Quantum Electrodynamics: The Path Integral
In classical optics, light appears to follow Fermat's Principle of Least Time, taking a single, straight path to bounce off the center of a mirror. However, Richard Feynman's Sum Over Histories formulation of quantum mechanics reveals a deeper reality: a photon doesn't just take one path; it explores every possible path simultaneously.
The probability \( P \) of a photon traveling from the Source (\(S\)) to the Detector (\(D\)) is the squared magnitude of the total probability amplitude \( \Psi \). This total amplitude is the sum of contributions from every conceivable path \( x \):
Each path contributes a vector (a phasor) of identical length. Its phase angle is determined by the classical action \( S[x] \) (proportional to the travel time). At the edges of the mirror, the travel time changes drastically between adjacent paths. Their phasors spin wildly and cancel each other out (destructive interference). At the center, the travel time is stationary (\( \delta S = 0 \)). The phasors point in the exact same direction and stack up perfectly (constructive interference), generating the classical macroscopic path.
Interactive Sandbox: Drag the Source, Detector, or Grating below. Watch the paths, amplitudes, and time curves recalculate in real-time. Click Bake & Apply Diffraction Grating to scrape away the destructively interfering edge paths—physically proving the light was interacting with the entire mirror all along!