Regular Weyl Representations and Uniqueness: Examples and Counterexamples

On basis vectors of the finite-support subspace of ${\mathrm{\ell }}^2(V)$ define $L_v{\delta }_x=m_{\lambda }(v,x){\delta }_{v+x}$. Since $\mathrm{\mid }{m}_{\lambda }(v,x)\mathrm{\mid }=1$ and $x\mapsto v+x$ is a bijection of $V$, $L_v$ preserves the inner product and maps finite-support vectors to finite-support vectors. It extends to ${\mathrm{\ell }}^2(V)$ by $L_v[(a_n)]=[(L_v{a}_n)]$ on Cauchy-sequence classes; the extension is unitary with inverse $L_{-v}$. The cocycle identity gives $L_v{L}_w{\delta }_x=m_{\lambda }(w,x)m_{\lambda }(v,w+x){\delta }_{v+w+x}=m_{\lambda }(v,w)m_{\lambda }(v+w,x){\delta }_{v+w+x}=m_{\lambda }(v,w)L_{v+w}{\delta }_x\mathrm{.}$ Hence $L_v{L}_w=m_{\lambda }(v,w)L_{v+w}$ on the dense basis span and then everywhere. Also $L_0=I$ and $L_v^{\ast }=L_{-v}$, so ${\delta }_v\mapsto L_v$ extends linearly to a unital $\ast$-representation of the twisted finite-support algebra.

By [F1], this unital $\ast$-representation is contractive for the universal seminorm and extends to a representation $\mathrm{\Pi }$ of $C_{\mathrm{W}}^{\ast }(V,\omega ,\lambda )$ with $\mathrm{\Pi }({\delta }_v)=L_v$.

Fix any nonzero $v\in V$. For every real $t\mathrm{\ne }0$, $L_{tv}{\delta }_0={\delta }_{tv}$, while $L_0{\delta }_0={\delta }_0$. Since $tv\mathrm{\ne }0$, these are distinct orthonormal basis vectors, and therefore $\mathrm{\parallel }{L}_{tv}{\delta }_0-L_0{\delta }_0{\mathrm{\parallel }}_2=\sqrt{2}$ for every $t\mathrm{\ne }0$. The orbit map $t\mapsto \mathrm{\Pi }({\delta }_{tv})$ is not strongly continuous at zero. This proves that $\mathrm{\Pi }$ is nonregular although it is a representation of the universal Weyl $C^{\ast }$-algebra.

Since $\mathrm{\ell }$ is real-linear, $W_{\mathrm{\ell }}(0)=1$ and $W_{\mathrm{\ell }}(v)W_{\mathrm{\ell }}(w)=e^{i(\mathrm{\ell }(v)+\mathrm{\ell }(w))}=e^{i\mathrm{\ell }(v+w)}=W_{\mathrm{\ell }}(v+w)$. This is exactly the Weyl relation at $\lambda =0$, and each scalar has modulus one.

For every $v$, the map $t\mapsto W_{\mathrm{\ell }}(tv)=e^{it\mathrm{\ell }(v)}$ is continuous, so the system is regular. Its Hilbert space is one-dimensional, hence [F2] shows it is irreducible.

If a unitary intertwines $W_{{\mathrm{\ell }}_0}$ with $W_{{\mathrm{\ell }}_1}$, it is a nonzero scalar on $\mathbb{C}$, and the intertwining identity forces $e^{i{\mathrm{\ell }}_0(v)}=e^{i{\mathrm{\ell }}_1(v)}$ for every $v$. At $(q,p)=(1,0)$ these values are $1$ and $e^{i\pi }=-1$ by [F3], so they differ. Thus the two regular irreducible systems are inequivalent.

By [F1] and [F2], $U_q$ and $V_p$ are unitary. Direct evaluation gives $(U_q{V}_{p}f)(x)=e^{i\lambda p(x-q)}f(x-q)$ and $(V_p{U}_{q}f)(x)=e^{i\lambda px}f(x-q)$, so $U_q{V}_p=e^{-i\lambda qp}{V}_p{U}_q$.

The formula in the Statement yields $(W_{\lambda }(q,p)f)(x)=e^{i\lambda qp\mathrm{/}2}{e}^{i\lambda p(x-q)}f(x-q)=e^{i\lambda p(x-q\mathrm{/}2)}f(x-q)$, proving the asserted Schrödinger formula directly.

The commutation identity from step 1.1 gives $V_p{U}_{q^{\mathrm{\prime }}}=e^{i\lambda p{q}^{\mathrm{\prime }}}{U}_{q^{\mathrm{\prime }}}{V}_p$. Therefore $W_{\lambda }(q,p)W_{\lambda }(q^{\mathrm{\prime }},p^{\mathrm{\prime }})=e^{i\lambda ((qp+q^{\mathrm{\prime }}{p}^{\mathrm{\prime }})\mathrm{/}2+p{q}^{\mathrm{\prime }})}{U}_{q+q^{\mathrm{\prime }}}{V}_{p+p^{\mathrm{\prime }}}$. The exponent coefficient satisfies $((qp+q^{\mathrm{\prime }}{p}^{\mathrm{\prime }})\mathrm{/}2+p{q}^{\mathrm{\prime }})=((q+q^{\mathrm{\prime }})(p+p^{\mathrm{\prime }})-(q{p}^{\mathrm{\prime }}-p{q}^{\mathrm{\prime }}))\mathrm{/}2$, so this is $e^{-i\lambda (q{p}^{\mathrm{\prime }}-p{q}^{\mathrm{\prime }})\mathrm{/}2}{W}_{\lambda }(q+q^{\mathrm{\prime }},p+p^{\mathrm{\prime }})$, as required by [F3].

The sum defining $\omega$ is finite for $v,w\in V$; each term is bilinear and alternating, so $\omega$ is bilinear and alternating. If $v\mathrm{\ne }0$, choose $j$ with $q_j^2+p_j^2>0$ and let $w$ be supported only at $j$ with $(q_j^{\mathrm{\prime }},p_j^{\mathrm{\prime }})=(p_j,-q_j)$. Then $\omega (v,w)=-(q_j^2+p_j^2)\mathrm{\ne }0$, proving nondegeneracy. For $a\in \{1,2\}$, rescaling [F5] gives $\mathrm{\parallel }{\mathrm{\Omega }}_a{\mathrm{\parallel }}_2^2=(a\mathrm{/}\pi {)}^{1\mathrm{/}2}\int e^{-a{x}^2}dx=1$. Hence [F2] constructs the Hilbert space ${\mathcal{H}}_a$ and its unit reference vector ${\mathrm{\Omega }}_a^{\mathrm{\infty }}$.

If $v$ has support in $\{1,\dots ,n\}$, define on $H_n=L^2({\mathbb{R}}^n)$ the finite-coordinate operator $W_1(q_1,p_1)\otimes \cdots \otimes W_1(q_n,p_n)$. Each factor is unitary by [F1], so the product is unitary. Appending reference factors in the embeddings of [F2] commutes with this operator, because all acted-on coordinates are among the first $n$; the operators therefore define a unitary on the dense compatible union and extend to ${\mathcal{H}}_a$. Its inverse is the operator for $-v$.

For each finite $m\ge 1$, the span of the vectors ${⨂}_{j=1}^m{W}_1(q_j,p_j){\mathrm{\Omega }}_a$ is dense in $H_m=L^2({\mathbb{R}}^m)$. By [F4], finite linear combinations of indicators of finite-measure boxes $B=I_1\times \cdots \times I_m$ are dense. A box of zero measure has zero $L^2$ class; for a positive finite-measure box, ${\mathbf{1}}_B={⨂}_{j=1}^m{\mathbf{1}}_{I_j}$. By [F3], each ${\mathbf{1}}_{I_j}$ can be approximated in $L^2(\mathbb{R})$ by finite linear combinations of the one-mode orbit vectors $W_1(q,p){\mathrm{\Omega }}_a$. Fubini gives $\mathrm{\parallel }{⨂}_j{h}_j{\mathrm{\parallel }}_2={\prod }_j\mathrm{\parallel }{h}_j{\mathrm{\parallel }}_2$, and the finite telescoping estimate $\mathrm{\parallel }{⨂}_j{h}_j-{⨂}_j{k}_j{\mathrm{\parallel }}_2\le {\sum }_{r=1}^m({\prod }_{j<r}\mathrm{\parallel }{h}_j{\mathrm{\parallel }}_2)\mathrm{\parallel }{h}_r-k_r{\mathrm{\parallel }}_2({\prod }_{j>r}\mathrm{\parallel }{k}_j{\mathrm{\parallel }}_2)$ shows that products of these approximants converge to ${\mathbf{1}}_B$. Expanding the finite tensor sums gives finite linear combinations of the stated product-orbit vectors, each the restriction of ${\mathcal{W}}_a(v){\mathrm{\Omega }}_a^{\mathrm{\infty }}$ for a finitely supported $v$ by step 2.1. Hence their span is dense in $H_m$. For $m=0$, $H_0=\mathbb{C}$ is spanned by ${\mathrm{\Omega }}_a^{\mathrm{\infty }}$.

At each of the finitely many coordinates in the union of the supports of $v$ and $w$, [F1] gives $W_1(q_j,p_j)W_1(q_j^{\mathrm{\prime }},p_j^{\mathrm{\prime }})=e^{-i(q_j{p}_j^{\mathrm{\prime }}-p_j{q}_j^{\mathrm{\prime }})\mathrm{/}2}{W}_1(q_j+q_j^{\mathrm{\prime }},p_j+p_j^{\mathrm{\prime }})$. Multiplication of these finite equations gives ${\mathcal{W}}_a(v){\mathcal{W}}_a(w)=e^{-i\omega (v,w)\mathrm{/}2}{\mathcal{W}}_a(v+w)$; outside that finite union every factor is the identity. For fixed finitely supported $v$, choose $n$ containing its support. On each box indicator from $H_m$, embed into $H_r$ with $r=\max (m,n)$; the orbit $t\mapsto {\mathcal{W}}_a(tv){\mathbf{1}}_B$ is then a finite tensor product of strongly continuous one-mode orbits, hence is continuous by the finite telescoping estimate. Linear combinations of box indicators are dense in each $H_m$ by [F4], and the operators have norm $1$, so continuity extends to every vector in the compatible union. That union is dense in ${\mathcal{H}}_a$ by [F2]; the same norm bound extends continuity to the completion.

Fix a coordinate $j$ and a parameter $c>0$. The local family ${\rho }_{a,j}(q,p):={\mathcal{W}}_a(v_j(q,p))$, where $v_j(q,p)$ is supported only at coordinate $j$, is a strongly continuous one-mode Weyl family on ${\mathcal{H}}_a$: this holds on finite-coordinate elementary tensors by [F1] and extends by density from box-step functions [F4] and the unitary norm bound. Thus for every vector $\xi$ the weighted vector field $e^{-c{q}^2\mathrm{/}4-p^2\mathrm{/}(4c)}{\rho }_{a,j}(q,p)\xi$ is Bochner integrable by [F6]. On a finite-coordinate elementary tensor, its normalized integral acts on coordinate $j$ by the one-mode average $A_c$ and leaves the other coordinates fixed; [F3] identifies $A_c$ with the projection onto ${\mathrm{\Omega }}_c$. Therefore this coordinate average equals $P_j^{{\mathrm{\Omega }}_c}$ on the dense compatible union and, by boundedness, on ${\mathcal{H}}_a$. If a bounded map $T$ intertwines two such local families, [F6] lets $T$ pass through the vector-valued integral, so it intertwines the corresponding $P_j^{{\mathrm{\Omega }}_c}$. In particular, any bounded operator commuting with all global ${\mathcal{W}}_a(v)$ commutes with $P_j^{{\mathrm{\Omega }}_a}$ by taking $c=a$. The projections for distinct coordinates commute by their finite-coordinate formulas.

The orbit of ${\mathrm{\Omega }}_a^{\mathrm{\infty }}$ under finite-support ${\mathcal{W}}_a(v)$ is total in ${\mathcal{H}}_a$. For each $m$, step 3.1 gives density of its finite-coordinate product-orbit vectors in $H_m=L^2({\mathbb{R}}^m)$; these are restrictions of the full finite-support orbit. Since the compatible union of the $H_m$ is dense by [F2], the full orbit span is dense.

The joint fixed space ${\bigcap }_{j\ge 1}\mathrm{ran}{P}_j^{{\mathrm{\Omega }}_a}$ is exactly $\mathbb{C}{\mathrm{\Omega }}_a^{\mathrm{\infty }}$. Let $Q_a\xi :=⟨\xi ,{\mathrm{\Omega }}_a^{\mathrm{\infty }}⟩{\mathrm{\Omega }}_a^{\mathrm{\infty }}$. For $\xi$ in a finite-coordinate space $H_m$ and $n\ge m$, $R_n:={\prod }_{j=1}^n{P}_j^{{\mathrm{\Omega }}_a}$ satisfies $R_n\xi =⟨\xi ,{\mathrm{\Omega }}_a^{\otimes m}⟩{\mathrm{\Omega }}_a^{\mathrm{\infty }}=Q_a\xi$. Since the finite-coordinate union is dense and $R_n,Q_a$ are contractions, $R_n\to Q_a$ strongly on ${\mathcal{H}}_a$. Every vector fixed by all $P_j^{{\mathrm{\Omega }}_a}$ is fixed by every $R_n$, hence by $Q_a$ and lies in the reference line; conversely each $P_j^{{\mathrm{\Omega }}_a}$ fixes ${\mathrm{\Omega }}_a^{\mathrm{\infty }}$.

Let $P_j=P_j^{{\mathrm{\Omega }}_1}$ act on ${\mathcal{H}}_2$. For a vector $\xi \in H_m$ embedded using reference ${\mathrm{\Omega }}_2$ and any $j>m$, the $j$th coordinate is ${\mathrm{\Omega }}_2$, so the local rank-one formula gives $⟨P_j\xi ,\xi ⟩=\mathrm{\mid }⟨{\mathrm{\Omega }}_2,{\mathrm{\Omega }}_1⟩{\mathrm{\mid }}^2\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$. For arbitrary $\eta \in {\mathcal{H}}_2$, approximate by such a finite-coordinate $\xi$. Since $\mathrm{\parallel }{P}_j\mathrm{\parallel }\le 1$, for every $j>m$ the difference between these expectations is at most $\mathrm{\parallel }\eta -\xi \mathrm{\parallel }(\mathrm{\parallel }\eta \mathrm{\parallel }+\mathrm{\parallel }\xi \mathrm{\parallel })$, and the difference between the squared norm terms is bounded by the same quantity. Density therefore gives ${\lim }_{j\to \mathrm{\infty }}⟨P_j\eta ,\eta ⟩=\mathrm{\mid }⟨{\mathrm{\Omega }}_2,{\mathrm{\Omega }}_1⟩{\mathrm{\mid }}^2\mathrm{\parallel }\eta {\mathrm{\parallel }}^2$. Direct substitution and [F5] give $⟨{\mathrm{\Omega }}_2,{\mathrm{\Omega }}_1⟩=\frac{2^{1\mathrm{/}4}}{\sqrt{\pi }}{\int }_{\mathbb{R}}{e}^{-3x^2\mathrm{/}2}dx=2^{3\mathrm{/}4}\mathrm{/}\sqrt{3}$, so the limit coefficient is $2\sqrt{2}\mathrm{/}3<1$.

Let $M$ be a nonzero closed subspace invariant under every ${\mathcal{W}}_a(v)$. For $\xi \in M$, the vector field defining the local average in step 3.3 takes values in $M$; its Bochner integral also lies in $M$, since the Bochner approximants are $M$-valued and $M$ is closed. Thus every $P_j^{{\mathrm{\Omega }}_a}$ and every $R_n$ maps $M$ into $M$. Choose $0\mathrm{\ne }\eta \in M$. Totality in step 4.1 gives a finitely supported $v$ with $⟨\eta ,{\mathcal{W}}_a(v){\mathrm{\Omega }}_a^{\mathrm{\infty }}⟩\mathrm{\ne }0$. Then $\zeta :={\mathcal{W}}_a(-v)\eta \in M$ satisfies $⟨\zeta ,{\mathrm{\Omega }}_a^{\mathrm{\infty }}⟩=⟨\eta ,{\mathcal{W}}_a(v){\mathrm{\Omega }}_a^{\mathrm{\infty }}⟩\mathrm{\ne }0$. By step 4.2, $R_n\zeta \to Q_a\zeta =⟨\zeta ,{\mathrm{\Omega }}_a^{\mathrm{\infty }}⟩{\mathrm{\Omega }}_a^{\mathrm{\infty }}$; closedness and linearity of $M$ give ${\mathrm{\Omega }}_a^{\mathrm{\infty }}\in M$. Its total Weyl orbit forces $M={\mathcal{H}}_a$, proving irreducibility for each $a$.

If a unitary $U:{\mathcal{H}}_1\to {\mathcal{H}}_2$ intertwined the two global Weyl representations, it would intertwine their local coordinate families for every $j$. By the Bochner-integral argument in step 3.3 with $c=1$, it therefore intertwines the common averages $P_j^{{\mathrm{\Omega }}_1}$ on the two spaces. Since $P_j^{{\mathrm{\Omega }}_1}{\mathrm{\Omega }}_1^{\mathrm{\infty }}={\mathrm{\Omega }}_1^{\mathrm{\infty }}$, the unit vector $\eta =U{\mathrm{\Omega }}_1^{\mathrm{\infty }}$ would satisfy $P_j^{{\mathrm{\Omega }}_1}\eta =\eta$ for all $j$ and hence $⟨P_j^{{\mathrm{\Omega }}_1}\eta ,\eta ⟩=1$ for every $j$. Step 4.3 gives limit $2\sqrt{2}\mathrm{/}3<1$, a contradiction. Thus the two irreducible line-regular Weyl representations are inequivalent.