another by a Lorentz scalar: 4-velocity , where is the proper time 4-momentum , where is the rest mass 4-wavevector , which is the 4-vector version of the Planck–Einstein relation & the de Broglie matter wave relation 4-gradient , which is the 4-gradient version of complex-valued plane waves Now, just apply the standard Lorentz scalar product rule to each one: The last equation (with the 4-gradient scalar product) is a fundamental quantum relation. When applied to a Lorentz scalar field , one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations: The Schrödinger equation is the low-velocity limiting case () of the Klein–Gordon equation. If the quantum relation is applied to a 4-vector field instead of a Lorentz scalar field , then one gets the Proca equation: If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation: More complicated forms and interactions can be derived by using the minimal coupling rule: As a component of the RQM covariant derivative (internal particle spaces) In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist. The version known from classical EM (in natural units) is: The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is: or where the scalar product summations () here refer to the internal spaces, not the tensor indices: corresponds to U(1) invariance = (1) EM force gauge boson corresponds to SU(2) invariance = (3) weak force gauge bosons (i = 1, …, 3) corresponds to SU(3) invariance = (8) color force gauge bosons (a = 1, …, 8) The coupling constants are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the are fixed for one representation, they are known for all representations. These internal particle spaces have been discovered empirically. Derivation In three dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to 4 dimensions should be: which is incorrect. However, a line integral involves the application