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4.14.6.4 Spinor metric tensor
The function spinor_metric() creates an antisymmetric tensor with
two indices that is used to raise/lower indices of 2-component spinors.
It is output as ‘eps’:
{
symbol psi("psi");
spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
e = spinor_metric(A, B) * indexed(psi, B_co);
cout << e.simplify_indexed() << endl;
// -> psi~A
e = spinor_metric(A, B) * indexed(psi, A_co);
cout << e.simplify_indexed() << endl;
// -> -psi~B
e = spinor_metric(A_co, B_co) * indexed(psi, B);
cout << e.simplify_indexed() << endl;
// -> -psi.A
e = spinor_metric(A_co, B_co) * indexed(psi, A);
cout << e.simplify_indexed() << endl;
// -> psi.B
e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
cout << e.simplify_indexed() << endl;
// -> 2
e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
cout << e.simplify_indexed() << endl;
// -> -delta.A~C
}
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The matrix representation of the spinor metric is [[0, 1], [-1, 0]].