A1 The 2spinor notation Maxwell equations

The 2-spinor formalism employs quantities with abstract spinor indices (for the complex 2-dimensional spin-space) for which I use italic capital Latin letters, either unprimed (A, B, C, ... ) or primed (A', B', C', ... ), these being interchanged under complex conjugation. The (complexified) tangent space at each space-time point is the tensor product of the unprimed with the primed spin-space. This enables us to adopt the abstract-index identification a=AA', b=BB', c = CC', ...

where the italic lower-case Latin index letters a, b, c, ... refer to the space-time tangent spaces. More specifically, the tangent spaces refer to indices in upper position and the cotangent spaces to indices in the lower position.

The anti-symmetric Maxwell field tensor Fab ( = - Fba) can be expressed in 2-spinor form in terms of a symmetric 2-index 2-spinor <ab ( = <ba) by

where Sab ( = - Sba = Sab) is the quantity defining the complex symplectic structure of spin-space and is related to the metric by the abstract-index equation gab = £AB £a 'B', spinor indices being raised or lowered according to the following prescriptions (where index-ordering on the epsilons is important!)

= £ab Sb, ^B = Sab, nB'= £A'BnB', ^b' = nA' £a b'.

The Maxwell field equations (denoted collectively by VF=4nJ in §3.2), with source as the charge-current vector Ja, are

(where square brackets around indices denote anti-symmetrization; round brackets, symmetrization), the charge-current conservation equation being

These take the respective 2-spinor forms (P&R 5.1.52, P&R 5.1.54)

When there are no sources (Ja = 0), we get the free Maxwell equations (denoted by VF=0 in §3.2)