Editing Stone–von Neumann theorem (section)

== Uniqueness of representation ==
One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, ''up to unitary equivalence''. By [[Stone's theorem on one-parameter unitary groups|Stone's theorem]], there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups.

Let {{mvar|Q}} and {{mvar|P}} be two self-adjoint operators satisfying the canonical commutation relation, {{math|1=[''Q'', ''P''] = ''i''}}, and {{mvar|s}} and {{mvar|t}} two real parameters. Introduce {{math|''e<sup>itQ</sup>''}} and {{math|''e<sup>isP</sup>''}}, the corresponding unitary groups given by [[functional calculus]]. (For the explicit operators {{math|''x''}} and {{math|''p''}} defined above, these are multiplication by {{math|''e<sup>itx</sup>''}} and pullback by translation {{math|''x'' → ''x'' + ''s''}}.)   A formal computation{{r|Hall 2013|p=Section 14.2}} (using a special case of the [[Baker–Campbell–Hausdorff formula]]) readily yields
<math display="block">e^{itQ} e^{isP} = e^{-i st} e^{isP} e^{itQ} .</math>

Conversely, given two one-parameter unitary groups {{math|''U''(''t'')}} and {{math|''V''(''s'')}} satisfying the braiding relation
{{Equation box 1
|indent =:
|equation =  <math>U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t,</math>
|ref=E1
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the '''Weyl form of the CCR'''.

It is important to note that the preceding derivation is purely formal. Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions. Indeed, there exist operators satisfying the canonical commutation relation but not the Weyl relations ({{EquationNote|E1}}).{{r|Hall 2013|p=Example 14.5}} Nevertheless, in "good" cases, we expect that operators satisfying the canonical commutation relation will also satisfy the Weyl relations.

The problem thus becomes classifying two jointly [[Irreducible representation|irreducible]] one-parameter unitary groups {{math|''U''(''t'')}} and {{math|''V''(''s'')}} which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the '''Stone–von Neumann theorem''': ''all such pairs of one-parameter unitary groups are unitarily equivalent''.{{r|Hall 2013|p=Theorem 14.8}} In other words, for any two such {{math|''U''(''t'')}} and {{math|''V''(''s'')}} acting jointly irreducibly on a Hilbert space {{mvar|H}}, there is a unitary operator {{math|''W'' : ''L''<sup>2</sup>('''R''') → ''H''}} so that
<math display="block">W^*U(t)W = e^{itx} \quad \text{and} \quad W^*V(s)W = e^{isp},</math>
where {{mvar|p}} and {{mvar|x}} are the explicit position and momentum operators from earlier. When {{mvar|W}} is {{mvar|U}} in this equation, so, then, in the {{mvar|x}}-representation,  it is evident that {{mvar|P}} is unitarily equivalent to {{math|''e''<sup>−''itQ''</sup>&thinsp;''P''&thinsp;''e<sup>itQ</sup>'' {{=}} ''P'' + ''t''}}, and the spectrum of {{mvar|P}} must range along the entire real line. The analog argument holds for {{mvar|Q}}.

There is also a straightforward extension of the Stone–von Neumann theorem to {{mvar|n}} degrees of freedom.{{r|Hall 2013|p=Theorem 14.8}}
Historically, this result was significant, because it was a key step in proving that [[Werner Heisenberg|Heisenberg]]'s [[matrix mechanics]], which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to [[Erwin Schrödinger|Schrödinger]]'s wave mechanical formulation (see [[Schrödinger picture]]),
<math display="block"> [U(t)\psi ] (x)=e^{itx} \psi(x), \qquad [V(s)\psi ](x)= \psi(x+s) .</math>
{{see also|Generalizations of Pauli matrices#Construction: The clock and shift matrices}}

=== 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.

=== Reformulation via Fourier transform ===
Let {{mvar|G}} be a [[locally compact abelian group]] and {{math|''G''<sup>^</sup>}} be the [[Pontryagin dual]] of {{mvar|G}}. The [[Fourier transform|Fourier–Plancherel transform]] defined by
<math display="block">f \mapsto {\hat f}(\gamma) = \int_G  \overline{\gamma(t)} f(t) d \mu (t)</math>
extends to a C*-isomorphism from the [[group algebra of a locally compact group|group C*-algebra]] {{math|C*(''G'')}} of {{mvar|G}} and {{math|C<sub>0</sub>(''G''<sup>^</sup>)}}, i.e. the [[Spectrum of a C*-algebra|spectrum]] of {{math|C*(''G'')}} is precisely {{math|''G''<sup>^</sup>}}. When {{mvar|G}} is the real line {{math|'''R'''}}, this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language.

The group {{mvar|G}} acts on the {{mvar|C}}*-algebra {{math|C<sub>0</sub>(''G'')}} by right translation {{mvar|ρ}}: for {{mvar|s}} in {{mvar|G}} and {{mvar|f}} in {{math|C<sub>0</sub>(''G'')}},
<math display="block">(s \cdot f)(t) = f(t + s).</math>

Under the isomorphism given above, this action becomes the natural action of {{mvar|G}} on {{math|C*(''G''<sup>^</sup>)}}:
<math display="block"> \widehat{ (s \cdot f) }(\gamma) = \gamma(s) \hat{f} (\gamma).</math>

So a covariant representation corresponding to the {{mvar|C}}*-[[crossed product]]
<math display="block">C^*\left( \hat{G} \right) \rtimes_{\hat{\rho}} G </math>
is a unitary representation {{math|''U''(''s'')}} of {{mvar|G}} and {{math|''V''(''γ'')}} of {{math|''G''<sup>^</sup>}} such that
<math display="block">U(s) V(\gamma) U^*(s) = \gamma(s) V(\gamma).</math>

It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all [[irreducible representation]]s of
<math display="block">C_0(G) \rtimes_\rho G </math>
are unitarily equivalent to the <math>{\mathcal K}\left(L^2(G)\right)</math>, the [[compact operator on Hilbert space|compact operators]] on {{math|''L''<sup>2</sup>(''G''))}}. Therefore, all pairs {{math|{''U''(''s''), ''V''(''γ'')} }} are unitarily equivalent. Specializing to the case where {{math|1=''G'' = '''R'''}} yields the Stone–von Neumann theorem.