DifferentialGeometry/Tensor/SpinConnection - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

Home : Support : Online Help : DifferentialGeometry/Tensor/SpinConnection

Tensor[SpinConnection] - compute the spin connection defined by a solder form

Calling Sequences

     SpinConnection()

Parameters

       - a solder form

 

Description

Examples

See Also

Description

• 

The DifferentialGeometry Tensor package supports general computations with connections on vector bundles (Connection, Example 3; CovariantDerivative, Example 3; DirectionalCovariantDerivative, Example 3; and CurvatureTensor, Example 3).  This functionality naturally provides for covariant differentiation of spinors.

• 

The command SpinConnection() computes the connection compatible with the solder form and the epsilon spinors.

• 

Given a solder form , let  be the associated metric. There is a unique spin connection  such that  and , where  denotes either of the epsilon spinors (EpsilonSpinor). In the definition of  the tensorial argument (or index) is covariantly differentiated with respect to the Christoffel connection for . It is this connection  which is computed by the command SpinConnection(sigma).

• 

Note that a generic connection for the differentiation of spinors can be constructed using the Connection command.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SpinConnection(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SpinConnection.

Examples

 

Example 1.

First create a vector bundle  with base coordinates  and fiber coordinates .

(2.1)

 

Define a spacetime metric  on .

E > 

(2.2)

 

Define an orthonormal frame on  with respect to the metric .

E > 

(2.3)

 

Calculate the solder form  from the frame F.

E > 

(2.4)

 

Calculate the spin-connection for the solder form .

E > 

(2.5)

 

Example 2.

Define a rank 1 spinor . Calculate the covariant derivative of . Calculate the directional derivatives of .

E > 

(2.6)
E > 

(2.7)
E > 

(2.8)
E > 

(2.9)
E > 

(2.10)
E > 

(2.11)

 

Example 3.

Check that the covariant derivative of  vanishes. Because  is a spin-tensor, two connections are required. Calculate the Christoffel connection for the metric .

E > 

(2.12)
E > 

(2.13)

 

Define an epsilon spinor and check that its covariant derivative vanishes.

E > 

(2.14)
E > 

(2.15)

 

Example 4.

Calculate the curvature spin-tensor for the spin-connection Gamma2.

E > 

(2.16)

 

The curvature tensor  for the Christoffel connection can be expressed in terms of the curvature spin-tensor  and the bivector solder forms by the identity

 

 

Let's check this formula for the Christoffel connection Gamma1 and the spin-connection Gamma2. First calculate the curvature tensor for Gamma1.

E > 

(2.17)

 

Calculate the complex conjugate of the spinor curvature F.

E > 

(2.18)

 

Calculate the bivector soldering forms S and barS.

E > 

(2.19)
E > 

(2.20)

 

The first term on the right-hand side of (*) is

E > 

(2.21)

 

The second term on the right-hand side of (*) is

E > 

(2.22)
E > 

(2.23)
E > 

(2.24)
E > 

(2.25)

See Also

DifferentialGeometry, Tensor, BivectorSolderForm, Connection, Physics[Christoffel], CovariantDerivative, Physics[D_], DirectionalCovariantDerivative, CurvatureTensor, Physics[Riemann], EnergyMomentumTensor, EpsilonSpinor, MatterFieldEquations


Download Help Document