Composition law of amplitudes and quantum theory

Mathematical methods in optics show that the overall amplitude resulting from the infinite summation over all the various alternative paths that the light can take inside a Fabry-Pérot interferometer can directly be obtained by using a complex generalization of Einstein’s composition law for parallel velocities. Moreover, this result appears to be quite general. In fact, the reflection coefficient for any number of interfaces or quantum wells can be easily obtained by using the composition law. This suggests that a more general way to obtain the probability amplitude for an event which usually needs infinite summation on virtual paths (as in the Fabry-Pérot interferometer) is to compose amplitudes following Einstein’s addition law. I explore this problem and show how the composition law avoids divergences and guarantees the modulus of any total wave function to be less than (or equal to) unity. By examining its physical meaning in the light of probability theory, I also show that it provides an interesting expression for nonlocality and nonseparability.