 Solution

For one coordinate with Hamiltonian $H={\hat{p}}^2/(2m)+V(\hat{x})$, the propagator is $K(x_f,t_f;x_i,t_i)=⟨x_f|e^{-iHT/\hslash }|x_i⟩$, 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:Multiplying the short-time kernels gives the time-sliced configuration-space path integral,with $x_0=x_i$, $x_M=x_f$. Continuum notation abbreviates this limit as ${\int }_{x_i}^{x_f}\mathcal{D}x{e}^{iS[x]/\hslash }$, 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.^[ref]

When actions vary rapidly relative to $\hslash$, stationary phase selects classical paths. Put $x=x_{\mathrm{cl}}+\eta$, with $\eta$ zero at the endpoints. Vanishing first variation gives $m{\ddot{x}}_{\mathrm{cl}}+V^{\prime }(x_{\mathrm{cl}})=0$. The quadratic term isThe resulting Gaussian fluctuation determinant is the semiclassical propagator prefactor. In one dimension, away from caustics,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 $S_{\mathrm{cl}}=m(x_f-x_i{)}^2/(2T)$ gives the exact free-particle kernel $\sqrt{m/(2\pi i\hslash T)}{e}^{i{S}_{\mathrm{cl}}/\hslash }$. 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 $e^{-\tau H/\hslash }$, with Euclidean actionTaking a thermal trace identifies and integrates the endpoint positions: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 $\beta \hslash$. 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 $\hslash /(2m)$, 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 becomesFor 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 $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$ tends to $1/(\beta \hslash \omega )$ as $\beta \hslash \omega \to 0$, recovering the classical phase-space integral. 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.