Editing Stone–von Neumann theorem (section)

== 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{aligned}[]
[x \psi](x_0) &= x_0 \psi(x_0) \\[]
[p \psi](x_0) &= - i \hbar \frac{\partial \psi}{\partial x}(x_0)
\end{aligned}</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.)