Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 5

Split the time-evolution operator into $N$ factors with $ϵ=T/N$, and insert $N-1$ position completeness relations between them. Each factor is evaluated by inserting momentum states normalized by $⟨x|p⟩=(2\pi \hslash {)}^{-1/2}{e}^{ipx/\hslash }$:Multiplying the $N$ kernels and integrating the intermediate positions produces the stated time-sliced path integral, including its $N/2$ normalization power. For a free particle this Gaussian short-time kernel is exact, so the convolution identity already holds at finite $N$. Oscillatory integrals are defined with a vanishing convergence regulator, consistently fixing the square-root phase.

The exponent is $i{S}_N/\hslash$, whereis the classical action of the piecewise-linear interpolating path. For smooth paths its limit is $S={\int }_0^{T}m{\dot{x}}^{2}dt/2$. The discretization defines the quantum integral; it does not assert that typical quantum paths are differentiable. Weighting histories by their action phase is the central formulation of the original path-integral paper.^[ref]

When action differences on the experimental scales are large compared with $\hslash$, nearby nonstationary phases cancel. stationary phase selects $\delta S=0$ at fixed endpoints, hence $m\ddot{x}=0$. The surviving free classical trajectory is $x_{\mathrm{cl}}(t)=x_0+(x_T-x_0)t/T$, with action $m(x_T-x_0{)}^2/(2T)$. This is the semiclassical mechanism; it does not mean each individual path becomes a classical path.

The Fresnel convolution of free-particle time slices follows by completing the square:The second term is independent of $x_k$. The shifted Fresnel integral in the first term givesHere $k$ is the positive slice index, not a wave number. The phase is fixed by the same convergence prescription as the short-time kernel.

After integrating $x_1,\dots ,x_{N-1}$, the remaining exponent contains $(x_T-x_0{)}^2/N$. The square-root factors telescope:Since $Nϵ=T$, the free-particle propagator isFor $T>0$, the square root has phase $e^{-i\pi /4}$. Its short-time limit is the position delta function in the distributional sense, consistent with unitary evolution.