The simplest representation of a particle’s momentum–whether or not it has rest mass–is given by the wavelength operator. The need for a separate momentum operator arises only in formulas that use the rest mass operator.
The momentum operator has a value between 0 and 1 representing the inverse number of natural length units comprising the span between a particle’s Compton and de Broglie wavelengths.
The rest mass operator only quantifies the span from the Planck length to a particle’s Compton wavelength–the wavelength of a particle moving at the speed of light. For a particle with rest mass, the Compton wavelength represents its maximum length potential–its shortest possible wavelength at the limit of its velocity potential, 𝑐. Moving downward from the maximum potential, the particle’s wavelength increases in the same proportion as the decrease in its velocity. The interval between the particle’s Compton wavelength and its physical wavelength (the de Broglie wavelength) is quantified by the momentum operator.
The New Foundations Model shows the momentum operator as the ratio between the wavelength operator and the rest mass operator in the three unit dimensions.
The three correlated unit dimensions give three equivalent forms of the momentum operator–one in each unit dimension:
By itself, the momentum operator does not produce observable quantities of length, mass, and time since it only quantifies a portion of a particle’s wavelength and is not related to the Planck length, Planck mass, and Planck time. Combining the momentum operator with the rest mass operator produces the equivalent of the wavelength operator.
Relationship with other mechanical operators
The rest mass and momentum operators create an expanded form of the wavelength operator. Using these two operators in place of the wavelength operator is pragmatic because wavelength is a difficult property to measure in particles that have rest mass. It is even more difficult in large systems of particles; it is much easier to measure the mass and velocity of the system.
The relationship between wavelength, rest mass, and momentum operators is
Returning to the relational diagram of the mechanical operators, the relationship between the three operators is shown in blue, red, and green (also notated as β𝜆, β𝑚, and β𝑝 respectively). The momentum operator fills the span between the Compton wavelength–quantified by the rest mass operator–and the de Broglie wavelength.
The following illustrations express the relationship between the wavelength, rest mass, and momentum operators in each unit dimension.