### mathematical physics

**Infinitesimal to finite special conformal transformations**

Last modified: 2014-10-17

An investigation of conformal transformations on $\mathbb R^n$ is important in conformal classical and quantum field theories for which these transformations are symmetries. Among all possible conformal transformations, there is a particular type, called special conformal transformations such that for each $b\in\mathbb R^n$, there is a corresponding special conformal transformation given by the flow generated by the vector field $G_b:\mathbb R^n\to\mathbb R^n$ given explicitly by

\begin{align} G_b(x) = 2(b\cdot x)x - x^2b. \end{align} A in the language of physicists, specifying this vector field amounts to specifying the "infinitesimal generator" of special conformal transformations. Determine the corresponding "finite" conformal transformations. In other words, determine the flow generated by the vector field $G_b$.