Commutator (math)

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

Commutators in groups

The commutator of two elements and a group is the element ${\displaystyle [g,h]}$ ${\displaystyle g}$ ${\displaystyle h}$

${\displaystyle [g,h]=g^{-1}h^{-1}gh=(hg)^{-1}gh.}$

Sometimes the commutator is also called the element

${\displaystyle [g,h]=ghg^{-1}h^{-1}}$

Are defined. In particular, the commutator of two invertible matrices is the matrix . ${\displaystyle A,B\in GL(n,\mathbb {R} )}$ ${\displaystyle ABA^{-1}B^{-1}}$

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 . ${\displaystyle gh=hg}$ ${\displaystyle [g,h]}$

Commutators in algebras

Commutators are also defined for rings and associative algebras . Here is the commutator of two elements and is defined as ${\displaystyle [a,b]}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle [a,b]=ab-ba.}$

It is equal to 0 if and only if and "commute" (swap), i.e. if : ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab=ba}$

${\displaystyle [a,b]=0\Leftrightarrow ab=ba}$

Let , and be elements of an associative algebra and , scalars (elements of the primitive). Then: ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle \lambda }$ ${\displaystyle \mu }$

1. The commutator is alternating ( antisymmetric ):

   ${\displaystyle [a,b]=-[b,a].}$

2. The commutator is bilinear :

   ${\displaystyle [\lambda a+\mu b,c]=\lambda [a,c]+\mu [b,c],}$

   ${\displaystyle [a,\lambda b+\mu c]=\lambda [a,b]+\mu [a,c].}$

3. The commutator satisfies the Jacobi identity :

   ${\displaystyle [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0.}$

4. The commutator satisfies the product rule :

   ${\displaystyle [a,bc]=[a,b]c+b[a,c],}$

   ${\displaystyle [ab,c]=a[b,c]+[a,c]b.}$

Due to the properties 1, 2 and 3, every associative algebra with the commutator as a Lie bracket becomes a Lie algebra . ${\displaystyle A}$

Because the commutator is linear and satisfies the product rule, the self-mapping adjoint to each element is algebra ${\displaystyle a}$

${\displaystyle a_{\text{adjungiert}}\colon b\mapsto [a,b]}$

a derivation or derivation .

Anti-commutator

The anti-commutator or two elements and is the sum of their products in both orders: ${\displaystyle \{a,b\}}$ ${\displaystyle [a,b]_{+}}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \{a,b\}=ab+ba.}$

It is equal to 0 if and only if and "anticommutate", i.e. if : ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab=-ba}$

${\displaystyle \{a,b\}=0\Leftrightarrow ab=-ba}$

The anti-commutator is symmetrical:

${\displaystyle \{a,b\}=\{b,a\}.}$

The relationship with the commutator follows:

${\displaystyle \Rightarrow [a,bc]=\{a,b\}c-b\{a,c\}.}$

The defining relations of a Clifford algebra or Dirac algebra concern anti-commutators.

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: ${\displaystyle |A\rangle }$ ${\displaystyle \zeta }$ ${\displaystyle \eta }$

${\displaystyle \zeta |A\rangle =\zeta ^{'}|A\rangle }$

${\displaystyle \eta |A\rangle =\eta ^{'}|A\rangle }$

with the generally complex eigenvalues and . It follows ${\displaystyle \zeta ^{'}}$ ${\displaystyle \eta ^{'}}$

${\displaystyle \zeta \eta |A\rangle =\zeta \eta ^{'}|A\rangle =\zeta ^{'}\eta |A\rangle =\eta \zeta ^{'}|A\rangle =\eta \zeta |A\rangle }$

${\displaystyle [\zeta ,\eta ]|A\rangle \equiv (\zeta \eta -\eta \zeta )|A\rangle =0}$

If the condition is fulfilled, the two observables and are commuting and have simultaneous eigenstates. ${\displaystyle [\zeta ,\eta ]=0}$ ${\displaystyle \zeta }$ ${\displaystyle \eta }$

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: ${\displaystyle x}$ ${\displaystyle p}$

${\displaystyle [x_{j},p_{k}]=i\hslash \delta _{j,k},}$

where and denote the components of the vector operators. ${\displaystyle j}$ ${\displaystyle k}$

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 ).

According to Heisenberg's uncertainty relation , the expectation value of the commutator of two operators gives a lower bound to the product of the uncertainties of the corresponding observables.

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 .

See also

• Lie bracket
• Associator

literature

• Siegfried Bosch : Algebra . 5th edition. Springer, Berlin 2003, ISBN 3-540-40388-4 , pp. 255 f ., doi : 10.1007 / 978-3-540-92812-6 .