Gauge gravitation theory
Encyclopedia
In quantum field theory
, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity
.
The first gauge model of gravity was
suggested by R. Utiyama in 1956 just two years after birth of the
gauge theory
itself. However, the initial attempts to construct the
gauge theory of gravity by analogy with the
gauge models of internal symmetries encountered a problem of treating general covariant transformations and
establishing the gauge status of a pseudo-Riemannian metric (a
tetrad field).
In order to overcome this drawback, representing tetrad
fields as
gauge fields of the translation group was
attempted. Infinitesimal generators
of general covariant transformations were considered as those of
the translation gauge group, and a tetrad (coframe) field was
identified with the translation part of an affine connection
on a world manifold
. Any such connection is a sum 
of a linear
world connection
and a soldering form
where
is a non-holonomic frame. For instance, if 
is the Cartan
connection, then
is
the canonical soldering form
on
.
There are different physical interpretations of the translation
part
of affine connections
. In
gauge theory of dislocations, a field
describes a distortion. At the
same time, given a linear frame
, the
decomposition
motivates many authors to treat a coframe
as a translation gauge field.
Difficulties of constructing gauge gravitation theory by analogy
with the Yang-Mills one result from the gauge
transformations in these theories belonging to different classes.
In the case of internal symmetries, the gauge transformations are
just vertical automorphisms of a principal bundle
leaving its base 
fixed. On the other hand,
gravitation theory
is built on the principal
bundle
of the tangent frames to 
.
It belongs to the category of natural bundles
for which diffeomorphisms of the base
canonically
give rise to automorphisms of
.
These automorphisms are called general covariant transformations.
General covariant transformations are sufficient in order to
restate Einstein's General Relativity
and metric-affine
gravitation theory as the gauge ones.
In terms of gauge theory
on natural bundles, gauge fields are
linear connections on a world manifold
, defined
as principal connections
on the
linear frame bundle
, and a metric (tetrad) gravitational field plays the role of a Higgs field
responsible for spontaneous symmetry breaking
of general covariant transformations.
Spontaneous symmetry breaking is a quantum effect when the vacuum
is not invariant under the transformation group. In classical
gauge theory
, spontaneous symmetry breaking occurs if the
structure group
of a principal bundle
is reducible to a closed
subgroup
, i.e., there exists a principal subbundle
of
with the structure group
. By virtue of the well-known theorem, there exists
one-to-one correspondence between the reduced
principal subbundles
of
with the structure
group
and the global sections of the quotient
bundle
. These sections are treated as
classical Higgs fields.
The idea of the pseudo-Riemannian
metric
as a Higgs field
appeared while
constructing non-linear (induced)
representations
of the general linear group
, of which the Lorentz group
is a Cartan
subgroup. The
geometric equivalence principle
postulating the existence of a reference frame in which Lorentz invariants are defined on the
whole world manifold is the theoretical justification of that the
structure group
of the
linear frame bundle
is reduced to the
Lorentz group
. Then the very definition of a
pseudo-Riemannian metric
on a manifold
as a global section of the quotient bundle
leads to its physical interpretation
as a Higgs field
. The physical reason for world
symmetry breaking is the existence of Dirac
fermion matter, whose symmetry group is the universal
two-sheeted covering
of the restricted
Lorentz group
,
.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
.
The first gauge model of gravity was
suggested by R. Utiyama in 1956 just two years after birth of the
gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
itself. However, the initial attempts to construct the
gauge theory of gravity by analogy with the
gauge models of internal symmetries encountered a problem of treating general covariant transformations and
establishing the gauge status of a pseudo-Riemannian metric (a
tetrad field).
In order to overcome this drawback, representing tetrad
Tetrad
Tetrad may refer to:* Tetrad , Bivalents or Tetrad of homologous chromosomes consisting of four synapsed chromatids that become visible during the Pachytene stage of meiotic prophase...
fields as
gauge fields of the translation group was
attempted. Infinitesimal generators
of general covariant transformations were considered as those of
the translation gauge group, and a tetrad (coframe) field was
identified with the translation part of an affine connection
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
on a world manifold
. Any such connection is a sum of a linearworld connection
Linear connection
In the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts:* a connection on a vector bundle, often viewed as a differential operator ;* a principal connection on the frame bundle of a manifold or the induced connection...
and a soldering form where is a non-holonomic frame. For instance, if is the Cartanconnection, then
isthe canonical soldering form
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
on
.There are different physical interpretations of the translation
part
of affine connectionsAffine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
. In
gauge theory of dislocations, a field
describes a distortion. At thesame time, given a linear frame
, thedecomposition
motivates many authors to treat a coframe
as a translation gauge field.
Difficulties of constructing gauge gravitation theory by analogy
with the Yang-Mills one result from the gauge
transformations in these theories belonging to different classes.
In the case of internal symmetries, the gauge transformations are
just vertical automorphisms of a principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
leaving its base fixed. On the other hand,gravitation theory
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
is built on the principal
bundle
of the tangent frames to .It belongs to the category of natural bundles
for which diffeomorphisms of the base
canonicallygive rise to automorphisms of
.These automorphisms are called general covariant transformations.
General covariant transformations are sufficient in order to
restate Einstein's General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
and metric-affine
gravitation theory as the gauge ones.
In terms of gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
on natural bundles, gauge fields are
linear connections on a world manifold
, definedas principal connections
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
on the
linear frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
, and a metric (tetrad) gravitational field plays the role of a Higgs fieldHiggs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
responsible for spontaneous symmetry breaking
of general covariant transformations.
Spontaneous symmetry breaking is a quantum effect when the vacuum
is not invariant under the transformation group. In classical
gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, spontaneous symmetry breaking occurs if the
structure group
of a principal bundlePrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
is reducible to a closedsubgroup
, i.e., there exists a principal subbundleof
with the structure group. By virtue of the well-known theorem, there existsone-to-one correspondence between the reduced
principal subbundles
G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
of
with the structuregroup
and the global sections of the quotientbundle
. These sections are treated asclassical Higgs fields.
The idea of the pseudo-Riemannian
metric
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
as a Higgs field
Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
appeared while
constructing non-linear (induced)
representations
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
of the general linear group
, of which the Lorentz groupLorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
is a Cartan
subgroup. The
geometric equivalence principle
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
postulating the existence of a reference frame in which Lorentz invariants are defined on the
whole world manifold is the theoretical justification of that the
structure group
of thelinear frame bundle
is reduced to theLorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
. Then the very definition of a
pseudo-Riemannian metric
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
on a manifold
as a global section of the quotient bundle leads to its physical interpretationas a Higgs field
Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
. The physical reason for world
symmetry breaking is the existence of Dirac
fermion matter, whose symmetry group is the universal
two-sheeted covering
of the restrictedLorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
,
.See also
- Einstein–Cartan theory
- Spontaneous symmetry breakingSpontaneous symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
- TeleparallelismTeleparallelismTeleparallelism , was an attempt by Einstein to unify electromagnetism and gravity...
- Reduction of the structure groupReduction of the structure groupIn mathematics, in particular the theory of principal bundles, one can ask if a G-bundle "comes from" a subgroup H In mathematics, in particular the theory of principal bundles, one can ask if a...