The Factor 2 in Fundamental Physics

A brief history is given of the factor 2, starting in the most elementary considerations of geometry and the kinematics of uniform acceleration, and moving to relativity, quantum mechanics and particle physics. The basic argument is that in all the significant cases in which the factor 2 or 1⁄2 occurs in fundamental physics, whether classical, quantum or relativistic, the same physical operation is taking place. 1 Geometry and kinematics We probably first come across the factor 2 in the formula for the triangle: area = length of base × perpendicular height 2 . This is an ancient formula, well-known to Egyptian, Babylonian and Chinese mathematicians. In the case of a right-angled triangle, it is clearly created by bissecting a rectangle along a diagonal. If we now take this as representing a straightline graph, of, say, velocity against time, under a uniform acceleration, the area under the graph becomes the distance travelled. For an object increasing its velocity uniformly from 0 to a value v in time interval t, the area under the graph, or distance travelled, using the triangle formula, becomes vt / 2. By comparison, if the object had travelled at steady speed v throughout the time interval t, the distance travelled would be the area of the rectangle under the horizontal straight line representing steady v, that is, vt. In effect, the factor 2 distinguishes here between steady conditions and steadily changing conditions. It was by this means that the factor first entered into physics from pure mathematics, via the Merton mean speed theorem, evolved in fourteenth-century Oxford. This result, which ultimately proved to be the foundation theorem of modern dynamics, showed that the total distance moved by a body during uniform acceleration was the same as that covered during the same time interval by a body travelling uniformly at the speed measured at the middle instant of the accelerated motion. In more modern terms, the total distance travelled under uniform acceleration must equal the product of the mean speed and the time. Mathematically, if we start with initial speed u and steadily accelerate to a final speed v over the time interval t, then the total distance travelled will be given by