View source for Stone–von Neumann theorem

{{Short description|Mathematical theorem}}
In [[mathematics]] and in [[theoretical physics]], the '''Stone–von Neumann theorem''' refers to any one of a number of different formulations of the [[uniqueness quantification|uniqueness]] of the [[canonical commutation relation]]s between [[position (vector)|position]] and [[momentum]] [[operator (physics)|operator]]s. It is named after [[Marshall Stone]] and [[John von Neumann]].<ref>{{Citation | last1=von Neumann | first1=J. | author1-link=John von Neumann | title=Die Eindeutigkeit der Schrödingerschen Operatoren | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01457956 | year=1931 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=104 | pages=570–578| s2cid=120528257 }}</ref><ref>{{Citation | last1= von Neumann | first1=J. | author1-link=John von Neumann | title=Ueber Einen Satz Von Herrn M. H. Stone | jstor=1968535 | publisher=Annals of Mathematics | language=German | series=Second Series | year=1932 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=33 | issue=3 | pages=567–573 | doi= 10.2307/1968535}}</ref><ref>{{Citation | last1=Stone | first1=M. H. | authorlink=Marshall Harvey Stone | title=Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory | jstor=85485 | publisher=National Academy of Sciences | year=1930 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=16 | issue=2 | pages=172–175 | bibcode=1930PNAS...16..172S | doi=10.1073/pnas.16.2.172| pmc=1075964 | pmid=16587545| doi-access=free }}</ref><ref>{{citation|first=M. H. |last=Stone|authorlink=Marshall Harvey Stone|jstor=1968538|title= On one-parameter unitary groups in Hilbert Space|journal= Annals of Mathematics |volume=33|issue= 3|pages= 643–648|year=1932|doi=10.2307/1968538}}</ref>

== Representation issues of the commutation relations ==
In [[quantum mechanics]], physical [[observable]]s are represented mathematically by [[linear operator]]s on [[Hilbert space]]s.

For a single particle moving on the [[real line]] <math>\mathbb{R}</math>, there are two important observables: position and [[momentum]].  In the  Schrödinger representation quantum description of such a particle, the [[position operator]] {{mvar|x}} and [[momentum operator]] <math>p</math> are respectively given by
<math display="block">\begin{align}[]
[x \psi](x_0) &= x_0 \psi(x_0) \\[]
[p \psi](x_0) &= - i \hbar \frac{\partial \psi}{\partial x}(x_0)
\end{align}</math>
on the domain <math>V</math> of infinitely differentiable functions of compact support on <math>\mathbb{R}</math>.  Assume <math>\hbar</math> to be a fixed ''non-zero'' real number—in quantum theory <math>\hbar</math> is the [[reduced Planck constant]], which carries units of action (energy ''times'' time).

The operators <math>x</math>, <math>p</math> satisfy the [[canonical commutation relation]] Lie algebra,
<math display="block"> [x,p] = x p - p x = i \hbar.</math>

Already in his classic book,<ref>[[Hermann Weyl|Weyl, H.]] (1927), "Quantenmechanik und Gruppentheorie", ''Zeitschrift für Physik'', '''46''' (1927) pp.&nbsp;1–46, {{doi|10.1007/BF02055756}}; Weyl, H., ''The Theory of Groups and Quantum Mechanics'',  Dover Publications, 1950, {{isbn|978-1-163-18343-4}}.</ref> [[Hermann Weyl]] observed that this commutation law was ''impossible to satisfy'' for linear operators {{mvar|p}}, {{mvar|x}} acting on [[finite-dimensional]] spaces unless {{math|''ħ''}} vanishes. This is apparent from taking the [[Trace (linear algebra)|trace]] over both sides of the latter equation and using the relation {{math|Trace(''AB'') {{=}} Trace(''BA'')}}; the left-hand side is zero, the right-hand side is non-zero. Further analysis shows that any two self-adjoint operators satisfying the above commutation relation cannot be both [[Bounded operator|bounded]] (in fact, a theorem of [[Helmut Wielandt | Wielandt]] shows the relation cannot be satisfied by elements of ''any'' [[normed algebra]]<ref group=note>{{math|1=[''x<sup>n</sup>'', ''p''] = ''i'' ℏ ''nx''<sup>''n'' − 1</sup>}}, hence {{math|2{{norm|''p''}}&thinsp;{{norm|''x''}}<sup>''n''</sup> ≥ ''n'' ℏ {{norm|''x''}}<sup>''n'' − 1</sup>}}, so that, {{math|∀''n'': 2{{norm|''p''}}&thinsp;{{norm|''x''}} ≥ ''n'' ℏ}}.</ref>). For notational convenience, the nonvanishing square root of {{math|ℏ}} may be absorbed into the normalization of {{mvar|p}} and {{mvar|x}}, so that, effectively, it is replaced by 1. We assume this normalization in what follows.

The idea of the Stone–von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples.<ref name="Hall 2013">{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians |series=Graduate Texts in Mathematics|volume=267 |publisher = Springer | year = 2013|isbn=978-1461471158}}</ref>{{rp|Example 14.5}} To obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations. The exponentiated operators are bounded and unitary.  Although, as noted below, these relations are formally equivalent to the standard canonical commutation relations, this equivalence is not rigorous, because (again) of the unbounded nature of the operators. (There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space,{{r|Hall 2013|p=Chapter 14, Exercise 5}} namely [[James Joseph Sylvester|Sylvester]]'s [[Generalizations of Pauli matrices#Construction: The clock and shift matrices|clock and shift matrices]] in the finite Heisenberg group, discussed below.)

== 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.

== Heisenberg group ==
The above canonical commutation relations for {{mvar|P}}, {{mvar|Q}} are identical to the commutation relations that specify the [[Lie algebra]] of the general [[Heisenberg group]] {{math|''H''<sub>2''n''+1</sub>}} for {{mvar|n}} a positive integer. This is the [[Lie group]] of {{math|(''n'' + 2) × (''n'' + 2)}} square matrices of the form
<math display="block"> \mathrm{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1_n & b \\ 0 & 0 & 1 \end{bmatrix}. </math>

In fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory. 

Note that the center of {{math|''H<sub>2n+1</sub>''}} consists of matrices {{math|M(0, 0, ''c'')}}. However, this center is ''not'' the [[identity operator]] in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for {{math|''n'' {{=}} 1}}, are
<math display="block">\begin{align}
  P &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, &
  Q &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, &
  z &= \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},
\end{align}</math>
and the central generator {{math|1=''z'' = log ''M''(0, 0, 1) = exp(''z'') − 1}} is not the identity.

{{math theorem | For each non-zero real number {{mvar|h}} there is an [[irreducible representation]] {{math|''U<sub>h</sub>''}} acting on the Hilbert space {{math|[[Lp space|''L''<sup>2</sup>]]('''R'''<sup>''n''</sup>)}} by
<math display="block"> \left [U_h(\mathrm{M}(a,b,c)) \right ] \psi(x) = e^{i (b \cdot x + h c)} \psi(x+h a). </math>}}

All these representations are [[Unitary representation|unitarily inequivalent]]; and any irreducible representation which is not trivial on the center of {{math|''H<sub>n</sub>''}} is unitarily equivalent to exactly one of these.

Note that {{math|''U<sub>h</sub>''}} is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the ''left'' by {{math|''ha''}} and multiplication by a function of [[absolute value]] 1.  To show {{math|''U<sub>h</sub>''}} is multiplicative is a straightforward calculation.  The hard part of the theorem is showing the uniqueness; this claim, nevertheless, follows easily from the Stone–von Neumann theorem as stated above.  We will sketch below a proof of the corresponding Stone–von Neumann theorem for certain [[finite set|finite]] Heisenberg groups.

In particular, irreducible representations {{mvar|π}}, {{mvar|π′}} of the Heisenberg group {{math|''H<sub>n</sub>''}} which are non-trivial on the center of {{math|''H<sub>n</sub>''}} are unitarily equivalent if and only if {{math|1=''π''(''z'') = ''π′''(''z'')}} for any {{mvar|z}} in the center of {{math|''H<sub>n</sub>''}}.

One representation of the Heisenberg group which is important in [[number theory]] and the theory of [[modular form]]s is the '''[[theta representation]]''', so named because the [[Jacobi theta function]] is invariant under the action of the discrete subgroup of the Heisenberg group.

=== Relation to the Fourier transform ===
For any non-zero {{mvar|h}}, the mapping
<math display="block"> \alpha_h: \mathrm{M}(a,b,c) \to \mathrm{M} \left( -h^{-1} b,h a, c -a\cdot b \right) </math>
is an [[automorphism]] of {{math|''H<sub>n</sub>''}} which is the identity on the center of {{math|''H<sub>n</sub>''}}.  In particular, the representations {{math|''U<sub>h</sub>''}} and {{math|''U<sub>h</sub>α''}} are unitarily equivalent.  This means that there is a unitary operator {{mvar|W}} on {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} such that, for any {{mvar|g}} in {{math|''H<sub>n</sub>''}},
<math display="block"> W U_h(g) W^* = U_h \alpha (g).</math>

Moreover, by irreducibility of the representations {{math|''U<sub>h</sub>''}}, it follows that [[scalar multiplication|up to a scalar]], such an operator {{mvar|W}} is unique (cf. [[Schur's lemma]]).  Since {{mvar|W}} is unitary, this scalar multiple is uniquely determined and hence such an operator {{mvar|W}} is unique.

{{math theorem | The operator {{mvar|W}} is the [[Fourier transform]] on {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}}.}}

This means that, ignoring the factor of {{math|(2''π'')<sup>''n''/2</sup>}} in the definition of the Fourier transform,
<math display="block"> \int_{\mathbf{R}^n} e^{-i x \cdot p} e^{i (b \cdot x + h c)} \psi (x+h a) \ dx = e^{ i (h a \cdot p + h (c - b \cdot a))} \int_{\mathbf{R}^n} e^{-i y \cdot ( p - b)} \psi(y) \ dy.</math>

This  theorem has the immediate implication that the Fourier transform is [[unitary operator|unitary]], also known as the [[Plancherel theorem]].  Moreover,  
<math display="block"> (\alpha_h)^2 \mathrm{M}(a,b,c) =\mathrm{M}(- a, -b, c). </math>

{{math theorem | The operator {{math|''W''<sub>1</sub>}} such that
<math display="block"> W_1 U_h W_1^* = U_h \alpha^2 (g)</math>
is the reflection operator
<math display="block"> [W_1 \psi](x) = \psi(-x).</math>}}

From this fact the [[Fourier inversion formula]] easily follows.

== Example: Segal–Bargmann space ==
The [[Segal–Bargmann space]] is the space of holomorphic functions on {{math|'''C'''<sup>''n''</sup>}} that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators
<math display="block"> a_j = \frac{\partial}{\partial z_j}, \qquad a_j^* = z_j, </math>
acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,
<math display="block"> \left [a_j,a_k^* \right ] = \delta_{j,k}. </math>

In 1961, Bargmann showed that {{math|''a''{{su|b=''j''|p=&lowast;}}}} is actually the adjoint of {{math|''a<sub>j</sub>''}} with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of {{math|''a<sub>j</sub>''}} and {{math|''a''{{su|b=''j''|p=&lowast;}}}}, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations. It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly.{{r|Hall 2013|p=Section 14.4}} The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators {{math|''a<sub>j</sub>''}} and {{math|''a''{{su|b=''j''|p=&lowast;}}}}. This unitary map is the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]].

== Representations of finite Heisenberg groups ==
The Heisenberg group {{math|''H<sub>n</sub>''(''K'')}} is defined for any commutative ring {{mvar|K}}.  In this section let us specialize to the field {{math|''K'' {{=}} '''Z'''/''p'''''Z'''}} for {{mvar|p}} a prime.  This field has the property that there is an embedding {{mvar|ω}} of {{mvar|K}} as an [[abelian group|additive group]] into the circle group {{math|'''T'''}}.  Note that {{math|''H<sub>n</sub>''(''K'')}} is finite with [[cardinality]] {{math|{{!}}''K''{{!}}<sup>2''n'' + 1</sup>}}.  For finite Heisenberg group {{math|''H<sub>n</sub>''(''K'')}} one can give a simple proof of the Stone–von Neumann theorem using simple properties of [[Character theory|character function]]s of representations.  These properties follow from the [[orthogonality relations]] for characters of representations of finite groups.

For any non-zero {{mvar|h}} in {{mvar|K}} define the representation {{math|''U<sub>h</sub>''}} on the finite-dimensional [[inner product space]] {{math|{{ell}}<sup>2</sup>(''K''<sup>''n''</sup>)}} by
<math display="block">\left[U_h \mathrm{M}(a, b, c) \psi\right](x) = \omega(b \cdot x + h c) \psi(x + ha). </math>

{{math theorem | For a fixed non-zero {{mvar|h}}, the character function {{mvar|χ}} of {{math|''U<sub>h</sub>''}} is given by:
<math display="block">\chi (\mathrm{M}(a, b, c)) = \begin{cases}
  |K|^n\, \omega(hc) & \text{if } a = b = 0 \\
                   0 & \text{otherwise}
\end{cases}</math>}}

It follows that
<math display="block"> \frac{1}{\left|H_n(\mathbf{K})\right|} \sum_{g \in H_n(K)} |\chi(g)|^2 = \frac{1}{|K|^{2n+1}} |K|^{2n} |K| = 1. </math>

By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups {{math|''H<sub>n</sub>''('''Z'''/''p'''''Z''')}}, particularly:
* Irreducibility of {{math|''U<sub>h</sub>''}}
* Pairwise inequivalence of all the representations {{math|''U<sub>h</sub>''}}.
Actually, all irreducible representations of {{math|''H<sub>n</sub>''(''K'')}} on which the center acts nontrivially arise in this way.{{r|Hall 2013|p=Chapter 14, Exercise 5}}

== Generalizations ==
The Stone–von Neumann theorem admits numerous generalizations.  Much of the early work of [[George Mackey]] was directed at obtaining a formulation<ref>Mackey, G. W.  (1976). ''The Theory of Unitary Group Representations'', The University of Chicago Press, 1976.</ref> of the theory of [[induced representation]]s developed originally by [[Ferdinand Georg Frobenius|Frobenius]] for finite groups to the context of unitary representations of locally compact topological groups.

== See also ==
{{div col}}
* [[Oscillator representation]]
* [[Wigner–Weyl transform]]
* [[CCR and CAR algebras]] (for bosons and fermions respectively)
* [[Segal–Bargmann space]]
* [[Moyal product]]
* [[Weyl algebra]]
* [[Stone's theorem on one-parameter unitary groups]]
* [[Hille–Yosida theorem]]
* [[C0-semigroup]]
{{div col end}}

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}
* {{Citation | last1=Kirillov | first1=A. A. | title=Elements of the theory of representations | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-07476-4 | mr=0407202  | year=1976 | volume=220}}
* Rosenberg,  Jonathan  (2004) [https://www.math.umd.edu/~jmr/StoneVNart.pdf "A Selective History of the Stone–von Neumann Theorem"]   Contemporary Mathematics '''365'''.  American Mathematical Society.
* Summers, Stephen J. (2001). "On the Stone–von Neumann Uniqueness Theorem and Its Ramifications." In ''John von Neumann and the foundations of quantum physics'', pp. 135-152. Springer, Dordrecht, 2001, [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.1051&rep=rep1&type=pdf online].
{{Functional analysis}}

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