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^{iS}. The paths which are very close to the classical path are the ones that contribute to the path integral. All these paths have approximately the same amplitude e

^{iSclassical}. This means that the final amplitude is equal to Ne

^{iSclassical}where N is the "number of paths" close to the classical one which constructively contribute to the path integral. This number is known to be G(0, 0), so we have

path integral = G(0, 0) e

^{iSclassical}

This is called the semi-classical approximation. As a function of the time β between the endpoints, G(0, 0) ~ β

^{-1/2}. Similarly, we have p

_{i}= 1/Z e

^{-βEi}. 1/Z can be thought of as a multiplicity, and the probability of i is the product of this multiplicity and the probability factor e

^{-βEi}. The analog of this equation in terms of the density operator is ρ = 1/Z e

^{-βH}Since Z = e

^{W[j]}, we have the expression e

^{-W[j]}e

^{-βH}for the density operator. The number density of particles per unit volume is 2E. This becomes 1/2E in momentum space, so there are dp/2E states in range dp of momenta. Instead of talking about paths, we can say that the particle "propagates from the initial to the final point with momentum p", and the corresponding amplitude would be e

^{ipx}e

^{-iEt}. This would have a multiplicity dp/2E in the path integral

∫ dp/2E e

^{ipx}e

^{-iEt}

which is a sum over all possible momenta with which the particle can propagate. In the path integral picture, the correlator of n operators is

$$ \langle O_1, ..., O_n \rangle = \frac{\int_{map(\Sigma, R)} O_1...O_n e^{-S}}{\int_{map(\Sigma, R)} e^{-S} } $$

The corresponding expression in the hamiltonian picture is the following, which is called the feynman-kac formula. The denominators in these two formulas are like Z in 1/Z e

^{-βH}.

$$ \langle \phi(x_1), ..., \phi(x_n) \rangle = \frac{\mathrm{tr} \ e^{-x^0 H}\phi(x_1) e^{(x^0_1 - x^0_2)H} \phi(x_2)...\phi(x_n) e^{(x^0_n - \beta)H}}{\mathrm{tr} e^{-\beta H}}$$

what is the logic behind this formula, and how is it connected to the path integral version?