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

Question

When going over my lecturer's notes on String Theory and trying to understand a particle as a theory of gravity in 1D, it is mentioned that

the action $(1)$ is regularisation invariant, $$S=-m\int d\tau \sqrt{-\dot x^\mu \dot x^\nu \eta_{\mu\nu}} \tag{1}$$ where $\dot x ^\mu = \frac{dx^\mu}{d \tau}$.

Later it is stated that:

"Instead of only thinking of $x^\mu(\tau)$ as functions parameterising abstract embeddings of a 1D object into D-dimensions, we can equivalently think of them as fields in a 1D theory,"

and that

If we think of $x^\mu(\tau)$ as fields in a 1D theory, then $(1)$ will be a complicated action for these fields because the action includes a square root term, which make quantisation difficult.

Why does having a square root difficult the quantisation?

My lecturer's notes are not online but they are similar to David Tong's.

Answer 1

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.

Answer 2

The answer of Qmechanic says everything can be said. But I think is worth to mention, that there are references where some details of relativistic quantum strings are discussed with the Nambu-Goto action as the starting point. Zwiebach's textbook on string theory is a well known example of this.

But I want to recommend the book "Introduction to The Relativistic String" by Barbashov and Nesterenko. Taking a look on this books you can see how difficult this route is and possibly you may learn something useful in the course. Although the methods haven't shown nothing too relevant for string theory, they seem to teach useful aspects of general non polynomial actions in classical mechanics if for some reason, you're interested on that.

Other references:

1) "General solutions of nonlinear equations in the geometric theory of the relativistic string" https://projecteuclid.org/euclid.cmp/1103921284

2) "Classical dynamics of the relativistic string with massive ends" https://iopscience.iop.org/article/10.1088/0305-4470/24/11/013