# Where did Einstein propose interpreting the square of amplitude as probability density (Born's rule)?

In Max Born's Nobel lecture, he alludes to Einstein's proposed interpretation of EM wave amplitude (squared) as being the probability density of detecting a photon:

Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons.

In which of his papers did Einstein first propose that interpretation?

There are no published papers where such interpretation is explicitly made. Born's Nobel lecture mention first appears in his celebrated Quantenmechanik der Stossvorgange (1926, Zeitschrift fur Physik 38, 803-827), also with no reference. According to Pais, well-known Einstein's biographer, the inspiration came from Einstein's never published remarks about the "ghost field" from early 1920-s, which are not extant. Their existence is known from Wigner, who does not give many details either. As Pais writes in Inward Bound:

"Nevertheless, it is true that Born's inspiration came from Einstein: not Einstein's statistical papers bearing on light, but his never published speculations during the early 1920s on the dynamics of light-quanta and wave fields. Born states this explicitly in his second paper: 'I start from a remark by Einstein on the relation between [a] wave field and light-quanta; he said approximately that the waves are only there to show the way to the corpuscular light-quanta, and talked in this sense of a "ghost field" [Gespensterfeld] [which] determines the probability [my italics] for a light-quantum ... to take a definite path'.

[...] Little concrete is known about his ideas of a ghost field or guiding field (Führungsfeld). The best description we have is from Wigner, who knew Einstein personally in the 1920s: '[Einstein's) picture has a great similarity with the present picture of quantum mechanics. Yet Einstein, though in a way he was fond of it, never published it. He realized that it is in conflict with the conservation principles... This Einstein never could accept and hence never took his idea of the guiding field quite seriously ... The problem was solved as we know, by Schroedinger's theory'.

Born was even more explicit about his source of inspiration in a letter to Einstein written in November 1926 (for reasons not clear to me this letter is not found in the published Born–Einstein correspondence): 'About me it can be told that physicswise I am entirely satisfied since my idea to look upon Schroedinger's wave field as a "Gespensterfeld" in your sense proves better all the time. Pauli and Jordan have made beautiful advances in this direction.'..."

Of course, another connection that comes to mind is to Einstein's 1905 "photoeffect paper" (historians consider this nickname a misnomer). But it has neither "photoeffect" nor "photons" in it, and certainly no complex-valued waves. It studies radiation in an enclosure, and even the photoelectric equation is derived in passing when treating radiation thermodynamically, see A Revolution in Physics: Einstein’s Papers of 1905 Made Simple. Pais is also skeptical of this connection:

"On the face of it, this appears to be a perfectly natural explanation. Had Einstein not stated that light of low intensity behaves as if it consisted of energy packets $$h\nu$$? And is the intensity of light not a function quadratic in the electromagnetic fields?... Recall that Born initially thought, however briefly, that $$|\psi|$$ rather than $$|\psi|^2$$ was a measure of the probability. I find this impossible to understand if it were true that, at that time, he had been stimulated by Einstein's brilliant discussions of the fluctuations of quadratic quantities (in terms of fields) referring to radiation".

In Creating Modern Probability, von Plato makes a somewhat tenuous connection to Einstein's Quantum Theory of a Monoatomic Ideal Gas, Second Communication (1925, p.9). This paper, and its 1924 predecessor, are best known for developing a theory based on what is now called the Bose-Einstein statistics. In the second one, Einstein derives a formula for the mean-square fluctuation in the number of molecules,

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \overline{\left(\frac{\Delta_\nu}{n_\nu}\right)^2}=\frac1{n_\nu}+\frac1{z_\nu}$$,

where $$n_\nu$$ is the number of molecules, and, as Einstein writes, "...a scalar wave field can be adjoined to a gas, and I have assured myself through computation that $$\frac1{z_\nu}$$ is the average square of fluctuation of this wave field". Pais's objection would still apply to this connection, but, perhaps, this field is an echo of the aforementioned "ghost field", as von Plato speculates (p.146):

"By adjoining a scalar wave field to matter, Einstein arrived at a computation of his 'interference fluctuation' $$\frac1{z_\nu}$$ , writing presumably with interference in mind that 'here more than a mere analogy is involved' (1925, p. 9). What did he use matter waves for? He used them for determining the mean-square fluctuation of $$n_\nu$$. But this can only be if the wave field determines a statistical law that the number of particles in the volume $$V$$ obeys. As we shall see, in some way this connection between matter wave and probability was used in Born's probabilistic interpretation of Schrodinger's wave function. At the same time it pointed at a characteristic feature of quantum probability: the appearance of interference terms."

Other papers of Einstein's that feature both statistics and quantum mechanics are Zum gegenwartigen Stand des Strahlungsproblems (1909, Physikalische Zeitschrift, 10, 185-193) and Zur Quantentheorie der Strahlung (1917, Physikalische Zeitschrift, 18, 121-128), but their relevance seems even more remote.