08/01/2015
Classical unified field
theories
Since the 19th century, some physicists
have attempted to develop a single
theoretical framework that can account for
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 World War I and World War
II. This work spurred the purely
mathematical development of differential
geometry . Albert Einstein is the best known
of the many physicists who attempted to
develop a classical unified field theory.
This article describes various attempts at a
classical (non- quantum ), relativistic unified
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, Theodor
Kaluza , and R. Bach also attempted to
develop approaches that could unify these
interactions. [1][2] 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 and Ernst
Reichenbacher in 1916. [3][4] 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 Forster, involved making the metric
tensor (which had previously been assumed
to be symmetric and real-valued) into an
asymmetric and/or 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 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 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
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. 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; 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. 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 , 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. [5]
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 one
would expect from Einstein's famous
equation E=mc 2 ) 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
vierbeins 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 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.[6]
In field theories of this kind, particles
appear as limited regions in space-time in
which the field strength or the energy
density are particularly high. Einstein and
coworker Leopold Infeld managed to
demonstrate that, in Einstein's final theory
of the unified field, true 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 (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.[7]
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 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 term
arising automatically .[8]
Skepticism from Einstein and published
criticisms from other physicists discouraged
Schrödinger, and his work in this area has
been largely ignored.
Later work
After the 1930s, progressively 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 unified field theories of gravity and
electromagnetism, but he became
increasingly isolated in this research, which
he pursued until his death. Despite the
publicity of this work due to Einstein's
celebrity status, it never resulted in a
resounding success.
Most scientists, though not Einstein,
eventually abandoned classical theories.
Current mainstream research on unified
field theories focuses on the problem of
creating a quantum theory of 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.) With four
fundamental forces now identified, gravity
remains the one force whose unification
proves problematic.
Although new "classical" unified field
theories continue to be proposed from time
to time, often involving non-traditional
elements such as spinors , none has been
generally accepted by physicists.
