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V. Discussion

Is there any need for quantum in Physics? Apparently, the only need comes from the convenience offered by statistical meaning of the density of blackbody radiation energy as a mean of the energy density of an ensemble of oscillators. This idea has penetrated the theory of blackbody radiation from the very beginning and lasts to this very day. It did not emerge without questions, as it does not dwell today without questions. In an attempt to discern the nature of these questions we traced back the procedure that led to quantum idea, and the conclusions (as well as the question marks) all seem related to the idea of process representing radiation as follows:

If the blackbody radiation is a stochastic Gaussian process, then the fact is that it is composed of two Gaussian processes. If these are independent, then we can have a quantum interpretation and aposteriori explanation of ε₀ as a quantum of energy, as indeed historically was the case. This however, leaves open the identity of the ensemble of oscillators to which the formula for mean is referred. When it comes the time to attach an identity to this ensemble, as in the process of calculation of specific heats of solids for instance, then the problems show up. Case in point: the necessity of introduction of the zero point energy, which has been a challenge for quite some time. Nowadays nobody seems to dispute the idea of quantum, and the reason is clear: there is no fault in the mind process that led to its idea! This process is conceptually clean. It is however the fact that this quantum has a carrier - the photon - that has always been challenged. Here it helps to recall the fact that this carrier does not qualify as a particle in the classical sense of this last concept, for the original identification was done on the determination of ensemble for the particle, not as individual in the mechanical connotation of the word. This fact has always been left aside in critical discussions, in spite of its overwhelming importance for, had it been taken explicitly into consideration, the concept of photon would have never made its appearance in Physics. Anyway, carrier or not carrier, the quantum itself had to be explained. The explanation is not easy but, whenever is produced, it always involves an ensemble. The best known instances are the De Broglie’s thermodynamics of isolated particle (De Broglie, 1964) and the theory of hidden parameters (Bohm, 1952), this last being a remote descendant of the original solution to the problem of radiation.

Still regarding the process the kind of rationale that takes us clearly outside the realm of the Gaussian distribution has apparently been inaugurated by Poincaré (Poincaré, 1911), with the result that the quantum has to be seriously taken into consideration. Born’s route makes it plain that the blackbody radiation ensemble has to be characterized by a quadratic variance distribution function. This fact could not be recognized at the beginning of the last century, for by then there was not even the faintest notion of such distribution functions. It is worth mentioning that the variance property has already been recognized as such by Einstein at the Solvay Congress in 1911 (De Broglie, 1922) or even earlier. Einstein made even the derivative connection for the relationship mean-variance. However, because in those times, as today for that matter, the quadratic variance equation has not been seen as an essential property, no effort has been made towards describing such distributions. Only slowly and by particular chances had these distributions made their entrance, mainly in the field of Statistics, until 1982 when the fundamental work of Carl Morris appeared (Morris, 1982). That work allows us to say that, with respect to blackbody radiation, there is nothing special about Gaussian distributions, they are just as natural in this field as any other distribution. It is a matter of experimental chance that they are just limit distributions for Gamma and Poisson which, in their turn are limiting distributions for Negative Binomials, which ought to be Planck’s original distributions. In witness thereof we have today the plethora of intermediate states (Fu, 1996), descendants of coherent and numerical states. There is an intimate connection between these intermediate states and the probability distributions having quadratic variance functions. Indeed, it has been proved by Fu and Sasaki (Fu, Sasaki, 1996) that all the intermediate states can be generated by a process of ’square-rooting’ from the probability distributions that give their name. As expected, the Negative Binomial states are ‘intermediate’ between thermal states and coherent states.

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