Editing Stone–von Neumann theorem (section)

== 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.