Spin rotations and projective $SO(3)$

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.