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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.

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The boundary conditions of a classical string theory are constraints that are an integral part of fully specifying the actual physical theory, not some sort of ad-hoc conditions you are allowed to just forget about after quantization - these constraints materially alter the content of the theory, as you can easily see by the different degrees of freedom: Right- and left-movers are independent on a closed string, but not on an open string. This is not different from quantization of any other constrained theory: Classical constraints must be either solved or implemented in the quantum theory.

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  • $\begingroup$ 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? $\endgroup$ Sep 19, 2022 at 17:14
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    $\begingroup$ @johnnydines I mean...a theory without solutions to the equations of motion isn't a very good theory, is it? We might even consider it the definition of a physical theory that its equations of motion have solutions; I for one certainly don't know how to quantize such a theory consistently. $\endgroup$
    – ACuriousMind
    Sep 19, 2022 at 17:16

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