content_image/372618.jpg content_image/372617.jpg content_image/372615.jpg content_image/372614.jpg content_image/372613.jpg
for every mode. 

Integrating the Gaussian integral for every mode and using the ζ -function regularization, we get (disregarding the logs to this order) a pre-factor of the type 

<img src='content_image/372618.jpg'>

where the product runs over 4 sets of modes, which results in a factor of 4 in the exponent. This coincides with the L¨ uscher term (6) for a rectangle of the size R × T in d = 26 dimensions provided 2 

<img src='content_image/372617.jpg'>

In Appendices C and D we confirm this by an explicit calculation of the determinant of the Laplace operator and the conformal anomaly for an outstretched ellipse. 

## V. A GENERALIZATION TO ARBITRARY DIMENSIONS 

The results of two previous sections demonstrate the already mentioned fact that the ansatz (1) has to be modified in order to describe the L¨ uscher term in d = 4 dimensions. 

A simple modification is based on the form of the path integral for a rectangle 

with quadratic action A$_{2}$ [ β ( t )] given by Eq. (36). 

<img src='content_image/372615.jpg'>

For an arbitrary curve this path integral can be expressed through the determinant of the corresponding operator, that enters A$_{2}$ , which we denote as O : 

<img src='content_image/372614.jpg'>

It is now clear that the following modification of the ansatz (1) will provide the correct value of the L¨ uscher term (6) in d dimensions: 

<img src='content_image/372613.jpg'>

This modification of Eq. (1) does not effect the classical limit, leading to the area law, and has the meaning of altering a pre-exponential in the semiclassical approximation. 

