It is well known that if you compute the expectation value of the Wilson loop along a suitable rectangle you can get the Coulomb potential

$$\langle W(\mathcal{C})\rangle\sim e^{TV(R)} , \ \ V(R)\sim \frac{1}{R}$$

In fact, it is problem 15.3 on page 503 from Peskin and Schroeder$^1$. After hard work I have done it.

In the same spirit, can we obtain (without resorting to numerics or lattice theory) the linear potential $V(R)\sim \kappa R$ ? If yes, please explain how this can be achieved.

$^1$ Peskin,Schroeder; An Introduction To Quantum Field Theory


I don't think there is a way of deriving this linear potential from first principles by calculating the expectation value of Wilson loop.

In this case, we must eventually end up with : $ W(\mathcal{C}) \sim e^{-\sigma A}$, then writing A = TR, we see that $V(R) \sim \sigma R$, here $\sigma$ is the string tension which vanishes for conformal theories. The non-zero $\sigma$ (quark confinement) in QCD lacks rigorous analytical arguments.

  • $\begingroup$ If you could do this rigorously, you would be in the running for the Millenium Prize. $\endgroup$ – mike stone Nov 10 at 22:23

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