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| In [[mathematics]] and in [[theoretical physics]], the '''Stone–von Neumann theorem''' is any one of a number of different formulations of the [[uniqueness]] of the [[canonical commutation relation]]s between [[position (vector)|position]] and [[momentum]] [[operator (physics)|operator]]s. The name is for [[Marshall Harvey Stone|Marshall Stone]] and John {{harvs|txt|authorlink=John von Neumann|last=von Neumann|year=1931}}.<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}}</ref>
| | Making your computer run swiftly is very simple. Most computers run slow considering they are jammed up with junk files, that Windows has to look from each time it wants to find something. Imagine needing to find a book in a library, but all the library books are inside a big big pile. That's what it's like for the computer to locate something, whenever a program is full of junk files.<br><br>Windows Defender - this does come standard with numerous Windows OS Machines, yet otherwise will be download from Microsoft for free. It may aid safeguard against spyware.<br><br>System tray icon makes it simple to launch the system and displays "clean" status or the number of mistakes in the last scan. The ability to locate and remove the Invalid class keys and shell extensions is one of the leading blessings of the program. That is not usual function for the additional Registry Cleaners. Class keys plus shell extensions that are not functioning can really slow down the computer. RegCure scans to obtain invalid entries and delete them.<br><br>The 1328 error is a usual issue caused by your program being unable to properly process different updates for a system or Microsoft Office. If you have this error, it usually signifies that a computer is either unable to read the actual update file or your computer has problems with the settings it's utilizing to run. To fix this issue, we first require to change / fix any problems that the computer has with its update files, and then repair any of the issues which a system could have.<br><br>Another option whenever arresting the 1328 error is to clean out your PC's registry. The registry is important because it happens to be where settings and files selected by Windows for running are stored. As it happens to be frequently employed, breakdowns and situations of files getting corrupted are not uncommon. Additionally due to the way it happens to be configured, the "registry" gets saved in the wrong fashion continually, which makes the system run slow, ultimately causing a PC to suffer from a series of errors. The best system 1 can use in cleaning out registries is to utilize a reliable [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014] program. A registry cleaner may seek out plus repair corrupted registry files and settings allowing one's computer to run usually again.<br><br>The initially thing you should do is to reinstall any system which shows the error. It's typical for various computers to have certain programs which require this DLL to show the error when we try and load it up. If you see a specific system show the error, we must first uninstall which system, restart a PC and then resinstall the system again. This must replace the damaged ac1st16.dll file and cure the error.<br><br>The 'registry' is just the central database which stores all your settings and choices. It's a truly significant part of the XP system, which means that Windows is constantly adding and updating the files inside it. The problems happen when Windows really corrupts & loses certain of these files. This makes your computer run slow, because it tries difficult to locate them again.<br><br>What I would suggest is to look on the own for registry cleaners. You can do this with a Google search. If you find treatments, look for reviews and reviews about the product. Next you are able to see how others like the product, and how perfectly it works. |
| <ref>{{Citation | last1= von Neumann | first1=J. | 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>
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| <ref>{{Citation | last1=Stone | first1=M. H. | 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}}</ref>
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| <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>
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| ==Representation issues of the commutation relations==
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| In [[quantum mechanics]], physical [[observable]]s are represented mathematically by [[linear operator]]s on [[Hilbert space]]s.
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| For a single particle moving on the [[real line]] {{math|'''R'''}}, there are two important observables: position and [[momentum]]. In the quantum-mechanical description of such a particle, the [[position operator]] {{mvar|x}} and [[momentum operator]] {{mvar|p}} are respectively given by
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| :<math> [x \psi](x) = x \psi(x) \quad </math>
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| :<math> [p \psi](x) = - i \hbar \frac{\partial \psi(x)}{\partial x} </math>
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| on the domain {{mvar|V}} of infinitely differentiable functions of compact support on {{math|'''R'''}}. Assume {{math|ℏ}} to be a fixed ''non-zero'' real number — in quantum theory {{math|ℏ}} is (up to a factor of {{math|2π}}) [[Planck's constant]], which is not [[dimensionless]]; it takes a small numerical value in terms of units of the macroscopic world.
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| The operators {{mvar|x}}, {{mvar|p}} satisfy the [[canonical commutation relation]] Lie algebra,
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| :<math> [x,p] = x p - p x = i \hbar.</math>
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| Already in his classic book,<ref>[[Hermann Weyl|Weyl, H.]] (1927), "Quantenmechanik und Gruppentheorie", ''Zeitschrift für Physik'', '''46''' (1927) pp. 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|Q}} 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 Trace(AB) = Trace (BA); the left-hand side is zero, the right-hand side is non-zero. Some analysis<ref>Note {{math|1=[''x<sup>n</sup>'', ''p''] = ''i'' ℏ ''n x''<sup>''n'' − 1</sup>}}, hence {{math|<big>2 ‖ ''p'' ‖ ‖ ''x'' ‖<sup>''n''</sup> ≥ ''n'' ℏ ‖ ''x'' ‖<sup>''n'' − 1</sup></big>}}, so that, {{math|∀''n'': 2 ‖ ''p'' ‖ ‖ ''x'' ‖ ≥ ''n'' ℏ}}.</ref> shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both [[Bounded operator|bounded]]. For notational convenience, the nonvanishing square root of {{math|ℏ}} may be absorbed into the normalization of {{mvar|Q}} and {{mvar|P}}, so that, effectively, it amounts to 1 below.
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| 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>{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians | page = 245 |publisher = Springer | year = 2013}}</ref> 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. 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 on a finite-dimensional space,<ref>{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians | page = 302 |publisher = Springer | year = 2013}}</ref> 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.
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| == Uniqueness of representation ==
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| 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.
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| 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>''i t Q''</sup>}} and {{math|''e''<sup>''i s P''</sup>}}, the corresponding unitary groups given by [[functional calculus]]. A formal computation<ref>{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians | page = 282 |publisher = Springer | year = 2013}}</ref> (using a special case of the [[Baker–Campbell–Hausdorff formula]]) readily yields
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| :<math>e^{itQ} e^{isP} - e^{-i st} e^{isP} e^{itQ} = 0.</math>
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| Conversely, given two one-parameter unitary groups {{math|''U''(''t'')}} and {{math|''V''(''s'')}} satisfying the braiding relation
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| {{Equation box 1
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| |indent =:
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| |equation = <math>U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t \qquad ,</math>
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| |ref=E1
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| |bgcolor=#F9FFF7}}
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| formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. With care, these formal calculations can be made rigorous.
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| Therefore, there is a one-to-one correspondence between representations of the canonical commutation relation and two one-parameter unitary groups {{math|''U''(''t'')}} and {{math|''V''(''s'')}} satisfying ({{EquationNote|E1}}). This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the '''Weyl form of the CCR'''.
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| The problem thus becomes classifying two [[jointly 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''. 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
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| :<math>W: L^2(\mathbb{R}) \rightarrow H </math>
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| so that
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| :<math>W^*U(t)W = e^{itQ} \quad \mbox{and} \quad W^*V(s)W = e^{isP},</math>
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| where {{mvar|P}} and {{mvar|Q}} are the position and momentum operators from above. There is also a straightforward extension of the Stone–von Neumann theorem to ''n'' degrees of freedom.<ref>{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians |publisher = Springer | year = 2013}}</ref>
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| 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]]).
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| Taking {{mvar|W}} to be {{mvar|U}}, one sees that {{mvar|P}} is unitarily equivalent to {{math| e<sup>−''itQ''</sup> ''P'' e<sup>''itQ''</sup> {{=}} ''P''+''t''}}, and the spectrum of {{mvar|P}}
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| must range along the entire real line. The analog argument holds for {{mvar|Q}}.
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| === Representation theory formulation ===
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| 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.
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| 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''}}.
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| More formally, there is a '''unique''' (up to scale) non-trivial central strongly continuous unitary representation.
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| This was later generalized by [[Mackey theory]] – and was the motivation for the introduction of the Heisenberg group in quantum physics.
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| In detail:
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| * 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'''}},
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| * 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'''}},
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| * 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
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| * 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'''}}.
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| These are thus all [[semidirect product]], and hence relatively easily understood.
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| In all cases, if one has a representation {{math|''H'' → ''A''}}{{clarify|date=March 2013|what is ''A''?}} where 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]].
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| If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to ''central'' representations. | |
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| 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, Planck's constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).
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| More formally, the [[group algebra]] of the Heisenberg group {{math|''K''[''H'']}} has center {{math|''K''['''R''']}}, so rather than simply thinking of the group algebra as an algebra over the field of [[scalar (mathematics)|scalars]] {{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.
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| Then the Stone–von Neumann theorem is that, given a quantization value, every strongly continuous unitary representation is unitarily equivalent to the standard representation as position and momentum.
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| === Reformulation via Fourier transform ===
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| 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
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| :<math>f \mapsto {\hat f}(\gamma) = \int_G \overline{\gamma(t)} f(t) d \mu (t)</math>
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| extends to a C*-isomorphism from the [[group algebra|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.
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| The group {{mvar|G}} acts on the 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'')}},
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| :<math>(s \cdot f)(t) = f(t + s).</math>
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| Under the isomorphism given above, this action becomes the natural action of {{mvar|G}} on {{math|C*(''G''<sup>^</sup>)}}:
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| :<math> \widehat{ (s \cdot f) }(\gamma) = \gamma(s) \hat{f} (\gamma). </math>
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| So a [[covariant representation]] corresponding to the C*-[[crossed product]]
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| :<math>C^*( \hat{G} ) \rtimes_{\hat{\rho}} G </math>
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| is a unitary representation {{math|''U''(''s'')}} of {{mvar|G}} and {{math|''V''(''γ'')}} of {{math|''G''<sup>^</sup>}} such that
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| :<math>U(s) V(\gamma) U^*(s) = \gamma(s) V(\gamma).\;</math>
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| 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
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| :<math>C_0(G) \rtimes_{\rho} G </math>
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| are unitarily equivalent to the <math>{\mathcal K}(L^2(G))</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.
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| ==The Heisenberg group==
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| 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>''n''</sub>}} for {{mvar|n}} a positive integer. This is the [[Lie group]] of {{math|(''n'' + 2) × (''n'' + 2)}} square matrices of the form
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| :<math> \mathrm{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1_n & b \\ 0 & 0 & 1 \end{bmatrix}. </math>
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| In fact, using the Heisenberg group, one can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the center of {{math|''H<sub>n</sub>''}} consists of matrices {{math|M(0, 0, ''c'')}}.
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| However, this center is ''not'' the [[identity operator]] in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for ''n''=1, are
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| :::<math> P = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \qquad Q = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \qquad
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| z= \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, </math>
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| and the central generator {{math|''z'' {{=}} log''M''(0,0,1) {{=}} exp(''z'') −1}} is not the identity.
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| '''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
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| :<math> [U_h(\mathrm{M}(a,b,c))]\psi(x) = e^{i (b \cdot x + h c)} \psi(x+h a). </math>
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| 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.
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| 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|''h a''}} 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 which is beyond the scope of the article. However, below we sketch a proof of the corresponding Stone–von Neumann theorem for certain [[finite set|finite]] Heisenberg groups.
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| 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>}}.
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| 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.
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| === Relation to the Fourier transform ===
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| For any non-zero {{mvar|h}}, the mapping
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| :<math> \alpha_h: \mathrm{M}(a,b,c) \rightarrow \mathrm{M}(-h^{-1} b,h a, c -a b) </math>
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| 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
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| {{mvar|W}} on {{math|L<sup>2</sup>('''R'''<sup>''n''</sup>)}} such that, for any {{mvar|g}} in {{math|H<sub>''n''</sub>}},
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| :<math> W U_h(g) W^* = U_h \alpha (g). \quad </math>
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| 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]]).
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| '''Theorem'''. The operator {{mvar|W}} is, up to a scalar multiple, the [[Fourier transform]] on {{math|L<sup>2</sup>('''R'''<sup>''n''</sup>)}}.
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| This means that, ignoring the factor of {{math|(2π)<sup>''n''/2</sup>}} in the definition of the Fourier transform, | |
| :<math> \int_{\mathbb{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_{\mathbb{R}^n} e^{-i y \cdot ( p - b)} \psi(y) \ dy. </math>
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| This theorem can actually be used to prove the [[unitary operator|unitary]] nature of the Fourier transform, also known as the [[Plancherel theorem]]. Moreover,
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| :<math> (\alpha_h)^2 \mathrm{M}(a,b,c) =\mathrm{M}(- a, -b, c). </math>
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| '''Theorem'''. The operator {{math|''W''<sub>1</sub>}} such that
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| :<math> W_1 U_h W_1^* = U_h \alpha^2 (g) \quad </math>
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| is the reflection operator | |
| :<math> [W_1 \psi](x) = \psi(-x).\quad </math>
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| From this fact the [[Fourier inversion formula]] easily follows.
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| ==Example: The Segal–Bargmann space==
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| The [[Segal–Bargmann space]] is the space of holomorphic functions on ''C''<sup>''n''</sup> that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators
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| :<math> a_j = \partial /\partial z_j </math>
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| and
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| :<math> a_j^* = z_j, </math>
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| acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,
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| :<math> [a_j,a_k^*] = \delta_{j,k}. </math>
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| In 1961, Bargmann showed that ''a''<sub>j</sub><sup>*</sup> is actually the adjoint of ''a''<sub>j</sub> with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of ''a''<sub>j</sub> and ''a''<sub>j</sub><sup>*</sup>, 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.<ref>Hall, B. C. (2013). ''Quantum Theory for Mathematicians'', Springer, Section 14.4.</ref>
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| The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from <math> L^2(\mathbb{R}^n) </math> to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators ''a''<sub>j</sub> and ''a''<sub>j</sub><sup>*</sup>. This unitary map is the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]].
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| == Representations of finite Heisenberg groups ==
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| 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|1=''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|{{!}}{{mvar|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.
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| 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
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| :<math> [U_h \mathrm{M}(a,b,c) \psi](x) = \omega(b \cdot x + h c) \psi(x+ h a). </math>
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| '''Theorem'''. For a fixed non-zero {{mvar|h}}, the character function {{mvar|χ}} of {{math|''U''<sub>''h''</sub>}} is given by:
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| :<math> \chi (\mathrm{M}(a,b,c)) = \left\{ \begin{matrix} |K|^n \ \omega( h c) & \mbox{ if } a = b = 0 \\ 0 & \mbox{ otherwise}. \end{matrix} \right. </math>
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| It follows that
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| :<math> \frac{1}{|H_n(\mathbf{K})|} \sum_{g \in H_n(K)} |\chi(g)|^2 = \frac{1}{|K|^{2 n+1}} |K|^{2 n} |K| = 1. </math>
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| 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:
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| * Irreducibility of {{math|''U''<sub>''h''</sub>}}
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| * Pairwise inequivalence of all the representations {{math|''U''<sub>''h''</sub>}}.
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| == Generalizations ==
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| 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.
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| ==See also==
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| {{Div col}}
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| * [[Oscillator representation]]
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| * [[Weyl quantization]]
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| * [[CCR algebra]]
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| * [[Segal–Bargmann space]]
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| * [[Moyal product]]
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| * [[Weyl algebra]]
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| * [[Stone's theorem on one-parameter unitary groups]]
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| * [[Hille–Yosida theorem]]
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| {{Div col end}}
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| == References ==
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| {{Reflist}}
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| *{{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}}
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| * Rosenberg, Jonathan (2004) [http://www2.math.umd.edu/~jmr/StoneVNart.pdf "A Selective History of the Stone-von Neumann Theorem"] Contemporary Mathematics '''365'''. American Mathematical Society.
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| *{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians | page = 279 |publisher = Springer | year = 2013}}
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| {{DEFAULTSORT:Stone-von Neumann theorem}}
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| [[Category:Mathematical quantization]]
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| [[Category:Theorems in functional analysis]]
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| [[Category:Theorems in mathematical physics]]
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