Switch (operator)

A switch is an operator introduced in Mathématiques and extended to the quantum Mécanique.

In mathematics

In Mathematical, the switch gives a rather vague idea in the way in which one law is not commutative. There exist several definitions used in Théorie of the groups and Théorie of the rings.

In theory of the groups

That is to say $(G, \ star)$ a group and is $g$ and $h$ two elements of the group. One calls switch of $g$ and $h$ the element of the group defined by:

=g \ star H \ star g^ {- 1} \ star h^ {- 1}

Remark : A switch represents in fact the defect of “permutability” of two elements of the group:

g \ star h= \ star H \ star g

The switch is equal to the neutral element of the group if and only if $g$ and $h$ are permutable (i.e. if $g \ star h=h \ star g$ ).

In addition, the Sous-groupe generated by the whole of the switches is called the Groupe derived noted $D (G)$ or the sub-group from the switches of $G$ .

If $D (G)$ is tiny room to the neutral element then the $G$ group abelian Groupe is a .

Let us notice that we must consider the sub-group generated by the switches because in general the whole of the switches is not closed for this law. The switches are used to define the groups nilpotents.

Note: Certains authors prefers to define the switch of $g$ and $h$ by

H = g^ {- 1} \ star h^ {- 1} \ star G \ star h

Identities

In the continuation, the law $\ star$ is noted multiplicativement and the expression $a^x$ indicates combined (by $x$ ) of the element $a$ i.e. $x^ {- 1} has x$ .

$= ^ {- 1}$
$X, y^ {- 1}], ^ {there} Y, z^ {- 1}], ^ {Z} Z, x^ {- 1}], ^ {X} = 1$
$= ^ {there}$
$= ^ {Z}$

The second identity is also known under the name D identity of Hall-Witt . It is about an identity of the theory of the groups similar to the Identité of Jacobi of the theory of the switches in the rings (see the following section).

In theory of the rings

The switch of two elements

a

and

b

of a ring or a associative Algèbre is defined by

b=ab-ba~

It is null if and only if $a$ and $b$ are permutable. In Linear algebra, if two matrices commutate relative with a bases, then they commutate relative at any base.

By using the switch as a Hook of Dregs, any algebra associative can be regarded as a Algèbre of Dregs. The switch of two operators on a Espace of Hilbert is an important concept in quantum Mécanique since it measures at which item two descriptions of Observable S by operators can be measured simultaneously. The Principe of uncertainty is finally a Théorème on the switches.

In the same way, the anticommutator definite like $ab+ba$ , written is often noted $\ {has, B \}$ . See also Hook of Poisson.

Identities

A switch checks the following properties:

Relation of algebra of Dregs:

$= -$
$= 0$
$] +] +] = 0$

Relations of addition:

$= C + b$
$= has + b$
$= +$
$= ab + ac + bc$

If $a$ is a given element of a ring $A$ , the first relation of addition can also be interpreted like the rule of derivation of a product of a $D_a application: With \ to A$ given by $B \ mapsto$ . In other words, the $D_a$ application defines a Dérivation on the ring $A$ .

In quantum mechanics

In Mechanical quantum, the switch of two operators

A

and

B

is:

=AB-BA

. Applied to a function of wave, a switch makes it possible to know if it is possible simultaneously to measure two sizes relative to this function of wave.

See too

Algebra of Dregs, Hook of Dregs
Hook of Poisson

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