Planck Uncertainties

by Steven Gusman

My understanding is that the Planck units were discovered when Max Planck simply took the fundamental constants, c, G, and h, and algebraically manipulated them until they returned a unique length, mass, and duration (in other words, he reverse-engineered them through unit analysis).¹ Utilizing the fact that p = mv and E = mc², one can further manipulate the Planck units to include a momentum and an energy.

       l_p = (hG/c³)^1/2
       m_p = (hc/G)^1/2
       t_p = (hG/c⁵)^1/2

       p_p = (hc³/G)^1/2
       E_p = (hc⁵/G)^1/2
The small values, length and time, are then naturally assumed by many physicists to be quanta of those variables (or at least of the right order-of-magnitude). That combining the relativistic constants G and c with the quantum of action, h, appears to yield a quantum of space and time leads most physicists attempting to discover a theory of quantum gravity to believe that the Planck units have something to do with it. Yet these constants are hypothetical, not theoretical; that they are borne only out of unit analysis of constants without any contact with physical law leads most physicists to treat them as estimates of the values they are interested in.

Yet if we take Heisenberg's Uncertainty Principle to be ∆x∆p ≈ h (and ∆t∆E ≈ h), which seems to be how Heisenberg initially formulated his principle,² we find that plugging in one of Planck's units (∆x or ∆p; ∆t or ∆E) produces the other! This suggests that Planck's units are not quanta, but uncertainties (what is special about them, as opposed to others, I do not know). Interpreting the Planck units as uncertainties allows one to predict complements using Heisenberg's uncertainty principle (given you plug one in), and thereby explains both why Planck units are proportional to √h, and why complementary Planck units are inversely related in terms of scale (l_p << p_p and t_p << E_p).

       Further, using the proper uncertainty principle, (∆x∆p)_min = ħ/2, we can slightly alter the Planck constants to their proper values (if this interpretation is correct):
       l_p = (ħG/2c³)^1/2
       m_p = (ħc/2G)^1/2
       t_p = (ħG/2c⁵)^1/2

       p_p = (ħc³/2G)^1/2
       E_p = (ħc⁵/2G)^1/2
       This suggests that rather than, “Planck units,” these values ought to be called, “Planck uncertainties,” and their physical significance investigated in that context (for example, interpreting the Planck length as an uncertainty rather than as a quantum no longer implies that space-time is quantized, or non-continuous, in a unification of quantum physics and the general theory of relativity).
       ħ/2 = ħ/2
       (ħ/2)^1/2 × (ħ/2)^1/2 = ħ/2
       ((ħ/2)^1/2 × (1/1)) ((ħ/2)^1/2 × (1/1)) = ħ/2
       ((ħ/2)^1/2 × (G/c³)^1/2) ((ħ/2)^1/2 × (c³/G)^1/2) = ħ/2
       l_pp_p= ħ/2

       ħ/2 = ħ/2
       (ħ/2)^1/2 × (ħ/2)^1/2 = ħ/2
       ((ħ/2) × (1/1))^1/2 ((ħ/2) × (1/1))^1/2 = ħ/2
       ((ħ/2) × (G/c⁵))^1/2 ((ħ/2) × (c⁵/G))^1/2 = ħ/2
       t_pE_p= ħ/2

The remaining problem is that there is still some Planckian fine-tuning in the derivation of the Planck uncertainties from the uncertainty principle, and this shrouds again our interpretation of the values. We can see that each is proportional to √h because we are getting at the minimum combined uncertainty (ħ/2), and each multiplicand (∆x and ∆p or ∆t and ∆E) is sharing the uncertainty. But if they were truly sharing the produced uncertainty, their individual uncertainties would not be only proportional to √h, they would be √h, as 1/1 is the only self-reciprocal. So how do we get the reciprocal values of (G/c³)^1/2 and (c³/G)^1/2 (or (G/c⁵)^1/2 and (c⁵/G)^1/2)? We get them by imposing Planckian unit analysis (only worse, because the equation already imposes that whatever we plug in for the terms, they will be of units length and momentum, before we play any game of unit analysis), given we already know from physical law that the equal division of the minimum produced uncertainty, (ħ/2)^1/2, is part of each term (meaning that it is only this part that is predicted from the equation, and even then, it's not clear why the total uncertainty ħ/2 should be shared evenly between terms if the reciprocal part of their terms should be so uneven). Perhaps I am ignorant, and (G/c³)^1/2 and (G/c⁵)^1/2 already have some known meaning in the general theory of relativity. (Further, these relativistic terms would be reciprocal, and therefore cancel out, with or without taking the square root of them, so it is not clear why this is done, either). Otherwise, assuming these values are special as Planck supposed, to the degree that Heisenberg's uncertainty principle can be derived from the Schrödinger equation, perhaps working backwards from the uncertainty relation, l_pp_p= ħ/2 (or t_pE_p = ħ/2), up to a wave-function will provide some hint for a theory uniting quantum physics and the general theory of relativity. My hope would be that it points to some dual between the matter-wave and the gravitational waves given off by accelerating matter, offering a natural gravitational pilot-wave.