Physics:Affine gauge theory

Short description: Gauge theory with affine connections

Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold [math]\displaystyle{ X }[/math]. For instance, these are gauge theory of dislocations in continuous media when [math]\displaystyle{ X=\mathbb R^3 }[/math], the generalization of metric-affine gravitation theory when [math]\displaystyle{ 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]\displaystyle{ TX }[/math] of an [math]\displaystyle{ n }[/math]-dimensional manifold [math]\displaystyle{ X }[/math] admits a natural structure of an affine bundle [math]\displaystyle{ ATX }[/math], called the affine tangent bundle, possessing bundle atlases with affine transition functions. It is associated to a principal bundle [math]\displaystyle{ AFX }[/math] of affine frames in tangent space over [math]\displaystyle{ X }[/math], whose structure group is a general affine group [math]\displaystyle{ GA(n,\mathbb R) }[/math].

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

There is the canonical imbedding of [math]\displaystyle{ FX }[/math] to [math]\displaystyle{ AFX }[/math] onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle [math]\displaystyle{ TX }[/math] as the affine one.

Given linear bundle coordinates

[math]\displaystyle{ (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]\displaystyle{ TX }[/math], the affine tangent bundle can be provided with affine bundle coordinates

[math]\displaystyle{ (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]\displaystyle{ ATX }[/math] admits an affine connection [math]\displaystyle{ A }[/math] which is associated to a principal connection on an affine frame bundle [math]\displaystyle{ AFX }[/math]. In affine gauge theory, it is treated as an affine gauge field.

Given the linear bundle coordinates (1) on [math]\displaystyle{ ATX=TX }[/math], an affine connection [math]\displaystyle{ A }[/math] is represented by a connection tangent-valued form

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

on [math]\displaystyle{ TX }[/math], which is associated to a principal connection on [math]\displaystyle{ FX }[/math].

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

[math]\displaystyle{ 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]\displaystyle{ A }[/math] (3) on [math]\displaystyle{ ATX\to X }[/math] is represented by a sum

[math]\displaystyle{ A=A\Gamma +\sigma \qquad\qquad (5) }[/math]

of the extended linear connection [math]\displaystyle{ A\Gamma }[/math] and a basic soldering form

[math]\displaystyle{ \sigma=\sigma_\lambda^\mu(x^\alpha)dx^\lambda\otimes\partial_\mu \qquad\qquad (6) }[/math]

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

Relative to the linear coordinates (1), the sum (5) is brought into a sum [math]\displaystyle{ A=\Gamma +\sigma }[/math] of a linear connection [math]\displaystyle{ \Gamma }[/math] and the soldering form [math]\displaystyle{ \sigma }[/math] (6). In this case, the soldering form [math]\displaystyle{ \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]\displaystyle{ TX }[/math]) is well defined only on a parallelizable manifold [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ u^k }[/math], [math]\displaystyle{ k = 1,2,3 }[/math], of small deformations are determined only with accuracy to gauge translations [math]\displaystyle{ u^k \to u^k + a^k(x) }[/math].

In this case, let [math]\displaystyle{ X=\mathbb R^3 }[/math], and let an affine connection take a form

[math]\displaystyle{ 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]\displaystyle{ A^j_l }[/math] describe plastic distortion, covariant derivatives [math]\displaystyle{ D_j u^i =\partial_ju^i- A^i_j }[/math] coincide with elastic distortion, and a strength [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \lambda }[/math] are the Lamé parameters of isotropic media. These equations however are not independent since a displacement field [math]\displaystyle{ 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]\displaystyle{ X }[/math], one can consider an affine, but not linear connection on the tangent bundle [math]\displaystyle{ TX }[/math] of [math]\displaystyle{ X }[/math]. Given bundle coordinates (1) on [math]\displaystyle{ TX }[/math], it takes the form (3) where the linear connection [math]\displaystyle{ \Gamma }[/math] (4) and the basic soldering form [math]\displaystyle{ \sigma }[/math] (6) are considered as independent variables.

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

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

[math]\displaystyle{ s: TX\ni \partial_\lambda\to \partial_\lambda\rfloor (\theta +\sigma) =(\delta_\lambda^\nu+ \sigma_\lambda^\nu)\partial_\nu\in TX, }[/math]

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

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

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

[math]\displaystyle{ T^\alpha{}_{\nu\mu}=D_\nu\sigma^\alpha{}_\mu -D_\mu\sigma^\alpha{}_\nu }[/math]

is the torsion of a linear connection [math]\displaystyle{ \Gamma }[/math] with respect to a soldering form [math]\displaystyle{ \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
• 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.