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I am wondering what happens if we quantize open strings in the presence of Dp-branes. I'm sure lots of people did all the calculation before, but my question is about the physical meaning behind those commutation relations found after the quantization. So what's the importance behind those boundary conditions (the presence of Dp-branes) rather than just preventing the strings to go away?

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The question is phrased slightly wrong, at least historically: It's not that we put the $D_p$-brane there and then quantize the string. It's that when we look at possible boundary conditions for an open string, we can choose either Neumann or Dirichlet boundary conditions for the endpoints, and we "invent" the fiction of the brane to represent those boundary conditions. The "physical meaning" of this brane is not in the commutation relations themselves, but in the spectrum that appears after quantization.

Each direction for which we choose a Dirichlet boundary condition will mean the string endpoints are fixed in that direction, while a Neumann boundary condition allows the string to move freely in that direction while imposing the absence of a momentum flow onto or off the string. A string with $n$ Dirichel boundary conditions will look as if it moves freely along a ${d-n}$-dimensional hypersurface inside the $d$ dimensional spacetime. The conventional naming is to write $D_p$-brane for a $p+1$-dimensional such hypersurface. Note that branes which don't arise from boundary conditions in this way should not be called $D_p$-branes, since the $D$ stands for "Dirichlet".

In the standard quantization of the (super)string, the brane itself doesn't really play much of a role - what we are quantizing is the string with certain boundary conditions. If one now examines the first level massless excitations of the bosonic string, then one finds they transform in a way under the reduced Lorentz $\mathrm{SO}(1,p)$ symmetry of the $D_p$-brane's directions that organizes them into a massless vector and $d-p$ massless scalars. This is usually interpreted as a vector ($\mathrm{U}(1)$ gauge) field living "on" the brane and a bunch of Goldstone bosons due to the breaking of the full Lorentz symmetry.

However, it gets really interesting once we allow the string to end on different branes. A string stretched between two different branes suddenly creates massive excitations at first level, whose mass gets larger the farther the two branes are apart. Making a set of $N$ branes coincide (taking the limit of zero separation for them having non-zero distance) creates the final standard ingredient for QFTs: $\mathrm{U}(N)$ gauge theories

Finally, Polchinski realized during the "second superstring revolution" that the $D_p$-branes represent dynamical objects in themselves, and possess their own Nambu-Goto actions and "worldvolume" theories. This, however, has not directly to do with the quantization of the string.

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When quantizing strings on Dp-branes, you find that there are additional restrictions placed on $x$ and $p$.

Firstly, you will find that strings can only move in directions along the brane surface: there can be no center of mass momentum in any directions other than along the brane. For the commutation relations you will find that they are largely unchanged, except that now $p^a = 0$ for $a = p+1,p+2,...,d-1$.

However, strings are still free to oscillate in any of the $D-1$ spatial directions.

Secondly, you find that both Lorentz symmetry and translational symmetry is broken by choosing preferred coordinates with different boundary conditions. What is preserved is a subgroup of the form:

$$ \left(T(p)\times SO(p,1)\right)\times SO(d-1-p) $$

Which is to say that Poincare symmetry is preserved along the brane, while only rotational symmetry is preserved in directions transverse to the brane.

This means that any particles that arise must be a representation of this subgroup, rather than the full Lorentz group in $d$ dimensions. This broken symmetry gives rise to Goldstone bosons, which in this case are massless scalar fields that characterize the dynamics of the Dp-brane in the transverse direction. You will also find that a massless particle living on the brane, a gauge field $A_\mu$ (i.e. a photon).

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