Commutators in groups
The commutator of two elements and a group is the element
Sometimes the commutator is also called the element
Are defined. In particular, the commutator of two invertible matrices is the matrix .
If and only then, the commutator is the neutral element of the group. Generated by all commutators subgroup is commutator called. Commutators are used, for example, in the definition of nilpotent and solvable groups .
Commutators in algebras
It is equal to 0 if and only if and "commute" (swap), i.e. if :
Let , and be elements of an associative algebra and , scalars (elements of the primitive). Then:
- The commutator is alternating ( antisymmetric ):
- The commutator is bilinear :
- The commutator satisfies the Jacobi identity :
- The commutator satisfies the product rule :
Because the commutator is linear and satisfies the product rule, the self-mapping adjoint to each element is algebra
a derivation or derivation .
The anti-commutator or two elements and is the sum of their products in both orders:
It is equal to 0 if and only if and "anticommutate", i.e. if :
The anti-commutator is symmetrical:
The relationship with the commutator follows:
Application in physics
In quantum mechanics , every measuring apparatus has a Hermitian operator . Its eigenvalues are the possible measured values, its eigenvectors correspond to those physical states of the system to be measured, in which the associated measured value occurs with certainty.
If two of these operators commute, there is a complete set of common eigenvectors, more precisely two mutually commuting spectral decompositions. Physically this means that both measurements can be carried out together and that conditions can be prepared in which both measurements have reliable results. One speaks then of commuting , compatible or tolerable observables . Let it be given: a state in Dirac notation and the observables (operators) and . Then the following applies for the condition of simultaneous eigenstates:
with the generally complex eigenvalues and . It follows
If the condition is fulfilled, the two observables and are commuting and have simultaneous eigenstates.
With canonical quantization of a physical system, the phase space coordinates place and momentum, which characterize the state of the classical system, are replaced by the position operator and the momentum operator , for which the fundamental canonical commutator relation applies:
where and denote the components of the vector operators.
In Heisenberg's equation of motion , the commutator replaces the Poisson bracket in the equation of the corresponding, classical equation of motion of Hamiltonian mechanics (see applications of the Poisson bracket ).
The commutator indicates the algebraic properties of those operators that generate or annihilate bosons in quantum mechanical multi-particle states . Since the creation operators commute with each other, the individual particles in multi-particle states are indistinguishable in the sense that exchanging two particles does not result in a different state, but the same state with the same phase.
In quantum mechanics, the anti-commutator is used to specify the algebraic properties of those operators that generate or destroy fermions in multi-particle states . Since the creation operators anti- commutate with each other , the individual particles in multi-particle states are indistinguishable in the sense that exchanging two particles does not result in a different state, but the same state with opposite phase .