Commutator

In mathematics, the commutator measures (Latin commutare swap ), how two elements of a group or an associative algebra violate the commutative.

Commutators in groups

The commutator of two elements, and a group is the element

Sometimes the commutator is also called the element

Defined. In particular, the commutator of two matrices is the matrix invertierterer.

Just when true, the commutator is the identity element of the group. The subgroup generated by all commutators is called the commutator subgroup. Commutators are used for example in the definition of nilpotent and solvable groups.

Commutators in algebras

Commutators are also defined for rings and associative algebras. Here, the two commutator elements, and is defined as

He is exactly equal to 0 if and " commute " ( swap ), so if applies.

Be, and elements of an associative algebra and scalars ( elements of the body ). Then:

Due to the properties 1, 2 and 3, any associative algebra with the commutator as Lie bracket to a Lie algebra.

Because the commutator is linear and satisfies the product rule, is the adjoint to each element of self-mapping of the algebra

A derivative or derivation.

Applications in physics

In quantum mechanics, one of each measurement apparatus, a Hermitian operator. Its eigenvalues ​​are the possible values ​​, its eigenvectors corresponding to those physical states of the system to be measured, in which the associated measurement value occurs with certainty.

Commutation of two of these operators, so there is a complete set of common eigenvectors, more precisely two mutually commuting spectral decompositions. Physically, this means that you can make both measurements together and that we can prepare states in which both measurements have reliable results. One then speaks of commuting, compatible or compatible observables.

According to the heisenberg 's uncertainty relation of the expectation value of the commutator of two operators is a lower bound on the product of the uncertainties of the corresponding observables.

In canonical quantization of a physical system to take the place of the phase space coordinates, the location and the momentum that characterize the state of the classical system, operators with canonical commutation relations. Between local operator and operator called the fundamental pulse Kommutatorrelation denote wherein or the corresponding components of the respective vector operators.

In the Heisenberg equations of motion, the Poisson bracket of the commutator replaced in the formula image of the corresponding classical equation of motion of Hamiltonian mechanics (see applications of the Poisson bracket ).

With the commutator, the algebraic properties of those operators are specified, the create or destroy bosons in quantum mechanical Mehrteilchenzuständen. Since the creation operators commute with each other, the individual particles are in Mehrteilchenzuständen indistinguishable in the sense that an interchange of two particles results in no other, but the same state.

Anticommutator

The anticommutator or two elements and is the sum of its products in either order,

It follows the relationship with the commutator:

The defining relations of a Clifford algebra or Dirac algebra relate anticommutators.

With the anticommutator the algebraic properties of those operators are given in quantum mechanics that create or destroy in Mehrteilchenzuständen fermions. Since the creation operators anticommute with each other, the individual particles are in Mehrteilchenzuständen indistinguishable in the sense that an interchange of two particles results in no other, but the same state with opposite phase.