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. 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 connection, whereas the affine connection (and so the affine spin connection) may contain torsion. Let e μ a {displaystyle e_{mu }^{;,a}} be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthogonal space time vector fields that diagonalize the metric tensor where g μ ν {displaystyle g_{mu u }} is the spacetime metric and η a b {displaystyle eta _{ab}} is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that g μ ν {displaystyle g_{mu u }} , when written in terms of the basis e μ a {displaystyle e_{mu }^{;,a}} , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. g μ ν {displaystyle g^{mu u }} or g μ ν {displaystyle g_{mu u }} . The Latin or 'Lorentzian' vierbein indices can be raised or lowered by η a b {displaystyle eta ^{ab}} or η a b {displaystyle eta _{ab}} respectively. For example, e μ a = g μ ν e ν a {displaystyle e^{mu a}=g^{mu u }e_{ u }^{;,a}} and e ν a = η a b e ν b {displaystyle e_{ u a}=eta _{ab}e_{ u }^{;,b}} The torsion-free spin connection is given by where Γ μ ν σ {displaystyle Gamma _{mu u }^{sigma }} 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 single, unique connection on a manifold that is torsion-free. In general, there is no restriction: the spin connection may also contain torsion. Note that ω μ   a b = e ν   a ∂ ; μ e ν b = e ν   a ( ∂ μ e ν b + Γ   σ μ ν e σ b ) {displaystyle omega _{mu }^{ ab}=e_{ u }^{ a}partial _{;mu }e^{ u b}=e_{ u }^{ a}(partial _{mu }e^{ u b}+Gamma _{ sigma mu }^{ u }e^{sigma b})} using the gravitational covariant derivative ∂ ; μ e ν b {displaystyle partial _{;mu }e^{ u b}} of the contravariant vector e ν b {displaystyle e^{ u b}} . The spin connection may be written purely in terms of the vierbein field as which by definition is anti-symmetric in its internal indices a , b {displaystyle a,b} . The spin connection ω μ   a b {displaystyle omega _{mu }^{ ab}} defines a covariant derivative D μ {displaystyle D_{mu }} on generalized tensors. For example, its action on V ν   a {displaystyle V_{ u }^{ a}} is