An elementary approach to quantum length of SLE

We present an elementary proof establishing the equality of the right and left-sided √κ-quantum lengths for an SLE_κ curve, where κ∈ (0,4]. We achieve this by demonstrating that the√κ-quantum length is equal to the (√κ/2)-Gaussian multiplicative chaos with reference measure given by half the conformal Minkowski content of the curve, multiplied by 2/(4-κ) for κ∈ (0,4) and by $1$ for κ=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.