ウィグナーの分類のソースを表示

'''ウィグナーの分類'''（{{lang-en-short|Wigner's classification}}） とは、[[数学]]と[[理論物理学]]において、[[ポアンカレ群]]の、質量の鋭敏な[[固有値]]を持つ、[[正の数と負の数#負の整数と負でない整数の形式的な構成|非負の]]エネルギー ''E'' &ge; 0  の[[既約表現|既約]][[ユニタリ表現]]の分類である。<!--In [[mathematics]] and [[theoretical physics]], '''Wigner's classification''' is a classification of the [[nonnegative]] (''E'' ≥ 0) [[energy]] [[Irreducible representation|irreducible unitary representation]]s of the [[Poincaré group]], which have sharp mass [[eigenvalue]]s. -->物理学における素粒子論での素粒子や場の量子論での場の[[表現 (数学)|数学的表現]]を分類するために、[[ユージン・ウィグナー]]によって提唱された。<!--It was introduced by [[Eugene Wigner]], to classify particles and fields in physics—see the article [[particle physics and representation theory]]. -->分類はポアンカレ群の[[安定化部分群]]に依拠し、さまざまな質量状態の<!--[[群作用]]をもたらす。この安定化部分群を-->'''ウィグナー小群'''（Wigner little groups）と呼ぶ。<!--It relies on the stabilizer subgroups of that group, dubbed the [[Group_action#Fixed_points_and_stabilizer_subgroups|Wigner little groups]] of various mass states.-->

質量 <math>m \equiv \sqrt {P^2}</math> はポアンカレ群の{{仮リンク|カシミール不変量|en|Casimir invariant}}であり、その表現を名づけるのには役に立つかもしれない。
<!--The mass {{math| ''m'' ≡ √{{overline|''P'' ² }}}}   is a [[Casimir invariant]] of the Poincaré group, and may thus serve to label its representations. -->

この表現は ''m'' > 0 の場合、''m'' = 0 だが ''P''<sub>0</sub> > 0 の場合、''m'' = 0 で ''P''<sup>''μ''</sup> = 0 の場合、の3つの場合に応じて分類される。
<!--The representations may thus be classified according to whether {{math|''m'' > 0}} ;  {{math|''m''  {{=}} 0}}  but {{math|''P''<sub>0</sub> > 0}};  and {{math|''m'' {{=}} 0}} with {{math| ''P''<sup>μ</sup> {{=}} 0}}.-->
<!-- 翻訳途中 -->
<!-- 
最初の場合では、<math>P_0 =m</math> と <math>P_i =0</math> に関連付けられた[[固有空間]]（[[w:generalized eigenspaces of unbounded operators]]<注 [[Rigged Hilbert space]]にリダイレクト > を参照）は[[特殊直交群]] '''SO(3)''' の[[リー群の表現]]である。[[半直線]]<注「ray」>の解釈ではあるものは代わりに[[スピノル群]] '''Spin(3)''' へと移行できる。よって、重い状態は'''Spin(3)'''ユニタリ表現によって分類され、正の質量 <math>m</math> を持つ。-->
<!--
For the first case, note that the [[eigenspace]] (see [[generalized eigenspaces of unbounded operators]]) associated with {{math|''P''<sub>0</sub> {{=}} ''m''}} and {{math|''P''<sub>i</sub> {{=}} 0}}  is a [[Representations of Lie groups/algebras|representation]] of [[Special orthogonal group|SO(3)]]. In the ray interpretation, one can go over to [[Spin group|Spin(3)]] instead. So, massive states are classified by an irreducible Spin(3) [[Unitary representation|unitary]] and a positive mass, ''m''.  -->
<!--
For the second case, look at the [[stabilizer (group theory)|stabilizer]] of   {{math|''P''<sub>0</sub> {{=}} ''k''}},  {{math|''P''<sub>3</sub> {{=}} –''k''}},  {{math|''P''<sub>i</sub> {{=}} 0}}, {{math| ''i'' {{=}} 1,2}}. This is the [[Double covering group|double cover]] of [[Euclidean group|SE(2)]] (see [[unit ray representation]]). We have two cases, one where [[irrep]]s are described by an integral multiple of 1/2, called the [[helicity (particle physics)|helicity]] and the other called the "continuous spin" representation.

The last case describes the [[vacuum]]. The only finite-dimensional unitary solution is the [[trivial representation]] called the vacuum.

The double cover of the [[Poincaré group]] admits no non-trivial [[Group extension%23Central extension|central extension]]s.

Left out from this classification are [[tachyon]]ic solutions, solutions with no fixed mass, [[infraparticle]]s with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states. A celebrated example is the case of [[Deep inelastic scattering]], in which a virtual space-like [[photon]] is exchanged between the incoming [[lepton]] and the incoming [[hadron]]. This justifies the introduction of transversaly and longitudinally-polarized photons, and of the related concept of transverse and longitudinal structure functions, when considering these virtual states as effective probes of the internal quark and gluon contents of the hadrons. From a mathematical point of view, one considers the SO(2,1) group instead of the usual [[SO(3)]] group encountered in the usual massive case discussed above. This explain the occurrence of two transverse polarization vectors <math>\epsilon_T^{\lambda=1,2}</math> and <math>\epsilon_L</math>
which satisfy <math>\epsilon_T^2=-1</math> and <math>\epsilon_L^2=+1</math>, to be compared with the usual case   of a free <math>Z_0</math> boson which has three polarization vectors <math>\epsilon_T^{\lambda=1,2,3}</math>, each of them satisfying <math>\epsilon_T^2=-1</math>.
-->

== 関連項目 ==
* {{仮リンク|誘導表現|en|Induced representation}}
* {{仮リンク|微分同相写像群の表現論|en|Representation theory of the diffeomorphism group}}
* {{仮リンク|ガリレイ群の表現論|en|Representation theory of the Galilean group}}
* {{仮リンク|ポワンカレ群の表現論|en|Representation theory of the Poincaré group}}
* {{仮リンク|System of imprimitivity|en|System of imprimitivity}}

== 参考文献 ==
* {{citation|first=E. P.|last=Wigner|authorlink=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]|issue=1|volume=40|pages=149–204|year=1939|doi=10.2307/1968551|mr=1503456 }}.
* Bargmann, V., & Wigner, E. P. (1948). "Group theoretical discussion of relativistic wave equations", ''Proceedings of the National Academy of Sciences of the United States of America'' '''34'''(5), 211. [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079095/pdf/pnas01706-0041.pdf online]

{{physics-stub}}

{{DEFAULTSORT:ういくなあのふんるい}}
[[Category:リー群論]]
[[Category:表現論]]
[[Category:場の量子論]]
[[Category:ユージン・ウィグナー]]
[[Category:数学に関する記事]]