The Stone–von Neumann Theorem

Derives Lemma A49.4. Unitarity and \(W(0)=\identity\) are immediate from Equation (A49.3), a phase times a product of unitaries. Rearranging Equation (A49.2) gives the form in which the relation will be used,

Now compute, using Equation (A49.6) once to move \(U(\vect{\alpha}')\) to the left of \(V(\vect{\beta})\), and then the group laws of \(U\) and of \(V\) separately:

On the other hand \(W(z+z')=\ee^{-\ii(\vect{\alpha}+\vect{\alpha}')\cdot (\vect{\beta}+\vect{\beta}')/2\hbar} U(\vect{\alpha}+\vect{\alpha}')V(\vect{\beta}+\vect{\beta}')\), so the ratio of Equation (A49.7) to \(W(z+z')\) is a phase whose exponent, multiplied by \(\hbar/\ii\), is

every term in \(\vect{\alpha}\cdot\vect{\beta}\) and in \(\vect{\alpha}'\cdot\vect{\beta}'\) cancelling. That is Equation (A49.5). Since \(\sigma(z,-z)=0\) we get \(W(z)W(-z)=W(0)=\identity\), so \(W(z)^{-1}=W(-z)\), and unitarity turns the inverse into the adjoint. Interchanging \(z\) and \(z'\) in Equation (A49.5) and using \(\sigma(z',z)=-\sigma(z,z')\) gives the commutation form. Setting \(\vect{\beta}=\vect{0}\) or \(\vect{\alpha}=\vect{0}\) in Equation (A49.3) recovers \(U\) and \(V\); conversely \(W(z)\) is a phase times a product of one \(U\) and one \(V\), so the two families have the same invariant subspaces.

Derives Lemma A49.5. The norm statement is unitarity. For continuity it is enough, by Equation (A49.3) and continuity of the phase, to show that \(\vect{\alpha}\mapsto U(\vect{\alpha})x\) and \(\vect{\beta}\mapsto V(\vect{\beta})x\) are continuous. Take \(U\); the argument for \(V\) is identical. By hypothesis the \(f\) one-parameter groups \(\alpha_{j}\mapsto U(\alpha_{j}\vect{e}_{j})\) commute and each is strongly continuous, and \(U(\vect{\alpha})=U(\alpha_{1}\vect{e}_{1})\cdots U(\alpha_{f}\vect{e}_{f})\). Telescoping the difference of two such products and using that every factor is an isometry,

with \(y_{j}=U(\alpha_{j+1}'\vect{e}_{j+1})\cdots U(\alpha_{f}'\vect{e}_{f})x\) a fixed vector for each \(j\) once \(\vect{\alpha}'\) is fixed. Each summand tends to \(0\) as \(\alpha_{j}\to\alpha_{j}'\) by strong continuity of the \(j\)-th group, which is continuity at \(\vect{\alpha}'\).

Derives Lemma A49.6. On each ball the integrand is a continuous \(\mathcal{H}\)-valued function of compact support, so the Riemann construction of Lemma A25.2 applies verbatim in \(2f\) variables — it uses only uniform continuity on a compact set and completeness of \(\mathcal{H}\), neither of which cares how many variables there are. For \(R'>R\) the difference of the two truncated integrals is bounded in norm by \(\norm{x}\int_{R<\abs{\zeta}\leq R'}\abs{G}\), which tends to zero as \(R\to\infty\) because \(\abs{G}\) is integrable; the truncations therefore form a Cauchy net and converge. The bound on \(\norm{\Pi_{G}}\) and the first identity in Equation (A49.13) pass to the limit from Equation (A25.2). For the adjoint, use that identity twice together with \(W(\zeta)^{\dagger}=W(-\zeta)\): \(\braket{\Pi_{G}w}{x}=\overline{\braket{x}{\Pi_{G}w}} =\int\overline{G(\zeta)}\braket{W(\zeta)w}{x}\dd^{2f}\zeta =\int\overline{G(\zeta)}\braket{w}{W(-\zeta)x}\dd^{2f}\zeta\), and substituting \(\zeta\to-\zeta\) identifies the result as \(\braket{w}{\Pi_{G^{\sharp}}x}\). Invariance of a closed invariant subspace is inherited from the truncated integrals, whose Riemann sums lie in it, the subspace being closed.

For Equation (A49.14), take \(w,x\in\mathcal{H}\) and expand \(\braket{w}{\Pi_{G}\Pi_{G'}x}\) by the first identity of Equation (A49.13), applied once to \(\Pi_{G}\) acting on the vector \(\Pi_{G'}x\) and once inside. The result is the scalar double integral

absolutely convergent because the matrix element is bounded by \(\norm{w}\norm{x}\) and \(G\), \(G'\) are integrable. Substituting Equation (A49.10) and changing variables to \(u=\zeta+\zeta'\) at fixed \(\zeta\) — a translation, which preserves \(\dd^{2f}\zeta'\) — and then exchanging the order of integration, which absolute convergence and Equation (11.151) on an exhausting sequence of boxes permit, turns Equation (A49.15) into \(\int G''(u)\braket{w}{W(u)x}\dd^{2f}u\), using \(\sigma_{0}(\zeta,u-\zeta)=\sigma_{0}(\zeta,u)\). Since \(w\) was arbitrary this is \(\braket{w}{\Pi_{G''}x}\), and \(G''\) is continuous and integrable because \(\abs{G''}\leq\abs{G}\ast\abs{G'}\).

Derives Lemma A49.7. Write \(\hat h(\eta)=(2\pi)^{-n/2}\int h(\xi)\ee^{-\ii\eta\cdot\xi} \dd^{n}\xi\), which vanishes identically by hypothesis, with \(\eta\) replaced by \(-\eta\). For \(\varepsilon>0\) put

Inserting the definition of \(\hat h\) and exchanging the order of integration — legitimate because \(\abs{h(\xi')}\ee^{-\varepsilon\abs{\eta}^{2}/2}\) is integrable over \(\R^{n}\times\R^{n}\), so Equation (11.151) applies on an exhausting sequence of boxes — gives \(I_{\varepsilon}(\xi)=\int h(\xi')g_{\varepsilon}(\xi-\xi') \dd^{n}\xi'\) with

the inner \(\eta\)-integral factorising into \(n\) one-dimensional Gaussian integrals, each evaluated by Equation (21.42). The family \(g_{\varepsilon}\) is an approximate identity exactly as in the proof of Theorem 21.32: non-negative, of unit mass, and concentrating on \(\abs{u}<\delta\) as \(\varepsilon\to0\). Since \(h\) is continuous at the fixed point \(\xi\), bounded near it and integrable at infinity, \(I_{\varepsilon}(\xi)\to h(\xi)\). But \(I_{\varepsilon}\equiv0\), so \(h(\xi)=0\).

Derives Lemma A49.9. Split \(v=p+\ii q\) with \(p,q\in\R^{n}\) and complete the square in the real part, \(-\abs{\zeta}^{2}/2+\zeta\cdot p =\abs{p}^{2}/2-\abs{\zeta-p}^{2}/2\). Substituting \(u=\zeta-p\),

The remaining integral factorises into \(n\) one-dimensional integrals by Equation (11.151), and each is Equation (21.42) with \(a=1\), giving \((2\pi)^{n/2}\ee^{-\abs{q}^{2}/2}\). Collecting the three exponents, \(\abs{p}^{2}/2+\ii p\cdot q-\abs{q}^{2}/2=v\cdot v/2\), which is Equation (A49.19).

Derives Theorem A49.10. Self-adjointness. The weight \(G(\zeta)=(2\pi)^{-f}\ee^{-\abs{\zeta}^{2}/4}\) is real and even, so \(G^{\sharp}=G\) and the second identity of Equation (A49.13) gives \(\Pi^{\dagger}=\Pi\).

Idempotence. By Equation (A49.14) it suffices to show that

Expand the product of the two Gaussians,

and write the phase as \(\ii\sigma_{0}(\zeta,u)/2=\ii\,\zeta\cdot Ju/2\). The integral in Equation (A49.22) is therefore \((2\pi)^{-2f}\ee^{-\abs{u}^{2}/4}\) times the integral Equation (A49.19) in \(n=2f\) variables with

because \(u\cdot Ju=\sigma_{0}(u,u)=0\) and \(Ju\cdot Ju=\abs{u}^{2}\) by orthogonality of \(J\). Hence the integral equals \((2\pi)^{-2f}\ee^{-\abs{u}^{2}/4}(2\pi)^{f} =(2\pi)^{-f}\ee^{-\abs{u}^{2}/4}=G(u)\), which is Equation (A49.22). The exponent \(-\abs{\zeta}^{2}/4\) and the coefficient \((2\pi)^{-f}\) are fixed by this requirement and by nothing else; that is where they come from.

The absorption identity. Expanding as in Equation (A49.15), and with the same justification,

and two applications of Equation (A49.10) give

Substitute \(u=\zeta+\zeta'\) at fixed \(\zeta\). Using \(\sigma_{0}(\zeta,u-\zeta)=\sigma_{0}(\zeta,u)\) and \(\sigma_{0}(\eta,u-\zeta)=\sigma_{0}(\eta,u)+\sigma_{0}(\zeta,\eta)\), the bracket becomes \(2\sigma_{0}(\zeta,\eta)+\sigma_{0}(\zeta,u)+\sigma_{0}(\eta,u)\), so

The inner integral is Equation (A49.19) again, with the same Gaussian Equation (A49.23) but with \(v=\bigl[u+\ii J(u+2\eta)\bigr]/2=p+\ii q\), where \(p=u/2\) and \(q=J(u/2+\eta)\). Now

using \(\abs{J\cdot}=\abs{\cdot}\) and \(p\cdot q=(u/2)\cdot J(u/2+\eta)=\sigma_{0}(u/2,\eta)\). Hence the inner integral equals \(G(u)\,\ee^{-u\cdot\eta/2-\abs{\eta}^{2}/2}\, \ee^{\ii\sigma_{0}(u,\eta)/2}\), and its phase cancels the prefactor \(\ee^{\ii\sigma_{0}(\eta,u)/2}\) of Equation (A49.27) exactly, because \(\sigma_{0}(\eta,u)+\sigma_{0}(u,\eta)=0\). What remains is a pure Gaussian in \(u\),

so that, substituting \(w=u+\eta\), \(\Pi W(\eta)\Pi=\ee^{-\abs{\eta}^{2}/4} (2\pi)^{-f}\int\ee^{-\abs{w}^{2}/4}W(w)\dd^{2f}w =\ee^{-\abs{\eta}^{2}/4}\Pi\). That is Equation (A49.21), and at \(\eta=0\) it reproduces \(\Pi^{2}=\Pi\), a check on every sign above.

Non-vanishing. Suppose \(\Pi=0\). Conjugating a single Weyl operator by another, Equation (A49.10) twice gives \(W(\eta)W(\zeta)W(\eta)^{\dagger} =\ee^{\ii\sigma_{0}(\eta,\zeta)}W(\zeta)\), hence

which vanishes for every \(\eta\) if \(\Pi\) does. Fix \(x,y\in\mathcal{H}\) and set \(h(\zeta)=G(\zeta)\braket{y}{W(\zeta)x}\), continuous by Lemma A49.5 and absolutely integrable because \(\abs{h}\leq G\norm{x}\norm{y}\). Taking the matrix element of Equation (A49.30) gives \(\int h(\zeta)\ee^{\ii\eta\cdot J\zeta}\dd^{2f}\zeta=0\) for every \(\eta\); since \(\eta\cdot J\zeta=-(J\eta)\cdot\zeta\) and \(J\) is a bijection of \(\R^{2f}\), that is the hypothesis of Lemma A49.7, so \(h\equiv0\). At \(\zeta=0\), \(h(0)=(2\pi)^{-f}\braket{y}{x}\), so \(\braket{y}{x}=0\) for all \(x\) and \(y\) — impossible unless \(\mathcal{H}=\set{0}\).

Derives Proposition A49.12. (1) \(M_{\Omega}\) is invariant under every \(W(\eta)\), since \(W(\eta)W(\zeta)\Omega\) is a multiple of \(W(\eta+\zeta)\Omega\) by Equation (A49.10), and the family is self-adjoint by Lemma A49.4; so \(M_{\Omega}^{\perp}\) is invariant too and \(M_{\Omega}\) reduces the system in the sense of Proposition 16.89. The restricted groups are again strongly continuous unitary groups on \(M_{\Omega}\) satisfying Equation (A49.2), and the restricted average is given by the same formula Equation (A49.18), so it is \(\Pi\) restricted to \(M_{\Omega}\), which by the last clause of Lemma A49.6 maps \(M_{\Omega}\) into itself.

(2) \(\Omega=\Pi\Omega\in M_{\Omega}\) lies in the range of the restricted average. Let \(u\in M_{\Omega}\) satisfy \(\Pi u=u\) and \(\braket{\Omega}{u}=0\). Then, for every \(\eta\),

by Equation (A49.21) and \(\Pi=\Pi^{\dagger}\). So \(u\) is orthogonal to every generator of \(M_{\Omega}\), hence to \(M_{\Omega}\), hence to itself: \(u=0\). The range is therefore the line \(\C\Omega\). For irreducibility, let \(N\subseteq M_{\Omega}\) be a closed subspace reducing the restricted system, and let \(N'\) be its orthogonal complement inside \(M_{\Omega}\). By the last clause of Lemma A49.6 both \(N\) and \(N'\) are invariant under \(\Pi\), and their images are orthogonal subspaces of the line \(\C\Omega\), so at most one of them is non-zero; and at least one is, because \(\Pi\Omega=\Omega\neq0\). Say \(\Pi N'=\set{0}\) and write \(\Omega=m+m'\) with \(m\in N\), \(m'\in N'\); then \(\Omega=\Pi\Omega=\Pi m\in N\), so \(N\) contains every \(W(\zeta)\Omega\) and equals \(M_{\Omega}\). In the other case the same argument gives \(N'=M_{\Omega}\), i.e. \(N=\set{0}\).

(3) \(M_{\Omega}\neq\set{0}\) contains \(\Omega\), so irreducibility of the original system forces \(M_{\Omega}=\mathcal{H}\), and then (2) is the statement about \(\Pi\) itself.

Derives Lemma A49.13. First, \(\Pi\Omega=\Omega\) and Equation (A49.21) give \(\braket{\Omega}{W(\eta)\Omega} =\braket{\Omega}{\Pi W(\eta)\Pi\Omega} =\ee^{-\abs{\eta}^{2}/4}\). Then, by Equation (A49.10) and \(W(\zeta)^{\dagger}=W(-\zeta)\),

which is Equation (A49.32).

Derives Proposition A49.14. Choose unit vectors \(\Omega\in\operatorname{ran}\Pi\) and \(\Omega'\in\operatorname{ran}\Pi'\), which exist and span those ranges by Theorem A49.10 and Proposition A49.12. On the dense subspace \(D\subset\mathcal{H}\) of finite linear combinations \(x=\sum_{k=1}^{N}c_{k}W(\zeta_{k})\Omega\) — dense because \(M_{\Omega}=\mathcal{H}\) — define

By Lemma A49.13 applied in each space,

the two Gram matrices being equal entry by entry. In particular the right-hand side of Equation (A49.34) vanishes whenever \(x\) does, so \(T\) is well defined on \(D\), and it is isometric. Its range contains every \(W'(\zeta)\Omega'\) and is therefore dense in \(\mathcal{H}'\), again by Proposition A49.12. An isometry with dense domain and dense range extends by continuity to a unitary of \(\mathcal{H}\) onto \(\mathcal{H}'\). Intertwining holds on \(D\) by construction — \(TW(\eta)W(\zeta_{k})\Omega\) and \(W'(\eta)TW(\zeta_{k})\Omega\) are both \(\ee^{\ii\sigma_{0}(\eta,\zeta_{k})/2}W'(\eta+\zeta_{k})\Omega'\) — and extends by continuity; by the last clause of Lemma A49.4 intertwining the \(W\)'s is the same as intertwining the \(U\)'s and the \(V\)'s.

For uniqueness, let \(T_{1}\) and \(T_{2}\) both intertwine. Then \(S=T_{2}T_{1}^{-1}\) is a unitary of \(\mathcal{H}'\) commuting with every \(W'(\zeta)\). The family \(\set{W'(\zeta)}\) is self-adjoint and acts irreducibly, so Theorem 16.91 gives \(S=c\,\identity\) with \(c\in\C\), and \(\abs{c}=1\) because \(S\) is unitary.

Derives Proposition A49.15. The kernel. By Equation (16.72) and Equation (A49.3),

In the dimensionless variables Equation (A49.9), \(\vect{\alpha}\cdot\vect{\beta}/2\hbar=\vect{a}\cdot\vect{b}/2\), \(\vect{\alpha}\cdot\vect{x}/\hbar=\vect{a}\cdot\vect{x}/s\) and \(\vect{\beta}=s\vect{b}\), so

Do the \(\vect{a}\) integral first, at fixed \(\vect{b}\). It is Equation (A49.19) in \(n=f\) variables, after the rescaling \(\vect{a}=\sqrt{2}\,\vect{u}\), and gives

Substituting \(\vect{x}'=\vect{x}-s\vect{b}\), so that \(\vect{b}=(\vect{x}-\vect{x}')/s\) and \(\dd^{f}b=s^{-f}\dd^{f}x'\), the two exponents combine:

because \(\vect{x}/s-\vect{b}/2=(\vect{x}+\vect{x}')/2s\) and the parallelogram identity turns the sum of the two squares into twice the sum of the squares of \(\vect{x}\) and \(\vect{x}'\). Hence

and the constant is \((4\pi)^{f/2}(2\pi)^{-f}s^{-f}=(\pi s^{2})^{-f/2}\), which is exactly what turns the two bare Gaussians into the normalised Equation (A49.36) twice over. That is Equation (A49.37), and it is manifestly an orthogonal projector of rank one, in agreement with Theorem A49.10.

Totality. Let \(K\) be the closed linear span of \(\set{W(\zeta)\Omega_{s}}\); it reduces the system by Proposition A49.12(1), so \(K^{\perp}\) is invariant and carries a Weyl system of its own, whose Gaussian average is the restriction of \(\Pi\) to \(K^{\perp}\). But that restriction is zero: for \(x\in K^{\perp}\), Equation (A49.37) gives \(\Pi x=\braket{\Omega_{s}}{x}\Omega_{s}=0\), because \(\Omega_{s}=W(0)\Omega_{s}\in K\). By the last clause of Theorem A49.10 a Weyl system on a non-zero space has a non-zero average, so \(K^{\perp}=\set{0}\) and \(K=L^{2}(\R^{f})\).

Irreducibility. \(\Omega_{s}\) spans \(\operatorname{ran}\Pi\) and \(M_{\Omega_{s}}=K=L^{2}(\R^{f})\), so Proposition A49.12(2) applied to \(M_{\Omega_{s}}\) says that the system is irreducible.

Proof of Theorem A49.2. Derives Theorem A49.2. Part (1) is Proposition A49.14 applied to the given system and to the Schrödinger system, which is an irreducible Weyl system by Proposition A49.15.

Part (2). Let \(\mathcal{H}\neq\set{0}\) carry a Weyl system, not assumed irreducible. By Theorem A49.10 the operator \(\Pi\) is a non-zero orthogonal projector; choose a unit vector \(\Omega_{1}\) in its range and put \(M_{1}=M_{\Omega_{1}}\). By Proposition A49.12, \(M_{1}\) reduces the system and the restriction to it is an irreducible Weyl system, hence carries a copy of the Schrödinger system by part (1). Now \(M_{1}^{\perp}\) is invariant and carries a Weyl system of its own; if it is not \(\set{0}\) its average is again non-zero, and the construction produces \(M_{2}\subseteq M_{1}^{\perp}\), and so on. The subspaces so produced are mutually orthogonal and each is infinite-dimensional — it is a copy of \(L^{2}(\R^{f})\) — so separability of \(\mathcal{H}\) (Definition 16.2) allows at most countably many of them, and the process exhausts \(\mathcal{H}\): the orthogonal complement of their closed sum is invariant and carries a vanishing average, hence is \(\set{0}\). That is the direct-sum statement, the sum being the one of Definition 16.84.