Classical unified field theories
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The goal of creating a unified field theory based on classical physics was initially most closely identified in the public mind with Albert Einstein. He was, however, by no means the only researcher to attempt such a unification of the physical forces. in the first half of the twentieth century. Unification of gravitation and electromagnetism was an actively pursued by several physicists and mathematicians in the years between the two world wars.
Most scientists, though not Einstein, eventually abandoned classical theories. Current research on unified field theories focuses on the problem of creating quantum gravity and unifying such a theory with the other fundamental theories in physics, which are quantum theories. (Some programs, most notably string theory, attempt to solve both of these problems at once.)
This article describes various attempts at a single classical, relativistic field theory. This work spurred the purely mathematical development of differential geometry.
For a survey of classical relativistic field theories of gravitation (which have been motivated by various theoretical concerns other than unification), see Classical theories of gravitation. For a survey of current work toward creating a quantum theory of gravitation, see quantum gravity.
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Overview
The early attempts at creating a unified field theory began with the geometry of general relativity, and attempted to incorporate electromagnetic fields into a generalized geometry. Despite popular misconceptions, Einstein was not alone in his attempts to unify electromagnetism and gravity. Hermann Weyl, Arthur Eddington, Theodor Kaluza and R. Bach also attempted to unify these interactions. These scientists pursued four different avenues of generalization: generalizing geometry, dynamics, number field, and the addition of an extra spatial dimension. These avenues were explored separately or together.
Early work
The first attempts to provide a unified theory were contributed by G. Mie, in 1912, and Ernst Reichenbacher, in 1916. However, these theories were incorrect, as they did not incorporate general relativity - in the former case, because general relativity had yet to be formulated, and in the latter due to an appararent lack of understanding. These efforts, along with those of Forster, involved making the metric tensor, which had been symmetric and real, into an asymmetric and/or complex tensor, and also attempted to create a field theory for matter as well.
Differential geometry and field heory
From 1918 until 1923, there were three distinct approaches to field theory: the gauge theory of Weyl, Kaluza's five-dimensional theory and Eddington's development of affine geometry. Einstein corresponded with these researchers, and collaborated with Kaluza, but was not yet fully involved in the unification.
Weyl's infinitesimal geometry
In order to include electromagnetism into the geometry of general relativity, Hermann Weyl worked to generalize the Riemannian geometry upon which general relativity is based. His idea was to create a more general infinitesimal geometry. This geometry generalized Riemannian geometry in that there was a vector field Q, in addition to the metric g, which together gave rise to both the electromagnetic and gravitational fields. This theory was mathematically sound, albeit complicated, resulting in difficult and high-order field equations. The critical mathematical ingredients in this theory - the Lagrangians, and curvature tensor - were worked out by Weyl and colleagues. Then Weyl carried out an extensive correspondence with Einstein and others as to its physical validity, and the theory was ultimately found to be physically unreasonable.
Kaluza's fifth dimension
Kaluza's approach to unification was to embed space-time into a five-dimensional cylindrical world; one of four space dimensions and one of time. Unlike Weyl's approach, Riemannian geometry was maintained, and the extra dimension allowed for the incorporation of the electromagnetic field vector into the geometry. Unfortunately, despite the relative mathematical elegance of this approach, in collaboration with Einstein, and Einstein's aide Grommer, it was determined that this theory did not admit a non-singular, static, spherically symmetric solution. Although, this theory influenced Einstein's later work and was further developed later by Klein, in an attempt to incorporate relativity into quantum theory, in what is now know as Kaluza-Klein theory.
Eddington's Affine Geometry
Einstein's Geometric Approaches
Later Work
After the 1930's, fewer and fewer scientists worked on classical unification, due to the continual development of quantum theory, and the difficulties encountered in developing a quantum theory of gravity. Einstein continued to work on a theory to incorporate electromagnetism, but he became increasingly isolated in his research over a unified field theory of gravity and electromagnetism until his death. Despite the publicity of this work, due to Einstein's celebrity status, it was a series of unsuccessful attempts. Even now with four fundamental forces, gravity remains the one force whose unification proves problematic.