View source for Affine gauge theory

{{Short description|Gauge theory with affine connections}}
'''Affine gauge theory''' is [[gauge theory|classical gauge theory]] where gauge fields are [[connection (affine bundle)|affine connections]] on the [[tangent bundle]] over a [[smooth manifold]] <math>X</math>. For instance, these are gauge theory of [[dislocation]]s in [[continuum mechanics|continuous media]] when <math>X=\mathbb R^3</math>, the generalization of [[metric-affine gravitation theory]] when <math>X</math> is a [[world manifold]] and, in particular, gauge theory of the [[fifth force]].

==Affine tangent bundle==

Being a [[vector bundle]], the tangent bundle <math>TX</math> of an <math>n</math>-dimensional manifold <math>X</math> admits a natural structure of an [[affine bundle]] <math>ATX</math>, called the ''affine tangent bundle'', possessing bundle atlases with affine transition functions. It is associated to a [[principal bundle]] <math>AFX</math> of affine frames in tangent space over <math>X</math>, whose [[fiber bundle|structure group]] is a [[affine group|general affine group]] <math>GA(n,\mathbb R)</math>.

The tangent bundle <math>TX</math> is associated to a principal [[frame bundle|linear frame bundle]] <math>FX</math>, whose structure group is a [[general linear group]] <math>GL(n,\mathbb R)</math>. This is a subgroup of <math>GA(n,\mathbb R)</math> so that the latter is a semidirect product of <math>GL(n,\mathbb R)</math> and a group <math>T^n</math> of translations.

There is the canonical imbedding of <math>FX</math> to <math>AFX</math> onto a [[reduction of the structure group|reduced principal subbundle]] which corresponds to the canonical structure of a vector bundle <math>TX</math> as the affine one.

Given linear bundle coordinates

: <math>(x^\mu,\dot x^\mu), \qquad \dot x'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}\dot x^\nu, \qquad\qquad (1)</math>

on the tangent bundle <math>TX</math>, the affine tangent bundle can be provided with affine bundle coordinates

: <math>(x^\mu,\widetilde x^\mu=\dot x^\mu +a^\mu(x^\alpha)), \qquad  \widetilde x'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}\widetilde x^\nu + b^\mu(x^\alpha). \qquad\qquad (2) </math>

and, in particular, with the linear coordinates (1).

==Affine gauge fields==

The affine tangent bundle <math>ATX</math> admits an affine connection <math>A</math> which is associated to a [[connection (principal bundle)|principal connection]] on an affine frame bundle <math>AFX</math>. In affine gauge theory, it is treated as an ''affine gauge field''.

Given the linear bundle coordinates (1) on <math>ATX=TX</math>, an affine connection <math>A</math> is represented by a [[connection (fibred manifold)|connection tangent-valued form]]

: <math> A=dx^\lambda\otimes[\partial_\lambda + (\Gamma_\lambda{}^\mu{}_\nu(x^\alpha)\dot x^\nu+\sigma_\lambda^\mu(x^\alpha))\dot\partial_\mu].\qquad \qquad (3)</math>

This affine connection defines a unique [[connection (vector bundle)|linear connection]]

: <math> \Gamma =dx^\lambda\otimes[\partial_\lambda + \Gamma_\lambda{}^\mu{}_\nu(x^\alpha)\dot x^\nu\dot\partial_\mu] \qquad\qquad (4)</math>

on <math>TX</math>, which is associated to a principal connection on <math>FX</math>.

Conversely, every linear connection <math>\Gamma</math> (4) on <math>TX\to X</math> is extended to the affine one <math>A\Gamma</math> on <math>ATX</math> which is given by the same expression (4) as <math>\Gamma</math> with respect to the bundle coordinates (1) on <math>ATX=TX</math>, but it takes a form

: <math> A\Gamma =dx^\lambda\otimes[\partial_\lambda + (\Gamma_\lambda{}^\mu{}_\nu(x^\alpha)\widetilde x^\nu + s_\lambda^\mu(x^\alpha))\widetilde\partial_\mu], \qquad s_\lambda^\mu = - \Gamma_\lambda{}^\mu{}_\nu a^\nu +\partial_\lambda a^\mu,  </math>

relative to the affine coordinates (2).

Then any affine connection <math>A</math> (3) on <math>ATX\to X</math> is represented by a sum

: <math>A=A\Gamma +\sigma \qquad\qquad (5) </math>

of the extended linear connection <math>A\Gamma</math> and a [[connection (affine bundle)|basic soldering form]]

: <math>\sigma=\sigma_\lambda^\mu(x^\alpha)dx^\lambda\otimes\partial_\mu \qquad\qquad (6) </math>

on <math>TX</math>,  where <math>\dot \partial_\mu= \partial_\mu</math> due to the canonical isomorphism <math>VATX=ATX\times_X TX</math> of the [[vertical bundle|vertical tangent bundle]]  <math>VATX</math> of <math>ATX</math>.

Relative to the linear coordinates (1), the sum (5) is brought into a sum <math>A=\Gamma +\sigma </math>
of a linear connection <math>\Gamma</math> and the soldering form <math>\sigma</math> (6). In this case, the soldering form <math>\sigma</math> (6) often is treated as a ''translation gauge field'', though it is not a connection.

Let us note that a true translation gauge field (i.e., an affine connection which yields a flat linear connection on <math>TX</math>) is well defined only on a [[parallelizable manifold]] <math>X</math>.

==Gauge theory of dislocations==

In field theory, one meets a problem of physical interpretation of translation gauge fields because there are no fields subject to gauge translations <math>u(x) \to u(x) + a(x)</math>.  At the same time, one observes such a field in gauge theory of dislocations in continuous media because, in the presence of dislocations, displacement vectors <math>u^k</math>, <math> k = 1,2,3</math>, of small deformations are determined only with accuracy to gauge translations <math> u^k \to u^k  + a^k(x)</math>.

In this case, let <math>X=\mathbb R^3</math>, and let an affine connection take a form

: <math> A=dx^i\otimes(\partial_i + A^j_i(x^k)\widetilde\partial_j)</math>

with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients <math> A^j_l</math> describe plastic [[dislocation|distortion]], covariant derivatives <math>D_j u^i =\partial_ju^i- A^i_j</math> coincide with elastic distortion, and a strength <math> F^k_{ji}=\partial_j A^k_i - \partial_i A^k_j</math> is a dislocation density.

Equations of gauge theory of dislocations are derived from a gauge invariant [[Lagrangian system|Lagrangian density]]

: <math> L_{(\sigma)} = \mu D_iu^kD^iu_k + \frac{\lambda}{2}(D_iu^i)^2 - \epsilon F^k{}_{ij}F_k{}^{ij}, </math>

where <math>\mu</math>  and <math>\lambda</math> are the [[Lamé parameters]] of isotropic media. These equations however are not independent since a displacement field <math>u^k(x)</math> can be removed by gauge translations and, thereby, it fails to be a dynamic variable.

==Gauge theory of the fifth force==

In [[gauge gravitation theory]] on a world manifold <math>X</math>, one can consider an affine, but not linear connection on the tangent bundle <math>TX</math> of <math>X</math>. Given bundle coordinates (1) on <math>TX</math>, it takes the form (3) where the linear connection <math>\Gamma</math> (4) and the basic soldering form <math>\sigma</math> (6) are considered as independent variables.

As was mentioned above, the soldering form <math>\sigma</math> (6) often is treated as a translation gauge field, though it is not a connection. On another side, one mistakenly identifies <math>\sigma</math> with a [[frame fields in general relativity|tetrad field]]. However, these are different mathematical object because a soldering form is a section of the tensor bundle <math>TX\otimes T^*X</math>, whereas a tetrad field is a local section of a [[metric-affine gravitation theory|Lorentz reduced subbundle]] of a frame bundle <math>FX</math>.

In the spirit of the above-mentioned gauge theory of dislocations, it has been suggested that a soldering field <math>\sigma</math> can describe ''sui generi'' deformations of a world manifold <math>X</math> which are given by a bundle morphism

: <math> s: TX\ni \partial_\lambda\to \partial_\lambda\rfloor (\theta +\sigma) =(\delta_\lambda^\nu+ \sigma_\lambda^\nu)\partial_\nu\in TX, </math>

where <math>\theta=dx^\mu\otimes \partial_\mu</math> is a [[tautological one-form]].

Then one considers metric-affine gravitation theory <math>(g,\Gamma)</math> on a deformed world manifold as that with a deformed pseudo-Riemannian metric <math>\widetilde g^{\mu\nu}=s^\mu_\alpha s^\nu_\beta g^{\alpha\beta}</math> when a Lagrangian of a soldering field <math>\sigma</math> takes a form

: <math> L_{(\sigma)}=\frac12[a_1T^\mu{}_{\nu\mu} T_\alpha{}^{\nu\alpha}+
a_2T_{\mu\nu\alpha}T^{\mu\nu\alpha}+a_3T_{\mu\nu\alpha}T^{\nu\mu\alpha} +a_4\epsilon^{\mu\nu\alpha\beta}T^\gamma{}_{\mu\gamma}
T_{\beta\nu\alpha}-\mu\sigma^\mu{}_\nu\sigma^\nu{}_\mu+
\lambda\sigma^\mu{}_\mu \sigma^\nu{}_\nu]\sqrt{-g} </math>,

where <math>\epsilon^{\mu\nu\alpha\beta}</math> is the [[Levi-Civita symbol]], and

: <math>T^\alpha{}_{\nu\mu}=D_\nu\sigma^\alpha{}_\mu  -D_\mu\sigma^\alpha{}_\nu
</math>

is the [[connection (fibred manifold)|torsion]]  of a linear connection <math>\Gamma</math> with respect to a soldering form <math>\sigma</math>.
 
In particular, let us consider this gauge model in the case of small gravitational and soldering fields whose matter source is a point mass. Then one comes to a modified [[Newtonian potential]] of the fifth force type.

==See also==
* [[Connection (affine bundle)]]
* [[Dislocations]]
* [[Fifth force]]
* [[Gauge gravitation theory]]
* [[Metric-affine gravitation theory]]
*[[Classical unified field theories]]

==References==
* A. Kadic, D. Edelen, ''A Gauge Theory of Dislocations and Disclinations'', Lecture Notes in Physics '''174'''  (Springer, New York, 1983), {{ISBN|3-540-11977-9}}
* [[Sardanashvily|G. Sardanashvily]], O. Zakharov, ''Gauge Gravitation Theory'' (World Scientific, Singapore, 1992), {{ISBN|981-02-0799-9}}
* C. Malyshev, The dislocation stress functions from the double curl T(3)-gauge equations: Linearity and look beyond, ''Annals of Physics'' '''286''' (2000) 249.

==External links==
* [[Sardanashvily|G. Sardanashvily]], Gravity as a Higgs field. III. Nongravitational deviations of gravitational field, {{arxiv|gr-qc/9411013}}.

[[Category:Gauge theories]]
[[Category:Theories of gravity]]