# Background

In textbooks (e.g. see around page 14 of [1]), string theory is introduced classically with the Polyakov action: $$S_P = \int d\tau d\sigma L :=-T \int d\tau d\sigma \sqrt{-h} h^{\alpha\beta}(\partial_\alpha X^\mu \partial_\beta X_\mu)\,,$$ where the indices $$\alpha,\beta$$ runs over worldsheet coordinates $$\tau,\sigma$$ and $$\tau_i\leq\tau\leq \tau_f,\qquad 0\leq \sigma \leq \ell\,.$$

Then, the boundary conditions are deduced using the principle of least action $$\delta S_P = \int_{\tau_i}^{\tau_f} d\tau \left(\frac{\delta L}{\delta X'^\mu}\right)\delta X^\mu\bigg\rvert^{\sigma=\ell}_{\sigma=0}+\int_{0}^{\ell} d\sigma \left(\frac{\delta L}{\delta \dot X^\mu}\right)\delta X^\mu\bigg\rvert^{\tau=\tau_f}_{\tau=\tau_i}\\+\int d\tau d\sigma \left(\frac{\partial L}{\partial X^\mu}-\partial_\alpha\frac{\partial L}{\partial (\partial_\alpha X^\mu)}\right)X^\mu \approx 0\,.$$ Here and from now on, I denote on-shell equalities with $$\approx$$.

Each term must individually vanish. The term at the two temporal boundaries vanishes by definition of the problem, i.e. we determined the string states at $$\tau_i,\tau_f$$ at the outset so $$\delta X^\mu=0$$ there.

However, the spatial boundary term does not vanish by itself, so we require $$\left(\frac{\partial L}{\partial X'^\mu}\right)\delta X^\mu \approx 0\,.$$ This is where the Dirichlet/Neumann boundary conditions are required.

# Question

Given that Dirichlet/Neumann boundary conditions are only needed on-shell in a classical mechanical setting, why do we still use them after quantizing the string? Is it necessary to use these boundary conditions after quantizing, or could we explore quantized strings with different boundary conditions or none at all?

References:

[1] Polchinski. String Theory, Volume I. Cambridge University Press, 1998.

• That makes sense, thanks. Then, is there any problem with starting with a classical string theory without any constraints and then quantizing it? The boundary term does not vanish any more, so there will be no solution to $\delta S=0$ classically. Would this problem necessarily plague the quantization of such a theory? Sep 19, 2022 at 17:14