The Stone Von-Neumann Theorem

Weyl form of the Heisenberg commutation relations

Consider the Heisenberg commutation relations, for simplicity, for a free particle with one degree of freedom, is $$ PQ-QP = -i\hbar $$ where $P,Q$ should be self-adjoint operators on a Hilbert space $\mathcal{H}$.

Several observations:

  1. $\mathcal{H}$ should be finite-dimensional if $\hbar\neq 0$, since the trace of any commutator vanish, while $\operatorname{Tr}(-i\hbar)$ doesn’t.
  2. Both $P,Q$ should be unbounded.

Theorem. The operators $P$ and $Q$ in the Heisenberg commutation relations are unbounded.
Proof:  First of all, $P$ and $Q$ should be both bounded or unbounded, since $P$ and $Q$ play symmetric roles via the fact that interchanging them amounts to replacing $i$ by $-i$ or to selecting a different square root of $-1$. Since $P$ and $Q$ are self-adjoint, by Stone's theorem, they can be exponentiated to one-parameter unitary groups $$ U_t = \exp(itP), V_t=\exp(itQ). $$ Then by expansion (which can be justified pointwise, applied to a vector in the range of a spectral projection of $Q$), $$ U_t Q U_{-t} = \operatorname{Ad}(\exp(itP))Q = Q+t\hbar $$ meaning that $Q$ is unitarily equivalent to $Q+t\hbar,\forall t\in \mathbb{R}$. The spectrum of $Q$ must consist of the whole real line, and hence $Q$ is unbounded.

Using the notation introduced in the proof, in a similar manner, functional calculus yields $$ U_t f(Q) U_{-t} = f(Q+t\hbar) $$ for any real analytic function $f$ on the spectrum of $Q$. In particular, $$ \begin{equation}\label{weyl-form} U_t V_\sigma U_{-t} = U_t e^{i\sigma Q}U_{-t} = \exp(i\sigma(Q+t\hbar))=e^{i\sigma t}V_\sigma \end{equation} $$ This is the multiplicative form of the commutation relations, discovered by Weyl, which is given the name Weyl integrated form of the commutation relation.

Remark. The equation $$ U_t Q U_{-t} = \operatorname{Ad}(\exp(itP))Q = Q+t\hbar $$ given in the proof should be seen as an integrated form of the commutation relation that gives the evolution of $V_\sigma$ along the flow of $U_t$, while the commutation relation itself, similar to Lie derivative, should be seen as a particular kind of equation of evolution.

The Stone-von Neumann Theorem

Let $P_j,Q_k$ be self-adjoint operators, $1\leq j,k\leq n$, satisfying the canonical comutation relations $$ [P_j,P_k] = [Q_j,Q_k]=0, [P_j,Q_k] = -i\delta_{jk}\hbar, $$ and ignore the difficulties presented by the unboundedness of the operators by going to the Weyl integrated form, that is, assume that there are representations $U,V$ of $\mathbb{R}^n$, then the theorem of Stone and von Neumann is

Theorem(Stone-von Neumann) . Given pairs $(U,V)$ of unitary representations of $\mathbb{R}^n$ on a Hilbert space $\mathcal{H}$ satisfying $$ U(x)V(y) = \exp(i\omega(x,y))V(y)U(x) $$ where $\omega:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ is nondegenerate and bilinear. Such pairs are all equivalent to multiples of the standard Schroedinger representation on $L^2(\mathbb{R}^n)$.
Proof:  This is a sketch. Exchanging $U,V$, it is easy to see that $\omega$ is skew-symmetric. Without lose of generality, the relations can be reduced to $$ U(x)V(y) = e^{i\langle x,y\rangle}V(y)U(x) $$ where $\langle\cdot,\cdot\rangle$ is the usual Euclidean inner product on $\mathbb{R}^n$ since any nondegenerate bilinearity form is equivalent to its standard form. Given any representation of the Weyl integrated relation on a Hilbert space $\mathcal{H}$, there's a large number of self-adjoint projections on $\mathcal{H}$ of the form $$ P_\psi = \iint U(x)V(y)\psi(x,y)dxdy $$ for $\psi\in\mathcal{S}(\mathbb{R}^2n)$ (the Schwartz space). It can be proved that $P_\psi$ vanish only if $\psi\equiv 0$. In particular, if one put $$ \psi(x,y)=\frac{1}{(2\pi)^n} e^{-i\langle x,y\rangle/2}e^{-(|x|^2+|y|^2)/4} $$ then $P_\psi$ is a self-adjoint projection. For this $\psi$, $P_\psi U(x) P_\psi$ and $P_\psi V(y) P_\psi$ agree with $P_\psi$ up to scalar factors depending on $x,y$. Thus if the representation of $U,V$ is irreducible, $P_\psi$ must be of rank one, otherwise $P_\psi$ would be able to be written as the sum of two proper subprojections that generate proper invariant subspaces of $\mathcal{H}$. Given $\mathcal{H},\mathcal{H}'$ and two irreducible representations of the Weyl integrated relations, correspondingly there are $P_\psi$ and $P_{\psi'}$, and a map sending a unit vector in $\operatorname{Range}P_\psi$ to that of $\operatorname{Range}P_{\psi'}$ extends uniquely to a unitary intertwining operator, and thus intertwining the representations of $U,V$.

Mackey’s version

Recall that a covariant representation of a covariant system is a pair $(\pi,\rho)$ where $\rho$ is a unitary representation of a locally compact group $G$ on $\mathcal{H}$ and $\pi$ is a $\ast$-representation of a $C^\ast$-algebra $A$ on $\mathcal{H}$. There is furthermore a homomorphism $\alpha: G\to \operatorname{Aut}(A)$, s.t. $$ \rho(g)\pi(a)\rho(g)^\ast = \pi(\alpha_g(a)). $$ When $A=C_0(G)$, there is a canonical $\alpha$ given by $\alpha_g(f(h)) = g\cdot f =f(g^{-1}h)$ where $f\in C_0(G)$. Then the covariance relation becomes $$ \rho(g)\pi(f)\rho(g)^\ast = \pi(g\cdot f) $$

Theorem(Mackey) . Let $G$ be a locally compact group. Any covaiant pair of representations of $G$ and $C_0(G)$ on a Hilbert space $\mathcal{H}$ is a multiple of the standard representation on $L^2(G)$.$\label{mac}$
Via Fourier transform, the unitary representation of a locally compact abelian group $G$ can be identified with the representation of $C_0(\hat{G})$ where $\hat{G}$ is the Pontrjagin dual. For $G=\mathbb{R}^n$, this is the Stone-von Neumann theorem. Generally, for $G$ abelian, a covariant pair is $(\pi,\rho)$, s.t. $$ \rho(g)\pi(\hat{g})\rho(g)^\ast = \langle g,\hat{g}\rangle \pi(\hat{g}). $$

Generalization: Morita Equivalence

This can be generalized to the theorem

Theorem(Mackey's Imprimitivity) . Let $G$ be a locally compact group and $H$ a closed subgroup. Any covariant pair $(\rho,\pi)$ of representations $\rho$ of $G$ and $\pi$ of $C_0(G/H)$ on a Hilbert space $\mathcal{H}$ is induced from a unitary representation $\sigma:H\to B(\mathcal{H}_\sigma)$.
If $G/H$ has a $G$-invariant measure (this is not important, just for simplicity), then this means $\mathcal{H}$ can be identified with the Hilbert space of measurable functions $f:G\to \mathcal{H}_\sigma$ s.t. $$ f(gh) = \sigma(h)^{-1}f(g),\quad \int_{G/H}|f(gH)|^2d\tilde{g}<\infty, $$ and $g$ acts by left translation $\alpha_g f(x)=f(g^{-1}x)$.

This in a more modern language means the following. The covariant pairs of representations of $G$ and that of $C_0(G/H)$, or systems of imprimitivity based on $G/H$, can be identified with representations of a crossed product algebra $C_0(G/H)\rtimes G$. The content of the theorem says that $C_0(G/H)\rtimes G$ is (strongly) Morita equivalent to the group $C^\ast$-algebra $C^\ast(H)$. The correspondence of representations matches induced representations $\operatorname{Ind}(\sigma)$ of $G$ with the inducing representations $\sigma$ of $H$.

Generalization: Takai Duality

Stated in terms of crossed product, Theorem 3 can be read as

Corollary. For a locally compact group $G$, $C_0(G)\rtimes G$ is isomorphic to $\mathcal{K}(L^2(G))$.

This turns out to be a special case of Takai duality and its noncommutative generalization.

Theorem(Takai duality) . Let $G$ be a locally compact abelian group, whose dual is $\hat{G}$. Then $$ A\rtimes_\alpha G\rtimes_{\hat{\alpha}}\hat{G} \cong A\otimes \mathcal{K}(L^2(G)). $$ The second dual action $\hat{\hat{\alpha}}$ of $G$ on $A\otimes \mathcal{K}(L^2(G))$ becomes $\alpha\rtimes \lambda$ where $\lambda$ is the action on $\mathcal{K}(L^2(G))$ consisting of conjugating by the representation of left translation on $L^2(G)$.

This can be generalized to nonabelian groups and even quantum groups.