In Sect. V we derive the generalization given by Eq. (8) and in Sect. VI we make some conclusions. A number of more technical points have been discussed in some Appendices. The path integral over reparametrizations (1) reproduces the exponential of the minimal area in the classical limit KS$_{min}$ → ∞ . To calculate the semiclassical correction, we expand where or and expand the Douglas integral to quadratic order in β around the classical trajectory θ ∗ ( σ ). This expansion makes sense because typical trajectories in the path integral over reparametrizations (1) are smooth as KS$_{min}$ → ∞ and have Hausdorff dimension one [10]. Substituting (9) into Douglas’ integral (2) and expanding in β , we find that the linear term vanishes because θ ∗ ( σ ) is the minimum, while the quadratic part reads The function β ( θ ) obeys for the derivative of the reparametrizing function to be positive. This is always satisfied if β is small and smooth enough. In order to calculate the semiclassical correction to the area law, we need to do the Gaussian integral with A$_{2}$ [ β ( θ )] given by Eq. (12). The typical values of β , which are essential in the path integral over β in (14), are β ∼ 1 $_{/}$√$_{KS}$$_{min}$$_{, i.e.}$ small for √$_{KS}$$_{min}$ ≫ 1. Hence, the higher terms of an expansion 4 ∫ D β ( θ ) e − A 2 [ β ( θ )] (14) ˙ β ( θ ) ≥ − ˙ σ ∗ ( θ ) (13) A$_{2}$ [ β ( θ )] = K 8 π ∫ 2 π 0 d θ$_{1}$ ∫ 2 π 0 d θ$_{2}$ ˙ x ( θ$_{1}$ ) · ˙ x ( θ$_{2}$ ) ( β ( θ$_{1}$ ) − β ( θ$_{2}$ )) 2 1 − cos( σ ∗ ( θ$_{1}$ ) − σ ∗ ( θ$_{2}$ )) . (12) β (0) = β (2 π ) = 0 (11) σ ( θ ) = σ ∗ ( θ ) + β ( θ ) , (10) θ ( σ ) = θ ∗ ( σ ) + β ( σ ) (9) II. A SEMICLASSICAL CORRECTION