Build a path integral from short evolution steps

Take a particle on the real line with Hamiltonian H=p²/(2m)+V(q), with m positive and a self-adjoint realization that generates time evolution. Divide a total interval T into N equal pieces. Between successive evolution operators insert the identity expressed as an integral of position eigenstates. The endpoints q₀ and qN are prescribed, so there are N−1 independent position integrations, not N+1. This finite composition identity is exact before any short-time approximation.

Split kinetic and potential evolution within each interval using a Trotter product. For suitable potentials the product converges as the step shrinks; a particular finite-N ordering still carries a discretization error. Evaluating the momentum Gaussian supplies a normalization factor for each slice. Omitting those factors changes the kernel’s composition law and its delta-function limit. We use a left-endpoint potential here and ordinary Cartesian coordinates; other orderings or curved configuration spaces require their own careful definitions.

The product of finitely many integrations motivates the notation Dq, but that symbol is shorthand for a defined limiting construction. Real-time weights have unit magnitude and are oscillatory rather than positive probabilities. Boundary prescriptions or a small damping regulator specify how the oscillatory integrals are approached. A stationary-action path organizes a semiclassical approximation, while the integral also contains fluctuations about it and possibly multiple relevant saddles.

For comparison, imaginary-time evolution of a free particle has a positive Gaussian kernel. Its variance grows with Euclidean time, and kernels at successive times compose by convolution. This provides a particularly transparent normalization test. The resulting path measure is not concentrated on differentiable classical paths: typical Euclidean free-particle paths are continuous but rough. Reading every regulated difference quotient as the velocity of a smooth trajectory would therefore misinterpret the continuum construction.

A transition kernel is an amplitude density in its endpoint variables. Obtaining a detection probability also requires an initial wavefunction, propagation, and the appropriate squared modulus or measurement rule. The intermediate integrations do not represent position measurements; inserting actual measurements would change the process. Fixed-endpoint kernels and traces with identified endpoints also define different quantities, a distinction that will reappear when constructing thermal partition functions.

Use the waves lab for a limited interference comparison. Reset and set phase difference to 0° and amplitude ratio to 1; predict resultant amplitude 2. Reset and set phase difference to 180° and ratio to 1; predict zero. These are two equal-frequency traveling sine waves. Their cancellation illustrates coherent addition, but there is no time-slice integration, kinetic action, path measure, or quantum transition probability in this instrument.