Regular Weyl systems, projective symmetry and massive particles

The stated $W$ has exactly the nonzero-space, fixed-parameter, Weyl multiplication, regularity, and irreducibility hypotheses in [F1], and [A1] supplies its choice assumption. Applying [F1] gives a unitary intertwiner from $W$ to the Schrödinger Weyl system.

If $W_1$ and $W_2$ satisfy the same hypotheses, step 1.1 gives unitaries $U_1$ and $U_2$ intertwining each with the same Schrödinger system. The composite $U_2^{-1}{U}_1$ is unitary and intertwines $W_1$ with $W_2$, so the equivalence class is unique.

The two parenthesizations of a triple product in $G_d$ have the same central coordinate, since both give $\theta +{\theta }^{\mathrm{\prime }}+{\theta }^{\mathrm{\prime }\mathrm{\prime }}-v^{\mathrm{\prime }}\cdot a-v^{\mathrm{\prime }\mathrm{\prime }}\cdot a-v^{\mathrm{\prime }\mathrm{\prime }}\cdot a^{\mathrm{\prime }}$. The identity is $(0,0,0)$ and the inverse of $(a,v,\theta )$ is $(-a,-v,-\theta -v\cdot a)$. Multiplying by $(0,0,\eta )$ on either side adds $\eta$ to the central coordinate and leaves $(a,v)$ fixed, so $Z$ is central; the quotient forgets $\theta$ and is the additive group ${\mathbb{R}}^{2d}$. The product and inverse maps are continuous in the product Euclidean topology: their coordinates are sums and the polynomial expression $\theta +{\theta }^{\mathrm{\prime }}-v^{\mathrm{\prime }}\cdot a$ or $-\theta -v\cdot a$. Thus $G_d$ is a topological group.

By [A2] and [F1], $\mathcal{H}=L^2({\mathbb{R}}^d,dx;\mathbb{C})$ with the stated pairing is a complex Hilbert space, so the operator definitions act on the required Hilbert space.

Translation is well-defined on almost-everywhere classes by [F2], and $\mathrm{\parallel }T(a)f{\mathrm{\parallel }}_2^2=\int \mathrm{\mid }f(x-a){\mathrm{\mid }}^{2}dx=\int \mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx$ by [F3]. The map is linear with inverse $T(-a)$; hence it is unitary by [F5] and step 1.2.

The multiplier $e^{imv\cdot x\mathrm{/}\mathrm{\hslash }}=\exp (imv\cdot x\mathrm{/}\mathrm{\hslash })$ is continuous, hence measurable, by [F15], and has modulus one by [F16]. Therefore $B_m(v)$ preserves the norm. It is linear, and the addition law [F14] gives $B_m(v)B_m(-v)=I$; hence $B_m(-v)$ is its inverse and it is unitary by [F5] and step 1.2.

For almost every $x$, $(B_m(v)T(a)f)(x)=e^{imv\cdot x\mathrm{/}\mathrm{\hslash }}f(x-a)$ while $(T(a)B_m(v)f)(x)=e^{imv\cdot (x-a)\mathrm{/}\mathrm{\hslash }}f(x-a)$. By the addition law [F14], $e^{imv\cdot x\mathrm{/}\mathrm{\hslash }}=e^{imv\cdot (x-a)\mathrm{/}\mathrm{\hslash }}{e}^{imv\cdot a\mathrm{/}\mathrm{\hslash }}$, proving the commutation identity. Multiplying it on the right by $B_m(v{)}^{\ast }T(a{)}^{\ast }$ gives the displayed loop identity.

Let $g=u+iw\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$. For the unitary families established in steps 2.1 and 2.2, [F10] gives $u,w\in C_c({\mathbb{R}}^d)$; applying [F9] to each component and using $\mathrm{\parallel }h+ik{\mathrm{\parallel }}_2^2=\mathrm{\parallel }h{\mathrm{\parallel }}_2^2+\mathrm{\parallel }k{\mathrm{\parallel }}_2^2$ yields $\mathrm{\parallel }T(a)g-T(a_0)g{\mathrm{\parallel }}_2\to 0$ as $a\to a_0$. If $v_n\to v_0$, continuity of the scalar exponential [F15] gives pointwise convergence $\mathrm{\mid }{e}^{im(v_n-v_0)\cdot x\mathrm{/}\mathrm{\hslash }}-1{\mathrm{\mid }}^2\mathrm{\mid }g(x){\mathrm{\mid }}^2\to 0$; by [F16] and the triangle inequality [F4], this is bounded by $4\mathrm{\mid }g(x){\mathrm{\mid }}^2$, which is integrable. Dominated convergence [F12] gives $\mathrm{\parallel }{B}_m(v_n)g-B_m(v_0)g{\mathrm{\parallel }}_2\to 0$. Since Euclidean parameter spaces are metric, both orbit maps are continuous on this dense smooth subspace.

For arbitrary $f\in \mathcal{H}$, approximate it in $L^2$ by $g\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$ using [F8]. Unitarity in steps 2.1 and 2.2 gives $\mathrm{\parallel }T(a)f-T(a_0)f{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }f-g{\mathrm{\parallel }}_2+\mathrm{\parallel }T(a)g-T(a_0)g{\mathrm{\parallel }}_2$, and the analogous estimate for $B_m(v)$; letting the approximation error be arbitrarily small extends both continuities to every $f$. The scalar phase $e^{im\theta \mathrm{/}\mathrm{\hslash }}$ is continuous in $\theta$ by [F15]. Moreover, $\mathrm{\parallel }{\mathrm{\Pi }}_m(a,v,\theta )f-{\mathrm{\Pi }}_m(a_0,v_0,{\theta }_0)f{\mathrm{\parallel }}_2\le \mathrm{\mid }{e}^{im\theta \mathrm{/}\mathrm{\hslash }}-e^{im{\theta }_0\mathrm{/}\mathrm{\hslash }}\mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }T(a)f-T(a_0)f{\mathrm{\parallel }}_2+\mathrm{\parallel }(B_m(v)-B_m(v_0))T(a_0)f{\mathrm{\parallel }}_2,$ so the orbit map of ${\mathrm{\Pi }}_m$ is norm-continuous on the product Euclidean group.

Using $T(a)B_m(v^{\mathrm{\prime }})=e^{-im{v}^{\mathrm{\prime }}\cdot a\mathrm{/}\mathrm{\hslash }}{B}_m(v^{\mathrm{\prime }})T(a)$ from step 3.1 and the translation law together with the exponential addition law [F14], multiplication gives ${\mathrm{\Pi }}_m(a,v,\theta ){\mathrm{\Pi }}_m(a^{\mathrm{\prime }},v^{\mathrm{\prime }},{\theta }^{\mathrm{\prime }})={\mathrm{\Pi }}_m(a+a^{\mathrm{\prime }},v+v^{\mathrm{\prime }},\theta +{\theta }^{\mathrm{\prime }}-v^{\mathrm{\prime }}\cdot a)$. By step 1.1 this is the group law; the operators are unitary by steps 2.1 and 2.2, and their orbit maps are continuous by step 4.1. Thus ${\mathrm{\Pi }}_m$ is a unitary representation by [F5], and ${\mathrm{\Pi }}_m(0,0,\eta )=e^{im\eta \mathrm{/}\mathrm{\hslash }}I$ is its central character.

For a finite direct sum the central subgroup acts as ${⨁}_j{e}^{i{m}_j\eta \mathrm{/}\mathrm{\hslash }}{I}_j$. If $m_j\mathrm{\ne }{m}_k$, equality of the two central characters as functions would imply $\exp (i{m}_j\eta \mathrm{/}\mathrm{\hslash })=\exp (i{m}_k\eta \mathrm{/}\mathrm{\hslash })$ for every real $\eta$. By [F13], $\exp (0)=1$; by the derivative at zero from [F15], for every nonzero real $\alpha$ we have ${\lim }_{\eta \to 0}(\exp (i\alpha \eta )-1)\mathrm{/}\eta =i\alpha$, since factoring $i\alpha$ reduces this limit to the complex derivative of $\exp$ at $0$. Applying this with $\alpha =m_j\mathrm{/}\mathrm{\hslash }$ and $\alpha =m_k\mathrm{/}\mathrm{\hslash }$ would give $i{m}_j\mathrm{/}\mathrm{\hslash }=i{m}_k\mathrm{/}\mathrm{\hslash }$, contrary to $m_j\mathrm{\ne }{m}_k$. Therefore the central characters are pairwise distinct.

The displayed form on $V={\mathbb{R}}^d\oplus {\mathbb{R}}^d$ is bilinear and alternating. If $\omega ((a,p),(a^{\mathrm{\prime }},p^{\mathrm{\prime }}))=0$ for every $(a^{\mathrm{\prime }},p^{\mathrm{\prime }})$, choosing $a^{\mathrm{\prime }}=0$ and arbitrary $p^{\mathrm{\prime }}$ forces $a=0$, while choosing $p^{\mathrm{\prime }}=0$ and arbitrary $a^{\mathrm{\prime }}$ forces $p=0$; hence it is nondegenerate on the $2d$-dimensional real space. Since $\mathrm{\hslash }>0$, the parameter $\lambda =1\mathrm{/}\mathrm{\hslash }$ is nonzero. Substituting $p=mv$ into the displayed Weyl formula gives $W(a,mv)=e^{-imv\cdot a\mathrm{/}(2\mathrm{\hslash })}{B}_m(v)T(a)$. Direct multiplication, using the exponential addition law [F14], gives $W(a,p)W(a^{\mathrm{\prime }},p^{\mathrm{\prime }})=e^{-i\omega ((a,p),(a^{\mathrm{\prime }},p^{\mathrm{\prime }}))\mathrm{/}(2\mathrm{\hslash })}W(a+a^{\mathrm{\prime }},p+p^{\mathrm{\prime }})$. The published model result [F7] verifies this system is irreducible and regular under Countable Choice [A2], and [F6] gives unitary equivalence for every other system with the same finite phase space and nonzero parameter $1\mathrm{/}\mathrm{\hslash }$ under AC [A1].

We use real-parameter differentiation in the finite orthonormal basis supplied by [F1]. For a complex scalar $f=a+ib$ with real components, its derivative is $a^{\mathrm{\prime }}+i{b}^{\mathrm{\prime }}$: convergence of the complex difference quotient is equivalent to convergence of its two real coordinates, since $\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }y\mathrm{\mid }\le \mathrm{\mid }x+iy\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }$. Expanding $(a+ib)(c+id)=(ac-bd)+i(ad+bc)$ transfers [F11] to complex scalar products, and conjugation differentiates by conjugating the derivative. Finite sums then give the scalar-vector and matrix product rules entry by entry, and $(M^{\ast }{)}^{\mathrm{\prime }}=(M^{\mathrm{\prime }}{)}^{\ast }$. These entry derivatives are the finite-dimensional vector or operator derivatives: in an orthonormal basis the squared vector norm is the finite sum of squared coordinate moduli, while for a $d\times d$ matrix $A$, each coordinate of $Ax$ has modulus at most $d{\max }_{j,k}\mathrm{\mid }{A}_{jk}\mathrm{\mid }\mathrm{\parallel }x\mathrm{\parallel }$ by the finite triangle inequality and $\mathrm{\mid }{x}_k\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }$. Thus $\mathrm{\parallel }Ax{\mathrm{\parallel }}^2\le d^3({\max }_{j,k}\mathrm{\mid }{A}_{jk}\mathrm{\mid }{)}^2\mathrm{\parallel }x{\mathrm{\parallel }}^2$. Thus entrywise difference-quotient remainders tending to zero also tend to zero in the corresponding norm. Direct calculation for each displayed Pauli matrix gives ${\sigma }_i^{\ast }={\sigma }_i$, ${\sigma }_i^2=I$, $\mathrm{tr}{\sigma }_i=0$, and $\det {\sigma }_i=-1$. With $c=\cos (t\mathrm{/}2)$ and $s_t=\sin (t\mathrm{/}2)$, direct $2\times 2$ matrix calculations and the Pythagorean identity [F8] give $A_i(t{)}^{\ast }{A}_i(t)=(c^2+s_t^2)I=I$ and $\det A_i(t)=c^2+s_t^2=1$, so each $A_i(t)$ lies in $SU(2)$. Its entries are differentiable by [F9]. The addition formulas [F10] and ${\sigma }_i^2=I$ give $A_i(t+h)=A_i(t)A_i(h)$ for each $i$, and [F9] gives $A_i(0)=I$; thus each map $t\mapsto A_i(t)$ is a homomorphism from $(\mathbb{R},+)$ to $SU(2)$. Differentiating $D(A_i(t){)}^{\ast }D(A_i(t))=I$ at $0$ and using $\frac{d}{dt}D(A_i(t)){\mathrm{\mid }}_{t=0}=-i{J}_i$ gives $J_i^{\ast }=J_i$. For $i=z$, the Euler formula [F7] and parity [F8] give $A_z(t)=\mathrm{diag}(\exp (-it\mathrm{/}2),\exp (it\mathrm{/}2))$. Set $U(t):=D(A_z(t))$. Since $D$ is a differentiable representation, $U$ is differentiable, $U(t+h)=U(t)U(h)=U(h)U(t)$, and the generator convention gives $U^{\mathrm{\prime }}(0)=-i{J}_z$. Differentiating the two products with respect to $h$ at $0$ yields $U^{\mathrm{\prime }}(t)=U(t)(-i{J}_z)=(-i{J}_z)U(t)$.

By [F1], the irreducibility hypothesis leaves one spin-$s$ block, with orthonormal basis $v_k$, $k=0,\dots ,2s$, and $J_z{v}_k=(s-k)v_k$. Write ${\mu }_k=s-k$ and $u_k(t)=U(t)v_k$. Step 1.1 gives $u_k^{\mathrm{\prime }}(t)=-i{\mu }_k{u}_k(t)$. From [F5] and [F6], the scalar map $t\mapsto \exp (i{\mu }_{k}t)$ has derivative $i{\mu }_k\exp (i{\mu }_{k}t)$: its difference quotient at $t$ factors as $\exp (i{\mu }_{k}t)(\exp (i{\mu }_{k}h)-1)\mathrm{/}h$ and the second factor tends to $i{\mu }_k$. The coordinate product rule established in step 1.1 gives $\frac{d}{dt}(\exp (i{\mu }_{k}t)u_k(t))=0$. To justify constancy, write this vector in the fixed finite orthonormal basis of [F1]. Every real and imaginary coordinate has zero derivative on $\mathbb{R}$ and is continuous (its increment is its difference quotient times the real increment). For arbitrary $a<b$, apply [F12] separately to these real coordinate functions on $[a,b]$; each has equal endpoint values. All coordinates, and hence the vector, are therefore constant on $\mathbb{R}$ and equal $v_k$ at $t=0$. Consequently $U(t)v_k=\exp (-i{\mu }_{k}t)v_k$.

Put $N=2s\in {\mathbb{N}}_0$. For every $k=0,\dots ,2s$, $2(s-k)=N-2k$ is an integer and [F5], [F7] give $\exp (-i2\pi (s-k))=\exp (-i\pi N)\exp (i2\pi k)=(-1{)}^N$. Indeed, $\exp (-i\pi )=-1$ follows from $\exp (i\pi )=-1$ and $\exp (-i\pi )\exp (i\pi )=\exp (0)=1$ by the power-series definition [F4]; also $\exp (i2\pi )=\exp (i\pi {)}^2=1$. Repeated use of [F5] gives the displayed integer powers. Thus step 2.1 implies $U(2\pi )v_k=(-1{)}^{2s}{v}_k$ for every basis vector.

By the diagonal form in step 1.1 and [F7], $A_z(2\pi )=\mathrm{diag}(\exp (-i\pi ),\exp (i\pi ))=-I$. Hence $D(-I)=U(2\pi )$; step 3.1 evaluates this operator on an orthonormal basis and proves $D(-I)=(-1{)}^{2s}{I}_V$.

By [F2], every $R\in SO(3)$ has exactly two lifts $W$ and $-W$. The representation law gives $D(-W)=D(-I)D(W)=(-1{)}^{2s}D(W)$ by step 4.1. The scalar has modulus one by [F7], so the projective class $[D(W)]\in PU(V)$ is independent of the lift. If $R_1=\rho (W_1)$ and $R_2=\rho (W_2)$, then $W_1{W}_2$ lifts $R_1{R}_2$, and $[D(W_1{W}_2)]=[D(W_1)][D(W_2)]$; hence the ray map is a homomorphism. The map is continuous: on any evenly covered neighborhood for $\rho$, its local section $s$ is continuous and the descended map there is $q\circ D\circ s$, a composition of continuous maps.

If $s$ is integer, $(-1{)}^{2s}=1$, so $D(W)$ is independent of the lift; defining $\tilde{D}(R)=D(W)$ gives a unitary representation because products of lifts lift the product, as in step 5.1, and it is continuous by the same local-section argument. If $s$ is half-integer, $(-1{)}^{2s}=-1$, so the two lifts $I$ and $-I$ of the identity have images $I_V$ and $-I_V$; these differ since $V\mathrm{\ne }\{0\}$. Therefore $D$ cannot factor as a unitary representation through $\rho$, although its projective class from step 5.1 does factor.