Editing Classical unified field theories

{{Short description|Theoretical attempts to unify the forces of nature}}
Since the 19th century, some physicists, notably [[Albert Einstein]], have attempted to develop a single theoretical framework that can account for all the [[fundamental forces]] of nature – a [[unified field theory]]. '''Classical unified field theories''' are attempts to create a unified field theory based on [[classical physics]]. In particular, unification of [[gravitation]] and [[electromagnetism]] was actively pursued by several physicists and mathematicians in the years between the two World Wars. This work spurred the purely mathematical development of [[differential geometry]].

This article describes various attempts at formulating a classical (non-[[Quantum mechanics|quantum]]), [[theory of relativity|relativistic]] unified [[field theory (physics)|field theory]]. For a survey of classical relativistic field theories of gravitation that have been motivated by 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]].

==Overview==

The early attempts at creating a unified field theory began with the [[Riemannian geometry]] of [[general relativity]], and attempted to incorporate [[electromagnetic fields]] into a more general geometry, since ordinary Riemannian geometry seemed incapable of expressing the properties of the electromagnetic field. Einstein was not alone in his attempts to unify electromagnetism and gravity; a large number of mathematicians and physicists, including [[Hermann Weyl]], [[Arthur Eddington]], and [[Theodor Kaluza]] also attempted to develop approaches that could unify these interactions.<ref>{{cite journal |author=Weyl, H. |title=Gravitation und Elektrizität |journal=Sitz. Preuss. Akad. Wiss. |year=1918 |page=465}}</ref><ref>{{cite book |author=Eddington, A. S. |title=The Mathematical Theory of Relativity, 2nd ed. |publisher=Cambridge Univ. Press |year=1924 }}</ref> These scientists pursued several avenues of generalization, including extending the foundations of geometry and adding an extra spatial dimension.

== Early work ==

The first attempts to provide a unified theory were by [[G. Mie]] in 1912<ref>{{cite journal |author=Mie, G. |title=Grundlagen einer Theorie der Materie |journal=Annalen der Physik |year=1912 |volume=37 |pages=511–534 |doi=10.1002/andp.19123420306 |issue=3|bibcode = 1912AnP...342..511M |url=https://zenodo.org/record/1424223 }}</ref><ref name=MehraEinsteinHilbert>{{Cite book |last=Mehra |first=Jagdish |chapter=Einstein, Hilbert, and the Theory of Gravitation |editor-last=Mehra |editor-first=Jagdish |title=The physicist's conception of nature |date=1987 |publisher=Reidel |isbn=978-90-277-2536-3 |edition=Reprint |location=Dordrecht}}</ref>{{rp|115}} and Ernst Reichenbacher in 1916.<ref>{{cite journal |author=Reichenbächer, E. |title=Grundzüge zu einer Theorie der Elektrizität und der Gravitation |journal=Annalen der Physik |year=1917 |volume=52 |pages=134–173 |doi=10.1002/andp.19173570203 |issue=2|bibcode = 1917AnP...357..134R |url=https://zenodo.org/record/1424315 }}</ref> However, these theories were unsatisfactory, as they did not incorporate general relativity because general relativity had yet to be formulated. These efforts, along with those of Rudolf Förster, involved making the [[metric tensor]] (which had previously been assumed to be symmetric and real-valued) into an asymmetric and/or [[Complex number|complex-valued]] tensor, and they also attempted to create a field theory for matter as well.

== Differential geometry and field theory ==

From 1918 until 1923, there were three distinct approaches to field theory: the [[gauge theory]] of Weyl, Kaluza's [[Five-dimensional space#Physics|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 effort.

==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. He noted that in addition to a [[metric tensor|metric]] field there could be additional degrees of freedom along a path between two points in a manifold, and he tried to exploit this by introducing a basic method for comparison of local size measures along such a path, in terms of a [[gauge field]]. 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 [[Lagrangian (field theory)|Lagrangian]]s and [[Riemann curvature tensor|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. However, Weyl's principle of [[gauge invariance]] was later applied in a modified form to [[quantum field theory]].

==Kaluza's fifth dimension ==
Kaluza's approach to unification was to embed space-time into a five-dimensional cylindrical world, consisting of four space dimensions and one time dimension. Unlike Weyl's approach, Riemannian geometry was maintained, and the extra dimension allowed for the incorporation of the electromagnetic field vector into the geometry. 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. This theory did have some influence on Einstein's later work and was further developed later by Klein in an attempt to incorporate relativity into quantum theory, in what is now known as [[Kaluza–Klein theory]].

==Eddington's affine geometry ==

Sir [[Arthur Stanley Eddington]] was a noted astronomer who became an enthusiastic and influential promoter of Einstein's general theory of relativity. He was among the first to propose an extension of the gravitational theory based on the [[affine connection]] as the fundamental structure field rather than the [[metric tensor]] which was the original focus of general relativity. Affine connection is the basis for ''parallel transport'' of vectors from one space-time point to another; Eddington assumed the affine connection to be symmetric in its covariant indices, because it seemed plausible that the result of parallel-transporting one infinitesimal vector along another should produce the same result as transporting the second along the first. (Later workers revisited this assumption.)

Eddington emphasized what he considered to be [[epistemological]] considerations; for example, he thought that the [[cosmological constant]] version of the general-relativistic field equation expressed the property that the universe was "self-gauging". Since the simplest cosmological model (the [[De Sitter universe]]) that solves that equation is a spherically symmetric, stationary, closed universe (exhibiting a cosmological [[red shift]], which is more conventionally interpreted as due to expansion), it seemed to explain the overall form of the universe.

Like many other classical unified field theorists, Eddington considered that in the [[Einstein field equations]] for general relativity the [[stress–energy tensor]] <math> T_{\mu\nu} </math>, which represents matter/energy, was merely provisional, and that in a truly unified theory the source term would automatically arise as some aspect of the free-space field equations. He also shared the hope that an improved fundamental theory would explain why the two [[elementary particles]] then known (proton and electron) have quite different masses.

The [[Dirac equation]] for the relativistic quantum electron caused Eddington to rethink his previous conviction that fundamental physical theory had to be based on [[tensors]]. He subsequently devoted his efforts into development of a "Fundamental Theory" based largely on algebraic notions (which he called "E-frames"). Unfortunately his descriptions of this theory were sketchy and difficult to understand, so very few physicists followed up on his work.<ref name="Kilmister">{{cite book |author=Kilmister, C. W. |title=Eddington's search for a fundamental theory |url=https://archive.org/details/eddingtonssearch00kilm |url-access=registration |publisher=Cambridge Univ. Press |year=1994 }}</ref>

==Einstein's geometric approaches ==

When the equivalent of [[Maxwell's equations]] for electromagnetism is formulated within the framework of Einstein's theory of [[general relativity]], the electromagnetic field energy (being equivalent to mass as defined by Einstein's equation E=mc<sup>2</sup>) contributes to the stress tensor and thus to the curvature of [[space-time]], which is the general-relativistic representation of the gravitational field; or putting it another way, certain configurations of curved space-time ''incorporate'' effects of an electromagnetic field. This suggests that a purely geometric theory ought to treat these two fields as different aspects of the same basic phenomenon. However, ordinary [[Riemannian geometry]] is unable to describe the properties of the electromagnetic field as a purely geometric phenomenon.

Einstein tried to form a generalized theory of gravitation that would unify the gravitational and electromagnetic forces (and perhaps others), guided by a belief in a single origin for the entire set of physical laws. These attempts initially concentrated on additional geometric notions such as [[vierbein]]s and "distant parallelism", but eventually centered around treating both the [[metric tensor]] and the [[affine connection]] as fundamental fields. (Because they are not independent, the [[metric-affine gravitation theory|metric-affine theory]] was somewhat complicated.) In general relativity, these fields are [[symmetric]] (in the matrix sense), but since antisymmetry seemed essential for electromagnetism, the symmetry requirement was relaxed for one or both fields. Einstein's proposed unified-field equations (fundamental laws of physics) were generally derived from a [[variational principle]] expressed in terms of the [[Riemann curvature tensor]] for the presumed space-time [[manifold]].<ref>{{cite book |author=Einstein, A. |title=The Meaning of Relativity. 5th ed. |publisher=Princeton Univ. Press |year=1956 }}</ref>

In field theories of this kind, particles appear as limited regions in space-time in which the field strength or the energy density is particularly high. Einstein and coworker [[Leopold Infeld]] managed to demonstrate that, in Einstein's final theory of the unified field, true [[mathematical singularity|singularities]] of the field did have trajectories resembling point particles. However, singularities are places where the equations break down, and Einstein believed that in an ultimate theory the laws should apply ''everywhere'', with particles being [[soliton]]-like solutions to the (highly nonlinear) field equations. Further, the large-scale topology of the universe should impose restrictions on the solutions, such as quantization or discrete symmetries.

The degree of abstraction, combined with a relative lack of good mathematical tools for analyzing nonlinear equation systems, make it hard to connect such theories with the physical phenomena that they might describe. For example, it has been suggested that the [[torsion of connection|torsion]] (antisymmetric part of the affine connection) might be related to [[isospin]] rather than electromagnetism; this is related to a discrete (or ''"internal"'') symmetry known to Einstein as "displacement field duality".

Einstein became increasingly isolated in his research on a generalized theory of gravitation, and most physicists consider his attempts ultimately unsuccessful. In particular, his pursuit of a unification of the fundamental forces ignored developments in quantum physics (and vice versa), most notably the discovery of the [[strong nuclear force]] and [[weak nuclear force]].<ref>{{cite web |author=Gönner, Hubert F. M. |title=On the History of Unified Field Theories |work=Living Reviews in Relativity |url=http://relativity.livingreviews.org/open?pubNo=lrr-2004-2 |access-date=August 10, 2005 |archive-url=https://web.archive.org/web/20060209142203/http://relativity.livingreviews.org/open?pubNo=lrr-2004-2 |archive-date=February 9, 2006 }}</ref>

==Schrödinger's pure-affine theory ==

Inspired by Einstein's approach to a unified field theory and Eddington's idea of the [[affine connection]] as the sole basis for [[differential geometry and topology|differential geometric]] structure for [[space-time]], [[Erwin Schrödinger]] from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational theory. Although he initially assumed a symmetric affine connection, like Einstein he later considered the nonsymmetric field.

Schrödinger's most striking discovery during this work was that the [[metric tensor]] was ''induced'' upon the [[manifold]] via a simple construction from the [[Riemann curvature tensor]], which was in turn formed entirely from the affine connection. Further, taking this approach with the simplest feasible basis for the [[variational principle]] resulted in a field equation having the form of Einstein's general-relativistic field equation with a [[cosmological constant|cosmological term]] arising ''automatically''.<ref name="Schrödinger">{{cite book |author=Schrödinger, E. |title=Space-Time Structure |url=https://archive.org/details/spacetimestructu00esch |url-access=registration |publisher=Cambridge Univ. Press |year=1950 }}</ref>

Skepticism from Einstein and published criticisms from other physicists discouraged Schrödinger, and his work in this area has been largely ignored.

==Later work ==
{{unreferenced small|section|date=August 2018}}
After the 1930s, progressively fewer scientists worked on classical unification, due to the continued development of quantum-theoretical descriptions of the non-gravitational fundamental forces of nature and the difficulties encountered in developing a quantum theory of gravity. Einstein pressed on with his attempts to theoretically unify gravity and electromagnetism, but he became increasingly isolated in this research, which he pursued until his death. Einstein's celebrity status brought much attention to his final quest, which ultimately saw limited success.

Most physicists, on the other hand, eventually abandoned classical unified theories. Current mainstream research on [[unified field theory|unified field theories]] focuses on the problem of creating a [[quantum gravity|quantum theory of gravity]] and unifying with the other fundamental theories in physics, all of which are quantum field theories. (Some programs, such as [[string theory]], attempt to solve both of these problems at once.) Of the four known fundamental forces, gravity remains the one force for which unification with the others proves problematic.

Although new "classical" unified field theories continue to be proposed from time to time, often involving non-traditional elements such as [[spinor]]s or relating gravitation to an electromagnetic force, none have been generally accepted by physicists yet.

==See also==
*[[Affine gauge theory]]
*[[Classical field theory]]
*[[Gauge gravitation theory]]
*[[Metric-affine gravitation theory]]

==References==
{{reflist}}

{{DEFAULTSORT:Classical Unified Field Theories}}
[[Category:History of physics]]
[[Category:Theoretical physics]]