Three routes to relativistic kinematics and time dilation and their connection to quantum-mechanical path integrals

The first calculation, similar to that of Lewis of 1908, takes as postulates Newton's Second Law of Motion and work-energy equivalence, the second makes use of the Hamilton-Jacobi equation for motion in free space and the third identifies the velocity of a freely moving particle with the group velocity of its associated de Broglie wave. In all cases the equivalence of mass and energy is also postulated. How the Path Integral formulation of quantum mechanics relates all the different calculations is demonstated. The essential roles of relativity, quantum mechanics and the particulate nature of light in the establishment of the current standard units of length, time and mass, are discussed, as well as the similar relations of the fundamental constants $c$, $h$ and $k$ to energy at the microscopic physics level.