Affine gauge theory

Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold $X$. For instance, these are gauge theory of dislocations in continuous media when $X=\mathbb {R} ^{3}$, the generalization of metric-affine gravitation theory when $X$ is a world manifold and, in particular, gauge theory of the fifth force.

Affine tangent bundle

Being a vector bundle, the tangent bundle $TX$ of an $n$-dimensional manifold $X$ admits a natural structure of an affine bundle $ATX$, called the affine tangent bundle, possessing bundle atlases with affine transition functions. It is associated to a principal bundle $AFX$ of affine frames in tangent space over $X$, whose structure group is a general affine group $GA(n,\mathbb {R} )$.

The tangent bundle $TX$ is associated to a principal linear frame bundle $FX$, whose structure group is a general linear group $GL(n,\mathbb {R} )$. This is a subgroup of $GA(n,\mathbb {R} )$ so that the latter is a semidirect product of $GL(n,\mathbb {R} )$ and a group $T^{n}$ of translations.

There is the canonical imbedding of $FX$ to $AFX$ onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle $TX$ as the affine one.

Given linear bundle coordinates

$(x^{\mu },{\dot {x}}^{\mu }),\qquad {\dot {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\dot {x}}^{\nu },\qquad \qquad (1)$

on the tangent bundle $TX$, the affine tangent bundle can be provided with affine bundle coordinates

$(x^{\mu },{\widetilde {x}}^{\mu }={\dot {x}}^{\mu }+a^{\mu }(x^{\alpha })),\qquad {\widetilde {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\widetilde {x}}^{\nu }+b^{\mu }(x^{\alpha }).\qquad \qquad (2)$

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

Affine gauge fields

The affine tangent bundle $ATX$ admits an affine connection $A$ which is associated to a principal connection on an affine frame bundle $AFX$. In affine gauge theory, it is treated as an affine gauge field.

Given the linear bundle coordinates (1) on $ATX=TX$, an affine connection $A$ is represented by a connection tangent-valued form

$A=dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }+\sigma _{\lambda }^{\mu }(x^{\alpha })){\dot {\partial }}_{\mu }].\qquad \qquad (3)$

This affine connection defines a unique linear connection

$\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }{\dot {\partial }}_{\mu }]\qquad \qquad (4)$

on $TX$, which is associated to a principal connection on $FX$.

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

$A\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\widetilde {x}}^{\nu }+s_{\lambda }^{\mu }(x^{\alpha })){\widetilde {\partial }}_{\mu }],\qquad s_{\lambda }^{\mu }=-\Gamma _{\lambda }{}^{\mu }{}_{\nu }a^{\nu }+\partial _{\lambda }a^{\mu },$

relative to the affine coordinates (2).

Then any affine connection $A$ (3) on $ATX\to X$ is represented by a sum

$A=A\Gamma +\sigma \qquad \qquad (5)$

of the extended linear connection $A\Gamma$ and a basic soldering form

$\sigma =\sigma _{\lambda }^{\mu }(x^{\alpha })dx^{\lambda }\otimes \partial _{\mu }\qquad \qquad (6)$

on $TX$, where ${\dot {\partial }}_{\mu }=\partial _{\mu }$ due to the canonical isomorphism $VATX=ATX\times _{X}TX$ of the vertical tangent bundle $VATX$ of $ATX$.

Relative to the linear coordinates (1), the sum (5) is brought into a sum $A=\Gamma +\sigma$ of a linear connection $\Gamma$ and the soldering form $\sigma$ (6). In this case, the soldering form $\sigma$ (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 $TX$) is well defined only on a parallelizable manifold $X$.

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 $u(x)\to u(x)+a(x)$. 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 $u^{k}$, $k=1,2,3$, of small deformations are determined only with accuracy to gauge translations $u^{k}\to u^{k}+a^{k}(x)$.

In this case, let $X=\mathbb {R} ^{3}$, and let an affine connection take a form

$A=dx^{i}\otimes (\partial _{i}+A_{i}^{j}(x^{k}){\widetilde {\partial }}_{j})$

with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients $A_{l}^{j}$ describe plastic distortion, covariant derivatives $D_{j}u^{i}=\partial _{j}u^{i}-A_{j}^{i}$ coincide with elastic distortion, and a strength $F_{ji}^{k}=\partial _{j}A_{i}^{k}-\partial _{i}A_{j}^{k}$ is a dislocation density.

Equations of gauge theory of dislocations are derived from a gauge invariant Lagrangian density

$L_{(\sigma )}=\mu D_{i}u^{k}D^{i}u_{k}+{\frac {\lambda }{2}}(D_{i}u^{i})^{2}-\epsilon F^{k}{}_{ij}F_{k}{}^{ij},$

where $\mu$ and $\lambda$ are the Lamé parameters of isotropic media. These equations however are not independent since a displacement field $u^{k}(x)$ 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 $X$, one can consider an affine, but not linear connection on the tangent bundle $TX$ of $X$. Given bundle coordinates (1) on $TX$, it takes the form (3) where the linear connection $\Gamma$ (4) and the basic soldering form $\sigma$ (6) are considered as independent variables.

As was mentioned above, the soldering form $\sigma$ (6) often is treated as a translation gauge field, though it is not a connection. On another side, one mistakenly identifies $\sigma$ with a tetrad field. However, these are different mathematical object because a soldering form is a section of the tensor bundle $TX\otimes T^{*}X$, whereas a tetrad field is a local section of a Lorentz reduced subbundle of a frame bundle $FX$.

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

$s:TX\ni \partial _{\lambda }\to \partial _{\lambda }\rfloor (\theta +\sigma )=(\delta _{\lambda }^{\nu }+\sigma _{\lambda }^{\nu })\partial _{\nu }\in TX,$

where $\theta =dx^{\mu }\otimes \partial _{\mu }$ is a tautological one-form.

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

$L_{(\sigma )}={\frac {1}{2}}[a_{1}T^{\mu }{}_{\nu \mu }T_{\alpha }{}^{\nu \alpha }+a_{2}T_{\mu \nu \alpha }T^{\mu \nu \alpha }+a_{3}T_{\mu \nu \alpha }T^{\nu \mu \alpha }+a_{4}\epsilon ^{\mu \nu \alpha \beta }T^{\gamma }{}_{\mu \gamma }T_{\beta \nu \alpha }-\mu \sigma ^{\mu }{}_{\nu }\sigma ^{\nu }{}_{\mu }+\lambda \sigma ^{\mu }{}_{\mu }\sigma ^{\nu }{}_{\nu }]{\sqrt {-g}}$,

where $\epsilon ^{\mu \nu \alpha \beta }$ is the Levi-Civita symbol, and

$T^{\alpha }{}_{\nu \mu }=D_{\nu }\sigma ^{\alpha }{}_{\mu }-D_{\mu }\sigma ^{\alpha }{}_{\nu }$

is the torsion of a linear connection $\Gamma$ with respect to a soldering form $\sigma$.

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
• 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

• G. Sardanashvily, Gravity as a Higgs field. III. Nongravitational deviations of gravitational field, arXiv:gr-qc/9411013.