content_image/298174.jpg content_image/298175.jpg content_image/298178.jpg content_image/298176.jpg content_image/298177.jpg
We want a static solution to the equations of motion, so it is convenient to set σ 0 = t and σ 1 = x , and assume y = z = 0. We also assume that the string is at a constant angular position in S $^{5}$. Then the action becomes 

with 

<img src='content_image/298174.jpg'>

<img src='content_image/298175.jpg'>

Equation (4.13) is very similar to the integral found when performing Fer- mat’s Principle for the path of light rays [102]. For large separation of endpoints, the trajectory which minimizes S locates itself very nearly at the minimum of n ( r ) for most of its length. Hence the complicated pro- cess of optimizing S is reduced to simply optimizing n ( r ). Furthermore, for small separations Δ x about its minimum, S becomes 

It is easy to show that n ( r ) is optimized by 

which gives 

<img src='content_image/298178.jpg'>

<img src='content_image/298176.jpg'>

<img src='content_image/298177.jpg'>

with Σ = Δ − δ Δ+ $_{δ}$. (4.16) shows us that this geometry gives us a linear quark- antiquark potential. In other words, it exhibits confinement. 

