Path-Integral Propagator

Formula

For $H=p^2/(2m)+V(x)$ and $T=t_f-t_i$ , the formal fixed-endpoint kernel is

K(x_f,t_f;x_i,t_i) =\int_{x(t_i)=x_i}^{x(t_f)=x_f}\mathcal Dx\, e^{iS[x]/\hbar}.

Its regulated time-sliced meaning is

\begin{aligned} K &=\lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int\prod_{j=1}^{N-1}dx_j\\ &\quad\times \exp\!\left\{ \frac{i\epsilon}{\hbar} \sum_{j=0}^{N-1} \left[ \frac{m}{2} \left(\frac{x_{j+1}-x_j}{\epsilon}\right)^2 -V(x_j) \right]\right\}, \end{aligned}

where $\epsilon=T/N$ , $x_0=x_i$ , and $x_N=x_f$ .

Assumptions and Conventions

The displayed formula is one-dimensional and Cartesian.
The endpoint, operator-splitting, and potential-evaluation prescription are part of the regulator.
The real-time continuum symbol is shorthand for the regulated limit, not an ordinary measure on smooth paths.
A phase-space form uses $\exp[(i/\hbar)\int(p\dot x-H)dt]$ with its own ordering prescription.

Symbols

Symbol	Meaning
$K$	coordinate propagator kernel
$S[x]$	classical action evaluated on a path
$\epsilon$	short time step
$\mathcal Dx$	formal continuum notation defined by slicing
$x_i,x_f$	fixed endpoint coordinates

Validity and Warnings

Keep every short-time normalization factor.
Do not omit endpoint conditions or assume all paths are differentiable.
Curved spaces, gauge fields, nonquadratic momenta, and nontrivial operator orderings require modified measures or counterterms.
Real-time convergence is oscillatory; Euclidean continuation has separate analytic assumptions.
The kernel must recover a delta function as $T\to0^+$ and satisfy the composition law.

Canonical Treatment

The composition derivation, phase-space form, free and Euclidean checks, mathematical cautions, exercises, and references are at From Propagators to Path Integrals.

Regulator mechanics are at Time Slicing.