In the entire history of physics there is absolutely no equation more famous than e = mc2. This relationship amongst mass (m) and energy (e) was discovered in 1905 by Albert Einstein from his Principle of Relativity. The derivation wasn’t simple and warranted a paper by itself, referred to as “Does the inertia of a body depend upon its energy content?”. The equation continues to baffle and mystify ordinary people, since in the usual particle picture of nature, it is tough to see exactly why there is an equivalence between mass and energy.

Meanwhile, a new theory called Quantum Field Theory was created. QFT was perfected in the 1950s by Julian Schwinger in five papers called “Theory of Quantized Fields”. In QFT there are absolutely no particles, there are only fields– quantized fields. Schwinger succeeded in positioning matter fields (leptons and hadrons) on an equal footing with force fields (gravity, electromagnetic, strong and weak), in spite of the obvious differences between them. Moreover, Schwinger developed the theory from fundamental axioms, as opposed to Richard Feynman’s particle picture, which he validated because “it works”. Unfortunately it was Feynman who won the battle, and today Schwinger’s method (and Schwinger himself) are mainly forgotten.

Yet QFT has many advantages. It has a stronger basis than the particle picture. It describes many things that the particle picture does not, including the many paradoxes associated with Relativity Theory and Quantum Mechanics, that have puzzled so many people. Philosophically, lots of people can accept fields as basic properties of space, as opposed to particles, whose composition is unknow. Or if there visualized as point particles, one can only ask “points of what?” And most of all, QFT provides a simple derivation and understanding of e = mc2, as follows.

**Mass.** In classical physics, mass is a measure of the inertia of a body. In QFT some of the field equations include a mass term that impacts the rate at which quanta of these fields evolve and propagate, slowing it down. Thus mass takes on the same inertial role in QFT that it does in classical physics. But this is not all it does; this exact same phrase causes the fields to oscillate, and the greater the mass, the higher the frequency of oscillation. The result, if you’re picturing these fields as a color in space (as in my book “Fields of Color”), is a sort of glimmer, and the greater the mass, the faster the glimmer. It might seem unusual that the same term that slows the spatial development of a field also causes it to oscillate, but it is actually straightforward mathematics to show from the field equations that the frequency of oscillation is given by f = mc2/h, where h is Planck’s constant.

**Energy.** In classical physics, energy means the capability to do work, which is defined as applying a force over a distance. This interpretation, however, doesn’t provide much of an image, so in classical physics, energy is a somewhat abstract idea. In QFT, on the other hand, the energy of a quantum is established by the oscillations in the field that makes up the quantum. As a matter of fact, Planck’s famous relationship e = hf, where h is Planck’s constant and f is frequency, found in the centennial year of 1900, follows directly from the equations of QFT.

Well, both mass and energy are associated with oscillations in the field, it doesn’t take an Einstein to see that there must be a relationship between the 2. In fact, any schoolboy can combine the 2 equations and find (big drum roll, please) e = mc2. Not only does the equation topple right out of QFT, its meaning can be visualized in the oscillation or “shimmer” of the fields. Nobel laureate Frank Wilczek calls these oscillations “a marvelous bit of poetry” that create a “Music of the Grid” (Wilczek’s term for space viewed as a lattice of points):.

“Instead of plucking a string, blowing through a reed, banging on a drumhead, or clanging a gong, we play the instrument that is empty space by plunking down various combinations of quarks, gluons, electrons, photons, … and allow them to settle until they get to stability with the spontaneous activity of Grid … These resonances represent particles of different mass m. The masses of particles sound the Music of the Grid.”.

This QFT derivation of e = mc2 is not typically known. Actually, I have never seen it in the publications I’ve read. And still I consider it one of the great accomplishments of QFT.

We take a closer look on the blog at Fields of Color!