Technical Notes 2006

A Revised Planck Scale [November 2006]

I. The Planck Length

An important scale factor in high-energy physics and quantum field theories is the Planck length.  This is the length scale at which General Relativity and Quantum Mechanics are both necessary for adequately modeling natural phenomena.

The equation for the Planck length (R_Pl) is:

R_Pl = (ħG/c³)^{1/2 ,}

where ħ is Planck’s constant divided by 2π, G is the gravitational constant and c is the velocity of light.

The standard (pre-SSCP) calculation of R_Pl uses the familiar Newtonian gravitational constant G, and gives:

R_Pl  ≈  1.62 x 10^-33 cm   ≈  10^-33 cm.

There is definitely a problem with this calculation as it stands.  The SSCP emphasizes that dimensional “constants” are different for each cosmological Scale.  Because c has dimensions of L/T, it scales as Λ/Λ = 1, so that it is the same on all Scales.  Planck’s constant (ħ) is an Atomic Scale value, so that is ok as it is.  However, it is not correct to use the Stellar Scale G_Ψ=0 @ 6.67 x 10^-8 cm³/g sec² for calculations involving Atomic Scale “constants”.  The correct G_Ψ=-1 value to use is:

G_Ψ=-1   ≈   Λ^2.174 G_Ψ=0   ≈   2.18 x 10³¹ cm³/g sec².

Substituting the correct values into the Planck length equation gives:

R_Pl   ≈   [(1.054 x 10^-27 cgs)(2.18 x 10³¹ cgs)/2.673 x 10³¹ cgs]^1/2   ≈   2.934 x 10^-14 cm.

Since the charge radius of the proton (r_p) is roughly 8 x 10^-14 cm, we have:

R_Pl   ≈   0.4 r_p .

This represents a huge change (19 orders of magnitude!) in thinking about field theory descriptions of nature.

(a)    The length scale at which quantum and gravitational physics are of comparable importance is not “way down” at 10^-33 cm, but rather already occurs on the size scale of the proton, at ≈ 10^-13 cm.

(b)   According to the SSCP, this balancing of quantum interactions and general relativistic interactions occurs at discrete intervals throughout nature’s infinite hierarchy and is intimately related to the division of the hierarchy into discrete Scales.  This, however, is largely unexplored territory.

II. The Planck Mass

The standard formula for the Planck mass is:

M_Pl = (ħc/G)^1/2 .

Using ħ, c and G_Ψ=0, i.e., 6.67 x 10^-8 cm³/g sec², M_Pl ≈ 2.17 x 10^-5 g, which corresponds to nothing I know of in nature.

On the other hand, if we follow the reasoning of the SSCP, then we realize that the previous calculation should have used G_Ψ=-1, i.e., 2.18 x 10³¹ cm³/g sec².  When the correct gravitational “constant” is used, we get:

M_Pl  ≈  1.2 x 10^-24 g  ≈  0.72 m_p.

As was the case with the Planck length, the revised SSCP Planck mass is fairly close to a basic property of the proton, whose mass (m_p) is ≈ 1.6 x 10^-24 g.  Therefore, the revised Planck mass proposed by the SSCP appears to be closely associated with one of the fundamental building blocks of nature: the proton.

III. The Planck Time

The Planck time (t_PL) is found by the following formula:

t_PL = (ħG/c⁵)^1/2.

Using the correct G_Ψ=-1 value gives:

t_PL ≈ 9.81 x 10^-25 sec,

which is about 0.4 times the time required for light to transit the radius of the proton.

IV. The Planck Charge

The equation for the Planck charge (q_Pl) is:

q_Pl = (ħc4πε_o)^1/2.

Since this calculation does not involve G, the standard value is correct, as is.

Note that once again the Planck value is closely associated with the proton:

q_Pl  ≈  11.7 e ≈ α^-1/2e  ≈ α^-1/2 q_proton,

where a is the fine structure constant and e is the unit of charge for the proton or the electron.  If a is included in the derivation of the Planck charge, i.e.,

q_Pl = (αħc4πε_o)^1/2,

then:

q_Pl = q_proton .

V. Conclusions

The SSCP shows us that the correctly scaled Planck length and Planck mass are very close to the radius and mass of the proton.  These are the length and mass scales at which quantum electrodynamics and general relativity are both crucial to modeling nature.

This result will have a revolutionary impact upon high-energy physics and efforts to create a quantum field theory, once the revised Planck Scale is acknowledged.

Addendum: Mass, Radius and Angular Momentum of the proton

Elsewhere on the website it has been emphasized that the Schwarschild radius for the proton, when the correct G_Ψ=-1 value is used, gives:

R_Sch,p  ≈  2G_Ψ=-1 m_p/c²  ≈  0.82 x 10^-13 cm.

This is very close to the radius of the proton: charge radius (r_p) is about 0.8 x 10^-13 cm, so:

R_Pl [0.3 x 10^-13 cm]  ≈  r_p [0.8 x 10^-13 cm]  ≈  R_Sch,p [0.8 x 10^-13 cm].

Since (ħG_Ψ=-1/c³)^1/2  ≈  2G_Ψ=-1 m_p/c²,

ħ  ≈  k(m_p)²,

where k  ≈  4G_Ψ=-1/c   ≈   2.92 x 10²¹ cm²/g sec.