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: <math>\langle x_f|e^{-\frac{i}{\hbar}\hat{H}(t-t_1)} F_1(\hat{x}) e^{-\frac{i}{\hbar}\hat{H}(t_1-t_2)} F_2(\hat{x}) e^{-\frac{i}{\hbar}\hat{H}(t_2)}|x_i\rangle =
\int_{x(0)=x_i}^{x(t)=x_f} \mathcal{D}[x] F_1(x(t_1)) F_2(x(t_2)) e^{\frac{i}{\hbar}\int dt L(x(t),\dot{x}(t))}</math>,
and to the general vacuum expectation value (in the large time limit)
: <math>\langle F\rangle=\frac{\int \mathcal{D}[\phi] F(\phi) e^{\frac{i}{\hbar}S[\phi]}}{\int \mathcal{D}[\phi] e^{\frac{i}{\hbar}S[\phi]}}</math>.
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