Kommutator (matematik)

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I matematik indikerer[hvordan?] kommutatoren hvor dårligt en bestemt binær operation kommuterer. Der anvendes forskellige definitioner i gruppeteori og ringteori.

Gruppeteori

Kommutatoren af to elementer g og h i en gruppe G er elementet

[g, h] = g−1h−1gh

Den er lig med gruppens identitet hvis og kun hvis g og h kommuterer (dvs. hvis og kun hvis gh = hg).

N.B. Nogle steder vælges kommutatoren at defineres som

[g, h] = ghg−1h−1

Identiteter

I det følgende angiver ax det x-konjugerede element x−1a x.

  • [y,x] = [x,y] −1
  • [[x,y−1],z] y [[y,z−1],x] z [[z,x−1],y]x = 1
  • [xy,z] = [x,z]y [y,z]
  • [x,yz] = [x,z] [x,y]z

Den anden identitet er også kendt under navnet Hall-Witt identiteten. Den er en gruppe-teoretisk analog af Jacobi-identiteten for den ring-teoretiske kommutator (se næste sektion).

Ringteori

Kommutatoren af to elemeter a og b i en ring eller associativ algebra er defineret ved

[a, b] = ab − ba

Den er nul hvis og kun hvis a og b kommuterer. I lineær algebra haves at hvis to matricer kommuterer i en basis, vil de kommutere i enhver anden basis.[bør uddybes]

Kommutatoren af to operatorer defineret i et Hilbertrum er et vigtigt koncept i kvantemekanik, da den angiver hvor godt de to målbare størrelser beskrevet af operatorerne kan måles samtidigt. Usikkerhedsprincippet er i bund og grund en sætning om disse kommutatorer.

Ligeledes er antikommutatoren defineret som ab + ba, ofte skrevet { a, b }.

Identiteter

Kommutatoren har de følgende egenskaber:

Lie-algebra relationer:

  • [A,B] = − [B,A]
  • [A,A] = 0
  • [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0

Yderligere relationer:

  • [A,BC] = [A,B]C + B[A,C]
  • [AB,C] = A[B,C] + [A,C]B
  • [A,BC] = [AB,C] + [CA,B]
  • [ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC

Kilder

  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2. udgave). Prentice Hall. ISBN 013805326X.
  • Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0805387145.

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