Editing Stone–von Neumann theorem (section)

=== 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.