Six-dimensional space

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

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

Its infinitesimal generator is

$K_{\mu }=-i(2x_{\mu }x^{\nu }\partial _{\nu }-x^{2}\partial _{\mu })\,.$

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

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