In natural unit of [cm] (the natural unit is enclosed within the bracket [ ]) this equation is in the form :
2E - 
E = 0Conversion back to cgs units in this case is very simple by substituting the natural unit [cm] with the cgs conversion factor c for the time variable. There is no conversion involved with the spatial variables. Then the equation in cgs units is recovered as:
2E - (1/c2)E = 0E = (m2 + p2)1/2
In natural unit of [erg], m is converted back to cgs by c2, while the conversion for p is c. Therefore the formula in cgs is :
E = (m2c4 + p2c2)1/2
In natural unit it has the form : -(1/2m)
= i
In converting from the natural unit of [erg], the left-handed side produces a factor of (1/c2)(c
)2=2. On the right-handed side the converting factor is . Combining everything together, we recovered the equation in cgs :-(
2/2m) = iIncidentally, for all cases, the same form in cgs units will be recovered whether the natural unit is in [cm] or [erg].
In natural unit with the energy E in [erg] :
= exp[i(kx-Et)]. Backward conversion to cgs becomes = exp[i(kx-Et/)].- Planck Scale - EPl = (8
G)-1/2 which has a dimension of [erg] in natural unit. Substituting the numerical values and converting the result from erg to ev yields EPl = 0.8x103 [erg] = 5x1014 [ev] which is incorrect. The proper way is to perform a reversed conversion by multiplying (c5)1/2 (see Table) to arrive at the correct answer of EPl = 2x1027 ev. This is the "Reduced Planck Energy" with a factor of (8)-1/2 difference from the conventional definition (see "Planck Scale"). The natural unit can also be expressed in term of [cm-1] so that EPl = 0.8x103 [cm-1]. It becomes EPl = 0.8x103x(c4/c)1/2 cm-1 = 1.5x1032 cm-1 by reversed conversion, the reciprocal of which is the length scale LPl = 0.7x10-32 cm (see "Scale of the Universe"). - Fermi Scale - EF = 1/(GF)1/2 = [Gev]/(1.17x10-5)1/2 = 3x1011 ev. The length scale is obtained by converting ev to erg, and then divided by c (changing the unit to cm-1), the reciprocal of which is LQCD = 0.6x10-16 cm. The conversion for this one and the next is straight forward because the natural unit is in unit of energy already.
- QCD Scale - The natural unit for this one is just the coupling constant EQCD =
QCD = 2x108 ev. The length scale is obtained by converting ev to erg, and then divided by c to change the dimension to cm-1, the reciprocal of which is LQCD = 10-13 cm. - Vacuum Scale - Evac = (
vac)1/4 [erg]. Conversion back to cgs units involves multiplying (c53) to vac = 0.7x10-29, taking the 1/4 root, and translating to ev. The result is Evac = 1.5x10-3 ev. The length scale is obtained by multiplying vac (in the alternate natural dimension of [cm-1]) with (c/), then takes the 1/4 root, the reciprocal of which yields Lvac = 0.7x10-2 cm. - Hubble Scale - EH = (8G)1/2 in natural unit of [erg]. Conversion to cgs units involves taking the 1/2 root of 8G (with = 10-29), multiplying , then translating to ev. The result is EH = 2.5x10-33 ev. The multiplication factor is 1/c (when it is in natural unit of [cm-1]) in deriving the inverse length scale, and then takes the reciprocal LH = 0.8x1028 cm.
BTW, the Hubble equation is originally derived from General Relativity, i.e., H =(dR/dt)/R = (8G/3)1/2 for k = 0 in cgs unit of sec-1. It has nothing to do with , which comes from quantum theory. However, in an effort to express this equation in unit of energy, it is artificially multiplied by and declared that (8G)1/2 is in natural unit of [erg] (also taking away a factor of 1/3). It is rather doubtful whether the uncertainty principle is applicable to such macroscopic object. Anyway, in a similar vein, it is multiplied by 1/c to pretend that it is in natural unit of [cm-1].