m},\,J_{n}-iK_{n}]=0}$ permits reduction of the Lorentz subalgebra to ${\textstyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}$ and efficient treatment of its associatedrepresentations. In terms of the physical parameters, we have

{\begin{aligned}\left[{\mathcal {H}},p_{i}\right]&=0\\\left[{\mathcal {H}},L_{i}\right]&=0\\\left[{\mathcal {H}},K_{i}\right]&=i\hbar cp_{i}\\\left[p_{i},p_{j}\right]&=0\\\left[p_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}p_{k}\\\left[p_{i},K_{j}\right]&={\frac {i\hbar }{c}}{\mathcal {H}}\delta _{ij}\\\left[L_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}L_{k}\\\left[L_{i},K_{j}\right]&=i\hbar \epsilon _{ijk}K_{k}\\\left[K_{i},K_{j}\right]&=-i\hbar \epsilon _{ijk}L_{k}\end{aligned}}

TheCasimir invariants of this algebra are ${\textstyle P_{\mu }P^{\mu }}$ and ${\textstyle W_{\mu }W^{\mu }}$ where ${\textstyle W_{\mu }}$ is thePauli–Lubanski pseudovector; they serve as labels for the representations of the group.

The Poincaré group is the full symmetry group of anyrelativistic field theory. As a result, allelementary particles fall inrepresentations of this group. These are usually specified by thefour-momentum squared of each particle (i.e. its mass squared) and the intrinsicquantum numbers ${\textstyle J^{PC}}$ , where $J {\displaystyle J}$ is thespin quantum number, $P {\displaystyle P}$ is theparity and $C {\displaystyle C}$ is thecharge-conjugation quantum number. In practice, charge conjugation and parity are violated by manyquantum field theories; where this occurs, $P {\displaystyle P}$ and $C {\displaystyle C}$ are forfeited. SinceCPT symmetry isinvariant in quantum field theory, atime-reversal quantum number may be constructed from those given.

As atopological 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.^[9]

Other dimensions

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The definitions above can be generalized to arbitrary dimensions in a straightforward manner. Thed-dimensional Poincaré group is analogously defined by the semi-direct product

\operatorname {IO} (1,d-1):=\mathbf {R} ^{1,d-1}\rtimes \operatorname {O} (1,d-1)

with the analogous multiplication

(\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)

.^[6]

The Lie algebra retains its form, with indicesµ andν now taking values between0 andd − 1. The alternative representation in terms ofJ_i andK_i has no analogue in higher dimensions.

Notes

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^Poincaré, Henri (1905-12-14),"Sur la dynamique de l'électron" ,Rendiconti del Circolo Matematico di Palermo,21:129–176,Bibcode:1906RCMP...21..129P,doi:10.1007/bf03013466,hdl:2027/uiug.30112063899089,S2CID 120211823 (Wikisource translation:On the Dynamics of the Electron). The group defined in this paper would now be described as the homogeneous Lorentz group with scalar multipliers.
^Minkowski, Hermann,"Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" ,Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse:53–111 (Wikisource translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies).
^Minkowski, Hermann,"Raum und Zeit" ,Physikalische Zeitschrift,10:75–88
^"Survey of Symmetry and Conservation Laws: More Poincare"(PDF).frankwilczek.com. Retrieved2021-02-14.
^Barnett, Stephen M (2011-06-01)."On the six components of optical angular momentum".Journal of Optics.13 (6) 064010.Bibcode:2011JOpt...13f4010B.doi:10.1088/2040-8978/13/6/064010.ISSN 2040-8978.S2CID 55243365.
^^a ^bOblak, Blagoje (2017-08-01).BMS Particles in Three Dimensions. Springer. p. 80.ISBN 978-3-319-61878-4.
^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 978-1-13950-4324.
^"Topics: Poincaré Group".www.phy.olemiss.edu. Retrieved2021-07-18.

References

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The WikibookAssociative Composition Algebra has a page on the topic of:Poincaré group

Wu-Ki Tung (1985).Group Theory in Physics. World Scientific Publishing.ISBN 9971-966-57-3.
Weinberg, Steven (1995).The Quantum Theory of Fields. Vol. 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.