Hendrik A. LorentzHendrik A. Lorentz was giant in theoretical physics who bridged the gap between classical electromagnetic field theory and modern relativity theories. He put forward a theory of the electron and he developed the famous Lorentz transformations that describe how objects appear contracted in the direction of their motion to observers in a frame at rest. Lorentz's equations provided the basis for Albert Einstein's theory of special relativity. Lorentz had many unpublished conversations with Einstein, some of which provide insight into Einstein's thoughts on the mysterious relationship between discrete light quanta (particles) and the continuous waves of classical electromagnetic theory. It shows that Einstein had a statistical view of the quanta. The probability of finding quanta is determined by the continuous wave, which controls the interference even for one quantum at a time. Lorentz also describes the two-slit experiment.
Excerpt from Problems of Modern Physics (1922 Lectures at Cal Tech)50. Interference and the Quantum Theory. I tried to explain to you how the production of light by quantum jumps can perhaps be reconciled with our old views concerning radiation, so that these would hold as to the constitution of the emitted radiation. But the question arises, Can this constitution be really just what we have thought; that is, can there be a propagation according to Maxwell's laws, with a tendency to spread out in all directions and the impossibility of a lasting concentration of energy? You know that phenomena like those of photo-electricity have led Einstein to his hypothesis of light-quanta. According to this, quantities of energy equal to hν would be concentrated in small spaces, moving with the speed of light; they would even be light and would produce all optical effects. In this way we can understand that even very feeble light can give to an electron the amount of energy hv, for the smallness of the intensity would be due to the small number of quanta which it contains, the magnitude of each remaining the same. So we should escape the difficulty which, in the case of wave-motion, arises from the continual spreading out and weakening of the energy. The hypothesis of light-quanta, however, is in contradiction with the phenomena of interference. Can the two views be reconciled? I should like to put forward some considerations about this question, but I must first say that Einstein is to be given credit for whatever in them may be sound. As I know his ideas concerning the points to be discussed only by verbal communication, however, and even by hearsay, I have to take the responsibility for all that remains unsatisfactory. Let us suppose that in the emission and propagation of light there is something that conforms wholly to Maxwell's equations, but that it has practically no energy at all, the electric and magnetic forces being infinitely small. Then in this, let us say, Fresnel radiation we shall have the ordinary laws of reflection, interference, and refraction, but we shall see nothing of it. On a screen you will have something like an undeveloped photographic image. We can now imagine that in the production of light this Fresnel radiation is accompanied by the emission of certain quanta of energy that are of a different nature. Although their precise nature is unknown, we may suppose that energy is concentrated in small spaces and remains so. These quanta move in such a way in our "pattern" that they can never come to a place where in this pattern there is darkness. In thus traveling from the source outward each quantum has a choice between many paths. The probability of following different paths is proportional to the intensity of the radiation along these paths in Fresnel's radiation. Now in all real cases the act of emission is repeated a great many times. Suppose it is repeated N times, and let the Fresnel radiation be the same in these different cases. Then we shall have N quanta moving in this pattern, and if their number is very great and the probability of following different paths as stated, the number of quanta coming on different parts of a screen on which we observe an interference phenomenon will be proportional to the intensity which we have in Fresnel's pattern. These considerations can easily be extended. Take, for instance, polarization. The polarization will be in the Fresnel pattern, not in the quanta, but the quanta will illuminate a screen or a photographic plate or our retina to exactly the degree determined by the classical theory. When light falls on the surface of a piece of glass, there is a partition between the reflected and refracted parts. The probability of the quantum's following one path or another is determined by the well-known formulae of Fresnel for the intensities of the reflected and the refracted light. Suppose that in an elementary act of radiation there are a million waves; these exist in Fresnel's pattern; but the quantum of energy can have any place in the train of waves, either near the front or near the rear of these waves. If we have an ordinary beam of light consisting of the superposition of a great number of elementary beams, we have quanta in great number distributed all through the space occupied by the beam.