Spinor-helicity variables for massless momenta

Use natural units with hbar and c equal to one and metric signature plus minus minus minus. Throughout this module every external momentum flows inward, and their sum vanishes. Physical final-state momenta therefore enter with reversed sign. Form the momentum matrix with energy plus longitudinal momentum in its upper-left entry. Its determinant equals the Minkowski norm, so a nonzero null momentum has rank one and admits an undotted-column times dotted-row factorization.

Each spinor has units of the square root of energy. For a real future-directed momentum one may choose the dotted spinor as the complex conjugate of the undotted spinor. A negative-energy real momentum instead needs a relative sign, and general complex momenta do not obey a conjugation relation. Enforcing positive-energy conjugation during a complex deformation would incorrectly restrict the analytic variables on which recursion relies.

Define an angle bracket as the determinant of the two undotted columns. Define the square bracket with the reverse determinant order for the dotted columns. Both are antisymmetric, and the product of an angle bracket with the oppositely ordered square bracket equals twice the momentum dot product. Stating the component definitions is safer than importing a remembered conjugation rule from a source using different index positions or metric signs.

For the future-directed real choice just described, the oppositely ordered square bracket is the complex conjugate of the angle bracket. Their product is consequently nonnegative for two such null momenta. With crossed energy signs or complex kinematics that positivity inference no longer applies. The invariant still follows algebraically from the determinant identity, even when neither bracket is the complex conjugate of the other and the invariant itself is complex.

Multiplying one undotted spinor by a nonzero number and dividing its dotted partner by the same number leaves momentum unchanged. This is the spinor version of the massless little-group freedom. Real positive-energy conjugation restricts that transformation to a phase, while complexified kinematics permits a general nonzero complex scale. Helicity amplitudes transform with definite weights under this freedom rather than being scalar functions of the momentum matrices alone.

Crossing reverses momentum, so it requires a different operation. One convenient analytic prescription multiplies both spinors by i, producing minus the original matrix. It also introduces phases into every bracket involving that leg. Reinterpreting a physical particle between initial and final states requires the corresponding antiparticle and helicity assignment; momentum replacement alone is insufficient. These conventions organize amplitudes before squaring, so convention phases cancel only when all interfering contributions use the same external-state choices.