Universality of minimal length

We present an argument reinterpreting the generalized uncertainty principle (GUP) and its associated minimal length as an effective variation of Planck constant ($\hbar$), complementing Dirac's large number hypothesis of varying $G$. We argue that the charge radii (i.e. the minimal length of a scattering process) of hadrons/nuclei along with their corresponding masses support an existence of an effective variation of $\hbar$. This suggests a universality of a minimal length in measurement of scattering process. Varying $\hbar$ and $G$ explains the necessity of Von Neumann entropy correction in Bekenstein-Hawking entropy-area law. Lastly, we suggest that the effective value of $\hbar$ derived from various elements may be related to the epoch of their creation via nucleosynthesis.

where c is the speed of light, and P and M P are the Planck length and Planck mass, respectively. The latter two quantities represent a minimal length and maximal mass imposed by Nature. These unique scales do not exist mathematically in relativistic quantum theories. This problem has been resolved in doubly special relativity (DSR) that was proposed by Magueijo and Smolin [36]. DSR suggests the existence of an invariant length/energy scale in addition to the invariance of the speed of light. The corresponding uncertainty principle of doubly special relativity was introduced and investigated in [9,37] and it was found that it implies discreteness of space [9,10], which can also be represented by the generic Equation (3). This model of GUP is written as follows: where α = α 0 l p / , and α 0 is a dimensionless constant. We now seek to establish a version of Equation (4), which defines a constant , that incorporates Equation (3) to define a varying . One can imagine the Planck length as the radius of a fundamental "nucleus" from which every particle is scattered. This interpretation is similar to Rutherford's historic discovery of the nucleus by analysing α−particle scattering on a very thin gold foil [38] using the data of Geiger and Marsden (as shown in Figure 1(a)). The gold nucleus was treated as a point particle that is sufficiently massive relative to the mass of incident α−particle, so that any nuclear recoil could be ignored. The key idea in this experiment was the existence of a specific distance of closest approach, D, at which the α−particle is obligated to completely reverse its direction (i.e. the scattering angle θ equals π as shown in Figure 1(b)). This specific distance can be obtained by equating the Coulomb energy with the initial kinetic energy of α− particle. In the context of quantum gravity, the analogue of this distance is a minimal length that may be similar to the Planck length P but at different energy scale. The scattering relation is : where b is the impact parameter. In atomic, nuclear, and particle physics, the concept of charge radius measures the minimal length size of a composite hadron or nuclei. For example, a charge radius has been measured for the proton [39][40][41], neutron [42,43], pion [44], deuteron [45] and kaon [46]. In that sense, we can set a generalization of Equation (4) in terms of the charge radius and mass of the corresponding system. We take the same form of Equation (4), replacing Planck length P by the charge radius (r) [47] and the Planck mass M P by the mass of the hadron/nuclei (m). This would take the following form: where m is the particle's mass and is the effective Planck's constant. Here, c is chosen to be constant to maintain consistency with the theory of relativity. For the proton, the charge radius is r p = 0.831 fm [47] and the mass is m p = 1.672 × 10 −27 kg [48]. Surprisingly, we obtain a value for that is around four times the accepted value of Planck constant's, i.e.
Proton case: r p m p c = 3.95 , Let us repeat the same computations for the pion π, which has charge radius r π = 0.657 ± 0.003 fm [44,49] and mass m π = 2.488 × 10 −28 kg [48]: Here, the result obtained is of the order of magnitude of Planck's constant. In the case of the Kaon, where r K = 0.56 ± 0.031 fm [50] and m K = 8.8 × 10 −28 kg [48]: Once again, the result we obtained is roughly equal to Planck's constant. Lastly for the Deuteron [45] r D = 2.130 ± 0.003 fm, and m D = 3.343 × 10 −27 [45,48], we find: which is larger than Planck's constant by a factor of 20. We note that the effective Planck's constant fluctuates around the true value of Planck's constant by few orders of magnitude, which may suggest that effective value of is actually dependent on the given physical system. This is in fact consistent with the GUP Equation (2), and implies that GUP parameter bounds would be varying for different physical systems [21,32]. We conclude that a particle's charge radius and the mass introduce a consistent picture between Equation (3) and Equation (4).
Furthermore, a recent result that may support this variation is the direct measurements of a varying fine structure constant [51] due to gravitational effects. It is well-known that the wide array of GUP models are motivated by different approaches to quantum gravity and each introduce correspondingly different corrections to quantum systems. It is therefore logical to suggest that is varying due to the same quantum gravity effects. Moreover, a recent study sets bounds on varying in terms of violation of local position invariance [52].
We thus extend our argument to the nuclei of elements in the periodic table and their corresponding radii [53]. All values are shown in data table in the appendix. The relation between / versus M/Z is plotted in Figure 2 for Z < 30 and in Figure 3 for Z ≥ 30. In the former Figure, it is apparent the ratio /h is fluctuating for small values of atomic mass M , while in the latter it starts to become linear for large values. The combined plots are collectively represented in Figure 4 that shows approximately exponential behaviour for varying .
The ratio between the effective Planck constant to the standard Planck's constant / as a function of the atomic mass M of elements is shown in Figure 5 This would tell us how the uncertainty principle loses its meaning by increasing the bound in the inequality of Equation (3). If one assumes h eventually becomes infinite, this would imply that variance of momentum (∆p) or variance of position (x) would also be bounded by infinity.  Alternatively, one can postulate that a GUP-based variation of may be connected with Dirac's Large Number Hypothesis, which itself implies a varying gravitational constant G [54,55]: which is supported by several astrophysical studies [56][57][58]. Conversely, we found in our analysis that the effective is varying and increasing with increasing the complexity of matter, or simply with increasing atomic mass. If complexity increases over the evolution of time [59], it is logical to conclude that varying is increasing with time evolution. This is shown in Figure 4 Determining the exact function of proportionality between and t is left for future investigation. We propose that it can be determined based determining the creation time of chemical elements based on Big bang nucleosynthesis [60,61]. By comparing Equation (16) with Equation (17), we find the following fundamental relation between the effective and effective G as follows, where g(t) = f (t)/t, which we dub the quantum gravity complementary relation. It is obvious that when g(t) → 1, G → G . Similar relation between and G was found in the timeless state of gravity where it was found it is related to fifth power of distance measured from the gravitational source [62]. It is worth mentioning that a similar function to g(t) was suggested in [63] as f ± to introduce variation of Planck quantities due to GUP.
One may additionally wonder how G is defined in terms of geometry. To answer this question, we appeal to the Bekenstein-Hawking entropy area law [64,65] and note this contains the product G, i.e.
Recently, it was suggested that this relation should be modified by a Von Neumann entropy term in order to preserve the second law of thermodynamics, as well as to preserve information inside and outside the horizon [66,67], i.e.
where Eq. (20) can be rearranged as follows: When comparing Eq. (18) with Eq. (21), we obtain an expression for the function g(t) that implies a timedependent variation of both and G : For a pure state case in which S matter = 0 one finds g(t) = 1, implying fixed values for and G according to Eq. (18). For the case of mixed states, i.e S matter = 0, the expression implies a time dependence of g(t). This may establish a correlation between mixed states in quantum mechanics and time as an emergent concept. In other words, time emerges from mixing states. Conversely, it may also be the case that a mixing of states is a result of the arrow of time. Moreover, the fundamental function g(t) may be a manifestation of the holographic principle between matter and information [68,69]. We further suggest that g(t) explains why the Von Neumann entropy correction is needed [66,67] using our new perspective on varying and G. Using Eq. (18), we simply can write the effective variation of the Planck constant and gravitational constant as follows: This forms a crucial bridge between the quantum and gravitational worlds, and discloses several interpretations. First, a varying is found to complement Dirac's Large Number Hypothesis and varying G [54,55]. In addition, Equation (23) supports the thermodynamic explanation of gravity and matter [70][71][72][73]. Implications on classical and quantum information theory can be also investigated [74].
In conclusion, we offer an outlook on a variety of additional novel implications of a varying . First, modifications of gravity should follow from Eq. (23), from which a varying G. can be inferred [75]. Furthermore, Eq. (23) can be interpreted to suggest that the Von Neumann entropy term is itself the source of modified gravity.
A varying could also replace the renormalization functions in quantum field theories, due to the fact that GUP or minimal length scenarios imply a natural cutoff scale that yields finite measurable values [34,76,77]. Since this cutoff (i.e. the minimal length) is included in the definition of , it may also resolve the renormalization problem in approaches to quantum gravity [78]. Recently, it was shown that entanglement entropy is varying in scattering processes [79,80], which supports our proposal on the correlation between minimal length of scattering that is included in and entropy. Lastly, it is expected that Equation (23) could have implications in condensed matter systems where the entropy-area law has wide applications [81]. As a final thought, it is worth mentioning that function g(t) may be determined by comparing it with the element formation timeline in the context of big bang nucleosynthesis [60,61,82]. If we determine the form of g(t) from the astrophysical observations and nucleosynthesis, we expect that the function g(t) will play a significant role in re-interpretation of renormalization functions in quantum field theories in terms of astrophysical data and black hole physics. We hope to report further investigations on these topics in the future.