Appendix: Planck Scale

This discussion is presented as a basic primer related to the Planck scale and the constants [c, h, G] defined below, which are used to form a system of units known as ‘Planck Units’. Collectively, these units provide a framework that operates on the Planck scale on which the perceived ‘weirdness’ of the quantum universe operates. However, this discussion will only present a minimal subset of all Planck parameters, but will first introduce the fundamental units of classical physics defined in terms of length and time plus mass and charge.

Unit Symbol MKS Symbol
length x meter m
time t second s
mass m kilogram kg
charge q Coulomb C

Note: Whether mass and charge really represent fundamental units may be questioned by any form of WSE model, where the existence of a mass particle is replaced by some sort of energy-wave structure. If so, then mass might be replaced by energy in Joules, while charge could be an emergent property of a wave field between two specific types of energy-wave structure, which are generally perceived as particles with charge,

Using the MKS units defined above, but initially restricted to metres (M), kilograms (K) and seconds (S), we might initially quantify the 3 physical constants mentioned in the introduction above, i.e.

Constant Symbol Value Units
speed of light c 2.998E+08 m/s
Planck constant h 6.626E-34 m2kg/s
gravitational constant G 6.673E-11 m3/s2kg

While we might understand that the physical constant [c=m/s] can be interpreted as the propagation velocity of light, it may also be seen as a conversion factor linking energy [E] and mass [m] in Einstein’s equation:

[1]             

In this respect, [h] might also be introduced as the conversion factor between energy [E] and the frequency [f] of radiation, which implicitly suggests some form of wave structure with both frequency [f] and wavelength [λ]. This concept is essentially encapsulated in Planck’s energy equation [E=hf] rather than Einstein’s [E=mc2], although it is possible that the constant [h] is hiding some of the attributes of a wave, i.e. the relationship between energy [E] and the amplitude [A] of the wave.

Note: Again, we will simply outline how the constant [h] might be ‘hiding’ wave attributes, if we follow the logic of simple harmonic oscillation, its potential energy [Ep] is equated to an elastic constant [k] and the amplitude [A] offset. In turn, [k] is equated to the perceived mass [m] in oscillation and the angular frequency [ω], which for the purposes of this note might be presented as the frequency [f] of oscillation.

For now, [2] below simply shows how energy [E] is related to [hf] as per Planck’s equation without any indication of how energy, as a scalar quantity, is propagating through space [in metres] as a function of time [seconds].

[2]             

Finally, there is the physical constant [G] that also acts as a conversion factor between the gravitational energy [Ep] and the masses [M,m] as defined by Newton’s law of gravitation.

[3]             

As described, the 3 physical constants [c, h, G] can form the basis of a system of Planck units to be outlined below. However, before addressing this system of units, we might also introduce two more physical constants that will feature in the description of many model, as they are seen as physical constants of spacetime:

Vacuum permeability µ0 1.257E-06 kg.m/C2
Vacuum permittivity ε0 8.854E-12 C2s2/kg.m3

In terms of Maxwell’s equations, there is a defined relationship between vacuum permeability [μ0], vacuum permittivity [ε0] and the velocity of EM wave propagation [c], which we might simply note at this point:

[4]             

Note: As a simply speculative comment, might the constants defined above actually be variables that might have changed over time? If so, might the speed of light [c] also be a variable rather than a constant? Clearly, such an idea might make extrapolating the laws of physics back in time to the early universe increasingly problematic.

Within this expanding framework, the standard particle model might be described in terms of just 3 particles, i.e. electrons, protons and neutrons with an associated mass [me, mp, mn] and possibly Compton wavelengths [λe, λp, λn]. The electron and proton are also said to have equal, but opposite signed charge [e], although some of these assumptions may be questioned further within any wave model.

rest mass of electron me 9.109E-31 kg wavelength of electron λe 2.426E-12 m
rest mass of proton mp 1.673E-27 kg wavelength of proton λp 1.321E-15 m
rest mass of neutron mn 1.675E-27 kg wavelength of neutron λn 1.320E-15 m

Note: In 1897, J. J. Thomson discovered a negatively charged particle, i.e. the electron, with a mass about 1840 times smaller than that of a hydrogen atom, when measured in terms of its mass-to-charge ratio. However, it was only after Robert Millikan observed the movement of oil droplets in an electric field and determined the charge of an electron [e] was its mass [me] estimated.

Based on equating the Einstein and Planck energy equations, we might perceive a relationship between the particle mass and the Compton wavelength can be defined by combining equations [1] and [2] as follows:

[5]             

In [5], we see the definition of the Compton wavelength for an electron based on the substitution of the electron mass [me]. Of course, from a classical perspective, or even special relativity, we might question what aspect of a particle has an associated velocity of the speed of light [c], although it might simply be assumed that [c] is only being used as an energy conversion factor.

But does [c] have any physical significance?

From the perspective of any wave model, we might assume that the waves involved in the particle structure are actually propagating with velocity [c] because this is a property if the wave media of spacetime. However, this then leads to another problem associated with the deBroglie wavelength of a matter wave, e.g. an electron, given in [6] below,

[6]             

In terms of [6], the assumption is that the energy is now associated with the kinetic energy of the wave-particle, i.e. it is a function of [v] not [c]. Likewise, we might assume that the deBroglie matter wave also has a propagation velocity [v=fλ]. Of course, unlike the specific value of the Compton wavelength given in [5], the deBroglie wavelength can only be defined for the electron, when its kinetic velocity [v] is known, although this statement might be challenged by the Quantum Wave Interpretation and the perception of Wave Dispersion.