On the law of composition of probabilities

J. M. Vigoureux

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-P\'erot 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-P\'erot 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.

Peter D. Finch

P. D. Finch

J. M. Vigoureux

The most general mathematical law for summing bounded quantities is not the arithmetic law, but a composition law of which the summation law for velocities in special relativity is only one particular example. We believe that this composition law, should also be used in quantum theory. We present it with a few examples, we discuss its physical meaning and we show how it can be useful in quantum theory. We show in particular that its use may open the way for a new interpretation of the use of probabilities in quantum theory.

Giorgio Parisi A

J. M. Vigoureux

I show that the reflection coefficient of any stratified planar structure can be obtained by using a complex generalization of Einstein's addition theorem for parallel velocities. This result also applies to multiple quantum wells. It provides a new mathematical tool in optics and in quantum theory and may lead to useful algorithms in computing. It may also give a new insight into special relativity. The composition law of velocities, in fact, no longer appears as a specific result of special relativity but rather as the expression, in the particular case of kinematics, of a more general law of physics. The possible use of the composition law of probability amplitude in quantum theory is also presented.