Editing Canonical commutation relation (section)

== Weyl relations ==
The [[Lie group|group]] <math>H_3(\mathbb{R})</math> generated by [[exponential map (Lie theory)|exponentiation]] of the 3-dimensional [[Lie algebra]] determined by the commutation relation <math>[\hat{x},\hat{p}]=i\hbar</math> is called the [[Heisenberg group]]. This group can be realized as the group of <math>3\times 3</math> upper triangular matrices with ones on the diagonal.<ref>{{harvnb|Hall|2015}} Section 1.2.6 and Proposition 3.26</ref>

According to the standard [[mathematical formulation of quantum mechanics]], quantum observables such as <math>\hat{x}</math> and <math>\hat{p}</math> should be represented as [[self-adjoint operator]]s on some [[Hilbert space]].  It is relatively easy to see that two [[operator (mathematics)|operator]]s satisfying the above canonical commutation relations cannot both be [[bounded operator|bounded]]. Certainly, if <math>\hat{x}</math> and <math>\hat{p}</math> were [[trace class]] operators, the relation <math>\operatorname{Tr}(AB)=\operatorname{Tr}(BA)</math> gives a nonzero number on the right and zero on the left.

Alternately, if <math>\hat{x}</math> and <math>\hat{p}</math> were bounded operators, note that <math>[\hat{x}^n,\hat{p}]=i\hbar n \hat{x}^{n-1}</math>, hence the operator norms would satisfy
<math display="block">2 \left\|\hat{p}\right\| \left\|\hat{x}^{n-1}\right\| \left\|\hat{x}\right\|  \geq n \hbar \left\|\hat{x}^{n-1}\right\|,</math> so that, for any ''n'',
<math display="block">2 \left\|\hat{p}\right\| \left\|\hat{x}\right\| \geq n \hbar</math>
However, {{mvar|n}} can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite. If the operators satisfy the Weyl relations (an exponentiated version of the canonical commutation relations, described below) then as a consequence of the [[Stone–von Neumann theorem]], ''both'' operators must be unbounded.

Still, these canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) [[unitary operator]]s <math>\exp(it\hat{x})</math> and <math>\exp(is\hat{p})</math>. The resulting braiding relations for these operators are the so-called [[Stone–von Neumann theorem|Weyl relations]]
<math display="block">\exp(it\hat{x})\exp(is\hat{p})=\exp(-ist\hbar)\exp(is\hat{p})\exp(it\hat{x}).</math>
These relations may be thought of as an exponentiated version of the canonical commutation relations; they reflect that translations in position and translations in momentum do not commute. One can easily reformulate the Weyl relations in terms of the [[Stone–von Neumann theorem#The Heisenberg group|representations of the Heisenberg group]].

The uniqueness of the canonical commutation relations—in the form of the Weyl relations—is then guaranteed by the [[Stone–von Neumann theorem]].

For technical reasons, the Weyl relations are not strictly equivalent to the canonical commutation relation <math>[\hat{x},\hat{p}]=i\hbar</math>. If <math>\hat{x}</math> and <math>\hat{p}</math> were bounded operators, then a special case of the [[Baker–Campbell–Hausdorff formula]] would allow one to "exponentiate" the canonical commutation relations to the Weyl relations.<ref>See Section 5.2 of {{harvnb|Hall|2015}} for an elementary derivation</ref> Since, as we have noted, any operators satisfying the canonical commutation relations must be unbounded, the Baker–Campbell–Hausdorff formula does not apply without additional domain assumptions. Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations.<ref>{{harvnb|Hall|2013}} Example 14.5</ref> (These same operators give a [[Uncertainty principle#A counterexample|counterexample]] to the naive form of the uncertainty principle.) These technical issues are the reason that the [[Stone–von Neumann theorem]] is formulated in terms of the Weyl relations.

A discrete version of the Weyl relations, in which the parameters ''s'' and ''t'' range over <math>\mathbb{Z}/n</math>, can be realized on a finite-dimensional Hilbert space by means of the [[Generalizations of Pauli matrices#Construction: The clock and shift matrices|clock and shift matrices]].