Affine gauge theory explained

X . For instance, these are gauge theory of dislocations in continuous media when X = 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, R) .

The tangent bundle

TX is associated to a principal linear frame bundle FX , whose structure group is a general linear group GL(n, R) . This is a subgroup of GA(n, R) so that the latter is a semidirect product of GL(n, 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, x} ^{\mu),} • x' ^{\mu = (\partial x' ^{\mu})/(\partial x ^{\nu})} • x ^{\nu,} ( 1 )

on the tangent bundle

TX , the affine tangent bundle can be provided with affine bundle coordinates(x ^{\mu,\widetilde} x ^{• \mu = x} ^{\mu} + a ^{\mu(x} ^{\alpha)),} \widetilde x' ^{\mu = (\partial x' ^{\mu})/(\partial x ^{\nu})} \widetilde x ^{\nu} + b ^{\mu(x} ^{\alpha).} ( 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 formA = dx ^{λ ⊗ [\partial} _{λ} + (\Gamma ^{• \alpha) x} _{\nu(x} ^{\mu(x} ^{λ} ^{• \alpha)) \partial} ^{\mu].} ( 3 )

This affine connection defines a unique linear connection

\Gamma = dx ^{λ ⊗ [\partial} _{λ} + \Gamma ^{• \alpha) x} _{\nu(x} ^{• \nu \partial} ^{\mu]} ( 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 formA\Gamma = dx ^{λ ⊗ [\partial} _{λ} + (\Gamma ^{\alpha)\widetilde} _{\nu(x} x ^{\nu} + s ^{\mu(x} _{λ} ^{\alpha))\widetilde\partial} ^{\mu],} s ^{\mu} _{λ} = - \Gamma ^{\mu{}} _{\nu} a ^{\nu} + \partial _{λ} a ^{\mu,}

relative to the affine coordinates (2).

Then any affine connection

A (3) on ATX\to X is represented by a sumA = A\Gamma + \sigma ( 5 )

of the extended linear connection

A\Gamma and a basic soldering form\sigma = \sigma ^{\mu(x} _{λ} ^{\alpha)dx} ^{λ ⊗ \partial} ^{\mu} ( 6 )

on

TX , where • \partial _{\mu=} \partial _{\mu} due to the canonical isomorphism VATX = ATX x _{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 = R ^{3} , and let an affine connection take a formA = dx ^{i ⊗ (\partial} _{i} + A ^{k)\widetilde\partial} _{j)}

with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients

A ^{j} _{l} describe plastic distortion, covariant derivatives D _{j} u ^{i} = \partial ^{i -} _{ju} A ^{i} _{j} coincide with elastic distortion, and a strength F ^{k} _{ji} = \partial _{j} A ^{k} _{i} - \partial _{i} A ^{k} _{j} is a dislocation density.

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

L _{(\sigma)} = \mu D ^{kD} _{iu} ^{iu} ^{k} + (λ)/(2) (D ^{i)} _{iu} ^{2} - \epsilon F ^{k{}} _{ij} F ^{ij} _{k{}} ,

where

\mu and λ 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 ⊗ 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 morphisms: TX\ni \partial _{λ\to} \partial _{λ\rfloor} (\theta + \sigma) = (\delta ^{\nu +} _{λ} \sigma ^{\nu)\partial} _{\nu\in} TX,

where

\theta = dx ^{\mu ⊗} \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 ^{\mu} _{\alpha} s ^{\nu} _{\beta} g ^{\alpha\beta} when a Lagrangian of a soldering field \sigma takes a formL _{(\sigma)} = (12 [a)/(_{1T}) ^{\mu{}} ^{\nu\mu} T ^{\nu\alpha} _{\alpha{}} + a _{2T} _{\mu\nu\alpha} T ^{\mu\nu\alpha} + a _{3T} _{\mu\nu\alpha} T ^{\nu\mu\alpha} + a ^{\mu\nu\alpha\beta} _{4 \epsilon} T ^{\gamma{}} _{\mu\gamma} T _{\beta\nu\alpha} - \mu\sigma ^{\mu{}} _{\mu} \sigma ^{\nu{}} _{\nu]\sqrt{ - g}} ,

where

\epsilon ^{\mu\nu\alpha\beta} is the Levi-Civita symbol, andT ^{\alpha{}} _{\nu\mu} = D ^{\alpha{}} _{\mu} - D ^{\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),
• G. Sardanashvily, O. Zakharov, Gauge Gravitation Theory (World Scientific, Singapore, 1992),
• 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, .