content_image/345797.jpg content_image/345796.jpg content_image/345795.jpg content_image/345794.jpg content_image/345793.jpg content_image/345792.jpg
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. 

## II. A SEMICLASSICAL CORRECTION 

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 

or 

where 

<img src='content_image/345797.jpg'>

<img src='content_image/345796.jpg'>

<img src='content_image/345795.jpg'>

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 

<img src='content_image/345794.jpg'>

The function β ( θ ) obeys 

<img src='content_image/345793.jpg'>

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). 

<img src='content_image/345792.jpg'>

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 

