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Wikipedia / Spin connection

In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.

The spin connection occurs in two common forms: the Levi-Civita spin connection, when it is derived from the Levi-Civita connection, and the affine spin connection, when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion-free and metric-preserving connection, whereas the affine connection (and so the affine spin connection) may contain torsion.

Definition

be the local Lorentz frame fields or vierbein is the Minkowski metric. Here, Latin letters denote the local Lorentz

The torsion-free are the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.

. The spin connection may be written purely in terms of the vierbein field as.

Cartan's structure equations

In the Cartan formalism, the spin connection is used to define both torsion and curvature. These are easiest to read by working with differential forms, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the connection form and the curvature form. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them. More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an affine connection to a homogeneous space; it is not yet as general as the idea of a principal connection on a fiber bundle. It serves as a suitable half-way point between the narrower setting in Riemannian geometry and the fully abstract fiber bundle setting, thus emphasizing the similarity to gauge theory. Note that Cartan's structure equations, as expressed here, have a direct analog: the Maurer–Cartan equations for Lie groups (that is, they are the same equations, but in a different setting and notation).

for the orthonormal coordinates on the cotangent bundle The torsion while the curvature 2-form These two equations, taken together are called Cartan's structure equations. Consistency requires that the Bianchi identities be obeyed. The first Bianchi identity is obtained by taking the exterior derivative The covariant derivative for a generic differential form of degree p The difference between a connection with torsion, and the unique torsionless connection is given by the contorsion tensor. Connections with torsion are commonly found in theories of teleparallelism, Einstein–Cartan theory, gauge theory gravity and supergravity.

Derivation

Metricity

It is easy to deduce by raising and lowering indices as needed that the frame fields. This is sometimes called the metricity condition; Note that this condition holds only for the Levi-Civita spin connection, and not for the affine spin connection in general.

where antisymmetrization of indices has an implicit factor of 1/2.

By the metric compatibility

Then, do a cyclic permutation

From this we obtain the same formula as before.

Applications

The spin connection arises in the Dirac equation when expressed in the language of curved spacetime, see Dirac equation in curved spacetime. Specifically there are problems coupling gravity to spinor fields: there are no finite-dimensional spinor representations of the general covariance group. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime. The Dirac matrices

We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation

is a function of space-time. This means that the partial derivative

The generally covariant fermion action couples fermions to gravity when added to the first order tetradic Palatini action is the curvature of the spin connection.

The tetradic Palatini formulation of general relativity which is a first order formulation of the Einstein–Hilbert action.

The spatial spin connection appears in the definition of Ashtekar–Barbero variables Yang–Mills is the extrinsic curvature is the Immirzi parameter Yang–Mills gauge theory, it allows the importation of non-perturbative techniques used in Quantum chromodynamics to canonical quantum general relativity.

See also

Notes

References