On the Fundamental Masses Derivable by Dimensional Analysis

Dimensional Analysis (DA) is utilized to identify fundamental masses in terms of four fundamental constants that describe, respectively, the relativistic, quantum, gravitational, and cosmological aspects of the Universe. Gaussian elimination is employed to obtain ten pairs of dimensionless products. Any two products paired together are independent of each other, and each of them serves as a regime characterized by a single output variable. In four out of these ten pairs, mass is one of the two output variables, and hence is expressible in terms of the other four constants taken three at a time. More importantly, we deduce that DA cannot rule out possibilities for fundamental masses that depend on all other four constants taken together. Techniques other than DA are needed to identify, interpret and calculate numerical values for such masses, and to demonstrate their fundamental physical significance.