Grupa Poincarégo

Grupa Poincarégogrupa izometrii zdefiniowanych w czasoprzestrzeni Minkowskiego. Grupa ta jest 10-wymiarową grupą Liego.

Generatorami grupy Poincarégo są elementy algebry Liego o następujących komutatorach:

gdzie:

– generator infinitezymalnej translacji,
– generator transformacji Lorentza.

Pełna grupa Poincaré jest iloczynem półprostym dwóch podgrup:

  • translacji w czasie,
  • translacji w przestrzeni,
  • transformacji Lorentza (grupy Lorentza).

Translacje tworzą grupę abelową, która jest podgrupą normalną grupy Poincaré.

Grupę Poincaré można wprowadzić poprzez rozszerzenie grupy Lorentza.

Zgodnie z programem z Erlangen geometrię czasoprzestrzeni Minkowskiego można zdefiniować jako geometrię, w której interwał czasoprzestrzenny jest niezmiennikiem transformacji grupy Poincarégo. Konsekwencją symetrii Poincarégo jest istnienie dodatkowych niezmienników: masy i całkowitego momentu pędu – stąd wynika m.in. istnienie spinu.

Grupa Poincarégo jest grupą symetrii każdej relatywistycznej teorii pola. Z tego powodu wszystkie cząstki elementarne są opisane za pomocą reprezentacji tej grupy.

Grupa została nazwana na cześć Henri Poincaré, jednego z twórców matematycznych podstaw teorii względności.

Symetrie Poincaré

Do symetrii grupy Poincaré należą:

  • translacje (tworzą abelową grupę Liego),
  • obroty (tworzą trójwymiarową nieabelową grupę Liego),
  • pchnięcia (boosty) – transformacje wiążące dwa układy, z których jeden porusza się względem drugiego, tzw. właściwe transformacje Lorentza.

Obroty i pchnięcia tworzą razem grupę Lorentza.

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