The first stage of unified physics occurred when Isaac Newton (x-y) unified planetary motion and falling bodies on earth by his gravitational law. Then William Clerk Maxwell (x-y) unified electricity and magnetism and optics by his electromagnetic equations, which predicted radio waves. Next, Max Planck (x-y), by his quantum theory unified optics and mechanics. The topological implications of the latter have never been explored in the literature, perhaps because of the neglect of topology in physics. It is possible to explain this by use of another field neglected by physicists, dimensional analysis. (At another Website, boltzmann.htm, I show that Ludwig Boltzmann (x-y) might have discovered "Planck's constant" if examined dimensions of a constant of integration in deriving his law for the most probable distribution of molecules.)The FORMS of dimensional analysis (PL) can be used an an algebra for deriving other FORMS from basic FORMS. Thus, the Planck-Einstein Law and the de Broglie Law of QUANTICS -- mixing terms such as momentum, energy from MECHANICS with terms such as frequency, wavelengthfrom OPTICS-ELECTROMAGNETICS -- can both be derived from the photonic law: ln = c. The intermediary is the concept of ACTION, h, from Lagrange's Principle, applicable both in MECHANICS and OPTICS-ELECTROMAGNETICS.
To derive deB, we start with the Photonic Law (PL), which dimensionally is [ln] = [c] = [L T-1]. Compare with dimensional FORM of ACTION: [h] = [M L2 T-1] . Now, a dimensional part of the latter is [M L T-1], equivalent to that for momentum: [p] = [M LT-1]; and this becomes dimensionally equivalent to ACTION by bring in length: [h] = [p L]. And we can obtain a term of length via the wavelength of the photonic law: [l] = [L]. Hence, we have h = pl , which is de Broglie's Law .
To derive PE, we start with the knowledge that light involves kinetic energy , which we wish to introduce into some part of PL. Now, dimensionally [E] = [M L2 T-2]. Using Planck's CONSTANT as intermediary: [h] = [(M L2)T-1]. Thus: [E] = [hT-1]. Taking WAVELENGTH from PL, we have: [El] = [hLT-1]. On the right, we can use the PL VACCUM LIGHT SPEED, c: [El] = [hc], for the ANTITONIC PE formula, El = hc. Converting to the standard form of PE, first take E = h(c/l), and the parenthetical expression is Photonically c/l = n, yielding the standard PE form, E = hn.
So, we find, by DIMENSIONAL ALGEBRA, that DB and PE are IMPLICIT in PL, mixing MECHANICS with OPTICS and ELECTROMAGNETICS. (What else is possible by DIMENSIONAL ALGEBRA? And what are the topological implications of this?)