The product of length and mass is conserved in both particles and fields. Accordingly, the wavelength of an elementary particle has a corresponding inverse quantity of inertial mass.
Length-mass symmetry is shown in the illustration below as the correlation of length and mass potentials, where length potential is the inverse of its unit value. The equivalence of length and mass potential means that a dimensionless proportionality operator in one unit dimension can be applied to the other.
The wavelength operator quantifies a particle’s wavelength and inertial mass in the ratios
Derivation
Elementary forms of the Compton and de Broglie wavelength formulas give the following equality for all elementary particles, whether or not they have rest mass
Using a natural unit value of 1 for the Planck length and Planck mass gives the inverse relationship between wavelength and inertial mass
Since ππππ is constant, the formula can be stated in SI units as
= 
= 3.52 x 10-43 kgm
The conserved quantity 3.517 672 94… x 10-43 kgm has an exact value under the May 2019 redefinition of the International System of Units, in which Planck’s constant and the speed of light were given exact values. The exact value of ππππ is given by the formula
The inverse relationship between wavelength and inertial mass means that the product of Planck length and Planck mass is not only the maximum length-mass potential, but is a conserved quantity for every combination of wavelength and inertial mass. The following table summarizes the length-mass symmetry
Symmetry
Ζπ = ππππ
Potential
Invariant
Conserved quantity
3.52 x 10-43 kgm
The length-mass symmetry is reflected in non-relativistic quantities of particle wavelength and inertial mass, shown in the following table using the mechanical properties of photons. The table applies the wavelength operator ππ / Ζ to the Planck length and mass, producing observable quantities of wavelength and inertial mass proportional to the Planck scale. Quantities calculated using New Foundation Model operators are equivalent to historical formulas in which Planck’s constant is used.
| Potential: |  |  |  | |
| Operator: |  | | | |
| Observable: |  |  |  | |
| Natural formula: | ππ / Ξ²π | Ξ²π ππ | ||
| Equivalent historical formula: | β / π0π£ | β / Ζπ | ||
| Photon | dimensionless | m | kg | kgm |
| Ζπ | 1 | 1.62 x 10-35 | 2.18 x 10-8 | 3.52 x 10-43 |
| ΖπΆ, π | 1.46 x 10-19 | 1.11 x 10-16 | 3.17 x 10-27 | 3.52 x 10-43 |
| ΖπΆ, π | 8.65 x 10-21 | 1.87 x 10-15 | 1.88 x 10-28 | 3.52 x 10-43 |
| ΖπΆ | 4.19 x 10-23 | 3.86 x 10-13 | 9.11 x 10-31 | 3.52 x 10-43 |
| Ζ1π -2π | 1.33 x 10-28 | 1.22 x 10-7 | 2.89 x 10-36 | 3.52 x 10-43 |
| Ζ, Ξπ£πΆπ | 4.96 x 10-34 | 3.26 x 10-2 | 1.08 x 10-41 | 3.52 x 10-43 |
** The table shows the length form of the wavelength operator but mass and time forms can also be used
** Natural formulas shown in the table are the same for particles with and without rest mass
The length-mass symmetry is used in a number of equations to find wavelength from mass, and mass from wavelengthβincluding formulas for the Compton wavelength, de Broglie wavelength, and Schwarzschild radius.
The quantity 3.52 x 10-43 kgm is an important physical constant in the conservation of length-mass. The consistency of this relationship across the different elementary particles of the standard model suggests that a standard particle unit may underlie each of the elementary particles of the standard model. Because the length-mass relationship is conserved for individual particles–reflecting the quantization of particle fieldsβthe New Foundation Model refers to this constant as the quantum constant.
A more natural representation of physics quantifies momentum in unit dimensions of M and energy in unit dimensions LMT-1. In this representation, the quantum constant is equal to Planck’s constant.