User:Pao.christ/sandbox2

UNIFICATION OF COUPLING CONSTANTS AT THE PLANCK SCALE

The Plank charge turns out to differ considerably from  the elementary one.

$$ q_{P} \simeq 11.7 \; e       $$

where $$e$$ stands for the electric charge $$\simeq 1.6 \times 10^{-19} C$$. A seemingly poor prediction. The implications of this result seem to have been however overlooked or misunderstood. Indeed by simply squaring the above equation one obtains

$$ \alpha_{P}= 137 \; \alpha = 1 $$

Thus at the Planck scale the fine structure constant is one. Notice that the result does not depend on the Planck length which implies that the same holds true also for the corresponding potentials.

$$ \frac{GM^2_{P}}{R_{P}^2} = \frac{1}{4\pi \epsilon_{0}}\frac{q_{P}^2}{R_{P}^2} $$

However from the explicit form of the Planck mass ($$ M_{P} =\sqrt{\hbar c/G} $$) one immediately obtains

$$ \hbar c = 1 = \frac{q_{P}^2}{4\pi \epsilon_{0}} $$

Thus the previous result is not at all fortuitous but indeed tell us that the fine structure constant at Planck scales is indeed unity. So, in a sense, $$ GM^2_{P}$$ plays the role of an effective Planck coupling constant which determines the fine structure one.   This provides interesting results also for the weak and strong coupling constants at the  Planck scale.

As a matter of fact the relevant quantity being

$$ \frac{g^2}{4\pi} \frac{1}{q^2 - M^2} \to\frac{g^2}{4\pi} \exp(-M/r) $$ where $$M$$ stands both the the pion and for the weak boson mass, it is obvious that by the same considerations which led to the previous equation (this time for the potential ) one straightforwardly obtains

$$ \frac{GM^2_{P}}{R_{P}} = \frac{g^2 }{4\pi }\exp(-M/r) $$

At the Planck radius the exponential can be safely approximated to 1 so that

$$ GM^2_{P}= \frac{g^2 }{4\pi }$$

Thus

$$ \hbar c = 1 = \frac{g^2 }{4\pi}$$



The statement "that at the Planck length scale it has been theorized that all the fundamental forces are unified  but the exact mechanism of this unification remains unknown"    gets here a partial rebuttal as well as that "comparing masses and charges is like comparing apples with oranges".

Therefore in addition to the unit Planck coupling constants $$ G, \hbar, c = 1 $$ also the coupling constants of the different interactions $$ \alpha$$ appear  in their own right with the same value. It is remarkable how the effect works respectively to increase and decrease the present values of the coupling constants. Needless to say the above results, as well as the description of a black hole, depend on the assumption of the validity of the Coulomb and Newton equations. This is anyhow in line with the theoretical instruments used in the derivation of Planck units. Similar conclusions have  also been explicitly obtained in a description of gravitation below CMB as a QED black body thus unifying gravitation and electromagnetism whith a combined  strong interaction effect.