# constructing string path integral through Markov chain procedure

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We all know the textbook way of constructing the path integral for a point particle. It goes approximately like this (if relativistic). Say we want the propagator $\Delta(x_\text{in},x_\text{out}) = \langle x_\text{in}|(-\partial^2+V(x))^{-1}|x_\text{out}\rangle$, then we can write this in Schwinger proper-time representation \begin{equation} \Delta(x_\text{in},x_\text{out}) = \int_0^\infty dT \langle x_\text{in}|e^{-T (-\partial^2+V(x))}|x_\text{out}\rangle \;, \end{equation} after which we divide the $T$ interval into a great many parts $d\tau$ (the number will go to infinity at the end) and introduce convolutions of unity (written both in momentum and in position states) in between all the parts. The result is the Hamiltonian path integral \begin{equation} \Delta(x_\text{in},x_\text{out}) = \int_0^\infty dT \int_{x(0)=x_\text{in}}^{x(T)=x_\text{out}} [x,p] e^{-\int d\tau (p^2+V(x) + i\dot x p)} \;. \end{equation} Integrating over $p$ gives the usual path integral. The final result can then be generalized to a two-dimensional worldsheet if we are interested in string theory.

Now I was wondering whether a similar derivation were possible for a string theory. Let us assume we somehow had access to the inverse propagator of some string theory. The above procedure goes through the Hamiltonian, and string theory can either not have a Hamiltonian (if written in Nambu-Goto style), or needs an extra metric (in Polyakov style). So it is obvious a worldsheet metric will have to be introduced at some point, but how would that go? Or is there another way to do the same? Are there any papers in existence doing analogous derivations?

This post imported from StackExchange Physics at 2015-09-08 17:53 (UTC), posted by SE-user David Vercauteren
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