Proof of a commutator

[Viona]

Hi, every one

How I can proof this commutator:
[AB,CD] = A[B, C]D + [A, C] BD +CA[B, D] + C[A, C]B


Thanks

 

[HallsofIvy]

You can't prove it -- it's not true! I suspect the last commutator is miswritten. If it were D[A, C]B rather than C[A, C]B then it would be true. Just expand each commutator and do a lot of cancelling.

 

[Otis]

I'm fairly well-versed in beginning algebra, yet I've never heard of commutators. I closed the Google search, after reading their snippet: "The commutator of two group elements and is, and two elements and are said to commute when their commutator is the identity element…".

 

[topsquark]

I'm fairly well-versed in beginning algebra, yet I've never heard of commutators. I closed the Google search, after reading their snippet: "The commutator of two group elements and is, and two elements and are said to commute when their commutator is the identity element…".

Wait until you get to non-commutative imaginary numbers! (Grassmann algebra.)

-Dan

 

[HallsofIvy]

"Commutators" are used extensively in Quantum Physics where physical properties are not given by numbers but by "operators" that are typically non-commutative. That is, if A and B are two such operators then \(\displaystyle AB\ne BA\) and their "commutator" is [A,B]= AB- BA.