Commutator

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For an electrical switch that periodically reverses the current see commutator (electric)


In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

Group theory

The commutator of two elements g and h of a group G is the element

[g, h] = g−1h−1gh

It is equal to the group's identity if and only if g and h commute (i.e. if and only if gh = hg). The subgroup generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent groups.

N.B. Some authors choose to define the commutator as

[g, h] = ghg−1h−1

Identities

In the sequel the expression ax denotes the conjugated (by x) 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

The second identity is also known under the name Hall-Witt identity. It is a group-theoretic analogue of the Jacobi-identity for the ring-theoretic commutator (see next section).

Ring theory

The commutator of two elements a and b of a rings or associative algebra is defined by

[a, b] = abba

It is zero if and only if a and b commute. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about these commutators.

Likewise, the anticommutator is defined as ab + ba, often written { a, b }. See also Poisson algebra.

Notice that the "commutator anticommutes", whereas the "anticommutator commutes". (Sounds paradoxical, does it not?)

Identities

The commutator has the following properties:

Lie-algebra relations:

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

Additional relations:

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

If A is a fixed element of a ring R, the first additional relation can also be interpreted as a Leibniz rule for the map given by In other words: the map DA defines a derivation on the ring R.

See also

References

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

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