Einstein spent the end of his career in a fruitless pursuit of a mathematical formulation that would unify the theories of electromagnetism and gravitation. His search was motivated by a simple proposition: the known interactions appear to occur simultaneously, and therefore there should be a mathematical description that allows us to calculate all of the effects at once. The challenge comes in developing a theory that conforms to mathematical transformations that correspond to assumed equivalences between physical systems (known as invariances in the jargon).
For example, in developing General Relavitity, Einstein observed that a man standing in a constantly accelerating elevator cannot prove that he is not standing on a planet. Since a light beam piercing the side of the elevator would appear to curve minutely as it traversed the elevator, this must mean that the beam would also follow a curved path in a gravitational field.
Remarkable mathematical consequences were evolved by Einstein from such simple postulates. That those consequences were validate by physical measurement is a testament to the importance of such postulates in guiding theoretical developments.
The mathematical constrains evolved from physical invariances are essential guides to the development of new theories. They constrain the imagination, guiding theorists away from magical thinking.
Among the most important of the guiding principles of modern particle theory are the following:
Causality - meaning, in effect, that all of the observable behavior of fermions is mediated by gauge fields, which, if massless, move at the speed of light.
Symmetry. The presumption that, at root, all fermions are interchangeable, and all gauge fields are interchangeable. The theory presumes that movement along spatial dimensions is mediated by gauge fields, so we have:
A mathematical foundation in group theory, which explores the dimensionality of mathematical objects, and the nature of the mathematical objects that regulate their inter-transformations.
Fermions in families of 16.
26-dimensional gauge fields, 10 of which correspond to spatial dimensions.
Symmetry breaking - the presumption that the diversity of physics arises because some of the gauge fields have mass.
This is a prejudice that arises solely out of theoretical success in applying group theory to field theory.
There are certain mathematical constraints that were adopted in achieving a successful unification. These include:
A particle's charges are constrained to the same position, but oscillate on a one-dimensional topological "string".
Predictions for energetically suppressed physics - both particles and fields - at least one order of magnitude more complex than the actual observable physics.
Neither of these last two assertions is directly falsifiable.