Can Physical Constants Which Are Obtained with Combinations of Fundamental Physical Constants Have a More Fundamental Nature?
Planck Scales (Planck's 'units of measurement') are different combinations of the three physical constants h, c, G, Planck Scales=f(c,h,G):
Planck Time: tp=√ℏG/c^5=5.31x10^-44s ......(1)
Planck Length: Lp=√ℏG/c^3=1.62x10^-35m ......(2)
Planck Mass: Mp=√ℏc/G=2.18x10^-8 kg ......(3)
“These quantities will retain their natural meaning for as long as the laws of gravity, the propagation of light in vacuum and the two principles of the theory of heat hold, and, even if measured by different intelligences and using different methods, must always remain the same.”[1] And because of the possible relation between Mp and the radius of the Schwarzschild black hole, the possible generalized uncertainty principle [2], makes them a dependent basis for new physics [3]. But what exactly is their natural meaning?
However, the physical constants, the speed of light, c, the Planck constant, h, and the gravitational constant, G, are clear, fundamental, and invariant.
c: bounds the relationship between Space and Time, with c = ΔL/ Δt, and Lorentz invariance [4];
h: bounds the relationship between Energy and Momentum with h=E/ν = Pλ, and energy-momentum conservation [5][6];
G: bounds the relationship between Space-Time and Energy-Momentum, with the Einstein field equation c^4* Gμν = (8πG) * Tμν, and general covariance [7].
The physical constants c, h, G already determine all fundamental physical phenomena‡. So, can the Planck Scales obtained by combining them be even more fundamental than they are? Could it be that the essence of physics is (c, h, G) = f(tp, Lp, Mp)? rather than equations (1), (2), (3). From what physical fact, or what physical imagination, are we supposed to get this notion? Never seeing such an argument, we just take it and use it, and still recognize c,h,G fundamentality. Obviously, Planck Scales are not fundamental physical constants, they can only be regarded as a kind of 'units of measurement'.
So are they a kind of parameter? According to Eqs. (1)(2)(3), c,h,G can be directly replaced by c,h,G and the substitution expression loses its meaning.
So are they a principle? Then who are they expressing? What kind of behavioral pattern is expressed? The theory of quantum gravity takes this as a " baseline ", only in the order sense, not in the exact numerical value.
Thus, Planck time, length, mass, determined entirely by h, c, G, do they really have unquestionable physical significance?
-----------------------------------------
Notes
‡ Please ignore for the moment the phenomena within the nucleus of the atom, eventually we will understand that they are still determined by these three constants.
-----------------------------------------
References
[1] Robotti, N. and M. Badino (2001). "Max Planck and the 'Constants of Nature'." Annals of Science 58(2): 137-162.
[2] Maggiore, M. (1993). A generalized uncertainty principle in quantum gravity. Physics Letters B, 304(1), 65-69. https://doi.org/https://doi.org/10.1016/0370-2693(93)91401-8
[3] Kiefer, C. (2006). Quantum gravity: general introduction and recent developments. Annalen der Physik, 518(1-2), 129-148.
[4] Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der Physik, 17(10), 891-921.
[5] Planck, M. (1900). The theory of heat radiation (1914 (Translation) ed., Vol. 144).
[6] Einstein, A. (1917). Physikalisehe Zeitschrift, xviii, p.121
[7] Petruzziello, L. (2020). A dissertation on General Covariance and its application in particle physics. Journal of Physics: Conference Series,
[8] https://www.researchgate.net/post/No28The_Relation_Between_Mathematics_and_Physics_4-Could_All_Physical_Constants_be_Unified_to_a_set_of_Fundamental_Constants_of_Nature
[9] Preprint The Relationship Between the Theory of Everything and the Co...