= Solution
For one coordinate with <Hamiltonian> $H=\hat p^2/(2m)+V(\hat x)$, the propagator is $K(x_f,t_f;x_i,t_i)=\langle x_f|e^{-iHT/\hbar}|x_i\rangle$, where $T=t_f-t_i>0$. Divide $T$ into $M$ intervals $\delta t$, use the <Trotter product formula> to separate kinetic and potential evolution, and insert complete <position> and <momentum> bases at each slice. The <momentum> integral is Gaussian:
$$
\langle x_{j+1}|e^{-i\delta t\hat p^2/(2m\hbar)}|x_j\rangle
=\int\frac{dp_j}{2\pi\hbar}\exp\left[\frac i\hbar p_j(x_{j+1}-x_j)-\frac{i\delta t}{2m\hbar}p_j^2\right]
=\left(\frac m{2\pi i\hbar\delta t}\right)^{1/2}
e^{im(x_{j+1}-x_j)^2/(2\hbar\delta t)}.
$$
Multiplying the short-time kernels gives the <time-sliced configuration-space path integral>,
$$
K=\lim_{M\to\infty}\left(\frac m{2\pi i\hbar\delta t}\right)^{M/2}
\int\prod_{j=1}^{M-1}dx_j\,
\exp\left\{\frac i\hbar\sum_{j=0}^{M-1}
\left[\frac{m(x_{j+1}-x_j)^2}{2\delta t}-\delta t V(x_j)\right]\right\},
$$
with $x_0=x_i$, $x_M=x_f$. Continuum notation abbreviates this limit as $\int_{x_i}^{x_f}\mathcal D x\,e^{iS[x]/\hbar}$, where $S=\int dt\,[m\dot x^2/2-V(x)]$. The measure includes the displayed normalization; it is not an unweighted count of paths. A small damping or analytic-continuation prescription defines the oscillatory integrals. The paths contribute complex amplitudes, so alternative histories interfere before <probabilities> are calculated. This is the central difference from a classical statistical mixture of trajectories.<Space-Time Approach to Non-Relativistic Quantum Mechanics by Richard Feynman>[pp.367–374, action phases and construction of the propagator]{ref}
When <actions> vary rapidly relative to $\hbar$, <stationary phase> selects <classical paths>. Put $x=x_{\rm cl}+\eta$, with $\eta$ zero at the endpoints. Vanishing first variation gives $m\ddot x_{\rm cl}+V'(x_{\rm cl})=0$. The quadratic term is
$$
S[x]=S[x_{\rm cl}]+\frac12\int dt\,
\eta\left(-m\frac{d^2}{dt^2}-V''(x_{\rm cl})\right)\eta+O(\eta^3).
$$
The resulting Gaussian fluctuation <determinant> is the <semiclassical propagator> prefactor. In one dimension, away from caustics,
$$
\boxed{K_{\rm sc}=\sum_{\rm cl}
\left(\frac{D_{\rm cl}}{2\pi i\hbar}\right)^{1/2}
\exp\left[\frac{iS_{\rm cl}}\hbar-\frac{i\pi\nu_{\rm cl}}2\right],\qquad
D_{\rm cl}=\left|-\frac{\partial^2S_{\rm cl}}{\partial x_f\partial x_i}\right|.}
$$
The branch convention makes $\nu=0$ for a direct free-particle path before any caustic. The <Maslov index> tracks the phase changes as fluctuations pass through a focal singularity. In several dimensions replace the endpoint <derivative> by the <Van Vleck determinant>. The approximation sums all relevant <classical paths>; real-time <action> need only be stationary, not a minimum. It fails in this simple form at a caustic or coalescing saddles, where a uniform approximation is needed. Quadratic <actions> have no higher fluctuation terms, making their Gaussian evaluation exact. For example $S_{\rm cl}=m(x_f-x_i)^2/(2T)$ gives the exact free-particle kernel $\sqrt{m/(2\pi i\hbar T)}e^{iS_{\rm cl}/\hbar}$. Tunnelling involves classically forbidden regions and can be described by imaginary-time saddles rather than real classical trajectories.
Quantum <statistical mechanics> follows by <Wick rotation> $t=-i\tau$. The imaginary-time evolution operator is $e^{-\tau H/\hbar}$, with <Euclidean action>
$$
S_E[x]=\int_0^{\beta\hbar}d\tau\left[\frac m2(\partial_\tau x)^2+V(x)\right].
$$
Taking a <thermal trace> identifies and integrates the endpoint <positions>:
$$
\boxed{Z=\operatorname{Tr}e^{-\beta H}
=\int dx\int_{x(0)=x}^{x(\beta\hbar)=x}\mathcal D x\,e^{-S_E[x]/\hbar}.}
$$
This <thermal trace as a closed Euclidean path integral> explains the periodic imaginary-time direction. A density-matrix element instead retains open endpoints. For many identical particles the <trace> also sums over endpoint permutations, with a plus sign for <bosons> and permutation parity for <fermions>. In coherent-state language this becomes periodic bosonic fields and antiperiodic <Grassmann fields>, with corresponding <Matsubara frequencies>.
The Euclidean weight resembles a classical <Boltzmann weight>. A time-sliced local <quantum field> in $d$ spatial dimensions becomes a classical-looking field model in $d+1$ dimensions, with anisotropic couplings and extra extent $\beta\hbar$. Thus zero-temperature quantum criticality can be treated using an infinitely extended imaginary-time direction, while nonzero <temperature> imposes a finite thickness. The dynamical scaling of <time> need not equal the spatial scaling: the <dynamical critical exponent> specifies their relationship. This <quantum-to-classical imaginary-time mapping> requires qualification when weights have signs or imaginary terms; such a <functional integral> is not automatically an ordinary positive classical <probability> <distribution>. Even with a positive weight, the mapping concerns equilibrium statistics, not literal dissipative real-time trajectories. For a free particle the Euclidean kernel is a <diffusion> kernel with <diffusion> coefficient $\hbar/(2m)$, illustrating the mathematical correspondence.
The classical high-temperature limit is a further approximation, rather than simply another name for imaginary <time>. If quantum level spacings and exchange effects are negligible, a one-particle <partition function> in $d$ dimensions becomes
$$
Z_{\rm cl}\simeq\int\frac{d^dx\,d^dp}{(2\pi\hbar)^d}
e^{-\beta[p^2/(2m)+V(x)]}.
$$
For <indistinguishable particles> there is a factor $1/N!$ in the dilute classical gas, and exchange is negligible when the <number density> <times> the <thermal de Broglie wavelength> to power $d$ is small. For a <harmonic oscillator>, the exact quantum result $Z=[2\sinh(\beta\hbar\omega/2)]^{-1}$ tends to $1/(\beta\hbar\omega)$ as $\beta\hbar\omega\to0$, recovering the classical phase-space integral. \b[<stationary phase> explains <classical mechanics>; the short imaginary-time, weak-exchange limit explains classical <statistical mechanics>.] These are related but distinct limits of the same quantum formulation.
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