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Use the north-pole stereographic projection and the cross-ratio convention $[a,b;c,d]=(a-c)(b-d)/[(a-d)(b-c)]$. Antipodes have coordinates $-1/\bar{u}$. Rotate the first point to the south pole, whose coordinate is zero: its antipode becomes infinity, while a point at spherical distance $d$ has projected modulus $\tan (d/2)$. Möbius invariance of the cross-ratio then gives the displayed identity. Antipodal endpoints require the extended-value interpretation.

Project from the north pole $N=(0,0,1)$ onto the equatorial plane, identifying that plane with the complex plane. The line from $N$ to $P=(X,Y,Z)$ meets the plane inThis is stereographic projection. Its inverse isRotation through $\theta$ about the $z$-axis changes $X+iY$ to $e^{i\theta }(X+iY)$ and preserves $Z$, so it becomes the Möbius transformation $u\mapsto e^{i\theta }u$.

Let $B$ be the rotation matrix supplied in the question. It sends the $z$-axis to the $x$-axis, and therefore $R_x(\theta )=B{R}_z(\theta )B^{-1}$. If $b=\varphi B{\varphi }^{-1}$ is its given Möbius transformation, thenwhich is again a Möbius transformation. Such $z$- and $x$-axis rotations can send any chosen point of the sphere to the south pole: first rotate its horizontal projection onto the $y$-axis, then rotate in the $yz$-plane. This supplies the rotations needed below.

The antipodal stereographic coordinate relation is $\varphi (-P)=-1/\overline{\varphi (P)}$, as follows immediately from the inverse formula. Fix the cross-ratio conventionWe claim the required ordering isFor the antipodal cross-ratio and spherical distance identity, rotate $P$ to the south pole. Its coordinate becomes zero and its antipode becomes infinity. A point at angular spherical distance $d$ from the south pole has vertical coordinate $-\cos d$ and horizontal magnitude $\sin d$, so its new projected coordinate $v^{\prime }$ has modulus $\sin d/(1+\cos d)=\tan (d/2)$. By Möbius invariance of the cross-ratio, the displayed expression equalsFor finite nonsingular coordinates the same ratio is $-|u-v{|}^2/|1+u\bar{v}{|}^2$. Zero or infinite antipodal coordinates are interpreted by limits. When $Q=-P$, $d=\pi$ and the ratio has the extended value infinity; the finite real formula applies for $0<d<\pi$.

Let $G_n(z_1,\dots ,z_n)$ denote the ordered vacuum correlator. Inserting the vacuum-preserving Möbius transformation operator and its inverse givesFor $\gamma (z)=z+ϵv(z)$, the three vector fields $v=1,z,z^2$ yield the global conformal Ward identities for chiral correlatorsAll formulas below concern separated insertion points and a consistent local choice of branches for fractional powers.

For one insertion, the translation identity makes $G_1$ constant and the dilation identity gives $h_1{G}_1=0$. ThusFor two insertions, translation invariance gives $G_2=f(z_{12})$, where $z_{ij}=z_i-z_j$. Dilations imply $z_{12}{f}^{\prime }+(h_1+h_2)f=0$, so $f=C_{12}{z}_{12}^{-h_1-h_2}$. Substituting into the special conformal identity leaves $(h_1-h_2)z_{12}f=0$. Thereforewith $G_2=0$ when the weights differ.

For three insertions write $G_3=C_{123}{z}_{12}^{-a}{z}_{13}^{-b}{z}_{23}^{-c}$. Under a Möbius transformation, $\gamma (z_i)-\gamma (z_j)=z_{ij}\sqrt{{\gamma }^{\prime }(z_i){\gamma }^{\prime }(z_j)}$ with compatible local branches. The covariance condition therefore fixesSolving givesThis is the complete solution because Möbius transformations act transitively on ordered triples of distinct points: the ratio of any two covariant nonzero solutions has no remaining invariant variable and is constant. Three-point covariance imposes no equality or triangle inequality among the weights. Additional fusion rules or other symmetries may force $C_{123}=0$, but none follow from the stated covariance and vacuum assumptions alone.

For more than three generic points, three positions may be fixed by a Möbius transformation, leaving $n-3$ independent complex conformal cross-ratios. Correlators consequently have a covariant prefactor multiplied by an arbitrary function of these invariants. For example one convenient four-point form isThe Möbius invariance of the cross-ratio guarantees covariance for any $F$. Thus global conformal symmetry fixes one-, two- and three-point functions up to constants, but does not fix higher-point functions. Their remaining functions encode dynamics.

Cross-ratio conventions differ by permutations. Use the following ordering, which gives the signs requested in part (c):Values involving infinity are defined by limits. This is the cross-ratio with second point mapped to zero: the Möbius transformationsends $z_1,z_2,z_4$ to $1,0,\infty$ and sends $z_3$ to $r$.

For $M(z)=(az+b)/(cz+d)$, $ad-bc\ne 0$, the identityshows that all factors cancel in the cross-ratio. This proves Möbius invariance of the cross-ratio, with the cases involving infinity following by limits. Algebraic subtraction also givesThe latter expression is another common ordering for the cross-ratio; keeping the convention explicit prevents an incorrect reciprocal or sign.