According to Feynman:
“A real field is a mathematical function we use for avoiding the idea of action at a distance.” (Feynman, Leighton, & Sands, 1977), Vol. II, p15-7.
For example, the electro-magnetic field based on Maxwell’s equations is a mathematical function that describes how electro-magnetic waves travel through empty space to create electrical and magnetic effects. Yet it isn’t a theory of how that happens because those equations require a complex plane that doesn’t exist physically, so it is a law of physics not a theory of physics.
In science, an equation like E=mc² is a mathematical law that relates data facts, usually by an equals sign, while a scientific theory explains those facts. For example, the law of gravity is an equation that relates gravity to mass but it isn’t a theory of gravity, so it puzzled even Newton who wrote it. Scientific laws specify how observed facts work while scientific theories explain why, so they aren’t the same. For example, the germ theory of disease has no equations but still works as a theory. In general, theories are theories and laws are laws, and they are always different.
Maxwell’s equations then describe how light travels but don’t explain it in theory terms, and the same is true for quantum electrodynamics (QED), its quantized extension. But when the standard model adds that the equations work because the electro-magnetic field emits photons, this is a theory. The theory that gravitons cause gravity, gluons cause nuclear bonds, and weak particles cause neutron decay must then stand on its own, apart from the equations it explains.
Theories survive by predicting new facts while equations just have to interpolate between the facts they are based on. This ability, to produce new knowledge, is how theories and equations differ, but the theory that virtual photons cause charge effects didn’t reveal anything new about charge, it just avoided the idea of action at a distance.
Attempts to develop field theory mathematics into a Theory of Everything led to superstring theory, then Witten’s M-theory, which assumes our space has eight extra dimensions, curled up so we can’t see them. Unfortunately, the result was equations that can predict anything, and so predict nothing, as Woit explained decades ago:
The possible existence of, say, 10500 consistent different vacuum states for superstring theory probably destroys the hope of using the theory to predict anything. If one picks among this large set just those states whose properties agree with present experimental observations, it is likely there still will be such a large number of these that one can get just about whatever value one wants for the results of any new observation. (Woit, 2006), p242.
M-theory can predict whatever you want, so it can’t be falsified, which is bad news in science. That a universe of eleven dimensions somehow collapsed into our three-dimensional world is untestable because no experiment can deny it. Good science is both fruitful and falsifiable but M-theory was neither, so that it led nowhere is no surprise, yet thousands of scientific papers were written on it!
A field that extends across all space adds a degree of freedom to it, so adding a field to space equates to adding a dimension to it. Based on field theory, gravity adds one dimension, electro-magnetism adds two, the strong force three, and the weak force two. Eight extra dimensions, plus three of space, require M-theory to have eleven dimensions, which interact to make anything possible, so M-theory failed because the standard model invented fields. M-theory became a theory of nothing because in science, assuming a fact to explain a fact isn’t profitable, just as borrowing $100 to make a $100 isn’t a profit in business. Yet this strategy of explaining by assuming has a long history in particle physics.