On the thermodynamic origin of the uncertainty principle

: The spatial resolution measurement limitation of the position-momentum uncertainty principle is shown to mathematically originate from the Bekenstein entropy bound and the associated second law of thermodynamics, as a special case in which the average energy of a statistical thermodynamic distribution of energies is specialized to a fixed, definite probe energy. This is used in combination with the Wein displacement law to predict an ultraviolet cutoff for Planck blackbody radiat ion at about ⅛ of the Wein peak. A new UV photon counting experiment is proposed to test for this. A physical understanding of these results may be provided by a UV-complete, intelligible theory of general relativistic quantum mechanics in which the observation of a blackbody spectrum is simply a remote observation of Hawking radiation emitted from black hole fluctuations in the gravitational vacuum.


Introduction
The position-momentum uncertainty principle first formulated by Heisenberg [1] and implied by Born's commutation relations [2] has long been part of the canonical foundation of quantum mechanics. We demonstrate how the spatial resolution measurement limitation of the uncertainty principle can be mathematically derived as a special case of a deeper relation between temperature and length arising from the Bekenstein entropy bound and associated second law of thermodynamics. When combined with the Wein displacement law, this predicts an ultraviolet cutoff at about ⅛ of the Wein peak for black holes and ordinary blackbodies alike. Because Planck's law [3] predicts, statistically speaking, that for every 245.61 billion photons emitted near the Wein peak there will be only one photon emitted near ⅛ of that peak, a new experiment to detect this UV cutoff is proposed to see whether in fact, after cataloguing trillions of photon emissions, there are in fact no emissions at all below about ⅛ of the Wein peak, as predicted here. Because Planck-scale black holes are conjectured to permeate the gravitational vacuum first described by Wheeler [4], [5], this appears, with UV-completeness, to ground canonical quantum mechanics in the general theory of relativity which ab initio predicts black holes and their event horizons beyond which -as with the uncertainty principle -external observations cannot penetrate.

The Bekenstein bound, Bekenstein-Hawking black hole entropy, and Hawking black hole radiation
In 1981 Bekenstein [6] found that any system enclosing a total energy E within a sphere of radius R has an entropy S with an upper limit rooted in the second law of thermodynamics:  [7] and later generalized in [6] as shown in (1.1), which is that black hole entropy varies linearly with surface area not volume. 't Hooft [8] subsequently characterized this as a holographic principle.
For a system with a large number N of independent degrees of freedom, and labeling the energy of each degree of freedom by n E , this average is Then, instead of applying a temperature to the system to impart a collection of energies n E which average out to E , apply a fixed, definite energy  to probe the system. This is precisely what is done in particle physics collision experiments to probe molecules and atoms and nuclei and nucleons, see e.g. section 1.4 of [11] and the conversion constant

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GeV fm . c = in the 2019 PDG data [12]. We then obtain a special case of (2.3) in which the average energy is specialized to a fixed, This, of course, is the widely-recognized spatial resolution measurement limitation rooted in the uncertainty principle

Prediction of an ultraviolet cutoff in Hawking and Planck blackbody radiation for wavelengths below about one-eighth of the Wein displacement peak
The Wein displacement law for the peak wavelength of a Planck blackbody distribution is: Listed in [12], this is obtained by the well-known calculation of ( ) equal to the diameter of the black hole, which likewise connects ordinary spatial lengths with photon radiation wavelengths. This very same approach is used by Susskind in his lecture [14] to derive Bekenstein's black hole entropy relation (1.2).
But (3.2) makes this even more general: It tells us that a generalized diameter 2 DR = from the Bekenstein bound (1.1) for any system emitting blackbody radiation similarly acts as the diameter of a "box," trapping all photon emissions with wavelengths below about ⅛ of the Wein peak. The inequality in (3.2) which signifies this ultraviolet cutoff is thus seen to be rooted in the inequality of the Bekenstein bound (1.1), and so in turn, in the second law of thermodynamics. Consequently, at bottom, the second law of thermodynamics forms the foundation of the ultraviolet cutoff in (3.2), and also of the uncertainty principle relation (2.4). Eq. (3.2) is simply how the uncertainty principle measurement limitation manifests more generally in thermodynamic blackbody systems. Possible theoretical implications of this will be reviewed in section 5. But first, we propose a new experiment by which this predicted UV cutoff might be detected.

Proposed photon counting experiment for detecting the predicted UV cutoff in Planck blackbody radiation
An experiment to directly detect the ultraviolet cutoff predicted at  2). As a predicate, we need to calculate how many photons are predicted by Planck's law in the vicinity of 1 8 peak  -which we expect will be a very small number -relative to the number of photons predicted right near peak  . The experiment is then to detect whether this very small number really is observed, or whether, as predicted in (3.2), no photons at all are observed.
A simple way to calculate this is to reparameterize Planck's law along dimensionless x and y axes re-scaled so the Wein peak is always at the point ( ) ( ) , 1,1 xy = regardless of temperature, then find the height of y at x~⅛. This provides the relative energy output which Planck's law predicts near the UV cutoff of (3.2). Then, because the energy of a single photon varies inversely with its wavelength, we must divide the energy result through by a factor of ~8 to obtain photon numbers rather than energies. Finally, take the ratio of (4.2) to (4.3), then define the desired dimensionless function: It will be appreciated that the peak wavelength  This says that the photon energy emitted at peak 1 xx == is predicted by Plank's law to be stronger than that at  and counting thousands of trillions of photons before 5-sigma confidence could be established that this cutoff truly exists in nature. The equipment needed for this is reasonably uncomplicated and inexpensive: a heated blackbody and a photon detector tuned to both peak  and to wavelengths under 1 peak 8  , coupled with a device that can count the detected photons into the quadrillions.

Outlines of a UV-complete general relativistic theory of quantum mechanics
As noted following (3.2), using "particle in a box" analysis, it is straightforward to envision how a black hole would trap wavelengths shorter than or equal to its diameter, producing the UV cutoff (3.2) for its Hawking radiation spectrum. Again, this is the approach Susskind uses in [14] to derive Bekenstein's (1.2). It is less straightforward to envision this for a generalized blackbody. Thus, we now examine why the gravitational vacuum may well provide the best explanation for why the UV cutoff (3.2) applies not only to black holes, but to blackbodies in general.
If one was only aware of the uncertainty principle measurement limitation (2.4) and not its thermodynamic generalization (2.2) based on the Bekenstein bound (1.1), then one could conclude that for a blackbody generally, the lower wavelength boundary based on uncertainty would be And these make clear that there is a single lower wavelength bound arising from the average energy of the degrees of freedom in the systemthat is, from the temperature of the system -rather than there being differing lower bounds arising from the statistically differing energies of each degree of freedom.
The way to explain this may be to start with the Planck-scale vacuum first articulated by Wheeler in his work on geometrodynamics [4], [5], conjectured to contain innumerable fluctuations which are themselves black holes, ultra-densely-packed with an average separation of P , and establishing a collective event horizon with a Schwarzschild radius 2 S P P r = and thus diameter 4 P . From (1.3) this vacuum (V) will emit its own blackbody spectrum at a temperature  (3.2). This is the very same as the collective event horizon diameter of the vacuum.
So, even when a photon with a statistically-high energy is emitted from a low-energy black hole fluctuation behind the horizon, that photon will be captured by one of the closely-neighboring higherenergy black hole fluctuations before it ever escapes the vacuum as a whole, because of the ultradense black hole packing behind the collective horizon. This capture by neighboring black holes enables us to envision how it is even possible to have The challenge, then, is to envision how this community cutoff carries over as predicted by (3.2) to a cutoff near ⅛ of the Wein peak for ordinary blackbodies, when there is no evident involvement of densely-packed black holes. The only apparent explanation is that observing Planck-1901  when we probe with energy  in particle physics scattering experiments. In particle physics collisions, we use focused energies to scatter atoms or molecules or nuclei or nucleons and observe the particle debris. But there is a spatial resolution limit given by (2.4). When we apply a temperature T to a blackbody and allow it to reach thermal equilibrium, we are scattering photons out of the vacuum and the observed debris is a blackbody spectrum. But here too there is a lower wavelength limit given by (3.2). Like the meter stick, no matter how "far away" the observational T might be

Conclusion
The derivation of the spatial measurement limitation (2.4) and the predicted UV cutoff (3.2) from the Bekenstein entropy bound (1.1), for a thermodynamic energy distribution specialized to a definite fixed probe energy, is strictly mathematical and non-speculative. So, if this predicted cutoff (3.2) for an ordinary blackbody is uncontradicted by the photon counting experiment quantified in (4.6), this would establish on observational in addition to mathematical grounds that the spatial measurement limitation (2.4) of the uncertainty principle is simply a special case of the statistical thermodynamics of the Bekenstein bound and the second law of thermodynamics. And, absent some compelling alternative explanation, this would provide empirical support for the physics of the cutoff (3.2) originating from the extreme black hole curvatures of the gravitational vacuum.
Specifically, if this cutoff is confirmed in ordinary blackbody radiation, there would appear to be no plausible explanation for this other than that when we observe this cutoff in an ordinary blackbody, we are really remotely observing the community cutoff of black hole fluctuations in the gravitational vacuum, analogously to the meter stick observed from afar. Theoretically, this would mean that the spatial measurement limitation of the uncertainty principle beyond which external observations cannot penetrate originates in the black hole event horizon of a gravitational vacuum beyond which external observations likewise cannot penetrate.
Consequently, experimental confirmation of this UV cutoff could help establish that the general theory of relativity which ab initio predicts black holes, provides a UV-complete, theoreticallyintelligible foundation not only for the entropic thermodynamics of the Bekenstein bound, but also for the uncertainty-based spatial resolution measurement limitation at the heart of quantum