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I) The substitution $f=r\psi$ is the standard substitution to get a radial 3D problem to resemble a 1D problem, see e.g. Ref. 1. II) From the perspective of the normalization of the wavefunction $\psi(r)$, a $1/r$ singularity of $\psi(r)$ at $r=0$ is fine because $|\psi(r)|^2$ is suppressed by a Jacobian factor $r^2$ coming from the measure in 3D spherical ...

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Starting from the boundary condition $$\partial^{\sigma}X^{\mu}(\tau,0)=0$$ and lowering the index on the derivative using the metric gives $$\gamma^{\sigma\tau}\partial_{\tau}X^{\mu}(\tau,0)+\gamma^{\sigma\sigma}\partial_{\sigma}X^{\mu}(\tau,0) =0.$$ Apparently Polchinski wants to express this in terms of the metric $\gamma_{ab}$ with its indices ...

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In a viscous fluid the shear stress is proportional to the velocity gradient. $\sigma=\eta \frac{dv}{dy}$ where $\eta$ is the viscosity, and $v$ is the fluid speed at right angles to the $y$ axis. Therefore as the small distance $dy$ tends to zero, the change of fluid speed $dv$ also tends to zero, for any non-zero viscosity. Let us now follow Navier ...

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Usually we think of friction as something like this (not a formal definition, but I think it's close enough to be understood as "friction"): when two objects move past one another and are in contact, the differential velocity between them leads to a force we call "friction" At the boundary between a liquid and a solid, if we permit a different velocity ...

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The little group is the subgroup of the Lorentz group that leaves an arbitrary four-momentum vector invariant, i.e. for an element of the group $g$ and momentum $V$ we have $gV=V$. This group is in general different for massive and massless particles. If you now find that the little group of your holomorphic primaries corresponds to that of massive states, ...

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