In what sense a path integral can be approximated by the classical contribution $\exp{[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}}]$?

Question

People often say that the amplitude $K(b,a)$ to go from $a$ to $b$ can be approximated by $$K(b,a) \sim \exp{\left[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}(b,a)\right]},\tag{1}$$ where $S_{\text{cl}}(b,a)$ is the corresponding classical action, and use this to calculate the amplitude (or partition function in some other sense) [1], paying no attention to the possible prefactor. But I want to know, in a general case, what's hidden in the '~' symbol.

In the example of a free particle, the exact path integral can be evaluated [2]: $$ K(b, a)=\left(\frac{m}{2 \pi i \hbar\left(t_b-t_a\right)}\right)^{1 / 2} \exp \left\{\frac{i m\left(x_b-x_a\right)^2}{2 \hbar\left(t_b-t_a\right)}\right\}.\tag{2} $$ Then it's clear that there's a function as a prefactor. Therefore my first question is: How can we just ignore a prefactor which doesn't have to be close to $1$ when we approximate something? What does it actually mean when we use the classical contribution to approximate the amplitude?

Then I think I'll just calculate the contribution from the classical path. But as Feynman said in his book, we have to provide a normalizing factor to let the path integral has a limit, which, in the case of free particle, is $A=\left(\frac{2\pi\mathrm i \hbar\epsilon}{m}\right)^{1/2}$. But what's the normalizing factor when I don't have to integrate over intermediate positions $x_1$, $x_2$, and so on. Or is there some other way to exactly calculate the contribution from the classical path?

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References

1. Liu, H. String Theory and Holographic Duality, Lec22, Eq. (5)
2. Feynman & Hibbs, Quantum Mechanics and Path Integrals.

Answer 1

1. The expansion (1) is the WKB/stationary phase approximation for the semi-classical limit $\hbar\to 0$ (which is just the Wick-rotated version of the formula for the method of steepest descent), see e.g. this related Phys.SE post.

2. The $1/\sqrt{\hbar}$ in the prefactor (2) in the propagator $$K(x_2,t_2 ; x_1,t_1) =\langle x_2,t_2 | x_1,t_1\rangle$$ is caused by the standard normalization of instantaneous position eigenstates $|x,t\rangle$, cf. e.g. my Phys.SE answer here.

3. Excluding the downstairs $\hbar$-dependence caused by the choice of normalization, the rest of the prefactor (2) is a 1-loop$^1$ effect from calculating a functional determinant$^2$, which is subleading/subdominant for $\hbar\to 0$ (as compared to the classical contribution).

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$^1$ See the $\hbar$/loop expansion, cf. e.g. my Phys.SE answer here.

$^2$ The functional determinant is perhaps more apparent for the case of a harmonic oscillator, cf. e.g. this Phys.SE post.