The Planck DistributionPlanck distribution, a Necessary Consequence of the Fluctuating Zero-Point Field

The historical problem of the spectral distributionDistribution of the radiation field in equilibrium at a given temperature is revisited in this chapter. Taking into account the inescapable presence of the fluctuatingFluctuations!zero-point zero-point radiation field, a derivation of Planck’s formula is presented that does not introduce any quantum postulate. The starting point is Wien’sWien!law law, which dictates that the mean energy of a Harmonic oscillator!and spinharmonic oscillator of frequency \(\omega \)—or rather, the mean energy of a monochromatic mode of the radiation field—must be proportional to the frequency. The description obtained through a standard thermodynamic analysis is shown to be incomplete, as it does not provide for the existence of fluctuations at zero temperature. This limitation is lifted by means of a statistical analysis that allows to determine the variance of the Fluctuationsfluctuations of the energy at \(T=0.\) A combination of the outcomes of the two analyses leads to a differential equation, which upon integration gives the Planck distribution Planck distributionat any temperature. The different terms contributing Commutator!and correlationto the energy fluctuations acquire thus a new meaning, different from those assigned to it by Planck and by EinsteinEinstein, A.. The implications of the results regarding the question of continuity or discontinuity of the field energy are discussed, and the chapter concludes with a brief discussion on the reality Realityof the zero-point fluctuations and the origin of quantum fluctuations.