The Wiener-Khinchin theorem
The Wiener-Khinchin theorem is one of the central results of statistical and stochastic physics and of signal processing. It states that the power spectrum $S(\omega)$ and the autocorrelation function $\langle\ud{a}(t+\tau)a(\tau)\rangle$ form a Fourier transform pair.
To appreciate the importance of this result, one needs to first understand what is a power spectrum, what is an autocorrelation function and why their connection as established by the theorem is, beyond useful for its practical purpose, insightful
The power spectrum $S(\omega)$ is the distribution of frequency or energy of a system, which has $a(t)$. I have used here notations of quantum optics for a quantized field with annihilation operator $a$.
The idea was first proposed by Einstein at a meeting of the Swiss "Société de Physique" whose compte-rendu lists the people who first heard about this connection (as well as what they discussed about themselves)
The communication was saved for posterity in a two-pages memo[1], written in French. Neither Einstein nor other—several prestigious—scientists seem to have appreciated or understood the significance of the statement at the time. Einstein himself discusses at length the technical feasibility of the integration, for which he consigns that his friend M. P. Habicht (not Conrad?) assured him:
M. P. Habicht, m'a montré en outre que la détermination des moyennes de (1) peuvent se faire aisément à l'aide d'un intégrateur mécanique de maniement facile.
One comment is recorded in the minutes of the meeting from the president Pierre Weiss who, despite his input to mean-field theory, also lacks a proper valuation of the result:
M. Weiss fait remarquer combien la méthode de M. Einstein rendra de services à la météorolog-ie très riche en matériaux qui, jusqu'à présent, étaient à peu près inutilisables.
Einstein drafted a longer version, that was, however, never published (it is available as Doc. 30 of the 4th volume of his collected papers (the two-page memo appears there as the previous Doc. 29).
The result comes in full bloom with Norbert Wiener, who, on opposite, writes almost a book on the topic of "Generalized harmonic analysis"[2].
References
- I talked about the topic at the Mathematics & Computer Science Research Seminar of the University of Wolverhampton on 28 April (2022): Why we don't need the Wiener-Khinchin theorem.