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I'm going to stick my neck out here and summarize the comment threads above as an answer of sorts. I'll delete it if it pulls down my reputation score :-/ The universe in which we happen to find ourselves is, as near as we can tell, not structured in a way that would support different values of its fundamental parameters in different locations within it- ...

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I know that a lot of time has passed, but maybe I can give my own answer to your question. Start by slightly modifying your relation in the following way \begin{align} :\mathcal{F}::\mathcal{G}:= exp\left(\int d^2z_1d^2z_2\frac{1}{z_{12}}\left[\overrightarrow{\frac{\delta}{\delta b_{\mathcal{F}}(z_1)}}\overleftarrow{\frac{\delta}{\delta c_{\mathcal{G}}(z_2)}...

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Perhaps the simplest derivation that the string stiffens $X^{\prime}\to 0$ in the point particle limit with infinite string tension $T_0\to \infty$ comes from the corresponding Hamiltonian formulations: On one hand, the Nambu-Goto Hamiltonian Lagrangian is  L_H~:=~\int_0^{\ell}\! d\sigma~{\cal L}_H, \qquad {\cal L}_H ~:=~ P\cdot \dot{X}-{\cal H}, \qquad {...

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From Polchinski's String Theory, Chapter 1: We want to study the classical and quantum dynamics of a one-dimensional object, a string. The string moves in $D$ flat spacetime dimensions, with metric $\eta_{\mu \nu} = \mathrm{diag}(-,+,+,\cdots,+)$. So all additional dimensions are spacelike. Strictly speaking Polchinski is only talking about bosonic ...

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FWIW, eq. (2.1.14) is in the Euclidean formulation, while eq. (A.1.17) is in the Minkowskian formulation. The operators inside the expectation value on the LHS of eq. (2.1.14) are implicitly assumed to be radially ordered.

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