Editing Stone–von Neumann theorem (section)

=== Representation theory formulation ===
In terms of representation theory, the Stone–von Neumann theorem classifies certain unitary representations of the [[Heisenberg group]]. This is discussed in more detail in [[#The Heisenberg group|the Heisenberg group section]], below.

Informally stated, with certain technical assumptions, every representation of the Heisenberg group {{math|''H''<sub>2''n'' + 1</sub>}} is equivalent to the position operators and momentum operators on {{math|'''R'''<sup>''n''</sup>}}. Alternatively, that they are all equivalent to the [[Weyl algebra]] (or [[CCR algebra]]) on a symplectic space of dimension {{math|2''n''}}.

More formally, there is a '''unique''' (up to scale) non-trivial central strongly continuous unitary representation.

This was later generalized by [[Mackey theory]] – and was the motivation for the introduction of the Heisenberg group in quantum physics.

In detail:
* The continuous Heisenberg group is a [[Central extension (mathematics)|central extension]] of the abelian Lie group {{math|'''R'''<sup>2''n''</sup>}} by a copy of {{math|'''R'''}},
* the corresponding Heisenberg algebra is a central extension of the abelian Lie algebra {{math|'''R'''<sup>2''n''</sup>}} (with [[trivial algebra|trivial bracket]]) by a copy of {{math|'''R'''}},
* the discrete Heisenberg group is a central extension of the free abelian group {{math|'''Z'''<sup>2''n''</sup>}} by a copy of {{math|'''Z'''}}, and
* the discrete Heisenberg group modulo {{mvar|p}} is a central extension of the free abelian {{mvar|p}}-group {{math|('''Z'''/''p'''''Z''')<sup>2''n''</sup>}} by a copy of {{math|'''Z'''/''p'''''Z'''}}. 

In all cases, if one has a representation {{math|''H''<sub>2''n'' + 1</sub> → ''A''}}, where {{math|''A''}} is an algebra{{clarify|date=March 2013|reason=What analytic restriction?}} and the [[center of a group|center]] maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is [[Fourier theory]].{{clarify|reason=This statement appears too loose to be true. Abelian groups are Fourier theory, just like that?|date=May 2015}}

If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to ''central'' representations.

Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the [[center of an algebra|center of the algebra]]: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the [[scalar matrices]]. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the '''quantization''' value (in physics terms, the Planck constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).

More formally, the [[group ring|group algebra]] of the Heisenberg group over its field of [[scalar (mathematics)|scalars]] ''K'', written {{math|''K''[''H'']}}, has center {{math|''K''['''R''']}}, so rather than simply thinking of the group algebra as an algebra over the field {{mvar|K}}, one may think of it as an algebra over the commutative algebra {{math|''K''['''R''']}}. As the center of a matrix algebra or operator algebra is the scalar matrices, a {{math|''K''['''R''']}}-structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of {{math|''K''['''R''']}}-algebras {{math|''K''[''H''] → ''A''}}, which is the formal way of saying that it sends the center to a chosen scale.

Then the Stone–von Neumann theorem is that, given the standard quantum mechanical scale (effectively, the value of ħ), every strongly continuous unitary representation is unitarily equivalent to the standard representation with position and momentum.