= Solution
For $H=\hat p^2/(2m)+V(\hat x)$, divide elapsed time $t$ into $J$ slices $\Delta t=t/J$ and apply the <Trotter product formula>. Inserting complete position and momentum bases gives the regulated <phase-space path integral>
$$
K(x_f,x_i;t)=\lim_{J\to\infty}\int\prod_{j=1}^{J-1}dx_j\prod_{j=0}^{J-1}\frac{dp_j}{2\pi\hbar}\exp\left\{\frac i\hbar\sum_{j=0}^{J-1}\left[p_j(x_{j+1}-x_j)-\Delta t\left(\frac{p_j^2}{2m}+V(x_j)\right)\right]\right\}.
$$
The endpoint values are $x_0=x_i$, $x_J=x_f$. Each <Gaussian momentum integration in a phase-space path integral> can be performed explicitly, leaving the <configuration-space path integral>
$$
\boxed{K=\int_{x_i}^{x_f}\mathcal Dx\,e^{iS[x]/\hbar},\qquad S[x]=\int_0^t ds\,[m\dot x^2/2-V(x)].}
$$
Its measure is the time-sliced limit of $(m/(2\pi i\hbar\Delta t))^{J/2}\prod_{j=1}^{J-1}dx_j$, with the usual infinitesimal convergence prescription. The integral sums amplitudes, not positive classical probabilities. Operator ordering remains part of the time-slice definition. Expanding a short-time kernel recovers the <Schrödinger equation>; composing kernels reproduces the <time-evolution operator>.
In the <classical limit>, rapid phase cancellation suppresses paths away from stationary <action>. Varying $S$ with fixed endpoints gives $m\ddot x=-V'(x)$. The semiclassical result sums the classical-path phases multiplied by fluctuation <functional determinants>; for a quadratic <Hamiltonian> the fluctuation integral is Gaussian and this expansion is exact.
For <quantum statistical mechanics>, continue $t=-i\tau$ by a <Wick rotation>. The <Euclidean action> and thermal <trace> give
$$
S_E=\int_0^{\beta\hbar}d\tau\,[m(\partial_\tau x)^2/2+V(x)],\qquad
\boxed{Z=\operatorname{Tr}e^{-\beta H}=\int_{x(\beta\hbar)=x(0)}\mathcal Dx\,e^{-S_E/\hbar}.}
$$
The trace closes position-space paths. In a many-particle trace one also sums over endpoint permutations with the appropriate exchange signs. In a <coherent state> field formulation, bosonic fields are periodic and <fermionic coherent states> instead give antiperiodic <Grassmann fields>, hence bosonic and fermionic <Matsubara frequencies> $2\pi n/(\beta\hbar)$ and $(2n+1)\pi/(\beta\hbar)$. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§§3.1–3.2, coherent-state time slicing and trace boundary conditions]{ref}
The <quantum-to-classical imaginary-time mapping> views a local $d$-dimensional quantum problem as a classical-looking field theory in $d+1$ dimensions, with imaginary-time extent $\beta\hbar$. A genuine classical probability interpretation additionally requires a nonnegative weight; fermion signs and Berry terms can obstruct it.
For a smooth potential in the high-temperature or small-$\hbar$ limit, imaginary-time paths fluctuate only over the <thermal de Broglie wavelength> $\lambda_T=\sqrt{2\pi\beta\hbar^2/m}$. If this is short compared with the potential's variation scale, the diagonal free kernel gives
$$
\boxed{Z\simeq\frac1{2\pi\hbar}\int dx\,dp\,e^{-\beta[p^2/(2m)+V(x)]}=\frac1{\lambda_T}\int dx\,e^{-\beta V(x)}.}
$$
This <classical phase-space limit of the thermal kernel> is the classical <phase space> partition function for one degree of freedom. A dilute identical-particle gas also needs its $1/N!$ factor; when exchange is appreciable, classical statistics fails. As a check, a single <quantum harmonic oscillator> gives $Z=[2\sinh(\beta\hbar\omega/2)]^{-1}\to1/(\beta\hbar\omega)$, precisely the classical phase-space integral.
Back to article page
