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 can be based on Maxwell’s equations, a set of mathematical functions that describe how electro-magnetic waves travel through empty space to create electrical and magnetic effects from afar. Yet they aren’t a theory of how that happens, as they assume a complex dimension that doesn’t physically exist.
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 the mass of a body like the earth, but it isn’t a theory of gravity, so it puzzled even Newton, and he wrote it. Scientific laws describe how observed facts work while scientific theories explain why, so theories aren’t laws. For example, a germ theory of disease doesn’t need equations to work as a theory. In general, theories are theories and laws are laws, so they are always different.
Maxwell’s equations then describe how light waves travel 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 gravity is caused by gravitons, nuclear binding by gluons, and neutron decay by weak particles must then stand as such, apart from the equations it explains.
The difference is that theories should predict new facts but equations just 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 tell us anything new about charge, it just avoided the idea of action at a distance.
Attempts were then made to develop a mathematical model based on field theory, hoping for a Theory of Everything, or TOE. The result was 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 a theory that could predict anything, and so it predicted 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.
Even worse, a theory that predicts everything 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. Thousands of scientific papers were then written on a theory of dubious scientific merit, so how did this happen?
A field that extends across all space adds a degree of freedom to it, so adding a new field equates to adding a dimension to space. Based on field theory, gravity then adds one dimension, electro-magnetism adds two, the strong force three, and the weak force two. These eight extra dimensions plus the three of space require M-theory to assume eleven dimensions. The problem then lies in the standard model’s invention of new fields, not M-theory itself.
The answer seems to be that in science, assuming a fact to explain a fact isn’t progress. In business, borrowing $100 to make a $100 isn’t profitable, and the same is true for science, but the strategy of assuming a new field to explain a new force has a long history in particle physics.