QED: The Strange Theory of Light and Matter

1. Waves reflect from a boundary in two basic ways depending on whether the boundary is "hard"($\Delta \phi=\pi$) or "soft"($\Delta\phi=0$).

2. Diffraction gratings disperse white light into its component colors because different wavelengths produce bright fringes at different angles. $d \sin \theta = m\lambda$.

3. The standard formulation of Quantum Mechanics is well known. It was built and created by Schrödinger, Heisenberg and Dirac, plus many others, between 1925-1931. Later, it was shown to be equivalent and analogue to the common Classical Mechanics (Dirac, Von Neumann,…). In 1933, Dirac made the remarkable observation that the action plays a central role in classical mechanics but that it seemed to have no important role in Quantum Mechanics so far. However, he speculated on how this situation could be solved, and he suggested that the propagator (Green’s function), essentially the matrix elements of the scattering matrix (S-matrix) in momentum space, in Quantum Mechanics should be proportional to the phase factor $$e^{iS/\hbar }$$

and where $S$ is the classical action evaluated along the classical path.

In 1948, Feynman developed Dirac’s idea, and he succeeded in his new approach: he was able to provide a third formulation of Quantum Mechanics, based on the fact that the propagator can be written as a sum over ALL (not a single one!) the possible paths (not only the classical path!) between the initial and final points. Each path contributes with a weight and the probability, the squared amplitude, reads:

$$P(A\rightarrow B)=\sum_\text{paths} \omega(A\rightarrow B) e^{iS/\hbar }$$

There is a difference between Dirac idea and the final output by Feynman. Dirac considered only the classical path, while Feynman showed that ALL paths contribute. In some sense, the quantum particle “smells” and “takes” all paths and interferes with itself, so the amplitudes for every path add according to the usual quantum mechanical rule for combining amplitudes, i.e., the rule of superposition and then it gets squared.