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III. First Thing Left Behind

Let us limit first to the Gaussian realm of statistical distributions: according to previous plot of Born everything is based on the normal distribution. Moreover, it seems that the Planck’s pursuit was for statistically independent processes. One can easily be induced to think that the two processes might not be independent. Assume that we have indeed two Gaussian processes as representing the limit cases of radiation, but they are in a general relationship, i.e. not statistically independent. One may further assume that the general radiation is actually a linear combination between the two processes, but we limit here the line of reasoning to just the sum of the two processes. The general bivariate normal distribution is given by

(13)

In terms of the variances σ_x, σ_y of the two processes and their correlation coefficient r, the coefficients a, b, c can be written as

(14)

Now we can write the probability density of the compound process (X + Y), which is

(15)

i.e. a Gaussian with the variance (a + c - 2b)/(ac - b²) or, in terms of variances and correlation coefficient of the two component processes

(16)

Still maintaining the philosophy above, instead of equation (9) we should have

(17)

This equation can be integrated to give

(18)

Here something is immediately obvious, which shows that, in the past, our focus might have been misled by the mirage of quick interpretation already in hand. Namely the energy ε₀, which has been introduced from dimensional considerations, and which has subsequently been explained as a quantum of energy to be carried by an invented particle whose existence is nowadays challenged and which etc…, had to be a priori explained. Here is a scenario: in the good old fashion of Statistical Mechanics, we correlate this energy with an exponential factor, which can play the role of a partition function over a certain ensemble, and which can be easily extracted from equation (18) as

(19)

The left hand side of this equation represents a thermal ensemble for the energy ε₀, having the mean β. The odd thing here is that the right hand side also depends on ε₀. However, this dependence occurs through the intermediary of the ratio w, which allows us to say that a statistical interpretation actually depends upon a sort of ε₀-content of the density of energy of the thermal radiation. This conclusion sounds quite normal: an experimentalist knows exactly how to characterize the radiation depending on its density. Should this density be of the order of ε₀ then w ≈ 1, and the right hand side of equation (19) does not depend but on the correlation coefficient between the two processes:

(20)

Three decades ago, Ioannidou (Ioannidou, 1982) tried to explain the quantum through the correlation of ensembles associated with oscillators, based on the uncertainty relation. The attempt has been forgotten, probably due to the connection it suggested. Well, that connection seems sound, for here it is again, in equation (20), which explicitly puts down a relationship between the quantum and the correlation coefficient of the two processes representing the radiation. Further on, if the correlation of the two processes is faint, which is the Planck’s case, then

(21)

independently of any other consideration. Thus, in this limit, the “quantum”, and therefore the frequency, is directly proportional to the temperature. This fact has been discussed at length by Louis de Broglie (De Broglie, 1964), who skillfully identified the action with the entropy.

In the general case, when the ε₀-content units and the correlation of the two processes are both arbitrary, it helps noticing that the right hand side of (19) is actually the generating function of a particular class of Pollaczek polynomials (Chihara, 1978). Specifically, we can write (19) in the form

(22)

The orthogonality relation of the polynomials involved here is given by

(23)

with the weight function ρ given by

(24)

where Γ is the Euler function of the first kind, generalization of the factorial. Should we agree to interpret ε₀ as the energy of a photon, as has historically been the case, then the formula (22) would be the source of constructions of some modern quantum states related to the coherence properties of radiation.

The bottom line is that we ought to supply for ε₀ an a priori explanation; otherwise we have to face problems of a posteriori explanations, as indeed was historically the case. One of these explanations aims the very existence of the photon, or of any other ‘ons’ for that matter. The ongoing discussion upon the legitimacy of the quantum precepts just proves this conclusion.

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