Special conformal transformation

In mathematics, a special conformal transformation is a member of the conformal symmetry group. It is given by^[1]

$x'^\mu = \frac{x^\mu-b^\mu x^2}{1-2b\cdot x%2Bb^2x^2} \,.$

$K_\mu = -i(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu) \,.$

Such a transformation can also be written as sequence of an inversion (x^μ → x^μ/x²), a translation (x^μ → x^μ − b^μ), and an inversion:

$\frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu \,.$

References

^ Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. pp. 97–98. ISBN 9780387947853.