An elementary approach to quantum length of SLE

We present an elementary proof establishing the equality of the right and left-sided $\sqrt{\kappa }$-quantum lengths for an SLE${}_{\kappa }$ curve, where $\kappa \in (0,4]$. We achieve this by demonstrating that the$\sqrt{\kappa }$-quantum length is equal to the $(\sqrt{\kappa }\mathrm{/}2)$-Gaussian multiplicative chaos with reference measure given by half the conformal Minkowski content of the curve, multiplied by $2\mathrm{/}(4-\kappa )$ for $\kappa \in (0,4)$ and by $1$ for $\kappa =4$. Our proof relies on a novel “one-sided” approximation of the conformal Minkowski content, which is compatible with the conformal change of coordinates formula.