6. Discussion

Published: J. Stat. Mech. (2024) 093209

I have shown that the Bose-Einstein distribution follows mathematically from probability domain quantization, and that probability domain quantization is a consequence of the existence of a limit, ${h_?}$, on the precision with which a system's microstate can be determined experimentally. Probability domain quantization does not imply that the microstates of the underlying physical system are quantized. It implies a quantization of the information contained in probability distributions that possess the quality of being testable empirically.

I have not justified my working assumption that a lower bound ${h_?}$ exists, but only demonstrated that one of its consequences would be that all sufficiently-cold classical dynamical systems are described by Bose-Einstein statistics. Therefore I have demonstrated that the existence of such a lower bound would have many important implications.

One implication would be that there is no qualitative discrepancy between the experimentally-observed spectrum of a blackbody and what should be expected if light was a mechanical wave in a bounded medium. As discussed in Sec. 1.1.1, if light was such a wave, the boundedness of the medium would mean that the smallest energy difference between two light waves of frequency $\approx f$ would be $h_m f$, for some constant $h_m$. In Sec. 5 I showed that the lower bound that an uncertainty principle would place on observable energy differences would be $\Delta\left(\mathcal{I} \omega\right)=\left(\Delta\mathcal{I}\right)\omega = {h_?} f$. Therefore, if $h_m={h_?}$, the existence of an uncertainty principle in a classical universe could be explained by all observations being mediated by classical light waves.

Another implication of a lower bound ${h_?}$ would be that there is no qualitative discrepancy between the experimentally-observed temperature dependence of a crystal's heat capacity and what should be expected of classical lattice waves.

Another implication would be that classical oscillators and waves would have zero point energies that were simply an artefact of small energies being empirically-indistinguishable from no energy.

Another implication would be that, when a cluster of massive particles was cold enough, the classical expectation would be that almost all of its vibrational energy would be possessed by its lowest-frequency normal mode. Therefore, below a certain temperature, all but one of its degrees of freedom would be almost inactive and it would be a Bose-Einstein condensate.

For simplicity I have assumed that the limit on microstate measurement precision that leads to probability domain quantization is a limit on certain knowledge. In other words, I assumed that it is theoretically possible to know with certainty that a DOF's microstate is within a subset of its phase space if and only if the area of that subset is greater than ${h_?}$. If, instead, it is assumed that the result of the most precise microstate measurements possible are probability density functions of the form

\[\begin{aligned}\rho(\sigma_{\mathrm{Q}},\sigma_{\mathrm{P}}):\mathbb{Q}\times\mathbb{P}\to\mathbb{R}^+;\; (Q,P)\mapsto \rho(Q,P;\sigma_{\mathrm{Q}},\sigma_{\mathrm{P}}),\end{aligned}\]
where $\sigma_{\mathrm{Q}}$ and $\sigma_{\mathrm{P}}$ are the standard deviations along the coordinate axis $\mathbb{Q}$ and the momentum axis $\mathbb{P}$, respectively, a more general form of uncertainty principle would be $\sigma_{\mathrm{Q}}\sigma_{\mathrm{P}}>{h_?}$. This would be a limit on probabilistic knowledge. It may be possible to adapt the derivations presented in this work to uncertainty principles of this more general form.

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