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In Polyakov's book, he explains that one possible way to compute the propagator for a point particle is to compute the lattice sum $\sum_{P_{x,x'}}\exp(-m_0L[P_{x,x'}])$, where the sum goes over all paths between $x$ and $x'$. One then needs to compute this sum and choose the bare mass so that there's a good continuum limit.

Polyakov then goes on to say that this doesn't work for string theory. I skimmed the literature and couldn't find any explanation of this fact. Naively I would think that in order to find the propagator, you could just compute the sum $\sum_{W_{C,C'}}\exp(-T_0 A[W_{C,C'}])$, where the sum is over worldsheets that end on the curves $C$ and $C'$. What goes wrong? Is this just a hard sum to do?

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