# Why does a square root term make the quantisation of action difficult?

When trying to perturbatively expand the square root action around a classical solution, there are infinitely many higher-order fluctuation terms. Compare that with the non-square root action, which is just quadratic in $x^{\mu}$.

Another issue is how to obtain a consistent path integral measure for the theory. This is most easily done in the Hamiltonian formulation, cf. e.g. this Phys.SE post. The Hamiltonian formulation is often closer related to the non-square root action.