Affine gauge theory

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Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold <math>X</math>. For instance, these are gauge theory of dislocations in 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 structure group is a general affine group <math>GA(n,\mathbb R)</math>.

The tangent bundle <math>TX</math> is associated to a principal 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 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 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 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 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 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 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 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 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 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 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 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

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

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